PH507
Multi-wavelength
Professor Michael Smith
1
PART 1: Measurement
LECTURE 1
Trignometric parallax: based on triangulation – need three parameters to fully
define any triangle e.g. two angles and one baseline.
BASELINE: 2 x earth-sun distance = 2 Astronomical Units (AU)
(The average distance from the earth to the sun is called the Astronomical Unit.)
We define the trigonometric parallax of the star as the angle  subtended, as
seen from the star, by the Earth’s orbit of radius 1 AU. If the star is at rest with
respect to the Sun, the parallax is half the maximum apparent annual angular
displacement of the star as seen from the Earth.
360
 57.3 degrees = 206265 arc seconds , approximately. There are 2
2
rad in a circle (360˚), so that 1 radian equals 57˚17´44.81” (206, 264.81”).
1 radian =
Independent distance unit is the light year:
c  t ( year )  9.47 1015 m
The light year is not used much by professional astronomers, who work instead
with a unit of similar size called the parsec, where 1 parsec = 1 pc = 206265
AU = 3.086 x 1016 m = 3.26 light years.

The measurement and interpretation of stellar parallaxes are a branch of
astrometry. Consider that the nearest star, Proxima/Alpha Centauri (Rigil
Kent), at a distance of 1.3 pc, has a parallax of only 0.76”; all other stars have
smaller parallaxes.
PH507
Multi-wavelength
Professor Michael Smith
2
Formula:
tan p 
1AU
d
or
d 
1
AU
p
where p is in radians
for small angles.
To convert to arcseconds:
2.063 105
d 
AU
p ''
or
d 
1
pc .
p"
Technological advances (including the Hubble Space Telescope) have improved
parallax accuracy to 0.001” within a few years. Before 1990, fewer than 10,000
stellar parallaxes had been measured (and only 500 known well), but there are
about 1012 stars in our Galaxy. Space observations made by the European Space
Agency with the Hipparcos mission (1989-1993) accurately determined the
parallaxes of many more stars. Though a poor orbit limited its usefulness,
Hipparcos was expected to achieve a precision of about 0.002”. It actually
achieved 0.001” for 118,000 stars. The method of trigonometric parallax is
important because it is our only direct distance technique for stars.
The ground-based trigonometric parallax of a star is determined by
photographing a given star field from a number (about 20) of selected points in
the Earth’s orbit. The comparison stars selected are distant background stars of
nearly the same apparent brightness as the star whose parallax is being
measured. Corrections are made for atmospheric refraction and dispersion and
for detectable motions of the background stars; any motion of the star relative
to the Sun is then extracted. What remains is the smaller annual parallactic
motion; it is recognised because it cycles annually.
Because a seeing resolution of 0.25” is considered exceptional (more typical it is
1”), it may seem strange that a stellar position can be determined to ±0.01” in
one measurement; this accuracy is possible because we are determining the
centre of the fuzzy stellar image.
PH507
Multi-wavelength
Professor Michael Smith
3
In 2011 – 2013, Gaia will be set into orbit with a Soyuz rocket (and SIM Space
Interferometric Mission from the US). It will be able to measure parallaxes
of 10 micro-arcseconds. It consists of a rotating frame holding three
telescopes. Some aims:
…….Accurate distances even to the Galactic centre, 8000 parsecs away.
……..Photometry: accurate magnitudes.
……..Planet quest
……..Reference frame from distant quasars (3C273 is 800 Mpc away)
In the meantime, to go further, we construct the COSMIC LADDER.
If we can estimate the luminosity of a star from other properties, they can be
used as STANDARD CANDLES.
2 LUMINOSITY.
The measured flux f is in units of W/m2 , the flow of energy per unit area. The
radiated power L, ignoring extinction, is given by:
f 
d2 
L
4d 2
L
4f
’
showing that a standard candle can yield the distance.
The Stellar Magnitude Scale
Because an interval of five magnitudes corresponds to a factor of 100 in
brightness, a one-magnitude difference corresponds to a factor of 1001/5 =
2.512. (This definition reflects the operation of human vision, which converts
equal ratios of actual intensity to equal intervals of perceived intensity. In other
words, the eye is a logarithmic detector). The magnitude scale has been
extended to positive magnitudes larger than +6.0 to include faint stars (the 5-m
telescope on Mount Palomar can reach to magnitude +23.5) and to negative
magnitudes for very bright objects (the star Sirius is magnitude -1.4). The
limiting magnitude of the Hubble Space Telescope is about +30.
PH507
Multi-wavelength
Professor Michael Smith
4
Astronomers find it convenient to work with logarithms to base 10 rather than
with exponents in making the conversions from brightness ratios to magnitudes
and vice versa. Consider two stars of magnitude m and n with respective
apparent brightnesses (fluxes) lm and ln. The ratio of their fluxes fn / fm
corresponds to the magnitude difference m - n. Because a one-magnitude
difference means a brightness ratio of 1001/5, (m - n) magnitudes refer to a ratio
of (1001/5)m-n = 100(m-n)/5, or
fn / fm = 100(m-n)/5
Taking the log10 of both sides (because log xa = a log x and log 10a = a log 10 =
a),
log (fn / fm) = [(m - n)/5] log 100 = 0.4(m - n)
or
m - n = 2.5 log (fn / fm)
This last equation defines the apparent magnitude; note that m > n when fn >
fm, that is: brighter objects have numerically smaller magnitudes. Also note that
when the brightnesses are those observed at the Earth, physically they are
fluxes. Apparent magnitude is the astronomically peculiar way of talking
about fluxes.
Here are a few worked examples:
(a) The apparent magnitude of the variable star RR Lyrae ranges from 7.1 to 7.8
- a magnitude amplitude of 0.7. To find the relative increase in brightness from
mini-mum to maximum, we use
log (fmax / fmin) = 0.4 x 0.7 = 0.28
so that
PH507
Multi-wavelength
Professor Michael Smith
5
fmax / fmin = 100.28 = 1.91
This star is almost twice as bright at maximum light than at minimum.
(b) A binary system consists of two stars a and b, with a brightness ratio of 2;
however, we see them unresolved as a point of magnitude +5.0. We would like
to find the magnitude of each star. The magnitude difference is
mb - ma = 2.5 log (fa / fb) = 2.5 log 2 = 0.75
Since we are dealing with brightness ratios, it is not right to put ma + mb = +5.0.
The sum of the luminosities (fa + fb) corresponds to a fifth-magnitude star.
Compare this to a 100-fold brighter star, of magnitude 0.0 and luminosity l0:
ma+ b - m0 = 2.5 log [l0 / (fa + fb)]
or
5.0 - 0.0 = 2.5 log 100 = 5.
But
fa = 2 fb, so that fb = (fa + fb)/3.
Therefore
(mb - m0) = 2.5 log (f0 / fb) = 2.5 log 300 = 2.5 x 2.477 = 6.19.
The magnitude of the fainter star is 6.19, and from our earlier result on the
magnitude difference, that of the brighter star is 5.44.
Only visual magnitudes can be converted to photometric
units. U, B, R or I magnitudes are not easily convertible
to luxes, lumens and friends, because of the different
wavelengths intervals used. The conversion factors would be
strongly dependent on e.g. the temperature of the blackbody
radiation or, more generally, the spectral distribution of
the radiation. The conversion factors between V magnitudes
and photometric units are only slightly dependent on the
spectral distribution of the radiation.
What units are used in radiometry/infrared astronomy?
Here we're not interested in the photometric response of some detector with a
well-known passband (e.g. the human eye, or some astronomical photometer).
Instead we want to know the strength of the radiation in absolute units: watts
etc. Thus we have:
Radiance, intensity or specific intensity
W m-2 ster-1 [Å-1]
SI unit
Irradiance/emittance, or flux
W m-2 [Å-1]
SI unit
Note the [A-1] within brackets. Fluxes and intensities can
be total (summed over all wavelengths) or monochromatic
PH507
Multi-wavelength
Professor Michael Smith
6
("per Angstrom Å" or "per nanometer").
In Radio/Infrared Astronomy, the unit Jansky is often used as a measure of
irradiance at a specific wavelength, and is the radio astronomer's equivalence to
stellar magnitudes. The Jansky is defined as: 1 Jansky = 10-26 W m-2 Hz-1
Absolute Magnitude and Distance Modulus
Hence, distance links fluxes and luminosities.
The luminosity of a star relates to its absolute magnitude, which is the
magnitude that would be observed if the star were placed at a distance of 10 pc
from the Sun. The inverse-square law of radiative flux links the flux f of a star
at a distance d to the luminosity F it would have it if were at a distance D = 10
pc:
F / f = (d / D)2 = (d / 10) 2.
If M corresponds to L and m corresponds to luminosity l, then
m - M = 2.5 log (F / f ) = 2.5 log (d/10)2 = 5 log (d / 10)
Expanding this expression, we have useful alternative forms: since
m1  m2  2.5 log
d1
 5 log d1  5 log d2 ,
d2
defining the absolute magnitude m2 = M at d2 = 10 pc, so m1 = m and d2 = d,
PH507
Multi-wavelength
Professor Michael Smith
7
m - M = 5 log d - 5
M = m + 5 - 5 log d
In terms of the parallax,
M = m + 5 + 5 log p”
Here d is in parsecs and p” is the parallax angle in arc seconds.
The quantity m - M is called the distance modulus, for it is directly related to
the star’s distance. In many applications, we refer only to the distance moduli
of different objects rather than converting back to distances in parsecs or lightyears.
Magnitudes at Different Wavelengths
PH507
Multi-wavelength
Professor Michael Smith
8
The kind of magnitude that we measure depends on how the light is filtered
anywhere along the path of the detector and on the response function of the
detector itself. So that problem comes down to how to define standard
magnitude systems.
PH507
Multi-wavelength
Professor Michael Smith
9
Because the flux of starlight varies with wavelength, the magnitude of a star
depends upon the wavelength at which we observe. Originally, photographic
plates were sensitive only to blue light, and the term photographic magnitude
(mpg) still refers to magnitudes centred around 420 nm (in the blue region of
the spectrum). Similarly, because the human eye is most sensitive to green and
yellow, visual magnitude (mv) or the photographic equivalent photo visual
magnitude (mpv) pertains to the wave-length region around 540 nm.
Today we can measure magnitudes in the infrared, as well as in the ultraviolet,
by using filters in conjunction with the wide spectral sensitivity of photoelectric
photometers. So systems of many different magnitudes (colour combinations)
are possible.
A commonly used wide-band magnitude system is the UBV system: a
combination of ultraviolet (U), blue (B), and visual (V) magnitudes, developed
by H. L. Johnson. These three bands are centred at 365, 440, and 550 nm; each
wavelength band is roughly 100 nm wide. In this system, apparent magnitudes are
denoted by B or V and the corresponding absolute magnitudes are sub-scripted:
MB or MV.
To be useful in measuring fluxes, the photometric system must be calibrated in
energy units for each of its bandpasses. This calibration turns out to be the
hardest part of the job. In general, it relies first on a set of standard stars that
define the magnitudes, for a particular filter set and detector; that is, these stars
define the standard magnitudes for the photometric system to the precision with
which they can be measured.
Infrared Windows
The UBV system has been extended into the red and infrared (in part because of
the development of new detectors, such as CCDs, sensitive to this region of the
spectrum). The extensions are not as well standardised as that for the Johnson
UBV system, but they tend to include R and I in the far red and J, H, K, L, and
PH507
Multi-wavelength
Professor Michael Smith
10
M in the infrared.
As well as measuring the properties of individual stars at different
wavelengths, observing at loner wavelengths, particularly in the infrared,
allows us to probe through clouds of small solid dust particles, as seen below
Infrared passbands which allow transmission (low absorption):
J Band: 1.3 microns
H Band: 1.6 microns
K band: 2.2 microns
L band 3.4 microns
M band 5 microns
N band 10.2 microns
Q band 21 microns
Bolometric magnitudes can be converted to total radiant energy flux: One star
of Mbol = 0 radiates 2.97 1028 Watts.
System is defined by Vega at 7.76 parsecs from the Sun with an apparent
magnitude defined as zero.
With Lbol = 50.1 Lsolar and Mbol = 0.58.
Sun: mbol = -26.8
Full moon: -12.6
Venus: -4.4
Sirius: -1.55
Brightest quasar: 12.8
For Vega: mb = mv = 0. mk = +0.02
Sun: Mb = 5.48, Mv = 4.83, Mk = 3.28
Colour Index: B-V, J-H, H-K are differences in magnitude….flux ratios.
But cooler, redder objects possess higher values.
Extinction
Interstellar Medium modifies the radiation. Dust particles with size of order of
the wavelength of the radiation.
Blue radiation is strongly scattered compared to red: blue reflection nebulae
and reddened stars.
Colour Excess:
measures the reddening.
E(B-V) = B-V - (B-V)o
Modified distance modulus:
m() = M() + 5 log d – 5 + A()
PH507
Multi-wavelength
Professor Michael Smith
11
where A () is the extinction due to both scattering and absorption, strongly
wavelength dependent. The optical depth is given by
exp(  ) 
I
Io
.
Therefore A() = 1.086 

