SPECIAL
SPECIALSECTION:
SECTION:SOLAR
SOLAR PHYSICS
The physics laboratory in the sky
E. N. Parker
Enrico Fermi Institute, University of Chicago, 1323 Evergreen Road, Homewood, Illinois 60430, USA
The Sun, by virtue of its large mass, size, and temperature,
exhibits a variety of effects that are unknown in the
terrestrial laboratory, thereby challenging the physicist to
relate them to the basic principles of physics derived from
the terrestrial laboratory. A number of solar phenomena
are reviewed, with comments on the degree to which they
are presently understood.
1. Introduction
THE Sun has played a prominent role in the development of
physics since the times of Kepler and Newton, providing a
window into phenomena unknown to the restricted scale of
the terrestrial laboratory1. The Sun continues in that role
today, providing mystery and opportunity in such diverse
fields as lepton physics and magnetohydrodynamics. It
challenges the experimental physicist to develop
increasingly sophisticated tools for investigation and the
theoretical physicist to see both new facets and extensions
to basic laws of physics. With its large size and mass, it
serves in two distinct roles: (i) as a passive self-gravitating
thermonuclear object providing opportunities for testing
existing laws of physics on a grand scale and (ii) as a large
radiative, magnetic, fluid dynamics laboratory where nature
displays astonishing dynamical phenomena, previously
unknown and wholly outside conventional wisdom. The
precision measurements of gravitational time delays in radio
signals propagating close to the Sun are an example of the
former, the sunspot is an example of the latter.
To begin with a review of past triumphs, recall that
the gravitational field of the Sun is responsible for the
motions of the planets in the manner described by Kepler’s
laws. The theory of mechanics and gravitation was put forth
by Newton in 1686 with confidence because the theory
described the motions of the planets, the Moon, etc. with
remarkable precision.
Over the following two centuries, observations of the
motions of the planets were refined, using the Newtonian
theory of mechanics and gravitation to compute the slight
gravitational effects of the planets and their moons on each
other, and including the precession of the elliptical orbits,
the precession of the planetary spin axes, etc. By the middle
of the 19th century, the observations had become so
precise that it was possible to show that the precession of
e-mail: parker@odysseus.uchicago.edu
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
the orbit of Mercury is of the order of 30 arc seconds per
century greater than predicted by the Newtonian theory. By
the end of the 19th century, the extra precession was
established as close to 43 arc seconds per century. The
only planet showing a significant anomaly was Mercury,
closest to the Sun, where the presumably radial
gravitational field of the Sun is strongest. Simon Newcomb
suggested that the inverse square law of gravity might have
to be modified.
The modification of Newtonian theory in 1916 by Albert
Einstein’s geometrized reformulation of gravitational theory,
known as general relativity, brought theoretical mechanics
into agreement with the observation by providing a
theoretical precession larger than the Newtonian value by
precisely 43 arc seconds per century. General relativity also
predicted the small deflection of starlight passing near the
Sun to be 1.75 arc seconds for a light ray grazing the surface
of the Sun and quite different from the prediction of
Newtonian gravity and special relativity theory. It also
predicted a longer transit time for light or radio signals
propagating across the deep gravitational potential well
close to the Sun. Both of those effects have since been
verified, the latter with great precision.
Now an obvious question concerns the uniqueness of
Einstein’s geometrical formulation of gravitational theory,
based on the simplest lowest order terms of the metric
tensor. There is a variety of other mathematical formulations
that, like general relativity, reduce exactly to Newtonian
gravitation in the limit of weak fields. They are in some ways
more complicated than general relativity, involving an
admixture of scalar field along with the tensor field. The
Dicke–Brans–Jordan theory represents what was perhaps
the most popular alternative. The theory introduces an extra
free parameter defining the relative strength of the scalar
field, thereby providing a wide range of precession of the
perihelion of the orbit of Mercury. This is a qualitative
difference from the unique value of 43 arc seconds per
century predicted by general relativity. Dicke then pointed
out that there is a possibility that the inner core of the Sun
is spinning much more rapidly than the visible surface. The
idea is based on the fact that newly-formed stars are
observed to rotate rapidly, with periods of just a few days,
as distinct from the 25-day period for the equatorial surface
of the present middle-aged Sun (4.6 × 109 years). The initial
angular momentum of a star is then reduced over the first
3 × 108 years of life by the mass loss (stellar wind) flowing
out through the long arms of the extended magnetic field of
the star. There is no reason to think that the magnetic field
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penetrates through the core of the star, so the process
might well leave the core with the original spin. The point of
this speculation is that a rapidly spinning core would be
slightly flattened (oblate) by the centrifugal force, and the
gravitational field of the core would be correspondingly
nonspherical. It is readily shown that a consequence would
be a precession of the perihelion of Mercury produced by
the core alone. So, the error in the Newtonian theory would
be less than the 43 arc seconds per century, and the unique
prediction of general relativity would be in error, agreeing
with the observational result only by chance coincidence.
The experimental test of this idea lay in searching for the
associated very slight oblateness of the surface of the Sun.
Dicke developed an optical apparatus for scanning around
the rim of the Sun to see whether it might be slightly oblate.
The rim of the Sun is distorted by prominences, faculae, the
chromosphere, etc. However, by scanning just inside the
limb of the Sun, Dicke devised ingenious ways of reducing
the contributions of these effects, and, in the end, there has
been no conclusive demonstration of any interesting
deviation from a purely circular solar disk. What is more,
recent probing of the rotation of the Sun by helioseismology – on which more will be said later – shows no
rapidly spinning core. The verdict seems to be that the Sun
is round, general relativity is precisely correct,
and there is no significant presence of a scalar gravitational
field in addition to the tensor field of general relativity.
Note, then, that general relativity has become the theoretical basis for understanding the dynamics of the expanding universe, with recent observational indications that the
expansion may be accelerating and the bothersome
cosmological constant may not be identically zero.
2. Internal structure
Now if the gravitational field of the Sun and the rotation rate
of the core have played a central role in establishing the
correct formulation of gravity and mechanics, what can the
structure and behaviour of the solar interior tell us about its
physics? The development of the kinetic theory of gases
and the recognition of the critical temperature of liquids and
solids in the second half of the 19th century led to the
realization of the entirely atomic gaseous composition of the
Sun, too hot throughout the interior for any chemical
bonding to form molecules. Spectroscopy and
thermodynamics established the surface temperature at
5600 K. The virial theorem of mechanics made it clear that
the interior must be very much hotter (∼ 107 K) in order to
maintain the radius (7 × 105 km) in opposition to the
immense gravity (28 times the gravitational acceleration of
9.8 m/sec2 at the surface of Earth). Lane in 1869 and Emden
in 1907 constructed simple mathematical models of the
interior of the Sun by assuming that the temperature varies
as some power of the density, T ∼ ργ–1, where γ (> 1) is a
constant.
Thus,
for
instance,
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if a purely monatomic gas were to mix adiabatically
throughout the interior, γ would be 5/3. The essential point
is that the mathematical model could be made to fit the mass
and radius of the Sun by suitable choice of mean atomic
weight and γ; and one could see how the kinetic theory of
gases and Newtonian gravitation provide a satisfactory
representation of the Sun. To do more required some
knowledge of the physics of the outward radiative transfer
of heat within the Sun2.
An obvious question, then, was the heat source that
maintains the temperature of the Sun. Chemical combustion
is entirely inadequate, providing the energy of the Sun for
only a couple of thousand years. Helmholtz in 1854, and
later Kelvin, pointed out that the gravitational energy of the
Sun is more than a thousand times greater than any
hypothetical chemical energy. The concept is quite simple.
The surface of the Sun is hot, so that it cannot avoid
radiating energy away into space. So the internal thermal
energy would decline except for the fact that gravity would
cause the Sun to shrink, compressing the gas until it is even
hotter than before in order to oppose the increased gravity
of the shrinking Sun. So the temperature within, and at the
surface, increases as the radius of the Sun declines. To put
it in other words, the Sun is stable against radial
disturbances. An inward kick merely causes the Sun to
contract and then rebound. So the net result of the
continuing radiation from the surface of the Sun would be
slow contraction over several million years, with the
gravitational force compressing and heating the gas to even
higher temperatures. The Sun would grow brighter with the
passage of time as a consequence of the temperature
increase.
The rate of contraction of the Sun is easily computed to
be about 50 m/year and is entirely undetectable in the brief
span of human science. The Sun would gradually increase
in brightness over a couple of million years. It is interesting
to note that Kelvin took this scenario so seriously that he
rejected Darwin’s proposal of biological evolution on the
grounds that there simply was not enough time for
evolution to take place. Fortunately the geologists stepped
in with clear evidence that – Darwin or not – the rock
structures found at the surface of Earth require 108 to 109
years for their formation, and there was evidently plenty of
time for biological evolution. Thus the age of Earth vitiated
Kelvin’s gravitational contraction as the principal source of
energy for the Sun.
With the recognition by Einstein of the equivalence of
matter and energy, thinking turned to the annihilation of
mass by some unknown process as the sustaining source
for the Sun2. With the luminosity of 4 × 1033 ergs/sec, this
means that the mass of the Sun is declining at the
rate of 4 × 1012 g/sec (4 million tons/sec). Ultimately, Bethe
and others3–5 supplied the answer based on the newly
founded nuclear physics, pointing out both the proton–
proton chain and the carbon cycle. Both processes fuse
four hydrogen nuclei into one helium nucleus, converting
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about one per cent of the mass into energy. The proton–
proton chain is the more effective of the two in the Sun,
where the central density is of the order of 102 g/cm3 and the
temperature is 1.5 × 107 K. Hydrogen is converted into
helium at the rate of 4 × 1014 g/sec (400 m tons/sec).
Now the present age of the Sun is inferred from the
assumption that the Sun and Earth were formed more or less
simultaneously, and the uranium and lead isotope ratios in
the crust of Earth indicate an age of 4.6 × 109 years. In that
span of time, approximately 6 × 1031 g of hydrogen,
representing 3 per cent of the mass of the Sun, have been
converted to helium, with a total loss in solar mass of
6 × 1029 g.
The production of helium in the central core of the Sun
increases the mean atomic weight, of course, and the result
is a slow contraction of the core to increase the temperature
to support the core in opposition to gravity. The helium
also has the effect of diluting the hydrogen so that the core
contracts until the density and temperature increase to
maintain the thermonuclear energy supply. Thus the Sun is
now smaller, hotter, and approximately 30 per cent brighter
than it was in early times. The fact that the Sun was
substantially fainter 4 × 109 years ago raises interesting
questions about the temperate climate of Earth, and even
Mars, at that time.
3. Composition of the Sun
Before going into helioseismology and neutrino emission,
consider the knowledge of the interior of the Sun about
a hundred years ago. Spectroscopy showed a spectrum
dominated by the lines of such elements as carbon, calcium,
sodium, silicon, iron, magnesium, etc., suggesting that the
Sun was composed largely of the vapours of these
substances. The immediate problem was that such large
atomic weights required a much hotter Sun than the
mathematical models could accommodate to maintain the
Sun against its own gravity. Hydrogen and helium (which
was discovered spectroscopically on the Sun before it
was known in the laboratory), suggested by the atomic
weights needed for the mathematical models, were relatively
inconspicuous in the solar spectrum, and in any case were
too transparent at 5600 K to account for the opaque surface
layer of the Sun. The fuzziness of the visible surface of the
Sun in white light is limited to something of the order of
100 km at the center of the disk.
To make a long story short, the dilemma was resolved
by the negative hydrogen ion—a hydrogen atom with an
additional electron orbiting about it. Hans Bethe first
calculated the existence of this atomic structure, pointing
out in 1929 that the electron is attracted to the electrically
neutral hydrogen atom because the presence of the external
electron attracts the positively charged nucleus and repels
the negative electron cloud. Thus, the otherwise spherical
hydrogen atom is slightly distorted with a small net positive
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
charge in the end pointing toward the external electron.
Ten years later, Wildt6 pointed out the astrophysical
implications of the negative hydrogen ion. The loosely
bound external electron is nicely tuned to absorbing and reradiating light, so it serves very well at the 5600 K
temperature of the visible surface of the Sun in impeding the
passage of light and providing a relatively sharp visible
surface. Chandrasekhar7 and others undertook the difficult
task of precise calculations of the properties of the negative
hydrogen ion.
Spectroscopic observation of the ‘red flames’ or chromosphere of the Sun that peek around the limb of the Moon
during an eclipse indicated that hydrogen is the most
abundant element in the Sun. Hydrogen shows clearly in
the chromosphere because the temperatures are somewhat
higher (6000–8000 K) than at the visible surface. The 5600 K
at the surface only weakly excites the sturdy hydrogen
atom, where the hydrogen is able to hang on to that extra
electron to form the negative ion. The chromosphere is
better suited to exciting the hydrogen atom, but, of course,
is so hot as to knock the negative hydrogen ion to pieces.
So the dilemma was resolved! The Sun is mostly hydrogen,
with about one in ten atoms being helium. The heavier
elements are present only at the one per cent level, or less,
and their weaker electron structures are strongly excited by
the 5600 K at the surface so that they dominate the line
spectrum there.
With some knowledge of the composition of the Sun, the
next step in working out the structure of the interior requires
knowledge of the impediment of the gas to the passage of
electromagnetic radiation, dominated by soft X-rays in the
core where T ∼ 1.5 × 107 K, and by visible light at the
surface where T ∼ 5600 K. There is an ultraviolet domain
between. The impediment is called the opacity and its
calculation requires computing the absorption and reradiation by each of the atomic constituents. The high
temperature in the deep interior of the Sun means that the
atoms are highly ionized, with only the heavier elements,
e.g. Ca, Si, Fe, etc. able to hang on to one or two of their
innermost electrons. These few bound electrons make a
major contribution to the opacity so essentially all atomic
species have to be treated in their several states of
ionization.
One can begin to see the immensity of the task2: One
needs precise detailed models of the solar atmosphere in
order to determine the relative abundances of the elements
at the visible surface from spectroscopic observations. The
assumption is made that the relative abundances of the
elements is the same in the interior as at the surface. Then
the enormous calculation begins computing the probability
of each number of electrons knocked off each different
element at arbitrary temperature and density, and then
computing the absorption of radiation by the electrons still
attached. From this a table of opacity as a function of
temperature and density can be constructed. Then finally,
one can address the problem of computing inward from the
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visible surface where the temperature and density are
known. The temperature must increase inward sufficiently
rapidly to provide the observed outflow of heat
(4 × 1033 ergs/sec). At the same time the density must
increase inward sufficiently rapidly that the gas pressure
(the product of temperature and density) is sufficient to
support the weight of the overlying layers of gas against
the gravitational field of all the matter below. For this
reason, the procedure is usually reversed, starting with
an assumed temperature (∼ 1.5 × 107 K) and density
(∼ 100 g/cm3) at the center and calculating outward. The
local rate of thermonuclear energy production has to be
worked out too, of course, in order to know the outflow of
energy at each radius. When the calculation arrives at the
surface, where the temperature and density essentially
vanish, the total mass and radius of the theoretical model
are obtained. The calculation is repeated for different central
temperature and density until the result matches the Sun.
Then, once the Sun is properly modelled, the calculations
can be applied to other stars where the radius is not
precisely known, although the masses can be determined in
double-star systems. It is interesting to note, then, that the
deep interior of the Sun turns out to be stably stratified
against vertical mixing. That is to say, the central core is
hotter than the overlying layers, but not enough hotter that
it can exchange places. Any upward displacement of gas
from the core provides expansion and cooling to such a
degree that the uplifted gas is cooler than the ambient gas,
so that the uplifted gas is denser than its surroundings and
relaxes back into the core.
This is all based on well-known principles of physics, but
the computation of opacities, with hundreds of different
ions, is so complex and tedious that estimates are
introduced in place of detailed computations for many
classes of ions. The problem has been taken up in weapons
laboratories where the properties of nuclear bomb
explosions are explored theoretically, using many of the
same opacities. The tables of opacities have been improved
for this purpose, allowing more precise calculation of the
interior of the Sun. A number of refinements have been
introduced, such as the accumulation of helium throughout
the central core as the hydrogen burning progresses, and
the gravitational settling of the heavier ions relative to the
lighter constituents with the passage of time. The result is a
detailed quantitative model of the internal structure of the
Sun.
Then comes the ultimate question: Is there some way to
check this sophisticated theoretical model of the interior to
be sure that it is correct? Are the abundances of the
elements really the same as at the surface? Is there really no
vertical mixing of the elements in the core? Is the Sun really
as old, or as young, as the Earth?
4. Probing the interior
The first test for the theoretical model of the solar interior
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was based on the emission of neutrinos from the thermonuclear reactions in the core. Neutrinos pass so freely
through matter that they easily escape from the core, being
the only messengers coming directly from the
core to the outside world. Needless to say, their free
passage through matter makes them extremely difficult to
detect – difficult but not impossible8,9. With six-hundred
tons of cleaning fluid in a tank deep in the Homestake Mine
in South Dakota (to get away from the spurious effects
produced by cosmic rays) Ray Davis looked for neutrinos
above 0.8 MeV reacting with the chlorine in the cleaning
fluid to produce the radioactive isotope 40A. The argon
atoms are radioactive with a half-life of about a month, and
are swept out of the cleaning fluid about once a month to be
detected and counted by their radioactive decay. To make a
long story short, the theoretical models of the solar interior
forty years ago when the planning of the experiment was
initiated, suggested one or two argon atoms per day. In fact
Davis found nothing beyond statistical fluctuations in the
background count rate. A careful re-evaluation and
refinement of the theoretical model, leading to a small
downward
readjustment
of
the central temperature of the Sun, drastically reduced the
expected neutrino detection rate, to something of the order
of five per month. The continuing accumulation of data in
the Homestake detector gradually reduced the statistical
uncertainties, to where it began to appear that there really
were neutrinos from the Sun, but only about a third of the
expected number10,11. The question was whether the
theoretical model of the Sun was in error, or was there
something about the physics of neutrinos that was not
properly understood? In particular, neutrinos have
conventionally been assumed to have no rest mass, in the
same way that a photon of light has no rest mass. That is to
say, a neutrino travels through free space at the speed of
light and experiences no passage of time. However, it is not
impossible that a neutrino has a rest mass, so that it
experiences the passage of time. If so, it would open up the
possibility that the electron neutrinos emitted from the core
of the Sun oscillate through the mu-neutrino state and tau
neutrino state during their passage to Earth. In that way the
electron neutrinos would spend only one-third of their time
in the electron–neutrino state to which the Homestake
detector is sensitive. The result would be the detection of
only one third of the expected number of neutrinos.
This dilemma has been cleared up in favour of unknown
neutrino physics, because of the precise probing of the
interior of the Sun through helioseismology. It was first
noted by Leighton that the surface of the Sun is continually
agitated
by
fluctuations
with
periods
of
the general order of 5 min. Imagine a pan of water
sitting in a metal sink with some such machinery as a
garbage grinder running immediately below. The vibrations
set up a pattern of small-scale waves on the surface of the
water. Roger Ulrich12 and Leibacher and Stein13 recognized
that the oscillations in the Sun represent sound-waves
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trapped between the surface of the Sun and a refractive
turn-around far below the surface. The Sun rings like a bell,
but at a thousand different frequencies all at once. The
essential point is that the period of each mode of oscillation
is just the sound-transit time along the path of the sound
wave down and back up through the interior. Each acoustic
mode represents a sound wave travelling on a different path
down into the Sun, around, and back up to the surface.
Some paths go deep, some are only shallow, etc. The
theoretical model of the solar interior provides the speed of
sound as a function of depth from which one can calculate
the path of each acoustic wave mode, and the transit time
along that path for each acoustic mode. The transit time
determines the period of the oscillation at the surface of the
Sun. The calculated periods are then compared to the
observed periods for several hundred different modes,
sampling different depths. It is a precision test of the
theoretical model, and it is gratifying to find that, when all
the refinements of the theoretical model are included, the
periods provided by the theoretical model agree closely
with the observed periods, indicating that the speed of
sound in the theoretical model nowhere differs by more than
onepart in five-hundred from the actual speed of sound—
approximately the expected observational error11. It appears
then that the theoretical model of the interior of the Sun is
properly constructed and there are no anomalous
abundances of elements in deeper layers of the Sun. This is
a major triumph for theoretical physics.
Turning again to the low level of neutrino emission from
the Sun, the evident accuracy of the theoretical model of the
interior leaves only unknown neutrino physics as the
explanation for the discrepancy11. This challenge has
generated a vigorous response among physicists. The first
step was to build a neutrino-detecting system using gallium
instead of chlorine (in both Italy and the former Soviet
Union) to detect neutrinos down to 0.2 MeV, thereby
picking up the neutrinos emitted in the initial p–p reaction
of the proton chain. The huge water–Cerenkov detector,
Kamiokande, in Japan, was also applied to the task,
providing both the energy and direction of the incoming
neutrino. The data is from these three additional projects,
and they all detect neutrinos from the Sun, but only at
about a third or half of the predicted number, more or less
along the same lines as the Homestake detector. The next
generation of neutrino detectors is coming on line, designed
to give the energy and direction of each incoming solar
neutrino at substantial count rates of ten or more per day
(Super Kamiokande in Japan; the heavy-water Sudbury
Neutrino Observatory in Ontario (Canada), and Borexino
(Italy)). In the meantime, Super Kamiokande has shown that
neutrinos do, in fact, have a nonvanishing rest mass, based
on measurements of the difference in the up-and-down
fluxes of the neutrinos produced by cosmic rays colliding
with the terrestrial atmosphere. The essential point is that
the downward neutrinos are fresh from their creation in the
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
atmosphere overhead, while the upward moving neutrinos
have passed through the solid Earth in a time of the order of
30 milliseconds, allowing time for conversion to neutrino
states not detected by Super Kamiokande.
5. Convection and magnetic fields
There is a branch of classical physics that is challenged to
come up with new concepts, by the seemingly innocent fact
that the theoretical model of the interior of the Sun provides
an outer region of convective turnover, extending up to the
visible surface of the Sun from a depth of 2 × 105 km (solar
radius = 7 × 105 km)14. The continual turnover of the gas
arises because the radiative transfer of heat deep in the Sun
becomes less effective as the temperature diminishes
outward, with the result that it cannot handle the heat
transport when the temperature falls below about 2 × 106 K
at the depth of 2 × 105 km. The outward heat flow is
4 × 1033 erg/s, and to transport so much heat by radiation
the temperature would have to decline outward so rapidly
that the cooler gas above becomes too dense, and the
hotter gas below becomes too tenuous for stability. The hot
and cold gases prefer to exchange places, providing
convective turnover just like the rolling boil of the water in a
pot on a hot stove. The resulting vigorous vertical mixing of
the gas takes over the heat transport to the surface.
The small effect of the convection on the overall
theoretical model of the solar interior is easily handled by
approximate methods known from hydrodynamics as the
mixing length theory, and, as already noted, there seems to
be no problem there. The new physics arises because the
gas within the Sun is so hot as to be ionized, providing free
electrons so that the gas is an excellent conductor of
electricity – as good as cold copper in the central regions.
The high electrical conductivity over the broad Sun means
that there can be no significant electric field in the local
frame of reference of the moving fluid, because any attempt
to initiate an electric field would be met by a rush of free
electrons, neutralizing the attempt. It follows that any
magnetic field present in the gas is carried along bodily in
the swirling convection, always moving in the frame of
reference in which there is no electric field. The magnetic
field is swirled, stretched, and deformed just like a wisp of
smoke. Only when the magnetic field becomes so strong
that it can physically stop the convection does the mixing,
stretching, and intensifying of the field cease.
The effects that arise from this simple magnetic transport
property of hot ionized gas, or plasma, are legion, and
involve hitherto unfamiliar combinations of hydrodynamics,
magnetohydrodynamics, and local radiative transfer. First of
all, there is the convection itself, extending from where the
plasma density is 0.2 g/cm3 at the base of the convective
zone up to the visible surface where the plasma density is
0.2 × 10–6 g/cm3. An understanding of convection in an
atmosphere with such strong vertical stratification is only
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beginning to be developed. Numerical simulations are the
principal tool for exploring the subject, but, computers are
still some way from being able to handle so much
stratification. The convection has a number of
characteristics quite unlike convection in a pot of water,
where the fluid has essentially uniform density. For
instance, numerical simulations have shown that there are
downward plunging clumps of cold dense gases. The
nonuniform rotation of the Sun is presumed to be a direct
result of the convection, but it has not yet been possible to
simulate the convection with sufficient accuracy and detail
to show how this works. As another example, the formation
of a sunspot represents a systematic concentration of
magnetic field driven by the convection in opposition to the
enormous pressure of the magnetic field, and once again the
convective mechanism is not understood.
The explosive flare phenomenon is an example of
magnetohydrodynamic interaction where at least the
general principles seem to be in hand. Flares are observed
on all scales from the largest (∼ 1032 ergs over dimensions of
104 km) down to the limit of detection (∼ 1025 ergs over 102–
103 km), rapidly converting magnetic free energy into hot
plasma and fast particles. The basic effect appears to be
rapid dissipation and reconnection of nonparallel magnetic
fields. The discovery of this peculiar aspect
of classical physics was motivated by the otherwise
inexplicable explosive conversion of magnetic energy
required to explain the flare15,16. That is to say, the physicist
is lured onward by the mysteries presented by Nature, and
the observations provide some hint as to the nature of the
unknown physical effects.
Rapid reconnection involves such diverse phenomena as
plasma turbulence in the small and the Petschek effect in the
large, and the subject is active today, with attention
directed to the various forms the dynamics can take in three
dimensions and to the acceleration of ions and electrons to
very high energies in the central regions of the dissipation.
The ubiquitous nature of rapid reconnection in the
astronomical universe is demonstrated observationally by
the widespread astronomical appearance of million degree
tenuous
plasmas
and
fast
particles,
and
is to be understood in terms of the spontaneous appearance
of discontinuities (intense current sheets) in any magnetic
field embedded in a plasma undergoing slow continuous
deformation17,18. The effect arises from the nature of the
Maxwell stress tensor for the magnetic field. Each surface of
discontinuity, or current sheet, becomes a site for resistive
instabilities and rapid reconnection, i.e. explosive
dissipation of magnetic free energy. It appears that this
general theoretical property of the magnetic field is the
major heat source responsible for the X-ray emitting corona
of the Sun, on which more will be said later.
This brings us to the fact that the outer atmosphere of
the Sun – the corona, conspicuous when the dazzling disk
of the Sun is obscured by the Moon during an eclipse – is
heated to temperatures in excess of a million degrees. So
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high a temperature was suggested a hundred years ago by
the great outward extension of the corona in opposition to
the powerful gravitational field of the Sun (28 times the
acceleration of gravity at the surface of Earth). Then in the
1930s the temperature was confirmed by some clever
experiments in the terrestrial spectroscopy laboratory,
showing that the observed spectral lines are from 10, 11,
and more times-ionized iron, silicon, calcium, etc., occurring
only at million degree temperatures19–21.
The corona is so tenuous (typically 108 atoms/cm3 near
the Sun) that it cools only relatively slowly by radiation,
while it has an enormous thermal conductivity as a consequence of the high temperature and the associated
104 km/sec thermal velocities of the free electrons. Thus it is
not surprising that the million degree temperature extends
far out into space. The corona is strongly bound by gravity
near the Sun, where the mean thermal energy is only about a
tenth of the energy necessary to escape from the Sun. The
outer regions, far from the Sun, are not strongly bound and
escape into space. The surprise was that Newton’s
equation of motion showed that there is no static
equilibrium for such an atmosphere, the only steady state
being gradual outward acceleration, from negligible velocity
(∼ 1 km/sec) near the Sun to supersonic velocity (300–
1000 km/sec) at large distance. This is the origin of the solar
wind, which drags the weaker magnetic fields of the Sun
along with it, thereby filling all of interplanetary space with
an
outward
sweeping
spiral
(because
of
the rotation of the Sun) magnetic field22. The outward
sweeping wind and field push back the galactic cosmic rays
to some degree, impact and agitate the magnetic fields of
Earth, Jupiter, Saturn, etc., and generally determine the
dynamical state of space throughout the solar system23. The
outstanding question remaining is the heat source that
creates the million degree temperatures around the Sun.
6. Magnetic fields of the Sun
Ordinarily, in the terrestrial laboratory, we do not think of
magnetic fields in association with gases; we think of iron
magnets and coils of copper wire carrying electric currents
(electromagnets). But in fact, as already noted, a gas
sufficiently hot as to be fully ionized and an excellent
conductor of electricity over the large dimensions appropriate to the Sun carries with it whatever magnetic fields
happen to be present, on a more or less permanent basis.
Rapid reconnection at incipient discontinuities is the one
scheme that can quickly dissipate the magnetic field, by the
simple expedient of creating very small scales in the
structure of the field. But this rapid dissipation is effective
only in the tenuous outer atmosphere of the Sun, and it is
not obvious that it occurs in the tumbling convection below
the surface of the Sun, where the gas is so dense as to
respond only sluggishly to the magnetic forces.
Magnetic fields were first established on the Sun by
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
George Ellery Hale24 when he observed the Zeeman splitting
of some of the spectral lines in sunspots. The splitting
indicated magnetic fields of 2000–3000 G in the dark central
umbra of the spot. That is a very strong field, comparable to
what can be produced in a strong electromagnet. Hale went
on to point out that the magnetic field in the leading and
following spots of each bipolar sunspot group had opposite
sign, suggesting that the field emerges from one, arches
over and returns back into the Sun in the other. He noted,
too, that magnetic fields have opposite directions in the
northern and southern hemispheres, and the whole
magnetic system reverses with each successive 11-year
sunspot cycle, which is now often referred to as the
magnetic cycle since it is driven by the generation of
magnetic field deep in the convective zone.
Following World War II, the advent of electronics made it
possible to develop a much more sensitive magnetograph,
which soon detected magnetic fields of other stars and
showed that there is a general background dipole magnetic
field in the Sun of about 10 G, extending in at the north pole
and out at the south pole, or vice versa in the next 11-year
period25. The general field reverses near the maximum of the
sunspot cycle when most of the sunspots are popping up
within about 15° of the solar equator. The Babcocks also
found that the active regions, in which the sunspots and
large flares occur, lie in the midst of extended bipolar
regions of ∼ 100 G. The bipolar character of these regions of
magnetic activity indicates that the magnetic fields are part
of a general intense east–west magnetic field somewhere
deep in the convective zone. The individual bipolar active
region is created by the upward bulging of a segment of
that east–west field, the bulge having the form Ω of the
capital omega.
The first question is obviously the origin of the magnetic
fields, for which the answer seems to be that the
nonuniform rotation of the Sun (for instance, the equatorial
surface of the Sun has a 25-day rotation period, while the
polar regions rotate in approximately 30 days) continually
shears and stretches out the dipole component of the
magnetic field into an east–west field, while the cyclonic
rotation of the tumbling convection raises and rotates Ω
loops in the east–west field to reinforce the dipole field26–28.
Why, then, are sunspots formed in the otherwise
100 G surface regions of bipolar field? There one can only
say that ‘we do not know’. The magnetic field somehow
interacts with the convection and the convection somehow
interacts with the magnetic field to compress the field into
the 2000–3000 G, that is observed. This all takes place in
opposition to the enormous pressure of the 2000–3000 G
field.
It is observed that the tenuous gas trapped in the strong
(100 G) bipolar fields of the active regions is heated to
temperatures of 2–5 × 106 K, and along some bundles
of field lines the density becomes so high (109–
1010 atoms/cm3) as to produce strong thermal emission of Xrays (107 ergs/cm2 sec). In contrast, the broad regions of
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
weak field (5–10 G) are heated to temperatures
of 1.5 × 106 K and more, with densities limited to
108 atoms/cm3 by the free expansion of the gas to form the
solar wind, as already noted. It appears at the present time
that this expanding corona is heated largely by the
microflares (1025–1028 ergs) among the small magnetic
elements that appear at the surface of the Sun, while the
dense X-ray corona is heated by even smaller flares
(nanoflares, below the limit of detection by present
instruments) arising in the spontaneous discontinuities
created in the 100 G bipolar fields by the continual
intermixing of the footpoints of the field at the convecting
surface of the Sun18.
Speaking of the small magnetic elements brings us to
further subtleties in the mysterious behaviour of the Sun.
About thirty years ago it became clear from quantitative
observational studies that the magnetic fields of the Sun do
not form a continuum at the surface of the Sun, but instead
are made up of widely separated tiny concentra-ted
magnetic flux bundles or magnetic fibrils of 1000–2000 G
with diameters of the general order of 100 km. Each separate
fibril is held in the grip of the surrounding gas, for otherwise
it would expand and disperse. The typical observed mean
field of 10 G is merely a measure of the spacing of the many
unresolved individual magnetic fibrils. The 100 G regions
have their fibrils more closely spaced. Only recently has it
been possible to detect the individual fibrils and it is not
possible to resolve their internal structure nor to study their
interactions with ground-based telescopes because the
atmosphere above the telescope limits the resolution to
∼ 300 km even under the best seeing conditions.
One asks why the field is in this peculiar state with so
much ‘unnecessary’ free energy. New fibrils continually
bulge upward through the surface, forming small Ω loops
and jostling against each other to provide the microflaring
that seems to be the main energy source for the solar wind.
These small-scale fibrils and bipoles appear over almost the
entire surface of the Sun. These background fibrils show
only modest variation with the 11-year cycle, suggesting a
different origin from the main fibril fields of the active
regions.
A critical review of the standard explanation for generating the magnetic fields of the Sun, already mentioned,
offers no enlightenment, and, in fact, turns up another
serious puzzle. For the fact is that the generation of magnetic field by the cyclonic convection and nonuniform
rotation of the Sun requires that the magnetic field diffuse
across the surface of the Sun and the depth of the
convective zone during the 11-year magnetic cycle. No
adequate diffusion mechanism is known. We used to think
that the turbulent convection mixes the magnetic field over
these dimensions, the way turbulence mixes a puff of smoke
throughout a room. But it is now clear that the mean
magnetic field in the deep convective zone is
at least 103 G, and much too strong to submit to the
‘indignity’ of turbulent mixing. It may be that the answer to
1451
SPECIAL SECTION: SOLAR PHYSICS
the dilemma lies in the individual fibril being the basic
magnetic entity, rather than the mean fields usually
employed in calculating the generation of magnetic field.
The fibrils are capable of rapid reconnection where they
meet and may be transported more freely. But, all this has to
be worked out quantitatively before any claim to
understanding can be made.
It is clear that the first step in studying the fibril nature of
the magnetic field is to develop and construct a groundbased telescope that can resolve and study the detailed
properties of the magnetic fibrils, as the principal players in
the magnetic activity of the Sun. They appear to be the
microscopic architects of sunspots, flares, coronal heating
and the solar wind, and the generation of magnetic field
itself, and yet we cannot see them clearly from Earth. The
development of adaptive optics to correct for the blurring
by the atmosphere above the telescope has now progressed
to the point that the necessary resolution of
0.1″ (75 km), or better, should be possible. High-spectral
resolution combined with the necessary rapid-observing
cadence (∼ 10 sec) and the high-spatial resolution require a
large aperture (∼ 4 m) to gather enough photons. It is
an essential step if we are to advance the physics of the
active Sun. In fact the implications of the variable magnetic
activity of the Sun, the associated varying brightness of the
Sun, and the resulting climatic effects here at Earth, together
with the implications for the activity of all stars and the new
physics to be learned, place the successful construction of
such a solar telescope at the highest priority, which brings
us to the last section of this brief review.
7. The terrestrial challenge
NASA began monitoring the brightness of sunlight with
absolute radiometers on orbiting spacecraft in 1978, with the
startling discovery that the brightness varies by as much as
0.15 per cent with the 11-year variation of sunspots, flares,
and general magnetic activity of the Sun. More recent
monitoring of other solar-type stars shows that they do
much the same, with one such star, ominously, showing a
decline in brightness of 0.4 per cent over only six years as
its activity tumbled to low levels29. Jack Eddy30 emphasized
some years ago that the Sun was almost entirely without
activity over the seventy-year period of 1645 to 1715, called
the Maunder Minimum after its discoverer at the end of the
19th century. With modern 14C-production data, Eddy went
on to show that the Sun went through another extended
inactive period during the 15th century and a prolonged
state of hyperactivity during the 12th century. Zhang et
al.29 estimated that the brightness of sunlight was depressed by something of the order of 0.4 per cent during the
Maunder Minimum and probably enhanced above normal
by a comparable amount during the 12th century. Then
Eddy noted that the mean annual temperature in the
Northern Temperate Zone varied up and down 1–2°C, while
1452
tracking these variations in solar activity. The close tracking
of climate with solar activity has been investigated in detail
since that time and proves to be much closer than Eddy
could have imagined with the data available at that time.
Historically, the extreme cold periods had devastating
consequences for agriculture in northern Europe and China,
and the warm periods had devastating consequences
around the periphery of desert regions, e.g. what is now
southwestern United States.
The bottom line is that the great physics laboratory in the
sky not only extends our opportunities to study physics,
but some of its more mysterious demonstrations have
profound implications for the human population of Earth. In
particular, the Sun has become substantially more active
during the 20th century, and presumably brighter on the
average by as much as 0.1 per cent. The general warming of
the climate from 1900 to 1950 would appear to be a
consequence of this phenomenon. Since that time the
climate picture has been complicated by the substantial
increase in carbon dioxide in the atmosphere and the warmer
sea water temperatures which discourage the absorption of
the carbon dioxide into the oceans. It is a problem that
needs to be thoroughly investigated so that we can have
some idea of how to respond to these changes.
1. Parker, E. N., Solar Phys., 1997, 176, 219.
2. Eddington, A. S., The Internal Constitution of the Stars, Dover,
New York, 1926.
3. Bethe, H. A., Phys. Rev., 1939, 55, 434.
4. Bethe, H. A. and Critchfield, C. L., Phys. Rev., 1938, 54, 248.
5. Weizsacher, C. F., Phys. Z., 1938, 39, 633.
6. Wildt, R., Astrophys. J., 1939, 89, 295.
7. Chandrasekhar, S., Astrophys. J., 1944, 100, 176.
8. Davis, R., Phys. Rev. Lett., 1964, 12, 303.
9. Bahcall, J. N., Phys. Rev. Lett., 1964, 12, 300.
10. Bahcall, J. N., Calaprice, F., McDonald, A. B. and Totsuka, Y.,
Phys. Today, 1996, p. 30.
11. Bahcall, J. N., Pinsonneault, M. H., Basu, S. and ChristensenDalsgaard, J., Phys. Rev. Lett., 1997, 78, 171.
12. Ulrich, R., Astrophys. J., 1970, 162, 993.
13. Leibacher, J. W. and Stein, R. F., Astrophys. J. Lett., 1991, 7,
L191.
14. Schwarzschild, M., Structure and Evolution of the Stars,
Princeton University Press, Princeton, 1958.
15. Parker, E. N., J. Geophys. Res., 1957, 107, 830.
16. Sweet, P. A., Nuovo Cim. Suppl., 1958, 8, 188.
17. Parker, E. N., Astrophys. J., 1972, 174, 499.
18. Parker, E. N., Spontaneous Current Sheets in Magnetic Fields,
Oxford University Press, New York, 1994.
19. Grotrian, W., Naturwissenschaften, 1939, 27, 214.
20. Lyot, B., Mon. Not. R. Astron. Soc., 1939, 99, 580.
21. Edlen, B., Z. Astrophys., 1942, 22,30.
22. Parker, E. N., Astrophys. J., 1958, 128, 664.
23. Parker, E. N., Interplanetary Dynamical Processes, Interscience
Div, J. Wiley and Sons, New York, 1963.
