Summer School
Radiative Transfer and Numerical MHD
Institute of Theoretical Astrophysics
Oslo, 19 – 29 June 2007
Radiative transfer exercises
Rob Rutten
STELLAR SPECTRA
A. Basic Line Formation
R.J. Rutten
Sterrekundig Instituut Utrecht
February 6, 2007
c 1999 Robert J. Rutten, Sterrekundig Instuut Utrecht, The Netherlands.
Copyright Copying permitted exclusively for non-commercial educational purposes.
Contents
Introduction
1 Spectral classification
(“Annie Cannon”)
1.1 Stellar spectra morphology . . . . . . . . .
1.2 Data acquisition and spectral classification .
1.3 Introduction to IDL . . . . . . . . . . . . .
1.4 Introduction to LaTeX . . . . . . . . . . . .
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3
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4
4
2 Saha-Boltzmann calibration of the Harvard sequence
(“Cecilia Payne”)
2.1 Payne’s line strength diagram . . . . . . . . . . . . . . .
2.2 The Boltzmann and Saha laws . . . . . . . . . . . . . .
2.3 Schadee’s tables for schadeenium . . . . . . . . . . . . .
2.4 Saha-Boltzmann populations of schadeenium . . . . . .
2.5 Payne curves for schadeenium . . . . . . . . . . . . . . .
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Saha-Boltzmann populations of hydrogen . . . . . . . .
2.8 Solar Ca+ K versus Hα: line strength . . . . . . . . . . .
2.9 Solar Ca+ K versus Hα: temperature sensitivity . . . . .
2.10 Hot stars versus cool stars . . . . . . . . . . . . . . . . .
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3 Fraunhofer line strengths and the curve of growth
(“Marcel Minnaert”)
3.1 The Planck law . . . . . . . . . . . . . . . . . . . . .
3.2 Radiation through an isothermal layer . . . . . . . .
3.3 Spectral lines from a solar reversing layer . . . . . .
3.4 The equivalent width of spectral lines . . . . . . . .
3.5 The curve of growth . . . . . . . . . . . . . . . . . .
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Epilogue
37
References
38
Text available at \tthttp://www.astro.uu.nl/~rutten/education/rjr-material/ssa
(or via “Rob Rutten” in Google).
Introduction
These three exercises concern the appearance and nature of spectral lines in stellar spectra.
Stellar spectrometry laid the foundation of astrophysics in the hands of:
– Wollaston (1802): first observation of spectral lines in sunlight;
– Fraunhofer (1814–1823): rediscovery of spectral lines in sunlight (“Fraunhofer lines”); their
first systematic inventory. Also discussions of the spectra of Venus and some stars;
– Herschel (1823): realization that spectral lines must provide information on the constitution
of stellar matter;
– Kirchhoff & Bunsen (1860): absorption lines in stellar spectra are the reverse of emission
lines from the same particle species in laboratory flames. The strength of the absorption is a
measure of the concentration of the species (abundance);
– Pickering plus “harem” (Williamina Fleming, Antonia Maury, Annie Cannon, and a dozen
other women): large-scale spectral classification using photographic spectrograms taken with
objective prisms. Annie Cannon classifies over 200 000 spectrograms and fine-tunes the Harvard spectral classification sequence O – B – A – F – G – K – M. This is a purely morphological
division on the basis of spectral line appearances;
– Hertzsprung (1908) and Russell (1913) independently plot a diagram of stellar absolute magnitude against spectral type (Figure 1). It shows that stars occupy sharply defined locations
in this parameter space;
– Cecilia Payne (1925): demonstration that the Saha ionization law explains stellar line strength
variations. The Harvard spectral sequence is simply a measure of temperature. All stars
have about the same chemical composition — not significantly different from the earth’s
composition apart from hydrogen)1 ;
– Morgan (1938): introduction of the luminosity classification I – II – III – IV – V. This is “the
other axis” of the empirical HRD, roughly orthogonal to the main sequence;
– Minnaert and coworkers (1930–1965, Utrecht): introduction of the equivalent width of a line
and the curve of growth for quantitative abundance determination. Detailed inventory of
the solar spectrum, first graphically in the “Utrecht Atlas” (Minnaert et al. 1940), then in
tabular form in “The Solar Spectrum 2935Å– 8770Å” (Moore et al. 1966), listing equivalent
widths for 24 000 Fraunhofer lines;
– Unsöld and coworkers (1940-1970, Kiel): precise stellar abundance determinations using LTE
(Local Thermodynamic Equilibrium) modeling;
– Schwarzschild, Eddington, Milne, Thomas and many others throughout the twentieth century:
development of more general line-formation theory, especially radiation transport through
resonance scattering;
– Avrett, Auer, Mihalas, Hummer, Rybicki and many others, from 1965: application of computers to model non-LTE line formation numerically.
1
An update of this finding is that the solar composition is the same as that of the oldest meteorites (carbonaceous chondrites) to very high precision, except for the five lightest elements and carbon. The earth has lost much
of its hydrogen to space, while the sun has burned up 99% of its lithium — fortunately, not more than 6% of its
hydrogen yet.
1
In these three exercises you will retrace some steps of the early pioneers. They don’t require
prior astronomy courses. The topics are:
• spectral classification (you in the role of Annie Cannon);
• Saha-Boltzmann modeling of the Harvard classification (you in the role of Cecilia Payne);
• Schuster-Schwarzschild modeling of Fraunhofer line strengths (you in the role of Marcel Minnaert).
The sequel sets of exercises “Stellar Spectra B: LTE Line Formation” and “Stellar Spectra C:
NLTE Line Formation” treat more advanced spectral interpretation.
Figure 1: The Hertzsprung-Russell diagram (HRD). The Harvard spectral sequence along the x axis is
the ordering as to spectral type that you will rediscover yourself in the first exercise. Stars to the right
have red appearance, to the left they are blue. The photographically-determined absolute magnitude
(luminosity at standard distance on a reversed logarithmic scale) is plotted along the y axis. Obviously
the HRD is not filled at random but contains stars only in specific locations. Most stars lie on a diagonal
band that is therefore called the “main sequence”. Some other stars lie on a branch to the upper right
called the “giant branch”. Their presence implies that another parameter than only the spectral type is
needed to define a star, the “luminosity classification” I–II–III–IV–V. Stars with luminosity class V are on
the main sequence (“dwarfs”) while stars with lower luminosity class (higher luminosity) are increasingly
above it (“giants”). The point density in this HRD portrays stellar statistics: there are far more dwarfs
than giants — but it is likely that many more white dwarfs exist than we observe. The stellar locations
4
in the HRD correspond to the simple equation L = 4πR2 σTeff
but when you are Annie Cannon in the
first exercise you don’t know that yet. You believe that spectral classification gauges important intrinsic
stellar properties whatever they are — making it worthwhile to classify as many as as you can. From
Novotny (1973).
2
1
Spectral classification
(“Annie Cannon”)
In this exercise you classify stellar spectra without prior knowledge — just as Annie Cannon
did while setting up the Harvard sequence O – B – A – F – G – K – M.
Figure 2: Annie Jump Cannon (1863 – 1941) entered Harvard College Observatory in 1886 as an assistant
to Edward Pickering, director of Harvard Observatory during 42 years. He was devoted to large-scale
stellar spectrometry and set up the monumental “Henry Draper Catalague” of stellar spectra which was
mostly assembled by Annie Cannon and her assistants, funded by a series of gifts from Mrs. Draper who
wanted a memorial to her husband, the first spectroscopist to photograph a stellar spectrum (from a
private observatory on the Hudson River). The effort started in 1886 (Draper died in 1882) and was
concluded in 1924 with the publication of the 9th volume of the Catalogue in the Harvard Annals. By
then, Mrs. Draper’s quarter-million dollar bequest had yielded a quarter million stellar classifications.
More history of astronomical spectroscopy is found in “The analysis of starlight” by Hearnshaw (1986),
from which this photograph is copied.
1.1
Stellar spectra morphology
Figure 3 shows a set of photographic stellar spectrograms. They are rather like the spectrograms
classified at Harvard. Note that they are negatives; stellar spectra usually contain absorption
lines, dark on a brighter background (the “continuum”).
• Cut the page into strips, one per spectrum, and order them into a morphological sequence.
You are the first astronomer studying these spectra and you don’t know what the coding is,
except that the lines have something to do with the presence of specific elements.
• Try to explain all you see and speculate on the meaning of your ordering.
1.2
Data acquisition and spectral classification
This is a canned computer exercise for Windows from:
academics/physics/clea/CLEAhome.html.
3
http://www.gettysburg.edu/
• Get the CLEA-SPEC exercise (file http://www.astro.uu.nl/~rutten/education/
rjr-material/ssa/clea.zip).
• Unzip clea.zip and install the CLEA-SPEC exercise.
• Start the exercise.
• Try out the various options. Classify a number of stellar spectra. Do the observing first if
you prefer taking spectra before classifying them.
1.3
Introduction to IDL
We will use the interactive programming language IDL in the second and third exercise. The
Student Edition suffices. First familiarize yourself with IDL. Under Windows it may be installed
under Research Systems Inc. (RSI, an unfortunate acronym). Under Unix/Linux type idl in a
terminal window, or idlde for a mouse-oriented environment.
• Start IDL.
• Work through the brief IDL manual at http://www.astro.uu.nl/~rutten/education/
rjr-material/idl/manuals. You won’t need the more advanced plotting and the input/output commands for these exercises.
• Write an IDL function ADDUP(array) as the one in the manual in a file ADDUP.PRO and try it
out. You will need to put the file in a partition where you have write access, and to redirect
IDL to that location (under Windows through cd,’c:\yourdir\idl\’).
1.4
Introduction to LaTeX
You should write a report in which you include the pertinent graphs made with IDL. You should
explain everything seen in the graphs.
If you use LaTeX as text processor in order to gain experience in writing reports the astronomer’s
way then:
• Copy all files at http://www.astro.uu.nl/~rutten/education/rjr-material/latex/
student-report/ into your writing directory.
• Study file latex-bibtex-manual.txt and follow its instructions regarding file report.tex.
The latter is a template for your report. Under Windows you may open it in WinEdt
(http://www.winedt.com/) to process it with MikTex (http://www.miktex.org/). Under
Unix or Linux insert it in a plain-text editor such as Emacs.
• Start writing, and frequently inspect the result.
• Experiment with PostScript or pdf figure insertion.
4
61 Cyg A
β Vir
Coma T60
HD 36936
16 Cyg A
γ Uma
HD 95735
HD 36865
78 Uma
HD 37129
Coma T 183
61 Cyg b
45 Boo
σ Dra
β Com
61 Uma
HD 46149
HD 109011
Figure 3: Stellar spectrograms taken with a low-dispersion grating spectrometer. The wavelength increases to the right. From Abt et al. (1968).
5
6
2
Saha-Boltzmann calibration of the Harvard sequence
(“Cecilia Payne”)
In this exercise you will explain the spectral-type sequence that is studied morphologically in
the first exercise and that is summarized in Figure 5. You so re-act the work of Cecilia Payne at
Harvard. Her 1925 thesis was called “undoubtedly the most brilliant PhD thesis ever written
in astronomy” by Otto Struve. Its opening sentences are:
“The application of physics in the domain of astronomy constitutes a line of investigation that
seems to possess almost unbounded possibilities. In the stars we examine matter in quantities
and under conditions unattainable in the laboratory. The increase in scope is counterbalanced,
however, by a serious limitation — the stars are not accessible to experiment, only to observation,
and there is no very direct way to establish the validity of laws, deduced in the laboratory, when
they are extrapolated to stellar conditions.”
Extrapolation of terrestrial physics laws is precisely what Payne did in her thesis. She applied
the newly derived Saha distribution for different ionization stages of an element to stellar spectra,
finding that the empirical Harvard classification represents primarily a temperature scale. Her
work crowned efforts of Saha, Russell, Fowler, Milne, Pannekoek and others along the same lines.
It illustrates that detailed physics, in this case atomic physics, is usually needed to explain cosmic
phenomena.
Figure 4: Cecilia Payne (1900 – 1979) was educated at Cambridge by Milne and Eddington. She went to
the US in 1923 and spent the rest of her career at Harvard (Boston). Her 1925 thesis was the first one
in astronomy at Harvard University and remains highly readable as a wide review of stellar spectroscopy
at the time. The main conclusion was that stellar composition does not change much from star to star.
Russell had already suggested so a decade earlier, but her thesis under Russells’ guidance, published as
the first Harvard Observatory Monograph, brought the point home. Copied from Hearnshaw (1986).
2.1
Payne’s line strength diagram
The key graph in Payne’s thesis (page 131, earlier published in Payne 1924) is reprinted in
Figure 6. Clearly, the observed behavior in the upper panel is qualitatively explained by the
computed behavior in the lower panel. We will recompute the latter.
7
Figure 5: A selection of stellar spectrograms illustrating the Harvard spectral sequence. These example
spectra are printed positively, with the absorption lines dark on a bright background. Wavelengths in
Ångstrom (1 Å= 0.1 nm = 10−8 cm). The peak brightness shifts from left to right from the “early-type”
stars (O and B) to the “late-type” stars (G and lower). The sun has spectral type G2 V and is a late-type
star. The early-type stars display the hydrogen Balmer lines prominently, but these become weak in
solar-type spectra in which the Ca+ H and K resonance lines are strongest. The M dwarfs on the bottom
display strong molecular bands. From Novotny (1973).
2.2
The Boltzmann and Saha laws
In thermodynamical equilibrium (TE) macroscopic equipartition laws hold with the gas temperature as the major parameter. These are the Kirchhoff, Planck, Wien and Stefan-Boltzmann
laws for radiation, and the Maxwell, Saha and Boltzmann laws for matter. In this exercise
we are concerned with the latter two. They describe the division of the particles of a specific
element over its different ionization stages and over the discrete energy levels within each stage.
For example, the Saha law specifies the distribution of iron particles between neutral iron (Fe),
once-ionized iron (Fe+ ), twice-ionized iron (Fe2+ ), etc., whereas the Boltzmann law specifies
the sub-distribution of the iron particles per ionization stage over the discrete energy levels that
each of the Fe, Fe+ , Fe2+ etc.2 stages may occupy. Figure 7 illustrates the energy level structure
of neutral hydrogen.
Boltzmann law. In TE the partitioning of a specific atom or ion stage over its discrete energy
levels (“excitation equilibrium”) is given by the Boltzmann distribution
nr,s
gr,s −χr,s /kT
=
e
,
(1)
Nr
Ur
2
In astronomy one doesn’t write ions as Fe3+ but rather as Fe IV. More precisely: Fe I is the spectrum of
neutral iron Fe, Fe II the spectrum of once-ionized iron Fe+ , etc.
8
Figure 6: The strengths of selected lines along the spectral sequence. Upper panel: variations of observed
line strengths with spectral type in the Harvard sequence. The latter is plotted in reversed order on a
non-linear scale that was obtained by making the peaks coincide with the corresponding peaks in the
lower panel. The y-axis units are eye estimates on an arbitrary scale. Lower panel: Saha-Boltzmann
predictions of the fractional concentration Nr,s /N of the lower level of the lines indicated in the upper
panel, each labeled with its ionization stage, on logarithmic y-axis scales that are specified per species at
the bottom, against temperature T along the x axis given in units of 1000 K along the top. The pressure
was taken constant at Pe = Ne k T = 131 dyne cm−2 = 13.1 Pascal. From Novotny (1973) who took it
from Payne (1924).
with T the temperature, k the Boltzmann constant, nr,s the number of particles per cm3 in level
s of ionization stage r, gr,s the statistical weight of that level, and χr,s the excitation energy of
P
that level measured from the ground state (r, 1), Nr ≡ s nr,s the total particle density in all
levels of ionization stage r, and Ur its partition function 3 defined by
Ur ≡
X
gr,s e−χr,s /kT .
(2)
s
Thus, the neutral stage has r = 1, each ground state is at s = 1, and each ground state has
excitation energy χr,1 = 0. and ionization energy to the next stage χr . A radiative deexcitation
between levels (r, s) and (r, t), with level s “higher” than level t, releases a photon with energy
χr,s − χr,t = hν = hc/λ, with h the Planck constant, ν the photon frequency, c the velocity
of light and λ the wavelength. The excitation energy χr,s is the energy difference between
the excited level (r, s) and the ground state (r, 1). Astronomers usually call it “excitation
3
Dutch: toestandssom.
9
Figure 7: Energy level diagram for hydrogen. The bound levels are at excitation energies given by (5)
on page 19. They approach the ionization threshold at χH = 13.598 eV for n → ∞. The principal
quantum number n equals the level counter s in this simple structure. The fine structure of each level
(splitting in 2 n2 sublevels) is not shown. For each of the first four hydrogen series the principal boundbound transitions between bound levels are marked by vertical lines with the name and the wavelength
of the corresponding spectral line. The series limits (n = ∞) are also marked. A bound-free ionization/recombination transition is added to the Balmer series. The amount of energy above the ionization
threshold represents the kinetic energy that is gained or lost. A free-free transition (radiative encounter
between a bare proton and a free electron) is also marked. The bound-free and free-free transitions contribute to stellar continua, while the bound-bound transitions produce the hydrogen lines. The Lyman
lines are in the ultraviolet, the Balmer lines are in the visible and the Paschen and Brackett lines are in
the infrared. Some Balmer lines are present in the stellar spectrograms in Figure 5. The solar Balmer α
line (usually called Hα) is shown in Figure 9. From Novotny (1973).
potential” and measure it from the ground state up4 in electron volt, with 1 eV corresponding
to 1.6022 × 10−12 erg (1.6022 × 10−19 Joule). For example, the H I Balmer α line results from
photonic transitions between levels n = 2 and n = 3 of neutral hydrogen, with χ1,3 = 12.09 eV,
χ1,2 = 10.20 eV and wavelength λ = hc/(χ1,3 − χ1,2 ) = 656.3 nm (Figure 7).