The optical depth is

where N is the total column density of dust (m-2) between the star and the
observer and is the scattering/absorption cross-section (m2).
ISM Law related extinction to reddening:
Av / E(B-V) = 3.2 + - 0.2
Spectroscopic Parallax
Hertzsprung-Russell deduced the main-sequence stars for nearby objects,
relating their luminosity to their colour. Groups of distant stars should
also\line along the same main-sequence strip. However they appear very
dim,\of course due to their distance. On comparison of fluxes, we determine
the distance. This works out to about 100,000 pc, beyond which main-sequence
stars are too \dim.
Cepheids as Standard Candles: The Period-Luminosity Relationship
Cepheids show an important connection between period and luminosity: the
pulsation period of a Cepheid variable is directly related to its median
luminosity. This relationship was first discovered from a study of the variables
in the Magellanic Clouds, two small nearby companion galaxies to our Galaxy
that are visible in the night sky of the southern hemisphere. To a good
approximation, you can consider all stars in each Magellanic Cloud to be at the
same distance. Henrietta Leavitt, working at Harvard in 1912, found that the
brighter the median apparent magnitude (and so the luminosity, since the stars
are the same distance), the longer the period of the Cepheid variable. A linear
relationship was found.
Harlow Shapley recognised the importance of this period-luminosity (P-L)
relation-ship and attempted to find the zero point, for then a knowledge of the
period of Cepheid would immediately indicate its luminosity (absolute
magnitude).
This calibration was difficult to perform because of the relative scarcity of
Cepheids and their large distances. None are sufficiently near to allow a
trigonometric parallax to be determined, so Shapley had to depend upon the
PH507
Multi-wavelength
Professor Michael Smith
12
relatively inaccurate method of statistical parallaxes. His zero point was then
used to find the distances to many other galaxies.
Further work showed that there are two types of Cepheids, each with its own
separate, almost parallel P-L relationship.
The classical Cepheids are the more luminous, of Population I, and found in
spiral arms. Population II Cepheids, also known as W Virginis stars after their
prototype, are found in globular clusters and other Population II systems.
Classical Cepheids have periods ranging from one to 50 days (typically five to
ten days) and range from F6 to K2 in spectral class.
Population II Cepheids vary in period from two to 45 days (typically 12 to 20
days) and range from F2 to G6 in spectral class.
Population I and II Cepheids are both regular, or periodic, variables; their
change in luminosity with time follows a regular cycle.
PH507
Multi-wavelength
Professor Michael Smith
13
Cepheids are bright and distinct. They can be used to determine distances to
quite distant galaxies, to about 5 Mpc. HST stretched this to 18 Mpc (Virgo
cluster).
Tully-Fisher Relation
In a spiral galaxy, the centripetal force of gas and stars balances the
gravitational force:
mV2/R = GmM/R2.
If they have the same surface brightness ( L/R2 is constant) and the same massto-light ratio (M/L is constant), then
L ~ V4. So, provided we can measure V, certain galaxies can be used as standard
candles. (determine V through the 21 cm line of atomic hydrogen in the galaxy).
Type 1a Supernovae.
The peak light output from these supernovae is always about Mb = -19.33 +0.25. Therefore we can infer the distance from the inverse square law. Being so
bright , they act as standard cadles to large distances: to 1000 Mpc.
Why are they standard candles? White dwarfs I binary systems. Material from\
a companion red giant is dumped on the white dwarf surface until the WD
reaches a critical mass (Chandrasekhar mass) of 1.4 solar masses. Explosion
occurs with fixed rise and fall of luminosity.
Other methods: time delay of light rays due to gravitational lensing, cluster size
influences Compton scattering of CMB radiation and bremsstrahlung emission
PH507
Multi-wavelength
Professor Michael Smith
14
(X-rays). Combining, yields the size estimate (Sunyaev-Zeldovich effect). Or,
rotational properties of stars with starspots…….
New Method?
Reverse argument: knowing the Hubble constant is 72 km/s/Mpc, (WMAP
result), distances can be found directly from the redshift!
Questions
How do we scale the solar system?
How do we find the distance to gas clouds?
Lecture 4
Mass can be measured in two ways. We could count up the atoms, or count up
PH507
Multi-wavelength
Professor Michael Smith
15
the molecules and grains of dust and infer the number of atoms. This method
can be used if the object is optically thin and we have good tracer: a radiation
or scattering mechanism in which the number of photons is related to the
number of particles.
Otherwise, measuring the mass of an object relies upon its gravitational
influence….on nearby bodies or on itself (self-gravity).
Kepler's First Law: The orbit of each planet is an ellipse with the Sun at one
focus
p
b
F
r
S
a
C
Q
f q
S = Sun, F = other focus, p = planet.
r = HELIOCENTRIC DISTANCE.
f = TRUE ANOMALY
a = SEMI-MAJOR AXIS = mean heliocentric distance),
size of the orbit.
b = SEMI-MINOR AXIS.
e = ECCENTRICITY,
defines shape of orbit.
Ellipse:
SP + PF = 2a
e = CS / a
Therefore
which defines the
(1)
(2)
b2 = a2(1-e2)
(3)
•When CS = 0, e = 0, b = a, the orbit is a circle.
When CS = , e = 1,
the orbit is a parabola.
• q = PERIHELION DISTANCE = a - CS = a – ae
q = a(1-e)
(4)
• Q = APHELION DISTANCE = a + CS = a + ae
Q = a(1+e)
(5)
Second Law: For any planet, the radius vector sweeps out equal areas in
equal times
• Time interval t for planet to travel from p to p1 is the same as time taken for
PH507
Multiwavelength
16
planet to get from p2 to p3. Shaded areas are equal.
• Let the time interval t be very small. Then the arc from p to p1 can be
regarded as a straight line and the area swept out is the area of the triangle
S p p1. If f1 is the angle to p1, and f is the angle to p:
p
1
p2
p3
r1
p
r
f
S
i.e
Area = 1/2 r r1 Sin (f1-f).
Since t is very small, r ~ r1 and Sin (f1-f) ~ (f1-f) = f
Area = 1/2 r2 f
The rate this area is swept out is constant according to Kepler's second
law, so
r2 df/dt = h
(6)
where h, a constant, is twice the rate of description of area by the radius
vector. It is the orbital angular momentum (per unit mass.
The total area of the ellipse is πab which is swept out in the orbital period
P, so using eqn (7)
2ab/P = h.
The average angular rate of motion is n = 2/P, so
n a2(1-e2)1/2 = h
(7)
Kepler’s Third Law
Kepler's third law took another ten years to develop after the first two. This law
relates the period a planet takes to travel around the sun to its average distance and
the Sun. This is sometimes called the semi major axis of an elliptical orbit.
P 2 = ka3
PH507
Multiwavelength
17
where P is the period and a is the average distance from the Sun.
Kepler’s Third Law follows from the central inverse square nature of the law of
gravitation. First look at Newton's law of gravitation - stated
mathematically this is
F
Gm1 m2
r2
Newton actually found that the focus of the elliptical orbits for two bodies of masses m1 and
m2 is at the centre of mass. The centripetal forces of a circular orbit are
F1
r1
v2
X
Centre of M ass
m1
m2
v1
F2
r2
2
F1
and
m1 v1
4 2 m1 r1


r1
P2
2
F2
m2 v2
4 2 m2 r2


r2
P2
where
v
2r
P
and since they are orbiting each other (Newton’s 2nd law)
r1
m2

r2
m1
Let's call the separation a = r1 + r2. Then;
PH507
a  r1 
Multiwavelength
18
m1 r1
 m1

 r1 
 1 and multiplying both sides by m2 , am2  m1 r1  m2 r1
m2
m2
r1 
or, solving for r1 ,
am2
m1  m2 
Now, since we know that the mutual gravitational force;
Gm1 m2
Fgrav  F1  F2 
2
a
then substituting for r1,
3
a 
2
G m1  m2  P
4
2
Solving for P:
3
P  2
a
G M1  M2 
Third Law is therefore: The cubes of the semi-major axes of the planetary
orbits are proportional to the squares of the planets' periods of
revolution.
Example
Europa, one of the Jovian moons, orbits at a distance of 671,000 km from the centre
of Jupiter, and has an orbital period of 3.55 days. Assuming that the mass of Jupiter
is very much greater than that of Europa, use Kepler's third law to estimate the
mass of Jupiter.
Using Kepler's third law:
m jupiter  meuropa
4 2 a 3