24. Hale, G. E., Astrophys. J., 1908, 28, 100, 315.
25. Babcock, H. W. and Babcock, H. D., Astrophys. J., 1955, 121,
349.
26. Parker, E. N., Astrophys. J., 1955, 122, 293.
27. Parker, E. N., Proc. Natl. Acad. Sci., 1957, 43, 8.
28. Parker, E. N., Cosmical Magnetic Fields, Clarendon Press,
Oxford, 1979, pp. 532–815.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
29. Zhang, Q., Soon, W. H., Baliunas, S. L., Lockwood, G. W., Skiff,
B. A. and Radick, R. R., Astrophys. J. Lett., 1994, 427, L111.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
30. Eddy, J. A., Science, 1976, 192, 1189.
1453
SPECIAL SECTION: SOLAR PHYSICS
Seismic sun
S. M. Chitre and H. M. Antia
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
Even though the interior of the Sun is not directly
accessible to observations, it is nonetheless possible to
infer the physical conditions inside the Sun using the
theory of stellar structure and the accurately measured
frequencies of solar oscillations. The helioseismic data has
provided a powerful tool to probe the Sun and also to test
physical theories describing its internal constitution.
1. Introduction
THE Sun has been verily described as the Rosetta Stone of
astronomy. This is very apt since our nearest cosmic
laboratory is readily available for studying a variety of
physical processes operating both inside and outside the
object. Astrophysicists have an abiding hope that the
study of the Sun can serve as a guide for theory of
structure and evolution of stars in general, and pulsating
stars in particular. Clearly, its internal layers are not directly
accessible to observations. Nonetheless, it is possible to
construct a reasonable picture of the interior with the help
of structure equations governing its equilibrium, together
with the boundary conditions provided by observations.
The principal question concerning the structure of the Sun
is about checking the correctness of the theoretically
constructed solar models. Fortunately, it turns out that the
Sun is transparent to neutrinos released in the nuclear
reaction network operating in the energy-generating core
and also to seismic wave motions generated through the
solar body. These complementary probes enable us to see
inside the Sun and to infer the physical conditions
prevailing in the solar interior and relate them to larger
issues in astronomy and physics. The internal layers, in
fact, provide an ideal celestial laboratory for testing atomic
and nuclear physics, and high-temperature plasma physics
and neutrino physics.
on chain. The energy is transported outwards by radiative
processes except in the outer unstable zone, extending over
approximately a third of the solar radius below the surface
where the energy flux is carried largely by convection
modelled in the framework of a local mixing length theory.
There is supposed to be no mixing of material outside the
convection zone, save the slow gravitational diffusion of
helium and heavy elements beneath the convection zone
into the radiative interior, and there is no wave transport of
energy or material. The standard nuclear and neutrino
physics is adopted for constructing theoretical models
satisfying the observed constraints, namely,
Mass (M¤) = (1.9889 ± 0.0002) × 1033 g,
Radius (R¤) = (6.9599 ± 0.0007) × 1010 cm,
Luminosity (L¤) = (3.846 ± 0.006) × 1033 erg s –1,
Age (t¤) = (4.6 ± 0.1) × 109 yrs, and
Chemical composition (Z/X) = 0.0245 ± 0.002.
(1)
Here X and Z respectively, refer to the fractional abundance
by mass of hydrogen and elements heavier than helium.
The manner in which the pressure, density and temperature vary throughout the solar interior can be determined
by solving the equations of mechanical and thermal
equilibrium applicable to the spherically symmetric Sun1,
where these variables are all taken to be functions of the
radial coordinate, r. The structure equations are integrated
numerically, with appropriate boundary conditions, with the
auxiliary physical input of the opacity, nuclear energy
generation rate and equation of state to construct solar
models. The conventional approach to the theory of solar
structure is to adopt at zero-age, a homogeneous chemical
composition and the mass, and then to evolve the Sun over
the solar age to yield the present luminosity and radius by
adjusting the initial helium abundance and the mixing-length
parameter which determines the convective flux in the
convection zone.
2. Standard solar model
3. Seismic waves
The standard solar model (SSM) is constructed using a
variety of simplifying assumptions. The Sun is assumed
to be spherically symmetric maintaining mechanical and
thermal equilibrium, with negligible effects of rotation,
magnetic field, mass loss or accretion of material and tidal
forces on its overall structure. The energy generation takes
place in the central regions by thermonuclear reactions
converting hydrogen into helium mainly by the proton–
proton chain. The energy is transported outwards by
It has been observed since the early 1960s that the solar
surface undergoes a series of mechanical vibrations; these
manifest as Doppler shifts oscillating with a period centered
around 5 min2. The pulsations have now been identified as
acoustic modes of oscillation of the entire Sun3–5. Just like
any musical instrument, the Sun also oscillates in a number
of characteristic modes whose frequencies are determined
by the internal structure and dynamics. The solar surface is
seen to oscillate simultaneously in millions of modes, with
the amplitude of an individual mode of the order of a few
*For correspondence. (e-mail: chitre@astro.tifr.res.in)
1454
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Yl m (θ , φ ).
SPECIAL SECTION: SOLAR PHYSICS
centimeters per second. Remarkably, the frequencies of
many of these modes have been determined to an accuracy
of better than 0.01%. In much the same manner as the
geophysicists are able to study the internal layers of Earth
from seismic disturbances, the helioseismic tool furnished
by the rich spectrum of velocity fields observed at the solar
surface can probe the Sun’s internal layers to an
extraordinary degree of precision. The accurately measured
frequencies of oscillations, in fact, provide very stringent
constraints on the admissible solar models.
In order to determine the frequencies of these oscillations
to high accuracy, one needs continuous observations
extending over very long periods. From most observatories
on the surface of Earth it is not possible to observe the Sun
continuously for more than 15 h due to the day–night cycle.
Thus, to get longer coverage of the Sun various strategies
have been tried, which include observations from the
geographic south pole, from a network of sites located
around the Earth and observations from a suitably located
satellite. There are several ground-based networks
observing the Sun more or less continuously with a variety
of instruments. The most prominent among these is the
Global Oscillations Network Group (GONG) which includes
six stations located in contiguous longitudes around the
world6. GONG has been observing the Sun more or less
continuously since 1995 and frequencies of approximately
half a million modes have been calculated for different
periods of observations7. Apart from earth-based networks,
many instruments located on satellites have been observing
the solar oscillations. The most important among these is
the Michelson Doppler Imager (MDI) instrument8 on board
the Solar and Heliospheric Observatory (SOHO) satellite,
which was launched on 2 December 1995. The higher spatial
resolution provided from space has enabled MDI to study
oscillations with small-length scales.
Solar oscillations may be regarded as a superposition of
many standing waves, whose frequencies are controlled by
the physical properties of the solar interior. There
are two distinct types of wave-modes that the Sun can
support: high-frequency acoustic modes (p-modes) for
which pressure gradient provides the main restoring force;
and low-frequency gravity modes (g-modes) for which
buoyancy is the dominant restoring force, and separating
these two classes of modes are the fundamental modes
(f-modes) which are essentially the surface gravity modes.
The eigenmodes of oscillations can be characterized by
three quantum numbers: the angular degree, l; azimuthal
order, m; and radial order, n. The oscillation amplitudes are
small, and so they can be analysed using a linear
perturbation theory. Further, since the Sun is spherically
symmetric to a good approximation, the eigenmodes of
oscillations can be expressed in terms of the spherical
harmonics,
Thus, for example, the radial component of velocity can be expressed as
v ( r , θ, φ, t ) = v nl ( r ) Yl m (θ, φ) ei ω nlmt .
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
(2)
Here r is the radial distance from the center, θ the colatitude
and φthe longitude and ωnlm is the frequency of oscillations.
It is often convenient to express frequencies in terms of
mHz and define ν = ω/2πas the cyclic frequency. In absence
of rotation and magnetic field, the frequencies will be
independent of the azimuthal order m; the rotation and other
symmetry-breaking forces lift this degeneracy giving rise to
splitting of the modes for a given value of n, l. The mean
frequency of a given multiplet νnl is determined by the
spherically symmetric structure of the Sun, while the
frequency splittings are determined by the rotation rate,
magnetic field and other asphericities in solar interior.
Extensive observations from GONG and MDI instruments
have provided the mean frequencies of modes with degree,
l,
from
0
to
4000
and frequencies from 1 to 10 mHz (refs 7, 9–11). These
include the p- and f-modes, while the low-frequency
g-modes have not yet been unambiguously detected.
The p-modes are believed to be excited by turbulent
convection in the subsurface layers12. The propagation
characteristics of these seismic waves are affected by
sound speed in the solar material, which increases inwards
due to rising temperature with depth. Thus, a wave excited
near the surface and propagating inwards is refracted away
from the radial direction until at some depth it suffers a total
internal reflection and bounces back to the surface. Near the
solar surface the waves tend to get reflected because of
sharply declining density. In this way, the acoustic waves
are trapped within a cavity and the wave may travel around
the Sun several times, establishing a standing wave pattern,
in the process providing a global diagnostic of the solar
interior.
Acoustic waves propagate through the body of the Sun
along ray paths, as shown in Figure 1. The penetration
depth of a given wave depends on its horizontal wavelength, or the angle of inclination to the radial direction –
shorter waves (oscillations with large l) are confined within
relatively shallow cavities below the surface, while the
longer waves (small l) propagate deeper penetrating
practically to the central regions. The radial modes (l = 0)
propagate radially and hence suffer no refraction, thus
penetrating all the way to the center. As different modes are
trapped in different regions of solar interior, they sample
properties of the region where they are trapped. This
improves the diagnostic potential of solar oscillations since
by studying the properties of a large variety of modes it is
possible to infer the conditions over a sizeable fraction of
solar
interior.
The
disturbances
observed
at
the photosphere naturally encounter the ‘murky’ surface
layers which influence the oscillation frequencies to a
significant extent; these surface effects must be properly
filtered out while analysing the seismic data. Clearly, the
characteristics of p-modes are mainly determined by
the sound speed inside the Sun, but other properties like
density, rotation velocity, magnetic field also affect the
waves to smaller extent. Consequently, the accurately deter1455
SPECIAL SECTION: SOLAR PHYSICS
mined frequencies of solar oscillations provide a powerful
tool to probe the structure and dynamics inside the Sun.
4. Probes of the solar interior
The initial attempts to learn about the solar interior were
concerned with the boundary conditions at the surface. The
spectroscopic data was extensively collected for studying
the solar atmosphere, and the theory of solar structure was
widely used to surmise the physical conditions below the
surface for obtaining the observed temperature and
luminosity.
Since the 1960s, there have been valiant attempts to
measure the flux of neutrinos generated by the nuclear
reactions operating in the solar core (cf., Bahcall, this issue).
The neutrino flux is sensitive to the temperature and
composition profiles in the central regions of the Sun. It
was, therefore, expected that the steep temperature
dependence of some of the nuclear reaction rates will
determine Sun’s central temperature to better than a
few per cent. The persistent discrepancy between the
measured solar neutrino counting rates and the predictions
of standard models raised doubts about the reliability
of structure calculations, based on the assumption of
standard physical properties for neutrinos. This had
prompted solar physicists to look for some independent
means to explore conditions inside the Sun and the techniques of geo-seismology were adopted by using the
precisely measured eigenfrequencies of global oscillations
to determine the sound speed and density variations
through most of the solar body.
The helioseismic database of oscillation frequencies may
be analysed in two ways: (i) forward method, and
(ii) inverse method. In the forward method, an equilibrium
solar model, constructed using the structure equations, is
perturbed to obtain the eigenfrequencies of solar osci-
llations in a linearized theory, and these are compared with
the accurately measured oscillation frequencies. The fit is,
of course, seldom perfect; but the comparison suggested
that the thickness of the convection zone is close to
200,000 km, deeper than what was previously estimated, and
helium-abundance by mass in solar envelope was indicated
to be 0.25. The direct method has had only a limited
success, though, since it is not possible to produce a
perfect fit for the seismic data by merely fitting a set of
adjustable parameters characterizing the specific models. As
a result, the values for various parameters may be nonunique. A number of inversion techniques13 have, therefore,
been employed to extract more information about the solar
interior.
One of the major accomplishments of the inversion
techniques has been the effective use of the observed solar
oscillation frequencies for a reliable inference of the internal
structure of the Sun14,15. Thus, the profile of the sound
speed,
Γ1 = (∂ ln P/∂ ln ρ)S is the
c = Γ(where
1P/ρ
adiabatic index), has now been established through the bulk
of the solar interior to an accuracy of better than 0.1%, and
the profiles of
a density, adiabatic index and other
thermodynamic quantities are known to somewhat lower
accuracy. It also appeared from the variation of sound
speed beneath the convection zone that the adopted
opacities for solar modelling near the base of the convection zone, were low by about 15–20%. This was later
confirmed by the use of Livermore opacity calculations16. In
Figure 2 are shown the plots of the relative difference in sound speed, and density between the Sun, as
inferred from helioseismic inversions and a standard
solar model with gravitational settling of helium and heavy
elements17. There is a reasonably close agreement except for
a noticeable discrepancy near the base of the convection
zone and a smaller discrepancy in the energy-generating
core. The bump below 0.7 R¤ could be attributed to a sharp
change in the gradient of helium-abundance profile arising
from diffusion in the reference model. A moderate amount of
turbulent mixing (induced by say, a rotationally induced
b
instability) immediately
underneath the convection zone can
alleviate this discrepant feature. The dip in the relative
sound speed difference around 0.2 R¤ is not yet well
understood; it could be due to inaccurate composition
profile in the solar model, possibly due to use of incorrect
nuclear reaction rates.
The sound speed profile in ionization zones is affected by
the variation in Γ1 and it is possible to use this to determine
the helium-abundance in solar convection zone. The
inverted sound speed profile can be employed to compute
the quantity,
W (r ) =
Figure 1. Propagation of acoustic waves corresponding to
different values of l. The distance between two successive points at
which
a
ray intersects the surface is a measure of its horizontal wavelength.
Modes with low l or large horizontal wavelength penetrate into
deeper layers.
1456
r 2 dc 2
,
GM¤ dr
(3)
which is shown
3. The
peakmodel
around
Figure in
3. Figure
The function
W(r)small
for a solar
is shown by the
Figure
b.
Relative
difference
in
sound
speed
and
density
continuous
line,
while
the
dashed
line
represents
the same for the Sun
r = 0.98 2Ra,
in
this
curve
is
due
to
the
HeII
ioniza¤
17
profiles between
the inverted
Sun and sound
a standard
solar
model .
using
speed
profile.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
tion zone, which can be calibrated to measure the helium
abundance. The sharp change in its gradient around
r = 0.713 R¤ indicates the base of the convection zone and
this curve can thus be used to measure the depth of the
convection zone. The helium abundance in solar envelope
is found to be18 0.249 ± 0.003. This value is less than what
was used in the earlier standard solar models and the
discrepancy was attributed to the fact that some of the
helium would diffuse into the interior through gravitational
settling. An examination of the inverted sound speed
profile, below the convection zone, also reinforces this
conclusion19. The incorporation of gravitational settling in
radiative interior, indeed, results in a significant
improvement in solar models. This also reduces the life time
of main sequence stars and the estimated age of globular
clusters is also diminished. Clearly, this will have wider
implications for the age problem in the context of standard
big bang model of cosmology.
The dip in Γ1 inside the ionization zone is also determined
by the equation of state and the inverted sound speed in
this region provides a test for the equation of state20. It is
found that standard equations of state, which were widely
used in stellar evolution calculations, were not good
enough to model the solar interior. More sophisticated
equations of state, like the MHD (Mihalas, Hummer and
Dappen)21, or OPAL22 equation of state, are found to
produce good accordance with helioseismic data. Further,
the OPAL equation of state is found to be in better
agreement with solar data compared to the MHD equation
of state18. Even these equations of state show slight
discrepancy in the core and this discrepency has recently
been attributed to the neglect of relativistic correction for
electrons23.
In solar models the second derivative of temperature and
hence that of the sound speed is discontinuous at
the base of the convection zone. This discontinuity in the
function W(r), eq. (3) can be utilized to identify the position
of the base of the convection zone24. The sound speed as
well as the frequencies of p-modes are very sensitive to the
depth of the convection zone and therefore seismic
inversions enable a very accurate measurement of its
thickness. Using recent data the depth of the convection
zone is estimated to be25 (0.2865 ± 0.0005) R¤. Further, the
position of the base of the convection zone is controlled by
the opacity of solar material. We can then estimate the
opacity at the base of the convection zone26 and it has been
found that the current OPAL opacity tables27 are consistent
with helioseismic data to within an estimated error of 3%.
The convective eddies inside the convection zone are
expected to penetrate beyond the theoretical local boundary, but there is no satisfactory theory to describe this
overshoot. A significant overshoot can alter the stellarevolution calculations and so far the extent of penetration is
treated as a parameter in stellar evolution calculations. Now
with the availability of helioseismic data, it has become
possible to estimate this extent of overshoot below the base
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
of the solar convection zone. The discontinuity in the
derivatives of sound speed at the base of the convection
zone
introduces
an
oscillatory
component28
in the frequencies as a function of radial order n. The
amplitude of this signal depends on the magnitude of
discontinuity, which in turn depends on the extent of
overshoot below the solar convection zone. Thus by
measuring the amplitude of this oscillatory signal we can
determine the extent of overshoot below the convection
zone29,30. The measured oscillatory signal is found to be
consistent with no overshoot and on the basis of this result
an upper limit of 1/20 of the local pressure scale height has
been obtained31 for the overshoot distance. This is, of
course, too small to affect the stellar evolution calculations
significantly.
The frequencies of f-modes, which are surface gravity
modes, are largely independent of stratification in the solar
interior and are essentially determined by the surface gravity. These frequencies, which have now been
measured reliably by GONG and MDI data, provide an
important diagnostic of the near-surface regions, as well as
an accurate measurement of the solar radius32,33. Furthermore, the changes in solar radius with time by a few
km have also been recorded. Another application of the
accurately measured f-mode frequencies is their potential
use as a diagnostic of solar oblateness and of near-surface
magnetic fields, in addition to the possibility of investigating solar cycle variation in these quantities.
The primary inversions which have provided information
about the physical quantities like the sound speed, density
Figure 4. Fractional helium abundance by mass in the Sun as
obtained from inversions is shown by the continuous line. The dashed
line represents the abundance profile for a solar model without
diffusion, the dotted line shows that for a model incorporating
diffusion of helium and heavy elements.
1457
SPECIAL SECTION: SOLAR PHYSICS
and adiabatic index in the solar interior are based on the
equations of mechanical equilibrium. The equations of
thermal equilibrium are not used, because on time scales of
several minutes, no significant energy exchange is expected
to take place in moving elements. The frequencies of solar oscillations are, therefore, largely unaffected by
the thermal processes in the interior. However, having
obtained the sound speed and density profiles in solar
interior through primary inversions, we can employ the
equations of thermal equilibrium to determine the temperature and chemical composition profiles inside the
Sun34–36, provided input physics like the opacity, equation
of state and nuclear energy generation rates are known. In
general, the computed luminosity resulting from these
inferred profiles would not necessarily match the observed
solar luminosity. The discrepancy between the computed
and measured solar luminosity can, in fact, provide a test of
input physics, and using these constraints it has been demonstrated that the nuclear reaction cross-section for the
proton–proton reaction, needs to be increased slightly
to (4.15 ± 0.25) × 10–25 MeV barns36. This cross-section
has a controlling influence on the rate of nuclear energy
generation and neutrino fluxes, but it has never been
measured in the laboratory and all estimates are based on
theoretical computations. More recently, this cross-section
has been revised upwards37 to a value close to what was
estimated helioseismically.
The inferred helium-abundance profile agrees with that in
the standard solar model, incorporating diffusion of helium
and heavier elements, except in layers just below the solar
convection zone. This is the region where the solar rotation
rate has a sharp gradient in radial direction (ChristensenDalsgaard and Thompson, this issue). The inferred heliumabundance profile, for example, shown in Figure 4 is
essentially flat in this region. This indicates the presence of
some sort of mixing process, possibly by rotationally
induced instability which has not been properly accounted.
The mixing in this region can also explain the anamolous
low-lithium abundance in solar envelope. The destruction of
lithium by nuclear reactions can take place at temperatures
exceeding 2.5 × 106 K, but at the base of the solar
convection zone the temperature is still not high enough to
burn lithium. Thus, if the mixing extends a little beyond the
solar convection zone to a radial distance of 0.68 R¤, the
temperature can reach high enough value to explain the low
abundance of lithium. This is exactly the region where the
inferred composition profile is flat, indicating the operation
of a mixing process.
With an allowance of up to 10% uncertainty in opacity
values, the central temperature of the Sun is found to
be (15.6 ± 0.4) × 106 K (ref. 38). The inferred temperature and
composition profiles may be used to compute the neutrino
fluxes in the seismic solar models and the predicted neutrino
fluxes come close to what is obtained for the current
standard solar models. This suggests that the known
discrepancy between the observed and predicted neutrino
1458
fluxes is likely to be due to non-standard neutrino physics.
Thus, helioseismology has turned the Sun into a laboratory
to study properties of neutrinos.
Apart from spherically symmetric structure of solar
interior, it is also possible to determine helioseismically the
rotation rate inside the Sun from the accurately measured
rotational splittings (cf., Christensen-Dalsgaard and
Thompson, this issue), Dnlm = (νnlm–νnl – m)/2m. It turns out
that first-order effects of rotation yield splittings which
depend on odd powers of m, and these odd splitting
coefficients have been used to determine the rotation rate
as a function of depth and latitude. On the other hand,
magnetic
field
or other asphericities give only even order splittings and
hence these can be separated from the rotational effects.
These even splitting coefficients allow us to study the
departures from spherical symmetry inside the Sun. Further,
the local helioseismic techniques (cf., Kosovichev and
Duvall, this issue) allow us to study other large-scale flows,
including meridional flows in solar interior.
The Sun’s oblateness has been measured to be about
10–5 at the solar surface39, and there does not seem to be
any evidence of temporal variation in oblateness. The
oblateness is indeed, consistent with what is expected from
the helioseismically inferred rotation rate in solar interior.
The resulting quadrupole moment40 turns out to be
(2.18 ± 0.06) × 10–7, which yields a precession of perihelion
of planet Mercury’s orbit by about 0.03 arc sec/century,
validating the general theory of relativity.
The continuing efforts in helioseismology will hopefully
reveal the nature and strength of the magnetic field present
in the solar interior and will also help in ascertaining the
causes that drive the cyclic magnetic activity, and also
locate
the
seat
of
the
solar
dynamo
(cf., Choudhuri, this issue). The global and local
seismology of the Sun is clearly poised to reveal its interior
to a remarkably accurate detail. The uninterrupted accruing
of the seismic data, can enable us to study the temporal
variation of mode frequencies and amplitudes, which will
indicate what changes are taking place in solar structure
and dynamics. We may also learn how the Sun’s magnetic
field changes with the solar activity cycle and what causes
the Sun’s irradiance to vary with the sunspot cycle.
1. Cox, J. P. and Giuli, R. T., Principles of Stellar Structure,
Gordon and Breach, New York, 1968.
2. Leighton, R. B., Noyes, R. W. and Simon, G. W., Astrophys. J.,
1962, 135, 474–499.
3. Ulrich, R. K., Astrophys. J., 1970, 162, 993–1001.
4. Leibacher, J. and Stein, R. F., Astrophys. Lett., 1971, 7, 191–
192.
5. Deubner, F.-L., Astron. Astrophys., 1975, 44, 371–375.
6. Harvey, J. W. et al., Science, 1996, 272, 1284–1286.
7. Hill, F. et al., Science, 1996, 272, 1292–1295.
8. Scherrer, P. H. et al., Solar Phys., 1995, 162, 129–188.
9. Rhodes, E. J. Jr., Kosovichev, A. G., Schou, J., Scherrer, P. H.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
and Reiter, J., Solar Phys., 1997, 175, 287–310.
10. Rhodes, E. J. Jr., Reiter, J., Kosovichev, A. G., Schou, J. and
Scherrer, P. H., Structure and Dynamics of the Interior of the
Sun and Sun-like Stars (eds Korzennik, S. G. and Wilson, A.),
ESA SP-418, 1998, pp. 73–82.
11. Antia, H. M. and Basu, S., Astrophys. J., 1999, 519, 400–406.
12. Goldreich, P. and Keeley, D. A., Astrophys. J., 1977, 212, 243–
251.
13. Gough, D. O. and Thompson, M. J., in Solar Interior and
Atmosphere (eds Cox, A. N., Livingston, W. C. and Matthews,
M.), Space Science Series, University of Arizona Press, 1991,
pp. 519–561.
14. Gough, D. O. et al., Science, 1996, 272, 1296–1300.
15. Kosovichev, A. G. et al., Solar Phys., 1997, 170, 43–62.
16. Rogers, F. J. and Iglesias, C. A., Astrophys. J. Suppl., 1992, 79,
507–568.
17. Christensen-Dalsgaard, J. et al., Science, 1996, 272, 1286–1292.
18. Basu, S. and Antia, H. M., Mon. Not. R. Astron. Soc., 1995, 276,
1402–1408.
19. Christensen-Dalsgaard, J., Proffitt, C. R. and Thompson, M. J.,
Astrophys. J., 1993, 403, L75–L78.
20. Basu, S. and Christensen-Dalsgaard, J., Astron. Astrophys., 1997,
322, L5–L8.
21. Däppen, W., Mihalas, D., Hummer, D. G. and Mihalas, B. W.,
Astrophys. J., 1988, 332, 261–270.
22. Rogers, F. J., Swenson, F. J., Iglesias, C. A., Astrophys. J., 1996,
456, 902–908.
23. Elliott, J. R. and Kosovichev, A. G., Astrophys. J., 1998, 500,
L199–L202.
24. Christensen-Dalsgaard, J., Gough, D. O. and Thompson, M. J.,
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Astrophys. J., 1991, 378, 413–437.
25. Basu, S., Mon. Not. R.. Astron. Soc., 1998, 298, 719–728.
26. Basu, S. and Antia, H. M., Mon. Not. R.. Astron. Soc., 1996,
287, 189–198.
27. Iglesias, C. A. and Rogers, F. J., Astrophys. J., 1996, 464, 943–
953.
28. Gough, D. O., in Progress of Seismology of the Sun and Stars,
Lecture Notes in Physics (eds Osaki, Y. and Shibahashi, H.),
Springer, Berlin, 1990, vol. 367, pp. 283–318.
29. Monteiro, M. J. P. F. G., Christensen-Dalsgaard, J. and
Thompson, M. J., Astron. Astrophys., 1994, 283, 247–262.
30. Basu, S., Antia, H. M. and Narasimha, D., Mon. Not. R.. Astron.
Soc., 1994, 267, 209–224.
31. Basu, S., Mon. Not. R.. Astron. Soc., 1997, 288, 572–584.
32. Schou, J., Kosovichev, A. G., Goode, P. R. and Dziembowski,
W. A., Astrophys. J., 1997, 489, L197–L200.
33. Antia, H. M., Astron. Astrophys., 1998, 330, 336–340.
34. Gough, D. O. and Kosovichev, A. G., in Proceedings IAU
Colloquium No 121, Inside the Sun (eds Berthomieu G. and
Cribier M.), Kluwer, Dordrecht, 1990, pp. 327–340.
35. Takata, M. and Shibahashi, H., Astrophys. J., 1998, 504, 1035–
1050.
36. Antia, H. M. and Chitre, S. M., Astron. Astrophys., 1998, 339,
239–251.
37. Adelberger, E. C. et al., Rev. Mod. Phys., 1998, 70, 1265–1292.
38. Antia, H. M. and Chitre, S. M., Astrophys. J., 1995, 442, 434–
445.
39. Kuhn, J. R., Bush, R. I., Scheick, X. and Scherrer, P., Nature,
1998, 392, 155–157.
40. Pijpers, F. P., Mon. Not. R.. Astron. Soc., 1998, 297, L76–L80.
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Rotation of the solar interior
Jørgen Christensen-Dalsgaard* and Michael J. Thompson**
*Teoretisk Astrofysik Center, Danmarks Grundforskningsfond, and Institut for Fysik og Astronomi, Aarhus Universitet, DK-8000, Aarhus C,
Denmark
**Astronomy Unit, Queen Mary and Westfield College, University of London, UK
Helioseismology has allowed us to infer the rotation in the
greater part of the solar interior with high precision and
resolution. The results show interesting conflicts with
earlier theoretical expectations, indicating that the Sun is
host to complex dynamical phenomena, so far hardly
understood. This has important consequences for our ideas
about the evolution of stellar rotation, as well as for models
for the generation of the solar magnetic field. Here we
provide
an overview of our current knowledge about solar
rotation, much of it obtained from observations
from the SOHO spacecraft, and discuss the broader
implications.
SOLAR rotation has been known at least since the early
seventeenth century when, with the newly invented telescope, Fabricius, Galileo and Scheiner observed the motion
of sunspots across the solar disk. Indeed, at the last solar
maximum 10 years ago, one of us made naked-eye
observations of sunspots from the sunset walk at the TIFR,
Mumbai: over several evenings the day-to-day change in
position of sunspots, visible to the naked eye during the
haze just before sunset, clearly showed that the Sun was
rotating.
It is hardly surprising that the Sun and other stars are
observed to rotate. Stars are born of contracting interstellar
gas clouds which share the rotation of the Galaxy. As the
clouds contract, they rotate more rapidly, much as an ice
skater makes herself spin around faster by pulling in her
arms to her body, because she is reducing her moment of
inertia while her angular momentum is conserved. Although
the details of star formation within the contracting clouds
are uncertain and involve mass loss and interaction with
disks around the star which will transport angular
momentum from one part of the cloud to another, it is
plausible that newly formed stars should be spinning quite
rapidly. This is indeed observed: the rotation of the stellar
surface causes a broadening of the lines in the star’s
spectrum, owing to the Doppler effect,
and from
measurements of this effect it is inferred that many young
stars rotate at near the break-up speed, where the
centrifugal force at the equator almost equals gravity.
Stars tend to slow down when they get older. At least for
*For correspondence. (e-mail: jcd@obs.aau.dk)
1460
stars of roughly solar type, the observations show that the
rotation rate decreases with increasing age. The Sun’s
slowdown is thought to take place through angularmomentum loss in the solar wind, magnetically coupled to
the outer parts of the Sun. The extent to which the slowdown affects the deep interior of the Sun then depends on
the efficiency of the coupling between the inner and outer
parts. In fact, simple models of the dynamics of the solar
interior tend to predict that the core of the Sun is rotating
up to fifty times as rapidly as the surface. Such a rapidly
rotating solar core could have serious consequences for the
tests of Einstein’s theory of general relativity based on
observations of planetary motion: a rapidly rotating core
would flatten the Sun and hence perturb the gravitational
field around it. Even a subtle effect of this nature, difficult to
see directly on the Sun’s turbulent surface, might be
significant.
Very detailed observations have been carried out of the
solar surface rotation by tracking the motion of surface
features, such as sunspots and, more recently, by Dopplervelocity measurements. It was firmly established by the
nineteenth century, by careful tracking of sunspots at
different latitudes on the Sun’s surface, that the Sun is not
rotating as a solid body: at the equator the rotation period is
around 25 days, but it increases gradually towards the poles
where the period is estimated to be in excess of 36 days.
This differential rotation is not as surprising as it might
seem: since the Sun is a sphere of gas, it is not constrained
to rotate at a uniform rate. Nevertheless, the origin of the
differential rotation, and how it is continued in the solar
interior, are evidently interesting questions.
The origin of the differential rotation is almost certainly
linked to the otherwise dynamic nature of the outer parts of
the Sun. The Sun’s radius is 700 Mm (i.e. 700,000 km). In the
outer 200 Mm, energy is transported by convection, in
rising elements of warm gas and sinking elements of colder
gas: this region is called the convection zone. The
convection zone can be seen directly using high-resolution
observations of the solar surface, in the granulation with
brighter areas of warm gas just arrived at the surface,
surrounded by colder lanes of sinking gas. The gas motions
also transport angular momentum, and hence provide a link
between rotation in different parts of the convection zone.
Also, convection is affected by rotation, which may
introduce anisotropy in the angular momentum transport.
Indeed, it is likely that this transport is responsible for the
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
differential rotation, although the details are far from
understood.
Similarly complex dynamical interactions are also found in
the giant gaseous planets (Jupiter, Saturn, Uranus and
Neptune) which, like the Sun, are vigorously convecting as
they rotate. Here the interaction probably gives rise to the
banded structures immediately visible on Jupiter, and more
faintly on Saturn. Even closer to home, the Earth’s
atmosphere and oceans are rotating fluid systems and
exhibit, among other things, large-scale circulations and
meandering jets, such as the jet stream. In all these systems,
rotation plays a significant role in the observed dynamical
behaviour.
Helioseismic probes of the solar interior
In recent years, the observation that the Sun is oscillating
simultaneously in many small-amplitude global resonant
modes has provided a new diagnostic of the solar interior.
As discussed elsewhere in this issue (in the article by Chitre
and Antia), the frequencies of these global modes depend
on conditions inside the Sun, and so by measuring these
frequencies we are able to make deductions about the state
of the interior. This field is known as helioseismology. The
observed oscillations are sometimes called five-minute
oscillations, because they have periods in the vicinity of
five minutes. The modes are distinguished not only by their
different frequencies, but also by their different patterns on
the surface of the Sun. These patterns are described by
spherical
harmonics
(see
Figure 1 for some examples) which are characterized by two
integer numbers, their degree l and their azimuthal order m.
As Chitre and Antia explain (see also below), different
modes are sensitive to different regions of the Sun,
depending on their frequency, degree and azimuthal order.
By exploiting the different sensitivities of the modes,
helioseismology is able to make inferences about localized
conditions inside the Sun.
One of the factors that affect the mode frequencies is the
Sun’s rotation. The dominant effect of rotation on the
oscillation frequencies is quite simple: the oscillation
patterns illustrated in Figure 1 actually correspond to waves
running around the equator; if the images were animated,
they would essentially look like rotating beach balls.
Patterns travelling in the same direction as the rotation of
the Sun would appear to rotate a little faster, patterns
rotating in the opposite direction a little more slowly. When
Figure 1. Examples of spherical harmonic patterns for different values of the degree l and order m. Red and blue represent positive and
negative regions, black represents regions where the spherical harmonic is close to zero.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
observed at a given position on the Sun, the oscillations in
the former case would have slightly higher frequency, and
in the latter case slightly lower frequency, than if the Sun
had not been rotating. The frequency difference between
these two cases therefore provides a measure of the
rotation rate of the Sun.
In reality other effects must be taken into account to
describe the frequency shifts caused by rotation, which are
often referred to as the rotational frequency splitting. The
Coriolis force affects the dynamics of the oscillations and
hence their frequencies, although it turns out that for the
modes observed in the five-minute region this effect
is modest. However, the variation of rotation rate with
position in the Sun must be taken into account. Each mode
feels an average rotation rate, where the average is
determined by the mode’s frequency, degree and azimuthal
order: the precise form of this spatial average is described
by a weight function. These weight functions vary widely
from mode to mode. As already indicated in Figure 1, modes
with m = l are concentrated near the equator, increasingly
so with increasing l, whereas modes of lower azimuthal
order extend to higher latitudes. Thus modes with m = l feel
only an average of the rotation near the equatorial plane,
whereas modes of lower azimuthal order sense the average
rotation over a wider range of latitudes. In a similar manner,
as described by Chitre and Antia, the high-degree fiveminute modes (i.e. with large values of l) sense only
conditions near the surface of the Sun, whereas modes of
low degree feel conditions averaged over much of the solar
interior.
These properties can be illustrated by a few examples of
weight functions (as shown in Figure 2). The observed
modes include some that penetrate essentially to the solar
centre, others that are trapped very near the surface, and
the whole range of intermediate penetration depths, with a
similar variation in latitudinal extent. Thus the observed
frequency splittings provide a similarly wide range of
averages of the internal rotation. It is this wealth of data
which allows the determination of the detailed variation of
rotation with position in the solar interior. Modes of high
degree, trapped near the surface, provide measures of the
rotation of the superficial layers of the Sun. Having
determined that, its effect on the somewhat more deeply
penetrating modes can be eliminated, leaving just a measure
of rotation at slightly greater depths. In this way,
information about rotation in the Sun can be ‘peeled’ layer
by layer, much as one could an onion, in a way that allows
us to obtain a complete image of solar internal rotation.
It is fairly evident that this process gets harder, the
deeper one attempts to probe, since fewer and fewer modes
penetrate to the required depth; furthermore, the effect of
rotation decreases because of the smaller size of the region
involved. Thus the rotation of the solar core is difficult to
determine. Similarly, all modes are affected by the equatorial
rotation while only modes of low m extend to the vicinity of
the poles, and the polar regions have relatively little effect
on the oscillations, complicating the determination of the
high-latitude rotation. However, as we shall see, the quality
of current data is such that the rotation rate can be
determined quite near the poles, at least in the outer parts of
Figure 2. Weight functions determining the sensitivity of different modes to the solar internal rotation. Red indicates essentially no
sensitivity, whereas green and blue show regions of successively higher sensitivity. All modes have frequencies near 2 mHz; their degree l and
azimuthal order m are, from left to right: (l, m) = (5, 2), (28, 10), (28, 26), and (60, 50).
1462
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SPECIAL SECTION: SOLAR PHYSICS
the convection zone. We also note an intrinsic limitation of
the frequency splittings of global modes: the weight
functions are symmetrical around the solar equator, as can
be seen in Figure 2; thus we can infer only the similarly
symmetric component of rotation. This must be kept in mind
in the following, when interpreting the results. We note that
this restriction can be avoided by applying local
helioseismology techniques to the data: such techniques
are described elsewhere in this issue by Kosovichev and
Duvall.
The solar internal rotation
Early helioseismic data on rotational splittings provided
information only about the modes with m ~ ± l; as a result,
they were sensitive mainly to rotation near the equator.
Observations from the Kitt Peak National Observatory,
USA, showed around 1984 that there was relatively little
variation of rotation with depth; in particular, there were no
significant indications of a rapidly rotating core. A few
years later, initial data on the dependence of the splitting on
m were obtained at the Sacramento Peak and Big Bear Solar
Observatories. Strikingly, they indicated that the surface
latitudinal differential rotation persisted through the
convection zone, whereas there was little indication
of variation with latitude in the rotation beneath the
convection zone.
In the last few years, the amount and quality of helioseismic data on solar rotation have increased dramatically,
as a result of several ground-based and space-based
experiments. The LOWL instrument of the High Altitude
Observatory has provided high-quality data on modes of
low and intermediate degree over the past more than five
years. The BiSON and IRIS networks, observing low-degree
modes in Doppler velocity integrated over the solar disk,
have yielded increasingly tight constraints on the rotation
of the solar core, while the GONG six-station network
(including a station at the Udaipur Solar Observatory) is
setting a new standard for ground-based helioseismology.
Finally, the SOI-MDI experiment on the SOHO spacecraft
has yielded a wealth of data on modes of degree up to 300,
allowing for the first time a detailed analysis of the
properties of rotation in the convection zone. The results
we present below are the combined knowledge that has
emerged from these observational efforts.