The number densities nr,s and nr,t are called “level populations” and are usually measured per
4
Physicists often measure level energies as “binding energy” from the ground state of the next ion down in
wavenumbers (cm−1 ).
10
Figure 8: Energy level diagram for Schadee’s element E, showing the neutral stage (lefthand column,
r = 1) and the first three ionization stages (r = 2 − 4). The level energies increase in 1 eV steps. The
columns may be thought of as being stacked on top of each other since each ion requires the previous
stage to be ionized. The level counter s starts at 1 within each stage (but IDL starts at 0, as do the
level energies). In astronomical convention the spectra of neutral schadeenium E, ionized schadeenium
E+ and doubly ionized schadeenium E2+ are called E I, E II, and E III, respectively.
cm3 .
The statistical weights gr,s measure the degeneracy of levels due to magnetic fine splitting. The
latter occurs only in the presence of an external magnetic field; in its absence, magnetic finestructure levels coincide and may accommodate more particles than allocated per single level by
the Pauli exclusion principle. The weights measure such excess. For example, neutral hydrogen
atoms have g1,1 = 2 for their ground state because the electron and proton spins can be parallel
or anti-parallel5 .
• Inspect the hydrogen energy level diagram in Figure 7. Which transitions correspond to
the hydrogen lines in Figure 5? Which transitions share lower levels and which share upper
levels?
• Payne’s basic assumption was that the strength of the absorption lines observed in stellar
spectra increases with the population density of the lower level of the corresponding transition.
Why might this be a reasonable assumption (it is)?
• Use this expectation to give initial rough estimates of the strength ratios of the α lines in the
the H I Lyman, Balmer, Paschen and Brackett series.
5
The fine-structure transition between the two states produces the 21 cm radio line from interstellar gas.
11
Saha law. In TE the particle partitioning over the various ionization stages of an element
(“ionization equilibrium”) is given by the Saha distribution:
Nr+1
1 2Ur+1 2πme kT 3/2 −χr /kT
=
e
,
(3)
Nr
Ne Ur
h2
with Ne the electron density, me the electron mass, χr the threshold energy needed to ionize
stage r to stage r + 1, and Ur+1 and Ur the partition functions of ionization stages r + 1 and
r defined by (2). The ionization energy χr is the minimum photon energy that is absorbed at
ionization or emitted at recombination in a bound-free interaction. The factor two represents
the statistical weight of the freed electron, which has ge = 2 due to the two orientations that
its spin may take. The scaling with 1/Ne says that ionization is easier if there is room for the
resulting free electron or, reversedly, that recombination from stage r + 1 to stage r requires
catching a free electron. The kinetic energy of the free electron contributes the (. . .)3/2 term
through the Maxwell velocity distribution.
Ur
U1
U2 = U3 = U4
nr,s /Nr
s=1
2
3
4
5
6
7
10
15
Nr /N
r=1
2
3
4
5
ion
E
E+
E2+
E3+
E4+
5 000 K
1.11
1.11
5 000 K
0.90
0.09
0.01
(-3)
(-4)
(-5)
(-6)
[(-10)]
[(-15)]
10 000 K
1.46
1.46
10 000 K
0.69
0.22
0.07
0.02
0.01
(-3)
(-3)
(-5)
(-8)
5 000 K
0.91
0.09
(-11)
(-36)
(-82)
20 000 K
2.23
2.27
20 000 K
0.45–0.44
0.25
0.14
0.08
0.04
0.02
0.01
(-3)
(-4)
10 000 K
(-4)
0.95
0.05
(-11)
(-29)
20 000 K
(-10)
(-4)
0.63
0.37
(-6)
P
Table 1: Schadee’s Saha-Boltzmann population tables for element E. The quantity N = Nr is the total
density per cm3 of particles of element E. The notation (-i) stands for order of magnitude ≈ 10−i . The
bracketed values in the 5 000 K column of the second table are for levels that do not exist in the neutral
stage E.
2.3
Schadee’s tables for schadeenium
This section gives Saha-Boltzmann results for a hypothetical (but iron-like) element in conditions
similar to a stellar atmosphere. They are taken from old lecture notes by Schadee6 . He called
6
Aert Schadee (1936 – 1999) was a solar physicist at Utrecht University. He started under Minnaert’s guidance
as a spectroscopist concentrating on solar molecular line formation (Schadee 1964), then developed the theory
12
his element “E” but I call it “schadeenium” now. It has:
– ionization energies χ1 = 7 eV for neutral E, χ2 = 16 eV for E+ , χ3 = 31 eV for E2+ ,
χ4 = 51 eV for E3+ ;
– excitation energies that increase incrementally by 1 eV: χr,s ≡ s − 1 eV in each stage;
– statistical weights gr,s ≡ 1 for all levels (r, s).
Schadee evaluated the Saha and Boltzmann laws for element E, electron pressure Pe = Ne k T =
103 dyne cm−2 and temperature T = 5 000, 10 000 and 20 000 K, respectively. The corresponding
distributions specified by (1)–(3) are given in Schadee’s tables in Table 1.
• Note in the first table that the partition functions computed from (2) are of order unity and
barely sensitive to temperature.
• In the second table, note the steep Boltzmann population decay with χr,s given by (1). It is
less steep for higher temperature. The columns add up to unity because the values in this
table are scaled by Nr . They therefore depend on Ur , but the small variation between U1 and
U4 in the first table produces a difference at two-digit significance only for s = 1 at 20 000 K.
The partition function U1 of the neutral stage is the sum of only seven levels; the higher
levels present in stages r ≥ 2 contribute only marginally. The ground state always has the
largest population.
Thus, the lowest levels are the most important ones, due to the rapid decay of the Boltzmann
factor e−χ/kT with χr,s . This explains the insensitivity of Ur to temperature in the first table.
Real atoms and ions tend to have larger energy difference between levels 1 and 2, so that their
partition function is often well approximated by the statistical weight of the ground state.
• Inspect the third table, computed from (3). There are only two ionization stages significantly
present per column. For T = 5 000 K element E is predominantly neutral, for T = 10 000 K
it is once ionized (E+ ), for higher temperature stages E2+ and E3+ appear while E and E+
vanish.
There is a striking difference between the Boltzmann and the Saha dependencies on temperature.
Ionization may fully deplete the ground stage (third table), whereas excitation never depletes
the ground state by itself (second table) but only changes the steepness of the exponential decay.
• Explain from (1) and (3) why the Saha and Boltzmann distributions behave differently for
increasing temperature.
• Speculate why ionization can fully deplete a stage even though excitation puts only a few
atoms in levels below the ionization level. Hint: what parameter in the Saha distribution can
cause equality between high-level and next-ion population at a given temperature?
Summary: in TE one expects to find only at most two adjacent ionization stages to be present in
a gas of given temperature, with more or less steep exponential population decay with excitation
energy within each ionization stage.
of Zeeman broadening in molecular lines (Schadee 1978), and later worked on the analysis of solar X-ray images
taken with the Utrecht HXIS instrument in the Solar Maximum Mission (e.g., De Jager et al. 1983).
13
1 Å = 0.1 nm = 10−8 cm
1 erg = 10−7 Joule
1 dyne cm−2 = 0.1 Pascal = 10−6 bar = 9.8693 × 10−7 atmosphere
energy of 1 eV= 1.60219 × 10−12 erg
photon energy (in eV) E = 12398.55/λ (in Å)
speed of light c = 2.99792 × 1010 cm s−1
Planck constant h = 6.62607 × 10−27 erg s
Boltzmann constant k = 1.38065 × 10−16 erg K−1
= 8.61734 × 10−5 eV K−1
electron mass me = 9.10939 × 10−28 g
proton mass mp = 1.67262 × 10−24 g
atomic mass unit (amu, C=12) mA = 1.66054 × 10−24 g
first Bohr orbit radius a0 = 0.529178 × 10−9 cm
hydrogen ionization energy χH = 13.598 eV
Table 2: Selected units and constants. Precise values available at http://physics.nist.gov/cuu/Constants.
2.4
Saha-Boltzmann populations of schadeenium
We will now reproduce Schadee’s tables by writing IDL routines that compute Ur from (2),
nr,s /Nr from (1), and Nr /N from (3) for element E. Table 2 specifies various units and constants
(using cgs units in order to expose you to the real world — as seen by astronomers).
• Start a file SSA2.PRO to develop this exercise as an IDL main program. It should get the
form:
function something, inputparameter, inputparameter
IDL statement
IDL statement
return, outputparameter
end
pro something, inputparameter, inputparameter
IDL statement
IDL statement
end
IDL statement
IDL statement
STOP
; comment
; comment
; comment out or delete when fine
IDL statement
IDL statement
end
14
The IDL routines (functions and procedures) come at the top of this file or in separate NAME.PRO
files. Start with plain IDL statements in SSA2.PRO and convert these into a routine when you
are happy with them. Process the file to try out your command sequences by saving it and
typing .run ssa2.pro on the IDL command line.
Adding STOP statements to the file makes IDL stop right there, so that you can inspect intermediate results on the command line. Typing help,parameter lets you inspect your parameter
types. Typing print,parameter displays the current parameter value. You can also type individual IDL statements on the command line one by one to try them out. The up cursor arrow
brings back previous commands so that you don’t have to retype them. IDL continues the
program when you type .con (or just .c; IDL accepts non-ambiguous abbreviations). Typing a
question mark starts up the IDL help utility.
• Compute the partition functions Ur of the Schadee element:
u=fltarr(4)
; declare 4-element float array; set values 0
chiion=[7,16,31,51]
; Schadee ionization energies into integer array
k=8.61734D-5
; Boltzmann constant in eV/deg (double precision)
temp=5000.
; the decimal point makes it a float
for r = 0,3 do $
; a $ sign extends a command to the next line
for s = 0, chiion(r)-1 do u(r)=u(r) + exp(-s/(k*temp))
print,u
; print resulting values for U(0)...U(3)
Note that IDL starts counting at zero (just as the eV scale for the levels of element E). The
double precision of the Boltzmann constant forces double precision for the evaluation of the
exponential which gets very small for large s. (If a result exceeds the IDL numerical word
length IDL issues a warning, but it will not break off processing.)
• Compare your results for temp=5000, temp=10000 and temp=20000 to Schadee’s first table
on page 12.
• Now turn the above into a function “PARTFUNC_E,temp” for future use:
function partfunc_E, temp
; partition functions Schadee element
; input: temp (K)
; output: fltarr(4) = partition functions U1,..,U4
u=fltarr(4)
chiion=[7,16,31,51]
k=8.61734D-5
for r = 0,3 do $
for s = 0, chiion(r)-1 do u(r)=u(r) + exp(-s/(k*temp))
return,u
end
It has to go to the top of your SSA2.PRO file or to an independent PARTFUNC_E.PRO file and
it should produce:
15
IDL> .r ssa2.pro
% Compiled module: PARTFUNC_E.
IDL> print,partfunc_E(5000)
1.10887
1.10888
IDL> print,partfunc_E(10000)
1.45590
1.45634
IDL> print,partfunc_E(20000)
2.23243
2.27134
; .r suffices for .run
1.10888
1.10888
1.45634
1.45634
2.27155
2.27155
• Then write a Boltzmann routine which computes nr,s /Nr from (1):
function boltz_E,temp,r,s
; compute Boltzmann population for level r,s of Schadee element E
; input: temp (temperature, K)
;
r (ionization stage nr, 1 - 4 where 1 = neutral E)
;
s (level nr, starting at s=1)
; output: relative level population n_(r,s)/N_r
u=partfunc_E(temp)
keV=8.61734D-5
; Boltzmann constant in ev/deg
relnrs = 1./u(r-1)*exp(-(s-1)/(keV*temp))
return, relnrs
end
• Check its working by reproducing the second Schadee table on page 12 for the three temperatures:
IDL> for s=1,10 do print,boltz_E(5000,1,s)
0.90181500
0.088544707
0.0086937622
0.00085359705
8.3810428e-05
8.2289270e-06
8.0795721e-07
7.9329280e-08
7.7889454e-09
7.6475762e-10
• Then write a Saha routine to reproduce Schadee’s third table on page 12. It gives Nr /N
P
where N = Nr is the total element density. The simplest way to get this ratio is to set N1
to some value, evaluate the four next full-stage populations successively from (3), and divide
them by their sum = N in the same scale:
function saha_E,temp,elpress,ionstage
; compute Saha population fraction N_r/N for Schadee element E
; input: temperature, electron pressure, ion stage
; physics constants
kerg=1.380658D-16
; Boltzmann constant (erg K; double precision)
kev=8.61734D-5
; Boltzmann constant (eV/deg)
h=6.62607D-27
; Planck constant (erg s)
elmass=9.109390D-28
; electron mass (g)
16
; kT and electron density
kevT=kev*temp
kergT=kerg*temp
eldens=elpress/kergT
chiion=[7,16,31,51]
; ionization energies for element E
u=partfunc_E(temp)
; get partition functions U(0)...u(3)
u=[u,2]
; add estimated fifth value to get N_4 too
sahaconst=(2*!pi*elmass*kergT/(h*h))^1.5 * 2./eldens
nstage=dblarr(5)
; double-precision float array
nstage(0)=1.
; relative fractions only (no abundance)
for r=0,3 do $
nstage(r+1) = nstage(r)*sahaconst*u(r+1)/u(r)*exp(-chiion(r)/kevT)
ntotal=total(nstage)
; sum all stages = element density
nstagerel=nstage/ntotal ; fractions of element density
return,nstagerel(ionstage-1)
end
The double precision declarations again avoid error messages from too small exponentials.
• The IDL TOTAL(array) function used above sums the elements of an arry of any dimension.
Check it out in the Online Help (?).
• Check your Saha routine against Schadee’s third table on page 12:
IDL> for r=1,5 do print,saha_E(20000,1e3,r)
2.7277515e-10
0.00018027848
0.63200536
0.36781264
1.7197524e-06
IDL> for r=1,5 do print,saha_E(20000,1e1,r)
7.2875161e-16
4.8163564e-08
0.016884783
0.98265572
0.00045945253
The second example shows the larger degree of ionization that occurs at lower pressure.
2.5
Payne curves for schadeenium
If TE holds in a stellar atmosphere one may expect that the observed strength of a spectral line
involving level (r, s) scales with the Saha-Boltzmann prediction for the lower level population
nr,s — even if one doesn’t know how spectral lines are formed in detail. That was the underlying
premise of Payne’s analysis. We follow it by plotting curves as in the lower panel of Figure 6
for various levels of the neutral and ionization stages of element E.
• Write a function “SAHABOLT_E,temp,elpress,r,s” that evaluates nr,s /N for any level of E
as a function of T and Pe :
17
function sahabolt_E,temp,elpress,ion,level
; compute Saha-Boltzmann populaton n_(r,s)/N for level r,s of E
; input: temperature, electron pressure, ionization stage, level nr
return, saha_E(temp,elpress,ion) * boltz_E(temp,ion,level)
end
• Inspect a few values:
IDL> for s=1,5 do
0.81709346
0.080226322
0.0078770216
0.00077340538
7.5936808e-05
IDL> for s=1,5 do
1.2218751e-10
6.8397163e-11
3.8286827e-11
2.1431900e-11
1.1996980e-11
IDL> for s=1,5 do
0.64895420
0.20334646
0.063717569
0.019965573
0.0062561096
IDL> for s=1,5 do
0.16192141
0.090639095
0.050737241
0.028401295
0.015898254
print,sahabolt_E(5000,1e3,1,s)
print,sahabolt_E(20000,1e3,1,s)
print,sahabolt_E(10000,1e3,2,s)
print,sahabolt_E(20000,1e3,4,s)
These values represent multiplications of Schadee’s second and third tables. They illustrate
again that within a single ionization stage the lower levels always have higher population due
to the Boltzmann factor. The drop-off with s is less steep at higher temperature. The overall
population per stage is set by the Saha law.
• Compute the ground-state populations nr,1 /N for Payne’s pressure (Pe = 131 dyne cm−2 )
and a range of temperatures for each ion r, and plot them together in a Payne-like graph:
temp=1000*indgen(31)
; make array 0,...,30000 in steps of 1000 K
print,temp
; check
pop=fltarr(5,31)
; declare float array for n(r,T)
for T=1,30 do $
; $ continues statement to next line
for r=1,4 do pop(r,T)=sahabolt_E(temp(T),131.,r,1)
plot,temp,pop(1,*),/ylog,yrange=[1E-3,1.1], $
xtitle=’temperature’,ytitle=’population’
oplot,temp,pop(2,*)
; first ion stage in the same graph
oplot,temp,pop(3,*)
; second ion stage
oplot,temp,pop(4,*)
; third ion stage
18
• What causes the steep flanks on the left and the right side of each peak? What happens for
T ↓ 0 and for T ↑ ∞?
