[Mesa-users] degeneracy parameter

[Jason Wright]

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>
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