In discussing what we now know about the rotation
inside the Sun, we shall start from the near-surface layers
and work towards the centre. As we have already discussed, the outer 30 per cent of the Sun is convectively
unstable. Before helioseismology, models predicted that the
rotation inside the convection zone would organize itself on
cylinders aligned with the rotation axis. Thus the rotation at
depth at, say, equatorial latitudes would match the surface
rotation at high latitudes, rather than the faster equatorial
rotation at the surface, and so at a given
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
latitude the rotation in the convection zone would decrease
with depth. Helioseismology has shown that this is not so:
to a first approximation, it is more accurate to say that the
rotation at a given latitude is nearly constant with depth, or
to put it another way, the differential rotation seen at the
surface imprints itself through the convection zone. This
finding is clearly visible in Figure 3. In detail, the situation is
more complicated. At low latitudes, immediately beneath the
solar surface the rotation rate actually initially increases
with depth. The equatorial rotation reaches a maximum at a
depth of about 50 Mm (i.e. about 7 per cent of the way in
from the surface to the centre of the Sun): at this point the
rotation rate is about 5 per cent higher than it is at the
surface. This is consistent with a variety of surface
measurements of rotation. Tracking sunspots tends to give
a slightly higher rotation rate than that obtained by making
direct spectroscopic measurements of the velocity of the
surface. Probably the reason is that the sunspots extend to
some depth below the surface, and so are dragged along at
a rate that is similar to the subsurface rotation a few per cent
beneath the surface which helioseismology has revealed.
The rotational velocity at the surface of the Sun is about
2 km sec –1 (i.e. 7000 km h –1), dropping off rather smoothly
towards higher latitudes. However, it has now been found2
that superimposed on this are bands of faster and slower
rotation, a few m sec –1 higher or lower than the mean flow
(Figure
4).
The
origin
of
this
behaviour
is not understood, but it is reminiscent of the more
pronounced banded flow patterns seen on Jupiter and
Figure 3. Rotation of the solar envelope inferred from
observations by the SOI-MDI instrument on board the SOHO
satellite1 . Regions of faster rotation are red, regions of slower
rotation are blue and black. The values quoted on the colour key are
the frequency of the rotation (i.e. the reciprocal of the rotation
period), in nano-Hertz (nHz): 300 nHz corresponds to a period of
roughly 39 days, while 450 nHz corresponds to a period of about 26
days. The dashed line indicates the bottom of the convection zone.
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SPECIAL SECTION: SOLAR PHYSICS
Saturn. Evidence for such bands had been obtained
previously from direct Doppler measurements on the solar
surface. However, the seismic inferences have shown that
they extend to a depth probably exceeding 40 Mm beneath
the surface3. Moreover, these bands migrate from high
latitudes towards the equator over the solar cycle.
It has been customary to represent the directly measured
surface rotation rate in terms of a simple low-order
expansion in sin ψ, where ψ is latitude on the Sun. This in
fact quite successfully captured the observed behaviour;
however, since the solar rotation axis is close to the plane of
the sky, direct measurements of rotation near the poles are
difficult and uncertain. Strikingly, the helioseismic results
have shown a marked departure from this behaviour, at
latitudes above about 60°: relative to the simple fit, the
actual rotation rate decreases quite markedly there. The
origin or significance of this behaviour is not yet
understood. There is also evidence, hinted at in Figure 3, of
a more complex behaviour of rotation at high latitudes.
Some analyses have shown a ‘jet’, i.e. a localized region of
more rapid rotation, at a latitude around 75° and a depth of
about 35 Mm beneath the solar surface. Also, evidence has
been found that the rotation rate shows substantial
variations in time at high latitudes, over time scales of the
order of months. It is probably fair to say that the
significance of these results is still somewhat uncertain,
however. Also, it should be kept in mind, as mentioned
above, that the results provide an average of rotation in the
northern and southern hemispheres and, evidently, an
Figure 4. The evolution with time of the fine structure in the
near-surface solar rotation. The time-averaged rotation rate has been
subtracted from each of 11 independent inferences of rotation, for
consecutive 72-day time intervals. The result is represented as a
function of time (horizontal axis) and latitude (vertical axis), the
colour-coding at the right gives the scale in nHz; 1.5 nHz
corresponds to a speed of around 6/m sec–1 at the equator. The
banded structure, apparently converging towards the equator as time
goes by, should be noted. (From ref. 3.)
1464
average over the observing period of at least 2–3 months.
Thus the interpretation of the inferred rotation rates in terms
of the actual dynamics of the solar convection zone is not
straightforward.
At the base of the convection zone, a remarkable
transition occurs: the variation of rotation rate with latitude
disappears, so that the region beneath the convection zone
rotates essentially rigidly, at a rate corresponding to the
surface rate at mid-latitudes (Figure 5). The region over
which the transition occurs is very narrow, no more than a
few per cent of the total radius of the Sun. This layer has
been called the tachocline. Why the differential rotation
does not persist beneath the convection zone is not yet
known, but it is possible that a large-scale weak magnetic
field permeates the inner region and enforces nearly rigid
rotation by dragging the gas along at a common rate5. Such
a field is quite possible, as a relic from the original
collapsing gas cloud from which the Sun condensed.
The discovery of the tachocline, and of the form of the
rotation in the convection zone, has led to an adjustment of
our theories of the solar dynamo (see the article by
Choudhuri in this issue). The Sun displays a roughly
eleven-year cycle of sunspot activity, with the number of
spots and their latitudinal distribution on the Sun varying
over the cycle. Sunspots are formed where strong magnetic
fields poke through the Sun’s surface, and these magnetic
fields are widely believed to be generated by some kind of
dynamo action in the Sun. One idea is that the dynamo
action consists of two components: a twisting of the
magnetic field by the motion of convective elements, and a
shearing out of the field by differential rotation. Prior to the
Figure 5. The inferred rotation as a function of depth inside the
Sun at three solar latitudes: the equator (red), 30 degrees (orange) and
60 degrees (green). The vertical spread in the coloured bands shows
the statistical uncertainty on the rotation rate (± 1 standard
deviation). Note that the result becomes much more uncertain in the
deep interior. The values on the vertical axis are the rotation
frequency in nHz (see caption to Figure 3). The values on the
horizontal axis are the fractional radius inside the Sun, and run from
the centre of the Sun (r/R = 0.0) to the visible surface (r/R = 1.0).
The observations used to infer the rotation were from the LOWL
instrument and the BiSON network4 .
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
helioseismic findings, the simulations of rotation implied
that the radial gradient of differential rotation in the
convection zone could provide the second ingredient, so it
was thought that the dynamo action occurred in that region.
Now, however, the tachocline with its very substantial
radial gradient seems a more likely location for the dynamo.
Even deeper in the Sun, right down into the core where
the energy-releasing nuclear reactions take place, the
helioseismic results on the rotation are more uncertain
due to the fact that so few of the observed five-minute
modes (only the low-degree modes in fact) have any
sensitivity to this region. Indeed, the results have been
somewhat contradictory, some indicating rotation faster
than the surface rate and others indicating rotation slower
than or comparable to the rotation rate at the base of the
convection zone; an example is illustrated in Figure 5.
However, down to within 15 per cent of the solar radius
from the centre, which is the deepest point at which present
observations permit localized inferences to be made, all the
modern results agree that the rotation rate is not more than
a factor two different from the surface rate: thus early
models which predicted that the whole of the nuclearburning core was rotating much faster are firmly ruled out.
Again, this finding would be consistent with a magnetic
field linking the core to the bulk of the radiative interior.
Modelling solar rotation
Although helioseismology has provided us with a
remarkably detailed view of solar internal rotation, the
theoretical understanding of the inferred behaviour is still
incomplete. In the convection zone, the problem is to model
the complex combined dynamics of rotation and convection,
the latter occurring on scales from probably less than a few
hundred kilometres to the scale of the entire convection
zone and time scales from minutes to years. Viscous
dissipation is estimated to occur on even smaller spatial
scales, of the order 0.1 km or less. Capturing this range of
scales is entirely outside the possibility of current numerical
simulations; thus simplifications are required. Detailed
simulations of near-surface convection, on a scale of a few
Mm, have been remarkably successful in reproducing the
observed properties of the granulation6, but are evidently
not directly relevant to the question of rotation. Simulations
of the entire convection zone are necessarily restricted to
Figure 6. The results of two simulations of convection-zone rotation by Miesch, Elliott and Toomre, from the
paper by Miesch9 . The two simulations use different boundary conditions and parameter values, and illustrate some
of the range of possible responses of the differential rotation to the form of the convection. Also note that
Simulation A has a higher resolution and includes penetration into a stable region beneath the convection zone,
whereas the convective motions in Simulation B are more laminar and there is no penetration beneath the
convection zone.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
rather large scales and hence cannot capture the nearsurface details. Such simulations, therefore, typically
exclude the outer 30 Mm of the convection zone. Early
examples of such simulations by Gilman and Glatzmaier, of
fairly limited resolution, showed a tendency for rotation to
organize itself on cylinders7: the rotation rate depended
primarily on the distance to the rotation axis; such a
behaviour is predicted for simple systems by the Taylor–
Proudman theorem. The rotation rate on a given cylinder
would obviously be observable where the cylinder
intersected the surface; thus the observed decrease of
rotation rate with increasing latitude would correspond to a
similar decrease of rotation rate with depth at, say, the
equator. The convection itself in these simulations was
dominated by so-called banana cells – long, thin, largescale convection cells oriented in the north–south direction.
The actual behaviour of rotation, shown in Figure 3, is
obviously very different from these simulation results. The
overall variation of rotation within the convection zone is
evidently predominantly with latitude, with little variation in
the radial direction except in the tachocline. Given the
necessary simplifications of the calculations, their failure to
model
solar
rotation
is
perhaps
not
surprising. In particular, the effects of smaller-scale turbulence (beneath the smallest scale resolved in the
simulation) are typically represented as some form of
viscosity; it was suggested by Gough, and later by others,
that the effect of rotation on the small-scale motion might
render this turbulent viscosity non-isotropic, with important
effects on the transport of angular momentum within the
convection zone. In fact, simple models of convection- zone
dynamics, with parametrized anisotropic viscosity, have had
some success in reproducing the helioseismically inferred
rotation rate.
Recent advances in computing power have led to
improved numerical simulations8, which come closer to
representing turbulent convective flow regimes such as
exist in the Sun’s convection zone. Figure 6 shows
results from two such simulations by Miesch, Elliott and
Toomre. The simulations can yield a range of differential
rotation profiles, depending on the conditions imposed at
the top and bottom boundaries of the simulation region, and
on the parameter values adopted for the problem. Since it is
not obvious what are the most appropriate boundary
1466
conditions and parameter values to choose, it is necessary
to explore various possibilities and study the different
responses. Simulation B has rotation contours
at mid-latitudes which are nearly radial, as in the Sun
(compare Figure 3), but the contrast in rotation rate between
low and high latitudes is not as great as is observed in the
Sun (about 70 nHz, rather than 130 nHz). In case A, the
latitudinal variation of the Sun’s rotation is better
reproduced, but the mid-latitude contours do not look quite
as similar to those in Figure 3. Nonetheless, these results
are encouraging indications that we may be close to
reproducing theoretically the gross features of the solar
rotation inferred by helioseismology. There is still, though,
much work ahead, both observational and theoretical, in
getting a detailed understanding of the Sun’s rotation and
with that, we hope, a better understanding of the solar
activity cycle and of large-scale rotating fluid systems on
planets and stars.
1. Schou, J. et al., Astrophys. J., 1998, 505, 390–417.
2. Schou, J. and the SOI Internal Rotation Team, in Proceedings
IAU Symposium 185: New Eyes to See Inside the Sun and Stars
(eds Deubner, F.-L., Christensen-Dalsgaard, J. and Kurtz, D. W.),
Kluwer, Dordrecht, 1998, pp. 141–148.
3. Toomre, J., Christensen-Dalsgaard, J., Howe, R., Larsen, R. M.,
Schou, J. and Thompson, M. J., Solar Phys., 1999, 308, 405–
414.
4. Chaplin, W. J. et al., Mon. Not. R. Astron. Soc., 1999 (in press).
5. Gough, D. O. and McIntyre, M. E., Nature, 1998, 394, 755–
757.
6. Stein, R. F. and Nordlund, Å., Astrophys. J., 1989, 342, L95–
L98.
7. Gilman, P. A. and Miller, J., Astrophys. J. Suppl., 1986, 61,
585–608.
8. Elliott, J. R., Miesch, M. S., Toomre, J., Clune, T. and
Glatzmaier, G. A.,. in Structure and Dynamics of the Interior of
the
Sun
and Sun-like Stars; Proc. SOHO 6/GONG 98 Workshop (eds
Korzennik, S. G. and Wilson, A.), European Space Agency,
Noordwijk, The Netherlands, 1998, pp. 765–770.
9. Miesch, M. S., Solar Phys., 1999 (in press).
ACKNOWLEDGEMENTS. We are grateful to Dr M. Miesch for
providing Figure 6, and to Dr R. Howe for help with other Figure. We
acknowledge the financial support of the UK Particle Physics and
Astronomy Research Council, and the Danish National Research
Foundation through its establishment of the Theoretical
Astrophysics Center.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Solar tomography
A. G. Kosovichev*,† and T. L. Duvall Jr**
*W.W. Hansen Experimental Physics Laboratory, Stanford University, CA 95305-4085, USA
**Laboratory for Astronomy and Solar Physics, NASA Goddard Space Flight Center Greenbelt, MD 20771, USA
The solar tomography (or time-distance helioseismology)
is a new promising method for probing 3-D structures and
flows beneath the solar surface, which is potentially
important for studying the birth of active regions in the
Sun’s interior and for understanding the relation between
the internal dynamics of the active regions, and the
chromospheric and coronal activity. In this method, the
time for waves to travel along sub-surface ray paths is
determined from the temporal cross correlation of signals
at two separated surface points. By measuring the times
for many pairs of points from Dopplergrams, covering the
visible hemisphere, a tremendous quantity of information
about the state of the solar interior is derived. As an
example, we present the results on the internal structures
of supergranulation, meridional circulation, active regions
and sunspots. An active region which emerged on the solar disk in January 1998, was studied
from SOHO/MDI for nine days, both before and after its
emergence at the surface. The results show a complicated
structure of the emerging region in the interior, and
suggest that the emerging flux ropes travel very quickly
through the depth range of our observations.
Method of solar tomography
solar surface1
T
Ψ (τ, ∆) = ∫ f (t , r1 ) f * ( t + τ , r2 ) dt ,
0
(1)
where ∆ is the horizontal distance between the points with
coordinates r1 and r2, τ is the delay time, and T is the total
time of the observations. Because of the stochastic nature
of excitation of the oscillations, function Ψ must be averaged over some areas on the solar surface to achieve a
signal-to-noise ratio sufficient for measuring travel times τ.
The oscillation signal, f(t, r), is usually the Doppler velocity
or intensity. A typical cross-covariance function shown in
Figure 2 displays several sets of ridges which correspond to
the first, second, and third bounces of packets of acousticwave packets from the surface.
The origin of the multiple bounces is illustrated in Figure
1. Waves originated at point A may reach point B directly
(solid curve), or after one-bounce at point C (dashed curve),
or after two-bounces (dotted curve), and so on. Because the
sound speed is greater in the deeper layers, the direct
waves arrive first, followed by the second-bounce and
third-bounce waves.
The cross-covariance function represents a solar ‘seismogram’. Figure 3 shows the cross-covariance signal as a
Solar acoustic waves are excited by turbulent convection
near the solar surface and propagate through the interior
with the speed of sound. Because the sound speed increases with depth, the waves are refracted and reappear on the
surface at some distance away from the source. The wave
propagation is illustrated in Figure 1. The waves excited at
point A will reappear at the surface points B, C, D,
E, F, and others after propagating along the ray paths
indicated by curves.
The basic idea of solar tomography is to measure the
acoustic travel time between different points on the solar
surface, and then to use these measurements for inferring
variations of the structure and flow velocities in the interior
along the wave paths connecting the surface points. This
idea is similar to the Earth’s seismology. However, unlike in
Earth, the solar waves are generated stochastically by
numerous acoustic sources in the subsurface layer of
turbulent convection. Therefore, the wave travel-time is
determined from the cross-covariance function, Ψ(τ, ∆), of
the oscillation signal, f(t, r), between different points on the
†
For correspondence (e-mail: sasha@guake.stanford.edu)
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Figure 1. A cross-section diagram through the solar interior, illustrating the wave propagation inside the Sun.
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SPECIAL SECTION: SOLAR PHYSICS
function of time for a distance of 30 degrees. It consists of
three wave packets corresponding to the first, second, and
third bounces. Ideally, the seismogram should be inverted
to infer the structure and flows using a wave theory.
However, in practice, as in terrestrial seismology2, different
approximations are employed, the most simple and powerful
of which is the geometrical acoustic (ray) approximation.
Generally, the observed solar oscillation signal corresponds to displacement or pressure perturbation, and can be
represented in terms of normal modes, or standing waves.
Therefore, the cross-covariance function can be expressed
in terms of normal modes, and then represented as a
superposition of traveling wave packets3. An example of the
theoretical cross-covariance function of p-modes of the
standard solar model is shown in Figure 4. This model
reproduces the observational cross-covariance function
very well in the observed range of distances, from 0 to 90
degrees. The theoretical model was calculated for larger
distances, including points on the far side of the Sun, which
are not accessible for observation. A backward propagating
ridge originating from the second-bounce ridge at 180
degrees is a geometrical effect due to the choice of the
range of the angular distance from 0 to 180 degrees. This is
illustrated by the green curve in Figure 1. The waves
reaching point F after the reflection at point D propagate
more than 180 degrees, but are considered
as propagating the distance AF which is less than 180 degrees. In the theoretical diagram (Figure 4) one can notice a
weak backward ridge between 30 and 70 degrees and at
120 min. This ridge is due to reflection from
the solar core. However, it has not been detected in
observations.
By grouping the modes in narrow ranges of the angular phase velocity, ν = ωnl/L, where L = (l(l+ 1))1/2,
and applying the method of stationary phase, the cross-
Figure 3. The observed cross-covariance signal as a function of
time at the distance of 30 degrees.
Figure 2. The observational cross-covariance function as a
function of distance on the solar surface, ∆, and the delay time, τ.
The lowest set of ridges (first-bounce) corresponds to waves
propagated to the distance, ∆, without additional reflections from the
solar surface. The middle ridge (second-bounce) is produced by the
waves
arriving
to
the same distance after one reflection from the surface, and the
upper ridge (third-bounce) results from the waves arriving after two
bounces from the surface. The backward ridge associated with the
second-bounce ridge is due to the choice of the angular distance range
from
1468
Figure 4. The model cross-covariance function calculated from
the p-modes theoretical eigenfunctions.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
covariance function can be approximately represented in the
form3
2
 δω2 
 
∆ 
∆ 
Ψ (τ, ∆ ) ∝ ∑ cos ω0  τ −  exp −
τ −   ,
ν   4  u  
δv
 
(2)
where δv is a narrow interval of the phase speed,
u ≡ (∂ω/∂k h) is the horizontal component of the group
velocity, k h = L/R is the angular component of the wave
vector, R is the solar radius, ω0 is the central frequency of a
Gaussian frequency filter applied to the data, and δω is the
characteristic width of this filter. Therefore, the phase- and
group travel-times are measured by fitting individual terms
of eq. (2) to the observed cross-covariance function using a
least-squares technique. This technique measures both
phase (∆/v ) and group (∆/u) travel-time of the
p-mode wave packets. The previous time-distance measurements provided either the group-time4, or the phase times5.
It was found that the noise level in the phase-time
measurements was substantially lower than in the grouptime measurements. Therefore, we use the phase times in
this paper. We also employ the geometrical acoustic (ray)
approximation to relate the measured-phase times to the
internal properties of the Sun. More precisely, the variations
of the local travel-times at different points on the surface
relative to the travel-times averaged over the observed area
are used to infer variations of the internal structure and flow
velocities using a perturbation theory.
In the ray approximation, the travel-times are sensitive
only to the perturbations along the ray paths given by the
Hamilton equations. The variations of the travel-time obey
Fermat’s Principle6
δτ =
1
δ k d r,
ω ∫Γ
(3)
where δk is the perturbation of the wave vector due to
the structural inhomogeneities and flows along the unperturbed ray path, Γ. Using the dispersion relation for acous-
tic waves in the convection zone, the travel-time variations
can be expressed in terms of the sound speed, magnetic
field strength and flow velocity3.
The effects of flows and structural perturbations are
separated from each other by taking the difference and the
mean of the reciprocal travel-times
δτdiff ≈ −2 ∫
Γ
δτmean ≈ − ∫
Γ
( nU )
c2
ds ;
δc
S ds ,
c
(4)
(5)
where c is the adiabatic sound speed, n is a unit vector
tangent to the ray, S = k/ωis the phase slowness. Magnetic
field causes anisotropy of the mean travel-times, which
allows us to separate, in principle, the magnetic effects from
the variations of the sound speed (or temperature). So far,
only a combined effect of the magnetic fields and
temperature variations has been measured reliably. Onedimensional tests by Kosovichev and Duvall3 and twodimensional
numerical
simulations
by
Jensen
et al.7 have shown that eqs (4) and (5) provide a reasonable
approximation to the travel-time variations. The development of a more accurate theory for the travel-times,
based on the Born approximation is currently under way.
Typically, we measure times for acoustic waves to travel
between points on the solar surface and surrounding
quadrants symmetrical relative to the North, South, East and
West directions. In each quadrant, the travel-times are
averaged over narrow ranges of travel distance ∆. Then, the
times for northward-directed waves are subtracted from the
times for south-directed waves to yield the time,
NS
τ diff
which predominantly measures north–south motions.
EW
Similarly, the time differences, τ diff
, between westwardand eastward-directed waves yields a measure of eastward
oi
motion. The time, τ diff
, between outward- and inwarddirected waves, averaged over the full annuli, is mainly
sensitive to vertical motion and the horizontal divergence.
Figure 5. The regions of ray propagation (colour areas) as a function of depth, z, and the radial distance, ∆, from a point on the surface. The
rays are also averaged over circular regions on the surface, forming three-dimensional figures of revolution. The dashed lines show the
inversion grid.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
The time, τmean , which measures sound speed and magnetic
perturbations, is also averaged over the full annuli (see refs
3 and 8).
The next step is to determine the variations of the sound
speed and flow velocity from the observed travel-times
using eqs (4) and (5). It is assumed that the convective
structures and flows do not change during the observations
and can be represented by a discrete model. In this model,
the 3-D region of wave propagation is divided into
rectangular
blocks.
The
perturbations
of
the sound speed, and the three components of the flow
velocity are approximated by the linear functions of
coordinates within each block, e.g. for the flow velocity
U ( x, y, z ) =

| x − xi | 

i +1 − x i 

∑ U ijk 1 − x


| y − yj |
1 −
y

j +1 − y j




| z − zk |
1 −
z

k +1 − z k

,

flows. However, vertical flows in the deep layers are not
resolved because of the predominantly horizontal propagation of the rays in these layers. The vertical velocities
are also systematically underestimated by 10–20% in the
upper layers. Similarly, the sound speed variations are
underestimated in the bottom layers. These limitations of
the solar tomography should be taken into account in
interpretation of the inversion results.
Inversion results
Helioseismic tomography has been successfully used to
infer local properties of large-scale zonal and meridional
flows11, convective flows and structures (refs 3 and 8),
structure and dynamics of active regions12 and flows
around sunspots 5. Here we present some results of tomo-
(6)
where xi, yj, zk are the coordinates of the rectangular grid,
Uijk are the values of the velocity in the grid points, and
xi ≤ x ≤ xi + 1, yj ≤ y ≤ yj + 1, and zk ≤ z ≤ zk + 1. A part of the x – z
grid is shown in Figure 5 together with the ray system used
in inversion.
The travel-time measured at a point on the solar surface is
the result of the cumulative effects of the perturbations in
each of the traversed rays of the 3-D ray systems. Figure 5
shows, in the ray approximation, the sensitivity to
subsurface location for a certain point on the surface. This
pattern is then translated for different surface points in the
observed area, so that overall the travel-times are sensitive
to all subsurface points in the depth range
0–20 Mm, in this example.
We average the equations over the ray systems corresponding to the different radial distance intervals of the data,
using approximately the same number of ray paths as in the
observational procedure. As a result, we obtain two
systems of linear equations that relate the data to the sound
speed variation and to the flow velocity, e.g. for the velocity
field,
δτ diff; λµν = ∑ A ijk
λµν ⋅ U ijk ,
(7)
ijk
where vector-matrix A maps the structure properties into the
observed travel-time variations, and indices λ and µ label
the location of the central point of a ray system on the
surface, and index ν labels the annuli. These equations are
solved by a regularized least-squares technique using the
LSQR algorithm9. Jensen et al.10 suggested to speed up the
inversion by doing most of the calculation in
the Fourier domain.
The results of test inversions of Kosovichev and Duvall3
demonstrate a very accurate reconstruction of sound speed
variations and the horizontal components of subsurface
1470
Figure 6. The horizontal flow velocity field (arrows) and the
sound speed perturbation (red colour shows positive perturbations,
and blue colour shows negative perturbations) at the depths of (a)
1.4 Mm
and
(b) 5.0 Mm as inferred from the SOHO/MDI high-resolution data of
27 January 1996. The arrows at the South–North axis indicate the
location of the vertical cut in the East–West direction, which is
shown in Figure 7.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
graphic inversion for large-scale convective cells (supergranulation), meridional flow, sunspot and an emerging
active region.
Quiet-Sun convection
The data used were for 8.5 h on 27 January 1996 from the
high-resolution mode of the MDI instrument. The results of
inversion of these data are shown in Figure 6. We have
found that, in the upper layers, 2–3 Mm deep, the horizontal
flow is organized in supergranular cells, with outflows from
the centre of the supergranules. The characteristic size of
the cells is 20–30 Mm. Comparing with MDI magnetograms,
it was found that the cell boundaries coincide with the areas
of enhanced magnetic field. These results are consistent
with the observations of supergranulation on the solar
surface13. However, in the layers deeper than 5 Mm, we do
not see the supergranulation pattern. This suggests that
supergranulation is only 5 Mm deep. An alternative
interpretation suggesting a depth of 8 Mm was presented
by Duvall14.
The vertical flows (Figure 7) correlate with the supergranular pattern in the upper layers. Typically, there are
upflows in the ‘hotter’ areas and downflows in the ‘colder’
areas. In the hotter areas however, the sound speed is
higher than the average.
at low latitudes to higher latitudes and, therefore, contribute
to the cyclic polar-field reversal.
The meridional flows in the solar interior were detected
by the time-distance method. Figure 8 shows the differences
between the travel-times of acoustic waves propagating
poleward and equatorward at different latitudes λ. These
travel-time differences correspond to the mean
meridional flow averaged over the penetration depth of the
acoustic waves, which was 4–24 Mm in the measurements.
By using eq. (4) Giles et al.11 estimated that the maximum
mean speed of the flow is ~ 20 m s –1. They have also found
that the flow velocity is almost constant over the observed
range of depth.
Tomography of sunspots and active regions
An important problem of astrophysics is understanding the
mechanisms of solar activity. The solar tomography
provides a tool for studying the birth and evolution
of active regions and complexes of solar activity. In Figure
9, we show the results for the emerging active region
observed in January 1998. This was a high-
Meridional circulation
Meridional flows from the equator towards the north and
south poles have been observed on the solar surface in
direct Doppler-shift measurements15. The MDI observations
by Giles et al.11 have provided the first evidence that such
flows persist to great depths, and, thus, possibly play an
important role in the 11-year solar cycle. The poleward flow
can transport the magnetic remnants of sunspots generated
Figure 7. The vertical-flow field (arrows) and the sound speed
perturbation at the North–South position indicated by arrows in
Figure 6.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Figure 8. The average travel-time difference (south minus north)
as a function of latitude, λ, for surface separation of pairs of points
in the range 12–73 Mm. The individual points are shown (squares)
and the 1σ errors (vertical lines). The solid curve is the best-fit 2parameter model. The velocity scale on the right axis, in which
12.1 m/s flow corresponds to a 1 s time difference, is obtained from
eq. (3) (ref. 11).
1471
SPECIAL SECTION: SOLAR PHYSICS
latitude region of the new solar cycle which was started in
1997.
Figure 9 shows the distribution of the wave speed
variations in a vertical cross-section in the region of the
emerging flux and in a horizontal plane at a depth of 18 Mm,
for three 2-h intervals, a, at 17:00 UT, 11 January 1998; b,
3:00 UT, 12 January 1998; and c, 5:00 UT, 12 January 1998.
The perturbations of the magnetosonic speed shown in this
figure are associated with the magnetic field and
temperature variations in the emerging magnetic ropes. The
positive variations are shown in red, and the negative
variations are shown in blue.
Figure 9 a shows no significant variations in the region of
the emergence, which is at the middle of the vertical plane.
The MDI magnetogram shown at the top indicates only
very weak magnetic field above this region. Figure 9 b
shows a positive perturbation associated with the emerging
region. The strongest perturbation in this panel is at the
bottom of the observed region. During the next 2 h (Figure
9 c),
the
perturbation
is
propagated
to
the
top of the box. From these data, we conclude that the
emerging flux propagated through the characteristic depth
of 10 Mm during 2 h. This gives an estimate of the speed of
emergence ≈ 1.3 km/s. This speed is somewhat higher than
the speed predicted by theories of emerging flux. The
typical amplitude of the sound speed variation in the
perturbation is about 0.5 km/s. This may correspond to a
magnetic field strength of 500 G at the top of the box, or a
temperature variation of 800 K.
After the emergence we observed the gradual increase of
the perturbation in the subsurface layers with the formation
of sunspots. The observed development of the active
region suggests that the sunspots were formed as result of
the concentration of magnetic flux close to the surface.
Figure 10 is an example of the internal structure of a large
sunspot observed on 17 January 1998. An image of the spot
taken in the continuum is shown at the top. The sound
speed perturbations in the spot are much stronger than in
the emerging flux, and reach more than 3 km/s. It is
interesting that beneath the spot the perturbation is
negative in the subsurface layers and becomes positive in
the deeper interior. This data also shows the connection to
the spot of a small pore which is on the left side of the spot.
The negative perturbations beneath the spot are, probably
due to the lower temperature. However, the effects of
temperature and magnetic field have not been separated in
these inversions. Separating these effects is an important
problem of solar tomography.
Figure 9. The sound speed perturbation in the emerging
active region (a) on 11 January 1998, 17:00 UT; (b) 12
January 1998, 3:00; and (c) 12 January 1998, 5:00. The
horizontal size of the box is 415 Mm, the vertical size is
18 Mm. The panels on the top are MDI magnetogram
showing the surface magnetic field of positive (red) and
negative (blue) polarities. The perturbations of the sound
speed range from – 1.6 to 1.3 km/s. The positive variations
are shown in red, and the negative ones in blue.
1472
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Figure 10. The sound-speed perturbation in a sunspot region observed on 17 January 1998. The
horizontal size of the box is 158 Mm, the depth is 24 Mm.
mechanisms of solar activity.
Conclusion
Solar tomography, or time-distance helioseismology, provides unique information about three-dimensional structures and flows associated with magnetic field and turbulent
convection in the solar interior. This method is at the very
beginning of its development. In this paper, we have
reviewed some basic principles of this technique, based on
the geometrical ray approximation, and presented some
initial inversion results. Developing wave-form solar
tomography is one of the most challenging problems of
helioseismology.
Using time-distance seismology, we have been able to
measure the structure of supergranulation flows and detect
an active region before it appeared on the surface. The
inversion results also have shown interesting dynamics of
supergranulation, meridional circulation, emerging active
regions and the formation of sunspots in the upper
convection zone. Further studies of the Sun’s interior by
the time-distance seismology will shed light on the
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
1. Duvall, T. L., Jr., Jefferies, S. M., Harvey, J. W. and Pomerantz,
M. A., Nature, 1993, 362, 430–432.
2. Aki, K. and Richards, P., Quantitative Seismology. Theory and
Methods, Freeman, San Francisco, 1980.
3. Kosovichev, A. G. and Duvall, T. L. Jr., in SCORe’96: Solar
Convection and Oscillations and their Relationship (eds Pijpers,
F. P., Christensen-Dalsgaard, J. and Rosenthal, C. S.), Kluwer
Academic Publishers, 1997, pp. 241–260.
4. Jefferies, S. M., Osaki, Y., Shibahashi, H. et al., Astrophys. J.,
1994, 434, 795–800.
5. Duvall, T. L. Jr., D’Silva, S., Jefferies, S. M., Harvey, J. W. and
Schou, J., Nature, 1996, 379, 235–237.
6. Gough, D. O., in Astrophysical Fluid Dynamics (eds Zahn, J.-P.
and Zinn-Justin, J.), Elsevier Science Publ., 1993, p. 339.
7. Jensen, J. M., Jacobsen, B. H. and Christensen-Dalsgaard, J.,
1999, preprint.
8. Duvall, T. L. Jr., Kosovichev, A. G., Scherrer, P. H. et al., Solar
Phys., 1997, 170, 63–73.
9. Paige, C. C. and Saunders, M. A., ACM Trans. Math. Software,
1982, 8, 43–71.
10. Jensen, J. M., Jacobsen, B. H. and Christensen-Dalsgaard, J.,
Proceedings of the Interdisciplinary Inversion Workshop 5,
Aarhus, 1997, pp. 57–67.
1473
SPECIAL SECTION: SOLAR PHYSICS
11. Giles, P. M., Duvall, T. L. Jr. and Scherrer, P. H., Nature, 1997,
390, 52–54.
12. Kosovichev, A. G., Astrophys. J., 1996, 461, L55–L57.
13. Title, A. M., Tarbell, T. D., Topka, K. P., Ferguson, S. H. and
Shine, R. A., Astrophys. J., 1989, 336, 475–494.
14. Duvall, T. L. Jr., Proceedings of the SOHO-6/GONG-98 Workshop, ESA, 1998.
ACKNOWLEDGEMENTS. The authors acknowledge many years
of effort by the engineering and support staff of the MDI
development team at the Lockheed Palo Alto Research Laboratory
(now Lockheed–Martin Advanced Technology Center) and the SOI
development team at Stanford University. SOHO is a project of
international cooperation between ESA and NASA. This research is
supported by the SOI-MDI NASA contract NAG5-3077 at Stanford
University.
15. Duvall, T. L. Jr., Solar Phys., 1979, 63, 3–15.
1474
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
1475
SPECIAL SECTION: SOLAR PHYSICS
The solar dynamo
Arnab Rai Choudhuri
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
It is believed that the magnetic field of the Sun is produced
by the dynamo process, which involves nonlinear interactions
between the solar plasma and the magnetic field. After
summarizing the main characteristics of solar magnetic
fields, the basic ideas of dynamo theory are presented. Then
an appraisal is made of the current status of solar dynamo
theory.
1. Introduction
IN elementary textbooks on stellar structure, a star is usually
modelled as a spherically symmetric, non-rotating, nonmagnetic object. It is mainly the magnetic field which makes
our Sun much more intriguing than such a textbook star.
Several other reviews in this special section should
convince the reader of this. It comes, therefore, as no
surprise that one of the central problems in solar physics is
to understand the origin of the Sun’s magnetic field. The
solar dynamo theory attempts to address this problem. The
basic idea of this theory is that the solar magnetic fields are
generated and maintained by complicated nonlinear
interactions between the solar plasma and magnetic fields.
As we shall see in this review, there are still many
difficulties with this theory and we are still far from having a
completely satisfactory explanation of why the Sun’s
magnetic field behaves the way it does. However, no
alternate theory of the origin of solar magnetism has so far
been able to explain even a fraction of what dynamo theory
has explained. Some of us, therefore, are still struggling to
put the solar dynamo theory on firmer footing, with the fond
hope that we are probably approximately on the correct
path.
The aim of this special section is to make the readers of
Current Science aware of the present status of solar
physics. The solar dynamo theory is a fairly technical
subject. It is next to impossible to write a review that will
provide a comprehensive introduction to this subject for an
average reader of Current Science and, at the time, survey
the research frontiers. Still a partial attempt is made here at
this next-to-impossible task of presenting the subject in a
way which should be understandable – if not to a general
reader of Current Science – at least to a reader with some
familiarity in physics and fluid mechanics. It is left to the
readers to judge if the author has failed completely or only
moderately. Needless to say, no attempt is made at a
e-mail: arnab@physics.iisc.ernet.in
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
complete coverage of the fundamentals. After summarizing
the relevant observations in the following section, we write
just enough about the basics in the next two sections to
give a rough idea of what is going on. Then in the last three
sections we discuss some of the important issues from
current research frontiers.
Dynamo theory is based on the principles of magnetohydrodynamics (MHD), in which hydrodynamics equations
are combined with Maxwell’s electrodynamics equations.
Comprehensive introductions to MHD can be found in the
books by Alfvén and Fälthammar1, Cowling2, Parker3,
Priest4, and Choudhuri5. Some books devoted exclusively to
dynamo theory are by Moffatt6, Krause and Rädler7, and
Zeldovich et al.8. We also refer to the review articles on the
solar dynamo by Ruzmaikin9, Gilman10, Hoyng11,
Brandenburg and Tuominen12, and Schmitt13.
2. Relevant observations
In 1908 Hale14 discovered the first evidence of Zeeman
effect in sunspot spectra and made the momentous announcement that sunspots are regions of strong magnetic fields.
This is the first time that somebody found conclusive
evidence of large-scale magnetic fields outside the Earth’s
environment. The typical magnetic field of a large sunspot
is about 3000 G.
Even before it was realized that sunspots are seats of
solar magnetism, several persons have been studying the
occurences of sunspots. Schwabe15 noted that the number
of sunspots seen on the solar surface increases and
decreases with a period of about 11 years. Now we believe
that the Sun has a cycle with twice that period, i.e. 22 years.
Since the Sun’s magnetic field changes its direction after 11
years, it takes 22 years for the magnetic field to come back
to its initial configuration. Carrington16 found that sunspots
seemed to appear at lower and lower latitudes with the
progress of the solar cycle. In other words, most of the
sunspots in the early phase of a solar cycle are seen
between 30° and 40°. As the cycle advances, new sunspots
are found at increasingly lower latitudes. Then a fresh halfcycle begins with sunspots appearing again at high
latitudes. Individual sunspots live from a few days to a few
weeks.
After finding magnetic fields in sunspots, Hale and his
coworkers17 made another significant discovery. They
found that often two large sunspots are seen side by side
and they invariably have opposite polarities. The line
joining the centres of such a bipolar sunspot pair is usually
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SPECIAL SECTION: SOLAR PHYSICS
nearly parallel to the solar equator. Hale’s coworker Joy,
however, noted that there is a systematic tilt of this line with
respect to the equator and that this tilt increases with
latitude17. This result is usually known as Joy’s Law. It was
also noted17 that the sunspot pairs have opposite polarities
in the two hemispheres. In other words, if the left sunspot in
the northern hemisphere has negative polarity, then the left
sunspot in the southern hemisphere has positive polarity.
This is clearly seen in Figure 1, which is a magnetic map of
the Sun’s disk obtained with a magnetogram. The regions of
positive and negative polarities are shown in white and
black respectively. The polarities of the bipolar sunspots in
any hemisphere get reversed from one half-cycle of 11 years
to the next half-cycle.
After the development of the magnetograph by Babcock
and Babcock18, it became possible to study the much
weaker magnetic field near the poles of the Sun. This
magnetic field is of the order of 10 G and reverses its
direction at the time of solar maximum19 (i.e. when the
number of sunspots seen on the solar surface is maximum).