• Payne plotted her curves for the actual lower levels of the lines specified by their wavelengths
in the upper panel, including their Boltzmann factor. Study its influence on your E curves
by adding curves for higher values of s to your plot:
for T=1,30 do $
; repeat for s=2 (excitation energy = 1 eV)
for r=1,4 do pop(r,T)=sahabolt_E(temp(T),131.,r,2)
oplot,temp,pop(1,*)
oplot,temp,pop(2,*)
oplot,temp,pop(3,*)
oplot,temp,pop(4,*)
for T=1,30 do $
; repeat for s=4 (excitation energy = 3 eV)
for r=1,4 do pop(r,T)=sahabolt_E(temp(T),131.,r,4)
oplot,temp,pop(1,*)
oplot,temp,pop(2,*)
oplot,temp,pop(3,*)
oplot,temp,pop(4,*)
• Explain the changes between the three sets of curves. What happens for elements with
lower/higher ionization energies than E has?
2.6
Discussion
The E curves in your plot indeed resemble Payne’s curves in Figure 6. In order to reproduce her
lower panel in detail you would have to evaluate the partition functions for the actual elements
that she used and to enter the actual excitation energies of the lower levels of the lines that
she used. More work, but in principle not different from what you have done for element E.
So, you have confirmed Payne’s conclusion that the Harvard classification of stellar spectra is
primarily an ordering with temperature, controlled by Saha-Boltzmann population statistics.
In following her footsteps, you have crossed the border between morphological description and
physical modeling, from astronomy to astrophysics. Congratulations!
2.7
Saha-Boltzmann populations of hydrogen
It is easy to write an exact Saha-Boltzmann routine for hydrogen. Its ionization energy χ1 =
13.598 eV; the statistical weights gr,s and level energies χr,s of neutral hydogen are given by
g1,s = 2 s2
(4)
2
χ1,s = 13.598 (1 − 1/s )
eV
(5)
while the single ion stage (bare protons) has U2 = g2,1 = 1.
• Write a function “SAHABOLT_H,temp,elpress,s” that produces the population of hydrogen
level s (of course by copying bits and pieces with cut & paste from your routines for element
E):
function sahabolt_H,temp,elpress,level
; compute Saha-Boltzmann population n_(1,s)/N_H for hydrogen level
; input: temperature, electron pressure, level number
19
; physics constants
kerg=1.380658D-16
; Boltzmann constant (erg K; double precision)
kev=8.61734D-5
; Boltzmann constant (eV/deg)
h=6.62607D-27
; Planck constant (erg s)
elmass=9.109390D-28
; electron mass (g)
; kT and electron density
kevT=kev*temp
kergT=kerg*temp
eldens=elpress/kergT
; energy levels and weights for hydrogen
nrlevels=100
; reasonable partition function cut-off value
g=intarr(2,nrlevels)
; declaration weights (too many for proton)
chiexc=fltarr(2,nrlevels)
; declaration excitation energies (idem)
for s=0,nrlevels-1 do begin
; enclose multiple lines with begin...end
g(0,s)=2*(s+1)^2
; statistical weights
chiexc(0,s)=13.598*(1-1./(s+1)^2)
; excitation energies
endfor
; begin...end cannot go on command line!
g(1,0)=1
; statistical weight free proton
chiexc(1,0)=0.
; excitation energy proton ground state
; partition functions
u=fltarr(2)
u(0)=0
for s=0,nrlevels-1 do u(0)=u(0)+ g(0,s)*exp(-chiexc(0,s)/kevT)
u(1)=g(1,0)
; Saha
sahaconst=(2*!pi*elmass*kergT/(h*h))^1.5 * 2./eldens
nstage=dblarr(2)
; double-precision float array
nstage(0)=1.
; relative fractions only
nstage(1) = nstage(0) * sahaconst * u(1)/u(0) * exp(-13.598/kevT)
ntotal=total(nstage)
; sum both stages = total hydrogen density
; Boltzmann
nlevel = nstage(0)*g(0,level-1)/u(0)*exp(-chiexc(0,level-1)/kevT)
nlevelrel=nlevel/ntotal
; fraction of total hydrogen density
;stop
; in for parameter inspection
return,nlevelrel
end
• Computing the exact partition function this way is a bit overdone since it turns out that
U (1) (= u(0)) = g1,1 = 2.00000. You can see this by activating the STOP statement before
the RETURN statement and then listing internal subroutine parameters:
IDL> .r ssa2.pro
IDL> print,sahabolt_H(5000,1e2,1)
% Stop encountered: SAHABOLT_H
IDL> print,u
2.00000
1.00000
20
115 ssa2.pro
IDL> for s=0,5 do print,s+1,g(0,s),chiexc(0,s),$
IDL>
g(0,s)*exp(-chiexc(0,s)/kevT)
1
2
0.00000
2.0000000
2
8
10.1985
4.2020652e-10
3
18
12.0871
1.1803501e-11
4
32
12.7481
4.5249678e-12
5
50
13.0541
3.4757435e-12
6
72
13.2203
3.4032226e-12
IDL> for s=0,nrlevels-1,10 do print,s+1,g(0,s),chiexc(0,s),$
IDL>
g(0,s)*exp(-chiexc(0,s)/kevT)
1
2
0.00000
2.0000000
11
242
13.4856
6.1790222e-12
21
882
13.5672
1.8637147e-11
31
1922
13.5838
3.9070233e-11
41
3362
13.5899
6.7387837e-11
51
5202
13.5928
1.0357868e-10
61
7442
13.5943
1.4763986e-10
71
10082
13.5953
1.9957013e-10
81
13122
13.5959
2.5936970e-10
91
16562
13.5964
3.2703819e-10
The excitation energies were already illustrated in Figure 7 on page 10. The first excited level
s = 2 is at such high excitation energy that its small Boltzmann factor makes its population
negligible in comparison with the ground state population. However, the increase with s at large
values of s shows that the hydrogen partition function would get infinite if too many levels are
included in the summation, because g1,s ∼ s2 while χ1,s → 13.598. Actually, all atoms and ions
share in this behavior at very high excitation energy since they all get to be “hydrogenic” in
nature when the valence electron sits in a nearly detached orbit. This singularity has been a
cause of much debate, but real atoms are not worried by it. They are never alone7 and loose
their identity through interactions with neighbours long before they grow as large as this. A
reasonable cut-off value to the orbit size is set by the mean atomic interdistance N −1/3 (page 260
of Rybicki and Lightman 1979):
r
r
smax ≈
N −1/6 ,
(6)
a0
giving smax ≈ 100 for hydrogen at NH = 1012 cm−3 . With such a cut-off the partition functions
are generally not much larger than the ground state weights at all temperatures of interest —
those at which the pertinent stage of ionization is not devoid of population anyhow.
2.8
Solar Ca+ K versus Hα: line strength
Figure 5 on page 8 shows that in solar-type stars the hydrogen Balmer lines become much
weaker than the Ca+ K s = 2 − 1 line at λ = 3933.7 Å (called the calcium K line8 ). The
spectrograms in Figure 5 do not include the principal line of the Balmer sequence, the hydrogen
7
They are never alone where TE holds. In intergalactic space they may be quite lonely, but they won’t sit in
their high levels out there.
8
The astronomical notation is Ca II K. The K is an extension from Draper to the original alphabetic solar
spectrum feature list by Fraunhofer, who only named Ca II H.
21
s = 3 − 2 line at λ = 6563 Å which is called the Hα line (see the hydrogen term diagram in
Figure 7 on page 10), but even this line gets much weaker than Ca+ K in solar-type stars. This
is demonstrated in Figure 9 which shows high-precision tracings of the solar spectrum around
these two wavelengths. Obviously, the main line in the lefthand panel (Ca+ K) is much stronger
than the one at right (Hα). Since stars as the sun are mostly made up of hydrogen, it may
come as a surprise that a calcium line is much stronger than a hydrogen line. However, by now
(in your role of being Cecilia Payne) it is clear to you that line strength ratios between different
elements do not only depend on their abundance ratio but also on the temperature. We will
quantify this dependence for this solar line pair.
Figure 9: Two sections of the solar spectrum displaying the strongest lines in the visible region (except
that Ca+ H at λ = 3966 Å is very similar to its doublet twin Ca+ K). The lefthand segment contains
the central part of the Ca+ K line in the violet region of the spectrum, the righthand segment the Hα
line in the red. Each segment is 30 Å (3 nm) wide. The vertical axis measures disk-averaged solar
intensity so that these plots portray the solar spectrum as if it came from a non-resolved distant star.
The units are erg cm−2 s−1 cm−1 steradian−1 , the same as for the Planck function defined by equation
(7) on page 28. The line crowding is much larger in the violet than in the red. Most of the numerous
“blends” (overlapping lines) that are superimposed on the wings of Ca+ K are due to iron, an element
with extraordinary rich energy level structure. The strong blend at 3944 Å is due to aluminum atoms.
Very close to line center the Ca+ K line displays two tiny emission peaks which betray the presence of
magnetic fields in a way that is not understood. Stars that are more active than the sun have much
higher peaks in their Ca+ K line cores. The damping wings start just outside these peaks and extend
much further than the plotted range. The Hα line in the righthand panel is much weaker. The cores of
both lines are formed in the solar chromosphere, the regime where magnetic fields take over from the gas
pressure in dominating solar fine structure. There are hundreds of studies (including five Utrecht PhD
theses) of cool-star chromospheres using the Ca+ K line core because its emission is an indicator of stellar
magnetism. The Ca+ K line is also much used in recent studies of solar chromospheric dynamics. The
temperature sensitivity of Hα makes it complementary to Ca+ K as an atmospheric diagnostic, especially
of the magnetic field structures and their not-understood heating processes in the upper chromosphere.
These show up as thread-like brightenings and darkenings arranged as flower petals in images taken in
Hα line-center radiation. These plots are made from ASCII files of the “Kitt Peak Solar Flux Atlas”
by Kurucz et al. (1984), downloaded from ftp://ftp.noao.edu/fts/fluxatl. The atlas was made with the
Fourier Transform Spectrometer at Kitt Peak, a 1 m Michelson interferometer that produces unsurpassed
spectral resolution (λ/∆λ = 106 ), an order of magnitude better than the “Utrecht Atlas” of the solar
disk-center intensity spectrum by Minnaert et al. (1940).
• Explain qualitatively why the solar Ca+ K line is much stronger than the solar Hα line,
even though hydrogen is not ionized in the solar photosphere and low chromosphere (T ≈
22
nr.
element
solar abundance
χ1
χ2
χ3
χ4
1
2
6
7
8
11
12
13
14
20
26
38
H
He
C
N
O
Na
Mg
Al
Si
Ca
Fe
Sr
1
7.9 × 10−2
3.2 × 10−4
1.0 × 10−4
6.3 × 10−4
2.0 × 10−6
2.5 × 10−5
2.5 × 10−6
3.2 × 10−5
2.0 × 10−6
3.2 × 10−5
7.1 × 10−10
13.598
24.587
11.260
14.534
13.618
5.139
7.646
5.986
8.151
6.113
7.870
5.695
–
54.416
24.383
29.601
35.117
47.286
15.035
18.826
16.345
11.871
16.16
11.030
–
–
47.887
47.448
54.934
71.64
80.143
28.448
33.492
50.91
30.651
43.6
–
–
64.492
77.472
77.413
98.91
109.31
119.99
45.141
67.15
54.8
57
Table 3: Solar abundances (relative to H) and ionization energies (eV) for some elements. Mostly from
Allen (1976).
4000 − 6000 K) where these lines are formed, and even though the solar Ca/H abundance
ratio is only NCa /NH = 2 × 10−6 . Assume again that the observed line strength scales with
the lower-level population density (which it does, although nonlinearly through a “curve of
growth” as you will see in the next exercise).
• Prove your explanation by computing the expected strength ratio of these two lines as function
of temperature for Pe = 102 dyne cm−2 . Simply combine the actual calcium ionization
energies with the Schadee ad-hoc level structure. Since the Ca+ K line originates from the
Ca+ ground state the higher levels are only needed for the partition functions and those are
estimated to within a factor of two by the Schadee recipe. Therefore simply copy all your
routines for Schadee’s element E into routines for Ca and only adapt the ionization energies.
They are given in Table 3.
• Then apply these routines as in:
temp=indgen(191)*100.+1000.
; T = 1000-20000 in delta T = 100
CaH = temp
; declare ratio array
Caabund=2.E-6
; A_Ca = N_Ca / N_H
for i=0,190 do begin
NCa = sahabolt_Ca(temp(i),1e2,2,1)
NH = sahabolt_H(temp(i),1e2,2)
CaH(i)=NCa*Caabund/NH
endfor
plot,temp,CaH,/ylog,$
xtitle=’temperature’,ytitle=’Ca II K / H alpha’
• Estimate the solar line strength ratio Ca+ K/Hα. The temperature ranges over T = 4000 −
6000 K in the solar photosphere. You should get:
IDL> print,’Ca/H ratio at 5000~K = ’,CaH(where(temp eq 5000))
Ca/H ratio at 5000~K =
7649.39
23
2.9
Solar Ca+ K versus Hα: temperature sensitivity
The two lines also differ much in their temperature sensitivity in this formation regime.
• Show this by plotting the relative population changes (∆nCa /∆T )/nCa and (∆nH /∆T )/nH
for the two lower levels as function of temperature for a small temperature change ∆T :
; temperature sensitivity CaIIK and Halpha
temp=indgen(101)*100.+2000.
; T = 2000-12000, delta T = 100
dNCadT = temp
; declare array
dNHdt = temp
; declare array
dT=1.
for i=0,100 do begin
NCa = sahabolt_ca(temp(i),1e2,2,1)
; Ca ion ground state
Nca2 = sahabolt_ca(temp(i)-dT,1e2,2,1)
; idem dT cooler
dNCadT(i)= (NCa - NCa2)/dT/NCa
; fractional diff quotient
NH = sahabolt_H(temp(i),1e2,2)
; H atom 2nd level
NH2 = sahabolt_H(temp(i)-dT,1e2,2)
; idem dT cooler
dNHdT(i) = (NH-NH2)/dT/NH
; fractional diff quotient
endfor
plot,temp,abs(dNHdT),/ylog,yrange=[1E-5,1],$
xtitle=’temperature’,ytitle=’abs d n(r,s) / n(r,s)’
oplot,temp,abs(dNCadT),linestyle=2
; Ca curve dashed
• Around T = 5 600 K the Ca+ K curve dips down to very small values; the Hα curve does that
around T = 9 500 K. Thus, for T ≈ 5 600 K the temperature sensitivity of Ca+ K is much
smaller than the temperature sensitivity of Hα. Each dip has a ∆n > 0 and a ∆n < 0 flank.
Which is which?
• The dips can be diagnosed by overplotting the variation with temperature of each population
in relative units:
; recompute as arrays and overplot relative
NCa=temp
NH=temp
for i=0,100 do begin
NCa(i) = sahabolt_ca(temp(i),1e2,2,1)
NH(i) = sahabolt_H(temp(i),1e2,2)
endfor
oplot,temp,NH/max(NH)
oplot,temp,NCa/max(NCa),linestyle=2
populations
; declare array
; declare array
; Ca ion ground state
; H atom 2nd level
; Ca curve again dashed
• Explain each flank of the two population curves and the dips in the two temperature sensitivity curves.
2.10
Hot stars versus cool stars
• Final question: find at which temperature the hydrogen in stellar photospheres with Pe = 102
is about 50% ionized:
24
IDL> for T=2000,20000,2000 do print,T,sahabolt_H(T,1e2,1)
2000
1.0000000
4000
1.0000000
6000
0.99996480
8000
0.94991572
10000
0.17471462
12000
0.0096162655
14000
0.0010102931
16000
0.00017720948
18000
4.4167459e-05
20000
1.4132857e-05
or with a plot:
temp=indgen(191)*100.+1000.
; array 1000 - 20 000 in steps 1000
nH=temp
; declare same size array
for i=0,190 do nH(i)=sahabolt_H(temp(i),1e2,1)
plot,temp,nH,$
xtitle=’temperature’,ytitle=’neutral hydrogen fraction’
This transition divides the “hot” from the “cool” stars. It also represents a dividing line between
the mechanisms that cause the stellar continuum. In hot stars the hydrogen ionization produces
so many free electrons that Thomson scattering of photons off the free electrons dominates the
formation of the stellar continuum. In cool stars there are only a few free electrons, none from
hydrogen but only from the ionization of elements with lower first ionization energy (Si, Fe, Al,
Mg, Ca, Na; see Table 3). Interactions between these rare free electrons and the abundant neutral
hydrogen atoms, momentarily combining into “H-minus ions”, then dominate the formation of
the stellar continuum.
25
26
3
Fraunhofer line strengths and the curve of growth
(“Marcel Minnaert”)
In Exercise 1 you re-invented the Harvard spectral classification in order to typecast stellar
spectra morphologically. In Exercise 2 you interpreted the Harvard classification in terms of
physics by using the Saha and Boltzmann laws for the partitioning of the particles of an element
over its various modes of existence. The strength variations of spectral lines along the main
sequence were found to be primarily due to change in temperature.
We have not yet answered the question yet how spectral lines form. This issue is addressed here
following Minnaert’s work at Utrecht between World Wars I and II. This exercise introduces you
to some of the basic concepts introduced by Minnaert. They are still in use and are instructive
to re-develop yourself.
Figure 10: Marcel G.J. Minnaert (Brugge 1893 — Utrecht 1970) was a Flemish biologist who became a
physicist at Utrecht after World War I, picking up W.H. Julius’ interest in solar spectroscopy and taking
over the solar physics department after Julius’ death in 1925. In 1937 Minnaert succeeded A.A. Nijland as
director of “Sterrewacht Sonnenborgh” and revived it into a spectroscopy-oriented astrophysical institute.
In addition, he was a well-known physics pedagogue. His three books “De natuurkunde van het vrije
veld” (Outdoors Physics) are a delightful guide to outdoors physics phenomena. I took this photograph
in 1967.