GP 2
The semi-major axis, a = 6.71 x 105 km = 6.71 x 108 m, and
the period, P = 3.55 x 3600 x 24 = 3.07 x 105 seconds
Thus:
m jupiter  meuropa 
c
4 2 6.71  108
h
3
 19
.  1027 kg
c6.67  10 hc3.07  10 h
 11
5 2
and since mjupiter >> meuropa, then mjupiter ~ 1.9 x 1027 kg.
Summary: Measuring the mass of a planet
• Kepler’s third law gives G(M+m) =  a3/P2
Since M >> m
for all planets, it isn't possible to make precise enough
PH507
Multiwavelength
19
determinations of P and a to determine the masses m of the planets.
However, if satellites of planets are observed, then Kepler's law can be used.
• Let mp = mass of planet
Then:
ms = mass of satellite
Ps = orbital period of satellite
as = semi-major axis of satellite's orbit
about the planet.
G(mp+ms) = 42 as3/Ps2
If the mass of the satellite is small compared with the mass of the planet then
mp = 42 as3/(G Ps2)
• All the major planets have satellites except Mercury and Venus. Their
masses were determined from orbital perturbations on other bodies and
later, more accurately from changes in the orbits of spacecraft.
So: we can determine the masses of massive objects if we can detect and follow the
motion of very low mass satellites. That doesn’t lead very far. How can we determine
the masses of distant stars and exoplanets?
BASIC STELLAR PROPERTIES - BINARY STARS
• For solar type stars, single:double:triple:quadruple system ratios are
45:46:8:1.
• Binary nature of stars deduced in a number of ways:
VISUAL BINARIES:
- Resolvable, generally nearby stars (parallax likely to be available)
- Relative orbital motion detectable over a number of years
ASTROMETRIC BINARY: only one component detected
SPECTROSCOPIC BINARIES:
- Unresolved
- Periodic oscillations of spectral lines (due to Doppler shift)
- In some cases only one spectrum seen
SPECTRUM BINARY: 2 sets of lines but no apparent orbital motion but
spectrum is clearly combined from stars of differing spectral class.
ECLIPSING BINARY:
- Unresolved
- Stars are orbiting in plane close to line of sight giving eclipses
observable as a change in the combined brightness with time (‘’light
curves).
Some stars may be a combination of these.
Visual Binaries
• Angular separation ≥ 0.5 arcsec (close to Sun, long orbital periods - years) –
PH507
Multiwavelength
20
example Sirius:
• Observations:
Relative positions:
 = angular
 = position
Absolute positions: Harder to measure orbits of more massive star A and
separation less massive star B about centre
of mass C which has proper motion µ.
Declination
N
M otion of centre of mass
= proper motion µ
Secondary

E

Right Ascension
B
Primary
C
A
NB parallax and aberration must also be accounted for.
• RELATIVE ORBITS:
- TRUE orbit:
q = peri-astron distance (arcsec or km)
Q = apo-astron distance (arcsec or km)
a = semi-major axis (arcsec or km)
a = (q + Q)/2
- APPARENT orbits are projected on the celestial sphere
Inclination i to plane of sky defines relation between true orbit and apparent
orbit. If i≠0° then the centre of mass (e.g. primary) is not at the focus of the
elliptical orbit.
Measurement of the displacement of the primary gives inclination and true
semi-major axis in arcseconds a".
i
Incline by 45°
i
Apparent orbit
True orbit
• If the parallax p in arcseconds is observable then a can be derived from a".
PH507
Multiwavelength
21
Earth
B
radius
of Earth's
orbit
a
a"
Sun
For i=0°
p
A
r = distance of binary star
a = 1 AU . a"/p"
(In general correction for i≠0 required).
Now lets go back to Kepler’s Law …
• From Kepler's Law, the Period P is given by
2 3
4 a
P =
G (mA + mB)
2
(26)
For the Earth-Sun system P=1 year, a=1 A.U., mA+mB~msun so 4π2/G = 1
3
2
P =
a
(mA + mB)
P in years, a in AU, mA,mB in solar masses.
From (25) and (26),
a" 3 1
mA + mB = ( )
2
p
P
Sum of masses is determined
• ABSOLUTE ORBITS:
d
c
rA
*
B
rB
(27)
B
f
q
e
A
Q
A
Semi-major axes aA = (c+e)/2 Minimum separation = q = d + e
aB = (d+f)/2
Maximum separation = Q = c + f
So aA + aB = (c+d+e+f)/2 = (q + Q)/2 = a
a = aA + a B
(28)
(and clearly r =rA + rB)
From the definition of centre of mass, mA rA = mB rB ( mA aA = mB aB)
mA/mB = aB/aA = rB/rA
(29)
PH507
Multiwavelength
22
So from Kepler’s Third Law, which gives the sum of the masses, and Equation
(29) above, we get the ratio of masses, ==> mA, mB. Therefore, with both, we
can solve for the individual masses of the two stars.
Spectroscopic Binaries
• Orbital period relatively short (hours - months) and i≠0°.
• Doppler shift of spectral lines by component of orbital velocity in line of sight
(nominal position is radial velocity of system):
PH507
Multiwavelength
23
wavelength
wavelength
Time
Time
2 Stars observable
1 Star observable
• Data plotted as RADIAL VELOCITY CURVE:
recession
+
v
(km s-1)
recession
+
0
time
approach
4
1
3
2
4
1
3
4
2
RELATIVE
approach
3
2
time
-
ABSOLUTE
Relative Orbit
Observer
v
(km s-1) 0
Relative radial velocity curve
2• If the orbit is tilted to the line of
sight (i<90°), the shape is unchanged
1 reduced by a factor
but velocities are
v
1 sin 3i.
• Take a circular orbit with i=90°
a = rA + r4B
v = v A + vB
2
Orbital velocities:
vA = 2π rA / P 1
v
3
1
vB = 2π rB / P
4
Since mA rA = mB rB
2
v
1
1 A = vB/vA (31)
mA/mB = rB/r
rB
3
4
r
vA
=
1
• Shape of radial velocity curves
depends on orbital eccentricity
and orientation.
v
rA
v
B
PH507
Astrophysics
24
• In general, measured velocities are vB sin i and vA sin i, so sin i terms cancel.
• From Kepler's law
mA + mB = a3/P2 (in solar units).
Observed quantities: vA sin i => rA sin i
} a sin i
vB sin i => rB sin i
So can only deduce (mA + mB) sin3 i = (a sin i)3/P2
(32)
For a spectroscopic binary, only lower limits to each mass can be derived,
unless i is known independently.
Eclipsing Binaries
• Since stars eclipse i ~ 90°
4, 4' FOURTH CONTACT
4' 3'
2' 1'
v
1 2
3 4
Observer in plane
• For a circular orbit:
1, 1' FIRST CONTACT
2, 2' SECOND CONTACT
3, 3' THIRD CONTACT
PH507
Multiwavelength
25
• Variation in brightness with time is LIGHTCURVE.
• Timing of events gives information on sizes of stars and orbital elements.
• Shape of events gives information on properties of stars and relative
temperatures.
Case 1
Smaller star is hotter
Case 2
Larger star is hotter
F
or
magnitude
Secondary minimum
Primary minimum
time
Case 1 t'1
Case 2 t 1
t'2
t2
t'3
t3
t'
4
t4
t1
t'1
t2
t'2
•
t3
t'3
t4
t'4
If orbits are circular:
minima are half a period apart; eclipses are of same duration.
Assymetrical and/or unevenly spaced minima indicate eccentricity and
orientation of orbit.
• For a circular orbit:
t1 t2
t3 t 4
Distance = velocity x time
2RS = v (t2 - t1) (33)
2RL
and
2RS + 2RL = v (t4 - t1) => 2RL =
v(t4 - t2)
(34)
2RS
and ratio of radii
RS/RL = (t2 - t1) / (t4 - t2)
• Lightcurves are also affected by:
Non-total eclises
No flat minimum
Limb darkening
(non-uniform
brightness)
"rounds off"
eclipses
Ellipsoidal stars
(due to
proximity)
"rounds off"
maxima
Reflection effect
(if one star is
very bright)
minima are
(35)
PH507
Multiwavelength
26
Eclipsing-Spectroscopic Binaries
• For eclipsing binaries i ≥ 70°
(sin3i > 0.9)
• If stars are spectroscopic binaries then radial velocities are known.
So from eqns (31) and (32) masses are derived,
from eqns (33) and (34) radii are derived,
from (36) ratio of temperatures is derived
Examine spectra and lightcurve to determine which radius corresponds with which
mass and temperature:
+
v
-
B
From radial velocity curve
star A is more massive
A
Initially A is approaching (blue shift)
so first eclipse is A in front of B
Since first eclipse is primary eclipse
B is hotter than A
F
If 2 sets of lines are seen then B is larger
If 1 set of lines is seen then A is larger
time
• Since Luminosity L = 4 R2 T4, the ratio of Luminosities is derived from
(TO BE DISCUSSED LATER IN COURSE!)
LA
LB
=
Summary
Type
Visual
Spectroscopic
Eclipsing
Eclipsing/
LA/LB
spectroscopic
RA
RB
2
TA
4
TB
Observed
p, motion on sky
Apparent magnitudes
Derived
a, e, i, mA, mB
LA, LB
velocity curves
lightcurves
light + velocity curves
MA/MB, (MA+MB)sin3i, a sin i
e, i, RS/RL
MA, MB, RA, RB, TA/TB, a, e, i,
distance
LA, LB, TA, T
75 PH507
Multiwavelength
27
Lecture 6: Extrasolar Planets
134 other stars are now known to possess planetary systems. 157 planets have
been discovered. Although none of the stars has been directly imaged, the
effects of the gravity tugging at the stars, as well as the way that gravitation
affects can affect material close to the stars, has been clearly seen.
Disc of material around the star Beta Pictoris – the image of the bright central
star has been artificially blocked out by astronomers using a ‘Coronograph’
• How can we discover extrasolar planets?
• Characteristics of the exoplanet population
• Planet formation
• Explaining the properties of exoplanets
Rapidly developing subject - first extrasolar planet around an ordinary star only
discovered in 1995 by Mayor & Queloz. Observations thought to be secure, but theory
still preliminary...
Definition of a planet
Simplest definition is based solely on mass
• Stars: burn hydrogen (M > 0.075 Msun)
• Brown dwarfs: burn deuterium
• Planets: do not burn deuterium (M < 0.013 Msun)
Deuterium burning limit occurs at around 13 Jupiter masses (1 MJ = 1.9 x 1027 kg ~
0.001 Msun It is important to realise that for young objects, there is no large change in
properties at the deuterium burning limit. ALL young stars / brown dwarfs / planets
liberate gravitational potential energy as they contract
Types of planet
Giant planets (gas giants, `massive’ planets)
• Solar System prototypes: Jupiter, Saturn, Uranus...
• Substantial gaseous envelopes
• Masses of the order of Jupiter mass
• In the Solar System, NOT same composition as Sun
• Presence of gas implies formation while gas was still prevelant
Terrestrial planets
• Prototypes: Earth, Venus, Mars
• Primarily composed of rocks
• In the Solar System (ONLY) orbital radii less than giant planets
75 PH507
Multiwavelength
28
Much more massive terrestrial planets could exist (>10 Earth masses), though none are
present in the Solar System. The Solar system also has asteroids, comets, planetary
satellites and rings - we won’t discuss those in this course.
Detecting extrasolar planets
(1) Direct imaging - difficult due to enormous star / planet flux ratio
(2) Radial velocity
• Observable: line of sight velocity of star orbiting centre of mass of star - planet
binary system
• Most successful method so far - all detections to date
(3) Astrometry
• Observable: stellar motion in plane of sky
• Very promising future method: Keck interferometer, GAIA, SIM
(4) Transits
• Observable: tiny drop in stellar flux as planet transits stellar disc
• Requires favourable orbital inclination
• Jupiter mass exoplanet observed from ground HD209458b
• Earth mass planets detectable from space (Kepler (2007 launch. NASA
Discovery mission), Eddington)
(5) Gravitational lensing
• Observable: light curve of a background star lensed by the gravitational
influence of a foreground star. The light curve shape is sensitive to whether the
lensing star is a single star or a binary (star + planet is a special case of the
binary)
• Rare - requires monitoring millions of background stars, and also unrepeatable
• Some sensitivity to Earth mass planets
Each method has different sensitivity to planets at various orbital radii - complete
census of planets requires use of several different techniques
75 PH507
Multiwavelength
29
Planet detection method : Radial velocity technique
A planet in a circular orbit around star with semi-major axis a
Assume that the star and planet both rotate around the centre of mass with an angular
velocity:
Using a1 M* = a2 mp and a = a1 + a2, then the stellar speed (v* = a ) in an inertial
frame is:
(assuming mp << M*). i.e. the stellar orbital speed is small.
For a circular orbit, observe a sin-wave variation of the stellar radial velocity, with an
amplitude that depends upon the inclination of the orbit to the line of sight:
Hence, measurement of the radial velocity amplitude produces a constraint on:
mp sin(i)
(assuming stellar mass is well-known, as it will be since to measure radial velocity we
need exceptionally high S/N spectra of the star).
Observable is a measure of mp sin(i).
-> given vobs, we can obtain a lower limit to the planetary mass
In the absence of other constraints on the inclination, radial velocity searches
provide lower limits on planetary masses
Magnitude of radial velocity:
Sun due to Jupiter:
Sun due to Earth:
i.e. extremely small -
12.5 m/s
0.1 m/s
10 m/s is Olympic 100m running pace
75 PH507
Multiwavelength
30
Spectrograph with a resolving power of 105 will have a pixel scale ~ 10-5 c ~ few km/s
Therefore, specialized techniques that can measure radial velocity shifts of ~10-3 of
a pixel stably over many years are required
High sensitivity to small radial velocity shifts is achieved by:
• comparing high S/N = 200 - 500 spectra with template stellar spectra
• using a large number of lines in the spectrum to allow shifts of much less than
one pixel to be determined.
Absolute wavelength calibration and stability over long timescales is achieved by:
• passing stellar light through a cell containing iodine, imprinting large number
of additional lines of known wavelength into the spectrum
• with the calibrating data suffering identical instrumental distortions as the data
Error sources:
(1) Theoretical: photon noise limit
• flux in a pixel that receives N photons uncertain by ~ N1/2
• implies absolute limit to measurement of radial velocity
• depends upon spectral type - more lines improve signal
• around 1 m/s for a G-type main sequence star with spectrum recorded at
S/N=200
• practically, S/N=200 can be achieved for V=8 stars on a 3m class
telescope in survey mode
(2) Practical:
• stellar activity - young or otherwise active stars are not stable at the m/s
level and cannot be monitored with this technique
• remaining systematic errors in the observations
Currently, the best observations achieve:
 ~ 3 m/s
...in a single measurement. Thought that this error can be reduced to around 1 m/s with
further refinements, but not substantially further. The very highest Doppler precisions
of 1 m/s are capable\of detecting planets down to about 5 earth masses.
Radial velocity monitoring detects massive planets, especially those at small a, but
is not sensitive enough to detect Earth-like planets at ~ 1 AU.
Examples of radial velocity data
75 PH507
Multiwavelength
31
51 Peg b was the first known exoplanet with a 4 day, circular orbit: a hot Jupiter, \lying
close to the central star.
Example of a planet with an eccentric orbit: e=0.67
Summary: observables
(1) Planet mass, up to an uncertainty from the normally unknown inclination of
the orbit. Measure mp sin(i)
(2) Orbital period -> radius of the orbit given the stellar mass
(3) Eccentricity of the orbit
Summary: selection function
Need to observe full orbit of the planet: zero sensitivity to planets with P > Psurvey
75 PH507
Multiwavelength
32
For P < Psurvey, minimum mass planet detectable is one that produces a radial velocity
signature of a few times the sensitivity of the experiment (this is a practical detection
threshold)
Which planets are detectable?
1
m p sin i  a 2
Current limits:
• Maximum a ~ 3.5 AU (ie orbital period ~ 7 years)
• Minimum mass ~ 0.5 Jupiter masses at 1 AU, scaling with square root of semimajor axis
• No strong selection bias in favour / against detecting planets with different
eccentricities
Of the first 100 stars found to harbor planets, more than 30 stars host a Jupiter-sized
world in an orbit smaller than Mercury's, whizzing around its star in a matter of days.
Planet formation is a contest, where a growing planet must fight for survival lest it be
swallowed by the star that initially nurtured it.
Planet detection method : Astrometry
Conceptually identical to radial velocity searches. Light from a planet-star binary is
dominated by star. Measure stellar motion in the plane of the sky due to presence
of orbiting planet. Must account for parallax and proper motion of star.
Magnitude of effect: amplitude of stellar wobble (half peak displacement) for an orbit in
the plane of the sky is
 mp 
  a
a1  
 M* 
In terms of the angle:
 m p  a 
 
