The calculation of expected spectral line strengths and profiles is a
powerful tool for the analysis of the solar atmosphere, and other stellar
atmospheres. We present here a recipe in seven easy steps for the development
of such spectral synthesis software.Comment: 4 pages, 1 figure, 1 tabl

Anstee

Barklem

Holweger

J.E.R. Ross

Kurucz

Mihalas

Nieminen

Peach

Ross

T.A. Nieminen

Thompson

English

arXiv

Crossref

How to SYN in seven easy steps

The calculation of expected spectral line strengths and profiles is a powerful tool for the analysis of the solar atmosphere, and other stellar atmospheres. We present here a recipe in seven easy steps for the development of such spectral synthesis software

Nieminen, T. A.

Ross, J. E. R.

University of Queensland eSpace

1How to SYN in Seven Easy Steps
T. A. Nieminen and J. E. R. Ross a
aDepartment of Physics, The University of Queensland QLD 4072, Australia
Preprint version of:
T. A. Nieminen and J. E. R. Ross
“How to SYN in seven easy steps”
Computer Physics Communications 142, 160–163 (2001)
The calculation of expected spectral line strengths and profiles is a powerful tool for the analysis of the solar
atmosphere, and other stellar atmospheres. We present here a recipe in seven easy steps for the development of
such spectral synthesis software.
PACS codes: 97.10.Ex Stellar atmospheres; 95.30.Jx Radiative transfer; 96.60.Mz Photosphere, granulation
Keywords: spectral synthesis, opacity, line profile, asymmetry
1. Introduction
The solar spectrum can be calculated by (nu-
merically) solving the radiative transfer equation:
µ
dIλ
dτλ
= Iλ − Sλ (1)
If we know the source function Sλ, then finding
the emergent spectrum is straightforward. With a
model solar atmosphere, we have the LTE source
function, Sλ(z) = Bλ(z), the Planck radiation
function, as a function of physical depth. Un-
fortunately, it is not sufficient to know Sλ as a
function of physical depth z; we must convert the
physical depth scale to an optical depth scale:
τλ(z) =
∫
∞
z
ρ(z′)κλ(z
′)dz′ (2)
This requires calculation of the opacity κλ; this
is the core task of a spectral synthesis program.
The opacity can be considered to consist of a con-
tinuous opacity, and opacity due to atomic line
transitions. The main contribution to the contin-
uous opacity is due to bound-free and free-free H−
transitions; other major sources (H+2 molecule,
photoionisation of metals) also need to be con-
sidered. The line opacity can be found from the
line strength and line profile for each line in the
spectral region of interest.
2. Model atmospheres
The solar atmosphere is highly stratified; its
physical properties vary strongly with height
within the photosphere. Thus, the photosphere
can be well represented by a plane-parallel model
atmosphere. Many model atmospheres are avail-
able, with those by Holweger and Mu¨ller [1] and
Kurucz [2] being commonly used.
A model atmosphere tabulates temperature,
pressure, electron pressure, density, and opacity
at 5000 A˚.
With this data, and assuming LTE conditions,
then the source function is simply equal to the
Planck radiation function:
Sν(z) = Bν(T (z))
=
2hν3
c2
(exp(−hν/kT )− 1)−1 (3)
3. The continuous opacity
The continuous opacity (maximum at 8200 A˚)
is dominated by the H− ion, due to its large inter-
action cross-section. We can also consider other
abundant absorbers. If we are only considering
a small spectral region, we can assume that the
continuous opacity is uniform. In general, we use
approximation formulae and table lookup [3].
The H− opacity is due to both bound-free
(H− + photon ↔ H + e−) and free-free (H +
e− + photon ↔ H + e−) processes. Approxima-
tion formulae for both processes are given by Gin-
gerich [4]
Due to its high abundance, we should also con-
sider bound-free and free-free absorption due to
neutral hydrogen, which depend on the Gaunt
factors gII and gIII. Approximation formulae for
21. Calculate Sλ(z)
2. Calculate continuous opacity
3. Calculate line opacities.
This requires:
• line strengths
• damping
• thermal and mass motion
4. Calculate optical depth scale
5. Effect of intermediate scale mass motion
6. Numerical integration of radiative transfer
equation (1)
7. Non-plane parallel corrections – large scale
mass motion
Figure 1. Recipe for spectral synthesis
gII and gIII are given by Mihalas [5].
Other sources of opacity which can be consid-
ered include the H+2 molecule, for which cross-
sections have been calculated; a convenient tab-
ulation for interpolation is available in [4]. The
abundances of heavier elements much lower than
that of hydrogen, and they contribute far less to