3.1
The Planck law
For electromagnetic radiation the counterparts to the material Saha and Boltzmann distributions
are the Planck law and its relatives (the Wien displacement law and the Stefan-Boltzmann law)9 .
They also hold strictly in TE (‘Thermodynamical Equilibrium”) and reasonably well in stellar
photospheres. The Planck function specifies the radiation intensity emitted by a gas or a body
9
The Saha and Boltzman (and also the Maxwell) distributions have exp(−E/kT ) without the −1 that is
present in the denominator of the Planck function. The reason for this difference is a basic one: atoms, ions and
electrons are fermions that cannot occupy the same space-time-impulse slot, but photons are bosons that actually
prefer to share places, as in a laser.
27
in TE (a “black body”) as:
2hc2
1
(7)
λ5 ehc/λkT − 1
with h the Planck constant, c the speed of light, k the Boltzmann constant, λ the wavelength and T the temperature.
The dimension of Bλ in the cgs units used here is
erg cm−2 s−1 cm−1 steradian−1 (in standard units W m−2 m−1 steradian−1 ), which is the dimension of radiative intensity in a specific direction10 .
Bλ (T ) =
• Write an IDL function PLANCK,TEMP,WAV in cgs units. The required constants are given in
Table 2 on page 14. For TEMP=5000 and WAV=5000E-8 (5000 Ångstrom, in the yellow part of
the visible wavelength region and at about the sensitivity peak of your eyes) it should give:
IDL> .com planck
% Compiled module: PLANCK.
IDL> print,planck(5000,5000E-8)
1.2107502e+14
• Use it to plot Planck curves against wavelength in the visble part of the spectrum for different stellar-like temperatures, for example with the following statements in a main IDL file
SSA3.PRO:
wav=indgen(100)*200.+1000.
; produces wav(0,...99) = 1000 - 20800
print,wav
; check that
b=wav
; declare float array of the same size
for i=0,99 do b(i)=planck(8000,wav(i)*1E-8)
plot,wav,b,xtitle=’wavelength (Angstrom)’,ytitle=’Planck function’,$
xmargin=[15,5],$
; otherwise no place for y-axis label
charsize=1.2
; bigger characters
for T=8000,5000,-200 do begin
; step from 8000 K down to 5000 K
for i=0,99 do b(i)=planck(T,wav(i)*1E-8)
oplot,wav,b
; overplots extra curves in existing graph
endfor
; begin...end sequences can’t go on command line
• Study the Planck function properties. Bλ (T ) increases at any wavelength with the temperature, but much faster (exponentially, Wien regime) at short wavelengths then at long
wavelengths (linearly, Rayleigh-Jeans regime). The peak divides the two regimes and shifts
to shorter wavelengths for higher temperature (Wien displacement law). The spectrumintegrated Planck function (area under the curve in this linear plot) increases steeply with
temperature (Stefan-Boltzmann law).
• Add ,/ylog to the plot statement to make the y-axis logarithmic. Inspect the result. Then
make the x-axis also logarithmic and inspect the result. Explain the slopes of the righthand
part.
The quadratic length dimension (cm−2 ) represents a measurement area oriented perpendicular to the beam
direction. The linear length dimension (cm−1 ) represents the spectral bandwidth (∆λ = 1 cm). It becomes
∆ν = 1 Hz for Bν in frequency units. Steradians measure beam spreading over solid angle (a wedge of a sphere),
just as radians measure planar angles (a wedge of a circle). Sometimes Bλ is defined as flux, the energy radiated
outward by a surface, without steradian−1 and a factor π larger.
10
28
3.2
Radiation through an isothermal layer
We need another quantity next to the radiation produced by a gas of temperature T , namely
the amount of absorption. Take the situation sketched in Figure 11. A beam of radiation with
intensity I(0) passes through a layer in which it is attenuated. The weakened intensity that
emerges on the right is given by
I = I(0) e−τ ,
(8)
in which the decay parameter τ specifies the attenuation by absorption in the layer. It is a
dimensionless measure of the opaqueness that is usually called the “optical thickness” because
it measures how thick the layer is, not in cm but in terms of its effect on the passing radiation.
Nothing comes through if τ 1 and (almost) everything comes through if τ 1.
I(0) e
I(0)
-τ
∆ I(x)
0
x
τ
Figure 11: Radiation through a layer. The incident intensity at the left is attenuated by absorption in the
layer as specified by its total opaqueness τ (the “optical thickness” of the layer). The internal production
of radiation ∆I(x) in a thin sublayer with thickness ∆x that is added to the beam locally is given by
the product of the Planck function B[T (x)] and the sublayer opaqueness ∆τ (x); this contribution is then
attenuated by the remainder of the layer.
The next step is to add the radiation that originates within the layer itself. Its amount is locally
equal to ∆I = Bλ (T ) ∆τ . The scaling with ∆τ comes in through a Kirchhoff law which says
that a medium radiates better when it absorbs better (a “black” body radiates stronger than a
white one). This local contribution at a location x within the layer is subsequently attenuated
by the remainder of the layer to the right, so that its addition to the emergent beam is given
by:
∆Iλ = Bλ [T (x)] ∆τ (x) e−(τ −τ (x)) .
(9)
The total emergent intensity is:
Iλ = Iλ (0) e
−τ
Z
+
0
τ
Bλ [T (x)] e−(τ −τ (x)) dτ (x)
(10)
which for an isothermal layer (T and therefore also Bλ (T ) independent of x) simplifies to:
Iλ = Iλ (0) e−τ + Bλ 1 − e−τ .
• Derive (11) from (10).
• Make plots of the emergent intensity Iλ for given values Bλ and Iλ (0) against τ :
29
(11)
B=2.
tau=indgen(101)/10.+0.01
; set array tau = 0.01-10 in steps 0.01
int=tau
; declare float array of the same size
for I0=4,0,-1 do begin
; step down from I0=4 to I0=0
for i=0,100 do int(i)=I0 * exp(-tau(i)) + B*(1-exp(-tau(i)))
if i0 eq 4 then plot,tau,int,$
xtitle=’tau’,ytitle=’Intensity’,charsize=1.3
if i0 ne 4 then oplot,tau,int
endfor
• How does Iλ depend on τ for τ 1 when Iλ (0) = 0 (add ,/xlog,/ylog to study the behavior
at small τ )? And when Iλ (0) > Bλ ? Such a layer is called “optically thin”, why?
• A layer is called “optically thick” when it has τ 1. Why? The emergent intensity becomes
independent of τ for large τ . Can you explain why this is so in physical terms?
3.3
Spectral lines from a solar reversing layer
We will now apply the above result for an isothermal layer to a simple model in which the
Fraunhofer lines in the solar spectrum are explained by a “reversing layer”. Cecilia Payne had
this model in mind when she plotted her Saha-Boltzmann population curves. She thought that
her curves described the local density of the line-causing atoms and ions within stellar reversing
layers.
Tlayer
Tsurface
τλ
Figure 12: The Schuster-Schwarzschild or reversing-layer model. The stellar surface radiates an intensity given by Bλ (Tsurface ). The shell around the surface only affects this radiation at the wavelengths
where atoms provide a bound-bound transition between two discrete energy levels. These spectral line
transitions cause attenuation τλ . The layer has temperature Tlayer and gives a thermal contribution
Bλ (Tlayer ) [1 − exp(−τλ )] as in Eq. (11).
Schuster-Schwarzschild model. The basic assumptions are that the continuous radiation,
without spectral lines, is emitted by the stellar surface and irradiates a separate layer with the
intensity
Iλ (0) = Bλ (Tsurface ),
(12)
and that this layer sits as a shell around the star and causes attenuation and local emission only
at the wavelengths of spectral lines. Thus, the shell is thought to be made up exclusively by
line-causing atoms or ions.
30
The star is optically thick (any star is optically thick!) so that its surface radiates with the
τ 1 solution Iλ = Bλ (Tsurface ) of (11), but the shell may be optically thin or thick at the
line wavelength depending on the atom concentration. The line-causing atoms in the shell have
temperature Tlayer so that the local production of radiation in the layer at the line wavelengths
is given by Bλ (Tlayer ) ∆τ (x). The emergent radiation at the line wavelengths is then given by
(11) and (12) as:
Iλ = Bλ (Tsurface ) e−τλ + Bλ (Tlayer ) 1 − e−τλ .
(13)
Voigt profile. The opaqueness τ in (13) has gotten an index λ because it varies over the
spectral line. When atoms absorb or emit a photon at the energy at which the valence electron
may jump between two bound energy levels (bound electron orbits), the effect is not limited
to an infinitely sharp delta function at λ with hc/λ = χr,s − χr,t but it is a little bit spread
out in wavelength. An obvious cause for such “line broadening” consists of the Doppler shifts
given by individual atoms due to their thermal motions. Other broadening is due to Coulomb
interactions with neighboring particles. This broadening distribution is described by
τ (u) = τ (0) V (a, u)
(14)
where V is called the Voigt function and u measures the wavelength separation ∆λ = λ − λ(0)
from the center of the line at λ = λ(0) in dimensionless units
u ≡ ∆λ/∆λD ,
(15)
where ∆λD is the “Doppler width” defined as
λq
2kT /m
(16)
c
with m the mass of the line-causing particles (for example iron with mFe ≈ 56 mH ≈ 9.3 ×
10−23 g). The parameter a in (14) measures the amount of Coulomb disturbances (called
“damping”). Stellar atmospheres typically have a ≈ 0.01 − 0.5. The Voigt function V (a, u)
is defined as:
2
Z
1
a +∞
e−y
√
V (a, u) ≡
dy.
(17)
∆λD π π −∞ (u − y)2 + a2
It represents the convolution (smearing) of a Gauss profile with a Lorentz profile and therefore
has a Gaussian shape close to line center (u = 0) due to the thermal Doppler shifts (“Doppler
core”) and extended Lorentzian wings due to disturbances by other particles (“damping wings”).
A reasonable approximation is obtained by taking the sum rather than the convolution of the
two profiles:
a
1
2
√
(18)
V (a, u) ≈
e−u + √ 2 .
∆λD π
πu
∆λD ≡
• Start the IDL Online Help by typing ? on the command line. Inspect the description of
the VOIGT(a,u) function. We might have programmed approximation (18), but since IDL
furnishes the real thing we will use that instead.
• Plot the Voigt function against u from u = −10 to u = +10 for a = 0.1:
u=indgen(201)/10.-10.
vau=u
a=0.1
for i=0,200 do vau(i)=VOIGT(a,abs(u(i)))
plot,u,vau,yrange=[0,1]
31
;
;
;
;
;
u = -10 to 10 in 0.1 steps
declare same-size array
damping parameter
taking abs corrects IDL errors
yrange fixed to compare plots
• Cursor back up and vary the value of a between a = 1 and a = 0.001 to see the effect of this
parameter. Also add ,/ylog (without setting yrange) to inspect the far wings of the profile.
Use approximation (18) to explain what you see.
Emergent line profiles. You are now able to compute and plot stellar spectral line profiles
by combining (13) with (14). Again use the dimensionless u units for the wavelength scale so
that you don’t have to evaluate the Doppler width ∆λD .
• Write an IDL sequence that computes Schuster-Schwarzschild line profiles. Take Tsurface =
5700 K, Tlayer = 4200 K, a = 0.1, λ = 5000 Å. These values are good choices for the solar
photosphere as seen in the optical part of the spectrum. First plot a profile I against u for
τ (0) = 1:
Ts=5700
; solar surface temperature
Tl=4200
; solar T-min temperature = ‘reversing layer’
a=0.1
; damping parameter
wav=5000.D-8
; wavelength in cm
tau0=1
; reversing layer thickness at line center
u=indgen(201)/10.-10. ; u = -10 to 10 in 0.1 steps
int=u
; declare array
for i=0,200 do begin
tau=tau0 * VOIGT(a,abs(u(i)))
int(i)=PLANCK(Ts,wav) * exp(-tau) + PLANCK(Tl,wav)*(1.-exp(-tau))
endfor
plot,u,int
• Study the appearance of the line in the spectrum as a function of τ (0) over the range log τ (0) =
−2 to log τ (0) = 2. Example:
tau0=[0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100]
for itau=0,8 do begin
for i=0,200 do begin
tau=tau0(itau) * VOIGT(a,abs(u(i)))
int(i)=PLANCK(Ts,wav) * exp(-tau) + PLANCK(Tl,wav)*(1.-exp(-tau))
endfor
oplot,u,int
endfor
• How do you explain the profile shapes for τ (0) 1?
• Why is there a low-intensity saturation limit for τ 1?
• Why do the line wings develop only for very large τ (0)?
• Where do the wings end?
• For which values of τ (0) is the layer optically thin, respectively optically thick, at line center?
And at u = 5?
• Now study the dependence of these line profiles on wavelength by repeating the above for
λ = 2000 Å (ultraviolet) and λ = 10 000 Å (near infrared). What sets the top value Icont
32
and the limit value reached at line center by I(0)? Check these values by computing them
directly on the command line. What happens to these values at other wavelengths?
• Observed spectra that are measured in detector counts without absolute intensity calibration
(as in your Clea-Spec data gathering in Exercise 1) are usually scaled to the local continuum
intensity by plotting Iλ /Icont against wavelength. Do that for the above profiles at the same
three wavelengths:
for iwav=1,3 do begin
wav=(iwav^2+1)*1.D-5
; wav = 2000, 5000, 10000 Angstrom
for itau=0,8 do begin
for i=0,200 do begin
tau=tau0(itau) * VOIGT(a,abs(u(i)))
int(i)=PLANCK(Ts,wav) * exp(-tau) + PLANCK(Tl,wav)*(1.-exp(-tau))
endfor
int=int/int(0)
; convert into relative intensity
if iwav eq 1 and itau eq 0 then plot,u,int
if iwav eq 1 and itau gt 0 then oplot,u,int
if iwav eq 2 then oplot,u,int,linestyle=1
; dotted
if iwav eq 3 then oplot,u,int,linestyle=4
; dash dot dot dot
endfor
endfor
• Explain the wavelength dependences in this plot.
3.4
The equivalent width of spectral lines
Your profile plots demonstrate that the growth of the absorption feature in the spectrum for
increasing τ (0) is faster for small τ (0) then when it “saturates” for larger τ (0). Minnaert and
coworkers introduced the equivalent width Wλ as a line-strength parameter to measure this
growth quantitively. It measures the integrated line depression in the normalized spectrum:
Wλ ≡
Z
Icont − I(λ)
dλ
Icont
(19)
so that its value is the same as the width of a rectangular piece of spectrum that blocks the
same amount of spectrum completely (Figure 13). We will express it here in the dimensionless
wavelength units u.
• In order to add such profile integration it becomes handy to turn the profile computation
into an IDL function PROFILE(a,tau0,u):
33
Wλ
Iλ
0
λ
Figure 13: The equivalent width of a spectral line is the width of a rectangular piece of fully blocked
spectrum with the same spectral area as the integrated line depression.
function profile,a,tau0,u
; return a Schuster-Schwarzschild profile
; input: a = damping parameter
;
tau0 = SS layer thickness at line center
;
u = wavelength array in Doppler units
; output: int = intensity array
Ts=5700
Tl=4200
wav=5000.E-8
int=u
usize=SIZE(u)
; IDL SIZE returns array type and dimensions
for i=0,usize(1)-1 do begin
tau=tau0 * VOIGT(a,abs(u(i)))
int(i)=PLANCK(Ts,wav)*exp(-tau) + PLANCK(Tl,wav)*(1.-exp(-tau))
endfor
return,int
end
• Check your routine:
u=indgen(1001)/2.5-200.
a=0.1
tau0=1e2
int=profile(a,tau0,u)
plot,u,int
; u = -200 to +200 in steps of 0.4
• Continue by computing the equivalent width with the IDL TOTAL function (check it out in
the Online Help):
reldepth=(int(0)-int)/int(0)
plot,u,reldepth
eqw=total(reldepth)*0.4
print,eqw
; line depth in relative units
; integral = TOTAL times interval
The wide range of u specified above is needed to fully accommodate the extended line wings
that develop at large τ (0), otherwise int(0) will not equal Icont . However, the sampling should
be finely spaced around line center to get the proper summation for narrow lines at small τ (0).
34
Spectral line codes therefore often use equidistant wavelength spacing over the Doppler core but
logarithmic wavelength spacing in the damping wings.
Figure 14: Empirical curve of growth for solar Fe I and Ti I lines. Taken from Mihalas (1970) who took
it from Wright (1948). Wright measured the equivalent widths of 700 lines in the Utrecht Atlas. The
quantity Xf along the x axis scales with the product of the transition probability and the population
density of the lower level of each line. The populations were computed from the Saha-Boltzmann laws as
in Exercise 2. The transition probabilities were measured in the laboratory. The normalization of W by
λ removes the λ-dependence of the Doppler width defined by (16).
3.5
The curve of growth
The idea behind the equivalent width was obviously that the amount of spectral blocking should
be a direct measure of the number of atoms in the reversing layer. They should set the opaqueness
τ (0) of the layer. Your profile plots illustrate that the profile growth is only linear with τ (0) for
τ (0) 1. The “curve of growth” describes the full dependence: the growth of the line strength
with the line-causing particle density. Figure 14 shows an observed example.
• Compute and plot a curve of growth by plotting log Wλ against log τ (0):
tau0=10^(indgen(61)/10.-2.) ; 10^-2 to 10^4, 0.1 steps in the log
eqw=tau0
; same size array
for i=0,60 do begin
int=profile(a,tau0(i),u)
reldepth=(int(0)-int)/int(0)
eqw(i)=total(reldepth)*0.4
endfor
plot,tau0,eqw,xtitle=’tau0’,ytitle=’equivalent width’,/xlog,/ylog
35
• Explain what happens in the three different parts.