This shows that this weak, diffuse field of the Sun is in
some way coupled to the much stronger magnetic field of
the sunspots and is a part of the same solar cycle. Lowresolution magnetograms show the evidence of weak
magnetic field even in lower latitudes. The true nature of
this field is not very clear. It was found20 that the magnetic
field on the solar surface outside sunspots often exists in
the form of fibril flux tubes of diameter of the order of 300 km
with field strength of about 2000 G (large sunspots have
sizes
larger
than
10,000 km).
One
Figure 1. A magnetogram image of the full solar disk. The regions
with positive and negative magnetic polarities are respectively shown
in white and black, with grey indicating regions where the magnetic
field is weak. Courtesy: K. Harvey.
1476
is not completely sure if the field found in the lowresolution magnetograms is truly a diffuse field or a
smearing out of the contributions made by fibril flux tubes.
Keeping this caveat in mind, we should refer to the field
outside sunspots as seen in magnetograms as the ‘diffuse’
field. It was found21 that there were large unipolar matches
of this diffuse field on the solar surface which migrated
towards the pole. Even when averaged over longitude, one
finds predominantly one polarity in a belt of latitude which
drifts polewards22,23. The reversal of polar field presumably
takes place when sufficient field of opposite polarity has
been brought near the poles.
Figure 2 (taken from Dikpati and Choudhuri24) shows the
distribution of both sunspots and the weak, diffuse field in
a plot of latitude vs. time. The colour shades indicate values
of longitude-averaged diffuse field, whereas the latitudes
where sunspots were seen at a particular time are marked by
vertical black lines. The sunspot distribution in a timelatitude plot is often referred to as a butterfly diagram,
since the pattern (the vertical black lines in Figure 2)
reminds one of a butterfly. Such butterfly diagrams were
first plotted by Maunder25. Historically, most of the dynamo
models concentrated on explaining the distribution of
Figure 2. Colour-shades showing the latitude-time distribution of
longitudinally averaged weak, diffuse magnetic field (B is in Gauss)
with a ‘butterfly diagram’ of sunspots superimposed on it during the
interval from May 1976 to December 1985. Reproduced from
Dikpati and Choudhuri24 .
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
sunspots and ignored the diffuse field. Only during the last
few years, it has been realized that the diffuse fields give us
important clues about the dynamo process and they should
be included in a full self-consistent theory. The aim of such
a theory should be to explain diagrams like Figure 2 (i.e. not
just the butterfly diagram).
We have provided above a summary of the various
regular features in the Sun’s activity cycle. One finds lots of
irregularities and fluctuations superposed on the underlying
regular behaviour, as can be seen in Figure 2. These
irregularities are more clearly visible in Figure 3, where the
number of sunspots seen on the solar surface is plotted
against time. Galileo was one of the first persons in Europe
to study sunspots at the beginning of the 17th century.
After Galileo’s work, sunspots were almost not seen for
nearly a century26!
It may be noted that all the observations discussed
above pertain to the Sun’s surface. We have no direct
information about the magnetic field underneath the Sun’s
surface. The new science of helioseismology, however, has
provided us lots of information about the velocity field
underneath the solar surface. For an account of this subject,
the readers may turn to the reviews by Chitre and Antia,
and by Christensen-Dalsgaard and Thompson. We shall
have occasions to refer to some of the helioseismic findings
in our discussion later. It is to be noted that heat is
transported by convection in the outer layers of the Sun
from about 0.7 R¤ to R¤ (where R¤ is the solar radius). This
region is called the convection zone, within which the
plasma is in a turbulent state. The job of a theorist now is to
construct a detailed model of the physical processes in this
turbulent plasma such that all the surface observations of
magnetic fields are properly explained – a fairly daunting
problem, of which the full solution is still a distant dream.
3. Some basic magnetohydrodynamics
considerations
The velocity field v and the magnetic field B in a plasma
(regarded as a continuum) interact with each other according to the following MHD equations:
∂v
1 
B 2  (B ⋅ ∇ ) B
+
+ ( v ⋅ ∇ ) v = ∇  p +
+ g + ν∇ 2 v , (1)
∂t
ρ 
8π 
4πρ
∂B
2
= ∇ × ( v × B) + λ∇ B .
∂t
Here ρ is density, p is pressure, g is gravitational field, ν is
kinematic viscosity, and
λ=
c2
4πσ
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
(3)
is magnetic diffusivity (σ is electrical conductivity).
Equation (1) is essentially the Navier–Stokes equation, to
which magnetic forces have been added. It is clear from eq.
(1) that the magnetic field has two effects: (i) it gives rise to
an additional pressure B2/8π; and (ii) the other magnetic
term (B · ∇)B/4πρ is of the nature of a tension along
magnetic field lines.
Equation (2) is known as the induction equation and is
the key equation in MHD. It has the same form as the
vorticity equation in ordinary hydrodynamics (see, for
example, § 4.2 and § 5.2 of Choudhuri5). If V, B and L are the
typical values of velocity, magnetic field and length scale,
then the two terms on the RHS of eq. (2) are of order VB/L
and λB/L2. The ratio of these two terms is a dimensionless
number, known as the magnetic Reynolds number, given by
Rm =
Figure 3. The number of sunspots seen in a year plotted against
the year for the period 1610–1975. The original figure is due to John
A. Eddy. Reproduced from Moffatt6 .
(2)
VB /L
λB/L2
=
VL
.
λ
(4)
Since Rm goes as L, it is expected to be much larger in
astrophysical situations than it is in the laboratory. In fact,
usually one finds that Rm >> 1 in astrophysical systems and
Rm << 1 in laboratory-size objects. Hence the behaviours of
magnetic fields are very different in laboratory plasmas and
astrophysical
plasmas.
For
example,
it
is
not possible to have laboratory analogues of the selfsustaining magnetic fields of the Earth or the Sun. If Rm >> 1
in an astrophysical system, then the diffusion term in eq. (2)
is negligible compared to the term preceding it. In ordinary
hydrodynamics, when the viscous dissipation term in the
vorticity equation is neglected, we are led to the famous
1477
SPECIAL SECTION: SOLAR PHYSICS
Kelvin’s theorem of vorticity conservation (see, for example,
§ 4.6 of Choudhuri5). Exactly similarly, when the diffusion
term in eq. (2) is neglected, it can be shown that the
magnetic field is frozen in the plasma and moves with it.
This result was first recognized by Alfvén27 and is often
referred to as Alfvén’s Theorem of Flux-Freezing.
It is known that the Sun does not rotate like a solid body.
The angular velocity at the equator is about 20% faster than
that at the poles. Because of the flux freezing, this
differential rotation would stretch out any magnetic field
line in the toroidal direction (i.e. the φ direction
with respect to the Sun’s rotation axis). This is indicated in
Figure 4. We, therefore, expect that the magnetic field
inside the Sun may be predominantly in the toroidal
direction.
We have already mentioned in § 2 that energy is
transported by convection in the layers underneath the
Sun’s surface. To understand why the magnetic field
remains concentrated in structures like sunspots instead of
spreading out more evenly, we need to study the interaction
of the magnetic field with the convection in the plasma. This
subject is known as magnetoconvection. The linear theory
of convection in the presence of a vertical magnetic field
was studied by Chandrasekhar28. The nonlinear evolution of
the system, however, can only be found from numerical
simulations pioneered by Weiss29. It was found that space
gets separated into two kinds of regions. In certain regions,
magnetic field is excluded and vigorous convection takes
place. In other regions, magnetic field gets concentrated
and the tension of magnetic field lines suppresses
convection in those regions. Sunspots are presumably such
regions where magnetic field is piled up by surrounding
convection. Since heat transport is inhibited there due to
the suppression of convection, sunspots look darker than
the surrounding regions.
Although we have no direct information about the state
of the magnetic field under the Sun’s surface, it is expected
that the interactions with convection would keep the
magnetic field concentrated in bundles of field lines
throughout the solar convection zone. Such a concentrated
bundle of magnetic field lines is called a flux tube. In the
a
a
b
b
Figure 4. The production of a strong toroidal magnetic field
underneath
Figure
5. the
Magnetic
Sun’s surface.
buoyancy
a.ofAn
a flux
initial
tube.
poloidal
a. A nearly
field horizontal
line. b. A
sketch
flux
tubeofunder
the field
the solar
linesurface.
after itb.has
Thebeen
flux tube
stretched
after by
its upper
the faster
part
rotation
has
risennear
through
the equatorial
the solar surface.
region.
1478
regions of strong differential rotation, therefore, we may
have flux tubes aligned in the toroidal direction. If a part of
such a flux tube rises up and pierces the solar surface as
shown in Figure 5 b, we expect to have two sunspots with
opposite polarities at the same latitudes. But how can a
configuration like Figure 5 b arise? The answer to
this question was provided by Parker30 through his idea
of magnetic buoyancy. We have seen in eq. (1) that a
pressure B2/8π is associated with a magnetic field. If p in and
p out are the gas pressures inside and outside a flux tube,
then we need to have
p out = pin +
B2
8π
(5)
to maintain pressure balance across the surface of a flux
tube. Hence,
p in ≤ p out,
(6)
which often, though not always, implies that the density
inside the flux tube is less than the surrounding density. If
this happens in a part of the flux tube, then that part
becomes buoyant and rises against the gravitational field to
produce the configuration of Figure 5 b starting from Figure
5 a.
A look at Figure 4 now ought to convince the reader that
the sub-surface toroidal field in the two hemispheres should
have opposite polarity. If this toroidal field rises due to
magnetic buoyancy to produce the bipolar sunspot pairs,
we expect the bipolar sunspots to have opposite polarities
in the two hemispheres as seen in Figure 1.
We thus see that combining the ideas of flux freezing,
magnetoconvection and magnetic buoyancy, we can understand many aspects of the bipolar sunspot pairs.
We now turn our attention to the central problem – the
dynamo generation of the magnetic field.
4. The turbulent dynamo and mean field MHD
We now address the question whether it is possible
for motions inside the plasma to sustain a magnetic
field. Ideally, one would like to solve eqs (1) and (2) to
understand how velocity and magnetic fields interact
with each other. Solving these two equations simultaneously in any non-trivial situation is an extremely
challenging job. In the early years of dynamo research, one
would typically assume a velocity field to be given and then
solve eq. (2) to find if this velocity field would sustain a
magnetic field. This problem is known as the kinematic
dynamo problem. One of the first important steps was a
negative theorem due to Cowling31, which established that
an axisymmetric solution is not possible for the kinematic
dynamo problem. One is, therefore, forced to look for more
complicated, non-axisymmetric solutions.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
A major breakthrough occurred in 1955 when Parker32
realized that the turbulent motions inside the solar convection zone (which are by nature non-axisymmetric) may
be able to sustain the magnetic field. We have indicated in
Figure 4 how a magnetic field line in the poloidal plane may
be stretched by the differential rotation to produce a
toroidal component. Parker32 pointed out that the uprising
hot plasma blobs in the convection zone would rotate as
they rise because of the Coriolis force of solar rotation (just
like cyclones in the Earth’s atmosphere) and such helically
moving plasma blobs would twist the toroidal field shown in
Figure 6 a to produce magnetic loops in the poloidal plane
as shown in Figure 6 b. Keeping in mind that the toroidal
field has opposite directions in the two hemispheres and
helical motions of convective turbulence should also have
opposite helicities in the two hemispheres, we conclude that
the poloidal loops in both hemispheres should have the
same sense as indicated in Figure 6 c. Although we are in a
high magnetic Reynolds number situation and the magnetic
field is nearly frozen in the plasma, there is some diffusion
(especially due to turbulent mixing) and the poloidal loops
in Figure 6 c should eventually coalesce to give the largescale poloidal field as sketched by the broken line in Figure
6 c.
Figure 7 captures the basic idea of Parker’s turbulent
dynamo. The poloidal and toroidal components of the
magnetic field feed each other through a closed loop. The
poloidal component is stretched by differential rotation to
produce the toroidal component. On the other hand, the
helical turbulence acting on the toroidal component gives
back the poloidal component. Parker32 developed a heuristic
mathematical formalism based on these ideas and showed
by mathematical analysis that these ideas worked. However,
a more systemic mathematical formulation of these ideas
had to await a few years, when Steenbeck, Krause and
Rädler33 developed what is known as mean field MHD.
Some of the basic ideas of mean field MHD are summarized
below.
Since we have to deal with a turbulent situation, let us
split both the velocity field and the magnetic field into
average and fluctuating parts, i.e.
v = v + v ′, B = B + B ′.
is known as the mean e.m.f. and is the crucial term for
dynamo action. This term can be perturbatively evaluated
by a scheme known as the first-order smoothing approximation (see, for example, § 16.5 of Choudhuri5). If the
turbulence is isotropic, then this approximation scheme
leads to
ε = αB − β ∇ × B,
(10)
where
1
α = − v ′ ⋅ (∇ × v ′) τ ,
3
(11)
and
β=
1
v ′ ⋅ v ′τ .
3
(12)
Here τ is the correlation time of turbulence. On substituting
(10) in eq. (8), we get
∂B
= ∇ × ( v × B) + ∇ × (αB) + ( λ + β)∇ 2 B .
∂t
(13)
It should be clear from this that β is the turbulent diffusion.
This is usually much larger than the molecular diffusion λ
so that λ can be neglected in eq. (13). It follows from eq. (11)
that α is a measure of average helical motion in the fluid. It
is this coefficient which describes the production of the
poloidal component from the toroidal component by helical
turbulence. This term would go to zero if turbulence has no
net average helicity (which would happen in a non-rotating
a
b
(7)
Here the overline indicates the average and the prime
indicates the departure from the average. On substituting
eq. (7) in the induction eq. (2) and averaging term by term,
we obtain
∂B
= ∇ × ( v × B ) + ∇ × ε + λ∇ 2 B ;
∂t
(8)
on remembering that v ′ = B′ = 0 .Here,
ε = v ′ × B′
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
c
(9)
Figure 6.
7. Different
Schematicstages
representation
of the dynamo
of Parker’s
process. idea
See text
of the
for
explanation.
turbulent
dynamo.
1479
SPECIAL SECTION: SOLAR PHYSICS
frame).
Equation (13) is known as the dynamo equation and has
to be solved to understand the generation of magnetic field
by the dynamo process. A variant of this equation was first
derived by rather intuitive arguments in the classic paper of
Parker32. The mean field MHD developed by Steenbeck,
Krause and Rädler33 put this equation on a firmer footing. In
the kinematic dynamo approach, one has to specify a
velocity field v and then solve eq. (13). Using spherical
polar coordinates with respect to the rotation axis of the
Sun, we can write
v = Ω (r , θ ) r sin θ eφ + v P ,
(14)
where Ω(r, θ) is the angular velocity in the interior of the
Sun and v P is some possible average flow in the poloidal
plane. Until a few years ago, almost all the calculations of
the kinematic dynamo problem were done by taking v P
= 0. If this is the case, then one has to specify some
reasonable Ω(r, θ) and α(r, θ) before proceeding to solve
the dynamo eq. (13). In the 1970s almost an industry grew
up presenting solutions of the dynamo equation for
different specifications of Ω and α.
The first pioneering solution in rectangular geometry was
obtained by Parker32 himself. He showed that periodic and
propagating wave solutions of the dynamo equation are
possible. Presumably this offers an explanation for the solar
cycle. Sunspots migrate from higher to lower latitudes with
the solar cycle because sunspots are produced (by
magnetic buoyancy) where the crest of the propagating
dynamo wave lies. Parker32 found that the parameters α and
Ω have to satisfy the following condition in the northern
hemisphere to make the dynamo wave propagate in the
equatorward direction (so as to explain the butterfly diagram
of sunspots):
α
dΩ
≤ 0.
dr
(15)
Steenbeck and Krause34 were the first to solve the dynamo
equation in a spherical geometry appropriate for the Sun
and produced the first theoretical butterfly diagram
of the distribution of sunspots in time-latitude. Then
many dynamo solutions were worked out by Roberts35,
Köhler36, Yoshimura37, Stix38 and others. One might have felt
complacent about the varieties of butterfly diagrams
produced by these authors. However, it has to be admitted
that many basic physics questions remained unanswered.
Since nothing was known at that time about the conditions
in the interior of the Sun, different authors were choosing
different α and Ω subject only to the condition (15),
and thereby were trying to fit the observational data better.
Eventually it appeared that it was becoming a
game in which you could get solutions according to your
wishes by tuning your free parameters suitably. Further
progress in solar dynamo theory became possible only by
1480
asking fundamental questions about the basic physics in
the interior of the Sun, rather than by blindly solving the
dynamo equation. These efforts will be described in the next
section. It may be noted that all the authors of this period
focussed their attention on explaining the equatorward
propagation of sunspots, by assuming that sunspots were
produced in the regions where the toroidal component had
the peak value. No serious attempt was made to connect the
behaviour of the weak, diffuse magnetic field with the
dynamo process or to explain the poleward migration of this
field, although Köhler36 and Yoshimura37 presented some
models that show a polar branch, i.e. a region near the poles
where the dynamo wave propagates poleward.
5. Dynamo in the overshoot layer?
Where does the solar dynamo work? Since one needs
convective turbulence to drive the dynamo, it used to be
tacitly assumed in the early 1970s that the dynamo works in
the solar convection zone and the different researchers of
that period used to take α(r, θ) non-zero in certain regions
of the convection zone. This approach had to be
questioned when Parker39 started looking at the effect
of magnetic buoyancy on the solar dynamo. Magnetic
buoyancy is particularly destabilizing in the interior of the
convection zone, where convective instability and magnetic
buoyancy reinforce each other. On the other hand,
if a region is stable against convection, then magnetic
buoyancy can be partially suppressed there (see, for
example, § 8.8 of Parker3). Calculations of buoyant
rise by Parker39 showed that any magnetic field in the
convection zone would be removed from there by magnetic
buoyancy fairly quickly. Hence it is difficult to make the
dynamo work in the convection zone, since the magnetic
field has to be stored in the dynamo region for a sufficient
time to allow for dynamo amplification.
It is expected that there is a thin overshoot layer
(probably with a thickness of the order of 104 km) just below
the bottom of the convection zone. This is a layer which is
convectively stable according to a local stability analysis,
but convective motions are induced there due
to convective plumes from the overlying unstable layers
overshooting and penetrating there. Several authors
(Spiegel and Weiss40, van Ballegooijen41) pointed out that
this layer is a suitable location for the operation of
the dynamo. Although there would be enough turbulent
motions in this layer to drive the dynamo, magnetic
buoyancy would be suppressed by the stable temperature
gradient there. This idea turned out to be a really prophetic
theoretical guess, since helioseismology observations a few
years
later
indeed
discovered
a
region
of strong differential rotation at the bottom of the solar
convection zone. See the review by Christensen-Dalsgaard
and Thompson in this issue on this subject. So it is
certainly expected that a strong toroidal magnetic field
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
should be generated just below the bottom of the
convection zone due to this strong differential rotation. It
may be noted that there have been other ideas as well for
suppressing magnetic buoyancy at the bottom of the
convection zone. Parker42 suggested ‘thermal shadows’,
whereas van Ballegooijen and Choudhuri43 showed that
an equatorward meridional circulation at the base of
the convection zone can help in suppressing magnetic
buoyancy there.
For about a decade starting from the mid-1980s, most
researchers in this field believed that the whole dynamo
process in the Sun, as summarized in Figure 7, takes place in
the overshoot layer. Properties of such a dynamo operating
in the overshoot layer were studied by DeLuca and
Gilman44, Gilman et al.45, and Choudhuri46. If the dynamo
operates in the overshoot layer, some new questions arise.
Previously when the solar dynamo was supposed to work in
the convection zone, the sunspots seen on the solar surface
could be regarded as direct signatures of the dynamo
process. One could assume that sunspots appeared
wherever the dynamo produced strong toroidal fields just
underneath the surface. On the other hand, if the dynamo
works at the bottom of the convection zone, the whole
depth of the convection zone separates the region where
the magnetic fields are generated and the solar surface
where sunspots are seen. In order to understand the
relation between the solar dynamo and sunspots, one then
has to study how the magnetic fields generated at the
bottom of the convection zone rise through the convection
zone to produce sunspots.
The best way to study this is to treat it as an initial-value
problem. First an initial configuration with some magnetic
flux at the bottom of the convection zone is specified, and
then its subsequent evolution is studied numerically. The
evolution depends on the strength of magnetic buoyancy,
which is in turn determined by the value of the magnetic
field. If the dynamo is driven by turbulence, one would
expect an equipartition of energy between the dynamogenerated magnetic field and the fluid kinetic energy, i.e.
B2 1
2
≈ ρv .
8π 2
(16)
This suggests B ≈ 104 G on the basis of standard models of
convection. Because of the strong differential rotation, we
expect the magnetic field at the bottom of the convection
zone to be mainly in the toroidal direction. One, therefore,
has to take a toroidal magnetic flux tube going around the
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
rotation axis as the initial configuration. The evolution of
such magnetic flux tubes due to magnetic buoyancy was
first studied by Choudhuri and Gilman47 and Choudhuri48. It
was found that the Coriolis force due to the Sun’s rotation
plays a much more important role in this problem than what
anybody suspected before. If the initial magnetic field is
taken to have a strength around 104 G, the flux tubes move
parallel to the rotation axis and emerge at very high latitudes
rather than at latitudes where sunspots are seen. Only if the
initial magnetic field is taken as strong as 105 G, magnetic
buoyancy is strong enough to overpower the Coriolis force
and the magnetic flux tubes can rise radially to emerge at
low latitudes.
D’Silva and Choudhuri49 extended these calculations to
look at the tilts of emerging bipolar regions at the surface.
Figure 8 taken from their paper shows the observational tilt
vs. latitude plot of bipolar sunspots (i.e. Joy’s law) along
with the theoretical plots obtained by assuming different
values of the initial magnetic field. It is clearly seen that
theory fits observations only if the initial magnetic field is
about 105 G. Apart from providing the first quantitative
explanation of Joy’s law nearly three-quarters of a century
after its discovery, these calculations put the first stringent
limit on the value of the toroidal magnetic field at the bottom
of the convection zone. Several other groups50–53 soon
performed similar calculations and confirmed the result. The
evidence is now mounting that the magnetic field at the
bottom of the convection zone is indeed much stronger
than the equipartition value given by eq. (16) (see
Schüssler54 for a review of this topic).
If the magnetic field is much stronger than the equipartition value, it would be impossible for the helical
turbulence (entering the mathematical theory through the α
term defined in eq. (11)) to twist the magnetic field lines. The
dynamo process, as envisaged in Figure 7, is therefore, not
possible. We need a very different type of dynamo model.
Schmitt55 and Feriz-Mas et al.56 proposed that the buoyant
instability of the strong magnetic field itself may lead to a
magnetic configuration which was previously thought to be
created by helical turbulence. Parker57 suggested an
‘interface dynamo’ in which the helical turbulence acts in a
region above the bottom of the convection zone. This idea
has been further explored by Charbonneau and
MacGregor58. In the next section, we discuss what we
regard as the most promising approach to build a model of
the solar dynamo that can account for the very strong
toroidal magnetic field at the bottom of the convection zone.
1481
SPECIAL SECTION: SOLAR PHYSICS
6. The Babcock–Leighton approach and hybrid
models
We saw in § 4 and § 5 that one of the crucial ingredients in
turbulent dynamo theory is the role of helical turbulence in
generating the poloidal component from the toroidal
component, which is mathematically modelled through mean
field MHD. This approach to the dynamo problem will be
called the Parker–Steenbeck–Krause–Rädler or the PSKR
approach. In this approach, the dynamo is supposed to
operate within a region where convective turbulence exists
and no attention is paid to phenomena taking place at the
solar surface. Babcock59 and Leighton60 in the 1960s
developed a somewhat different approach, which we call the
Babcock–Leighton or the BL approach. Even in this
approach, the toroidal magnetic field is believed to be
produced by the differential rotation of the Sun. For the
production of the poloidal component, however, a totally
different scenario is invoked. The strong toroidal
component leads to bipolar sunspots due to magnetic
buoyancy. We have noted that these bipolar sunspots have
a
tilt
with
respect
to
latitudinal
lines
(i.e. Joy’s law). Therefore, when these bipolar sunspots
eventually decay, the magnetic flux spreads around in such
a way that the flux at the higher latitude has more
contribution from the polarity of the sunspot which was at
the higher latitude. In this way, a poloidal component arises.
Compared to the PSKR approach, the BL approach was
heuristic and semi-qualitative. The mean field MHD is, no
doubt, based on some assumptions and approximations,
and it is not clear whether these hold in the conditions
prevailing in the Sun’s interior. However, for an ideal
system satisfying these assumptions and approximations,
the mean field MHD is a rigorous mathematical theory. No
quantitative mathematical theory of comparable sophis-
Figure 8. Plots of sin (tilt) against sin (latitude) theoretically
obtained for different initial values of magnetic field indicated in kG.
The observational data indicated by the straight line fits the
theoretical curve for initial magnetic field 100 kG (i.e. 105 G).
Reproduced from D’Silva and Choudhuri49 .
1482
tication was developed for the BL approach. Nearly all
the self-consistent dynamo calculations in the 1970s and
1980s, therefore, followed the PSKR approach. The BL
approach was developed further by a group in NRL23,61–64
who were studying the spread of magnetic flux from the
decay of sunspots. There was growing evidence that there
is a general meridional flow with an amplitude of about a few
m s –1 near the Sun’s surface proceeding from the equator to
the pole65. The poloidal magnetic field produced from the
decay of tilted bipolar sunspots is carried poleward by this
meridional circulation. As we have already pointed out in
§ 2, the weak diffuse magnetic field on the solar surface
migrates towards the pole, in contrast to sunspots which
migrate equatorward. One presumably has to identify the
weak diffuse field as the poloidal component of the Sun’s
magnetic field, whereas sunspots form from the much
stronger toroidal component. The main aim of the NRL
group was to model the evolution of the weak diffuse field,
assuming that this was entirely coming from the decay of
bipolar sunspots. No attempt was made to address the full
dynamo problem. They even made the drastically simple
assumption that the magnetic field is a scalar residing on
the solar surface and the appropriate partial differential
equation was solved only on this two-dimensional surface.
Dikpati and Choudhuri66,24, and Choudhuri and Dikpati67
attempted to make a vectorial model of the evolution of the
weak diffuse field and to connect it to the dynamo problem.
Since the meridional flow at the surface is poleward, there
must be an equatorward flow in the lower regions of the
convection zone, rising near the equator. If the dynamo
operated at the base of the convection zone, then, in
accordance with the ideas prevalent a few years ago, the
poloidal component produced by this dynamo would be
brought to the surface by the meridional circulation. This
can be an additional source of the weak diffuse field at the
surface, apart from the contributions coming from the decay
of sunspots. Figure 9 from Choudhuri and Dikpati67 shows a
theoretical time-latitude distribution of the weak diffuse field
on the surface, obtained by assuming a dynamo wave at the
bottom of the convection zone as given. In other words, to
produce this figure – which should be compared with the
observational Figure 2 – the dynamo problem was not
solved self-consistently. The aim now should be to develop
a self-consistent model of the dynamo, which should be
able to explain the behaviours of both the sunspots (i.e. the
toroidal component) and the weak diffuse field (i.e. the
poloidal component).
With helioseismology establishing the existence of
strong differential rotation at the base of the convection
zone, there is little doubt that the toroidal magnetic field is
produced there and has a magnitude of the order of 105 G, a
result pinned down by the simulations of buoyant flux rise.
It will, however, be impossible for helical turbulence to twist
Figure
9. Amagnetic
theoreticalfields,
time-latitude
of the weak,
such strong
and it distribution
seems improbable
that
diffuse magnetic field on the solar surface, with ‘half-butterfly
the
generation
of
the
poloidal
diagrams’ obtained from a running dynamo wave assumed given at
the
of the
fromturbulence
Choudhuri–and
fieldbasefrom
theconvection
toroidal zone.
fieldReproduced
by helical
as
68
Dikpati .
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
envisaged in the PSKR approach – takes place at the base
of the convection zone. For the generation of the poloidal
field, we then invoke the BL idea that it is produced by the
decay of bipolar sunspots on the surface. The meridional
circulation can then carry this field poleward, to be
eventually brought to the bottom of the convection zone
where it is stretched by the differential rotation to produce
the strong toroidal field. If a mean field formulation is made
of the process of poloidal field generation near the surface
by invoking an α coefficient concentrated near the solar
surface, this hybrid model of the dynamo will incorporate
the best features of both the PSKR and the BL approaches.
On the one hand, detailed quantitative calculations will be
possible, as in the PSKR approach. On the other hand, the
surface phenomena emphasized in the BL approach, are
integrated in the dynamo problem. The meridional
circulation plays an important role in this hybrid model so
that a suitable form of
in eq. (14) is to be specified. It is
hoped that this hybrid model will account for both the
equatorward migration of the strong toroidal field at the
base of the convection zone and the poleward drift of the
poloidal field at the surface.
This hybrid model has one other attractive feature.
Researchers in 1970s built kinematic models of the solar
dynamo by arbitrarily specifying α(r, θ) and Ω(r, θ). These
were regarded as free parameters to be tuned suitably so as
to give solutions with desired characteristics. In the present
hybrid models, these important ingredients to the dynamo
process
are
directly
based
on
observations.
Helioseismology has given us Ω(r, θ), with its shear
concentrated at the base of the convection zone. The α
coefficient also arises out of the observed decay of bipolar
sunspots on the surface. Previously it used to be even
debated whether α is positive or negative. Researchers
used to fudge α such that the inequality (15) was satisfied.
The direction of tilt of bipolar sunspots on the surface,
however, clearly indicates that α arising out of their decay
has to be positive in the northern hemisphere: a point made
by Stix38 long ago. Once these key ingredients are fixed
directly by observation, we no longer have the freedom to
fudge them according to our wishes, which researchers in
1970s could do. This leads to one problem. Helioseismology
shows that dΩ/dr is positive in lower latitudes where
sunspots are seen. If α is also positive in the northern
hemisphere, then clearly inequality (15) is not satisfied and
dynamo waves are expected to propagate poleward!
The first calculations on the hybrid model were reported
by Choudhuri et al.68. Dynamo waves were indeed found to
propagate poleward, if meridional circulation was
switched off. The toroidal field and the poloidal field are
respectively produced in layers near the base of
the convection zone and near the solar surface. When
meridional circulation is switched off, any field can reach
from one layer to the other layer by diffusion with a time
scale of L2/β, where L is the separation between the layers
(i.e. the thickness of the convection zone). If the meridional
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
flow has a typical velocity of the order V, it takes time V/L
for the meridional circulation to carry some quantity
between the two layers. When the time scale V/L is shorter
than
the
diffusion
time
scale
L2/β,
the problem is dominated by meridional circulation and
Choudhuri et al.68 found that the strong toroidal component
at the bottom of the convection zone actually propagates
equatorward, overriding the inequality (15). Thus the
inequality (15), which was regarded as sacrosanct for four
decades since Parker32 obtained it, is found not to hold in
the presence of a meridional circulation having a time scale
shorter than diffusion time, thereby opening up the
possibility of constructing realistic hybrid models of the
solar dynamo. Further calculations on hybrid models have
been reported in a series of papers69–73. It should be
emphasized that all these studies are still of rather
exploratory nature. They demonstrate the viability of the
hybrid models and study their different characteristics. We
are, however, still far from building a sufficiently realistic
model, putting in all the details, that would account for the
observational data presented in Figure 2. Achieving this
should be our goal now.
7. Miscellaneous ill-understood issues
Since this is a review in a special section on solar physics,
we have primarily discussed those aspects of kinematic
dynamo models which directly pertain to the matching of
theory with observational data. It should, however, be kept
in mind that many fundamental issues of dynamo theory are
still very ill-understood. Until we have a better
understanding of these issues, the kinematic models can, at
best, be considered superficial attempts at a very deep
physics problem. The reader may look up the IAU Symposium volume on The Cosmic Dynamo74 for several articles
dealing with these fundamental issues. Here we make only
very brief comments on some of these issues.
We have seen in § 4 that the turbulent dynamo theory is
developed by averaging over turbulent fluctuations. The
existence of magnetic flux concentrations clearly indicates
that the fluctuations are much larger than the average
values (often by orders of magnitude). Does a mean field
theory make sense in such a situation? Can we trust the
perturbative procedures like the first-order smoothing
approximation? Hoyng75 raised some questions regarding
the interpretation of the averaged quantities. The dynamo
eq. (13) admits of several possible modes in spherical
geometry: the preferred mode seems to be the mode with
dipole symmetry, wherein the toroidal component is
oppositely directed in the two hemispheres. This mode
approximately corresponds to the observational data.
However, Stenflo and Vogel76 pointed out that one
hemisphere of the Sun often has more sunspots than the
other, indicating that there may be a superposition with
higher modes having different symmetry. Analysing the
1483
SPECIAL SECTION: SOLAR PHYSICS
statistics of sunspot data for several decades, Gokhale and
Javaraiah77 claimed to have found evidence for multiple
modes. If the fluctuations are so large, there is no reason
why a particular mode should be very stable, or why higher
modes should not be excited. The interference of modes
with different symmetry was theoretically studied by
Brandenburg et al.78 employing a nonlinear dynamo model.
Since the toroidal magnetic field is far stronger than the
equipartition value, it is certainly not justified to assume
that the magnetic fields do not back-react on the flow. One
should therefore ideally solve eqs (1) and (2) simultaneously, instead of proceeding with kinematic models.
Since this is a fairly difficult job even by the standard of
today’s computers, attempts are made to include the backreaction of the magnetic field within kinematic models
instead of going to fully dynamic models. One easy way to
incorporate the back-reaction in the dynamo eq. (13) is to
make the crucial quantity αdecrease with the magnetic field,
following some prescription like:
α=
α0
1 + ( B /B0 )2
.
(17)
The effect of such α-quenching on the dynamo process has
been extensively studied79–82. When the velocity field v is
specified, the dynamo eq. (13) is a linear equation for the
magnetic field
provided
B
we assume the various coefficients in the equation to be independent of B. If α
is quenched by B, in accordance with eq. (17), we have a
nonlinear problem. One important question is whether the
irregularities of the solar cycle, as seen in Figure 3, can be
explained with the help of nonlinear models. It seems that
the nonlinearity introduced through eq. (17) cannot cause
such chaotic behaviour. Since a sudden increase in the
amplitude of magnetic field would diminish the dynamo
activity by reducing α and thereby pull down the amplitude
again (a decrease in the amplitude would do the opposite),
the α-quenching mechanism tends to lock the system in a
stable mode once the system relaxes to it. In fact, Krause
and Meinel83 argued that nonlinearities must be what makes
one particular mode of the dynamo so stable. Only by
introducing more complicated kinds of nonlinearity (with
suppression of differential rotation) in some highly
truncated dynamo models, Weiss et al.84 were able to find
the evidence of chaos. Jennings and Weiss 85 presented a
study of symmetry-breakings and bifurcations in a
nonlinear dynamo model. Since α-quenching of the form
(17) cannot explain the irregularities of the solar cycle,
Choudhuri86 explored the effect of stochastic fluctuations
on the mean equations and obtained some solutions
resembling Figure 3. Several subsequent papers87–89
explored this possibility further.
Finally we comment on the efforts in building fully
dynamic models by solving both eqs (1) and (2) simultaneously. This is a highly complicated nonlinear problem
and can only be tackled numerically. Gilman90 and Glatz1484
maier91 presented very ambitious numerical calculations in
which convection, differential rotation and dynamo process
were all calculated together from the basic MHD equations.
These calculations, however, gave results which do not
agree with observational data. For example, angular velocity
was found to be constant on cylinders, whereas
helioseismology found it to be constant on cones. If
various diffusivities were set such that the surface rotation
pattern was matched, the dynamo waves propagated from
the equator to the pole. The codes of Gilman90 and
Glatzmaier91 naturally had finite grids, and the physics at the
sub-grid scales was modelled by introducing various eddy
diffusivities. Probably the physics at sub-grid scales is more
subtle and the details of it are crucially important in
determining
the
behaviour
of
the dynamo. This is generally believed to be the reason
why these massive codes did not produce agreement
with observations. The subsequent approach in numerical modelling has been to do dynamic calculations over
cubes which correspond to small regions of the Sun, rather
than trying to build models for the whole Sun. Brandenburg
et al.92 and Nordlund et al.93 have followed this approach.
8. Conclusion
It seems that the solar magnetic fields are generated and
maintained by the dynamo process. There is only a small
minority of solar physicists who would disagree with
this point of view. It is, however, not easy to build a
sufficiently detailed and realistic model of the dynamo
process to account for all the different aspects of solar
magnetism. The 1970s happened to be a period of optimism
in dynamo research when various researchers were
producing butterfly diagrams by choosing different forms of
α(r, θ) and Ω(r, θ). It was felt that further research would
narrow down the parameter space and establish a standard
model of the solar dynamo. As we discussed above, this did
not
happen.
In
1993
Schmitt13
wrote
in
his review on the solar dynamo: ‘The original hope that
detailed observational and theoretical information would
yield better results of the dynamo did not prove true, on the
contrary, they raised difficulties instead’. Today, a few
years afterwards, we can perhaps have a less pessimistic
outlook. The hybrid models, which are closely linked to
observations and which combine together some of the best
ideas that came out of dynamo research in the last few
decades, certainly do look promising. Although sufficiently
detailed models have not yet been worked out, we hope that
we are close to building kinematic models which are much
more realistic and sophisticated than the kinematic models
of the 1970s. Perhaps other researchers may regard this
point of view as a reflection of this author’s personal
prejudice. Only time will tell if this prejudice is justified.
Finally we end by cautioning the reader that this article
should not be regarded as a comprehensive review of the
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
solar dynamo problem. Since this article is primarily aimed
not at the experts but at more general readers, we have
emphasized those aspects of kinematic models which have
direct relevance to observations. A very incomplete
discussion of various fundamental issues is presented in
the § 7. There is no doubt that kinematic models can never
fully satisfy us. The ultimate challenge is to build fully
dynamic models starting from the basic equations, and then
to explain both the fluid flow patterns and magnetic patterns
in the interior of the Sun in a grand scheme. As we have
pointed out in § 7, the modern computers still seem
inadequate for handling this problem. The solar dynamo
problem will certainly remain alive for years to come.
1. Alfvén, H., and Fälthammar, C.-G., Cosmical Electrodynamics,
Oxford University Press, 1963, 2nd edn.
2. Cowling, T. G., Magnetohydrodynamics, Adam Hilger, 1976,
2nd edn.
3. Parker, E. N., Cosmical Magnetic Fields, Oxford University
Press, 1979.
4. Priest, E. R., Solar Magnetohydrodynamics, D. Reidel, 1982.
5. Choudhuri, A. R., The Physics of Fluids and Plasmas: An Introduction for Astrophysicists, Cambridge University Press, 1998.
6. Moffatt, H. K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, 1978.
7. Krause,
F.
and
Rädler,
K.-H.,
Mean-Field
Magnetohydrodynamics and Dynamo Theory, Pergamon, 1980.
8. Zeldovich, Ya B., Ruzmaikin, A. A. and Sokoloff, D. D.,
Magnetic Fields in Astrophysics, Gordon and Breach, 1983.
9. Ruzmaikin, A. A., Solar Phys., 1985, 100, 125.
10. Gilman, P. A., in Physics of the Sun (ed. Sturrock, P. A.),
D. Reidel, 1986, vol. 1, p. 95.
11. Hoyng, P., in Basic Plasma Processes on the Sun, IAU-Symp.
No. 142 (eds. Priest, E. R. and Krishan, V.), Kluwer, Dordrecht,
1990, p. 45.
