Equation of state: Vardya - Mihalas - Wittmann

These are my notes on equation of state derived initially by Vardya (1965), described by Mihalas (1967) and popularized by Wittmann (1974). It is widely used (or at least present as an option) in spectral synthesis codes like SIR or NICOLE. It is also prepared for the MANCHA code with several modifications and additionally computed quantities. However, the equation is derived in this particular form to be solvable on computing resources half a century ago. From today's perspective, this formulation is rather obsolete. While the results of the VMW EOS are still largely reliable, various tricks introduced to control numerical stability limit its usability to rather restricted range of the pressure and temperature.

Here I derive a simple equation of state for the solar atmosphere following the classical work of Vardya (1965), Mihalas (1967) and Wittmann (1974). The EOS is based on the Saha ionisation equilibrium and the instanteneous chemical equilibrium for the molecules. The main ingredient is hydrogen. It's included as atomic hydrogen (H), negative hydrogen ion (H-), positive hydrogen ion (H+), and as H2 and H2+ molecules. For all other atoms the neutral and the first two ionisation stages are included.

The equations were first published by Vardya (1967). Mihalas (1967) gave a simple numerical algorithm for an efficient solution of the system. Wittmann (1974) copied the equations and add a corrective factor that provides numerical stability at high temperature.

The derivation here follows Mihalas. However, in the original derivation there is a couple of inconsistencies that obscure the procedure. Here I write the equations in a correct and consistent way.

Definitions

Let's first define the pressures:
$p_{\mathrm{H}}$ - partial pressure of the neutral H atoms;
$p_{\mathrm{H^+}}$ - partial pressure of the positive H ions (protons);

Effective Lande g-factor (in IDL)

The effective Lande g-factor (Shenstone and Blaire, 1929) is defined for a spectral line. It shows how much the $\sigma$ components of the normal Zeeman triplet are separated from the line center.  The higher the Lande factor is, the line is more sensitive to the magnetic field. It is computed as a combination of g-factors for each of the involved atomic levels. Derivation of the equation can be found, for example, in Landi Degl'Innocenti (1982SoPh...77..285L, see it for more details and for an alternative expression useful in the polarized radiative transfer).

For each of the levels (u for upper, l for lower) in LS coupling holds:
$$g_\mathrm{LS} =\frac{3}{2}+\frac{S (S+1) - L (L+1)}{2J(J+1)}, $$ 
where $L$, $S$ and $J$ are the orbitatl, spin and total angular momentum quantum numbers of the atomic level (note that it is undefined for $J=0$). The effective Lande g-factor is then defined as:
$$\bar{g} = \frac{1}{2}(g_\mathrm{l} +g_\mathrm{u}) + \frac{1}{4}(g_\mathrm{l} - g_\mathrm{u})(J_\mathrm{l}(J_\mathrm{l}+1) - J_\mathrm{u}(J_\mathrm{u}+1)).$$

Here is my IDL code for computing the effective Lande g-factor of a spectral line in LS coupling. At the input the code takes the quantum number of the lower and the upper level of a transition. The numbers may be specified either as an array [L, S, J] or in spectroscopic notation.

Example:

IDL> PRINT, LANDE_FACTOR(lower  = [3., 1, 2.], upper = '3p1.0')
  Upper term: 3p1.0 => l = 1.00000s = 1.00000j = 1.00000
          0.250000

IDL> PRINT, LANDE_FACTOR(lower  = '5F1.0', upper = '5d0.0')
  Lower term: 5f1.0 => l = 3.00000s = 2.00000j = 1.00000
  Upper term: 5d0.0 => l = 2.00000s = 2.00000j = 0.00000
          0.00000

Download: lande_factor.pro

Kurucz' Atlas in GNU Linux

I have just found about great effort made by Sbordone, Bonifacio & Castelli to make the Atlas code of Bob Kurucz available at GNU Linux.

There is a webpage: http://atmos.obspm.fr/ providing the code and the documentation.