  
 M *  d 
for a star at distance d. Note we have again used mp << M*
75 PH507
Multiwavelength
33
Writing the mass ratio q = mp / M*, this gives:
Note:
• Units here are milliarcseconds - very small effect
• Different dependence on a than radial velocity method - astrometric planet
searches are more sensitive at large a
• Explicit dependence on d (radial velocity measurements also less sensitive for
distant stars due to lower S/N spectra)
• Detection of planets at large orbital radii still requires a search time comparable
to the orbital period
Detection threshold as function of semi-major axis
• Lack of units deliberate! Astrometric detection not yet achieved
• As with radial velocity, dependence on orbital inclination, eccentricity
• Very promising future: Keck interferometer, Space Interferometry Mission
(SIM), ESA mission GAIA, and others
• Planned astrometric errors at the ~10 microarcsecond level – good enough to
detect planets of a few Earth masses at 1 AU around nearby stars
Lecture 7:
Planet detection method : Transits
Simplest method: look for drop in stellar flux due to a planet
transiting across the stellar disc
Needs luck or wide-area surveys - transits only occur if the orbit is almost edge-on
75 PH507
Multiwavelength
34
For a planet with radius rp << R*, probability of transits is:
Close-in planets are more likely to be detected. P = 0.5 % at 1AU, P = 0.1 % at the
orbital radius of Jupiter
What can we measure from the light curve?
(1)
Depth of transit = fraction of stellar light blocked
This is a measure of planetary radius!
In practice, isolated planets with masses between ~ 0.1 MJ and 10 MJ, where MJ
is the mass of Jupiter, should have almost the same radii (i.e. a flat mass-radius
relation).
-> Giant extrasolar planets transiting solar-type stars produce transits
with a depth of around 1%.
Close-in planets are strongly irradiated, so their radii can be (detectably) larger.
But this heating-expansion effect is not generally observed for short-period
planets.
(2)
(3)
(4)
Duration of transit plus duration of ingress, gives measure of the orbital radius
and inclination
Bottom of light curve is not actually flat, providing a measure of stellar limbdarkening
Deviations from profile expected from a perfectly opaque disc could provide
evidence for satellites, rings etc
Photometry at better than 1% precision is possible (not easy!) from the ground.
HST reached a photometric precision of 0.0001.
Potential for efficient searches for close-in giant planets
Transit depth for an Earth-like planet is:
75 PH507
Multiwavelength
35
Photometric precision of ~ 10-5 seems achievable from space
May provide first detection of habitable Earth-like planets
NASA’s Kepler mission, ESA version Eddington
A reflected light signature must also exist, modulated on the orbital period,
even for non-transiting planets. No detections yet.
Planet detection method : Gravitational microlensing
Light is deflected by gravitational field of stars, compact objects, clusters of galaxies,
large-scale structure etc
Simplest case to consider: a point mass M (the lens) lies along the line of sight to a
more distant source
Define:
• Observer-lens distance
• Observer-source distance
• Lens-source distance
Azimuthal symmetry -> light from the source appears as a ring
...with radius R0 - the Einstein ring radius - in the lens plane
Gravitational lensing conserves surface brightness, so the distortion of the image of
the source across a larger area of sky implies magnification.
The Einstein ring radius is given by:
Dl
Ds
Dls
75 PH507
Multiwavelength
36
Suppose now that the lens is moving with a velocity v. At time t, the apparent distance
(in the absence of lensing) in the lens plane between the source and lens is r0.
Defining u = r0 / R0, the amplification is:
Note: for u > 0, there is no symmetry, so the pattern of images is not a ring and is
generally complicated. In microlensing we normally only observe the magnification A,
so we ignore this complication...
Notes:
(1) The peak amplification depends upon the impact parameter, small impact
parameter implies a large amplification of the flux from the source star
(2) For u = 0, apparently infinite magnification! In reality, finite size of source
limits the peak amplification
(3) Geometric effect: affects all wavelengths equally
(4) Rule of thumb: significant magnification requires an impact parameter
smaller than the Einstein ring radius
(5) Characteristic timescale is the time required to cross the Einstein ring radius:
Optical depth to microlensing
Define the optical depth to microlensing as:
This is just the integral of the area of the Einstein ring along the line of sight to the
source. For a uniform density of lenses, can easily show that the maximum contribution
comes from lenses halfway to the source.
75 PH507
Multiwavelength
37
Several groups have monitored stars in the Galactic bulge and the Magellanic clouds to
detect lensing of these stars by foreground objects (MACHO, Eros, OGLE projects).
Original motivation for these projects was to search for dark matter in the form of
compact objects in the halo.
Timescales for sources in the Galactic bulge, lenses ~ halfway along the line of sight:
• Solar mass star ~ 1 month
• Jupiter mass planet ~ 1 day
• Earth mass planet ~ 1 hour
The dependence on M1/2 means that all these timescales are observationally feasible.
However, lensing is a very rare event, all of the projects monitor millions of source
stars to detect a handful of lensing events.
Lensing by a single star
Lensing by a star and a planet
75 PH507
Multiwavelength
38
Planet search strategy:
• Monitor known lensing events in realtime with dense, high precision
photometry from several sites
• Look for deviations from single star lightcurve due to planets
• Timescales ~ a day for Jupiter mass planets, ~ hour for Earths
• Most sensitive to planets at a ~ R0, the Einstein ring radius
• Around 3-5 AU for typical parameters
Sensitivity to planets
Complementary to other methods:
Actual sensitivity is hard to evaluate: depends upon frequency of photometric
monitoring (high frequency needed for lower masses), accuracy of photometry (planets
produce weak deviations more often than strong ones)
Very roughly: observations with percent level accuracy, several times per night, detect
Jupiter, if present, with 10% efficiency
75 PH507
Multiwavelength
RV, Doppler technique (v = 3m/s)
Astrometry: angular oscillation
Photometry: transits - close-in planets
Microlensing:
39
75 PH507
Multiwavelength
40
Direct detection! Photometric :
2005 image of 2M1207 (blue) and its planetary companion, 2M1207b, one of the
first exoplanets to be directly imaged, in this case from the Very Large
Telescope array in Chile
Spectroscopic? The starlight scattered from the planet can be distinguished
from the direct starlight because the scattered light is Doppler shifted by virtue
of the close-in planet's relatively fast orbital velocity (~ 150 km/sec).
Superimposed on the pattern given by the planet's albedo changing slowly with
wavelength, the spectrum of the planet's light will retain the same pattern of
photospheric absorption lines as in the direct starlight.
Lecture 8 The extrasolar planet population
Radial Velocity Method (Doppler technique, gravitational wobble)
• 156 exoplanets hosted by134 stars discovered, with masses M.sin(i) as low as 6
Earth masses. Generally:~ 0.06 MJ and 10MJ…
and orbital radii from 0.02 AU to 6 AU.
• Planet fraction among ~ solar-type stars exceeds 7%
• Most are beyond 1 AU
75 PH507
Multiwavelength
41
• Around 1% of stars have hot Jupiters - massive planets at orbital radii a < 0.1
AU
• Four very low mass planets have been detected ….20 earth masses.
• Planet occurrence rises rapidly with stellar metallicity
• Multiple planets are common, often in resonant orbits
Microlensing: two strong detections, low detection rate imply upper limit of ~1/3 on the
fraction of lensing stars (~ 0.3 Msun) with Jupiter mass planets at radii to which lensing
is most sensitive (1.5 - 4 AU)
Transits: 7 known planets (5 found with OGLE photometrically – dimming).
Interesting upper limit from non-detection of transits in globular cluster 47 Tuc
Transits + Doppler yields mass and size, hence the density of the planet: 0.2 – 1.4
gm/cm3 : mainly gaseous. In addition, sodium and nitrogen found in their atmospheres.
Direct Imaging: reports of detections with HST and VLT.
Eccentricity: • Except at very small radii, typical planet orbit has significant
eccentricity
75 PH507
Multiwavelength
42
Eccentricity:
Eccentricity vs planet mass
75 PH507
Multiwavelength
43
75 PH507
Multiwavelength
44
Nothing very striking in these plots:
• Accessible region of mp - a space is fully occupied by detected planets
• Ignoring the hot Jupiters, no obvious correlation between planet mass and
eccentricity...
Results from radial velocity searches
(1) Massive planets exist at small orbital radii. Closest in planet is at a = 0.035 AU,
cf Mercury at ~ 0.4 AU. Less than 10 Solar radii.
(2) Hot Jupiters have close to circular orbits. All detected planets with semi-major
axis < 0.07 AU have low e. This is similar to binary stars, and is likely due to
tidal circularization.
(3) Remaining planets have a wide scatter in e, including some planets with large e.
Note that the distance of closest approach is a(1-e), and that the effect of
tidal torques scales as separation d-6. The very eccentric planet around
HD80606 (a = 0.438 AU, e = 0.93, a(1-e) = 0.03 AU) may pose some
problems for tidal circularization theory.
75 PH507
Multiwavelength
45
Account for this by considering only planets with masses large enough to be detectable
at any a < 2.7 AU.
-> dN / dlog(a) rises steeply with orbital radius
Implies that the currently detected planet fraction ~7% is likely to be a substantial
underestimate of the actual fraction of stars with massive planets.
Models suggest 15-25% of solar-type stars may have planets with masses 0.2 MJ < mp <
10 MJ.
Strong selection effect in favour of detecting planets at small orbital radii, arising from:
- lower mass planets can be detected there
- mass function rises to smaller masses
Observed mass function increases to smaller Mp:
75 PH507
Multiwavelength
46
Note: the brown dwarf desert!
Constraint from monitoring of 43 microlensing events. Typically, the lenses are
low mass stars.
At most 1/3 of 0.3 Solar mass stars have Jupiter mass planets between 1.5
AU and 4 AU.
Currently consistent with the numbers seen in radial velocity searches - not yet
known whether there is a difference in the planet fraction between 0.3 - 1 Solar
mass stars.
Transit lightcurve from Brown et al. (2001)
75 PH507
Multiwavelength
47
Consistent with expectations - the probability of a transiting system is ~10%.
Measured planetary radius rp = 1.35 J:
• Proves we are dealing with a gas giant.
• Somewhat larger than models for isolated (non-irradiated) planets effect of environment on structure.
• In detail, suggests planet reached its current orbit within a few x 10 Myr
after its formation.
Precision of photometry with HST / STIS impressive...
Metallicity distribution of stars with and without planets
Left plot: metallicity of stars with planets (shaded histogram) compared to a sample of
stars with no evidence for planets (open histogram) (data from Santos, Israelian &
Mayor, 2001)
Host star metallicity
Planets are preferentially found around stars with enhanced metal abundance.
75 PH507
Multiwavelength
48
Cause or effect? High metal abundance could:
(a) Reflect a higher abundance in the material which formed the star +
protoplanetary disc, making planet formation more likely.
(b) Result from the star swallowing planets or planetesimals subsequent to
planets forming. If the convection zone is fairly shallow, this can apparently
enrich the star with metals even if the primordial material had Solar abundance.
Detailed pattern of abundances can distinguish these possibilities, but results currently
still controversial.
Lack of transits in 47 Tuc
A long HST observation monitored ~34,000 stars in the globular cluster 47 Tuc looking
for planetary transits.
Locally: 1% of stars have hot Jupiters
~ 10% of those show transits
 Expect 10 -3 x 34,000 ~ few tens of planets
None were detected. Possible explanations:
• Low metallicity in cluster prevented planet formation
• Cluster environment destroyed discs before planets formed
• Stellar fly-bys ejected planets from bound orbits
All of these seem plausible - make different predictions for other clusters.
Lecture 9: Radiation processes
Blackbody Radiation
Planck’s Radiation Law