• The first part has slope 1:1, the third part has slope 1:2 in this log-log plot. Why?
• Which parameter controls the location of the onset of the third part? Give a rough estimate
of its value for solar iron lines through comparison with Figure 14.
• Final question: of which parameter should you raise the numerical value in order to produce
emission lines instead of absorption lines? Change it accordingly and rerun your programs
to produce emission profiles and an emission-line curve of growth. Avoid plotting negative
Wλ values logarithmically by:
plot,tau0,abs(eqw),$
xtitle=’tau0’,ytitle=’abs(equivalent width)’,/xlog,/ylog
36
Epilogue
In these three exercises “Stellar Spectra A” you have quickly gone through half a century of
stellar spectroscopy (the first half of the twentieth century), ending with a model of spectral line
formation that roughly explains the strengths and shapes of solar line profiles. Combination of
this model with Saha-Boltzmann routines as in Exercise 2 explains stellar spectra throughout
the Hertzsprung-Russell diagram. However, the model is very simple. Its two major deficiencies
are:
– use of the single-layer Schuster-Schwarschild description (Exercise 3).
It is obviously unrealistic to stick all line-causing particles together in a separate layer. The
gas in a stellar atmosphere is well mixed; the gas particles contribute to local absorption
and emission with continuum-causing processes and line-causing processes at the same time.
A much better description is therefore to have say a hundred layers instead of a single one,
each with its own τ c (i) for the continuum and additional τ l (i) for the spectral line. If the
temperature T (i) is then set to smoothly decline outwards, a fairly realistic model of a stellar
atmosphere results.
Setting the ratio τ l /τ c constant for all i would make this many-layer model a “MilneEddington” atmosphere, a much better description than the Schuster-Schwarzschild one.
Computing τ c (i) and τ l (i) in detail from the Saha-Boltzmann laws would make it an “LTE
model atmosphere”, where the “Local” in LTE = Local Thermodynamic Equilibrium alludes
to using the local temperature within a stratified atmosphere in the TE laws. Such localequilibrium modeling has been used for stellar abundance determination throughout the
second half of the twentieth century. It is treated in exercises “Stellar Spectra B: LTE line
formation”.
– use of the TE partitioning laws (Exercises 2 and 3).
We have assumed that the temperature defines both the particle populations (Saha and
Boltzmann laws; Exercise 2) and the production of radiation (Planck law; Exercise 3). These
TE laws are accurate in the stellar interior and hold reasonably well in the deeper parts of a
stellar atmosphere, but not in the outer parts. For example, they do not hold at all in the
solar corona. Its high temperature (Te ≈ 2 × 106 K) and low density (Ne ≈ 106 cm−3 ) cause
ionization to very high stages but without reaching thermodynamical equilibrium because
the X-ray radiation from the ions escapes directly from the corona and represents a limiting
energy loss. As a result, the production of coronal radiation falls very far below the coronal
Planck function, and the ion populations are far below the Saha prediction.
The TE laws hold much better for the stellar photospheres from which the visible radiation
escapes. Although that radiation leak contributes most of the total stellar energy loss, it
represents only a tiny loss in comparison to the local photospheric thermal energy content.
LTE is therefore a reasonable assumption for many photospheric lines and continua — but by
no means for all and certainly not for chromospheric lines such as Ca+ K and Hα in the solar
spectrum. Exercises “Stellar Spectra C: NLTE line formation” treat techniques appropriate
to those.
37
References
Abt, H. A., Meinel, A. B., Morgan, W. W., and Tapscott, J. W.: 1968, An Atlas of LowDispersion Grating Stellar Spectra, Kitt Peak National Observatory, Tucson, USA
Allen, C. W.: 1976, Astrophysical Quantities, Athlone Press, Univ. London
De Jager, C., Schadee, A., Svestka, Z., Van Tend, W., Machado, M. E., Strong, K. T., and
Woodgate, B. E.: 1983, Solar Phys. 84, 205
Hearnshaw, J. B.: 1986, The analysis of starlight. One hundred and fifty years of astronomical
spectroscopy, Cambridge Univ. Press, Cambridge UK
Kurucz, R. L., Furenlid, I., Brault, J. W., and Testerman, L.: 1984, Solar Flux Atlas from 296
to 1300 nm, NSO Atlas Nr. 1, National Solar Observatory, Sunspot, New Mexico
Mihalas, D.: 1970, Stellar Atmospheres, W. H. Freeman and Co., San Francisco (first edition)
Minnaert, M. G. J., Mulders, G. F. W., and Houtgast, J.: 1940, Photometric Atlas of the Solar
Spectrum 3332 Å to 8771 Å, Schnabel, Amsterdam
Moore, C. E., Minnaert, M. G. J., and Houtgast, J.: 1966, The Solar Spectrum 2935 Å to
8770 Å. Second Revision of Rowland’s Preliminary Table of Solar Spectrum Wavelengths,
NBS Monograph 61, National Bureau of Standards, Washington
Novotny, E.: 1973, Introduction to stellar atmospheres and interiors, Oxford Univ. Press, New
York
Payne, C. H.: 1924, Harvard Circular 256, 1
Payne, C. H.: 1925, Stellar Atmospheres, Harvard Observatory Monographs No. 1, Cambridge,
Mass.
Rybicki, G. B. and Lightman, A. P.: 1979, Radiative Processes in Astrophysics, John Wiley &
Sons, Inc., New York
Schadee, A.: 1964, The Formation of Molecular Lines in the Solar Spectrum, PhD Thesis,
Utrecht University
Schadee, A.: 1978, J. Quant. Spectrosc. Radiat. Transfer 19, 517
Wright, K. O.: 1948, Pub. Dominion Astrophys. Obs. 8, 1
38
STELLAR SPECTRA
B. LTE Line Formation
R.J. Rutten
Sterrekundig Instituut Utrecht
January 26, 2007
c 1999 Robert J. Rutten, Sterrekundig Instuut Utrecht, The Netherlands.
Copyright Copying permitted exclusively for non-commercial educational purposes.
http: // www. astro. uu. nl/ ~rutten/ education/ rjr-material/ ssb
Contents
Introduction
1 Stratification of the solar atmosphere
1.1 FALC temperature stratification . . . .
1.2 FALC density stratification . . . . . . .
1.3 Comparison with the earth’s atmosphere
1.4 Discussion . . . . . . . . . . . . . . . . .
1
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3
4
5
6
8
2 Continuous spectrum from the solar atmosphere
2.1 Observed solar continua . . . . . . . . . . . . . . .
2.2 Continuous extinction . . . . . . . . . . . . . . . .
2.3 Optical depth . . . . . . . . . . . . . . . . . . . . .
2.4 Emergent intensity and height of formation . . . .
2.5 Disk-center intensity . . . . . . . . . . . . . . . . .
2.6 Limb darkening . . . . . . . . . . . . . . . . . . . .
2.7 Flux integration . . . . . . . . . . . . . . . . . . .
2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . .
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11
11
13
14
15
16
16
17
18
3 Spectral lines from the solar atmosphere
3.1 Observed Na D line profiles . . . . . . . .
3.2 Na D wavelengths . . . . . . . . . . . . . .
3.3 LTE line formation . . . . . . . . . . . . .
3.4 Line extinction . . . . . . . . . . . . . . .
3.5 Line broadening . . . . . . . . . . . . . . .
3.6 Implementation . . . . . . . . . . . . . . .
3.7 Computed Na D1 line profile . . . . . . . .
3.8 Discussion . . . . . . . . . . . . . . . . . .
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21
21
21
23
23
24
25
25
25
References
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27
Introduction
These three exercises treat the formation of spectral lines in the solar atmosphere assuming
Local Thermodynamical Equilibrium (LTE). The topics are:
• stratification of the solar atmosphere;
• formation of the continuous solar spectrum;
• formation of Fraunhofer lines in the solar spectrum.
The method consists of experimentation using IDL. The exercises require knowledge of basic
radiative transfer.
These exercises are a sequel to the exercises “Stellar Spectra A: Basic Line Formation”. The
two sets are independent in the sense that you don’t have to have worked through the first
set to do the present set. However, various concepts needed in the present set are treated in
the second and third exercises of the first set and some of the programs constructed there are
useful here, so that it may be of interest to you to go through these quickly. They are found at
http://www.astro.uu.nl/~rutten/education/rjr-material/ssa.
Your report on these exercises should include pertinent graphs made with IDL. A LaTeX
template for writing a report and a brief manual are available at http://www.astro.uu.nl/
~rutten/education/rjr-material/latex/student-report/.
1
Table 1: Selected units and constants. Precise values available at http://physics.nist.gov/cuu/Constants.
1 Å = 0.1 nm = 10−8 cm
1 erg = 10−7 Joule
1 dyne cm−2 = 0.1 Pascal = 10−6 bar = 9.8693 × 10−7 atmosphere
energy of 1 eV= 1.60219 × 10−12 erg
photon energy (in eV) E = 12398.55/λ (in Å)
speed of light c = 2.99792 × 1010 cm s−1
Planck constant h = 6.62607 × 10−27 erg s
Boltzmann constant k = 1.38065 × 10−16 erg K−1
= 8.61734 × 10−5 eV K−1
electron mass me = 9.10939 × 10−28 g
proton mass mp = 1.67262 × 10−24 g
atomic mass unit (amu, C=12) mA = 1.66054 × 10−24 g
first Bohr orbit radius a0 = 0.529178 × 10−9 cm
hydrogen ionization energy χH = 13.598 eV
solar mass M = 1.9891 × 1033 g
solar radius R = 6.9599 × 1010 cm
distance sun – earth AE = 1.49598 × 1013 cm
earth mass M⊕ = 5.976 × 1027 g
earth radius R⊕ = 6.3710 × 108 cm
2
1
Stratification of the solar atmosphere
In this exercise you study the radial stratification of the solar atmosphere on the basis of the
standard model FALC by Fontenla et al. (1993). They derived this description of the solar
photosphere and chromosphere empirically, assuming that the solar atmosphere is horizontally
homogeneous (“plane parallel layers”) and in hydrostatic equilibrium (“time independent”).
Figure 1: Eugene H. Avrett (born 1933) represents the A in the VAL (Vernazza, Avrett and Loeser)
and FAL (Fontenla, Avrett and Loeser) sequences of standard models of the solar atmosphere. At the
Center for Astrophysics (Cambridge, Mass.) he developed, together with programmer Rudolf Loeser, an
enormous spectrum modeling code (called Pandora, perhaps a fitting name) which fits observed solar
continua and lines throughout the spectrum through the combination of trial-and-error adjustment of
the temperature stratification with simultaneous evaluation of the corresponding particle densities with
great sophistication, basically performing complete NLTE population analysis for all key transitions in
the solar spectrum. His VALIII paper (Vernazza, Avrett and Loeser 1981) stands as one of the three
prime solar physics papers of the second half of the twentieth century (with Parker’s 1958 solar wind
prediction and Ulrich’s 1970 p-mode prediction). Photograph from the web.
The FALC model is specified in Tables 3–4 and in file falc.dat1 . Copy this file to your work
directory.
The first column in Tables 3–4 specifies the height h which is the distance above τ500 = 1 where
τ500 , given in the second column, is the radial optical depth in the continuum at λ = 500 nm.
The quantity m in the third column is the mass of a column with cross-section 1 cm2 above the
given location.
In the photosphere (−100 ≤ h ≤ 525 km) and chromosphere (525 ≤ h ≤ 2100 km) the temperature T (fourth column) has been adjusted empirically so that the computed spectrum is
in agreement with the spatially averaged disk-center spectrum observed from quiet solar areas (away from active regions). The temperature distribution in the transition region (above
h ≈ 2100 km, up to T = 105 K) has been determined theoretically by balancing the downflow of
energy from the corona through thermal conduction and diffusion against the radiative energy
losses.
The microturbulent velocity vt roughly accounts for the Doppler broadening that is observed to
exceed the thermal broadening of lines formed at various heights. The total pressure Ptot is the
sum of the gas pressure Pgas and the corresponding turbulent pressure ρ vt2 /2 where ρ is the gas
density (last column).
The tables also lists the total hydrogen density nH , the free proton density np , and the free
1
At http://www.astro.uu.nl/∼rutten/education/rjr-material/ssb/, together with other IDL input files
and with my own graphical output of these exercises as files ssb_answers.
3
Figure 2: Temperature stratification of the FALC model. The height scale has h = 0 at the location with
τ500 = 1. The photosphere reaches until h = 525 km (temperature minimum). The chromosphere higher
up has a mild temperature rise out to h ≈ 2100 km where the steep increase to the 1–2 MK corona starts
(”transition region”).
electron density ne . Given the T and vt distributions with height, these number densities
and other quantities were determined by requiring hydrostatic equilibrium and evaluating the
ionization balances by solving the coupled radiative transfer and statistical equilibrium equations
(without assuming LTE). The adopted helium-to-hydrogen abundance ratio is NHe /NH = 0.1.
The relative abundances of the other elements came from Anders and Grevesse (1989).
1.1
FALC temperature stratification
Figure 2 shows the FALC temperature stratification. It was made with IDL code similar to the
following (supplied in file readfalc.pro):
; file: readfalc.pro = IDL main to read & plot falc.dat = FALC model
; last: May 7 1999
; note: file FALC.DAT from email Han Uitenbroek Apr 29 1999
;
= resampled Table 2 of Fontenla et al ApJ 406 319 1993
;
= 2 LaTeX tables in file falc.tex
; read falc model
close,1
openr,1,’falc.dat’
nh=80
falc=fltarr(11,nh)
; nr FALC height values
; 11 FALC columns
4
readf,1,falc
; this may be done more elegantly with IDL structure
h=falc(0,*)
; but I don’t know (yet) how
tau5=falc(1,*)
colm=falc(2,*)
temp=falc(3,*)
vturb=falc(4,*)
nhyd=falc(5,*)
nprot=falc(6,*)
nel=falc(7,*)
ptot=falc(8,*)
pgasptot=falc(9,*)
dens=falc(10,*)
; plot falc model
plot,h,temp,yrange=[3000,10000],$
xtitle=’height [km]’,ytitle=’temperature [K]’
; repeat plot on postscript file
set_plot,’ps’
; repeat plot on PostScript file
device,filename=’falc_temp_height.ps’
plot,h,temp,yrange=[3000,10000],$
xtitle=’height [km]’,ytitle=’temperature [K]’
device,/close
set_plot,’x’
; return to screen; set_plot,’win’ for Windows
end
• Pull this code over and make it work.
1.2
FALC density stratification
We will now study relations between various FALC model parameters. This is easiest done by
trying out plots on the IDL command line after running an IDL main program such as the
one above. You will often need to generate logarithmic axes with keywords /xlog and /ylog
in command plot, and you will also need keyword \ynozero to plot y-axes without extension
down to y = 0. Various constants are specified in Table 1 on page 2.
• Plot the total pressure ptotal against the column mass m, both linearly and logarithmically.
You will find that they scale linearly. Explain what assumption has caused ptotal = c m and
determine the value of the solar surface gravity gsurface = c that went into the FALC-producing
code.
• Fontenla et al. (1993) also assumed complete mixing, i.e., the same element mix at all heights.
Check this by plotting the ratio of the hydrogen mass density to the total mass density against
height (the hydrogen atom mass is mH = 1.67352 × 10−24 g, e.g., Allen 1976). Then add
helium to hydrogen using their abundance and mass ratios (NHe /NH = 0.1, mHe = 3.97 mH ),
and estimate the fraction of the total mass density made up by the remaining elements in
the model mix (the “metals”).
• Plot the column mass against height. The curve becomes nearly straight when you make the
5
y-axis logarithmic. Why is that? Why isn’t it exactly straight?
• Plot the gas density against height. Estimate the density scale height Hρ in ρ ≈
ρ(0) exp(−h/Hρ ) in the deep photosphere.
• Compute the gas pressure and plot it against height. Overplot the product (nH +ne ) kT . Plot
the ratio of the two curves to show their differences. Do the differences measure deviations
from the ideal gas law or something else? Now add the helium density NHe to the product
and enlarge the deviations. Comments?
• Plot the total hydrogen density against height and overplot curves for the electron density, the
proton density, and the density of the electrons that do not result from hydrogen ionization.
Explain their behavior. You may find inspiration in Figure 6 on page 13. The last curve is
parallel to the hydrogen density over a considerable height range. What does that imply?
And what happens at larger height?
• Plot the ionization fraction of hydrogen logarithmically against height. Why does this curve
look like the one in Figure 2? And why is it tilted with respect to that?
• Let us now compare the photon and particle densities. In thermodynamic equilibrium (TE)
the radiation is isotropic with intensity Iν = Bν and has total energy density (Stefan Boltzmann)
ZZ
1
4σ 4
u=
Bν dΩ dν =
T ,
(1)
c
c
so that the total photon density for isotropic TE radiation is given, with uν = du/dν, T in
K and Nphot in photons per cm3 , by
Z
Nphot =
∞
0
uν
dν ≈ 20 T 3 .
hν
(2)
This equation gives a reasonable estimate for the photon density at the deepest model location, why? Compute the value there and compare it to the hydrogen density. Why is the equa3 /2π
tion not valid higher up in the atmosphere? The photon density there is Nphot ≈ 20 Teff
3 = F + = πI + with
with Teff = 5770 K the effective solar temperature (since πB(Teff ) = σTeff
+
+
F the emergent flux and I the disk-averaged emergent intensity). Compare it to the hydrogen density at the highest location in the FALC model. The medium there is insensitive
to these photons (except those at the center wavelength of the hydrogen Lyα line), why?.