12. Brandenburg, A. and Tuominen, I., in The Sun and Cool Stars:
Activity, Magnetism, Dynamos (eds. Tuominen, I., Moss, D. and
Rüdiger, G.), Lecture Notes in Physics, 1990, 380, 223.
13. Schmitt, D., in The Cosmic Dynamo (eds. Krause, F., Rädler,
K.-H. and Rüdiger, G.), IAU-Symp. No. 157, Kluwer, 1993, p. 1.
14. Hale, G. E., Astrophys. J., 1908, 28, 315.
15. Schwabe, S. H., Astron. Nachr., 1844, 21, 233.
16. Carrington, R. C., Mon. Not. R. Astron. Soc., 1858, 19, 1.
17. Hale, G. E., Ellerman, F., Nicholson, S. B. and Joy, A. H.,
Astrophys. J., 1919, 49, 153.
18. Babcock, H. W. and Babcock, H. D., Astrophys. J., 1955, 121,
349.
19. Babcock, H. D., Astrophys. J., 1959, 130, 364.
20. Stenflo, J. O., Solar Phys., 1973, 32, 41.
21. Bumba, V. and Howard, R., Astrophys. J., 1965, 141, 1502.
22. Howard, R. and LaBonte, B. J., Solar Phys., 1981, 74, 131.
23. Wang, Y.-M., Nash, A. G. and Sheeley, N. R., Astrophys. J.,
1989, 347, 529.
24. Dikpati, M. and Choudhuri, A. R., Solar Phys., 1995, 161, 9.
25. Maunder, E. W., Mon. Not. R. Astron. Soc., 1904, 64, 747.
26. Eddy, J. A., Science, 1976, 192, 1189.
27. Alfvén, H., Ark. f. Mat. Astr. o. Fysik, 1942, B29.
28. Chandrasekhar, S., Philos. Mag., 1952, 43, 501.
29. Weiss, N. O., J. Fluid Mech., 1981, 108, 247.
30. Parker, E. N., Astrophys. J., 1955, 121, 491.
31. Cowling, T. G., Mon. Not. R. Astron. Soc., 1934, 94, 39.
32. Parker, E. N., Astrophys. J., 1955, 122, 293.
33. Steenbeck, M., Krause, F. and Rädler, K.-H., Z. Naturforsch.,
1966, 21a, 1285.
34. Steenbeck, M. and Krause, F., Astron. Nachr., 1969, 291, 49.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
Roberts, P. H., Philos. Trans. R. Soc. London, 1972, 272, 663.
Köhler, H., Astron. Astrophys., 1973, 25, 467.
Yoshimura, H., Astrophys. J. Suppl., 1975, 29, 467.
Stix, M., Astron. Astrophys., 1976, 47, 243.
Parker, E. N., Astrophys. J., 1975, 198, 205.
Spiegel, E. A. and Weiss, N. O., Nature, 1980, 287, 616.
van Ballegooijen, A. A., Astron. Astrophys., 1982, 113, 99.
Parker, E. N., Astrophys. J., 1987, 312, 868.
van Ballegooijen, A. A. and Choudhuri, A. R., Astrophys. J.,
1988, 333, 965.
DeLuca, E. E. and Gilman, P. A., Geophys. Astrophys. Fluid
Dyn., 1986, 37, 85.
Gilman, P. A., Morrow, C. A. and DeLuca, E. E., Astrophys. J.,
1989, 338, 528.
Choudhuri, A. R., Astrophys. J., 1990, 355, 733.
Choudhuri, A. R. and Gilman, P. A., Astrophys. J., 1987, 316,
788.
Choudhuri, A. R., Solar Phys., 1989, 123, 217.
D’Silva, S. and Choudhuri, A. R., Astron. Astrophys., 1993, 272,
621.
Fan, Y., Fisher, G. H. and DeLuca, E. E., Astrophys. J., 1993,
405, 390.
Schüssler, M., Caligari, P., Ferriz-Mas, A. and Moreno-Insertis,
F., Astron. Astrophys., 1994, 281, L69.
Caligari, P., Moreno-Insertis, F. and Schüssler, M., Astrophys. J.,
1995, 441, 886.
Longscope, D. and Fisher, G. H., Astrophys. J., 1996, 458, 380.
Schüssler, M., in The Cosmic Dynamo (eds. Krause, F., Rädler,
K.-H. and Rüdiger, G.), IAU-Symp. No. 157, Kluwer, Dordrecht,
1993, p. 27.
Schmitt, D., Astron. Astrophys., 1987, 174, 281.
Ferriz-Mas, A., Schmitt, D. and Schüssler, M., Astron.
Astrophys., 1994, 289, 949.
Parker, E. N., Astrophys. J., 1993, 408, 707.
Charbonneu, P. and MacGregor, K. B., Astrophys. J., 1997, 486,
502.
Babcock, H. W., Astrophys. J., 1961, 133, 572.
Leighton, R. B., Astrophys. J., 1969, 156, 1.
DeVore, C. R., Sheeley, N. R. and Boris, J. P., Solar Phys., 1984,
92, 1.
Sheeley, N. R., DeVore, C. R. and Boris, J. P., Solar Phys., 1985,
98, 219.
Sheeley, N. R., Wang, Y.-M. and Harvey, J. W., Solar Phys.,
1989, 119, 323.
Wang, Y.-M., Nash, A. G. and Sheeley, N. R., Science, 1989,
245, 712.
Duvall, T. L., Solar Phys., 1979, 63, 3; LaBonte, B. J. and
Howard, R., Solar Phys., 1982, 80, 361; Ulrich et al., Solar
Phys., 1988, 117, 291.
Dikpati, M. and Choudhuri, A. R., Astron. Astrophys., 1994,
291, 975.
Choudhuri, A. R. and Dikpati, M., Solar Phys., 1999, 184, 61.
Choudhuri, A. R., Schüssler, M. and Dikpati, M., Astron. Astrophys., 1995, 303, L29.
Durney, B. R., Solar Phys., 1995, 160, 213.
Durney, B. R., Solar Phys., 1996, 166, 231.
Durney, B. R., Astrophys. J., 1997, 486, 1065.
Dikpati, M. and Charbonneau, P., Astrophys. J., 1999 (in press).
Nandi, D. and Choudhuri, A. R., Astrophys. J., 1999 (submitted).
Krause, F., Rädler, K.-H. and Rüdiger, G. (eds.), The Cosmic
Dynamo, IAU Symp. No. 157, Kluwer, Dordrecht, 1993.
Hoyng, P., Astrophys. J., 1988, 332, 857.
Stenflo, J. O. and Vogel, M., Nature, 1986, 319, 285.
Gokhale, M. H. and Javaraiah, J., Mon. Not. R. Astron. Soc.,
1990, 243, 241.
Brandenburg, A., Krause, F., Meinel, R., Moss, D. and
Tuominen, I., Astron. Astrophys., 1989, 213, 411.
Stix, M., Astron. Astrophys., 1972, 20, 9.
Jepps, S. A., J. Fluid Mech., 1975, 67, 625.
1485
SPECIAL SECTION: SOLAR PHYSICS
81. Ivanova, T. S. and Ruzmaikin, A. A., Sov. Astron., 1977, 21,
479.
82. Yoshimura, H., Astrophys. J., 1978, 226, 706.
83. Krause, F. and Meinel, R., Geophys. Astrophys. Fluid Dyn.,
1988, 43, 95.
84. Weiss, N. O., Cattaneo, F. and Jones, C. A., Geophys. Astrophys.
Fluid Dyn., 1984, 30, 305.
85. Jennings, R. L. and Weiss, N. O., Mon. Not. R. Astron. Soc.,
1991, 252, 249.
86. Choudhuri, A. R., Astron. Astrophys., 1992, 253, 277.
87. Hoyng, P., Astron. Astrophys., 1993, 272, 321.
1486
88. Moss, D., Brandenburg, A., Tavakol, R. and Tuominen, I.,
Astron. Astrophys., 1992, 265, 843.
89. Ossendrijver, A. J. H., Hoyng, P. and Schmitt, D., Astron.
Astrophys., 1996, 313, 938.
90. Gilman, P. A., Astrophys. J. Suppl., 1983, 53, 243.
91. Glatzmaier, G. A., Astrophys. J., 1985, 291, 300.
92. Brandenburg, A., Nordlund, A., Pulkkinen, P., Stein, R. F. and
Tuominen, I., Astron. Astrophys., 1990, 232, 277.
93. Nordlund, A., Brandenburg, A., Jennings, R. L., Rieutord, M.,
Ruokolainen, J., Stein, R. F. and Tuominen, I., Astrophys. J.,
1992, 392, 647.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Solar neutrinos: An overview
J. N. Bahcall
Building E, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
I summarize the current state of solar-neutrino research.
1. Introduction
THE most important result from solar-neutrino research
is, in my view, that solar neutrinos have been detected
experimentally with fluxes and energies that are qualitatively
consistent with solar models that are constructed assuming that
the Sun shines by nuclear fusion reactions. The first experimental
result1,2 has now been confirmed by four other beautiful
experiments3–6.
The observation of solar neutrinos with approximately the
predicted energies and fluxes establishes empirically the theory7
that main-sequence stars derive their energy from nuclear fusion
reactions in their interiors, and has inaugurated what we all hope
will be a flourishing field of observational-neutrino astronomy.
The detection of solar neutrinos settle experimentally the debate
over the age and energy source of the Sun that raged for many
decades, beginning in the middle of the 19th century. The leading
theoretical physicists of the 19th century argued convincingly that
the Sun could not be more than 107 years old because that was the
maximum lifetime that could be fueled by gravitational energy (‘no
other natural explanation, except chemical action, can be
conceived’8). On the other hand, geologists and evolutionary
biologists argued that the Sun must be > 109 years old in order to
account for the observed geological features and for the evolutionary processes9. (The arguments of Lord Kelvin and his
theoretical physics associates were so persuasive that in later
editions Darwin removed all mention of time scales from The
Origin of the Species.) Today we know that the biologists and
geologists were right and the theoretical physicists were wrong,
which may be a historical lesson to which we physicists should
pay attention.
I will discuss predictions of the combined standard model in the
main part of this review. By ‘combined’ standard model I mean
the predictions of the standard solar model and the predictions of
the minimal electroweak theory. We need a solar model to tell us
how many neutrinos of what energy are produced in the Sun and
we need electroweak theory to tell us how the number and the
flavour content of the neutrinos are changed as they make their
way from the centre of the Sun to detectors on earth. For all
practical purposes, standard electroweak theory states that
nothing happens to solar neutrinos after they are created in the
deep interior of the Sun.
Using standard electroweak theory and fluxes from the
standard solar model, one can calculate the rates of neutrino
interactions in different terrestrial detectors with a variety of
energy sensitivities. The combined standard model also predicts
that the energy spectrum from a given neutrino source should be
the same for neutrinos produced in terrestrial laboratories and in
the Sun and that there should not be measurable time-dependences
(other than the seasonal dependence caused by the earth’s orbit
around the Sun). The spectral and temporal departures from
standard model expectations are expected to be small in all
currently operating experiments10 and have not yet yielded
definitive results. Therefore, I will concentrate here on inferences
that can be drawn by comparing the total rates observed in solarneutrino experiments with the combined standard model
predictions.
I will begin by reviewing in Section 2 the quantitative predictions of the combined standard solar model and then describe in
Section 3 the three solar-neutrino problems that are established by
the
chlorine,
Kamiokande,
SAGE,
GALLEX
and
Superkamiokande experiments. In Section 4 I detail the
uncertainties in the standard model predictions and then show in
Section 5 that helioseismological measurements indicate that the
standard solar model predictions are accurate for our purposes. In
Section 5 I discuss the implications for solar-neutrino research of
the precise agreement between helioseismological measurements
and the predictions of standard solar models. Next, ignoring all
knowledge of the Sun, I cite analyses in Section 6 that show that
one cannot fit the existing experimental data with neutrino fluxes
that are arbitrary parameters, unless one invokes new physics to
change the shape or flavour content of neutrino energy spectrum. I
summarize in Section 7 the characteristics of the best-fitting
neutrino oscillation descriptions of the experimental data. Finally,
I will discuss and summarize results in Section 8.
If you want to obtain numerical data or subroutines that are
discussed in this review, or to see relevant background
information, you can copy them from my Web site:
http://www.sns.ias.edu/~ jnb.
2. Standard model predictions
Table 1 gives the neutrino fluxes and their uncertainties for our
best standard solar model11, hereafter BP98. Figure 1 shows the
predicted neutrino fluxes from the dominant p–p fusion chain.
e-mail: jnb@ias.edu
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
The BP98-solar model includes diffusion of heavy elements and
helium, makes use of the nuclear reaction rates recommended by
the expert workshop held at the Institute of Nuclear Theory12,
recent (1996) Livermore OPAL radiative opacities13, the OPAL
equation of state14, and electron and ion screening as determined
Table 1. Standard model predictions (BP98): solar-neutrino fluxes
and neutrino capture rates, with 1σuncertainties from all sources
(combined quadratically)
Source
p–p
Flux
(10 10 cm–2 s–1 )
+ 0. 01
5.94 (1.00 − 0.01 )
Cl
(SNU)
Ga
(SNU)
0.0
69.6
pep
1.39 × 10 (1 .00 − 0.01 )
0.2
2.8
hep
2.10 × 10
–7
0.0
0.0
7
Be
4.80 × 10 –1 (1 .00 − 0.09 )
1.15
34.4
8
B
5.15 × 10 –4 (1 .00 − 0.14 )
5.9
12.4
–2
+ 0. 01
+ 0. 09
+ 0. 19
N
6.05 × 10 (1 .00 − 0.13 )
0.1
3.7
15
O
5.32 × 10 –2 (1 .00 − 0.15 )
0.4
6.0
F
+ 0. 12
10 –4 (1 .00 − 0.11 )
0.0
0.1
17
Total
–2
+ 0. 19
13
+ 0. 22
6.33 ×
+ 1. 2
7.7 − 1.0
129
+8
−6
SNU is a unit used to describe the measured rates of solar-neutrino
radiochemical experiments (10–36 interactions per target atom per
second).
1488
by the recent density matrix calculation15,16. The neutrino absorption cross-sections that are used in constructing Table 1 are the
most accurate values available17,18 and include, where appropriate,
the thermal energy of fusing-solar ions and improved nuclear and
atomic data. The validity of the absorption cross-sections has
recently been confirmed experimentally using intense radioactive
sources of 51Cr. The ratio, R, of the capture rate measured (in
GALLEX
and
SAGE)
to the calculated 51Cr-capture rate is R = 0.95 ± 0.07
(exp)+
(theory) and was discussed extensively at
Neutrino 98 by Gavrin and by Kirsten. The neutrino–electron
scattering cross-sections, used in interpreting the Kamiokande and
+ 0. 04
SuperKamiokande
experiments, now include electroweak radiative
− 0. 03
corrections19.
Figure 2 shows for the chlorine experiment all the predicted
rates and the estimated uncertainties (1σ) published by my
colleagues and myself since the first measurement by Ray Davis
and his colleagues in 1968. This figure should give you some
feeling for the robustness of the solar model calculations. Many
hundreds and probably thousands of researchers have, over three
decades, made great improvements in the input data for the solar
models, including nuclear cross-sections, neutrino cross-sections,
measured element abundances on the surface of the Sun, the solar
luminosity, the stellar radiative opacity, and the stellar equation of
state. Nevertheless, the most accurate predictions of today are
Figure 1. The energy spectrum of neutrinos from the p–p chain of interactions in the Sun, as
predicted by the standard solar model. Neutrino fluxes from continuum sources (such as p–p and
8
B) are given in the units of counts per cm2 per second. The p–p chain is responsible for more
than 98% of the energy generation in the standard solar model. Neutrinos produced in the carbon–
nitrogen–oxygen (CNO) chain are not important energetically and are difficult to detect
experimentally. The arrows at the top of the figure indicate the energy thresholds for the ongoing
neutrino experiments.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
essentially the same as they were in 1968 (although now they can
be made with much greater confidence). For the gallium
experiments, the neutrino fluxes predicted by standard solar
models, corrected for diffusion, have been in the range 120 SNU to
141 SNU since 1968 (ref. 17): A SNU is a convenient unit with
which to describe the measured rates of solar-neutrino
experiments; 10–36 interactions per target atom per second.
There are three reasons that the theoretical calculations of
neutrino fluxes are robust: (i) the availability of precision
measurements and precision calculations of input data; (ii) the
connection between neutrino fluxes and the measured solar
luminosity; and (iii) the measurement of the helioseismological
frequencies of the solar pressure- mode (p-mode) eigenfrequencies.
I have discussed these reasons in detail elsewhere20.
Figure 3 shows the calculated 7Be- and 8B-neutrino fluxes for all
19 standard solar models which have been published in the last 10
years in refereed science journals. The fluxes are normalized by
dividing each published value by the flux from the BP98-solar
model11: the abscissa is the normalized-8B flux and the ordinate is
the normalized-7Be neutrino flux. The rectangular box shows the
estimated 3σ uncertainties in the predictions of the BP98 solar
model.
All of the solar model results from different groups fall within
the estimated 3σ uncertainties in the BP98 analysis (with one not
understood exception that falls slightly outside). This agreement
demonstrates the robustness of the predictions, since the
calculations use different computer codes (which achieve varying
degrees of precision) and involve a variety of choices for the
nuclear parameters, the equation of state, the stellar radiative
opacity, the initial heavy element abundances, and the physical
processes that are included.
The largest contributions to the dispersion in values in Figure 3
Figure 2. The predictions of John Bahcall and his collaborators of
neutrino-capture rates in the 37 Cl experiment are shown as a function
of the date of publication (since the first experimental report1 in
1968). The event rate, SNU, is a convenient product of neutrinoflux times the interaction cross-section, 10–36 interactions per target
atom per sec. The format is from Figure 1.2 (ref. 40). The
predictions have been updated through 1998.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
are due to the choice of the normalization for S 17 (the production
cross-section factor for 8B neutrinos) and the inclusion or noninclusion of element diffusion in the stellar-evolution codes. The
effect in the plane of Figure 3 of the normalization of S 17 is shown
by the difference between the point for BP98 (1.0, 1.0), which
was computed using the most recent recommended normalization12, and the point at (1.18, 1.0) which corresponds to the BP98 result with the earlier (CalTech)
normalization21.
Helioseismological observations have shown11,22 that element
diffusion is occurring and must be included in solar models, so that
the most recent models shown in Figure 3 now all include helium
and heavy-element diffusion. By comparing a large number of
earlier models, it was shown that all published standard solar
models give the same results for solar-neutrino fluxes to an
accuracy of better than 10% if the same input parameters and
physical processes are included23,24.
Bahcall et al.10 have compared the observed rates
with the calculated standard-model values, combining quadratically
the theoretical solar model and experimental uncertainties, as well
as the uncertainties in the neutrino cross-sections. Since the
GALLEX and SAGE experiments measure the same quantity, we
treat the weighted average rate in gallium as one experimental
number.
We adopt the SuperKamiokande measurement as the
most precise direct determination of the higher-energy
8
B-neutrino flux.
Using the predicted fluxes from the BP98 model, the χ2 for the
Figure 3. Predictions of standard solar models since 1988. This
figure, which is Figure 1 of ref. 10, shows the predictions of 19 standard solar models in the plane defined by the 7 Be in and 8 B neutrino
fluxes. The abbreviations that are used in the figure to identify
different solar models are defined in the bibliographical item, ref. 45.
The figure includes all standard solar models, with which I am
familiar that were published in refereed journals in the decade 1988–
1998. All of Figure
the fluxes
4. are
Comparison
normalizedofto measured
the predictions
rates of
andthestandard-model
2–6
Bahcall–Pinsonneault
predictions
1998
for solar
five solar-neutrino
model, BP9811experiments
. The rectangular
. The unit for the
error box defines
radiochemical
the 3σ-errorexperiments
range of the
(chlorine
BP98 fluxes.
and gallium)
The best-fit
is SNU (see Figure
7
Be-neutrino flux
2 for
is negative.
a definition);
At thethe
99%
unit
C.L.,
forthere
the is
water-Cerenkov
no solution10 experiments
with all positive
(Kamiokande
neutrino fluxes
and (see
SuperKamiokande)
discussion in section
is the6).rate
Allpredicted
of
by the
the standard model
standard
solutions
solar model
lie farplus
from
thethe
standard
best-fitelectroweak
solution, even
theory11 .
far from the 3σ contour.
1489
SPECIAL SECTION: SOLAR PHYSICS
fit to the three experimental rates (chlorine, gallium, and
SuperKamiokande, see Figure 4) is:
2
χSSM
(3 experimental rates) = 61.
(1)
The result given in eq. (1), which is approximately equivalent to a
20σ discrepancy, is a quantitative expression of the fact that the
standard model predictions do not fit the observed solar-neutrino
measurements.
3. Three solar-neutrino problems
I will now compare the predictions of the combined standard
model with the results of the operating solar-neutrino experiments.
We will see that this comparison leads to three different discrepancies between the calculations and the observations, which I
will refer to as the three solar-neutrino problems.
Figure 4 shows the measured and the calculated event rates in
the five ongoing solar-neutrino experiments. This figure reveals
three discrepancies between the experimental results and the
expectations based upon the combined standard model: As we
shall see, only the first of these discrepancies depends in
significant measure upon the predictions of the standard solar
model.
Calculated vs observed absolute rate
The first solar-neutrino experiment to be performed
was the chlorine radiochemical experiment2, which detects
electron-type neutrinos that are more energetic than 0.81 MeV.
After more than a quarter of a century of operation of this experiment, the measured event rate is 2.56 ± 0.23 SNU, which is a
factor of three less than predicted by most detailed theoretical
calculations,
7.7 +−11..02 SNU (ref. 11). Most of the predicted rate in the chlorine
experiment is from the rare, high-energy 8B neutrinos, although the
7
Be neutrinos are also expected to contribute significantly.
According to standard-model calculations, the pep neutrinos and
the CNO neutrinos (for simplicity not discussed here) are
expected to contribute less than 1 SNU to the total event rate.
This discrepancy between the calculations and the observations
for the chlorine experiment was for more than two decades, the
only solar-neutrino problem. I shall refer to the chlorine
disagreement as the first solar-neutrino problem.
Incompatibility of chlorine and water experiments
The second solar-neutrino problem results from a comparison of
the measured event rates in the chlorine experiment and in the
Japanese
pure-water
experiments,
Kamiokande3
and
6
SuperKamiokande . The water experiments detect higher-energy
neutrinos, most easily above 7 MeV, by observing the Cerenkov
radiation from neutrino–electron scattering: ν + e → ν′ + e′.
According to the standard solar model, 8B-beta decay, and
1490
possibly the hep reaction25, are the only important source of
these higher-energy neutrinos.
The Kamiokande and SuperKamiokande experiments show that
the observed neutrinos come from the Sun. The electrons that are
scattered by the incoming neutrinos recoil predominantly in the
direction of the Sun–Earth vector; the relativistic electrons are
observed by the Cerenkov radiation they produce in the water
detector. In addition, the water Cerenkov experiments measure the
energies of individual scattered electrons and therefore provide
information about the energy spectrum of the incident solar
neutrinos.
The total event rate in the water experiments, about 0.5 the
standard-model value (see Figure 4), is determined by the same
high-energy 8B neutrinos that are expected, on the basis of a
combined
standard
model,
to
dominate
the event rate in the chlorine experiment. I have shown elsewhere26
that solar physics changes the shape of the 8B-neutrino spectrum
by less than 1 part in 105. Therefore, we can calculate the rate in
the chlorine experiment (threshold 0.8 MeV) that is produced by
the 8B neutrinos observed in the Kamiokande and
SuperKamiokande experiments at an order of magnitude higher
energy threshold.
If no new physics changes the shape of the 8B-neutrino energy
spectrum, the chlorine rate from 8B alone is 2.8 ± 0.1 SNU for the
SuperKamiokande
normalization (3.2 ± 0.4 SNU for the Kamiokande normalization), which
exceeds the total observed chlorine rate of 2.56 ± 0.23 SNU.
Comparing the rates of the SuperKamiokande and the chlorine
experiments, one finds – assuming that the shape of the energy
spectrum of 8Bνe ’s is not changed by new physics – that the net
contribution to the chlorine experiment from the pep, 7Be and
CNO
neutrino
sources
is negative: – 0.2 ± 0.3 SNU. The contributions from the pep,
7
Be, and CNO neutrinos would appear to be completely missing;
the standard model prediction for the combined contribution of
pep, 7Be, and CNO neutrinos is a relatively large 1.8 SNU (see
Table 1). On the other hand, we know that the 7Be neutrinos must
be created in the Sun since they are produced by electron capture
on the same isotope ( 7Be) which gives rise to the 8B neutrinos by
proton capture.
Hans Bethe and I pointed out27 that this apparent incompatibility of the chlorine and water-Cerenkov experiments
constitutes a second solar-neutrino problem that is almost
independent of the absolute rates predicted by solar models. The
inference that is usually made from this comparison is that the
energy spectrum of 8B neutrinos is changed from the standard
shape by physics not included in the simplest version of the
standard electroweak model.
Gallium experiments: No room for 7Be neutrinos
The results of the gallium experiments, GALLEX and SAGE,
constitute the third solar-neutrino problem. The average observed
rate in these two experiments is 73 ± 5 SNU, which is accounted
for in the standard model by the theoretical rate of 72.4 SNU that
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
the calculated 7Be-solar-neutrino flux. The required change in the
nuclear-physics cross-section would also increase the predicted
neutrino event rate by more than 100 in the Kamiokande
experiment, making that prediction completely inconsistent with
what is observed.
I conclude that either: (i) at least three of the five operating
solar-neutrino experiments (the two gallium experiments plus
either chlorine or the two water-Cerenkov experiments,
Kamiokande and SuperKamiokande) have yielded misleading
results, or (ii) physics beyond the standard electroweak model is
required to change the energy spectrum of νe after the neutrinos
are produced in the centre of the Sun.
is calculated to come from the basic p–p and pep neutrinos (with
only a 1% uncertainty in the standard solar model p–p flux).
The 8B neutrinos, which are observed above 6.5 MeV in
the Kamiokande experiment, must also contribute to the gallium
event rate. Using the standard shape for the spectrum of 8B
neutrinos and normalizing to the rate observed in Kamiokande, 8B
contributes another 6 SNU. (The contribution predicted by the
standard model is 12 SNU, see Table 1.) Given the measured rates
in the gallium experiments, there is no room for the additional
34 ± 3 SNU that is expected from 7Be neutrinos on the basis of
standard solar models (see Table 1).
The seeming exclusion of everything but p–p neutrinos in the
gallium experiments is the third solar-neutrino problem. This
problem is essentially independent of the previously-discussed
solar-neutrino problems, since it depends strongly upon the p–p
neutrinos that are not observed in the other experiments and
whose theoretical flux can be calculated accurately.
The missing 7Be neutrinos cannot be explained away by a
change in solar physics. The 8B neutrinos that are observed in the
Kamiokande experiment are produced in competition with the
missing 7Be neutrinos; the competition is between electron capture
on 7Be vs proton capture on 7Be. Solar model explanations that
reduce the predicted 7Be flux generically reduce much more (too
much) the predictions for the observed 8B flux.
The flux of 7Be neutrinos, φ( 7Be), is independent of
measurement uncertainties in the cross-section for the nuclear
reaction 7Be(p, γ)8B; the cross-section for this proton-capture
reaction is the most uncertain quantity that enters in an important
way in the solar model calculations. The flux of 7Be neutrinos
depends upon the proton-capture reaction only through the ratio
φ ( 7 Be) ∝
R (e)
,
R( e ) + R( p )
4. Uncertainties in the flux calculations
I will now discuss uncertainties in the solar-model-flux
calculations. Table 2 summarizes the uncertainties in the most
important solar-neutrino fluxes and in the Cl and Ga event rates
due to different nuclear fusion reactions (the first four entries), the
heavy element to hydrogen mass ratio (Z/X), the radiative opacity,
the solar luminosity, the assumed solar age, and the helium and
heavy element diffusion coefficients. The 14N + p reaction causes
a 0.2% uncertainty in the predicted p–p flux and a 0.1 SNU
uncertainty in the Cl (Ga) event rates.
The predicted event rates for the chlorine and gallium
experiments use recent improved calculations of neutrinoabsorption cross-sections17,18. The uncertainty in the prediction
for the gallium rate is dominated by uncertainties in the neutrinoabsorption cross sections, + 6.7 SNU (7% of the predicted rate)
and – 3.8 SNU (3% of the predicted rate). The uncertainties in the
chlorine-absorption cross- sections cause an error, ± 0.2 SNU (3%
of the predicted rate), that is relatively small compared to other
uncertainties in predicting the rate for this experiment. For nonstandard neutrino-energy spectra that result from new neutrino
physics, the uncertainties in the predictions for currently favoured
solutions (which reduce the contributions from the least welldetermined 8B neutrinos) will in general be less than the values
quoted here for standard spectra and must be calculated using the
appropriate cross-section uncertainty for each neutrino
(2)
where R(e) is the rate of electron capture by 7Be nuclei and R(p)
is the rate of proton capture by 7Be. With standard parameters,
solar models yield R(p) ≈ 10–3R(e). Therefore, one would have to
increase the value of the 7Be(p, γ)8B cross-section by more than
two orders of magnitude over the current-best estimate (which has
an estimated uncertainty of ~ 10%) in order to affect significantly
Table 2.
Average uncertainties in neutrino fluxes and event rates due to different input data. The flux
uncertainties
are expressed in fractions of the total flux, and the event-rate uncertainties are expressed in SNU. The 7 Be-electron
capture rate causes an uncertainty of ± 2% (ref. 44) that affects only the 7 Be-neutrino flux. The average
fractional uncertainties for individual parameters are shown
Fractional
uncertainty
p–p
0.017
3
He3 He
0.060
3
He4 He
0.094
7
Be + p
0.106
Z/X
Opacity
Luminocity
0.004
Age
0.004
0.003
0.003
0.0
0.003
0.028
0.014
0.003
0.018
0.052
0.028
0.006
0.040
Diffuse
0.033
Flux
p–p
0.002
0.002
0.005
0.000
0.002
7
8
Be
B
0.023
0.0155
0.040
0.080
0.000
0.019
0.021
0.075
0.105
0.042
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
1491
SPECIAL SECTION: SOLAR PHYSICS
energy17,18.
The nuclear fusion uncertainties in Table 2 were taken from
Adelberger et al.12, the neutrino cross-section uncertainties from
refs 17, 18, the heavy element uncertainty was taken from
helioseismological measurements28, the luminosity and age
uncertainties were adopted from BP95 (ref. 24), the 1σ-fractional
uncertainty in the diffusion rate was taken to be 15% (ref. 29),
which is supported by helioseismological evidence22, and the
opacity uncertainty was determined by comparing the results of
fluxes computed using the older Los Alamos opacities with fluxes
computed using the modern Livermore opacities23. To include the
effects of asymmetric errors, the now publicly-available code for
calculating rates and uncertainties (see discussion in previous
section) was run with different input uncertainties and the results
averaged. The software contains a description of how each of the
uncertainties listed in Table 2 were determined and used.
The low-energy cross-section of the 7Be + p reaction is the
most important quantity that must be determined more accurately
in order to decrease the error in the predicted event rates in solarneutrino experiments. The 8B-neutrino flux that is measured by
the Kamiokande3, Super-Kamiokande6, and SNO30 experiments is,
in
all
standard solar model calculations, directly proportional to the 7Be + p
cross-section. If the 1σ uncertainty in this cross-section can be
reduced by a factor of two to 5%, then it will no longer be the
limiting uncertainty in predicting the crucial 8B-neutrino flux (cf.
Table 2).
5. How large an uncertainty does
helioseismology suggest?
Could the solar model calculations be wrong by enough to explain
the discrepancies between predictions and measurements for solarneutrino experiments? Helioseismology, which confirms
predictions of the standard solar model to high precision, suggests
that the answer is probably ‘No’.
Figure 5 shows the fractional differences between the most
accurate available sound speed measured by helioseismology31 and
sound speed calculated with our best solar model (with no free
parameters). The horizontal line corresponds to the hypothetical
case in which the model predictions exactly match the observed
values. The root mean square (rms) fractional difference between
the calculated and the measured sound speeds is 1.1 × 10–3 for the
entire region over which the sound speeds are measured,
1492
0.05R¤ < R < 0.95R¤. In the solar core, 0.05R¤ < R < 0.25R¤ (in
which about 95% of the solar energy and neutrino flux is produced
in a standard model), the rms fractional difference between
measured and calculated sound speeds is 0.7 × 10–3.
Helioseismological measurements also determine two other
parameters that help characterize the outer part of the Sun (far
from the inner region in which neutrinos are produced): the depth
of the solar convective zone (CZ), the region in the outer part of
the Sun that is fully convective and the present-day surfaceabundance by mass of helium (Ysurf). The measured values,
RCZ = (0.713 ± 0.001)R¤ (ref. 32), and Ysurf = 0.249 ± 0.003 (ref.
28), are in satisfactory agreement with the values predicted by
the solar model BP98, namely, RCZ = 0.714R¤, and Ysurf = 0.243.
However, we shall see below that precision measurements of the
sound speed near the transition between the radiative interior (in
which energy is transported by radiation) and the outer convective
zone (in which energy is transported by convection) reveal small
discrepancies between the model predictions and the observations
in this region.
If solar physics were responsible for the solar-neutrino
problems, how large would one expect the discrepancies to be
between the solar model predictions and helioseismological
observations? The characteristic size of the discrepancies can be
estimated using the results of the neutrino experiments and scaling
laws for neutrino fluxes and sound speeds.
All recently published solar models predict essentially the same
fluxes from the fundamental p–p and pep reactions (amounting to
72.4 SNU in gallium experiments, cf. Table 1), which are closely
related to the solar luminosity. Comparing the measured gallium
rates and the standard predicted rate for the gallium experiments,
the 7Be flux must be reduced by a factor N if the disagreement is
not to exceed n standard deviations, where N and n satisfy
72.4 + (34.4)/N = 72.2 + nσ. For a 1σ (3σ) disagreement, N = 6.1
(2.05). Sound-speeds scale like the square root of the local
temperature divided by the mean molecular weight and the 7Beneutrino flux scales approximately as the 10th power of the
temperature33. Assuming that the temperature changes are
dominant, agreement to within 1σ would require fractional changes
of order 0.09 in sound speeds (3σ could be reached with 0.04
changes),
if all model changes were in the temperature.* This argument is
conservative because it ignores the 8B and CNO neutrinos which
contribute to the observed counting rate (cf. Table 1) and which, if
included, would require an even larger reduction of the 7Be flux.
*I have used in this calculation the GALLEX and SAGE measured
rates reported by Kirsten and Gavrin at Neutrino 98. The
experimental rates used in BP98 were not as precise and therefore
resulted in slightly less stringent constraints than those imposed here.
In BP98, we found that agreement to within 1σ with the then
available experimental numbers would require fractional changes of
order
0.08
in
sound
speeds
(3σcould be reached with 0.03 changes.)
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
I have chosen the vertical scale in Figure 5 to be appropriate
for fractional differences between measured and predicted sound
speeds that are of order 0.04 to 0.09, and that might therefore
affect solar-neutrino calculations. Figure 5 shows that the
characteristic agreement between the solar model predictions and
helioseismological measurements is more than a factor of 40 better
than would be expected if there were a solar model explanation of
the solar-neutrino problems.
6. Fits without solar models
Suppose (following the precepts of Hata et al.34, Parke35, and
Heeger and Robertson36) we now ignore everything we have
learned about solar models over the last 35 years and allow the
important p–p, 7Be, and 8B fluxes to take on any non-negative
values. What is the best fit that one can obtain to the solarneutrino
measurements
assuming
only that the luminosity of the Sun is supplied by
nuclear fusion reactions among light elements (the so-called ‘luminosity constraint’37)?
The answer is that the fits are bad, even if we completely ignore
what we know about the Sun; I quote the results from ref. 10.
If the CNO-neutrino fluxes are set equal to zero, there are no
acceptable solutions at the 99% C. L. (~ 3σ result). The best-fit is
worse if the CNO fluxes are not set equal to zero. All so-called
‘solutions’ of the solar-neutrino problems in which the
astrophysical model is changed arbitrarily (ignoring
helioseismology and other constraints) are inconsistent with the
observations at much more than a 3σ level of significance. No
fiddling of the physical conditions in the model can yield the
minimum value, quoted above, that was found by varying the
fluxes independently and arbitrarily.
Figure 3 shows, in the lower left-hand corner, the best-fit
solution and the 1σ – 3σ contours. The 1σ and 3σ limits were
obtained by requiring that χ2 = χ2min + δχ2, where for 1σ, δχ2 = 1
and for 3σ, δχ2 = 9. All of the standard model solutions lie far
from the best-fit solution and even lie far from the 3σ contour.
Since standard model descriptions do not fit the solar-neutrino
data, we will now consider models in which neutrino oscillations
change the shape of the neutrino energy spectra.
Table 3.
Neutrino oscillation solutions
Solution
SMA
LMA
LOW
VAC
5
2
8
8
∆m 2
(eV2 )
sin2 2θ
× 10 –6
× 10 –5
× 10 –8
× 10 –11
5 × 10 –3
0.8
0.96
0.7
production-reaction cross-section (3He + p → 4He + e+ + νe ) is
used10. However, for over a decade I have not given an estimated
uncertainty for this cross-section40. The transition matrix element
is essentially forbidden and the actual quoted value for the production cross-section depends upon a delicate cancellation between
two comparably sized terms that arise from very different and
hard to evaluate nuclear physics. I do not see any way at present
to determine from experiment or from first principles theoretical
calculations a relevant, robust upper limit to the hep-production
cross-section (and therefore the hep solar-neutrino flux).
The possible role of hep neutrinos in solar-neutrino
experiments is discussed extensively in ref. 25. The most
important unsolved problem in theoretical nuclear
physics related to solar neutrinos is the range of values allowed by
fundamental physics for the hep-production cross-section.
8. Discussion and conclusion
When the chlorine solar-neutrino experiment was first proposed41,
the only stated motivation was ‘. . . to see into the interior of a
star and thus verify directly the hypothesis of nuclear energy
generation in stars’. This goal has now been achieved.
The focus has shifted to using solar-neutrino experiments as a
tool for learning more about the fundamental characteristics of
7. Neutrino oscillations
The experimental results from all five of the operating solarneutrino experiments (chlorine, Kamiokande, SAGE, GALLEX,
and SuperKamiokande) can be fit well by descriptions involving
neutrino oscillations, either vacuum oscillations (as originally
suggested
by
Gribov
and
Pontecorvo38) or resonant matter oscillations (as originally discussed
by Mikeyhev, Smirnov, and Wolfenstein (MSW)39).
Table 3 summarizes the four best-fit solutions that
are found in the two-neutrino approximation10,25. Only
the SMA MSW solution fits well all the data – including
the recoil electron energy spectrum measured in the
SuperKamiokande experiment – if the standard value for the hep
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Figure 5. Predicted vs measured sound speeds. This figure shows
the excellent agreement between the calculated (solar model BP98,
Model) and the measured (Sun) sound speeds, a fractional difference
of 0.001 rms for all speeds measured between 0.05R¤ and 0.95R¤.