2h 3  1
I( )d   2  h 

 c   kT



 e  1 


where I(v)dv is the intensity (J/m2 . s . sr) of radiation from a blackbody at temperature
T in the frequency range between v and v + dv, h is Planck's constant, c is the
speed of light, and k is Boltzmann's constant. Note the exponential in the denominator.
75 PH507
Multiwavelength
49
Because the frequency v and wavelength of electromagnetic radiation are related by
v = c, we may also express Planck's formula in terms of the intensity emitted per
unit wavelength interval:
This is illustrated for several values of T:
Note that both I() and I(v) increase as the blackbody temperature increases - the
blackbody becomes brighter. This effect is easily interpreted when we note that I(v)∆v
is directly proportional to the number of photons emitted per second near the energy
hv. The Planck function is special enough so that its given its own symbol, B() or B(v),
for intensity.
Wien’s Law
A blackbody emits at a peak intensity that shifts to shorter wavelengths as its
temperature increases.
75 PH507

Multiwavelength
50
Wien's displacement law:
max = 2.898 x 10-3 / T
where max is in metres when T is in Kelvin. Note that because maxT =
constant, increasing one proportionally decreases the other.
The Law of Stefan and Boltzmann
The area under the Planck curve (integrating the Planck function) represents the
total energy flux, F (W/m2), emitted by a blackbody when we sum over all
wavelengths and solid angles:
where = 5.669 x 10-8 W/m2 . K4. The strong temperature dependence of this
formula was first deduced from thermodynamics in 1879 by Josef Stefan (18351893) and was derived from statistical mechanics in 1884 by Boltzmann.
Therefore we call the expression the Stefan-Boltzmann law. The brightness of
a blackbody increases as the fourth power of its temperature. If we
approximate a star by a blackbody, the total energy output per unit time of the
star (its power or luminosity in watts) is just L = 4R2T4 since the surface area
of a sphere of radius R is 4R2
To summarise: A blackbody radiator has a number of special characteristics.
One, a blackbody emits some energy at all wavelengths. Two, a hotter
blackbody emits more energy per unit area and time at all wavelengths than
does a cooler one. Three, a hotter blackbody emits a greater proportion of its
radiation at shorter wavelengths than does a cooler one. Four, the amount of
radiation emitted per second by a unit surface area of a blackbody depends on
the fourth power of its temperature.
Stellar Atmospheres
75 PH507
Multiwavelength
51
The spectral energy distribution of starlight is determined in a star’s atmosphere, the
region from which radiation can freely escape. To understand stellar spectra, we first
discuss a model stellar atmosphere and investigate the characteristics that determine
the spectral features.
Physical Characteristics
The stellar photosphere, a thin, gaseous layer, shields the stellar interior from view.
The photosphere is thin relative to the stellar radius, and so we regard it as a uniform
shell of gas. The physical properties of this shell may be approximately specified by
the average values of its pressure P, temperature T, and chemical composition µ
(chemical abundances).
We also assume that the gas obeys the perfect-gas law:
P =nkT
where k is Boltzmann’s constant. This relationship is also known as Boyle’s Law.
An important result that follows from it is that the kinetic energy of a particle, or
assemblage of particles, is given by the relationship;
KE 
3
kT
2
Thus temperature is just a measure of the kinetic energy of a gas, or an assemblage of
particles. This equation applies equally well to a star as a whole, as to a single particle,
and later we will look at the comparison between a star’s kinetic and gravitational
(potential) energies.
The kinetic energy is also a measure of the velocity that atoms or molecules are
moving about at - the hotter they are, the faster they move. Thus, for a cloud of gas
surrounding a hot star of temperature T = 15,000 K, which consists of hydrogen atoms
(mass = 1.67 10-27 kg);
3
1 2
kT  KE  mv
2
2
v
3kT
 19 km s 1  50,000 mph
m
The particle number density is related to both the mass density (kg/m3) and the
composition (or mean molecular weight) µ by the following definition of µ:
1


mH n

75 PH507
Multiwavelength
52
where mH = 1.67 x 10-27 kg is the mass of a hydrogen atom. For a star of pure atomic
hydrogen, µ = 1. If the hydrogen is completely ionised, µ = 1/2 because electrons and
protons (hydrogen nuclei) are equal in number and electrons are far less massive than
protons. In general, stellar interior gases are ionised and