the opacity. Only the most abundant species need
be considered, Mg and Si being the most impor-
tant. Calculations by Peach [6] can be used for
interpolation.
4. Line opacities
The opacity due to a spectral line depends
strongly on the wavelength – which is why it’s
a line – which makes it necessary to calculate the
line opacity on a sufficiently dense grid of wave-
length points. For a single (not too strong) line, a
total wavelength range of ≈ 1 A˚ is enough; a few
hundred wavelength points will be sufficient. The
use of a non-uniform grid of wavelength points
can greatly speed up line opacity calculation, at
the cost of a greater effort to implement. Note
that it is largely unimportant as to whether the
calculation is performed over a grid of wavelength
points or frequency points, since conversion from
one to the other is simple.
The opacity of a single line depends on
1. population of the absorber,
2. intrinsic strength of the transition,
3. and the line profile.
The populations of absorbers depend on the
atomic abundances and the equilibrium between
the atomic species, the rest of the atmosphere,
and the radiation field. We can simplify this
greatly by assuming LTE.
The strength of the transition – the oscilla-
tor strength, or f -value can be obtained from
a suitable source of data such as the NIST
Atomic Spectra Database at http://physics.
nist.gov/cgi-bin/AtData/main_asd or the Vi-
enna Atomic Line Database (VALD) at http:
//www.astro.univie.ac.at/~vald/.
4.1. The line profile
The line profile function of a stationary atom
is the Lorentz profile:
φ(ν) =
Γ/4pi2
(ν − ν0)2 + (Γ/4pi)2
(4)
Thermal motions will produce a profile that is
the convolution of the Lorentz profile and the
Maxwellian Doppler shift profile, giving the Voigt
profile:
φ(ν) =
a
pi3/2
∫
∞
−∞
exp(−x2)
(v − x)2 + a2 dx
=
1√
pi
H(a, v) (5)
where a = Γ/4pi∆νD and v = (ν − ν0)/∆νD,
where Γ is the natural (Lorentzian) line width,
and ∆νD is the Doppler shift due to the most
probable speed.
We can expect a spectral synthesis code to
spend a major part of the execution time re-
peatedly calculating the Voigt function. Various
3methods are available [7]. Fast methods usually
use asymptotic methods where possible (since the
Voigt profile approaches the Lorentz and Maxwell
profiles in the limit), and other approximate (eg
power series) methods elsewhere. Alternately, the
Voigt profile can be calculated by convolution of
the Lorentz and Maxwell Doppler profiles, using
Fourier transformation methods for speed.
4.2. Damping
The most important contribution to damping
is collisions with neutral hydrogen (almost all of
which is in the ground state). Damping due to
electrons is 50 times smaller, or even less, and
damping due to helium about 30 times smaller.
Other sources can be neglected.
The line broadening theory developed by
Anstee and O’Mara [8] has been shown to be ac-
curate. Collisions with hydrogen atoms in the
photosphere are fast; the impact approximation
can be used. The line width (HWHM) is
∆νcoll = N
∫
∞
0
vf(v)σ(v)dv (6)
where σ is the line broadening cross section, and
f(v) is the normalised Maxwellian velocity distri-
bution.
Data calculated using this theory is avail-
able [9], and code to calculate broadening has
been developed [10] (available at http://www.
astro.uu.se/~barklem) which can be used ei-
ther for stand-alone calculation, or can be incor-
porated into a spectral synthesis program. This
code calculates the linewidth per unit hydrogen
atom density for a given temperature.
The total Lorentzian width is the sum of the
collisional damping width, the natural line width,
and the stimulated absorption/emission width.
Only the collisional damping width will be im-
portant in the photosphere.
4.3. Small-scale mass motion
We can expect small-scale turbulent motion
to have a Gaussian velocity distribution (Kol-
mogorov turbulence). The effect will be the same
as that of thermal motion, and the two can be
combined. The average photospheric microtur-
bulence is 0.845 km/s. We can simply use this
average value as a uniform microturbulence, or a
depth-dependent value can be used.
5. Integrating the radiative transfer equa-
tion
All of the processes that affect the opacity have
been considered. The radiative transfer equa-
tion (1) can be integrated. The radiative transfer
equation (1) is well behaved, provided we solve
it in a reasonable manner. In order to avoid the
growth of errors, we integrate
Iλ(τ = 0) =
∫
∞
0
Sλ(τλ) exp(−τλ/µ)
1
µ
dτλ. (7)
In practice, we step through the layers of our
model atmosphere, from the top to the bottom,
with the step size for the numerical integration
being determined by the spacing in the model at-