The vertical scale is chosen so as to emphasize that the fractional
error is much smaller than generic changes in the model, 0.04 to
0.09, that might significantly affect the solar-neutrino predictions.
1493
SPECIAL SECTION: SOLAR PHYSICS
neutrinos as particles. Experimental effort is now concentrated on
answering the question: What are the probabilities for
transforming a solar νe of a definite energy into the other possible
neutrino states? Once this question is answered, we can calculate
what happens to νe ’s that are created in the interior of the Sun.
Armed with this information from weak interaction physics, we
can return again to the original motivation of using neutrinos to
make detailed, quantitative tests of nuclear fusion rates in the solar
interior. Measurements of the flavour content of the dominant low
energy neutrino sources, p–p and 7Be neutrinos, will be crucial in
this endeavour and will require another generation of superb solarneutrino experiments.
Three decades of refining the input data and the solar model
calculations has led to a predicted standard model event rate for
the chlorine experiment, 7.7 SNU, which is very close to 7.5 SNU,
the best-estimate value obtained in 1968 (ref. 42). The situation
regarding solar neutrinos is, however, completely different now,
thirty years later. Four experiments have confirmed the original
chlorine detection of solar neutrinos. Helioseismological measurements are in excellent agreement with the standard solar model
predictions and very strongly disfavour (by a factor of 40 or
more)
hypothetical
deviations
from
the
standard model that are required to fit the neutrino data (cf. Figure 5).
Just in the last two years, improvements in the helioseismological
measurements have resulted in a five-fold improvement in the
agreement between the calculated standard solar model sound
speeds and the measured solar velocities.
1. Davis, R. Jr., Harmer, D. S. and Hoffman, K. C., Phys. Rev. Lett.,
1968, 20, 1205–1209.
2. Davis, R. Jr., Prog. Part. Nucl. Phys., 1994, 32, 13; Cleveland,
B. T., Daily, T., Davis, R. Jr., Distel, J. R., Lande, K., Lee, C.
K., Wildenhain, P. S. and Ullman, J., Astrophys. J., 1998, 496,
505–526.
3. Kamiokande Collaboration, Fukuda, Y. et al., Phys. Rev. Lett.,
1996, 77, 1683–1686.
4. SAGE Collaboration, Gavrin, V. et al., in Proceedings of the
17th International Conference on Neutrino Physics and
Astrophysics (Helsinki) (eds Enqvist, K., Huitu, K. and
Maalampi, J.), World Scientific, Singapore, 1997, pp. 14–24;
SAGE collaboration, Abdurashitov, J. N. et al., Phys. Rev. Lett.,
1996, 77, 4708–4711.
5. GALLEX Collaboration, Anselmann, P. et al., Phys. Lett., 1995,
B342, 440–450; GALLEX Collaboration, Hampel, W. et al.,
Phys. Lett., 1996, B388, 384–396.
6. SuperKamiokande Collaboration, Suzuki, Y., in Proceedings of
the XVIII International Conference on Neutrino Physics and
Astrophysics, Takayama, Japan, 4–9 June 1998 (eds Suzuki, Y.
and Totsuka, Y.), to be published in Nucl. Phys. (Proc. Suppl);
the Super-Kamiokande Collaboration, Fukuda, Y. et al., Phys.
Rev. Lett., 1998, B81, 1562–1567; Fukuda, Y. et al., hepex/9812009; Totsuka, Y., in the Proceedings of the 18th Texas
Symposium on Relativistic Astrophysics and Cosmology, 15–20
December 1996, Chicago, Illinois (eds Olinto, A., Frieman, J.
and Schramm, D.), World Scientific, Singapore, 1998, pp. 114–
123.
7. Bethe, H. A., Phys. Rev., 1939, 55, 434–456.
8. Thompson, W., On the Age of the Sun’s Heat, MacMilan’s
Magazine, 1862, 5, 288–393; reprinted in Treatise on Natural
Philosphy (eds Thompson, W. and Gutherie, P.), Cambridge
Univ. Press, Cambridge, 1883, vol. 1, appendix E, pp. 485–494.
1494
9. Darwin, C., On the Origin of the Species, 1859 (Reprinted: Harvard University Press, Cambridge, MA, 1964).
10. Bahcall, J. N., Krastev, P. I. and Smirnov, A. Yu., Phys. Rev.,
1998, D58, 096016–096022.
11. Bahcall, J. N., Basu, S. and Pinsonneault, M. H., Phys. Lett.,
1998, B433, 1–8.
12. Adelberger, E. et al., Rev. Mod. Phys., 1998, 70, 1265–1292.
13. Iglesias, C. A. and Rogers, F. J., Astrophys. J., 1996, 464, 943–
953; Alexander, D. R. and Ferguson, J. W., Astrophys. J., 1994,
437, 879–891. These references describe the different versions
of the OPAL opacities.
14. Rogers, F. J., Swenson, F. J. and Iglesias, C. A., Astrophys. J.,
1996, 456, 902–908.
15. Gruzinov, A. V. and Bahcall, J. N., Astrophys. J., 1998, 504,
996–1001.
16. Salpeter, E. E., Austr. J. Phys., 1954, 7, 373–388.
17. Bahcall, J. N., Phys. Rev., 1997, C56, 3391–3409.
18. Bahcall, J. N., Lisi, E., Alburger, D. E., De Braeckeleer, L.,
Freedman, S. J. and Napolitano, J., Phys. Rev., 1996, C54, 411–
422.
19. Bahcall, J. N., Kamionkowski, M. and Sirlin, A., Phys. Rev.,
1995, D51, 6146–6158.
20. Bahcall, J. N., Astrophys. J., 1996, 467, 475–484.
21. Johnson, C. W., Kolbe, E., Koonin, S. E. and Langanke, K.,
Astrophys. J., 1992, 392, 320–327.
22. Bahcall, J. N., Pinsonneault, M. H., Basu, S. and ChristensenDalsgaard, J., Phys. Rev. Lett., 1997, 78, 171–174.
23. Bahcall, J. N. and Pinsonneault, M. H., Rev. Mod. Phys., 1992,
64, 885–926.
24. Bahcall, J. N. and Pinsonneault, M. H., Rev. Mod. Phys., 1995,
67, 781–808.
25. Bahcall, J. N. and Krastev, P. I., Phys. Lett., 1998, B436, 243–
250.
26. Bahcall, J. N., Phys. Rev., 1991, D44, 1644–1651.
27. Bahcall, J. N. and Bethe, H. A., Phys. Rev. Lett., 1990, 65,
2233–2235.
28. Basu, S. and Antia, H. M., Mon. Not. R. Astron. Soc., 1997, 287,
189–198.
29. Thoul, A. A., Bahcall, J. N. and Loeb, A., Astrophys. J., 1994,
421, 828–842.
30. McDonald, A. B., in The Proceedings of the 9th Lake Louise
Winter Institute (eds Astbury, A. et al.), World Scientific, Singapore, 1994, pp. 1–22.
31. Basu, S. et al., Mon. Not. R. Astron. Soc., 1997, 292, 234–251.
32. Basu, S. and Antia, H. M., Mon. Not. R. Astron. Soc., 1995, 276,
1402–1408.
33. Bahcall, J. N. and Ulmer, A., Phys. Rev., 1996, D53, 4202–
4210.
34. Hata, N., Bludman, S. and Langacker, P., Phys. Rev., 1994, D49,
3622–3625.
35. Parke, S., Phys. Rev. Lett., 1995, 74, 839–841.
36. Heeger, K. M. and Robertson, R. G. H., Phys. Rev. Lett., 1996,
77, 3720–3723.
37. Bahcall, J. N. and Krastev, P. I., Phys. Rev., 1996, D53, 4211–
4225.
38. Gribov, V. N. and Pontecorvo, B. M., Phys. Lett., 1969, B28,
493–496; Pontecorvo, B., Sov. Phys. JETP, 1968, 26, 984–
988.
39. Wolfenstein, L., Phys. Rev., 1978, D17, 2369–2374; Mikheyev,
S. P. and Smirnov, A. Yu., Yad. Fiz., 1985, 42, 1441–1448
(Sov. J. Nucl. Phys., 1985, 42, 913–917); Nuovo Cimento,
1986,
C9,
17–26.
40. Bahcall, J. N., Neutrino Astrophysics, Cambridge University
Press, Cambridge, 1989.
41. Bahcall, J. N., Phys. Rev. Lett., 1964, 12, 300–302; Davis, R. Jr.,
Phys. Rev. Lett., 1964, 12, 303–307; Bahcall, J. N. and Davis,
R. Jr., in Stellar Evolution (eds Stein, R. F. and Cameron, A. G.),
Plenum Press, New York, 1966, pp. 241–243 [proposal first
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
made in 1963 at this conference].
42. Bahcall, J. N., Bahcall, N. A. and Shaviv, G., Phys. Rev. Lett.,
1968, 20, 1209–1212.
43. Bahcall, J. N. and Pinsonneault, M. H., in Proceedings of the
17th International Conference on Neutrino Physics and
Astrophysics (Helsinki) (eds Enqvist, K., Huitu, K. and
Maalampi, J.), World Scientific, Singapore, 1997, pp. 56–70.
44. Gruzinov, A. V. and Bahcall, J. N., Astrophys. J., 1997, 490,
437–441.
45. (GONG) Christensen-Dalsgaard, J. et al., Science, 1996, 272,
1286–1292; (BP95) Bahcall, J. N. and Pinsonneault, M. H., Rev.
Mod. Phys., 1995, 67, 781–808; (KS94) Kovetz, A. and Shaviv,
G., Astrophys. J., 1994, 426, 787–800; (CDF94) Castellani, V.,
Degl’Innocenti, S., Fiorentini, G., Lissia, L. M. and Ricci,
B., Phys. Lett., 1994, B324, 425–432; (JCD94) ChristensenDalsgaard, J., Europhys. News, 1994, 25, 71–75; (SSD94) Shi,
X., Schramm, D. N. and Dearborn, D. S. P., Phys. Rev., 1994,
D50, 2414–2420; (DS96) Dar, A. and Shaviv, G., Astrophys. J.,
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
1996, 468, 933–946; (CDF93) Castellani, V., Degl’Innocenti, S.
and Fiorentini, G.. Astron. Astrophys., 1993, 271, 601–620;
(TCL93) Turck-Chièze, S. and Lopes, I., Astrophys. J., 1993,
408, 347–367; (BPML93) Berthomieu, G., Provost, J., Morel,
P. and Lebreton, Y., Astron. Astrophys., 1993, 268, 775–791;
(BP92) Bahcall, J. N. and Pinsonneault, M. H., Rev. Mod. Phys.,
1992, 64, 885–926; (SBF90) Sackman, I.-J., Boothroyd, A. I.
and Fowler, W. A., Astrophys. J., 1990, 360, 727–736; (BU88)
Bahcall, J. N. and Ulrich, R. K., Rev. Mod. Phys., 1988, 60,
297–372; (RVCD96) Richard, O., Vauclair, S., Charbonnel, C.
and Dziembowski, W. A., Astron. Astrophys., 1996, 312, 1000–
1011; (CDR97) Ciacio, F., Degl’Innocenti, S. and Ricci, B.,
Astron. Astrophys. Suppl. Ser., 1997, 123, 449–454.
ACKNOWLEDGEMENT.
#PHY95-13835.
I acknowledge support from NSF grant
1495
SPECIAL SECTION: SOLAR PHYSICS
The dynamics and heating of the quiet solar
chromosphere
Wolfgang Kalkofen*,† and Peter Ulmschneider**
*Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA02138, USA
**Institut für Theoretische Astrophysik, Universität Heidelberg, Tiergartenstr. 15, Germany
The solar chromosphere can be characterized by two
signatures: the spectroscopic signature is an emission
spectrum for all radiation originating in the chromosphere;
only NLTE effects in the cores of strong lines producing
absorption features. And the dynamical signature is in the
form of oscillations, with a period of 3 min in the
nonmagnetic chromosphere and a period of 7 min in
magnetic regions.
The paper explains these signatures in terms of waves:
The dynamics of the chromosphere is due to acoustic waves
in the magnetic-field-free atmosphere, which produce K2v
bright points, and to kink and longitudinal waves in
magnetic flux tubes, which produce network bright points.
The heating of the chromosphere is caused by acoustic waves
whose dissipation makes the kinetic temperature rise in the
outward direction, producing the emission spectrum. As far
as energy fluxes are concerned, the energy dissipated in
chromospheric heating outweighs the energy visible in
bright points by two orders of magnitude.
The paper interprets the observed oscillation periods in
the chromosphere as cutoff periods: for the 3 min period, as
the cutoff period of acoustic waves in a nonmagnetic,
stratified atmosphere; and for the 7-min period, as the cutoff
period of kink waves in magnetic flux tubes for field
strengths typical of the magnetic network.
preference for the blue peaks3. The bifurcation of the
chromosphere that characterizes optical lines is also found in the
ultraviolet,
where
SUMER
shows
3-min waves in emission lines and continua of neutral metals from
the internetwork chromosphere4, and 7-min waves in emission of
the Lyman continuum and several Lyman lines from the magnetic
network5.
In contrast to the dynamics, the differences in the appearance
of emission lines and continua are minor, especially those from the
layers of the low chromosphere6. It is therefore likely that the
nonmagnetic and magnetic chromospheres are heated by the same
mechanism. It is also noteworthy that the chromosphere produces
emission everywhere and all the time4 and never the absorption
spectrum predicted by the chromospheric bright point model of
Carlsson and Stein7,8.
The observations of chromospheric radiation thus suggest a
separation of the description of chromospheric phenomena into
(1) the dynamics of K2v
bright
points
with
3-min oscillations of large amplitude at a few, select points in the
internetwork
medium,
and
the
dynamics
of network bright points with 7-min oscillations of large amplitude
in magnetic flux tubes and (2) the general heating of the
chromosphere. The paper is structured accordingly: Section 2
discusses chromospheric dynamics on the basis of the
hydrodynamic and magnetohydrodynamic equations. Section 3
considers the general chromospheric heating requirements, while in
1. Introduction
THE most prominent lines in the visible spectrum of the
chromosphere are the H and K lines of Ca II. Their emission
shows a separation of the atmosphere into nonmagnetic and
magnetic regions, which are referred to as internetwork and
magnetic
network
or,
because
of
their connection to convection, as interior and boundary of supergranulation cells. This bifurcation of the medium is
further emphasized by the dynamical behaviour, which shows
oscillations with periods of 3 min in the internetwork
chromosphere and 7 min in the network1 (Figure 1). Furthermore,
the time variation of the intensity is different: The 3-min
oscillations are marked by a sharply-peaked time variation of the
intensity2 and a strong preference for the blue emission peaks, H2v
and K2v, and the 7-min oscillations are accompanied by a broad
time variation of the intensity and a much less pronounced
†
For correspondence. (e-mail: wolf@cfa.harvard.edu)
1496
Figure 1. Velocity power spectra from the Doppler motion of the
H3 -absorption minimum of the H line, at disk center. Network and
internetwork regions along the slit are averaged separately. From
Lites et al.1 .
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Section 4 the generation of acoustic waves in the convection zone
and the dissipation of this energy by weak acoustic shocks is
outlined. Section 5 summarizes the paper.
2. Chromospheric dynamics
Waves in the chromosphere are affected by the density
stratification due to gravity, which makes waves dispersive, and
introduces cutoffs that separate propagating from evanescent
frequencies. For acoustic waves in the nonmagnetic medium,
where gas pressure provides the restoring force in the wave
equation, the cutoff frequency, νac , at the temperature minimum
between
photosphere
and chromosphere is equal to 5 mHz (cutoff period
Pac = 1/νac = 3 min). Waves with frequencies lower than νac are
evanescent, i.e. their phase velocity is infinite and their group
velocity is zero; thus they transport no energy (in the lowamplitude limit). For internal-gravity waves in the nonmagnetic
medium, where gravity provides the restoring force in the wave
equation, the cutoff frequency is the Brunt-Väisälä frequency,
NBV. Waves with frequencies lower than NBV can propagate, but
they are excluded from the purely vertical direction. In a neutral,
monatomic gas, NBV and νac have practically the same value. Thus
the two cutoff frequencies separate the wave spectrum into the
regimes of low frequencies, where only internal-gravity waves can
propagate, and high frequencies, where only acoustic waves can.
In the magnetic chromosphere, two wave-modes play
a role, namely, longitudinal flux tube waves, where gas pressure
provides the main restoring force, and transverse flux tube waves,
or kink waves, where the magnetic field does9. For the magneticfield
strengths
encountered
in
the quiet network, the cutoff frequencies are νλ = 5 mHz
(Pλ = 3 min) for the longitudinal mode, and νκ = 2.5 mHz
(Pκ = 7 min) for the kink mode. For both wave-modes,
propagating waves have frequencies above the cutoff, and
evanescent waves, below the cutoff.
A dynamical model by Carlsson and Stein7 of the nonmagnetic
chromosphere combined a sophisticated hydrodynamic code,
incorporating NLTE radiative transfer of the Ca II ion, with
empirical driving, which was taken from the Doppler velocity
measured
in
a
photospheric
Fe-I line in an hour-long observing run1. The empirical approach
sacrificed the search for the underlying cause of the oscillations,
but, in return, allowed firm conclusions about the nature of the K2v
phenomenon. Since the simulation7 reproduced to high fidelity the
intricate intensity and velocity variations in the core of the H line
for two out of the four bright points from the same observing run,
it is clear that the waves powering K2v-bright points are
propagating acoustic waves.
In the network bright points on the supergranulation cell
boundary, the periods of oscillation, typically near 7 min, are
longward
of
the
acoustic
cutoff
period
and
of the Brunt-Väisälä period, where acoustic waves are evanescent
but internal-gravity waves propagate. Some observers have
therefore considered the waves in the magnetic network to be
internal-gravity waves10,11. A theoretical underpinning of this
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
hypothesis by Lou12 found the observed periods to be possible as
resonances of magneto-gravity waves in a chromospheric cavity.
But a heuristic picture of network bright points in terms of
internal-gravity waves by Deubner and Fleck13 was shown by
Kalkofen14 to fail in its intended purpose, namely, to dissipate the
wave energy in traveling downward in the narrowing flux-tube
funnels.
An explanation of network bright points not based on
oscillations in a cavity was proposed by Kalkofen14 who noted
the coincidence of the observed oscillation period with the cutoff
period of kink waves for magnetic field strengths found in the
magnetic network. Numerical modeling by Choudhuri et al.15 and
the analytic solution of the MHD equations for impulsive
excitation16 showed that oscillations at the cutoff could indeed be
produced. In order to dissipate, however, the transverse waves
have to transfer energy to the longitudinal mode, which forms
shocks17.
Wave excitation in a magnetic flux tube produces both
transverse and longitudinal waves. But the power spectrum of
waves in the network shows virtually no power at 5 mHz, the
cutoff frequency of the longitudinal mode. The scenario for
network-bright points therefore requires the wave excitation to
produce mainly transverse waves. This is indeed the case, as was
shown with numerical solutions18 and with analytic methods19.
A consequence of the almost exclusive excitation of transverse
waves is that a significant flux in longitudinal waves appears only
in the chromosphere, where the nonlinearity of the waves
facilitates the transfer of energy between modes. This transfer is
consistent with the low value of the observed velocity coherence
between the base and the middle of the network chromosphere1
(Figure 7), which indicates that the dissipating longitudinal waves
do not arrive from below but are formed in the chromosphere.
For the analytic modeling of chromospheric oscillations we consider the hydrodynamic equations. From a
small-amplitude expansion of the equations for a onedimensional, isothermal atmosphere we obtain the Klein–Gordon
equation20:
∂2
∂z 2
u−
∂2
∂t 2
u − u = 0,
(1)
which is written here for the ‘reduced’ velocity u and in terms of
the
dimensionless
depth
and
time
variables
z
and t. For acoustic waves, the ‘physical’ velocity v is obtained as
v (ζ, τ) = µ(ζ, τ)eζ /2H ,
(2)
in terms of the physical depth and time variables ζ and τ.
The derivation of the wave equation (1) for acoustic waves
assumes that the atmosphere is one-dimensional, isothermal and
stratified in plane-parallel layers with a constant scale height H. In
the vertical direction, only acoustic waves are allowed (i.e.
internal-gravity waves are excluded from the vertical direction).
The velocity v grows exponentially with height ζ with a scale
length of twice the density-scale height.
1497
SPECIAL SECTION: SOLAR PHYSICS
Magnetic waves propagating in the expanding geometry of a
thin magnetic flux tube, in pressure equilibrium with the
surrounding medium, satisfy the same wave equation20, but with different definitions of the dimensionless variables. An additional requirement (which is satisfied here) is a
constant ratio β(= 8πp/B2) of gas to magnetic pressure inside the
tube. The physical velocity v is now obtained as
v (ζ, τ) = u(ζ, τ)eζ /4H,
(3)
showing slower exponential growth for tube modes, with a scale
length of 4H because of exponential spreading of the tube cross
section.
The solution of the wave equation for a velocity impulse at
z = 0 and t = 0, in an infinite medium is given by Lamb21,
u ( z, t ) = 12 δ (t − | z |) −
2
2
t J1 ( t − z )
2
t 2 − z2
H
( t− | z |),
(4)
where δ is the Dirac δ-function; H is the Heaviside function and J1
is a Bessel function. The argument of both the δ-function and the
Heaviside function expresses the propagation of the head of the
wave (|z| = t), at the sound speed for acoustic waves and at the
tube speeds for the flux-tube waves. The tube speeds in
dimensioned units are given by cλ = cs [2/(2 + γβ)]1/2 for the
longitudinal mode, and cκ = cs [2/(γ(1 + 2β))]1/2 for the transverse
mode, where cs is the sound speed and γ (= 5/3) is the ratio of
specific heats.
Behind the head of the wave, the atmosphere oscillates initially
with
frequencies
in
a
broad
spectrum,
but
this narrows with time until it results in an oscillation at the
cutoff 22, as shown by the asymptotic solution of the Klein–
Gordon equation:
u ( z, t) ~
cos (t )
t
, t >> z ≥ 0 ,
(5)
in which the (reduced) velocity becomes independent of height z.
The solution implies that gas elements at all heights reach maximal
amplitude simultaneously. At the cutoff22 frequency, the phase
velocity, which is given by
ω
, v group = 1/ vφ ,
ω2 − 1
is infinite and the group velocity is zero.
In dimensioned units, the cutoff periods are given by
vφ =
(6)
Pac = 4πH/c s ,
Pλ = Pac ( 60 + 50 β) /(63 + 48 β ) ,
(7)
Pκ = Pac 2γ (1 + 2 β) ,
for acoustic waves and for longitudinal and transverse flux-tube
waves, respectively22.
It is interesting to note that the highest phase velocities are
1498
reached near the origin of the wave, and that they increase with the
order of the maximum behind the head of the wave16. For
comparison with observations it needs to be remembered that
these analytic results are for an isothermal atmosphere initially at
rest; even the observed values of the photospheric velocity
allowed to reproduce the H-line observations only when the
waves were launched into a disturbed atmosphere7.
The asymptotic solution of the Klein–Gordon equation shows
that an impulse can excite oscillations at the cutoff of the respective mode. Any sudden change in the forcing
function has the same consequence. An example of the former is
found in the seismic events following the collapse of an intergranular lane23, an example of the latter in stochastic excitation24.
Thus there are two possible models for the generation of K2vbright points: (1) Individual events at a few, discrete points in the
internetwork medium. They might account for the strong bright
points observed by Liu (1974)25 or by Brandt et al. (1992)26 with
a filling factor of about 1%, or modeled7. (2) Ubiquitous generation
of oscillations from the turbulence in the convection zone27,
leading to bright points that are visible anywhere in the K line2, or
in UV lines4 where they are observed in half the positions along
the slit of the SUMER instrument.
For network bright points Muller et al.26 have observed the
interaction of fast granules with magnetic-flux tubes. This process
was modeled by Choudhuri et al.15, as footpoint motion of a flux
tube, resulting in a kink wave in the tube.
While the various processes make plausible the excitation of
oscillations at the cutoff of a mode, observational confirmation of
a link between a photospheric event and, after a delay accounting
for the wave travel time, of a chromospheric brightening is still
lacking.
An interesting puzzle is posed by the dynamical modeling of
H2v-bright points7. On one hand, the simulation reproduced to
high accuracy the complex variation of the intensity and shape of
the H line, including the proper time delay between the motion of
the photospheric Fe-I line and that of the resulting H line, and on
the other hand, it predicted an absorption spectrum from any
location in the internetwork chromosphere and most of the time,
whereas the observations4 with SUMER show only an emission
spectrum. The high spatial and temporal resolution of the space
observations exclude the possibility of spatial and temporal
averaging to hide a chromospheric absorption spectrum behind the
observed emission. Thus the chromosphere must have a
permanent temperature rise in the outward direction, as shown by
empirical models. The flaw in the model7 was traced to the input
velocity spectrum by Kalkofen et al.28 who argue that
the dynamical model uses only about 1% of the acoustic energy
flux available. The hidden flux has frequencies in excess of
10 mHz. The flux critical for the bright point phenomenon is
emitted between 5 mHz and 10 mHz. That flux is found in the
observed velocity spectrum and accounts for the success of the
dynamical simulation. In the following section we address the
generation of the flux that is missing from the simulation7 and treat
its dissipation by means of the theory of weak shock waves.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Chromospheric energy losses and the heating
requirements
The empirical chromosphere model of Vernazza et al.6 has been
constructed
by
fitting
the
observed
ultraviolet
C I, Si I and H I continua with the simulated emission from a
hydrostatic equilibrium model with an arbitrary temperature
distribution. This temperature distribution was subsequently
modified until an optimal fit between the observed and simulated
emission was obtained. For the average sun model C, these authors
subsequently
computed the chromospheric energy-loss rate in H, H–,
Ca II and Mg II due to lines and continua. They found a total
radiative energy loss of FR = 4.6 × 106 erg cm–2 s–1. Anderson and
Athay29 later improved on this determination by including the
abundant line emission from Fe II and found a total chromospheric
radiative energy loss of FR = 1.4 × 107 erg cm–2 s–1. In addition,
they found (see Figure 2) that the cooling rate ΦR in most of the
chromosphere is proportional to the mass density ρ0, which leads
to a characteristic height dependence of the chromospheric
emission.
The question is how this persistent energy loss is balanced and
how the continuous supply of energy is provided. The time scale
in
which
an
excess
temperature
would
cool down to the boundary temperature if the mechanical heating
were suddenly disrupted is given by the radiative relaxation
time for which one has
t Rad =
∆E
ρcv ∆T
ρcv
=
=
≈ 1 .1 × 10 3 s.
Φ R 16κσT 3 ∆T 16 κσT 3
(8)
Here we used from the VAL81 model that at z = 1280 km,
T = 6200 K, ρ = 9.8 dyn/cm2, κ /ρ = 4.1 × 10–4 cm2/g, cv =
9.6 × 10–12 g/cm3, σ = 5.6 × 10–5 erg/cm2 s K4. It is seen that in
timescales of a fraction of an hour, the chromosphere would cool
down to the boundary temperature if mechanical heating would
suddenly stop.
In a purely hydrodynamic environment there are few
possibilities to persistently heat a medium: acoustic waves or
pulsational waves. For a review on heating mechanisms see Narain
and Ulmschneider30. Acoustic waves generated in the convection
zone are the most likely possibility. To see what acoustic wave
heating might do, consider a typical acoustic disturbance in the
solar chromosphere. Assume a characteristic perturbation of size
L = 200 km, temperature ∆T = 1000 K and velocity ∆v = 3 km/s.
Using appropriate values for the thermal conductivity κth = 105
erg/cm s K and viscosity ηvis = 5 × 10–4 dyn s/cm2 we find for the
thermal conductive and viscous heating rates
ΦC =
d
dT κ th ∆T
κ th
≈
≈ 3 × 10 − 7
dz
dz
L2
2
ΦV
 erg 
 3 ,
 cm s 
 dv 
n ∆v 2
= n vis   ≈ vis 2
≈ 1 × 10 − 7
L
 dz 
 erg 
 3 .
 cm s 
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
These heating are inadequate by a large margin to bal-ance the
typical empirical chromospheric cooling rate of 10–1 erg/cm3 s
found in the model6. Only when the length scale L is decreased by
several
orders
of
magnitude
can the heating rates be raised to acceptable levels. For acoustic
waves, this is accomplished by shock formation. This shows that
in pure hydrodynamic situations only shock heating has
sufficient power to balance the observed cooling rates. The same
very likely is the case for the heating of chromospheric magnetic
flux tubes, where the shock heating is by longitudinal tube waves.
Note as discussed in the review30, this is different in the transition
layer where in addition other types of heating, like microflare
heating become important.
4. Weak shock heating and the generation of
acoustic waves
Small-amplitude acoustic shock waves behave essentially like
acoustic waves, except that they dissipate at the shock front (see
below). In particular, the amplitude relations are identical.
Consider linear small-amplitude sawtooth waves with pressure
and
velocity
variations
p = p0 + pm
– 2p mt/P, v = v m – 2v mt/P, where P is the wave period, t the time
and subscript m indicates maximum amplitudes, the wave energy
flux (erg cm–2 s–1) is given by:
FM =
1
P
1
1
∫0 ( p − p0 ) v dt = 3 pmv m ≈ 12 γ p0 cSη ,
P
2
(11)
where p 0 is the unperturbed pressure, cS the sound speed, γ the
ratio of specific heats and where for weak shocks one has for the
total pressure, velocity, temperature and density jumps
(9)
lim
(10)
Figure 2. Comparison of the theoretical limiting acoustic flux
FM
(erg cm–2 s–1 ) with the solar chromospheric radiative loss flux Fr
determined empirically by Anderson and Athay29 . ΦR (erg cm–3 s–1 ) is
the empirical net radiative cooling rate. PA/5 and PA/10 label
different assumptions as to the acoustic frequency spectrum. Star
symbols show empirically determined acoustic fluxes by Deubner39 .
1499
SPECIAL SECTION: SOLAR PHYSICS
2p m ≈ γp 0η, 2v m ≈ cSη, 2Tm ≈ (γ – 1)T0η, 2ρm ≈ ρ0η. Here the
shock strength is defined as η = (ρ2 – ρ1)/ρ1, where ρ1, ρ2 are the
densities in front and behind the shock31. By expanding the
entropy jump ∆S per unit mass at the shock front, the shock
dissipation rate (erg cm–3 s–1) of the wave can be written
ΦMM =
≈
ρT ∆S
P
=
 p  ρ  −γ
ln  2  2 
γ (γ − 1) P  p 1  ρ1 

ρ0 c S2
1 γ (γ + 1)
p 0 η3 .
12
P




(12)
The approximate equality is only valid for weak shocks where the
entropy jump is small. Let us assume a gravitational atmosphere
and in analogy to ray optics that the quantity
is conserved.
FM cS2eq. (11) with respect to height z and using (12)
Differentiating
gives an equation for the shock strength
2
dη η  γ g
3 dcS (γ + 1)η 
=
−
−
,
dz 2  cS2
cS P 
2cS2 dz
term of eq. (13). Limiting strength is reached when η becomes
constant and the flux proportional to the gas pressure p 0.
Figure 2 shows limiting-strength acoustic-heating fluxes
lim
for wave F
periods
P = PA/5 and P = PA/10, where PA = 4πcS/(γg) is
M
the acoustic cut-off period. It is seen that the fluxes log
rise
linearly with the logarithm of the
FMlimmass column density m, since
m = p 0/g and from eq. (15) F lim ~ p 0. The figure therefore shows
that acoustic shock waves are able to explain the observed height
dependence of the chromospheric emission. As will be shown
below, acoustic wave generation calculations for the solar
convection zone by Musielak et al.32 also provide the correct
wave period for Figure 2. As shown below one finds that P = PA/5
is roughly the period of the maximum of the acoustic-wave
spectrum generated in the convection zone. Using this value one
has from the above equations
ηlim =
1 4π
= 0 .94 ,
5 γ +1
(13)
FMlim =
where g is the gravitational acceleration. The refractive term
dcisS2 /small
dz in the chromosphere and can be
involving
neglected. For an isothermal, nonionizing, gravitational atmosphere
eq. (13) can be solved for various initial shock strength’s η0 and
shows that irrespective of the η0 the shocks eventually reach a
limiting shock strength ηlim given by
γ gP
ηlim =
.
(14)
(γ + 1)cS
4π 2
γ
c S p 0 ≈ 0.123 cS p 0
75 (γ + 1) 2
1 γ3 g2 P 2
p0 .
12 (γ + 1) 2 cS
and a heating rate per gram
1 dFM
ρ dz
=
dFM
lim
dm
=
ΦM
ρ
=
γ
4π 2
cS g
75 (γ + 1) 2
(18)
9
erg gg–1-1 ss–1-1. .
≈≈22.4
.4 ⋅×10109 erg
(15)
Eq. (13) has the property that for small shock strengths one
initially has an exponential growth due to flux conservation which
results from the first term on the RHS of the equation. This
growth of the sawtooth wave is similar to that for acoustic waves
in
a
gravitational
atmos2 2
phere assuming flux conservation, ρ0v 2~ ρ 0 cS η = const.
The increase in shock strength is eventually balanced by the
increasing shock dissipation described by the last
1500
(17)
≈ 10 5 p 0 ≈ 32..74m
×10
, 9 m,
From eq. (11) one obtains a limiting wave energy flux
FMlim =
(16)
The computation of the generation of acoustic energy
in stellar convection zones dates to the theory of
quadrupole sound generation from turbulence developed by Lighthill33,34 and Proudman35. Considering a Kolmogorov
turbulence spectrum, Lighthill has derived an expression
FM = ∫ 38
ρ0 u 8
cS5 H
dz,
(19)
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Figure 3. Acoustic wave spectra for different values of the mixing-length parameter α (left panel), different contributions to the acoustic
spectrum (right panel), computed with the Lighthill–Stein theory, after Musielak et al.32 .
called Lighthill formula, where z is height, H is the scale height,
ρ0 the density, cS the sound speed and u the turbulent velocity.
This famous u 8-formula was found to be in excellent agreement
with
measurements
in
terrestrial applications. Lighthill’s theory was further extended
by Stein36,37 to allow for the computation of the acoustic frequency spectrum and was recently revisited32. To compute
the acoustic flux using the Lighthill–Stein theory, one must first
calculate a convection zone model where one uses the mixinglength theory, which depends on a free parameter, the mixinglength parameter α. Musielak et al.32 on the basis of a description
of the turbulence with an extended Kolmogorov spectrum and
a modified Gaussian frequency factor, found total acoustic fluxes
FM = 1.3 × 107 erg cm–2 s–1 for α= 1.0 and 1.7 × 108 erg cm–2 s–1
for α= 2.0.
Figure 3 shows the acoustic spectra obtained by these authors
and demonstrates that quadrupole generation (also used in the
Lighthill formula) is the most important contribution to the
acoustic-wave flux. Since the frequencies in Figure 3 are circular
frequencies, the spectra have a maximum at periods P = 79, 58 and
41 s for α= 1.0, 1.5, 2.0, respectively. Recent numerical
convection zone calculations show that α, the ratio of the mixinglength
L
to the scale height H, typically varies only in a narrow range of
α≈ 2.0–2.16 (ref. 38). Taking a temperature T = Teff = 5770 K,
the acoustic cut-off period is found to be 216 s such that
P = PA/5 = 43 s
is
a
good
estimate
for the maximum of the generated solar acoustic wave spectrum.
Note that Figure 2 shows also empirical acoustic fluxes by
Deubner39 which are in rough agreement with the empirical losses.
5. Conclusions
Waves are responsible for the characteristic features of the
chromosphere: the permanent outward temperature rise, which is
due to acoustic waves that dissipate in shocks, the 3-min
oscillations of the nonmagnetic chromosphere, whose period
matches the cutoff period of acoustic waves, and the 7-min
oscillations, whose period matches the cutoff period of transverse
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
flux tube waves for a pressure ratio of β ≈ 0.25.
The empirical radiative emission rate of the chromosphere can
be understood on the basis of the dissipation rate of weak-shock
theory, and the appearance of the cutoff periods in the
chromospheric oscillations can be understood as due to impulsive
or stochastic wave excitation, which causes the atmosphere behind
the head of a wave to oscillate at the cutoff period.
1. Lites, B. W., Rutten, R. J. and Kalkofen, W., Astrophys. J.,
1993, 414, 345.
2. von Uexküll, M. and Kneer, F., Astron. Astrophys., 1995, 294,
252.
3. Grossmann-Doerth, U., Kneer, F. and von Uexküll, M., Solar
Phys., 1974, 37, 85.
4. Carlsson, M., Judge, P. G. and Wilhelm, K., Astrophys. J., 1997,
486, L63.
5. Curdt, W. and Heinzel, P., Astrophys. J., 1998, 503, L95.
6. Vernazza, J. E., Avrett, E. H. and Loeser, R., Astrophys. J.
(Supplement), 1981, 45, 635.
7. Carlsson, M. and Stein, R. F., in Chromospheric Dynamics,
Proceedings of the Mini-Workshop, Institute for Theoretical
Astrophysics, Oslo, 1994.
8. Carlsson, M. and Stein, R. F., in IAU Symposium 185 (eds
Deubner, F.-L., Christensen-Dalsgaard, J. and Kurtz, D.), Kluwer,
Dordrecht, 1998, 435.
9. Spruit, H. C., in The Sun as a Star (ed. Jordan, S.), NASA
Monograph Series on Nonthermal Phenomena in Stellar Atmospheres, 1981, p. 385.
10. Damé, L., Thèse, Université de Paris, VII, 1983.
11. Damé, L., in Small-Scale Dynamical Processes in Quiet Stellar
Chromospheres (ed. Keil S. L.), NSO/SPO, Sunspot, NM, 1984,
p. 54.
12. Lou, Y.-Q., MNRAS, 1995, 276, 769.
13. Deubner, F.-L. and Fleck, B., Astron. Astrophys., 1990, 228,
506.
14. Kalkofen, W., Astrophys. J., 1997, 486, L145.
15. Choudhuri, A. R., Auffret, H. and Priest, E. R., Solar Phys.,
1993, 143, 49.
16. Kalkofen, W., Rossi, P., Bodo, G. and Massaglia, S., Astron.
Astrophys., 1994, 284, 976.
17. Zhugzhda, Y. D., Bromm, V. and Ulmschneider, P., Astron.
Astrophys., 1995, 300, 302.
18. Ulmschneider, P., Zähringer, K. and Musielak, Z. E., Astron.
Astrophys., 1991, 241, 625.
1501
SPECIAL SECTION: SOLAR PHYSICS
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
Hasan, S. S. and Kalkofen, W., Astrophys. J., 1999, (in press).
Rae, I. C. and Roberts, B., Astrophys. J., 1982, 256, 761.
Lamb, H., Proc. Lond. Math. Soc., 1909, 7, 122.
Spruit, H. C. and Roberts, B., Nature, 1983, 304, 401.
Goode, P. R., Strous, L. H., Rimmele, T. R. and Stebbins, R. T.,
Astrophys. J., 1998, 495, L27.
Sutmann, G. and Ulmschneider, P. Astron. Astrophys., 1995,
294, 232.
Liu, S.-Y., Astrophys. J., 1974, 189, 359.
Muller, R. and Roudier, Th., Vigneau, J. and Auffret, H., Astron.
Astrophys., 1994, 283, 232.
Theurer, J., Ulmschneider, P. and Kalkofen, W. Astron.
Astrophys., 1997, 324, 717.