1.3
Comparison with the earth’s atmosphere
Figure 3 shows the temperature distribution in our own atmosphere. Table 2 specifies its average
parameters (the original table in Allen 1976 continues to h = 50 000 km). It is available as file
earth.dat. Copy this file also to your work directory.
The particle density in the last column is the sum of the molecules, atoms and ions, but neglects
free electrons which are indeed negligible below h ≈ 100 km.
• Write IDL code to read file earth.dat.
• Plot the temperature, pressure, particle density and gas density against height, logarithmically where appropriate.
6
Figure 3: Temperature stratification of the mean earth atmosphere. The height scale has h = 0 at the
earth’s surface. Nomenclature: the troposphere ranges until h =11 km, the stratosphere over h = 11 –
47 km, the ionosphere above h = 50 km, with subdivision in the mesosphere (h = 52 – 79 km) and the
thermosphere (beyond h = 90 km). The temperature gradient change starting at h = 11 km is called
the tropopause, the temperature maximum around h = 50 km the stratopause, the minimum around
h = 85 km the mesopause.
• Plot the pressure and density stratifications together in normalized units in one graph. Comments?
• Plot the mean molecular weight µE ≡ m/mH = ρ/(N mH ) against height. Why does it
decrease in the high atmosphere?
• Estimate the density scales height of the lower terrestrial atmosphere. Which quantities make
it differ from the solar one? How much harder do you have to breathe on Mount Everest?
• Compare the terrestrial parameter values to the solar ones, at the base of each atmosphere.
What is the ratio of the particle densities at h = 0 in the two atmospheres?
• The standard gravity at the earth’s surface is gE = 980.665 cm s−2 . Use this value to estimate
the atmospheric column mass (g cm−2 ) at the earth’s surface and compare that also to the
value at the base of the solar atmosphere.
• Final question: the energy flux of the sunshine reaching our planet (“irradiance”) is
R=
4πR2 +
F
4πD2 (3)
+
with F
= πB(Teff
) the emergent solar flux, R the solar radius and D the distance sun–earth,
so that the sunshine photon density at earth is
Nphot = π
R2 top
N
D2 phot
(4)
top
with Nphot
the photon density at the top of FALC which you determined at the end of
Section 1.1. Compare Nphot to the particle density in the air around us, and to the local
thermal photon production derived from (2). Comments?
7
1.4
Discussion
The comparisons between particle density and column mass at the base of the solar photosphere
and the base of the earth’s atmosphere show that sunlight reaches the earth’s surface at a particle
density that is much higher than the density at the sunlight τ = 1 escape location within the
sun, i.e., the atmospheric depth where the solar gas becomes opaque. If, reversely, the earth
were irradiating the sun, that level is where the earthshine would stop already, having passed
through a much smaller amount of solar gas then it went through in the nearly transparant
earth atmosphere (transmission 80% at λ = 500 nm at sea level when the sky is clear, Allen
1976).
Thus, the solar gas is much more opaque per particle than air, even though the terrestrial air
particles (molecules) are much larger than the solar atmospheric particles (mostly hydrogen
atoms and electrons). In the first decades of the twentieth century the source of this very large
opacity was an unsolved riddle as important as the quest for the subatomic energy source that
makes the sun shine (for example, see the eloquent last paragraph of Eddington’s “The Internal
Constitution of the Stars”). That opacity is the topic of the next exercise.
Table 2: Earth atmosphere, from Allen (1976).
h
km
log P
dyn cm−2
T
K
log ρ
g cm−3
log N
cm−3
0
1
2
3
4
5
6
8
10
15
20
30
40
50
60
70
80
90
100
110
120
150
200
250
300
6.01
5.95
5.90
5.85
5.79
5.73
5.67
5.55
5.42
5.08
4.75
4.08
3.47
2.91
2.36
1.73
1.00
0.19
–0.53
–1.14
–1.57
–2.32
–3.06
–3.55
–4.00
288
282
275
269
262
256
249
236
223
217
217
230
253
273
246
216
183
183
210
260
390
780
1200
1400
1500
–2.91
–2.95
–3.00
–3.04
–3.09
–3.13
–3.18
–3.28
–3.38
–3.71
–4.05
–4.74
–5.39
–5.98
–6.50
–7.07
–7.72
–8.45
–9.30
–10.00
–10.62
–11.67
–12.50
–13.10
–13.60
19.41
19.36
19.31
19.28
19.23
19.19
19.14
19.04
18.98
18.61
18.27
17.58
16.92
16.34
15.82
15.26
14.60
13.80
12.98
12.29
11.69
10.66
9.86
9.30
8.90
8
Table 3: FALC model, part 1.
h
km
τ500
m
g cm−2
T
K
2218.20
2216.50
2214.89
2212.77
2210.64
0.00E+00
7.70E−10
1.53E−09
2.60E−09
3.75E−09
6.777E−06 100000
6.779E−06 95600
6.781E−06 90816
6.785E−06 83891
6.788E−06 75934
2209.57
2208.48
2207.38
2206.27
2205.72
4.38E−09
5.06E−09
5.81E−09
6.64E−09
7.10E−09
6.790E−06
6.792E−06
6.794E−06
6.797E−06
6.798E−06
2205.21
2204.69
2204.17
2203.68
2203.21
7.55E−09
8.05E−09
8.61E−09
9.19E−09
9.81E−09
2202.75
2202.27
2201.87
2201.60
2201.19
vt
km s−1
nH
cm−3
np
cm−3
ne
cm−3
Ptot
dyn cm−2
Pgas /Ptot
ρ
g cm−3
11.73
11.65
11.56
11.42
11.25
5.575E+09
5.838E+09
6.151E+09
6.668E+09
7.381E+09
5.575E+09
5.837E+09
6.150E+09
6.667E+09
7.378E+09
6.665E+09
6.947E+09
7.284E+09
7.834E+09
8.576E+09
1.857E−01
1.857E−01
1.858E−01
1.859E−01
1.860E−01
0.952
0.950
0.948
0.945
0.941
1.31E−14
1.37E−14
1.44E−14
1.56E−14
1.73E−14
71336
66145
60170
53284
49385
11.14
11.02
10.86
10.67
10.55
7.864E+09
8.488E+09
9.334E+09
1.053E+10
1.135E+10
7.858E+09
8.476E+09
9.307E+09
1.047E+10
1.125E+10
9.076E+09
9.718E+09
1.059E+10
1.182E+10
1.266E+10
1.860E−01
1.861E−01
1.862E−01
1.862E−01
1.863E−01
0.938
0.935
0.931
0.925
0.921
1.84E−14
1.99E−14
2.19E−14
2.47E−14
2.66E−14
6.800E−06
6.801E−06
6.803E−06
6.805E−06
6.807E−06
45416
41178
36594
32145
27972
10.42
10.27
10.09
9.90
9.70
1.233E+10
1.356E+10
1.521E+10
1.724E+10
1.971E+10
1.217E+10
1.332E+10
1.483E+10
1.667E+10
1.887E+10
1.365E+10
1.491E+10
1.657E+10
1.858E+10
2.098E+10
1.863E−01
1.863E−01
1.864E−01
1.864E−01
1.865E−01
0.916
0.910
0.903
0.894
0.883
2.89E−14
3.18E−14
3.56E−14
4.04E−14
4.62E−14
1.05E−08
1.13E−08
1.21E−08
1.27E−08
1.36E−08
6.809E−06
6.812E−06
6.815E−06
6.817E−06
6.820E−06
24056
20416
17925
16500
15000
9.51
9.30
9.13
9.02
8.90
2.276E+10
2.658E+10
3.008E+10
3.255E+10
3.570E+10
2.154E+10
2.483E+10
2.778E+10
2.979E+10
3.218E+10
2.389E+10
2.743E+10
3.049E+10
3.256E+10
3.498E+10
1.866E−01
1.866E−01
1.867E−01
1.868E−01
1.869E−01
0.871
0.856
0.843
0.834
0.823
5.33E−14
6.23E−14
7.05E−14
7.63E−14
8.36E−14
2200.85
2200.10
2199.00
2190.00
2168.00
1.44E−08
1.63E−08
1.90E−08
4.15E−08
9.85E−08
6.823E−06
6.830E−06
6.840E−06
6.936E−06
7.203E−06
14250
13500
13000
12000
11150
8.83
8.74
8.66
8.48
8.30
3.762E+10
4.013E+10
4.244E+10
4.854E+10
5.500E+10
3.343E+10
3.441E+10
3.456E+10
3.411E+10
3.619E+10
3.619E+10
3.699E+10
3.695E+10
3.663E+10
3.889E+10
1.869E−01
1.871E−01
1.874E−01
1.900E−01
1.974E−01
0.816
0.808
0.801
0.785
0.775
8.81E−14
9.40E−14
9.94E−14
1.14E−13
1.29E−13
2140.00
2110.00
2087.00
2075.00
2062.00
1.76E−07
2.62E−07
3.30E−07
3.66E−07
4.05E−07
7.588E−06
8.063E−06
8.483E−06
8.724E−06
9.005E−06
10550
9900
9450
9200
8950
8.10
7.87
7.70
7.61
7.52
6.252E+10
7.314E+10
8.287E+10
8.882E+10
9.569E+10
3.806E+10
3.923E+10
3.954E+10
3.956E+10
3.952E+10
4.095E+10
4.238E+10
4.291E+10
4.305E+10
4.314E+10
2.079E−01
2.209E−01
2.324E−01
2.390E−01
2.467E−01
0.769
0.760
0.753
0.748
0.743
1.46E−13
1.71E−13
1.94E−13
2.08E−13
2.24E−13
2043.00
2017.00
1980.00
1915.00
1860.00
4.62E−07
5.41E−07
6.53E−07
8.53E−07
1.03E−06
9.453E−06
1.014E−05
1.128E−05
1.387E−05
1.676E−05
8700
8400
8050
7650
7450
7.41
7.26
7.06
6.74
6.49
1.055E+11
1.203E+11
1.446E+11
1.971E+11
2.547E+11
3.937E+10
3.921E+10
3.908E+10
3.974E+10
4.100E+10
4.314E+10
4.313E+10
4.310E+10
4.351E+10
4.423E+10
2.590E−01
2.778E−01
3.092E−01
3.800E−01
4.593E−01
0.738
0.732
0.727
0.724
0.727
2.47E−13
2.82E−13
3.39E−13
4.62E−13
5.97E−13
1775.00
1670.00
1580.00
1475.00
1378.00
1.31E−06
1.69E−06
2.07E−06
2.59E−06
3.19E−06
2.298E−05
3.510E−05
5.186E−05
8.435E−05
1.363E−04
7250
7050
6900
6720
6560
6.12
5.69
5.34
4.93
4.53
3.788E+11
6.292E+11
9.900E+11
1.726E+12
2.970E+12
4.399E+10
4.922E+10
5.390E+10
6.037E+10
6.824E+10
4.630E+10
5.085E+10
5.535E+10
6.191E+10
7.007E+10
6.297E−01
9.616E−01
1.421E+00
2.311E+00
3.735E+00
0.736
0.752
0.767
0.787
0.809
8.87E−13
1.47E−12
2.32E−12
4.05E−12
6.96E−12
9
Table 4: FALC model, part 2.
h
km
τ500
m
g cm−2
T
K
1278.00
1180.00
1065.00
980.00
905.00
4.02E−06
5.19E−06
7.43E−06
1.03E−05
1.44E−05
2.312E−04
4.022E−04
8.074E−04
1.396E−03
2.314E−03
6390
6230
6040
5900
5755
855.00
805.00
755.00
705.00
650.00
1.85E−05
2.39E−05
3.09E−05
4.00E−05
5.55E−05
3.282E−03
4.710E−03
6.868E−03
1.022E−02
1.624E−02
600.00
560.00
525.00
490.00
450.00
8.53E−05
1.40E−04
2.39E−04
4.29E−04
8.51E−04
400.00
350.00
300.00
250.00
200.00
vt
km s−1
nH
cm−3
np
cm−3
ne
cm−3
Ptot
dyn cm−2
Pgas /Ptot
ρ
g cm−3
4.04
3.53
2.94
2.52
2.19
5.393E+12
1.002E+13
2.164E+13
3.931E+13
6.806E+13
7.768E+10
8.783E+10
9.992E+10
1.068E+11
1.078E+11
7.994E+10
9.083E+10
1.047E+11
1.142E+11
1.192E+11
6.335E+00
1.102E+01
2.212E+01
3.824E+01
6.341E+01
0.837
0.867
0.901
0.924
0.940
1.26E−11
2.35E−11
5.07E−11
9.21E−11
1.59E−10
5650
5490
5280
5030
4750
1.99
1.77
1.54
1.38
1.18
9.931E+13
1.481E+14
2.268E+14
3.560E+14
6.033E+14
1.051E+11
9.014E+10
6.493E+10
3.637E+10
1.375E+10
1.208E+11
1.122E+11
9.690E+10
8.387E+10
9.000E+10
8.993E+01
1.290E+02
1.882E+02
2.799E+02
4.451E+02
0.949
0.958
0.967
0.972
0.978
2.33E−10
3.47E−10
5.31E−10
8.34E−10
1.41E−09
2.538E−02
3.680E−02
5.125E−02
7.149E−02
1.044E−01
4550
4430
4400
4410
4460
1.00
0.89
0.80
0.72
0.65
9.895E+14
1.478E+15
2.078E+15
2.898E+15
4.192E+15
5.368E+09
2.825E+09
2.424E+09
2.618E+09
3.600E+09
1.255E+11
1.767E+11
2.413E+11
3.300E+11
4.714E+11
6.954E+02
1.008E+03
1.404E+03
1.959E+03
2.860E+03
0.983
0.986
0.989
0.991
0.993
2.32E−09
3.46E−09
4.87E−09
6.79E−09
9.82E−09
1.98E−03
4.53E−03
1.01E−02
2.20E−02
4.73E−02
1.664E−01
2.626E−01
4.103E−01
6.344E−01
9.705E−01
4560
4660
4770
4880
4990
0.55
0.52
0.55
0.63
0.79
6.549E+15
1.012E+16
1.545E+16
2.331E+16
3.476E+16
6.715E+09
1.267E+10
2.604E+10
5.605E+10
1.253E+11
7.344E+11
1.134E+12
1.737E+12
2.645E+12
4.004E+12
4.558E+03
7.194E+03
1.124E+04
1.738E+04
2.659E+04
0.995
0.996
0.995
0.994
0.990
1.53E−08
2.37E−08
3.62E−08
5.46E−08
8.14E−08
175.00
150.00
125.00
100.00
75.00
6.87E−02
9.92E−02
1.42E−01
2.02E−01
2.87E−01
1.195E+00
1.466E+00
1.790E+00
2.174E+00
2.625E+00
5060
5150
5270
5410
5580
0.90
1.00
1.10
1.20
1.30
4.211E+16
5.062E+16
6.024E+16
7.107E+16
8.295E+16
2.028E+11
3.579E+11
7.119E+11
1.485E+12
3.281E+12
4.945E+12
6.153E+12
7.770E+12
1.003E+13
1.353E+13
3.274E+04
4.017E+04
4.905E+04
5.957E+04
7.192E+04
0.988
0.985
0.983
0.980
0.977
9.87E−08
1.19E−07
1.41E−07
1.67E−07
1.94E−07
50.00
35.00
20.00
10.00
0.00
4.13E−01
5.22E−01
6.75E−01
8.14E−01
1.00E+00
3.148E+00
3.496E+00
3.869E+00
4.132E+00
4.404E+00
5790
5980
6180
6340
6520
1.40
1.46
1.52
1.55
1.60
9.558E+16
1.027E+17
1.098E+17
1.142E+17
1.182E+17
7.614E+12
1.439E+13
2.588E+13
3.926E+13
6.014E+13
1.980E+13
2.779E+13
4.064E+13
5.501E+13
7.697E+13
8.624E+04
9.578E+04
1.060E+05
1.132E+05
1.207E+05
0.975
0.973
0.972
0.971
0.971
2.24E−07
2.40E−07
2.57E−07
2.68E−07
2.77E−07
–10.00
–20.00
–30.00
–40.00
–50.00
1.25E+00
1.61E+00
2.14E+00
2.95E+00
4.13E+00
4.686E+00
4.975E+00
5.269E+00
5.567E+00
5.869E+00
6720
6980
7280
7590
7900
1.64
1.67
1.70
1.73
1.75
1.219E+17
1.246E+17
1.264E+17
1.280E+17
1.295E+17
9.269E+13
1.536E+14
2.597E+14
4.249E+14
6.668E+14
1.107E+14
1.730E+14
2.807E+14
4.480E+14
6.923E+14
1.284E+05
1.363E+05
1.444E+05
1.525E+05
1.608E+05
0.970
0.970
0.970
0.971
0.971
2.86E-07
2.92E-07
2.96E-07
3.00E-07
3.04E-07
–60.00
–70.00
–80.00
–90.00
–100.00
5.86E+00
8.36E+00
1.20E+01
1.70E+01
2.36E+01
6.174E+00
6.481E+00
6.790E+00
7.102E+00
7.417E+00
8220
8540
8860
9140
9400
1.77
1.79
1.80
1.82
1.83
1.307E+17
1.317E+17
1.325E+17
1.337E+17
1.351E+17
1.022E+15
1.515E+15
2.180E+15
2.942E+15
3.826E+15
1.050E+15
1.546E+15
2.215E+15
2.979E+15
3.867E+15
1.691E+05
1.776E+05
1.860E+05
1.946E+05
2.032E+05
0.972
0.972
0.973
0.973
0.974
3.06E-07
3.09E-07
3.10E-07
3.13E-07
3.17E-07
10
2
Continuous spectrum from the solar atmosphere
We now turn to the formation of the solar continuum radiation, concentrating on the visible
and near-infrared parts of its spectrum.