1
3
1
2X  Y  Z
4
2
where X is the mass fraction of hydrogen, Y is that of helium, and Z is that of all
heavier elements. The mass fraction is the percentage by mass of one species relative
to the total. Thus, for a pure hydrogen star (X=1.0, Y = 0.0, Z = 0.0),  ~ 0.5, and for a
white dwarf star (X = 0.0, Y = 1.0, Z = 0.0)  ~ 1.33.
Temperatures
The continuous spectrum, or continuum, from a star may be approximated by the
Planck blackbody spectral-energy distribution. For a given star, the continuum
defines a colour temperature by fitting the appropriate Planck curve. We can also
define the temperature from Wien’s displacement law: maxT = 2.898 x 10-3 m . K
which states that the peak intensity of the Planck curve occurs at a wavelength max
that varies inversely with the Planck temperature T. The value of max then defines a
temperature. Also note here that the hotter a star is, the greater will be its luminous
flux (in W/m2), in accordance with the Stefan-Boltzmann law: F = T4 where = 5.67
x 10-8 W/m2 . K4. Then the relation
L = 4πR2T4eff
defines the effective temperature of the photosphere.
A word of caution: the effective temperature of a star is usually not identical to its
excitation (Boltzmann eqn) or ionisation temperature (Saha eqn) because spectralline formation redistributes radiation from the continuum. This effect is called line
blanketing and becomes important when the numbers and strengths of spectral lines
are large.
75 PH507
Multiwavelength
53
When spectral features are not numerous, we can detect the continuum between them
and obtain a reasonably accurate value for the star’s effective surface temperature.
The line blanketing alters the atmosphere’s blackbody character.
Spectrophotometry
The goal of the observational astronomer to to make measurements of the EMR
from celestial objects with as much detail, or the finest resolution, possible. The
ideal goal of spectrophotometry is to obtain the spectral energy distribution
(SED) of celestial objects, or how the energy from the object is distributed in
wavelength. We want to measure the amount of energy received by an observer
outside the Earth's atmosphere, per second, per unit area, per unit wavelength
or frequency interval. Units of spectral flux (in cgs) look like:
f  = ergs s-1 cm-2 Å -1
if we measure per unit wavelength interval, or
f = ergs s-1 cm-2 Hz -1
Classifying Stellar Spectra
The Spectral-Line Sequence
At first glance, the spectra of different stars seem to bear no relationship to one
another. In 1863, however, Angelo Secchi found that he could crudely order the
spectra and define different spectral types. Alternative ordering schemes appeared in
the ensuing years, but the system developed at the Harvard Observatory by Annie J.
Cannon and her colleagues was internationally adopted in 1910. This sequence, the
Harvard spectral classification system, is still used today. (About 400,000 stars were
classified by Cannon and published in various volumes of the Henry Draper Catalogue,
1910-1924, and its Extension, 1949.
75 PH507
Multiwavelength
54
At first, the Harvard scheme was based upon the strengths of the hydrogen Balmer
absorption lines in stellar spectra, and the spectral ordering was alphabetical (A
through to P). Some letters were eventually dropped, and the ordering was rearranged
to correspond to a sequence of decreasing temperatures (see the effects of the
Boltzmann and Saha equations): OBAFGKMRNS. Stars nearer the beginning of the
spectral sequence (closer to O) are sometimes called early-type stars, and those closer
to the M end are referred to as late-type. Each spectral type is divided into ten parts
from 0 (early) to 9 (late); for example, . . . F8 F9 G0 G1 G2 . . . G9 K0 . . . . In this
scheme, our Sun is spectral type G2. In 1922, the International Astronomical Union
(IAU) adopted the Harvard system (with some modifications) as the international
standard.
Many mnemonics have been devised to help students retain the spectral sequence. A
variation of the traditional one is “Oh, Be a Fine Girl, Kiss Me Right Now, Smack.”
Comparison of spectra observed for seven different stars having a range of surface
temperatures. The hottest stars, at the top, show lines of helium and multiply-ionised
heavy elements. In the coolest stars, at the bottom, helium lines are not seen, but lines of
75 PH507
Multiwavelength
55
neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen
lines are strongest. The actual compositions of all seven stars are about the same.
The Temperature Sequence
The spectral sequence is a temperature sequence, but we must carefully qualify this
statement. There are many different kinds of temperatures and many ways to determine them.
Theoretically, the temperature should correlate with spectral type and so with the
star’s colour. From the spectra of intermediate-type stars (A to K), we find that the
(continuum) colour temperature does so, but difficulties occur at both ends of the
sequence. For O and B stars, the continuum peaks in the far ultraviolet, where it is
undetectable by ground-based observations. Through satellite observations in the far
ultraviolet, we are beginning to understand the ultraviolet spectra of O and B stars.
For the cool M stars, not only does the Planck curve peak in the infrared, but
numerous molecular bands also blanket the spectra of these low-temperature stars.
75 PH507
Multiwavelength
56
When the strengths of various spectral features are plotted against excitationionisation (or Boltzmann-Saha) temperature; the spectral sequence does correlate with
this temperature as seen below;
In practice, we measure a star’s colour index, CI = B - V, to determine the effective
stellar temperature. If the stellar continuum is Planckian and contains no spectral
lines, this procedure clearly gives a unique temperature, but observational
uncertainties and physical effects do lead to problems: (a) for the very hot O and B
stars, CI varies slowly with Teff and small uncertainties in its value lead to very large
uncertainties in T; (b) for the very cool M stars, CI is large and positive, but these faint
stars have not been adequately observed and so CI is not well determined for them; (c)
any instrumental deficiencies, calibration errors, or unknown blanketing in the B or V
bands affect the value of CI - and thus the deduced T. Hence, it is best to define the CI
versus T relation observationally.
SPECTROSCOPY
• Last year discussed stellar spectra and classification on an empirical basis:
Spectral sequence
O B A F G K M
Temperature
~40,000 K
---->
2500 K
Classification based on relative line strengths of He, H, Ca, metal, molecular
lines.
• We will now look a little deeper at stellar spectra and what they tell us about
stellar atmospheres.
75 PH507
Multiwavelength
57
Radiative Transfer Equation
• Imagine a beam of radiation of intensity I passing through a layer of gas:
Power passing into volume
Area
dA
E = I d dA d
Power passing out
of volume
E  + dE 
where I = intensity into
solid angle element d 
path length ds
NB in all these equations subscripts  can be replaced by 
In the volume of gas there is:
ABSORPTION - Power is reduced by amount
dE = -   E ds = -   I d dA d ds
where  is the ABSORPTION COEFFICIENT or OPACITY
= the cross-section for absorption of radiation of wavelength  (frequency )
per unit mass of gas. Units of are m2 kg-1
The quantity  is the fraction of power in a beam of radiation of wavelength
 absorbed by unit depth of gas. It has units of m-1. (NB in many texts  is
simply given the symbol  in the equations given here - beware!)
EMISSION - Power is increased by amount
dE = j  d dA d ds
(1)
where j = EMISSION COEFFICIENT = amount of energy emitted per second
per unit mass per unit wavelength into unit solid angle.
Units of j (j) are W kg-1 µm-1 sr-1 (W kg-1 Hz-1 sr-1) or m s-3 sr-1
(NB power production per unit volume per unit wavelength into unit solid
angle is  =j More confusion is possible here, since is also the
symbol used for total power output of a gas, units are W kg-1, - Beware!)
So total change in power is
dE = dI d dA d = -   I d dA d ds + j  d dA d ds
which reduces to
dI = -   I ds + j  ds
(2)
75 PH507
Multiwavelength
dI
58
= - I + j 
ds
(3)
This is a form of the radiative transfer equation in the plane parallel case.
Optical depth
• Take a volume of gas which only absorbs radiation (j = 0) at  :
dI = -   I ds
For a depth of gas s, the fractional change in intensity is given by
I (s)
s

dI
 =
-  ds

I
I (0) 
0



ln (
Integrating ==>
I (s)

I (0)
s
) = -

  ds
0

s
-
I (s) = I (0) e
==>

 ds
0

We define Optical depth 
s

   ds

So


(4)
-

I (s) = I (0) e


(5)
• Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its original value if optical
depth = 1.
• Optical depth is not a physical depth. A large optical depth can occur in a
short physical distance if the absorption coefficient  is large, or a large
physical distance if  is small.
Full Radiative Transfer Equation again
dI
= - I + j 
ds
divide by  
75 PH507
Multiwavelength
dI

j

= -I +

 ds
59



dI

d
= -I + S

(6)


As ds --> 0,  is constant over ds. 
This is the RADIATIVE TRANSFER EQUATION in the plane parallel case.
Define:
j
S = 
or
j =  S


 


where
and S is the SOURCE FUNCTION.
Radiative transfer in a blackbody
• Remember definition of a blackbody as a perfect absorber and emitter of
radiation. Matter and radiation are in THERMODYNAMIC EQUILIBRIUM,
ie gross properties do not change with time. Therefore a beam of radiation in
a blackbody is constant:
dI
= 0 = - I + j 
ds
==> 0 =  (I - S),
i.e. I = S.
but for a blackbody I = B
2
B =

2hc

the PLANCK FUNCTION
3
1
hc/kT
(e
from definition of source function, j =  S
- 1)
B =

2h
2
c
1
hkT
(e
- 1)
Summary: in complete thermodynamic equilibrium the source function equals the
Planck function,
i.e.
j =  B
• In studies of stellar atmospheres we make the assumption of LOCAL
THERMODYNAMIC EQUILIBRIUM (LTE), i.e. thermodynamic equilibrium
for each particular layer of a star.
• Note that if incoming radiation at a particular wavelength (e.g. in a spectral
line) enters a blackbody gas it is absorbed, but emission is distributed over all
wavelengths according to the Planck function. All information about the
original energy distribution of the radiation is lost. This is what happens in
interior layers of a star where the density is high and photons of any
wavelength are absorbed in a very short distance. Such a gas is said to be
optically thick (see below).
Emission and Absorption lines
(7)
(K
75 PH507
Multiwavelength
60
•  the absorption coefficient describes the efficiency of absorption of material
in the volume of gas. In a low density gas, photons can generally pass
through without interaction with atoms unless they have an energy
corresponding to a particular transition (electron energy level transition, or
vibrational/rotational state transition in molecules). At this particular
energy/frequency/wavelength the absorption coefficient  is large.
• Let's imagine the volume of gas shown earlier with both absorption and
emission:
I
I (0)