mosphere. By the time the bottom of the model
atmosphere is reached, the contributions to the
emergent intensity are negligible.
6. Large-scale mass motion – beyond the
plane-parallel approximation
Large-scale flows prevent the photosphere from
being strictly plane-parallel. The main motion at
this scale is that due to the granulation. A gran-
ular cell is typically about 1000km across, con-
sisting of a hot, slowly rising, central region, and
a cooler, more rapidly falling outer region.
Even a simple model of granulation can greatly
improve the results of spectral synthesis, com-
pared with assuming a Gaussian distribution of
large-scale motion (macroturbulence). The gran-
ule can be divided into three purely plane-parallel
regions: the rising core, the falling outer region,
and a transition region between the two. We can
assume that the flow velocity in each region varies
with height h as
V = V0 exp(−h/Vs), (8)
where Vs is the velocity scale height. We can
expect the microturbulence, since it is driven by
the large scale flow, to have the same depth de-
pendence. A Gaussian macroturbulent velocity
can also be associated with each region to model
4inter-granular variation. The emergent spectra
from each region are calculated separately, and
then combined to give the total spectrum.
A suitable granular model is shown in ta-
ble 1. Using this granular model, calculated spec-
tral lines closely match observed spectral lines
in width and asymmetry. The given granular
model improves the fit between observed and cal-
culated spectra by a factor of 2.5, as measured by
the squared deviation between the spectra, with
the improvement mostly due to matching the ob-
served asymmetry [3].
Parameter Upflow Transition Downflow
Area 0.45 0.40 0.15
Brightness 1.07 0.97 0.87
Weighting 0.48 0.39 0.13
V0 0.577 0 -1.072
Vs 368 368 368
micro 1.58 3.67 3.67
macro 1.6 1.6 3.5
Table 1
Granular model. All velocities are in km/s, and
heights in km
7. Spectral synthesis
The presented recipe allows the rapid develop-
ment of simple, modular, customisable spectral
synthesis software for the accurate calculation of
small regions of the solar spectrum, suitable for
the study of spectral line shapes and strengths,
and the photospheric processes affecting them –
abundance, damping, granulation, etc.
In particular, the granular model presented
here provides a convenient compromise between
computational speed and ease of implementation
on one hand, and accurate modelling of photo-
spheric motion on the other. It provides a suit-
able starting point for further investigation of
processes affecting line profiles.
Such codes can be readily implemented on a
typical PC. A number of spectral synthesis codes
are available [11,12].
REFERENCES
1. H. Holweger and E. Mu¨ller, The photospheric
barium spectrum: solar abundance and colli-
sion broadening of Ba II lines by hydrogen,
Solar Physics 39, (1974) 19–30.
2. R. L. Kurucz, Model atmospheres for G, F,
A, B, and O stars, Astrophys. J. Suppl. Ser.
40, (1979) 1-340.
3. T. A. Nieminen, Solar line asymmetries:
Modelling the effect of granulation on the so-
lar spectrum, PhD thesis, The University of
Queensland, (1995). http://www.physics.
uq.edu.au/people/nieminen/thesis.html
4. Harvard-Smithsonian Conference on Stellar
Atmospheres, Proceedings of the First Con-
ference, Smithsonian Astrophysical Observa-
tory Special Report, 167, (1964).
5. D. Mihalas, Statistical-equilibrium model at-
mospheres for early-type stars. I. Hydrogen
continua, Astrophys. J. 149, (1967) 169–190.
6. G. Peach, Total continuous absorption coef-
ficients for complex atoms, Memoirs of the
Royal Astron. Soc. 71, (1967) 29–45.
7. W. J. Thompson, Numerous neat algorithms
for the Voigt profile function, Computers in
Phys. 7, (1993) 627.
8. S. D. Anstee and B. J. O’Mara, An investiga-
tion of Brueckner’s theory of line broadening
with application to the sodium D lines, Mon.
Not. Royal Astron. Soc. 253, (1991) 549–560.
9. P. S. Barklem, N. Piskunov and B. J. O’Mara,
A list of data for the broadening of metallic
lines by neutral hydrogen collisions, Astron.
Astrophys. Suppl. Ser. 142, (2000) 467–473.
10. P. S. Barklem, S. D. Anstee and B. J. O’Mara,
Line broadening cross sections for the broad-
ening of transitions of neutral atoms by colli-
sions with neutral hydrogen, Pub. Astr. Soc.
Aust. 15, (1998) 336–338.
11. J. E. R. Ross, SYN spectral synthesis soft-
ware, http://www.uq.edu.au/~phjross/
syn.htm
12. T. A. Nieminen, MATSYN spec-
tral synthesis toolbox for MATLAB,
http://www.physics.uq.edu.au/people/
nieminen/syn.html


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