Kalkofen, W., Ulmschneider, P. and Avrett, E. H., Astrophys. J.
(Lett.), 1999, 521, (in press).
Anderson and Athay, Astrophys. J., 1989, 346, 1010.
Narain, U. and Ulmschneider, P., Space Sci. Rev., 1996, 75,
453.
Ulmschneider, P., Solar Phys., 1970, 12, 403.
1502
32. Musielak, Z. E., Rosner, R., Stein, R. F. and Ulmschneider, P.,
Astrophys. J., 1994, 423, 474.
33. Lighthill, M. J., Proc. R. Soc. London, 1952, A211, 564.
34. Lighthill, M. J., Proc. R. Soc. London, 1954, A222, 1.
35. Proudmann, I., Proc. R. Soc. London, 1952, A214, 119.
36. Stein, R. F., Solar Phys., 1967, 2, 385.
37. Stein, R. F., Astrophys. J., 1968, 154, 297.
38. Trampedach, R., Christensen-Dalsgaard, J., Nordlund, A., Stein,
R. F., in Solar Convection and Oscillations and their
Relationship (eds Pijpers, F. P., Christensen-Dalsgaard, J.,
Rosenthal, C. S.), 1997, p. 73.
39. Deubner, F.-L., in Pulsation and Mass Loss in Stars (eds Stalio,
R., Willson, L. A.), Kluwer, 1988, p. 163.
ACKNOWLEDGEMENTS. W.K. thanks his colleagues at the
Institut für Theoretische Astrophysik for their hospitality and the
University of Heidelberg for a Mercator guest professorship funded
by the DFG. Partial support by NASA is acknowledged.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Solar activity: An overview
v
Zdenek Š vestka
CASS UCSD, La Jolla, California, CA 92093-0424, USA and SRON Laboratory for Space Research, Sorbonnelaan 2, 3584 CA, Utrecht, The
Netherlands
This review article describes briefly the main characteristics of the active Sun: the different active
phenomena, the 11-year cycle of their appearance, and
their influence on the environment of our Earth.
1. Solar cycles
Although the Sun illuminating our Earth looks like a steadily
shining celestial body, its surface is actually the seat of
continuous changes and powerful activity. As the Solar and
Heliospheric Observatory (SOHO) spacecraft recently
revealed, in ultraviolet lines the solar surface looks like
‘boiling’ all the time and everywhere one can see variations
in brightness, plasma flows, and small ejections of gas,
indicating permanent changes of the structures in the solar
atmosphere.
But this is not what we call solar activity – all these
changes are still considered to occur on the ‘quiet Sun’.
The real processes, called solar activity, which have their
impacts also on the Earth environment, appear in limited
parts of the solar atmosphere, and their occurrence varies
quasi-periodically with time, creating 11-year cycles of solar
activity. Each new solar cycle is born close to the solar
poles and its activity then slowly propagates to lower
heliographic latitudes. The real length of one cycle is
actually about 22 years, but only the second half of it
begins to produce clearly visible active processes on the
Sun.
When the Sun is viewed in white light, one observes the
lowest level of the solar atmosphere, which is called the
photosphere. In the photosphere, solar activity manifests
itself as sunspots or groups of sunspots (Figures 1 and
2 a). Therefore, solar cycles were long characterized (and
still are) by the so-called relative sunspot numbers R: A
daily R is the sum of the number of sunspot groups on the
Sun plus the number of individual sunspots in all of them. A
monthly or yearly R is the average daily R during a month or
a year. Individual solar cycles differ in their lengths and
heights – lengths varying between 9 and 16 years, and
yearly maximum R varying between 46 and 190 have been
observed between the half of the 18th century and present
days.
Rare sunspot groups begin to appear first at high
latitudes (between 40 and 50 heliographic degrees) and as
the frequency of their occurrence increases, their positions
e-mail: z. svestka@sron.nl
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
move progressively closer to the equator. A few years after
the onset of an active cycle, when the spots appear mostly
at latitudes below 25°, the cycle reaches its maximum.
Thereafter it slowly declines, with the spot occurrence
approaching the equator, and eventually reaches a minimum when sunspots appearance becomes very rare and for
many days the Sun is without any sunspots. However,
usually before the last sunspots of the old cycle disappear,
new ones begin to appear at high latitudes.
2. Active regions
This solar activity is due to magnetic field which exists in
the Sun, generated by electric currents. Sunspots become
visible in the photosphere when ropes of magnetic flux
emerge on the solar surface. The magnetic flux in the central
dark umbra of a spot is usually between 1500 and
3000 gauss and, as a consequence of this strong field,
temperature in a spot is much lower (4000 K or less) than in
the surrounding photosphere. This causes the dark
appearance of sunspots in which the umbra is surrounded
Figure 1. A sunspot group in the photosphere (above) and the
photospheric magnetic field (below). Black and white denote the
opposite magnetic polarities.
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SPECIAL SECTION: SOLAR PHYSICS
by a less dark penumbra (Figure 1).
However, as Figure 2 b shows, the situation looks quite
differently if we observe sunspot groups in the higher layer
of the solar atmosphere, the chromosphere. Since about
1930, this layer could be observed in a spectrohelioscope
and through filters, which make it possible to observe the
Sun in a narrow monochromatic band, usually centered on
the hydrogen-Balmer-Hα line. In this line, sunspots are
surrounded by bright plages, and we call these groups of
plages and spots the active regions on the Sun. And since
1973, when Skylab orbited the Earth and imaged the Sun in
X-rays, we can also see the highest layer of the solar
atmosphere, the solar corona in which these active regions
appear extensive and bright (Figure 2 c).
Active regions appear where magnetic flux emerges from
subphotospheric layers to the solar chromosphere and
corona (see the magnetic map in Figure 1). There are many
flux emergences on the solar surface, all the time creating
ephemeral active regions which in X-rays are seen as
bright points on the Sun (examples – bright dots –can be
seen in Figure 2 c). However, only few of them grow further,
with newly emerging flux continuously added to them and
eventually developing into much larger active regions (four
of them are seen in Figures 2 a, b and c and many more on
the full-disk picture of the Sun in the Hαline in Figure 3).
Most active regions are bipolar, with two main spots and
surrounding plages of opposite magnetic polarities. The
two polarities are connected by chromospheric and coronal
loops, particularly well seen in X-rays and ultraviolet lines
(Figure 4), and are separated by a neutral line below the
tops of these loops where the longitudinal magnetic field is
zero. (Compare the schematic drawings in Figures 9 a and
b.) Solar atmospheric gas slowly accumulates along these
lines and cools. In the Hαline one can see them along these
neutral lines dark filaments (many are seen in Figure 3)
which on the solar limb look like bright prominences (Figure
5). Dark filaments often survive even after the active region
decays, and can form outside of active regions as well (like
that one to the southwest of the Sun’s center in Figure 3).
In some cases, irregular patches of magnetic flux emerge
in an active region and cause irregularities in its magnetic
structure. Then several different neutral lines exist in the
region and more dark filaments can form there. While all
active regions are seats of various kinds of active
processes, the most powerful events of solar activity occur
in these magnetically complex active regions, where both
magnetic polarities are mixed. Most active is the so-called δ
configuration, when umbrae of opposite polarities are
embedded in one common penumbra.
3. Complexes of activity and interconnecting
loops
Figure 2. Images of four active regions in (a) white light, (b) Hα
line, and (c) X-rays.
1504
Solar activity seems to prefer, sometimes for a period of
many months, selected active longitudes where active
regions form more frequently than elsewhere on the Sun,
creating there so-called complexes of activity (Figure 6).
This seems to be due to some irregularities in distribution of
the subphotospheric magnetic flux that emerges through
the photosphere. Once such an irregularity is formed, it
takes a long time before effects of the solar rotation, which
slightly varies with the latitude, remove it.
Many active regions in such a complex of activity are
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
connected by coronal loops that very often extend across
the equator. We call them interconnecting loops. They can
be best observed in soft X-rays, as was first done on
Skylab in 1973 (an example is shown in Figure 2 c) and
presently by Yohkoh. Due to magnetic field variations in the
interconnected active regions, these connections vary all
the time in their shape and brightness. In the new solar
cycle, when active regions begin to emerge at high latitudes, these interconnections have been found very long,
up to 60 heliographic degrees (700,000 km, i.e. 55 diameters
of the Earth, Figure 7).
It is quite impossible that such long connections could
possibly exist below the photosphere and emerge through it
into the corona. The most plausible explanation is that they
are formed by magnetic field-line reconnection in the solar
corona.
(Figure 8 b) which have some features common with surges,
but apparently are not exactly the same phenomena. Most
probably, as the new satellite TRACE recently revealed,
both have a similar cause, but the hot jets and cold surges
move along different trajectories through the corona. In the
past few years, the sophisticated spacecraft SOHO and
TRACE could detect some jets even in the quiet Sun
regions, reflecting the complexity of magnetic field structure
even outside the active regions.
A much more powerful phenomenon seen in Hαimages is
a spray (Figure 8 c), in which large amounts of active region
plasma are abruptly ejected into the corona, and often
escape into interplanetary space, possibly being one of the
sources of coronal mass ejections (cf., Section 6).
4. Surges, jets, and sprays
In most active regions, variations in brightness occur all the
time reflecting either new emergence of magnetic flux or oldflux decay or interactions between individual active region
loops. Small short-lived brightenings in active regions have
been called Ellerman bombs or moustages. In the
magnetically complex regions also ejection of material
occurs, mostly rooted inside tiny patches of magnetic
Figure 4. Image of the active Sun in soft X-rays (made on
Yohkoh). Note the sets of bright loops that cross the neutral lines in
active regions.
polarities embedded inside, or penetrating into, a region of
opposite polarity which the moustages prefer.
In the Hα line the most common ejection is a bright or
dark surge (Figure 8 a) in which solar material first moves
upward into the corona along magnetic field lines and
subsequently falls back to the chromosphere. In X-rays, the
Japanese satellite Yohkoh observed many bright jets
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Figure 6. High-resolution photograph of a complex of activity
near the western solar limb. Some sunspots and many dark filaments
can be seen and new magnetic flux is emerging in the middle
Figure
5. and
Prominence
observed
the See
limbalso
in the
(Big
foreground
at the limb
(upper on
right).
the Hα
fine line
structure
Bear
It would be seen
filament
in
of theObservatory
surroundingphotograph).
‘quiet’ chromosphere.
Northas isa dark
to the
left and
projection
the(Hα
solar
disk.
west to theon
top.
line,
Ottawa River Solar Observatory.)
1505
SPECIAL SECTION: SOLAR PHYSICS
5. Solar flares
However, the most powerful brightening in an active region
is a solar flare. In the optical range almost all flares can be
seen only in monochromatic light (most observations are
made in the Hα line), but the most powerful ones also emit
in the white light and the first flare ever detected was
discovered by Carrington on 1 September 1859 when he
observed a large sunspot group looking at the photosphere.
After Hale’s invention of the spectrohelioscope, which
made it possible to observe continuously the whole solar
surface in the Hα line, from early thirties flares have been
observed regularly at many solar observatories throughout
the world and listed in monthly reports.
As the resolving power of solar instruments was improving, smaller and smaller flares and flare-like phenomena
could be detected in active regions on the Sun. Thus first
the category of subflares has been added to the original
flares, and later the categories of microflares and still smaller nanoflares. Obviously, flare-like processes can be detected in an active region on all scales and an idea – originally
due to Parker – is that nanoflares occur everywhere on the
Sun and are the actual source of coronal heating.
Generally, flares are of two different kinds: Confined
flares, where preexisting loops in an active region suddenly
brighten and thereafter slowly decay; and eruptive flares,
where the whole configuration of loops crossing a neutral
line in an active region is disrupted and must be newly
rebuilt. Most flares, and essentially all small flares are
confined flares, and quite often they originate through an
interaction of two active region loops which magnetically
reconnect. But the most powerful and energetic phenomena
are the eruptive flares which are also one of the sources of
a
b
c
d
Figure 9. The interpretation of eruptive flares. a and b, Two
different views of the preflare situation when a dark filament
(prominence) extends along a neutral line and is embedded in a
system of loops forming a coronal helmet structure; c, Opening of
magnetic field lines; d, Subsequent closing of field lines, creating the
flare loops.
1506
coronal mass ejections (cf. section 6).
Figure 9 shows schematically the development of an
eruptive flare, following the original suggestion of Kopp
and Pneuman in 1976, later improved and further developed
by many other authors. The originally closed magnetic field
Figure 7. The longest transequatorial interconnecting loop of the
new solar cycle connecting active regions over the distance of
700,000 km. (Yohkoh soft X-ray image of the Sun.)
Figure 8. a. Bright surge in the Hα line, observed at Big Bear Solar
Observatory; b. X-ray jet observed by Yohkoh; and c. a spray,
observed at Wroclaw Observatory.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
in an active region, in which a filament (prominence) is
embedded, suddenly opens. Reasons for it can be a newly
emerging magnetic flux, a confined flare nearby, a wave
disturbance coming along the solar surface from another
source of activity, or some internal instability (e.g. a shear
of field lines which exceeds a certain critical limit). As field
lines open (Figure 9 c) plasma begins to flow from the dense
chromosphere upward to the corona, so that gas pressure
decreases and magnetic pressure begins to prevail. That
leads to sequential reconnections of the open field lines,
which begin to create new loops in the active region (Figure
9 d). The reconnection process produces intense heating at
the top of each new loop which is conducted downward to
the chromosphere, and it also accelerates particles which
flow along the loop to its footpoints. Thus the gas at the
chromospheric footpoint is strongly heated and evaporates
into the newly formed loop, making it visible in X-rays and
high-temperature lines as a flare loop. The loop then cools,
but in between other loops are formed above it through
reconnection of other field lines and the whole process is
repeated in each of them. Thus the loop system gradually
grows. After some time, the lowest loops cool to about
10000 K and begin to be visible in the Hα line. This takes
some time, so that earlier, when no X-ray observations of
eruptive flares were available, these structures were called
post-flare loops.
While in compact flares energy is suddenly released and
thereafter the flare structure cools and decays, in eruptive
flares energy is released during each reconnection and thus
this process of energy release, though decreasing in
efficiency, can continue for many hours. Because each
reconnection produces new X-ray flux, one can see
Figure 10. Composed image of an eruptive flare near the solar
limb. A quiescent prominence, like that in Figure 5, can be seen in
the background. (Images made and composed at Wroclaw
Observatory.)
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
enhanced emission in X-rays during the whole time of the
repeated reconnections. Therefore, based on these X-ray
records, eruptive flares are also often called long-duration
events.
When looking at the chromospheric image of an eruptive
flare in the Hαline, one observes at the footpoints of newly
formed coronal loops the heated chromosphere, in the form
of two bright ribbons, which slowly separate as the flare
loop system grows. Therefore, these phenomena were
earlier called (and often still are) two-ribbon flares. The two
ribbons are connected by bright or dark loops. Figure 3
shows an example of such a flare (to the southwest of the
center) and the bright patches between the bright ribbons
are tops of the loops that connect them. And a more
detailed photograph of an eruptive flare, composed from
three different images, is shown in Figure 10: we see there
the sunspot group in the photosphere, the two bright
ribbons in the chromosphere, and the ‘post’-flare loops
seen in the Hα line above the limb. Both in the Hα line and
in X-rays one can see how in all eruptive flares the loop
system, while decaying, slowly grows, sometimes for many
hours.
6. Coronal mass ejections
The opening of magnetic field lines which initiates an
eruptive flare is connected with ejection of material. This
was first recognized some 30 years ago on metric radio
waves, when strong outbursts of radio emission were seen
high in the corona, some continuously moving upward. But
only spacecraft observations (first by Skylab in 1973)
showed plasma ejections from the Sun, now called coronal
mass ejections (CMEs), which moved with high speeds into
interplanetary space. (See examples in Figure 11.) Eruptive
flares are one of the sources of these CMEs which, as we
know now, play the most important role in solar-terrestrial
relations. But eruptive flares are not the only source of the
CMEs. Everywhere on the Sun magnetic field lines, closed
across a neutral line, can be disrupted, open, and eject solar
plasma into space. Eruptive flares are only one special – and
apparently the most energetic – case of the field-line
opening when strong magnetic field inside an active region
is involved in the process.
Because in many cases the neutral line inside an active
region is marked in Hα line images by a dark filament, the
opening of magnetic field is often first made apparent by its
activation. The filament structure begins to change, parts of
the filament slowly rise into the corona, the speed of the rise
accelerates, and finally the whole filament erupts. Only later
on, when the open field lines begin to reconnect, bright flare
loops
begin
to
appear,
but
at
that time the ejected material (often with the filament embedded in it) already propagates with a speed of a
few hundred km/s high in the corona into interplanetary
space.
1507
SPECIAL SECTION: SOLAR PHYSICS
We can often see a very similar process far from any
active region. A quiescent filament (several are seen in
Figure 3) becomes activated and eventually erupts. Only, on
the quiet Sun, we do not see in Hα any eruptive flare,
because magnetic field there is not strong enough to
produce all the flare effects which we see in active regions.
These activations of quiescent filaments have been known
for some 50 years and called disparition brusques, but only
in the seventies space observations in X-rays revealed that
these disruptions can have equally important effects in
interplanetary space and at the Earth as major flares in
active regions.
However, field lines crossing neutral lines can open also
at places where no dark filament exists, so that no eruption
at all is visible in the Hα line. In X-rays or in spectral lines
corresponding to high temperatures, all these field openings
have some detectable effect, but such observations are not
carried out so often as in the Hα line, so that these
responses can easily be missed. This fact is one of the
reasons why for a long time the real sources of many CMEs
remained unknown. Another reason probably is that not all
CMEs originate in this way. Some may be connected with
ejections of material along magnetic field lines, for example
in sprays (cf., Section 4).
intense events deep into interplanetary space, up to the
Earth distance. And particles trapped in magnetic clouds
associated with CMEs produce powerful Type IV bursts,
partly stationary and partly moving upward.
Accelerated electrons also give rise to hard X-rays in
solar active regions by bremsstrahlung, and particles
accelerated to high energies produce in some flares γ-rays
which are partly continuum, originating through bremsstrahlung of relativistic electrons, and partly lines excited
through electron–positron annihilation, neutron capture by
protons and helium nuclei in the photosphere, or by
transitions in excited nuclei of heavier atoms.
Particles propagating through interplanetary space produce disturbances in the Earth environment, about which
we will talk more in the last section of this review.
8. Effects of solar activity at the Earth
The Sun, as the central body of our planetary system, has
very serious impacts, of various kinds, on interplanetary
space and the environments of planets. Near the solar poles
the magnetic field lines are open and solar plasma flows
continuously into space, creating there the fast solar wind
blowing around the Earth deep into outer regions of
7. Accelerated particles
Many active processes on the Sun are apparently due to
magnetic field-line reconnections, and every reconnection
process can accelerate electrons and atomic nuclei to higher
energies. In addition to that, eruptions in the chromosphere
and corona incite wave motions which propagate both
along the solar surface (these waves are often called
Moreton waves) and upward through the corona into
interplanetary space. Some of these disturbances develop
into shock waves, which can be another source of particle
acceleration, or can produce second-step acceleration of
particles accelerated earlier elsewhere on the Sun.
Therefore, the active Sun is a rich source of energetic
particles, mainly electrons, protons, and helium nuclei,
which in some events can reach energies of hundreds
of MeV, and exceptionally particles are recorded even in the
BeV range. The most energetic source of accelerated
particles are eruptive flares and CME-associated shocks,
but also confined flares are sometimes sources of intense
flows of accelerated particles, in particular electrons.
Particles accelerated in the solar atmosphere are responsible for various kinds of radio emission recorded from
the Sun. Clouds of particles, captured in magnetic traps
above active regions, cause radio noise storms on metric
waves which can last for many days. They have been called
Type I radio bursts. Shock waves from flares produce
travelling radio disturbances, called Type II bursts. Very
frequent radio events are short-lasting Type III bursts which
are caused by accelerated electrons streaming along open
magnetic field lines high into the corona and in particularly
1508
Figure 11. Above: the development of a coronal mass eject
Mission spacecraft. Below: a more recent observation of a co
board SOHO on 5 November 1996. The Sun is hidden behind th
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
Figure 12. Images of the solar corona in the maximum (above)
and minimum (below) of solar cycle. Many coronal helmet streamers
dominate the maximum corona.
the planetary system. At lower latitudes, coronal helmet
streamers (like that one shown in Figure 9 and those in
Figure 12) and active regions during periods of field-line
openings are sources of slow solar wind. Streams of
accelerated particles, both electrons and atomic nuclei,
propagate at various places through interplanetary space.
And in addition to these streams of plasma and particles,
coronal mass ejections send shock waves and plasma
clouds in various directions through interplanetary space
and eventually cause other particle accelerations there. All
this creates highly variable and very complex conditions in
the space between the Sun and the Earth and in the last
decade we began to speak about, and regularly study, the
space weather. In particular, during periods of high solar
activity weather in space is very stormy, as one can imagine
from a look at Figure 12, which shows the solar corona at
the maximum and minimum of the solar activity cycle. For
us, of course, most important is the impact at the Earth
itself.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
When a flare appears on the Sun, it is the source of
X-rays which influences Earth’s ionosphere and thus
causes disturbances in radio communications around the
Earth. A major eruptive flare can disturb radio contacts for
many hours. This disturbance is the same for flares
occurring everywhere on the visible solar surface.
If a flare (or another active phenomenon on the Sun)
accelerates particles, they also can arrive at the Earth,
but not from all positions on the solar surface, because
they are guided by interplanetary magnetic field lines
which – due to the solar rotation – are curved into
Archimedean spirals. The most intense particle events come
from about 45° west from solar central meridian, and
generally from the western solar hemisphere. The most
energetic flares emit protons with energy exceeding 500 MeV which arrive at the Earth some 15 min
after the flare onset, produce streams of neutrons in Earth’s
atmosphere, and cause the so-called ground level effect.
Flares that produce protons of such high energies are
sometimes called cosmic ray flares. Flares that
emit protons with energies higher than 10 MeV are often
called proton flares. Particles of lower energy are guided by
the Earth magnetic field to the polar regions and cause there
absorption of radio waves (polar cap absorption) and
intense aurorae. All these effects are delayed by tens of
minutes or several hours after the flare onset, depending on
the energy of the propagating particles.
And then, moving with much slower speeds of a few
hundred to 1000 km/s, a coronal mass ejection, often with a
shock wave, arrives at the Earth, if it propagates in the right
direction. This arrival – two or three days after its origin on
the Sun – has a strong impact at the Earth magnetosphere
and causes a geomagnetic storm which sometimes can last
for several days and has serious impact on communications
all around the Earth.
Without any doubt, active processes on the Sun also
influence the weather at the Earth, but these effects are
indirect – depending on the behaviour of the magnetosphere and ionosphere – and very complex: the same effect
on the Sun can have quite different consequences at
different places of the Earth. Therefore, we still know very
little about it. But the active Sun is surely a very important
factor in our life.
ACKNOWLEDGEMENTS. Illustrations used in this review were
obtained at the Big Bear Solar Observatory, California, USA (courtesy Dr H. Zirin); Ottawa River Solar Observatory, Canada (courtesy
Dr V. Gaizauskas); University Observatory Wroclaw, Poland (courtesy Prof. B. Rompolt); Skylab Mission (courtesy AS&E, Cambridge,
Massachusetts, USA); Yohkoh satellite (courtesy Yohkoh SXT
Team), Solar Maximum Mission satellite (courtesy High Altitude
Observatory, Boulder, Colorado, USA); and the Large-Angle Spectroscopic Coronagraph (LASCO) on board the SOHO spacecraft
(courtesy LASCO consortium). Most illustrations were reprinted
from
Solar
Physics published by Kluwer Academic Publishers in Dordrecht,
Holland.
1509
SPECIAL SECTION: SOLAR PHYSICS
Probing the Sun’s hot atmosphere
Kenneth J. H. Phillips* and Bhola N. Dwivedi**,†
*Space Science Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK
**Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India
The solar corona is an extremely hot (106 K), almost fully
ionized plasma which extends from a few thousand km above
the photosphere to where it freely expands into the solar
system as the solar wind. The exact reason for its high
temperature is still unknown, despite more than 50 years of
research, but magnetic fields are certainly involved. This
article reviews some recent progress in our understanding
using data from spacecraft (SOHO, Yohkoh, and TRACE) as
well as ground-based eclipse experiments.
1. History of coronal studies
Against the expectations of the Second Law of Thermodynamics, the outer atmosphere of the Sun is hotter than
the visible surface or photosphere, from which much of its
radiation is emitted. The chromosphere, that part of the
Sun’s atmosphere which is up to 900 km above the
photosphere, has a temperature which rises from a value of
4400 K, the temperature minimum (a region some 500 km
above the photosphere), to temperatures of up to 20,000 K,
where the dominant radiation is the Lyman lines of neutral
hydrogen. Extending above the chromosphere and into the
interplanetary space is the solar corona, an extremely hot,
tenuous part of the Sun’s atmosphere where the temperature is typically (1 to 2) × 106 K, locally much more.
Both the chromosphere and corona are highly structured,
with clear evidence of association of magnetic fields which
are revealed at the photospheric layers by the Zeeman
splitting of magnetically sensitive Fraunhofer lines. The
coronal structures are generally loops or large arches, with
footpoints apparently in the photosphere, but there are also
radial structures called streamers which extend out to very
large distances. Over the polar regions, almost radial
structures known as plumes are present, giving the whitelight corona during total eclipses the appearance of a bar
magnet’s field pattern. This is an important clue to the
physical properties of the solar corona.
The corona’s high temperature means that it is visible at
ultraviolet and X-ray wavelengths, and so many spacecraft
and rocket instruments over the years have studied its
character. However, it is also visible to the naked eye during
the rare circumstances of a total eclipse, when the Moon
covers the bright photosphere. The corona then appears as
a pearly white, often irregularly shaped, structure all round
†
the Moon’s limb (Figure 1). The white-light emission is
mostly due to Thomson-scattered photospheric light off
fast-moving free electrons in the corona. The spectrum of
this radiation – the so-called K corona – has the broad
characteristics of the photosphere’s spectrum but without
the Fraunhofer lines, since they have line profiles which are
so highly Doppler-broadened that they cannot be made out
against the continuous spectrum. There is a faint extra
component, the F corona, due to dust particles in
interplanetary space which scatter photospheric radiation
also. In this case, the cold dust particles faithfully
reproduce the photospheric spectrum including the
Fraunhofer lines.
The first clues that the corona might be an unusually hot
environment were obtained during total eclipses in the
nineteenth century. Astronomers were motivated to go to
eclipses, even if they were only visible in remote parts of
the Earth, as there was no other means available then of
studying the Sun’s outer atmosphere. The Americans
Young and Harkness studied the corona during the 1869
total eclipse and found a bright emission line at 530.3 nm (to
become known as the ‘green’ line) which was unknown in
laboratory spectra1. Several more unidentified lines became
evident in spectra obtained in subsequent eclipses, and a
new element named ‘coronium’ was suspected to be the
reason for these spectral lines. As the years passed,
however, the periodic table of elements began to be much
better understood and it was clear that coronium (as well as
‘nebulium’, discovered from spectral lines in certain
gaseous nebulae) could not be easily admitted into the
scheme. It was not until the 1930s and 1940s were the lines
of coronium and nebulium reproduced in the spectra of very
hot spark sources in the laboratory, so it was then realized
that the corona must have a temperature of at least that in
sparks, nearly a million degrees K. The clinching argument
was the discovery by Edlén that the green line was due, not
to an unknown element, but to 13-times-ionized iron.
Edlén’s work also led to the identification of other coronal
lines as being due to multiply ionized atoms of familiar
elements like Fe and Ni (see, Phillips1 for more details).
2. The spacecraft era
The solar corona is a strong radiator in the ultraviolet and
X-ray parts of the spectrum. This radiation is absorbed by
For correspondence. (e-mail: dwivedi@banaras.ernet.in)
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
Figure 1 a.
Computer-processed image of the total solar eclipse in India on 24 October 1995 (Courtesy E. Hiei).
Figure 1 b. Images of part of the west limb of the Sun as follows: (left) From the EIT instrument on the SOHO spacecraft, He II (304 Å)
image; (right) From the EIT instrument on the SOHO spacecraft, Fe XII (195 Å) image; (centre) One of 6364 images obtained with the SECIS
instrument in Shabla, Bulgaria, during the total solar eclipse of 11 August 1999. Prominences in the SECIS image can be seen in the EIT He II
image (emission temp. 20,000 K) while coronal loops above sunspot regions can be seen in the EIT Fe XII image (emission temp. 106 K).
1512
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
the Earth’s atmosphere so instrumentation has to be flown
on rockets or satellites to be able to view it. Observations
were begun shortly after World War II by US groups,
notably at the Naval Research Laboratory1. Tousey and
colleagues obtained the first ultraviolet spectra of the solar
atmosphere, finding that the hydrogen Lyman-alpha line,
emitted by the chromosphere, was a strong feature at
121.6 nm. Solar X-ray emission was first detected by
Burnight in 1949 using a pinhole camera on board a rocket.
Spacecraft built by the USA and Soviet Union in the 1960s
and 1970s dedicated to solar observations added much to
our knowledge of the solar atmosphere, with the manned
Skylab Mission of 1973–1974 providing an enormous boost.
Ultraviolet and X-ray telescopes on board gave the first
high-resolution images of the chromosphere and corona
and the intermediate transition region (temperatures
between 104 and 106 K). Images of active regions revealed a
complex of loops which varied appreciably over their
lifetimes, while ultraviolet images of the quiet Sun showed
that the transition region and chromosphere followed the
‘network’ character previously known from Ca II K-line
images. (The Ca K-line is a visible-wavelength line formed in
the chromosphere.) The X-ray images showed that the
quiet-Sun corona was characterized by diffuse large-scale
loops.
In more recent times, the spatial resolution of spacecraft
instruments have steadily improved to extremely impressive
levels, almost comparable to what can be achieved with
ground-based solar telescopes. The Japanese Yohkoh
spacecraft, launched in 1991, has on board the US/Japanese
soft X-ray telescope (SXT), which images X-rays from
active and quiet-Sun regions and flares (which are sudden
releases of energy in active regions) with a resolution of
about 2 arc seconds (1 arc second corresponds to 725 km at
mean solar distance: the mean solar diameter is 32 arc
minutes). X-rays with wavelengths in the range 0.2–2 nm are
sensed by the SXT. Yohkoh, which is in a low-Earth orbit,
continues to operate at the present time, and has obtained
many thousands of images from the SXT (Figure 2) as well
as considerable amounts of data from the other instruments
on board which are mainly for detecting X-ray emission
during flares.
The ESA/NASA Solar and Heliospheric Observatory
(SOHO) was launched in 1995 into an orbit about the inner
Lagrangian (L1) point situated some 1.5 × 106 km from the
Earth on the sunward side. Its twelve instruments therefore
get an uninterrupted view of the Sun, unlike the instruments
on Yohkoh. Apart from a period in 1998 when the spacecraft
was temporarily out of contact, there have been continuous
operations since launch. There are several imaging
instruments, sensitive from visible-light wavelengths to the
extreme-ultraviolet. The Extreme-ultraviolet Imaging
Telescope (EIT), for instance, uses normal-incidence optics
to
get
full-Sun
images
several times a day in the wavelength bands containing
lines emitted by the coronal ions Fe IX/Fe X, Fe XII, Fe XV
(emitted in the temperature range 600,000 K to 2,500,000 K)
as well as the chromospheric He II 30.4 nm line (Figure 3).
The spatial resolution is about 2 arc seconds (1500 km). The
set of three coronagraphs making up the Large Angle and
Spectrometric Coronagraph (LASCO) view the white-light
corona
with
high
resolution out to distances of 30 solar radii (1 solar radius is
700,000 km). Movies of the corona from LASCO show the
large-scale structures in the corona as they rotate with the
rest of the Sun (the solar rotation period as viewed from the
Earth is 27 days or so, with slight latitude dependence), but
more particularly they show the large ejections of coronal
Figure 2. Yohkoh soft X-ray telescope (SXT) image of the Sun’s corona on 8 May 1992
(Courtesy The Yohkoh SXT Team).
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mass in the form of huge bubbles, moving out with
velocities of several hundred kilometer per second
(typically) that, on colliding with the Earth and other planets
in the solar system, give the well-known magnetic storms
and associated phenomena that have become a matter of
widespread concern for telecommunications in recent years.
The Transition Region and Coronal Explorer (TRACE) is a
highly successful spacecraft observatory, operated by the
Stanford-Lockheed Institute for Space Research, which was
launched in 1998 and has imaging capabilities that are truly
staggering. The spatial resolution is of order 1 arc second
(725 km), and there are wavelength bands covering the
Fe IX, Fe XII, and Fe XV lines which EIT observes as well
as the Lyman-alpha line at 121.6 nm. Movies made by
stringing together many images of, for example, active
regions reveal a vast wealth of detail, with the coronal loops
showing continuous brightenings and motions. A
remarkable feature is that the loops often have widths which
are no more than the spatial resolution of TRACE, and so
they are probably thinner than 1 arc second.
Some of the SOHO instruments are able to obtain spectra
in the ultraviolet region, and such spectra are of great use in
‘diagnosing’ (i.e. deducing the prevailing physical
conditions in) the emitting plasma. The corona is an almost
fully ionized plasma, i.e. is composed of mostly protons and
electrons, with the density of heavier ions only 10–6 of the
proton density (hydrogen is by far the most abundant
element in the Sun). However, the atoms of heavier elements
like Fe or Si generally retain a few of their electrons, and
hence the spectrum of the corona in, e.g. the extreme-
1514
ultraviolet range is characterized by numerous emission
lines. The corona is generally optically thin in these lines,
and their intensities often give important information about
temperatures, densities, and flow speeds. Much work has
gone into deducing particle densities from the ratios of
particular lines which are density-sensitive, and as a result,
we are now able to overlay density maps onto images of
portions of the corona. We will discuss this further in
Section 5.
3. Heating of the corona: Theory
There is evidence that the corona is heated by its magnetic
fields, although the evidence is not direct. One vital piece of
information that we are still unable to measure is the
magnetic field in the corona. We are able to measure, with
considerable accuracy, the photospheric magnetic field,
using magnetographs that work on the Zeeman principle.
This can be done for small regions so that a complete
magnetic field map of the Sun’s visible hemisphere
(magnetogram) can be constructed. These are routinely
available in, for example, the Solar-Geophysical Data
Bulletin issued by NOAA. For vector magnetographs, all
three components of the magnetic field can be deduced. But
the measurements refer to magnetic field at the
photospheric level, not in the corona. Although eventually
infrared measurements may change this situation, the only
way at present in which the coronal field can be deduced is
through extrapolations of the photospheric field through
the assumption, e.g. of a potential (current-free) or force-
Figure 3. Image taken in the Fe XII 19.5 nm extreme-ultraviolet (EUV) line by the SOHO
extreme-ultraviolet imaging telescope (EIT) instrument on 7 June 1999 (Courtesy The SOHO
EIT Team).
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
free field. We do find, however, that the photospheric field
in active regions is more complex than in quiet regions, and
it is also known that the active region corona is appreciably
hotter (typically 4 × 106 K, depending on the nature of the
active region) than in quiet regions (2 × 106 K, less in coronal holes at the poles). So there does seem to be a relation
between field strength and heating.
A considerable theoretical problem with magnetic field
heating is the fact that it requires the diffusion, and
therefore reconnection, of magnetic field which implies
a resistive plasma. However, the coronal plasma is on
the contrary highly conducting. Using Spitzer’s expression for plasma resistivity, η = 103 T – 1.5 Ohm m (T is the
temperature in K), we find for T = 2 × 106 K, η = 4 × 10–7
Ohm m, only a factor 20 or so higher than solid copper, a
familiar example of an almost perfect conductor, at room
temperature! Using the induction equation of magnetohydrodynamics, we find that the diffusion time for a
magnetic field is extremely long unless the characteristic
distance over which diffusion occurs is as short as a few
meters when the diffusion time is a few seconds. Put
another way, the magnetic Reynolds number Rm, measuring
how tied the magnetic field is to the plasma, is typically 106
to 1012 for the corona, indicating that the field is completely
‘frozen in’ to the plasma. However, reconnection requires
Rm to be very small, much less than one in fact. Thus, only if
the length scales are very small can one achieve magnetic
reconnection. Very small length scales do occur in the
region of neutral points or current sheets, where there are
steep magnetic field gradients which give rise to large
currents. It is thought, then, that such geometries are
important for coronal heating if this is by very small energy
releases, known (from the original work on the subject, by
Parker2) as nanoflares. Some 1016 J are released in a
nanoflare, i.e. 10–9 of a large solar flare, and many energy
releases like this occurring all over the Sun, quiet regions as
well as active regions, could account for heating of the
corona. However, it is doubtful whether this mechanism
would apply to coronal hole regions where the field lines are
open to interplanetary space.
The above reasoning applies equally to the competing
wave heating hypothesis of coronal heating, in which
magnetohydrodynamic waves generated by photospheric
motions (e.g. granular or supergranular convection motions)
are dampened in the corona. In this case, we need conditions such that the magnetic field changes occur in a
shorter time than, say, the Alfvén wave transit time across a
closed structure like an active region or quiet Sun loop. The
literature for wave heating of the corona is very
considerable, but we may briefly summarize it by stating
that the waves, generated by turbulent motions in the solar
convection zone or at the photosphere, may be surface
waves in a loop geometry, or body waves which are guided
along the loops and are trapped. The work of Porter,
Klimchuk and Sturrock3 shows that short-period fast-mode
and slow-mode waves (periods less than 10 s) could be
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
responsible for heating since only for them are the damping
rates high enough.
It is worth mentioning that MHD waves need not
be generated by photospheric motions. Axford and
McKenzie4 have proposed a ‘furnace’ model in which small
loops convected into the chromospheric network undergo
reconnection and launch high frequency waves which can
heat the corona (to a few million degrees) through ioncyclotron resonance dissipation and rapidly accelerate the
wind up to high speeds (~ 750 km/sec) within a few solar
radii.
4. Observational evidence: Transient
brightenings
Early observations with the high resolution telescope
and spectrograph (HRTS) by Brueckner and colleagues5
showed that the profiles of the strong C IV (transition
region) ultraviolet (154.8, 155.1 nm) line pair, emitted at
100,000 K, showed much dynamic activity that could be
broadly classified into turbulent events (speeds up to
250 km/s in small areas) and jets (speeds up to 400 km/s).
The energy contained in the jets amounts to a ‘microflare’
(i.e. up to 1019 J), and it was considered that shock waves
generated by a jet could heat up the corona. Enough energy
and mass are contained in the jets, assumed to occur over
the whole Sun, to satisfy the requirements of the corona
and its dynamic extension, the solar wind.
Such phenomena are just an example of the many
transient events that occur in the solar atmosphere. Shimizu
and colleagues6 have been studying the Yohkoh SXT data
for active regions, and they find numerous
small brightenings in active region loop structures having
energies of the order 1020 J, i.e. comparable to microflares.
Similar X-ray flares have been noted at higher energies by
Lin and colleagues in 1984. Again there is a possibility that
the energy supplied by these small active region events is
sufficient to heat the corona outside coronal holes, though
present
indications
are
that
it
is
short
by a factor of about 5. Even smaller events – ‘network
flares’ – have been noted outside of active regions by Benz
and Krucker and later by Pres and Phillips7
(Figure 4) in studies of Yohkoh SXT and SOHO EIT data.
