[Mesa-users] degeneracy parameter

[Jason Wright]

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> 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> 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> 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/
>> > @Astro_Wright
>> >
>> > _______________________________________________
>> > mesa-users at lists.mesastar.org
>> > https://lists.mesastar.org/mailman/listinfo/mesa-users
>> >
>>
>>
>
> --
>                                                 -
> ---------------
> 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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