path length s
dI

= S - I

d


Multiply both sides by e and re-arrange
dI

==>
d



e + I e = S e



d
==>
d


(I e ) = S e 



integrate over whole volume, i.e. from 0 to s, or 0 to 


I e

==>


=
0

S e


0
assuming S = constant along path
I e - I(0) = S e - S
==>
==>
I
I(0) e- +
radiation left
over from light
entering box.
=
S (1 - e- )
light from radiation
emitted in the
box.
(8)
 >> 1: OPTICALLY THICK CASE
If  >> 1, then e- --> 0, and eqn (8) becomes I = S
In LTE
S = B, the Planck function.
• Case 1
(9)
75 PH507
Multiwavelength
61
So for an optically thick gas, the emergent spectrum is the Planck function,
independent of composition or input intensity distribution. True for stellar
photosphere (the visible "surface" of a star).
• Case 2
 << 1 OPTICALLY THIN CASE
If  << 1, then e- ≈ 1 - 
(first two terms of Taylor series expansion)
eqn (8) becomes
I =
==>I = I(0) +  ( S - I(0) )
I(0) (1 - )
+
S (1 - 1 + )
• If I(0) = 0 : no radiation entering the box (from direction of interest):
From eqn (8)
I =  S
(=  B in LTE)
Since  = ∫  , then
I =   s S
If  is large (true at wavelength of spectral lines) then I is large,
we see EMISSION LINES. This happens for example in gaseous
nebulae or the solar corona when the sun is eclipsed.
• If I(0) ≠ 0 , let's examine eqn (8)
I = I(0) +  ( S - I(0) )
If S > I(0) then right hand term is +ve
when  is large (ie  is large) we see higher intensity than I(0)
==>
EMISSION LINES ON BACKGROUND INTENSITY.
If S < I(0) then right hand term is -ve
when  is large (ie  is large) we see lower intensity than I(0)
==>
ABSORPTION LINES ON BACKGROUND INTENSITY.
For stars we see absorption lines. This means I(0) > S,
i.e. (intensity from deeper layers) > (source function for the top layers
Assuming LTE (S = B) the source function increases as temperature
increases:
I(0) = B(Tdeep layer) > S = B(Touter layer).
Therefore temperature must be increasing as we go into the star for
absorption lines to be observed.
• To summarise
- We see CONTINUUM RADIATION for an optically thick gas
(= PLANCK FUNCTION assuming LTE).
- We see EMISSION LINES for an optically thin gas.
- We see ABSORPTION LINES + CONTINUUM for an optically thick gas
overlaid by optically thin gas with temperature decreasing outwards.
- We see EMISSION LINES + CONTINUUM for an optically thick gas
overlaid by an optically thin gas with temperature increasing outwards.
(10)
75 PH507
Multiwavelength
62
Atomic Spectra - Absorption & Emission line series and continua
• There are 3 basic photon absorption mechanisms related to electrons. Using
Hydrogen as the example, the electron energy levels are given by the
principal quantum number n, as:
E(n) = - 2 2 me e4 Z2 / n2 h2
The lowest energy level of H (n = 1) is about -13.6 eV.
The next energy level (n = 2) is
-3.4 eV.
The third (n = 3) is
-1.51 eV
from Bohr Theory
Opacity. We first introduced the concept of opacity when deriving the
equation of radiative transport. Opacity is the resistance of material
to the flow of heat, which in most stellar interiors is determined by all
the processes which scatter and absorb photons. We will now look at
each of these processes in turn, of which there are four:

bound-bound absorption

bound-free absorption

free-free absorption

scattering
The first three are known as true absorption processes because they
involve the disappearance of a photon, whereas the fourth process
only alters the direction of a photon. All four processes are described
below and are shown pictorially in figure 1.
bound-bound absorption
Bound-bound absorptions occur when an electron is moved from
one orbit in an atom or ion into another orbit of higher energy due
to the absorption of a photon. If the energy of the two orbits is E1
and E2, a photon of frequency bb will produce a transition if
E2 - E1 = hvbb.
Bound-bound processes are responsible for the spectral lines visible
in stellar spectra, which are formed in the atmospheres of stars. In
stellar interiors, however, bound-bound processes are not of great
importance as most of the atoms are highly ionised and only a
small fraction contain electrons in bound orbits. In addition, most of
the photons in stellar interiors are so energetic that they are more
likely to cause bound-free absorptions, as described below.
75 PH507
Multiwavelength
63
• BOUND - BOUND transitions give rise to spectral lines.
• ABSORPTION LINE if a photon is absorbed, causing increase in energy of an
electron. This is RADIATIVE EXCITATION.
• Note energy can also be absorbed through collisions of a free particle
(COLLISIONAL EXCITATION) - no absorption line is seen in this case.
bound-free absorption
Bound-free absorptions involve the ejection of an electron from a
bound orbit around an atom or ion into a free hyperbolic orbit due to
the absorption of a photon. A photon of frequency bf will convert a
bound electron of energy E1 into a free electron of energy E3 if
E3 - E1 = hvbf.
Provided the photon has sufficient energy to remove the electron
from the atom or ion, any value of energy can lead to a bound-free
process. Bound-free processes hence lead to continuous absorption in
stellar atmospheres. In stellar interiors, however, the importance of
bound-free processes is reduced due to the rarity of bound electrons.
free-free absorption
Free-free absorption occurs when a free electron of energy E3 absorbs
a photon of frequency ff and moves to a state with energy E4, where
E4 - E3 = h
ff.
There is no restriction on the energy of a photon which can induce a
free-free transition and hence free-free absorption is a continuous
absorption process which operates in both stellar atmospheres and
stellar interiors. Note that, in both free-free and bound-free
absorption, low energy photons are more likely to be absorbed than
high energy photons.
scattering
In addition to the above absorption processes, it is also possible for a
photon to be scattered by an electron or an atom. One can think of
scattering as a collision between two particles which bounce of one
another. If the energy of the photon satisfies
h
<< mc2,
75 PH507
Multiwavelength
64
where m is the mass of the particle doing the scattering, the particle
is scarcely moved by the collision. In this case the photon can be
imagined to be bounced off a stationary particle. Although this
process does not lead to the true absorption of radiation, it does slow
the rate at which energy escapes from a star because it continually
changes the direction of the photons.
Bound-Bound Transitions
• Atom remains in excited state until
SPONTANEOUS EMISSION (photon is emitted typically after ~10-8 s)
or INDUCED EMISSION (Photon emitted at same energy and coherently
with incoming photon - as in lasers – stimulated emission).
Both produce EMISSION LINES.
(frequency/wavelength)
corresponding to difference in
energy levels.
 Narrow lines are seen since
transitions can only occur if
photon has energy
• Energy level diagram shows electron energy level changes for absorption of
a photon.
Lowest energy level set to zero energy. 1eV = 1.6 x 10-19 J.
13.6 eV
n=•
n=4
n=3
12.73 eV
12.07 eV
n=2
10.19 eV
n=1
Lyman
Series
Balmer Paschen
Series Series
0 eV
• Series of lines seen
-LYMAN SERIES transitions to/from n=1 lines seen in ultraviolet
-BALMER SERIES ""
n=2
""
visual
-PASCHEN SERIES""
n=3
""
infrared ...
• Energy of absorbed photon is
Bound-free transitions
h = ( - E(nl)) + mev2/2
• If photon has energy greater than
that required to move an electron in
an atom from its current energy
level to level n=∞, the electron will
be released, ionizing the atom.
• Ionization potential for Hydrogen is
13.6 eV.
PH507
n=•
n=4
n=3
n=2
Astrophysics
Dr. S.F. Green
65
1/2 m ev 2
13.6 eV
12.73 eV
12.07 eV
10.19 eV
n=1
0 eV
• Since one of the states (free electron) can have any energy, the transition can
have any energy and the photon any frequency (above a certain value
determined by  and E(nl)).
Thus BOUND-FREE transitions give an ABSORPTION CONTINUUM.
• RE-COMBINATION is a FREE-BOUND transition and results in an
EMISSION CONTINUUM.
• The
spectrum
produced
by
continuum • 
absorption from a single energy
level will therefore appear as a
series of lines of increasing energy


(increasing
frequency, decreasing wavelength) up to a limit defined by -E(nl), with an
absorption continuum shortward of this limit. the characteristic of a bound-f
ree transition in a
spectrum is an edge: no absorption below some energy, then a sharp onset in
the absorption above that critical energy. As we’ll see, the absorption
decreases above the critical energy.
• For nl=1 the Lyman series (Lyman-, Lyman- etc) is observed together with
the Lyman continuum shortward of =91.2 nm. (Since interstellar space is
populated by very low density and low temperature hydrogen (ie with n=1),
photons with <91.2nm are easily absorbed so it is opaque in the near-UV).
For nl=2 the Balmer series (H, H etc) is observed together with the Balmer
continuum shortward of =364.7 nm.
Free-free transitions
• Absorption of a photon by a free electron in the vicinity of an ion.
Electron changes from free energy state with velocity v 1 to one with velocity
v2
i.e. h = 1/2 me v22 - 1/2 me v12 The term means
The inverse process “braking radiation” occurs when an electron is
accelerated by passage near an ion, and hence radiates. Bremsstrahlung and
free-free absorption are basic radiative processes that show up in many
contexts.
PH507
Astrophysics
Professor Michael Smith
66
When X-rays and gamma-rays are considered, we’ll talk about the more
general process of Compton scattering (heating the electrons) and inverse
Compton cooling.
Cyclotron and Synchrotron Radiation When magnetic fields are present,
charges can interact with them and radiate or absorb radiation. For slowly
moving particles this happens at a single frequency, the cyclotron frequency.
For relativistically moving particles, the emission or absorption occurs over a
large range of frequencies, and is called in this case synchrotron radiation.
Determination of 
• The actual spectrum of a star depends on the physical conditions (notably
temperature) and composition of the stellar atmosphere. The intensity is
produced at a physical level in the star where  ~ 2/3. In order to
determine the total spectrum, the value of  needs to be determined at all
wavelengths. The overall  is the sum of the contributions from each
atomic/molecular species in the atmosphere. Each component of  depends
on the number of atoms/molecules with a given energy state capable of
absorbing radiation at that frequency and the absorption efficiency. We will
deal with the energy state populations first:
Boltzmann's equation (Excitation equilibrium)
• Boltzmann's equation describes the population distribution of energy states
for a particular atom in a gas. The ratio of number of atoms per m3 in energy
state B to energy state A:
NB
gB (EA - EB)/kT
=
e
(50)
N
g
A
A
where gA and gB are STATISTICAL WEIGHTS (number of different
quantum states of the same energy), k = Boltzmann const and T =
temperature of gas.
NB EB > EA so exponential power is -ve.
• The probability of finding an atom in an excited state decreases exponentially
with the energy of the excited state, but increases with increasing
temperature.
Saha Equation (Ionization Equilibrium)
• The Boltzmann eqn does not describe all the possible atomic states.
Excitation may cause electrons to be lost completely. There are therefore a
number of different ionization states for a given atom, each of which has one
or more energy states.
• The ratio of the number of atoms of ionization state i+1 to those of ionization
state i (i=I is neutral, i=II is singly ionized, etc) is given by
PH507
Astrophysics
Professor Michael Smith
67
3/2
Ni+1
Ni
=
Ui+1 2
Ui Ne
2 me k T
2
h
-i /kT
e
where Ne is the electron density (number of electrons per m3), i is the
ionization potential of the ith ionization state, Ui+1 and Ui are PARTITION
FUNCTIONS obtained from the statistical
•
Ui = gi1 +
weights:

