[Mesa-users] degeneracy parameter
Jason Wright
astrowright at gmail.com
Tue Feb 23 17:28:47 UTC 2021
Thanks for everyone's help so far! I'm mostly happy with things now; I
can't quite reproduce the degeneracy pressure in the core from eta but I'm
using the purely nonrelativistic formula and a first-order approximation
for Coulomb forces so I'm not too worried about that.
My main issue now is that I don't have a check for the number density of
electrons. I have been trying to use the parameter free_e, but I don't
think I understand it. It says it's the "mean number of electrons per
nucleon." I think this means that I should at least roughly be able to get
the number density of electrons as
n_e = rho/amu * free_e
where amu is the unit, but that's not even close to right. In fact, just
counting hydrogen and helium I get 0.86 free electrons per nucleon at the
center of the ZAMS sun, but free_e says there are 0.67. Indeed, in
complete ionization I would expect the number of free electrons to match
the number of protons, so free_e = ye, but again I get 0.86 for ye,
matching my expectation but inconsistent with free_e.
So my questions are:
1) What does free_e represent?
2) Is there a way to extract n_e from MESA output parameters precisely?
Perhaps an unlisted parameter like mu_e or n_e?
Thanks,
Jason
On Mon, Feb 22, 2021 at 4:00 PM Francis Timmes <fxt44 at mac.com> wrote:
>
> On Feb 22, 2021, at 1:35 PM, RICHARD H D TOWNSEND via Mesa-users <
> mesa-users at lists.mesastar.org> wrote:
>
> I think Frank may have mis-spoken earlier. In MESA, eta -> -infinity is
> the classical limit, and vice versa.
>
>
>
>
> oh, that has never happened before :o
>
> especially for a sign convention where both have been used in the
> literature (alpha = -eta).
>
> fxt
>
>
>
>
>
> On Feb 22, 2021, at 2:08 PM, mesa-users at lists.mesastar.org wrote:
>
> 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>
> 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
> he/him/his
> https://sites.psu.edu/astrowright/
> @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
> he/him/his
> https://sites.psu.edu/astrowright/
> @Astro_Wright
>
> _______________________________________________
> mesa-users at lists.mesastar.org
> https://lists.mesastar.org/mailman/listinfo/mesa-users
>
>
> _______________________________________________
> mesa-users at lists.mesastar.org
> https://lists.mesastar.org/mailman/listinfo/mesa-users
>
>
> _______________________________________________
> 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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