First observation of linear polarization in the forbidden [OI] 630.03 nm line

In a new paper (de Wijn, Socas-Navarro & Vitas, 2017, ApJ, 836, 29D) we present the first results of our observations of a sunspot and an active region using the SP/SOT instrument on board the Hinode satellite. The novelty in our observation is a trick that we used to double the standard wavelength range observed by the instrument. Thanks to that, we were able to see the sun not only in the two iron lines at 630.2 nm, but also in four other lines. One of those is particularly interesting: the forbidden ground-based line of neutral oxygen ([OI] 630.03 nm). It is one of only few oxygen lines in the solar spectrum and probably the best diagnostics of the solar oxygen abundance. For the first time ever we observed the linear polarization in this line! As an M2 (magnetic dipole) transition, it is predicted by the theory (Landi degl'Innocenti and Landi, 2004, Section 6.8) that this line produces the linear polarization signal with the opposite sign to the lines produced by E2 transitions. It is also the first time that linear polarization in M2 and nearby E2 lines is measured simultaneously, so that the flip in sign is obvious (see the left-most spectral line in the red circle in the Figure; in linear polarization it has "W" shape, while all other lines in the wavelength range have "M" shapes). This result may bring new light to the ongoing debate on the solar oxygen crisis.

  

More details of this unique observation will appear soon in a follow-up publication.

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);

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.

Use of "some useful atomic data"

To illustrate how the routines in the previous post can be used, I made a couple of plots.

Fig.1 The logarithmic solar abundances relative to hydrogen A (A(H) = 12) after Asplund et al (2009, 2009ARA&A..47..481A) versus the atomic number Z. The data are loaded using load_abundances.pro function.

Some useful atomic data in IDL

Here I list a couple of my IDL routines that load some useful atomic data (atomic numbers up to 92 are supported).


List of elements

This routine returns symbols and names of atomic elements for a given atomic number Z.

load_list_of_elements.pro

Example:

IDL> sym = LOAD_LIST_OF_ELEMENTS([6, 7, 8], element = names)
IDL> PRINT, sym
C N O
IDL> PRINT, names
Carbon Nitrogen Oxygen

Atomic weights (aka relative atomic masses)

This routine loads the atomic weights for a given atomic number Z. The data comes from
  http://www.nist.gov/pml/data/comp.cfm

The original source is Wieser & Berglund (2009, Pure Appl. Chem., Vol.81, No.11, p.2131-2156), published as an IUPAC Tecnical Report available at:
   http://pac.iupac.org/publications/pac/pdf/2009/pdf/8111x2131.pdf

load_atomic_weights.pro

Example:

IDL> a = LOAD_ATOMIC_WEIGHTS([1, 2, 6, 7, 8])
IDL> PRINT, a
1.00790      4.00260      12.0107      14.0067      15.9994