Figure 4: Subrahmanyan Chandrasekhar (1910–1995) is regarded by many as the greatest astronomer of
the twentieth century. He also played an important role in the topic of this exercise, undertaking a very
lengthy and difficult calculation (not a computation, no computers yet at that time!) of the bound-free
and free-free extinction cross-sections of the H− ion, together with F.H. Breen. Their result, published
in Chandrasekhar and Breen (1946), will be duplicated in one of your graphs below. The shape of the
extinction curve, with a peak near λ = 10000 Å and a minimum at the H− bound-free ionization limit at
λ = 16000 Å, reproduced the observed spectral variation of the unknown solar continuum extinction as
derived already by G.F.W. Mulders at Utrecht in 1935 for λ = 4000 – 25000 Å. The brilliant suggestion
that H− might cause this extinction, instead of the ensemble of unknown metal edges that was earlier
postulated, was given by Wildt (1939); the proof came with Chandrasekhar’s laborious quantummechanical evaluation. It revolutionized the understanding of cool-star atmospheres by much increasing their
hydrogen-to-metals ratio and gas pressure compared with the earlier metallic-absorption modeling of Biermann, Unsöld and Pannekoek. A more detailed description is found in Hearnshaw (1986). Picture copied
from the “Astronomy Picture of the Day” website (http://antwrp.gsfc.nasa.gov/apod/ap950901.html).
2.1
Observed solar continua
Table 5 specifies the continuum radiation emitted by the sun in the wavelength range λ =
0.2 − 5 µm, taken from Allen (1976). The table caption defines four different quantities
as a function of wavelength, respectively the radially emergent intensity and the astrophysical flux in the solar continuum with and without smoothed lines. The units are cgs with
∆λ = 1 µm for the spectral bandwidth. The table is available as file solspect.dat at
http://www.astro.uu.nl/∼rutten/education/rjr-material/ssb/.
• Write IDL code to read Table 5.
• Plot the four spectral distributions together in one figure over the range λ = 0 − 2 µm. Use
a statement such as
print,’ max(Ic) =’,max(Icont), ’ at ’,wav(where(Icont eq max(Icont)))
to check that the continuum intensity reaches Iλc = 4.6 × 1010 erg cm−2 s−1 ster−1 µm−1 at
λ = 0.41 µm. Explain why the four distributions share the same units and discuss the
differences between them.
11
• Convert these spectral distributions into values per frequency bandwidth ∆ν = 1 Hz. Plot
these also against wavelength. Check: peak Iνc = 4.21 × 10−5 erg cm−2 s−1 ster−1 Hz−1 at
λ = 0.80 µm.
• Write an IDL function planck.pro (or use your routine from Exercises “Stellar Spectra A:
Basic Line Formation”, or use mine) that computes the Planck function in the same units.
Try to fit a Planck function to the solar continuum intensity. What rough temperature
estimate do you get?
• Invert the Planck function analytically to obtain an equation which converts an intensity
distribution Iλ into brightness temperature Tb (defined by Bλ (Tb ) ≡ Iλ ). Code it as an
IDL function and use that to plot the brightness temperature of the solar continuum against
wavelength (with the plot,/ynozero keyword). Discuss the shape of this curve. It peaks
near λ = 1.6 µm. What does that mean for the radiation escape at this wavelength?
Figure 5: Continuous extinction coefficients κcν for hydrogen and helium, per neutral hydrogen atom
and per unit electron pressure, for the depth τ0 = 1 (continuum optical depth at λ = 500 nm) in the
photosphere of a solar-like dwarf star. The coefficients κ are here measured per neutral hydrogen atom in
whatever state of excitation, assuming Saha-Boltzmann population partitioning, and normalized by the
electron pressure Pe = ne k T . The cross-sections are in units of 10−26 cm2 (not cm2 as specified in the
y-axis label); stimulated emission was not included in the computation of these curves. The H− curve
shows the bound-free Balmer, Paschen and Brackett edges, plus part of the Pfund edge at right. The
curves do not extend beyond the Balmer edge at left where the neglected metal edges become important.
Thomson scattering is also neglected. From page 140 of Gray (1992).
12
Figure 6: Ionization edges for a selection of abundant elements. The triangular symbols depict boundfree continuum edges in the form of schematic hydrogenic ν −3 decay functions above each ionization
threshold. The plot shows the edge distribution over ionization energy χ1c (along the bottom) or threshold
wavelength (along the top) and the logarithmic product of the element abundance A12 ≡ log(nE /nH +
12) and the bound-free cross-section per particle at threshold σ/σH (vertically). Thus, it shows the
relative importance of the major bound-free edges throughout the spectrum. They are all at ultraviolet
wavelengths and do not contribute extinction in the visible and infrared. The plus signs indicate important
first-ion edges; higher ionization stages produce edges at yet shorter wavelengths. The abundance values
come from Engvold (1977), the ionization energies from Novotny (1973), the cross-sections from Baschek
and Scholz (1982). Thijs Krijger production following unpublished lecture notes by E.H. Avrett, taken
from the lecture notes available at http://www.astro.uu.nl/∼rutten.
2.2
Continuous extinction
−
We will assume that H (a hydrogen atom with an extra electron) is the major provider of
continuous extinction in the solar atmosphere. This is quite a good assumption for the solar
photosphere for wavelengths λ > 0.5 µm (5000 Å). The second-best extinction provider are H I
bound-free interactions, at only a few percent. This may be seen in Figure 5 taken from Gray
(1992).
Below λ = 500 nm there is heavy line crowding (not added in Figure 5) which acts as a quasicontinuum. Below λ = 365 nm the Balmer bound-free edge provides large extinction, and at
yet shorter wavelengths the bound-free ionizaton edges of various metals (Al I, Mg I, Fe I, Si I,
C I) provide steep extinction increase yet before the H I Lyman continuum sets is, as may be
expected from Figure 6. In this exercise we will neglect these contributions by evaluating only
−
the H extinction and the extinction due to scattering off free electrons (Thomson scattering).
−
IDL function exthmin.pro evaluates polynomial fits for H extinction that are given on
−
page 135 ff of Gray (1992). The routine delivers the total (bound-free plus free-free) H ex-
13
−
tinction in units of cm2 per neutral hydrogen atom (not per H ion!). LTE is assumed in the
−
computation of the H ion density relative to the neutral hydrogen density (through a Saha
equation where the neutral atom takes the place of the ionized state, see page 135 of Gray 1992).
• Pull exthmin.pro over and compare it to Gray’s formulation.
−
• Plot the wavelength variation of the H extinction for the FALC parameters at h = 0 km (see
Tables 3–4). This plot reproduces the result of Chandrasekhar and Breen (1946). Compare
it to Gray’s version in Figure 5.
• Hydrogenic bound-free edges behave just as H I with maximum extinction at the ionization
limit and decay ∼ λ3 for smaller wavelengths, as indeed shown by the H I curve in Figure 5.
−
The H bound-free extinction differs strongly from this pattern. Why is it not hydrogenic
although due to hydrogen?
• How should you plot this variation to make it look like the solar brightness temperature
variation with wavelength? Why?
• Read in the FALC model atmosphere (copy the reading code in readfalc.pro into your IDL
program). Note that the column nH in the FALC model of Tables 3–4 is the total hydrogen
density, summing neutral atoms and free protons (and H2 molecules but those are virtually
absent). This is seen by inspecting the values of nH and np at the top of the FALC table
−
where all hydrogen is ionised. Gray’s H extinction is measured per neutral hydrogen atom,
so you have to multiply the height-dependent result from exthmin(wav,temp,eldens) with
−
nneutral H ≈ nH (h) − np (h) to obtain extinction αλ (H ) measured per cm path length (or
−
cross-section per cubic cm) at every height h. Plot the variation of the H extinction per cm
with height for λ = 0.5 µm. This plot needs to be logarithmic in y, why?
• Now add the Thomson scattering off free electrons to the extinction per cm. The Thomson
cross-section per electron is the same at all wavelengths and is given by
σ T = 6.648 × 10−25 cm2 .
(5)
With which height-dependent quantity do you have to multiply this number to obtain extinction per cm? Overplot this contribution to the continuous extinction αλc (h) in your graph
and then overplot the total continuous extinction too. Explain the result.
2.3
Optical depth
Knowing the stratification (FALC model) and the continuous extinction as function of height,
we may now compute the corresponding optical depth scale given by:
τλ (h0 ) ≡ −
Z
h0
∞
αλc dh
(6)
at any height h0 . Note that FALC is tabulated in reverse order, corresponding to the −h
direction. The IDL library routine INT TABULATED supplies a closed five-point NewtonCotes integration routine that might be applied, but I found that the simple trapezium rule
gives nearly identical results. (It is often much safer to use trapezoidal integration than higherorder schemes. The latter may give wildly wrong answers by fitting the samples with large
excursions in between samples. A good recipe is to always stick to trapezoidal integration and
to refine the grid if higher precision is desired.)
14
• Integrate the extinction at λ = 500 nm to obtain the τ500 scale and compare it graphically
to the FALC τ500 scale. Here is my code, with ext(ih) denoting the continuous extinction
per cm at λ = 500 nm and at height h(ih):
; compute and plot tau at 500 nm, compare with FALC tau5
tau=fltarr(nh)
for ih=1,nh-1 do tau(ih)=tau(ih-1)+$
0.5*(ext(ih)+ext(ih-1))*(h(ih-1)-h(ih))*1E5
plot,h,tau,/ylog,$
xtitle=’height [km]’,ytitle=’tau at 500 nm’
oplot,h,tau5,linestyle=2
2.4
Emergent intensity and height of formation
We are now ready to compute the intensity of the radiation that emerges from the center of the
solar disk (in the radial direction from the solar sphere). It is given (assuming plane-parallel
stratification) by:
Z
∞
Iλ =
0
Sλ e−τλ dτλ .
(7)
It is interesting to also inspect the intensity contribution function
dIλ
= Sλ e−τλ αλ
dh
(8)
which shows the contribution of each layer to the emergent intensity. Its weighted mean defines
the “mean height of formation”:
R∞
R∞
h Sλ e−τλ dτλ
0R h (dIλ /dh) dh
<h> ≡
= R0 ∞
.
∞
−τ
0
(dIλ /dh) dh
0
Sλ e
λ
dτλ
(9)
Here is my code to compute and diagnose these quantities for a given wavelength (wl in µm)
assuming LTE and using trapezoidal integration:
; emergent intensity at wavelength wl (micron)
ext=fltarr(nh)
tau=fltarr(nh)
integrand=fltarr(nh)
contfunc=fltarr(nh)
int=0.
hint=0.
for ih=1,nh-1 do begin
ext(ih)=exthmin(wl*1E4,temp(ih),nel(ih))*(nhyd(ih)-nprot(ih))$
+0.664E-24*nel(ih)
tau(ih)=tau(ih-1)+0.5*(ext(ih)+ext(ih-1))*(h(ih-1)-h(ih))*1E5
integrand(ih)=planck(temp(ih),wl)*exp(-tau(ih))
15
int=int+0.5*(integrand(ih)+integrand(ih-1))*(tau(ih)-tau(ih-1))
hint=hint+h(ih)*0.5*(integrand(ih)+integrand(ih-1))*(tau(ih)-tau(ih-1))
contfunc(ih)=integrand(ih)*ext(ih)
endfor
hmean=hint/int
• The code above sits in file emergint.pro. Copy it into your IDL program and make it work
setting wl=0.5.
• Compare the computed intensity at λ = 500 nm with the observed intensity, using a statement
such as
print,’ observed cont int = ’,Icont(where(wl eq wav))
to obtain the latter.
• Plot the peak-normalized contribution function against height and compare its peak location
with the mean height of formation.
• Repeat the above for λ = 1 µm, λ = 1.6 µm, and λ = 5 µm. Discuss the changes of the
contribution functions and their cause.
• Check the validity of the LTE Eddington-Barbier approximation Iλ ≈ Bλ (T [τλ = 1]) by
comparing the mean heights of formation with the τλ = 1 locations and with the locations
where Tb = T (h).
2.5
Disk-center intensity
The solar disk-center intensity spectrum can now be computed by repeating the above in a big
loop over wavelength.
• Compute the emergent continuum over the wavelength range of Table 5.
• Compare it graphically with the observed solar continuum in Table 5 and file solspect.dat.
2.6
Limb darkening
The code is easily modified to give the intensity that emerges under an angle µ = cos θ, in
plane-parallel approximation given by:
Z
Iλ (0, µ) =
0
∞
Sλ e−τλ /µ dτλ /µ.
(10)
• Repeat the intensity evaluation using (10) within an outer loop over µ = 0.1, 0.2, . . . , 1.0.
• Plot the computed ratio Iλ (0, µ)/Iλ (0, 1) at a few selected wavelengths, against µ and also
against the radius of the apparent solar disk r/R = sin θ. Explain the limb darkening and
its variation with wavelength.
16
2.7
Flux integration
The emergent intensity may now be integrated over emergence angle to get the emergent astrophysical flux:
Z
1
Fλ (0) = 2
0
Iλ (0, µ) µ dµ.
(11)
The problem arises that (10) cannot be evaluated at µ = 0. The naive way to get F is simply
to integrate Iλ (0, µ) trapezoidally over the ten angles µ = 0.1, 0.2, . . . , 1.0 at which you have it
already, but that produces too much flux by ignoring the relatively low contribution from the
outer limb (µ = 0.0 − 0.1, sin θ = 0.995 − 1.0). This integral is therefore better evaluated with
“open quadrature”, an integration formula neglecting the endpoints. Classical equal-spacing
recipes are the Open Newton-Cotes quadrature formulae but it is much better to use nonequidistant Gaussian quadrature (see the chapter “Integration of functions” in Numerical Recipes
by Press et al. 1986). The defining formula is:
Z
+1
f (x) dx ≈
−1
n
X
wi f (xi )
(12)
i=1
and the required abscissa values xi and weights wi are tabulated for n = 2 − 10 and even higher
orders on page 916 of Abramowitz and Stegun (1964). Three-point Gaussian integration is
sufficiently accurate for the emergent flux integration.
• Compute the emergent solar flux and compare it to the observed flux in Table 5 and file
solspect.dat. Here is my code (file gaussflux.pro):
; ===== three-point Gaussian integration intensity -> flux
; abscissae + weights n=3 Abramowitz & Stegun page 916
xgauss=[-0.7745966692,0.0000000000,0.7745966692]
wgauss=[ 0.5555555555,0.8888888888,0.5555555555]
fluxspec=fltarr(nwav)
intmu=fltarr(3,nwav)
for imu=0,2 do begin
mu=0.5+xgauss(imu)/2.
; rescale xrange [-1,+1] to [0,1]
wg=wgauss(imu)/2.
; weights add up to 2 on [-1,+1]
for iw=0,nwav-1 do begin
wl=wav(iw)
@emergintmu.pro
; old trapezoidal integration I(0,mu)
intmu(imu,iw)=int
fluxspec(iw)=fluxspec(iw)+wg*intmu(imu,iw)*mu
endfor
endfor
fluxspec=2*fluxspec
; no !pi, AQ has flux F, not {\cal F}
plot,wav,fluxspec,$
xrange=[0,2],yrange=[0,5E10],$
xtitle=’wavelength [micron]’,ytitle=’solar flux’
oplot,wav,Fcont,linestyle=2
xyouts,0.5,4E10,’computed’
xyouts,0.35,1E10,’observed’
17
2.8
Discussion
You have succeeded in explaining the solar continuum at visible and infrared wavelengths as
−
largely due to H extinction. This is what Chandrasekhar and Breen (1946) accomplished after
−
Wildt (1939) suggested that H might be the long-sought source of photospheric extinction.
The bound-free contribution from H I is small since the required populations of n = 3 (Paschen
continuum) and n = 4 (Brackett continuum) are small (Boltzmann partitioning; cf. Figure 5).
Other elements do not contribute continuous extinction in the visible and infrared because their
major bound-free edges are all in the ultraviolet (Figure 6).
The large opacity of the solar photosphere is therefore due to the combination of abundant
neutral hydrogen atoms, the presence of free electrons (that come mostly from other particle
species with lower ionization energy), and the large polarization of the simple electron-proton
combination which produces a large cross-section for Coulomb interactions between hydrogen
atoms and free electrons. There are no free electrons in our own atmosphere and the molecules
making up the air around us possess much better Coulomb shielding; terrestrial air is therefore far
less opaque at visible wavelengths than the solar gas. In the infrared, air is similarly transparant
in some spectral windows correspondig to energy bands in which the molecules can’t rotate or
vibrate.
In general, radiative transfer in stellar atmospheres is a difficult subject because in our daily
physical experience, gases ought to be transparent. Even though stars are fully gaseous, they are
−
far from transparent. H extinction makes even the atmospheres of cool stars less transparent
than what we would expect for gases so tenuous.
−
A final note: while LTE is an excellent assumption for H bound-free processes and exactly
−
valid (as long as the Maxwell distribution holds) for H free-free processes, it is not at all
valid for Thomson scattering. The process source function for purely coherent (monochromatic)
R
scattering is not given by Bλ (T ) but by the angle-averaged intensity Jλ ≡ (1/4π) Iλ dΩ.