Notes on some basic quantities, units and constants: Part II

Mass fractions

The mass fraction of H, He and the metals is a quick way to specify the chemical composition of the stellar plasma. The mass fraction of hydrogen is defined as ratio of the mass of hydrogen particles and the total mass of all particles (in a given volume): $$ X = \frac{M_\mathrm{H}}{M}= \frac{\rho_\mathrm{H}}{\rho}, $$ where $\rho_\mathrm{H}$ and $\rho$ are the respective (mass) densities. The hydrogen mass density is equal to the hydrogen number density ($n_\mathrm{H}$) times the mass of one hydrogen particle $m_{\mathrm{H}} = A_{\mathrm{H}}\,m_{\mathscr{A}}$, thus: $$ X = \frac{M_\mathrm{H}}{M}= \frac{n_\mathrm{H}\,m_{\mathrm{H}}}{\rho}= \frac{n_\mathrm{H}\,A_{\mathrm{H}}\,m_{\mathscr{A}}}{\rho}. $$ Similar to that, for helium and the metals we define: $$ Y = \frac{M_\mathrm{He}}{M}= \frac{n_\mathrm{He}\,m_{\mathrm{He}}}{\rho}= \frac{n_\mathrm{He}\,A_{\mathrm{He}}\,m_{\mathscr{A}}}{\rho}, $$ $$ Z = \frac{M_\mathrm{metals}}{M}= \frac{n_\mathrm{metals}\,m_{\mathrm{metals}}}{\rho} = \frac{\sum_{i=3} n_\mathrm{i}\,A_{\mathrm{i}}\,m_{\mathscr{A}}}{\rho},. $$

Notes on some basic quantities, units and constants: Part I

There is a definition of mole as a unit that every student learns at very elementary level. Although mole is one of the seven base units of the International System, it is a bit specific and sometimes creates confusion. Here is my attempt to clarify the concept of mole and to derive in one place some useful relations used in the radiative transfer and atmospheric modeling. The first version of these notes I wrote for a group of students at the University of Belgrade many years ago. I still use them as a personal reminder. 

Counting "Elementary particles"

For the solar/stellar plasma, the "elementary" particles are atoms, ions (positive or negative), free electrons and molecules. In the very cool atmospheres there are dust particles as well. In the solar atmosphere dust can be completely neglected. Regarding the chemical composition, the atmospheric plasma is made out of hydrogen, helium and the metals (all other elements). There is no nuclear reactions and thus the total number density of nuclei per atomic specie is constant with time.

It is important to distinguish between the number of free atoms and the total number of atoms including those bound in the molecules. The former I denote as $N_{\mathrm{a}}^{\mathrm{free}}$, the latter as $N_{\mathrm{a}}^{\mathrm{tot}}$. The total number of atoms is identical to the number of atomic nuclei. The total number of molecules is $N_{\mathrm{m}}$.

The total number of particles $N$ is therefore:
$$N = N_\mathrm{e} + N_{\mathrm{a}}^{\mathrm{free}} + N_{\mathrm{m}},$$ or when there is no molecules $$N = N_\mathrm{e} + N_{\mathrm{a}}^{\mathrm{free}} = N_\mathrm{e} + N_\mathrm{H} + N_\mathrm{He} + N_\mathrm{\mathrm{metals}},$$ where $\mathrm{e}$, $\mathrm{m}$, $\mathrm{H}$ and $\mathrm{He}$ stand for the electrons, the molecules, hydrogen, helium and $\mathrm{metals}$ refer to all other elements together. The contribution of the metals can be further divided into the contributions of the individual elements.

1D Solar Models in IDL: Spruit's Convection Model

The semi-empirical models of the solar atmosphere rely on the observed intensities and, therefore, they cannot say much about the invisible convection zone below the surface. Spruit (1974SoPh...34..277S) constructed a 1D model of the solar convection zone using the mixing-length theory with 4 free parameters. The model is constructed so that it matches the HSRA model (Gingerich et al, 1971SoPh...18..347G) of the solar atmosphere. The model of Spruit is sometimes used to initiate the convection simulations with 3D numerical codes.

Fig 1. The HSRA model atmosphere (dashed) and Spruit's model of convective zone (solid). This is reproduced after Fig.3 of Spruit's paper.

Atlantis over Tenerife

Japanese astronaut Sochi Noguchi took this iconic picture of the Space Shuttle Atlantis over the Atlantic Ocean and the island of Tenerife. The picure was taken from the International Space Station on Sunday, May 16 2010 at 10:28 am EDT (1428 GMT) while the Atlantis was getting ready for docking. At that time the ISS was about 350 km above the ground. This was the flight before the last one for the shuttle. In the highest resolution, the white towers of the solar telescopes at the Observatorio del Teide (IAC) are visible between the clouds. To help your eyes, I took a snapshot from the Google Maps: look for the distinctive dark patches in the NASA picture, the towers are tiny white dots just above them.

Credit: NASA/Sochi Noguchi (click for hi-res)

Credit: Google Maps (click to go to the interactive map)