n=2
-Ein /kT
gin e
• The higher the Ionization Potential, i, the lower the fraction of atoms in the
upper ionization state,
The higher the Temperature, , the higher the fraction of atoms in the upper
ionization state, (Collisional excitation is more likely to ionize atom),
The higher the electron density, the lower the fraction of atoms in the upper
ionization state (due to re-combination).
• The Boltzmann and Saha Equations give the fraction of atoms in a given
ionization state and energy level allowing (when combined with
absorption/emission probabilities)  and hence the line strengths to be
related to abundances.
Example - Abundances in the Sun
• In line forming regions in the Sun:
Gas
Hydrogen
Calcium
I
13.6 eV
6.1 eV
II
11.9 eV
T ~ 6000 K, Ne ~ 7x1019 m3.
UII/UI
2
~2
UIII/UII
~0.5
g1
2
1
g2
2
6
From Saha Equation for Hydrogen, the ratio of ionized to un-ionized
H atoms
NII/NI ≈ 6x10-5
i.e. most of Hydrogen is un-ionized.
From Boltzmann equation, ratio of number of atoms with electrons
in level n=2 to those in level n=1 (E1-E2 = -10.19 eV) is
N2/N1 ≈ 3x10-9
i.e almost all H atoms are in the ground state.
The H Balmer lines which originate from level n=2 are strong only because the H
abundance is so high.
From Saha Equation for Calcium,
NII/NI ≈ 600 and NIII/NII ≈ 2x10-3
i.e. most of Calcium is in the singly ionized state.
From Boltzmann equation, ratio of number of atoms with
electrons in
energy states which contribute to the H and K lines
PH507
Astrophysics
Professor Michael Smith
68
to those in the ground state (E1-E2 = -3.15, -3.13 eV) is (NB/NA)II ≈ 10-2 i.e
most Ca atoms are in the ground state.
The H and K lines of Calcium are therefore strong because most Ca atoms in
the Sun are in an energy state capable of producing the lines.
• For stars cooler than the Sun more H is in the ground state so Balmer lines
will be weaker, for stars hotter than the Sun more H is in n=2 state so Balmer
lines will be stronger. (T ~ 85000 K needed for N2/N1 =1). But at this
temperature NII/NI = 105 so little remains un-ionized.
• Balmer line strength depends on excitation (function of T) and ionization
(function of T and Ne). Balance of effects occurs at T ~ 10,000 K so Balmer
lines are strongest in A0 stars.
• A similar effect occurs for other species but at different temperatures.
Transition probabilities
• Once we know the population of all energy states for a given gaseous species
we need to know the transition probabilities for each energy state change
before the absorption coefficient can be determined.
• The transition probabilities must be calculated from atomic theory or
determined by experiment - much time has been invested in this major
problem in astrophysics.
• The EINSTEIN TRANSITION PROBABILITY (inverse of lifetime):
for spontaneous emission, A21  2
for stimulated emission
B21  -1
for absorption
A12  -1
Total 
• We can now calculate  for a given gaseous species.
(removing spectral line opacities for clarity):
falls
off with decreasing 
Lyman continuum

absorption
due to  -1 dependence
Log 
For Hydrogen
T~25000K (B star)
Balmer
continuum
absorption
Paschen
continuum
absorption
T~5000K (G star)
(nm)
PH507
Astrophysics
Professor Michael Smith
69
• Similar diagrams exist for other species. The total  will be the sum for all
species in the star.
• The region of a star for which optical depth ~2/3 determines where
observed radiation originates. So if  is large, then = 2/3 at a high level
in the atmosphere and if  is low, = 2/3 deep in the atmosphere.
Solar photospheric opacity
• The solar atmosphere is dominated by hydrogen. The visible surface, the
photosphere, has a temperature ~5800 K. However, as can be seen from the
diagram above,  for hydrogen at low temperatures is low in the visible
region (~400-700nm). This is because the continuum absorption in the
visible is due to Paschen absorption (electrons originating in level n=3) and
most hydrogen is in ground state or n=2 level. We would therefore expect
the continuum to come from much deeper in the sun where temperatures are
higher. So what causes the high solar photospheric opacity?
The solar opacity comes from the H- ion. The ionization potential
For H- --> H + eis 0.75 eV (=1650nm).
From Boltzmann eqn
N3/N1 = 6 x 10-10
But from Saha eqn
N(H)/N(H-) ≈ 3 x 107
Therefore N(H-)/N3 ≈ 500.
Log 
T~25000K (B star)
H - bound-free H - free-free
T~5000K (G star)
(nm)
i.e. number of H- ions is greater than number of H atoms in level n=3, so
absorption of photons to dissociate H- to H dominates the continuum
absorption in the optical.
Limb darkening
• The Sun is less bright near the limb than at the centre of the disk.

The continuum spectrum of the entire solar disk defines a Stefan-Boltzmann
effective temperature of 5800 K for the photosphere, but how does the
PH507
Astrophysics
Professor Michael Smith
70
temperature vary in the photosphere? A clue is evident in a white-light
photograph of the Sun.


We see that the brightness of the solar disk decreases from the centre to the
limb - this effect is termed limb darkening.
Limb darkening arises because we see deeper, hotter gas layers when we look
directly at the centre of the disk and higher, cooler layers when we look near
the limb.
Assume that we can see only a fixed distance d through the solar atmosphere.
The limb appears darkened as the temperature decreases from the lower to the
upper photosphere because, according to the Stefan-Boltzmann law (Section 86), a cool gas radiates less energy per unit area than does a hot gas. The top of
the photosphere, or bottom of the chromosphere, is defined as height = 0 km.
Outward through the photosphere, the temperature drops rapidly then again
starts to rise at about 500 km into the chromosphere, reaching very high
temperatures in the corona.
At this point, you may have discerned an apparent paradox: how can the solar
limb appear darkened when the temperature rises rapidly through the
chromosphere? Answering this question requires an understanding of the
concepts of opacity and optical depth. Simply put, the chromosphere is almost
optically transparent relative to the photosphere. Hence, the Sun appears to
end sharply at its photospheric surface - within the outer 300 km of its 700,000
km radius.
PH507
Astrophysics
Professor Michael Smith
71
Our line of sight penetrates the solar atmosphere only to the depth from which
radiation can escape unhindered (where the optical depth is small). Interior
to this point, solar radiation is constantly absorbed and re-emitted (and so
scattered) by atoms and ions.
Y
Length of each solid bar is
approximately the same,
i.e. depth for which =2/3
R
y
X
Rx
Observer
Since R y > R x, radiation from the edge
of the disk, Y, originates from a higher
(cooler) region than at the centre of the
disk, X.
Assuming LTE, the continuum radiation
is described by the Planck function since
Y is at lower temperature, radiation is of
lower intensity
Spectral line formation
• Lines form higher in atmosphere than continuum. For optical lines this
corresponds to lower temperature than continuum and therefore lower
intensity (absorption lines) (see p21 where S < I).
PH507
Astrophysics
Professor Michael Smith
72
Spectral line strength
Spectral lines are never perfectly monochromatic. Quantum mechanical
considerations govern minimum line width, and many other processes cause
line broadening :
Shape of absorption line — line profile.
Natural broadening — consequence of uncertainty principle.
Doppler broadening — consequence of velocity distribution.
Pressure broadening — perturbation of energy levels by ions.
• For abundance calculations we want to know the total line strength. Total
line strength is characterised by EQUIVALENT WIDTH.
� Equivalent width: measure strength of lines.
� Rectangle with same area as line, i.e. same amount of absorption.
� EW is width in °A across rectangle
� Need EW to determine number of absorbing atoms
Stellar composition
• Derived from spectral line strengths in stellar atmospheres. In the solar
neighbourhood, the composition of stellar atmospheres is:
Element H
He
C,N,O,Ne,Na,Mg,Al,Si,Ca,Fe, others
% mass 70
28
~2.
Spectral line structure
• NATURAL WIDTH: Due to uncertainty principle, E=h/t, applied to
lifetime of excited state. For "normal" lines the atom is excited (by a photon
or collision) to an excited state which has a short lifetime t ~ 10-8 s. The
upper energy level therefore has uncertain energy E and the resultant
spectral line (absorption or emission) has an uncertain energy (wavelength).
The line has a Lorentz profile,  ~ 10-5 nm for visible light.
• COLLISIONAL/PRESSURE BROADENING:
Outer energy levels of atoms affected by presence of neighbouring charged
particles (ions and electrons). random effects lead to line broadening since
the energy of upper energy level changes relative to the unexcited state
energy level. This is the basis of the Luminosity classification for A,B stars.
Gaussian profile.  ~ 0.02 - 2 nm.
• DOPPLER BROADENING:
Due to motions in gas producing the line. Doppler shift occurs for each each
photon emitted (or absorbed) since the gas producing the line is moving
relative to the observer (or gas producing the photon).
Thermal Doppler broadening due to motions of individual atoms in the gas.
~0.01 - 0.02 nm for Balmer lines in the Sun. Gaussian profile.
PH507
Astrophysics
Professor Glenn White
73
Bulk motions of gas in convection cells. Gaussian profile.
• ROTATION:
If there is no limb darkening, then lines have hemispherical profile due to
combination of radiation from surface elements with different radial
velocities. Effect depends on rotation rate, size of star and angle of polar tilt.
In general, V sin i is derived from the profile.
_
V -1
(km s )
200
Receding
+V
A
B
C
F

A
B
Approaching
-V
C
100

o
0
O B A F G K
• ATMOSPHERIC OUTFLOW:
Many different types.
Star with expanding gas shell (result of outburst) gives P-CYGNI PROFILE.
Continuum (+ absorption lines) from star, emission or absorption lines from
shell:
F
Expanding
gas
shell

D
C
Star
D
B
D
A
o

Observer
C
B C
A
B
Radiation from star, A, passes through cooler cloud giving absorption line
due to shell material which is blue shifted relative to star. Elsewhere,
emission lines are seen.
Be STARS: Very rapid rotators with material lost from the equator:
Radiation from star, A, passes through cooler cloud giving absorption line.
Overall line structure is hemispherical rotation line (B,D). Emission lines
seen due to shell material (C,E).
PH507
Astrophysics
Professor Glenn White
C
F
Rotating
gas
shell
74
E
Star
B
A
o
Observer
C
B
A
D
E
D

PH507
Astrophysics
Professor Glenn White
75
Forbidden lines
• Only certain transitions are generally seen for two reasons:
1) Outer energy levels are far from the nucleus so in dense gases, levels are
distorted or destroyed by interactions.
2) Selection rules for change of quantum numbers restrict possible
transitions.
• In fact forbidden transitions are not actually forbidden. However, the
probability of a forbidden transition is very low, so an allowed transition will
generally occur. The lifetimes in an excited state for which there are no
allowed downward transitions are ~10-3 - 109 seconds (ie very low transition
probability). These are called METASTABLE STATES.
• De-excitation from a metastable state can be by:
1) Collisional excitation, or absorption of another photon to higher energy
state allowing another downward transition to the equilibrium state,
2) FORBIDDEN TRANSITION producing a FORBIDDEN LINE. Usually
denoted with [], e.g. [OII 731.99].
• Forbidden lines are usually much fainter than those from allowed transitions
due to low probability.
• In interstellar nebulae excited by UV from nearby hot stars, some elements'
excited states have no allowed downward transitions to the ground state. In
the absence of frequent collisions (due to low density) or high photon flux, a
forbidden transition is the only way to the ground state.
• These lines were not understood for a long while. A new element Nebulium
was invented to account for them.