[Mesa-users] degeneracy parameter

Mon Feb 22 20:08:08 UTC 2021

Hi Jason,

I'll just throw in my two cents to say that I agree with your interpretation of the sign of eta, and that's consistent with my experience using eta in MESA.

eta << -1 for conditions approaching ideal gas
eta > 1 for conditions where electron degeneracy starts to become significant

Cheers,
Evan

> On Feb 22, 2021, at 8:15 AM, Jason Wright via Mesa-users <mesa-users at lists.mesastar.org> wrote:
> 
> OK, after chewing on it a bit I think I am happy with your definition, except for one part:
> 
> I think the chemical potential of an ideal gas is negative infinity (the fugacity goes to zero) so eta goes to negative infinity.  
> 
> I think in the limit where the gas is degenerate the chemical potential approaches the Fermi energy, in which case eta is positive and eta ~ E_f/kT, as stated in the paper.
> 
> Jason
> 
> On Sun, Feb 21, 2021 at 4:53 PM Jason Wright <astrowright at gmail.com <mailto:astrowright at gmail.com>> wrote:
> Hmmm… I'll have to chew on that. I'm especially puzzled by you stating that negative values indicate degeneracy, because the first MESA instrument paper states (p. 15):
> 
> "As the electrons become degenerate (i.e., η > 0)"
> 
> which would seem to indicate the opposite? Also, I find that in the ZAMS Sun eta is indeed slightly negative in the core, but becomes much more negative in the envelope, where I would have naively expected electron degeneracy pressure to be less important there. 
> 
> Also: may I compute the electron density robustly from a MESA profile as:
> 
> free_e * 10**logRho / (mu * amu)
> 
> (where amu is the unit) or are there subtleties that make this less than exact?
> 
> The context is that I'm teaching a graduate stellar structure course using MESA now and trying to write some good homework problems to explore electron degeneracy in the sun, connecting the values they see in MESA to the equations in Kippenhahn Weigert & Weiss.  It's nice when the equations work out very precisely, and when they don't I like to explore what physics they are missing.
> 
> In KWW the electron degeneracy parameter is psi with approximate value exp(psi) = 8/(3*pi) (E_F/(kT))^(3/2). I think psi is the same as eta because it has very similar values when I compute it for the Sun, but I'm struggling to make the precise connection, especially in light of your response. 
> 
> Jason
> 
> On Sun, Feb 21, 2021 at 4:15 PM Francis Timmes <fxt44 at mac.com <mailto:fxt44 at mac.com>> wrote:
> hi jason,
> 
> eta, more properly eta_electron, is the electron chemical potential divided by kT.
> it is found from an equation of state by requiring that the number density of 
> free electrons as given by the fermi-dirac integrals equals that as given by the 
> saha equations (or the simpler zbar * N_A/abar * rho in the case of complete ionization).
> this equality can, for example, by enforced by doing a root find to determine eta_electron.
> mesa usually interpolates tables of eta_electron found from an equation of state to report a value.
> approximate analytic expressions are not used.
> 
> the sign convection is negative values of eta_electron indicate electron degeneracy.
> for example, for the center of the sun, eta_electron ~ -1.1 indicating a slight electron degeneracy.
> 
> clearer and sufficient?
> 
> fxt
> 
> 
> 
> 
> 
> 
> > On Feb 21, 2021, at 1:20 PM, Jason Wright via Mesa-users <mesa-users at lists.mesastar.org <mailto:mesa-users at lists.mesastar.org>> wrote:
> > 
> > How is eta, the degeneracy parameter, defined in MESA?
> > 
> > The first instrument paper claims (p.19) it is roughly eta = E_F/kT, but I think the proper definition is something more like (in the non-relativistic regime)
> > 
> > np.log ( 8/(3*pi) (E_F/(kT))^(3/2) )
> > 
> > Indeed, when I plot the above equation vs. eta I get pretty good agreement: eta is <1 for most of my star, so there must be a logarithm in there. 
> > 
> > But I'm also confused because I do not get very good agreement: I get differences of factors of a few in the atmosphere of my star, where the non-relativistic approximation should hold best!
> > 
> > Is there a good reference for how to interpret this parameter?
> > 
> > -- 
> >                                                 - 
> > ---------------
> > Jason T Wright
> > Professor of Astronomy & Astrophysics
> > Director, Penn State Extraterrestrial Intelligence Center
> > Acting Director, Center for Exoplanets and Habitable Worlds
> > NEID Instrument Team Project Scientist
> > he/him/his
> > https://sites.psu.edu/astrowright/ <https://sites.psu.edu/astrowright/>
> > @Astro_Wright
> > 
> > _______________________________________________
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> > 
> 
> 
> 
> -- 
>                                                 - 
> ---------------
> Jason T Wright
> Professor of Astronomy & Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> Acting Director, Center for Exoplanets and Habitable Worlds
> NEID Instrument Team Project Scientist <http://neid.psu.edu/>
> he/him/his
> https://sites.psu.edu/astrowright/ <https://sites.psu.edu/astrowright/>
> @Astro_Wright
> 
>  <http://twitter.com/Astro_Wright>
> 
> -- 
>                                                 - 
> ---------------
> Jason T Wright
> Professor of Astronomy & Astrophysics
> Director, Penn State Extraterrestrial Intelligence Center
> Acting Director, Center for Exoplanets and Habitable Worlds
> NEID Instrument Team Project Scientist <http://neid.psu.edu/>
> he/him/his
> https://sites.psu.edu/astrowright/ <https://sites.psu.edu/astrowright/>
> @Astro_Wright
> 
>  <http://twitter.com/Astro_Wright>_______________________________________________
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