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)

1D Solar Atmosphere Models in IDL: Gingerich et al (HSRA)

The Harvard-Smithsonian Reference Atmosphere (HSRA) by Gingerich et al (1971, 1971SoPh...18..347G) is another widely used semi-empirical model of the idealized plane-parallel solar atmosphere in hydrostatic equilibrium. This model also includes the chromosphere where the hydrogen ionization is solved using the statistical equilibrium equations. The model is still often used as the reference solar atmosphere in 1D and its number of citations steadily grows (> 820 so far). The paper is very clearly written and it's a very recommendable reading especially if you use this model. Beside the model itself, there is several tables with computed intensities, brightness temperature, optical depths at different wavelengths, etc.

1D Solar Atmosphere Models in IDL: Maltby et al (MACKKL)

Here I add files containing the one-dimensional models of the solar atmosphere from the famous paper of Maltby et al (1986, 1986ApJ...306..284M). There is a quiet sun model and three models of the sunspot atmosphere (L, M, E corresponding to the late, mid and early cycle of the solar activity). For the details of these models please check the paper.

The data in the files comes directly from Tables 7, 8, 9 and 11 of Maltby et al. The variables stored in the original models are:

• geometrical height scale, 
• column mass scale, 
• optical depth at 5000 A, 
• temperature, 
• turbulent velocity, 
• number density of hydrogen, 
• electron density, 
• total pressure, 
• ratio of gas pressure to total pressure, 
• density.

Fig. The temperature stratification of the three sunspot models (remake of Fig.8 of Maltby et al)

Theory vs. Observations, by Jean-Claude Pecker

A cartoon by the famous French astronomer Jean-Claude Pecker (published in the Proceedings of the 3rd European Solar Meeting, "Solar activity, April 13-15 1981, Oxford", ed. C. Jordan; a non-ADS volume).

No matter how much we pushed the frontier of the science and how much our theoretical machinery, diagnostic tools and large and expensive gadgets evolve, this is still in a way very actual. It is important not to forget that even the most sophisticated realistic 3D models of the sun are still mammals, pardon, models. On the other hand, the results of the observations should not be consider  as the real "elephant" either. They are pictures of the elephant made with imperfect instruments!

Eddington on Icarus

From the Presidential Address of Professor Arthur Eddington to Section A of the British Association in Cardiff, August, 24th, 1920 (1920Obs....43..341E, also in the book Stars & Atoms, 1927stat.book.....E):

"In ancient days two aviators procured to themselves wings. Daedalus flew safely through the middle air and was duly honoured on his landing. Icarus soared upwards to the sun till the wax melted which bound his wings and his flight ended in fiasco. The classical authorities tell us, of course, that he was only doing a stunt; but I prefer to think of him as the man who brought to light a serious constructional defect in the flying-machines of his day. So, too, in science. Cautious Daedalus will apply his theories where he feels confident they will safely go; but by his excess of caution their hidden weaknesses remain undiscovered. Icarus will strain his theories to the breaking-point till the weak joints gape. For the mere adventure? Perhaps partly, this is human nature. But if he is destined not yet to reach the sun and solve finally the riddle of its construction, we may at least hope to learn from his journey some hints to build a better machine."

Landscape with the Fall of Icarus by Pieter Bruegel the Elder (Musées royaux des Beaux-Arts de Belgique).