How SPECTRUM Carries out its Calculations

SPECTRUM carries out some preliminary calculations before getting down to the business of calculating the spectrum itself. SPECTRUM uses as input a stellar atmosphere model with the format of a Kurucz model, but actually only utilizes the columns corresponding to mass depth, temperature and gas pressure. SPECTRUM then computes for itself the electron number density and the number densities of important species, namely, hydrogen, helium, carbon, nitrogen, oxygen, their ions and the relevant diatomic molecules. Specifically, in this preliminary stage, SPECTRUM solves a system of 7 nonlinear equilibrium equations to determine those number densities. The exact details of these equations depend upon the effective temperature of the star. For $T_{\rm eff} > 8500$K, no molecules, except for H$_2$ are included in the equilibrium equations, but for lower temperatures molecules are included. The 7 equilibrium equations are:
1) Equilibrium equation for hydrogen, including neutral and ionized hydrogen, the H$^-$ ion, H$_2$, H$_2^+$, and, for $T_{\rm eff} < 8500$K, H$_2$O, CH, NH and OH.
2) Equilibrium equation for Helium, including two stages of ionization.
3) Equilibrium equation for Carbon, including up to four stages of ionization (i.e. C I, C II, C III, C IV and C V), and the molecules CH,CN,CO and C$_2$ (the number of stages of ionization depends on the effective temperature of the model. For instance, for $T_{\rm eff} > 8500$K, four stages of ionization are considered, but the molecules are left out; likewise for the Nitrogen and Oxygen equations below).
4) Equilibrium equation for Nitrogen, including up to four stages of ionization, and the molecules CN, NH, NO, N$_2$.
5) Equilibrium equation for Oxygen, including up to four stages of ionization, and the molecules CO, NO, OH, O$_2$.
6) Charge balance equation, including electron contributions from the ionization of hydrogen, helium, carbon, nitrogen and oxygen and the following species: sodium, magnesium, aluminum, silicon, sulfur, potassium, calcium and iron.
7) Total number density, including all species mentioned above, which must agree with $P{_{\rm gas}}/kT$ as read from the stellar atmosphere.

This system of equations is solved iteratively at each of the levels in the stellar atmosphere. Electron number densities so derived are always within 1% of those in the stellar atmosphere model. Convergence is rapid and the abundances of the different species and molecules are stored for each of the atmosphere levels in memory.

SPECTRUM then moves on to the computation of reference opacities and optical depths at each level in the atmosphere. Opacities currently included in SPECTRUM are: hydrogen bound-free and free-free, He I bound-free and free-free, He II bound-free and free-free, H$^-$ bound-free and free-free, H$_2^+$ opacity, hydrogen Rayleigh scattering, H$_2$ Rayleigh scattering, He I Rayleigh scattering, He I$^-$ free-free opacity, low temperature opacities (including continuous opacities due to C I, Mg I, Al I, Si I, Ca I, Fe I, CH, OH and MgH), intermediate temperature opacities (including opacities due to N I, O I, Mg II, Ca II and Si II) and electron scattering. With version 2.75, bound-free opacities for He I, C I, Mg I, Si I and Ca I have been updated with data from the Opacity project, although some of these have been modified by comparison with observations.

With these preliminaries completed, SPECTRUM can now begin to compute the synthetic spectrum. The numbers or words in parentheses correspond to the parameters used in the ultraviolet part of the spectrum. SPECTRUM basically computes the spectrum in 20 (5) Å blocks, although smaller segments of spectra can be synthesized. For each 20 (5) Å block, SPECTRUM first computes the continuum opacity and the emergent continuum flux at both ends of each block (the exception to this is if the 20 (5) Å block contains a continuous absorption edge. Then SPECTRUM modifies the red end of this block to 0.05 Å before the absorption edge; the next block straddles the absorption edge, and then the new block starts at 0.05Å past the absorption edge). The continuous opacity and continuum flux at every intermediate point is then estimated by interpolation. SPECTRUM then proceeds to the calculation of the line opacity at each spectrum point. SPECTRUM must decide, at every spectrum point, which spectral lines to include in the calculation of the line opacity. To do this SPECTRUM maintains two lists of lines. The first list includes all spectral lines in the line file which lie within 20Å (10Å in the optical) of the current spectrum point. For all the lines in this list, SPECTRUM assigns a ``computation radius'' based upon the distance from line center at which the line depth drops below 0.0001 continuum units. If the computation radius of a line includes the spectrum point being currently computed, it is moved from the first list into a second list which comprises all of the lines which contribute to the line opacity at that point. Once SPECTRUM has stepped entirely through the computation radius of a line, that particular line is dropped from the second list as well.

For each line in the second list, the abundance of the ion in question and the level population for that particular transition are computed using the Saha and Boltzmann equations. The broadening parameter is then computed; broadening mechanisms include natural, van der Waals (see specifics in § ) and quadratic Stark broadening (see, however, hydrogen and helium lines below). The line opacity at that point (for each of the levels in the atmosphere) is then computed using the Voigt function. The line opacity, line optical depth, continuum opacity and continuum optical depth are then included in a ``line contribution'' calculation which, when ratioed with the continuum flux, leads to the calculation of the normalized residual intensity at that spectral point. SPECTRUM calculates in its default mode the disk-integrated spectrum. This quantity, along with the wavelength is then written to the output file. As we saw above, this output can be modified by specifying certain SPECTRUM flags.

The hydrogen lines are computed by a routine adopted from Deane Peterson, used in the spectral synthesis code SYNTHE (Kurucz, 1993), which includes Stark and resonance broadening and fine structure in the core, as well as the Lyman-$\alpha$ quasi H$_2$ satellites. To save time, the full calculation of the hydrogen-line opacity is performed only at 1Å intervals, except within 5Å of the line core. The line opacity at intermediate spectral points is then obtained by 4 point interpolation (performed at each of the layers in the atmosphere). Hydrogen lines are included in the line opacity calculation for all points nominally within 2000Å of the line center (although this is modified in a complicated sort of way for the crowded lines in the vicinity of the Lyman, Balmer and Paschen convergences to avoid spurious line opacity in that region). The code for SPECTRUM includes 130 Lyman lines, 250 Balmer lines, 246 Paschen lines, 30 Brackett lines, 29 Pfund lines and 28 Humphreys lines. The computation for the Balmer jump is quite complex and includes computation of the lowering of the ionization energy using the Debye approximation, overlap of energy levels near to the continuum, etc. This more correct computation of the Balmer jump is also applied to the Paschen and Lyman convergences, but not yet to the Brackett, Pfund and Humphreys convergences.

The He I lines are computed on the basis of new line profile calculations of Beauchamp et al. (private communication & Beauchamp et al., 1997). These line profiles yield very accurate reproductions of the He I lines, including forbidden components. He II lines are currently not included in SPECTRUM, but will likely be included in an upcoming version upgrade. He I lines are computed only for models with $T_{\rm eff} > 8500$K. We note that certain helium lines in the red - especially 5875Å and 6678Å - are strongly affected by non-LTE and are thus not well reproduced by SPECTRUM.

Certain very strong lines (such as the Ca II K and H lines), and a handful of lines for which SPECTRUM's ``calculation radius'' is not adequate are not included in the luke.lst file, as they are calculated in a special routine. This routine allows a larger computation radius for those lines. This routine for strong lines might include, in the future, individualized source functions for certain strong lines, or even a more sophisticated calculation based on a multi-level atom.