Here, the energies of the events are much less (down to
only 100 times a nanoflare) but then the energy requirements of the quiet (i.e. non-active region) solar corona is
correspondingly less. Within coronal holes, where the soft
X-ray background is very small indeed, Koutchmy has seen
tiny coronal ‘flashes’ which have energies of order 10 times
a nanoflare.
In the extreme-ultraviolet (EUV), very small brightenings
have been noted by various authors using SOHO data in
quiet-Sun regions. It would appear that these were visible in
earlier spacecraft data such as those from the
Figure 4.
Examples of soft X-ray quiet Sun ‘network flares’, noted o
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SPECIAL SECTION: SOLAR PHYSICS
OSO series from the 1970s. A comparison by Pres of
network flares and these so-called ‘blinkers’ reveals a
strange lack of correlation, with the EUV transients in quiet
regions being apparently much more numerous than their Xray counterparts.
Table 1 gives some indication of the thermal energy rate
(W) deduced from observations of various transients in either X-rays or the EUV. As the total radiative
power of the entire quiet-Sun corona is several 1019 W, it
appears that the numerous EUV quiet-Sun transients might
offer the best possibility of heating, assuming that the
Parker nanoflare hypothesis is correct.
5. Physical characteristics of the corona
It is clear from eclipse or spacecraft images of the corona
(Figures 1 and 2) that the corona is highly structured, and
that the hot plasma making up the corona is confined to
intricate magnetic field patterns. Particle densities and
temperatures can be derived for different coronal regions
using a variety of methods. For example, measurements of
the surface brightness of the white-light corona yield
electron densities, since the emission is by Thomson
scattering of photospheric light off free electrons. Near the
base of the corona the measured electron densities are a few
times 1014 m–3, though this is strongly dependent on
whether the feature is a quiet region (smaller densities) or
within a complex of active region loops (larger densities).
The densities are further reduced within coronal holes. The
density falls off rapidly with height: at 5 solar radii from Sun
centre, the density is 1011 m–3, while at the distance of the
Earth’s orbit (where the corona is in the form of a freely
flowing
wind)
the
density
is
less
than
107 m–3. As indicated earlier, temperatures vary in the corona
from place to place, with maximum values in highly complex
active regions (up to 4 × 106 K) and minimum values in the
polar coronal holes (slightly less than 106 K). In general,
then, densities and temperatures are correlated.
To illustrate the fact that the coronal gas is strongly
dominated by the magnetic field, we find that the magnetic
pressure B2/2 µ is generally many times more than the gas
pressure NkT (N = particle number density, k = Boltzmann’s
constant). For a typical active region, magnetic pressure
might be as much as 50 Pa, but the gas pressure is perhaps
Table 1.
Energetics for EUV and X-ray transient events in the
quiet Sun corona
No. of events h –1
40,000 brightenings
1,200
100
Thermal energy rate
(W)
3 × 10
2 × 10 18
2 × 10 18
19
Event type
EUV transients
Small network flares
Large network flares
The data in the table are based on recent publications using SOHO
and Yohkoh observations (see References).
1516
a factor of almost 10 less.
Measurements of densities and temperatures using
X-ray or ultraviolet data are possible, particularly using line
intensities. Most regions in these lines are optically thin,
though there are notable exceptions like in the hydrogen
Lyman-alpha line and the other strong resonance lines. The
line intensity is a function of temperature, through the
ionization fraction of the emitting ion (a strongly peaked
curve) and the excitation rate from the ground energy level
of the ion to the upper level giving rise to the line emission.
Excitation of most ions emitting lines in the X-ray and
ultraviolet ranges is due to collisions of free electrons with
the
ions.
Very
roughly
we can assign a single temperature to an ion, e.g.
three-times-ionized carbon, which gives rise to the C IV
154.8/155.1 nm line doublet, corresponds to a temperature of
about 100,000 K, approximately the maximum fractional
abundance of the C+3 ion in ‘ionization equilibrium’, i.e. a
balance between collisional ionization processes and
radiative and dielectronic recombination processes which
occurs in the solar corona. These fractional abundances can
be calculated using atomic data, and there are many
publications which give values as a function of temperature
(in general there is practically no density dependence).
Excitation of the lines can be calculated to much higher
accuracy than was possible about 20 years ago as there are
sophisticated atomic codes which take into account
resonances in the collisional cross sections. Among these
is the close-coupling R-matrix code developed at Queen’s
University Belfast. As a result, there are a number of line
pairs recognized in especially the extreme-ultraviolet part of
the spectrum which are sensitive to electron densities. This
fact is very useful as nearly all other methods
of getting densities are indirect. Thus, a measured line
intensity of
N e2aV feature leads to a value for the ‘emission
measure’
(V = volume, Ne = electron number density).
This combined with a measured value for V (e.g. from image
data) gives Ne . However, this technique is often quite
imprecise since the presence of fine structure within the
feature, if unresolved by the instrument taking the
observations, renders the value of Ne to be merely a lower
limit. Account then has to be taken of a ‘filling factor’, often
much less than one. Spectral line diagnostics for solar
plasmas have been discussed in an accompanying article of
this issue by Dwivedi, Mohan and Wilhelm8.
The Coronal Diagnostic Spectrometer (CDS) on SOHO
has been widely used to get densities, and it is now
possible to construct maps of regions of the Sun showing
electron densities. Gallagher and colleagues9 at Queen’s
University Belfast and Rutherford Appleton Laboratory
have been active in this. Two examples of suitable densitysensitive lines in the wavelength range of CDS are those
due to Si IX and Si X, both emitted at around 1.3 × 106 K.
This value of temperature makes them ideal for studying the
quiet corona. Figure 5 shows the positions of scans using a
pair of lines emitted by each ion around the Sun’s limb with
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
the CDS instrument in February 1996, and Figure 6 shows
the resulting scan with light areas indicating measured
electron densities from each ion, the vertical scale being
position angle around the Sun’s limb. It shows clearly the
presence of higher densities in low-latitude regions (Ne
around 4 × 1014 m–3), while near the south pole (position
angle 180 degrees) the electron density is at least a factor 4
lower. The density profile with position angle at three radial
distances out to 1.2 solar radii agrees remarkably well with a
recent analytical model that has been developed for Sun-like
stars 10.
6. Observational evidence: Wave motions
Despite the fact that the nanoflare hypothesis seems to be
observationally plausible (as indicated in Section 4), MHD
waves may well be implicated in the heating of the corona,
and it is important to look for signatures of them. It is, for
example, unlikely that nanoflares could heat the corona in
the regions of open field lines such as occur in polar
coronal holes, yet it appears that the corona is still hot in
these open-field regions. A basic difficulty seems to be that
theoretical predictions indicate that MHD waves having
only a short period (less than 10 sec) are likely to be
effective in the heating, since only for such waves are the
damping rates sufficiently great. However, spacecraft
imaging, limited as it is by the rate at which data are
telemetered to the Earth, is necessarily rather slow. It takes
about 2 min for any instrument on SOHO to produce an
image of even relatively small portions of the Sun.
There is hence still considerable interest in observing the
visible-light corona during total eclipses from the ground,
since one can use high-speed electronic cameras to obtain
rapid imaging of particular coronal structures. A pioneer in
this work has been Pasachoff 11, who has tried this kind of
experiment at various eclipses around the world since the
1980s. Analysis of his best results indicate the presence of
a
slight
peak
in
Fourier
spectra
at
frequencies of 0.5–1 Hz. This has been seen in more recent
eclipses, including the 1998 eclipse in the Caribbean. Other
measurements using ground-based white-light coronagraphs have been taken, notably by Koutchmy12 at the US
National Solar Observatory/Sacramento Peak some years
ago. Here searches were made for periodic modulations in
both the intensity and velocity of the green line, with
evidence of periods equal to 43, 80 and 300 s (the last is
probably related to the familiar five-minute oscillation seen
with photospheric Fraunhofer lines).
While Pasachoff continues to develop his instrument
with colleagues at Williams College, Massachusetts, a
group including Rutherford Appleton Laboratory, Queen’s
University Belfast, and the Astronomical Institute, University of Wroclaw in Poland have been developing a fastimaging system with charge-couple device (CCD) cameras
that can image up to 50 frames a second with a specially
adapted computer that ‘grabs’ images, placing the data on
to large-capacity hard discs for later analysis. The cameras
were developed by EEV, a company specializing in CCD
cameras in Chelmsford, UK, and the computer hardware and
software were developed by Carr-Crouch Computer
Company, Maidenhead, UK. The system, called the solar
eclipse
coronal
imaging
system
(SECIS),
has been tested on a number of occasions already, the first
being during the 1998 eclipse and most recently on
11 August 1999, the last total solar eclipse of this millennium. Scientifically useful results were obtained during a
run on the Evans Coronagraph Facility at Sacramento Peak.
Figure 6. Electron density plot showing variation of electron
density with position angle round the Sun (see Figure 5). Density
Figure 5. SOHO
Figure
EIT7.image
Image
of of
thecoronal
Sun showing
loops on
positions
the solar
of scans
limb in September
scale is on1998
the right.
with The
preliminary
left panelFourier
is using
analysis
a Si IXof
line
tworatio,
points
thatinon
the image (A is
with the CDS instrument.
within the coronal active region, B is considered to be non-coronal
the rightorasky
Si Xbrightness)
ratio (Courtesy
(Courtesy
P. T. P.
Gallagher).
T. Gallagher).
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
1517
SPECIAL SECTION: SOLAR PHYSICS
Preliminary results, with one channel (using a green-line
filter), are shown in Figure 7, showing a system of active
region loops on the limb with initial Fourier analysis done
by Gallagher. Taken at face value, there is very slight
evidence for excess power at small (less than 5 Hz) at a
location (A) within the active region loops but none in the
more distant location (B) where the coronal signal is
negligible compared with sky (terrestrial) background.
7. Concluding remarks
A large amount of information concerning the physics of
the solar atmosphere has become available in recent years
and it is taking some time to digest the new data. The exact
reason for heating the solar corona is still not known for
certain, but there is much evidence from spacecraft
observations that, in regions where there are complex
magnetic field geometries or at least the presence of closed
loops, heating by numerous small flare-like releases of
energy is adequate to explain the energy requirements of
the corona. In coronal holes, i.e. regions of open field lines,
this is less likely to be true and many consider that heating
proceeds through the damping of MHD waves, which may
still have a role in the heating of the corona in closed-field
regions. At present this can only be investigated using
ground-based instruments since the periods of MHD waves
which have sufficiently large damping rates are likely to be
very small, of order a few seconds. Spacecraft imaging is
too slow to search for such periodicities.
1518
1. Phillips, K. J. H., Guide to the Sun, Cambridge University Press,
Chapter 5, 1995.
2. Parker, E. N., Astrophys. J., 1988, 330, 474–479.
3. Porter, L. J., Klimchuk, J. A. and Sturrock, P. A., Astrophys. J.,
1994, 435, 482–501.
4. Axford, W. I. and McKenzie, J. F., in Cosmic Winds and the
Heliosphere (eds Jokipii, J. R., Sonett, C. P. and Giampapa,
M. S.), University of Arizona Press, Tucson, 1997, pp. 31–66.
5. Brueckner, G. E. and Bartoe, J.-D. F., Astrophys. J., 1983, 272,
329–348.
6. Shimizu, T., Pub. Astron. Soc. Japan, 1995, 47, 251–263.
7. Pres, P. and Phillips, K. J. H., Astrophys. J., 1999, 510, L73–
L76.
8. Dwivedi, B. N., Mohan, A. and Wilhelm, K., Curr. Sci., 1999
(this issue).
9. Gallagher, P. T., Mathioudakis, M., Keenan, F. P., Phillips,
K. J. H. and Tsinganos, K., Astrophys. J. (submitted).
10. Lima, J. J. G., Priest, E. R. and Tsinganos, K., The Corona and
Solar Wind near Minimum Activity, ESA SP-404, 1997, pp.
521–526.
11. Pasachoff, J. and Ladd, E. F., Solar Phys., 1987, 109, 365–372.
12. Koutchmy, S., Zugzda, Y. D. and Locans, V., Astron. Astrophys.,
1983, 120, 185–191.
ACKNOWLEDGEMENTS. We thank colleagues for allowing us to
use their work to illustrate this article, in particular P. T. Gallagher,
F. P. Keenan, P. Pres, and members of the SOHO (EIT) and Yohkoh
experiment teams. We are indebted to K. Wilhelm for critical
reading of the manuscript.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
SPECIAL SECTION: SOLAR PHYSICS
A new Sun: Probing solar plasmas in the
extreme-ultraviolet light from SUMER
on SOHO
Bhola N. Dwivedi*,**,†, Anita Mohan* and Klaus Wilhelm**
*Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India
**Max-Planck-Institut für Aeronomie, 37191 Katlenburg-Lindau, Germany
We briefly outline the extreme-ultraviolet spectroscopy of
solar plasmas in the light of a wealth of high-resolution
observations, both in spectral and spatial regimes, from the
SUMER spectrograph (Solar Ultraviolet Measurements of
Emitted Radiation) on the spacecraft SOHO (Solar and
Heliospheric Observatory). We then present some of the new
results on a new Sun seen by the SUMER spectrograph. In
particular, we discuss coronal holes and the solar wind, the
‘red’/‘blue’ Sun, abundance anomalies, explosive events and
sunspot transition region oscillations. We conclude this
article by reiterating clues, obtained from SUMER, that
provide information essential for solving solar riddles of
coronal heating and the wind acceleration.
44 mÅ in first order of diffraction and 22 mÅ in second order.
The spectral resolution, which can be improved for relative
(and under certain conditions for absolute) measurements
to fractions of a pixel, allows to measure Doppler shifts
corresponding to plasma bulk velocities of about 1 km s –1
along the line of sight. Turbulent velocities are obtained by
determining line broadenings. Vast literature has resulted on
the basis of observations carried out using SUMER. But, we
have not reviewed this in the present article. Instead, we
present some new results from SUMER on solar plasmas
that have added a new dimension to a better understanding
of some of the solar mysteries, especially coronal heating
and the solar wind acceleration.
Introduction
Plasma diagnostics
THE Sun presents us with a thousand-fold face. Depending
on the ways we observe it, whether it be through a groundbased telescope, during an eclipse, or from a space
observatory, we see a different Sun. One can distinguish
the wavelengths in which it is observed: X-rays, ultraviolet,
visible, infrared, radio or even by non-photon instruments
(e.g. neutrino detectors). One can distinguish the time when
it is observed: near the maximum activity of its sunspot
cycle, or the minimum activity. Still this multi-faceted Sun of
ours is a single celestial object, and it is this idea of the Sun
as whole that has been described in the previous articles of
this special issue. A new Sun that has emerged from the
analysis and the interpretation of observations from the
Solar Ultraviolet Measurements of Emitted Radiation
(SUMER) spectrograph is briefly presented in this article.
Full descriptions of the SUMER spectrograph on the
spacecraft SOHO and its performance are available1–3.
Briefly, the SUMER spectrograph observes the Sun in the
extreme-ultraviolet (EUV) light from 465 Å to 1610 Å with
high spatial and spectral resolution. This wavelength range
contains EUV lines from the chromosphere, the transition
region, and the corona, thereby providing a unique
opportunity to probe plasmas of the solar atmosphere. The
spatial resolution is close to 1″ (about 715 km at the Sun),
while the spectral resolution element (one pixel) is about
Without a knowledge of the densities, temperatures, and
elemental abundances of space plasmas, almost nothing can
be said regarding the generation and transport of mass,
momentum and energy. Thus, since early in the era of
space-borne spectroscopy we have faced the task of
inferring plasma temperatures, densities, and elemental
abundances for hot solar and other astrophysical plasmas
from optically thin emission-line spectra4,5. A fundamental
property of hot solar plasmas is their inhomogeneity. The
emergent intensities of spectral lines from optically thin
plasmas are determined by integrals along the line of sight
(LOS) through the plasma. Spectroscopic diagnostics of the
temperature and density structures of hot optically thin
plasmas using emission-line intensities is usually described
in two ways. The simplest approach, the line-ratio
diagnostics, uses an observed line-intensity ratio to
determine density or temperature from theoretical density or
temperature-sensitive line-ratio curves, based on an atomic
model and taking account of physical processes for the
formation of lines. The line-ratio method is stable, leading to
well-defined values of Te or Ne , but in realistic cases of
inhomogeneous plasmas these are hard to interpret, since
each
line
pair
yields
a
different
value
of density or temperature. The more general differential
emission measure (DEM) method recognizes that observed
plasmas are better described by distributions of temperature
or density along the LOS, and poses the problem in inverse
†
For correspondence. (e-mail: dwivedi@banaras.ernet.in)
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SPECIAL SECTION: SOLAR PHYSICS
form. It is well known that the DEM function is the solution
to the inverse problem, which is function of Te , Ne , or both.
Derivation of DEM functions, while more generally
acceptable, is unstable to noise and errors in spectral and
atomic data. The exact relationship between the two
approaches has never been explored in depth, although
particular situations were discussed6. The mathematical
relationship between these two approaches has recently
been reported7.
Line shifts and broadenings give information about the
dynamic nature of the solar and stellar atmospheres. The
transition region spectra from the solar atmosphere are
characterized by broadened line profiles. The nature of this
excess broadening puts constraints on possible heating
processes. Systematic redshifts in transition region lines
have been observed in both solar spectra and stellar spectra
of late-type stars. On the Sun, outflows of coronal material
have been correlated with coronal holes. The excess
broadening of coronal lines above the limb provides
information on wave propagation in the solar wind. Figure 1
shows the full Sun raster image obtained by SUMER in the
Ne VIII (λ770) line on 2 February 1996. The emission line is
formed at 630000 K and observed in second order. The polar
coronal holes are clearly seen in this line as well as some
bright points and polar plumes8. In Figure 2 a Doppler
Figure 1. Full Sun raster image obtained by Solar Ultraviolet
Measurements of Emitted Radiation (SUMER) in the line Ne VIII
(λ770) on 2 February 1996. The emission line is formed at a temperature of 630000 K and observed in second order of diffraction on
the bare microchannel plate portion of detector A. The 1″ x 300″
slit was used with a step width of 1.88″ and an exposure time of 7.5 s.
The polar coronal holes can clearly be seen in this line as well as
some bright points and polar plumes (from Wilhelm et al.8 ).
1522
velocity map is shown, derived from the Ne VIII
observations under the assumption that the line shifts can
be interpreted as plasma flows. The range of the velocity
scale is from – 30 km s –1 (blue) to + 30 km s –1 (red). The zero
point is adjusted to give no Doppler shift just above the
limb at about 20 arc sec. The blue polar coronal holes stand
out with some white spots, which upon close inspection,
coincide with bright points in the intensity image. Strong up
and down motions can be seen near the active regions in
the western hemisphere.
Coronal holes and the solar wind
In 1950s Waldmeier9 first recognized persistent depressions
in the intensity of the monochromatic corona (outside the
polar caps) as observed by ground-based coronagraphs
and called them ‘holes’ (Löcher in German). He published
his coronal observations obtained between 1939 and 1952
(during the solar cycles 17 and 18). At the end of cycle 17,
coronal holes were identified in coronal maps. More than 20
years later, after theoretical predictions, Krieger et al.10
related
a
coronal
hole,
seen
on
an X-ray image taken on 24 November 1970 during a
sounding rocket flight, to a recurrent high-speed stream of
Figure 2. Doppler velocity map derived from the Ne VIII
observations shown in Figure 1. The range of the velocity scale from
– 30 km s–1 (blue) to + 30 km s–1 (red). The zero point is adjusted to
give no Doppler shift just above the limb (at ~ 20″). The blue polar
coronal holes stand out with some white spots, which upon close
inspection, coincide with bright points in the intensity image. Strong
up and down motions can be seen near the active regions in the
western hemisphere.
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SPECIAL SECTION: SOLAR PHYSICS
the solar wind observed by instruments on PIONEER-VI
and on the Vela satellite.
Observations from the Skylab mission further established
that the high-speed solar wind originates in coronal holes
which are well-defined regions of strongly reduced EUV and
soft X-ray emissions11. More recent data from Ulysses show
the importance of the polar coronal holes, particularly at
times near the solar minimum when a magnetic dipole
dominates the field configuration. Figure 3 shows a polar
coronal hole with bright points and polar plumes12 seen in
Mg X (λ625) with a formation temperature of ~ 1100000 K.
The mechanism for accelerating the wind to the high values
observed, of the order of 800 km s –1, is not understood
quantitatively. The Parker model13 is based upon a thermally
driven wind. To reach such high velocities, temperatures of
the order 3 to 4 MK would be required near the base of the
corona. However, other processes are available for
acceleration of the wind, for example the direct transfer of
momentum from magneto-hydrodynamic (MHD) waves,
with or without dissipation. This process results from the
decrease of momentum of the waves as they enter less
dense regions, coupled with the need to conserve
momentum of a system consisting of waves plus the local
plasma. If this transfer predominates, it may not be
necessary to invoke high coronal temperatures at the base
of the corona.
In reality, very little information was available on the
density and temperature structure in coronal holes prior to
the SOHO Mission. Data from Skylab is limited, due to the
very low intensities in holes and the poor spectral
resolution, leading to many line blends. Skylab was able to
follow temperatures up to nearly 1 MK and no further, and
the interpretation of the data is quite uncertain. Highresolution EUV observations from instruments on SOHO
provided the opportunity to infer the density and temperature profile in coronal holes. Wilhelm et al.14 observed
several polar coronal holes with SUMER spectrograph in
Si VIII lines at 1445.75 Å and 1440.49 Å and Mg IX lines at
706 Å and 750 Å to determine density and temperature
structure via line-ratio spectroscopic diagnostics. Figure 4
shows the electron densities deduced. Comparing the
electron temperatures with the ion temperatures, Wilhelm et
al. concluded that ions are extremely hot and the electrons
are relatively cool. This result is also in agreement with the
UVCS (Ultraviolet Coronagraph Spectrometer) SOHO
results15 at greater altitudes.
Using the two SOHO instruments CDS (Coronal Diagnostic Spectrometer) and SUMER David et al.16 have now
measured electron temperatures as a function of height
above the limb in a polar coronal hole. Observations of two
lines from the same ion stage O VI 1032 Å from SUMER and
O VI 173 Å from CDS/SOHO were made to determine the
electron temperature gradient in a coronal hole. They
deduced temperatures of around 0.8 MK close to the limb,
rising to a maximum of less than 1 MK at 1.15 R¤, then
falling to around 0.4 MK at 1.3 R¤. It seems that present
observations preclude the existence of temperatures over
1 MK at any height near the centre of a coronal hole. Wind
acceleration by temperature effects is therefore inadequate
as an explanation of the high-speed wind and it becomes
essential to look towards other effects, probably involving
the momentum and the energy of Alfvén waves.
The ‘red’/‘blue’ Sun
In the so-called transition region, the temperature rises
sharply from some 10000 K to more than 1 MK. It has been
known since the Skylab period that there is a net red shift in
the transition region lines17. It has recently been reported
that the redshift peaks at about 12 km s –1 near 150000 K and
extends into the hotter regions where Ne VIII emission line
is
formed18,19.
Ne VIII
belongs
to
the
Figure 3. Polar coronal hole with bright points and polar plumes seen in Mg X (λ625) with a
formation temperature of ~ 1100000 K (from Dammasch et al.12 ).
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Li-sequence and has a strong 2s–2p resonance line at 770 Å
which is observed by SUMER in both first- and secondorder. A laboratory wavelength of (770.409 ± 0.005) Å was
used for this line. Dammasch et al.20 made an accurate
wavelength measurement of this line recorded in second
order together with several S I and C I lines, which have
well-known wavelengths. Assuming that there is no net
Doppler flow along the line of sight at and above the limb
and eliminating other residual errors, a rest wavelength of
(770.428 ± 0.003) Å was derived. With 1 mÅ corresponding to
390 m s –1, this new result moves the Doppler shift of Ne VIII
towards
the
blue
by
(–)7.4 km s –1. This immediately implies that there is no net
downflow at this temperature in quiet Sun regions.
That the solar wind is coming from coronal holes (open
magnetic field regions in the corona) has been widely
accepted, although little additional direct observational
evidence has been obtained to support this view. Hassler et
al.21 found the Ne VIII emission blueshifted in the north
polar coronal hole along the magnetic network boundary
and at network boundary interfaces compared to the
average quiet Sun flow. We show in Figure 5 velocity map
of the Ne VIII (λ770) line as observed on 21 September 1996
(top panel). The areas of dark blue, corresponding to an
outflow velocity of more than 5 km s –1 (LOS), are enriched
by contours. A repeat of the velocity contours is overlaid
on the intensity diagram of the same area in the bottom
panel. It is to be noted that coronal hole conditions prevail
in the northern portion of the field of view (520″ × 300″),
whereas quiet Sun regions are present in the south. These
Ne VIII observations reveal the first two-dimensional
coronal images showing velocity structure in a coronal hole,
and provide strong evidence that coronal holes are indeed
the source of the fast solar wind. The apparent relationship
to the chromospheric magnetic network, as well as the
relatively large outflow velocity signatures at the
intersections of network boundaries at midlatitudes, is a
first step in better understanding the complex structure and
dynamics at the base of the corona and the source region of
the solar wind.
combination/forbidden lines, which provided good possibilities to study the relative element abundance of Ne (high
FIP) and Mg (low FIP) in transition region emission in the
corona. The observation of the FIP effect in transition
region emission in the corona is a new observational fact.
Laming et al.26 have also investigated the behaviour of the
FIP effect with height above the solar limb in a region of
diffuse quiet corona and found a low FIP bias of a factor of
3 to 4, with no significant height variation. Yet another
study confirmed the new observational facts about FIP
effects in transition region emission27 which is shown in
Figure 6 for the Mg/Ne FIP bias from four line ratios as a
function
of
height
above
the limb.
Explosive events
The universe abounds with explosive energy release that
may heat plasma to millions of degrees and accelerate
particles to relativistic velocities. Such occurrences are not
Abundance anomalies
The status of coronal abundances relative to hydrogen is
not entirely settled. It is often suggested that abundances
are correlated with the first ionization potential (FIP). Many
studies show that ‘FIP bias’ does exist22–24. Classically, a
step function increase by a factor of 4 is assumed for
elements with increasing FIP. The FIP effect should
eventually offer valuable clues to the process of heating,
ionization, and injection of material into coronal and flaring
loops for the Sun and other stars.
Dwivedi et al.25 presented results from a study of EUV
off-limb spectra obtained on 20 June 1996 with SUMER
spectrograph. They recorded Ne VI and Mg VI inter1524
Figure 5. Velocity map of the Ne VIII (λ770) line as observed on
21 September 1996 (top panel). The areas of dark blue,
corresponding to an outflow velocity of more than 5 km s–1 LOS
(line-of-sight), are enriched by contours. Single pixel contributions
have been eliminated by a floating average over 3″ in east–west and
north–south directions. A repeat of the velocity contours is overlaid
on the intensity diagram of the same area in the bottom panel. Note
that coronal hole conditions prevail in the northern portion of the
field of view (520″ × 300″), whereas quiet Sun regions are present in
the south.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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uncommon on our own star too. Examples include solar
flares, coronal mass ejections, chromospheric and coronal
microflares, etc. In many cases, the magnetic field seems to
be the only source of energy available to power these
cosmic explosions. While it is well established that the Sun
has a large reservoir of magnetic energy, the reason for its
release is still debated.
The widely accepted explanation for explosive energy
release is a process known as magnetic reconnection. This
effectively involves the cutting and reattachment of
magnetic lines of force. Many solar physicists feel that
reconnection is observationally and theoretically well
established. Sceptics, however, argue that there is no
definite proof for the reconnection taking place on the Sun.
It is for this reason that new results from SUMER for the
magnetic reconnection on the Sun is so important. They
provide the best evidence to date for the existence of bidirectional outflow jets, forming fundamental part of the
standard reconnection model.
Explosive events28 were first seen in the ultraviolet
spectra obtained with the NRL’s (Naval Research Laboratory) High Resolution Telescope and Spectrograph
Figure 6.
(HRTS) flown on several rocket flights and Spacelab 2.
They were found to be short-lived (60 sec), small-scale
(1500 km), high-velocity (± 150 km s –1) flows that occurred
very frequently over the entire surface of the Sun. There are
estimated to be 30,000 events at any one time on the Sun.
The energy involved in the events observed, however,
suggests that they are not the major source of mass or
energy in either chromosphere or corona. Their importance
lies in the fact that they probably represent the high energy
tail of a spectrum of network events that occur on scales
unobservable with current techniques. It is also noted that
explosive events are associated with freshly emerged
magnetic field and their Doppler velocities are roughly equal
to the Alfvén speed in the chromosphere. The suggestion is
that the events result from magnetic reconnection. Possible
evidence for bi-directional nature of the flows had been
noted in earlier spectroscopic data but the examples were
not very clear. This was so because the structure of the
flow could not be resolved due to limited time and space
coverage by previous space experiments.
The SUMER spectrograph has now made it possible to
observe the chromospheric network continuously over an
Mg/Ne FIP bias derived from four line ratios as a function of height above the limb (from Dwivedi et al.27 ).
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
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SPECIAL SECTION: SOLAR PHYSICS
extended period and to discern the spatial structure of the
flows associated with these explosive events. The observation that explosive events are bi-directional jets, provides
new evidence that they result from magnetic reconnection
above the solar surface. From simultaneous magnetic field
and ultraviolet measurements, it has already been
suggested that explosive events are often found on the
chromospheric network boundary and seem to be
associated with the cancellation of photospheric magnetic
fields. The network consists of curtains of very strong
magnetic flux tubes. All flux tubes are anchored by their
footpoints to the photosphere. The continual motions in the
photosphere mean that field lines of opposite-polarity are
naturally drawn together. If flux tubes with oppositepolarity field lines are pushed together, a current sheet
forms. In a finite resistivity plasma, a small region near the
neutral region may collapse and create a thin reconnection
region. From this region, plasma is ejected in both directions
along the field lines, with velocity of the order of the Alfvén
speed (the Alfvén speed depends on magnetic field
strength and plasma density). Electrical resistance to this
current flow liberates energy from the system to heat the
plasma, much in the same way as the filament of a light bulb
is heated.
Interpreting the evolution of the jets in the Si IV 1393 Å
line profile, Innes et al.29 have now shown that explosive
events have the bi-directional jets ejected from small sites
above the solar surface. In Figure 7, we show the explosive
events seen in the Si IV (λ1393) line. The SUMER
spectrograph slit shown in each section has a projected
length of 84000 km on the Sun and a width of 700 km. The
exposure time was 10 s each. The Doppler shift near bright
portions correspond to plasma motions with line-of-sight
velocities ± 150 km s –1. The structure of these plasma jets
evolves in the manner predicted by theoretical models of
magnetic reconnection. This lends support to the view that
magnetic reconnection is one of the fundamental processes
for accelerating plasma on the Sun. Such observations seem
to provide the best evidence to date for the existence of bidirectional outflow jets, a fundamental part of the standard
Figure 7. Explosive events seen in the Si IV (λ1393) line. The SUMER spectrograph slit shown in each section has a projected length of
84000 km on the Sun and a width of 700 km. The exposure time was 10 s each. The Doppler shift near bright portions (i.e. at chromospheric
network crossings of the slit) correspond to plasma motions with line-of-sight velocities ± 150 km s–1 .
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Sunspot transition region oscillations
The sunspot transition region between the chromosphere
and the corona oscillates. This may reveal crucial information about the structure and the physics of sunspots.
The first detailed study of the oscillations in the sunspot
transition region was presented by Gurman et al.30 mainly
based on observations of eight sunspots in the C IV (λ1548)
line with the UVSP (Ultraviolet Spectrometer and
Polarimeter) instrument on the Solar Maximum Mission.
They observed oscillations with periods of 129–173 s, with
no signs of shocks and suggested that the oscillations are
caused by upward-propagating acoustic waves.
Observations with instruments on SOHO have revived
the interest for the transition region oscillations. Intensity
oscillations in the 2 min range were recently reported31.
Based on observations with a 15 sec time resolution
Fludra32 has observed 3 min-intensity oscillations and
suggested that oscillations occur in sunspot plumes.
Combining observations of intensity and line-of-sight
velocity oscillations in three transition region lines,
Brynildsen et al.33 found that their observations of NOAA
8156 were compatible with the hypothesis that the 3 min
oscillations in sunspot transition region are caused by
linear, upward-propagating, progressive acoustical waves.
This appears to be in conflict with the upward propagating
shock waves observed in the sunspot chromosphere34.
Hence, either the wave amplitude decreases abruptly
between the chromosphere and the transition region or
considerable difference in the 3 min umbral oscillations exist
between different sunspots. Figure 8 shows sunspot
oscillations observed with SUMER in the O V (λ629) line. In
the
upper
panel,
the
position
of
the spectrometer slit analysed is shown with respect to
the sunspot of a TRACE image of NOAA 8487 on
18 March 1999. In the lower panels the intensity variations
and the line-of-sight velocity along the slit section are
displayed.
Figure 8. Sunspot oscillations observed with SUMER in the O V
(λ629) line. In the upper panel the position of the section of the
spectrometer slit analysed is shown with respect to the sunspot of a
TRACE image of NOAA 8487 on 18 March 1999. In the lower
panels the intensity variations and the line-of-sight velocity along
the slit section are displayed (courtesy, N. Brynildsen and the
TRACE team).
magnetic reconnection model. There exists a vast collection
of data obtained from the SUMER instrument on the
spacecraft SOHO and similar bi-directional jets are expected
to be seen in the solar atmosphere wherever reconnection
takes place. The present and future observations of this
kind are likely to provide new clues to a better
understanding of how the Sun’s magnetic energy feeds its
million-degree hot corona and the solar wind.
CURRENT SCIENCE, VOL. 77, NO. 11, 10 DECEMBER 1999
Concluding remarks
In conclusion, SUMER results on plasma density and
temperature structure, and evidence for hot ions and cool
electrons in coronal holes, the ‘blue’ Sun, observational
evidence for magnetic reconnection on the Sun, FIP effect
in the corona and sunspot transition region oscillations
presented in this article are crucial to a better understanding
of how the Sun’s magnetic energy feeds its million-degree
hot corona and the solar wind. And it will take some time to
digest SUMER data to decipher what tricks the Sun is
performing!
1. Wilhelm, K., Curdt, W., Marsch, E. et al., Solar Phys., 1995,
162, 189–231.
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2. Wilhelm, K., Lemaire, P., Curdt, W. et al., Solar Phys., 1997,
170, 75–104.
3. Lemaire, P., Wilhelm, K., Curdt, W. et al., Solar Phys., 1997,
170, 105–122.
4. Dwivedi, B. N., Space Sci. Rev., 1994, 65, 289–316.
5. Mason, H. E. and Monsignori-Fossi, B. C., Astron. Astrophys.
Rev., 1994, 6, 123–179.
6. Brown, J. C., Dwivedi, B. N., Almleaky, Y. M., Sweet, P. A.,
Astron. Astrophys., 1991, 249, 277–283.
7. McIntosh, S. W., Brown, J. C., Judge, P. G., Astron. Astrophys.,
1998, 333, 333–337.
8. Wilhelm, K., Lemaire, P., Dammasch, I. E., Hollandt, J.,
Schühle, U., Curdt, W., Kucera, T., Hassler, D. M. and Huber, M.
C. E., Astron. Astrophys., 1998, 334, 685–702.
9. Waldmeier, M., Die Sonnenkorona I (1951), Die Sonnenkorona
II (1957), Birkhäuser Verlag, Basel, Stuttgart.
10. Krieger, A. S., Timothy, A. F. and Roelof, E. C., Solar Phys.,
1973, 29, 505–525.
11. Zirker, J. B. (ed.), Coronal Holes and High Speed Wind Streams,
Colorado Associated Univ. Press, Boulder, 1977.
12. Dammasch, I. E., Hassler, D. M., Wilhelm, K. and Curdt, W., in
8th SOHO Workshop (ed. Kaldeich-Schürmann, B.), ESA, SP446, 1999, in press.
13. Parker, E. N., Astrophys. J., 1958, 128, 664–676.
14. Wilhelm, K., Marsch, E., Dwivedi, B. N., Hassler, D. M.,
Lemaire, P., Gabriel, A. H. and Huber, M. C. E., Astrophys. J.,
1998, 500, 1023–1038.
15. Kohl, J. L., Noci, G., Antonucci, E. et al., Solar Phys., 1997,
175, 613–644.
16. David, C., Gabriel, A. H., Bely-Dubau, F., Fludra, A., Lemaire, P.
and Wilhelm, K., Astron. Astrophys., 1998, 336, L90–L94.
17. Doschek, G. A., Feldman, U. and Bohlin, J. D., Astrophys. J.,
1976, 205, L177–L180.
18. Brekke, P., Hassler, D. M. and Wilhelm, K., Solar Phys., 1997,
175, 349–374.
19. Chae, J., Yun, H. S. and Poland, A. I., Astrophys. J. Suppl.,
1998, 114, 151–164.
20. Dammasch, I. E., Wilhelm, K., Curdt, W. and Hassler, D. M.,
1528
Astron. Astrophys., 1999, 346, 285–294.
21. Hassler, D. M., Dammasch, I. E., Lemaire, P., Brekke, P., Curdt,
W., Mason, H. E., Vial, J.-C. and Wilhelm, K., Science, 1998,
283, 810–813.
22. Widing, K. G. and Feldman, U., Astrophys. J., 1993, 416, 392–
397.
23. Sheeley, N. R., Astrophys. J., 1996, 469, 423–428.
24. Young, P. R. and Mason, H. E., Solar Phys., 1997, 175, 523–
539.
25. Dwivedi, B. N., Curdt, W. and Wilhelm, K., Astrophys. J., 1999,
517, 516–525.
26. Laming, J. M., Feldman, U., Drake, J. J. and Lemaire, P.,
Astrophys. J., 1999, 518, 926–936.
27. Dwivedi, B. N., Curdt, W. and Wilhelm, K., in 8th SOHO
Workshop (ed. Kaldeich-Schürmann, B.), ESA SP-446, 1999 (in
press).
28. Dere, K. P., Bartoe, J.-D. F. and Brueckner, G. E., J. Geophys.
Res., 1991, 96, 9399–9407.
29. Innes, D. E., Inhester, B., Axford, W. I. and Wilhelm, K.,
Nature, 1997, 386, 811–813.
30. Gurman, J. B., Leibacher, J. W., Shine, R. A., Woodgate, B. E.
and Henze, W., Astrophys. J., 1982, 253, 939–948.
31. Rendtel, J., Staude, J., Innes, D. E., Wilhelm, K. and Gurman,
J. B., in A Crossroads for European Solar and Heliospheric
Physics (ed. Harris, R. A.), ESA SP-417, 1998, pp. 277–280.
32. Fludra, A., Astron. Astrophys., 1999, 344, L75–L78.
33. Brynildsen, N., Leifsen, T., Kjeldseth-Moe, O., Maltby, P. and
Wilhelm, K., Astrophys. J., 1999, 511, L121–L124.
34. Bard, S. and Carlsson, M., in Fifth SOHO Workshop (ed. Wilson,
A.), ESA SP-404, 1997, pp. 189–191.
ACKNOWLEDGEMENTS. The SUMER project is financially
supported by DLR, CNES, NASA, and the ESA PRODEX Programme
(Swiss contribution). SUMER is a part of SOHO, the Solar and Heliospheric Observatory, of ESA and NASA. Anita Mohan is supported
by the DST, New Delhi, under the young scientist programme.
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