Obtaining Jλ from such an integration requires knowledge of the intensity Iλ (h, θ) locally and
in all directions θ, not just the emergent one at τ = 0 computed here. This evaluation will
be treated extensively in exercises “Stellar Spectra C: NLTE Line Formation”. For now, such
sophistication is too much work, and in any case Thomson scattering is not important for the
formation of the solar continuum intensity in this wavelength range (as you can see by deleting or
doubling its contribution in your code). However, Thomson scattering dominates the continuous
−
extinction in the photospheres of hot stars in which hydrogen is largely ionized so that the H
contribution vanishes while the electron contribution rises.
18
Table 5: Solar spectral distribution, from Allen (1976). Fλ = astrophysical flux at the solar surface with
spectral irregularities smoothed; Fλ0 = astrophysical flux at the solar surface for the continuum between
lines; Iλ = radially emergent intensity at the solar surface with spectral irregularities smoothed; Iλ0 =
radially emergent intensity at the solar surface for the continuum between lines. “Astrophysical” flux Fλ
is defined as π Fλ ≡ Fλ with Fλ the net outward flow of energy through a stellar surface element. The
astrophysical flux is often prefered because it has Fλ =< Iλ > with < Iλ > the intensity averaged over
the stellar disk received by a distant observer. It has the same dimension as intensity, whereas Fλ is not
measured per steradian.
λ
µm
Fλ
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.48
0.50
0.55
0.60
0.65
0.70
0.75
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
4.0
5.0
0.02
0.07
0.09
0.19
0.35
0.76
1.10
1.33
1.46
1.57
1.46
1.53
2.05
2.46
2.47
2.46
2.66
2.90
2.93
2.86
2.83
2.72
2.58
2.31
2.10
1.88
1.69
1.33
1.08
0.73
0.512
0.375
0.248
0.171
0.0756
0.0386
0.0130
0.0055
10
10
Fλ0
Iλ
erg cm−2 s−1 µm−1 ster−1
0.04
0.11
0.2
0.4
0.7
1.36
1.90
2.11
2.30
2.50
2.85
3.10
3.25
3.30
3.35
3.36
3.38
3.40
3.35
3.30
3.19
2.94
2.67
2.42
2.13
1.91
1.70
1.36
1.09
0.74
0.512
0.375
0.248
0.171
0.0756
0.0386
0.0130
0.0055
19
0.03
0.14
0.18
0.37
0.59
1.21
1.61
1.91
2.03
2.33
2.14
2.20
2.9
3.43
3.42
3.35
3.58
3.86
3.88
3.73
3.63
3.40
3.16
2.78
2.50
2.22
1.96
1.53
1.21
0.81
0.564
0.403
0.268
0.183
0.081
0.041
0.0135
0.0057
Iλ0
0.04
0.20
0.30
0.5
1.19
2.15
2.83
3.01
3.20
3.62
4.1
4.4
4.58
4.60
4.59
4.55
4.54
4.48
4.40
4.31
4.08
3.68
3.27
2.88
2.53
2.24
1.97
1.55
1.23
0.81
0.564
0.403
0.268
0.183
0.081
0.041
0.0135
0.0057
20
3
Spectral lines from the solar atmosphere
We now turn to the formation of spectral lines in the solar spectrum. We will concentrate on
the formation of the Na I D1 line at λ = 589.0 nm.
Figure 7: Albrecht O.J. Unsöld (1905 – 1995) was the first, in 1941 at Kiel, to study stellar composition
(“abundance analysis”) in the detail permitted by proper physical understanding of line formation (for
the B0 dwarf τ Scorpii, using spectrograms taken during a six-month visit to the US just before the
second world war). He used the so-called “coarse” analysis based on Minnaert’s curve of growth for a
Schuster-Schwarzschild single “reversing layer” as in the third exercise of “Stellar Spectra A: Basic Line
Formation”. Later, he led a school of astrophysicists at Kiel in “fine analysis”, combining LTE line
formation with non-grey stratified atmosphere modeling just as you do in this exercise. The technique
was developed by L.H. Aller, C. de Jager and others in the fifties, and has dominated stellar abundance
analysis throughout the second half of the twentieth century. Unsöld spelled out its physical basis in
great detail in his “Physik der Sternatmosphären” (1955). Picture copied from Hearnshaw (1986).
3.1
Observed Na D line profiles
The solar Na I D lines are shown in Figure 8.
• Pull the data that went into making Figure 8 over from ftp://ftp.noao.edu/fts/visatl.
Concatenate the files covering the two Na I D lines into a single file, deleting the overlaps.
• Write IDL code to read these data. You will need the first and third columns (see the
README explanation).
3.2
Na D wavelengths
According to the README file, the first column specifies frequencies but these numbers are
actually wavenumbers σ = 1/λvac in cm−1 . If you invert them into wavelengths they differ by
a few Ångstrom (in this wavelength region) from the standard-air wavelengths λair habitually
used for spectral lines with λ > 2000 Å.
• Plot the solar Na I D lines against vacuum wavelength at various dispersions.
• Find the vacuum wavelengths of their minima (use the IDL where and min functions).
21
Figure 8: A page out of the solar disk-center intensity atlas of Wallace et al. (1998). The intensity scale
(vertical) is in relative units, normalized to the local continuum intensity between the lines. The lower
horizontal scales specify wavenumbers in cm−1 , the upper ones vacuum wavelengths in Å. The atlas
was made with the Fourier Transform Spectrometer at the McMath-Pierce solar telescope at Kitt Peak
(Brault 1978). Telluric lines have been removed. Both the atlas pages and the input data are available
at ftp://ftp.noao.edu/fts/visatl.
• Check that the Na I D wavelengths tabulated in the solar spectrum line list of Moore
et al. (1966) (computer-readable at ftp://ftp.noao.edu/fts/linelist/Moore) are λ =
5895.94 Å for Na I D1 and λ = 5889.97 Å for Na I D2 , respectively. Check the identification
of a few blends (other lines) in Figure 8 with the entries in this table2 .
• The Astrolib3 routines airtovac and vactoair convert air into vacuum wavelengths and
vice versa. A reasonably accurate transformation is also given by
λair = 0.99972683 λvac + 0.0107 − 196.25/λvac
(13)
with both λ’s in Å, from Neckel (1999)4 . Use this equation or routine vactoair to plot the
Na I D lines against air wavelength.
2
The solar spectrum line list was constructed at Utrecht. All equivalent widths were measured by hand,
hard labour for many during two decades. The line identifications were made from laboratory wavelength tables
compiled by Mrs. Charlotte Moore and co-workers at the US National Bureau of Standards.
3
IDL astrolib at http://idlastro.gsfc.nasa.gov/homepage.html/.
4
This publication is the announcement of two solar atlases, respectively for disk-averaged and disk-center
intensity, constructed from the same FTS data from which the NSO atlases were made. These also specify absolute
intensities. They are available through anonymous ftp at ftp.hs.uni-hamburg.de, /pub/outgoing/FTS.
22
3.3
LTE line formation
We will now compute the solar Na I D1 line assuming the FALC model atmosphere and LTE for
the line source function. Since LTE holds already for the continuum processes at these wave−
lengths (being dominated by H bound-free transitions), the assumption of LTE line formation
implies that you can simply set Sλl = Sλc = Sλtotal = Bλ (T ). What remains is first to evaluate the
line extinction as a function of height and wavelength, and then to add that to the continuous
extinction in the integration loop of the previous exercise. The outer loop over wavelength then
has to sample the Na I D1 profile (but not as finely as the atlas data spacing).
3.4
Line extinction
The monochromatic line extinction per cm path lenght for a bound-bound transition between a
lower level l and an upper level u is given by:
√ 2 2
πe λ
nLTE
H(a, v)
bu −hc/λkT
l
l
αλ =
bl
NH AE flu
1−
e
,
(14)
me c c
NE
∆λD
bl
which holds generally when the line broadening is described by the Voigt function H(a, v), a
valid assumption for the Na I D lines (but wrong for hydrogen lines which are broadened with the
Holtsmark distribution). For LTE the population departure coefficients of the lower and upper
levels are bl = bu = 1. The LTE population fraction nLTE
/NE (lower level population scaled by
l
the total element population) is given by the combined Saha and Boltzmann distributions
Ur ≡
nr,s
Nr
Nr+1
Nr
=
=
X
gr,s e−χr,s /kT
s
gr,s
e−χr,s /kT
Ur
1 2Ur+1 2πme kT 3/2 −χr /kT
e
,
Ne Ur
h2
(15)
(16)
(17)
where s is the level counter and r the ionization stage counter.
Some numbers for the Na I D1 and Na I D2 lines:
– sodium ionization energies χ1 = 5.139 eV, χ2 = 47.29 eV, χ3 = 71.64 eV (Appendix D of
Gray 1992);
– the Na I D1 and Na I D2 lower-level excitation energy χ1,1 = 0 eV (shared ground state);
– the Na I D1 and Na I D2 lower-level statistical weight g1,1 = 2 (ground state 3s 2S1/2 , g =
2J + 1);
– the oscillator strengths are flu = 0.318 for Na I D1 , flu = 0.631 for Na I D2 ;
– the Na I partition function defined by (15) is given in Appendix D of Gray (1992) as
log UNa I (T ) ≈ c0 + c1 log θ + c2 log2 θ + c3 log3 θ + c4 log4 θ with θ ≡ 5040./T and c0 = 0.30955,
c1 = −0.17778, c2 = 1.10594, c3 = −2.42847 and c4 = 1.70721;
– the Na II and Na III partition functions are well approximated by the statistical weights of
the ion ground states, respectively UNa II = 1, UNa III = 6 (Allen 1976);
– the sodium abundance is ANa = NNa /NH = 1.8 × 10−6 (Allen 1976).
23
3.5
Line broadening
The Voigt function H(a, v) describes the extinction profile shape and is defined by:
H(a, v) ≡
y =
v =
a =
a
π
Z
+∞
−∞
2
e−y
dy
(v − y)2 + a2
ξ λ0
c ∆λD
λ − λ0
∆λD
λ2 γ
,
4πc ∆λD
(18)
(19)
(20)
(21)
where ξ is velocity along the line of sight and a the damping parameter. The Dopplerwidth
∆νD is not only set by the thermal broadening but includes also the microturbulent “fudge
parameter” vt (column vt in FALC Tables 3–4) through defining it as:
λ0
∆λD ≡
c
s
2kT
+ vt2
m
(22)
where m is the mass of the line-causing particle in gram, for sodium mNa = 22.99 × 1.6605 ×
10−24 g (http://physics.nist.gov/cuu/Constants/).
The Voigt function represents the convolution (smearing) of a Gauss profile with a Lorentz
profile and therefore has a Gaussian shape close to line center (v = 0) due to the thermal
Doppler shifts (“Doppler core”) and extended Lorentzian wings due to disturbances by other
particles (“damping wings”). A reasonable approximation is obtained by taking the sum rather
than the convolution of the two profiles:
a
.
π v2
(23)
√ H(a, v)
∆λD π
(24)
2
H(a, v) ≈ e−v + √
The area-normalized version
V (a, v) ≡
1
is available in IDL as function VOIGT(a,v) but it works correctly only for positive v, so use it
with abs(v).
The damping parameter a may be approximated by taking only Van der Waals broadening into
account in (21). Figure 11.6 of Gray (1992) shows this by comparing Van der Waals broadening
with natural broadening and Stark broadening for the Na I D lines throughout a solar model.
The classical evaluation recipe of Van der Waals broadening by Unsöld (1955) is (cf. Warner
1967):
log γvdW ≈ 6.33 + 0.4 log(ru2 − rl2 ) + log Pg − 0.7 log T,
(25)
where the mean square radii r2 of the upper and lower level are usually estimated from the
hydrogenic approximation of Bates and Damgaard (1949)
r2 =
n∗ 2 ∗ 2
5n
+
1
−
3l(l
+
1)
2Z 2
24
(26)
with r2 measured in atomic units, l the angular quantum number of the level and n∗ its effective
(hydrogen-like) principal quantum number given by
n∗ 2 = R
Z2
E∞ − En
(27)
in which the Rydberg constant R = 13.6 eV = 2.18 × 10−11 erg, Z is the ionization stage (Z = 1
for Na I, Z = 2 for Na II, etc) and E∞ − En is the ionization energy from the level (compute the
excitation energy of the upper level from the line-center wavenumber). The common Na I D1
O
O
and Na I D2 lower level (3s 2S1/2 ) has l = 0, the upper levels (3p 2 P1/2 and 3p 2 P3/2 ) have l = 1.
3.6
Implementation
• Split the above equations modularly in IDL functions, for example:
parfunc_Na(temp)
saha_Na(temp,eldens,chi_ion)
boltz_Na(temp,chi_level)
sahaboltz_Na(temp,eldens,chi_ion,chi_level)
dopplerwidth(wav,temp,vmicro,atmass)
gammavdw(temp,pgas,ru,rl)
rsq(r,l,chi_ion.chi_level).
• Combine calls of these functions into one that returns the Na I D1 line extinction, for example
NaD1_ext(wav,temp,eldens,nhyd,vmicro).
3.7
Computed Na D1 line profile
You are now ready to model the solar Na I D1 line.
• Add the Na I D1 line extinction to the continuous extinction in your integration code from
the previous exercise and compute the disk-center Na I D1 profile.
• Compare the computed line profile to the observed line profile and discuss the differences.
Explain why your computed profile has a line-center reversal.
• Traditionally, stellar abundance determiners vary a collisional enhancement factor E by which
γvdW is multiplied in ad-hoc fashion in order to obtain a better fit of the line wings. Try the
same5 .
3.8
Discussion
Your code constitutes a 1960–style solar line synthesis program which is actually quite good for
photospheric lines and is easily generalized to other atomic species. However, your computed
Na I D1 line core doesn’t reach as deep as the observed one which does not show an intensity
reversal. Its bad reproduction shows that the assumption of LTE breaks down for the core of
this line. No wonder, the solar Na I D lines are strong scatterers (small ε) with large NLTE
source function sensitivity to Jλ rather than to Bλ at heights around and above the temperature
5
A better recipe than the classical Unsöld one is now available for lines of neutral stages from Paul Barklem
at http://www.astro.uu.se/∼barklem/.
25
minimum where their cores originate. In fact, their formation closely follows the two-level atom
description for resonance scattering in which both Jλ and Sλ drop down to a value of only about
√
ε Bλ at the surface, displaying standard NLTE scattering behavior. Thus, it is time to turn
to the sequel exercises “Stellar Spectra C: NLTE Line Formation” in which you will apply more
advanced numerical techniques permitting deviations from LTE.
26
References
Abramowitz, M. and Stegun, I.: 1964, Handbook of Mathematical Functions, U.S. Dept. of
Commerce, Washington
Allen, C. W.: 1976, Astrophysical Quantities, Athlone Press, Univ. London
Anders, E. and Grevesse, N.: 1989, Geochim. Cosmochim. Acta 53, 197
Baschek, B. and Scholz, M.: 1982, in K.-H. Hellwege (Ed.), Landolt-Börnstein New Series, Stars
and Star Clusters, Group VI Vol. 2b, Astronomy and Astrophysics, Springer, Heidelberg,
p. 91
Bates, D. R. and Damgaard, A.: 1949, Phil. Trans. R. Soc. London 242, 101
Brault, J. W.: 1978, in G. Godoli, G. Noci, and A. Righini (Eds.), Future solar optical observations: needs and constraints, Procs. JOSO Workshop, Osservazioni e Memorie Oss. Astrof.
Arcetri, Florence, p. 33
Chandrasekhar, S. and Breen, F.: 1946, Astrophys. J. 104, 430
Eddington, A. S.: 1926, The Internal Constitution of the Stars, Dover Pub., New York
Engvold, O.: 1977, Physica Scripta 16, 48
Fontenla, J. M., Avrett, E. H., and Loeser, R.: 1993, Astrophys. J. 406, 319
Gray, D. F.: 1992, The Observation and Analysis of Stellar Photospheres, Cambridge Univ.
Press, U.K. (second edition)
Hearnshaw, J. B.: 1986, The analysis of starlight. One hundred and fifty years of astronomical
spectroscopy, Cambridge Univ. Press, Cambridge UK
Moore, C. E., Minnaert, M. G. J., and Houtgast, J.: 1966, The Solar Spectrum 2935 Å to
8770 Å. Second Revision of Rowland’s Preliminary Table of Solar Spectrum Wavelengths,
NBS Monograph 61, National Bureau of Standards, Washington
Neckel, H.: 1999, Solar Phys. 184, 421
Novotny, E.: 1973, Introduction to stellar atmospheres and interiors, Oxford Univ. Press, New
York
Parker, E. N.: 1958, Astrophys. J. 128, 664
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: 1986, Numerical Recipes,
Cambridge Univ. Press, Cambridge UK
Ulrich, R. K.: 1970, Astrophys. J. 162, 933
Unsöld, A.: 1955, Physik der Sternatmosphären, Springer Verlag, Berlin (second edition)
Vernazza, J. E., Avrett, E. H., and Loeser, R.: 1981, Astrophys. J. Suppl. Ser. 45, 635
Wallace, L., Hinkle, K., and Livingston, W.: 1998, An Atlas of the Spectrum of the Solar Photosphere from 13,500 to 28,000 cm−1 (3570 to 7405 Å), Technical Report 98-001, National
Solar Observatory, Tucson
Warner, B.: 1967, Mon. Not. R. Astron. Soc. 136, 381
Wildt, R.: 1939, Astrophys. J. 89, 295
27