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I am after the basic physical properties of the corona of a star, here some questions for which I would like to have an answer, even only for a subset of them:

  1. I know some things about the corona of our sun, but how typical is the solar corona?
  2. Is there any rule of thumb for the average dimensions of the corona, possibly depending on the mass of or other parameters of the star? Is it possible to estimate to at least an order of magnitude how thick a shell around a star would be which would be most likely be filled with plasma?
  3. What are the main constituents of the coronal plasma?
  4. If I intend to model the corona as an homogenous shell (which definitely is a strong assumption), what are the ranges of the number densities of the abundant particles?

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...how typical is the solar corona?

The Sun is remarkably "average" (e.g., see https://physics.stackexchange.com/a/262732/59023). So my best guess is that the corona is also remarkably "average." Technically the most abundant stars are red dwarfs and the Sun is a yellow dwarf, but dwarf stars are the most abundant and the sun is part of the main sequence line of stars.

Is there any rule of thumb for the average dimensions of the corona, possibly depending on the mass of or other parameters of the star? Is it possible to estimate to at least an order of magnitude how thick a shell around a star would be which would be most likely be filled with plasma?

This is tricky because the corona actually stops at the termination shock of the heliosphere. The size and shape of the Sun's region of influence in the interstellar medium (ISM) depends upon the strength/speed of the solar wind and the properties of the ISM. This would be true of all astrospheres. The location of the associated termination shocks would be determined largely by a balance of hydrodynamic and magnetic forces (much more heavily weighted toward the former than latter in many cases).

What are the main constituents of the coronal plasma?

The main particle species (order of particle number density from largest to smallest) are electrons, protons, and alpha-particles. Beyond alpha-particles, the number densities drop exponentially with CNO constituents plus sulphur and silicon picking up the tail. Iron is present as well, but in much lower amounts. I have provided some references in the following answer: https://astronomy.stackexchange.com/a/17899/13663.

If I intend to model the corona as an homogenous shell (which definitely is a strong assumption), what are the ranges of the number densities of the abundant particles?

You need to be a little more specific. At what altitude do you intend to model the corona? The number density varies with altitude. The paper by Gopalswamy and Yashiro [2011] discusses some simplified number density models that vary with radial distance, for instance. I would also glance at the references therein to see some examples of other coronal number density models.

In general, the corona and solar wind are comprised of ~95% protons, ~4% alpha-particles, and the remaining ~1% contains all the heavier ions. The plasma is quasi-neutral so the total electron number density will satisfy: $$ n_{e} = \sum_{s} \ Z_{s} \ n_{s} $$ where $Z_{s}$ is the charge state of species $s$ and $n_{s}$ is the number density of species $s$.

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