Notes on Thermodynamics
Derivatives With Respect to Composition
In the following we assume that the molar mass of species
The number density [cm:math:^-{3}] of isotope
where
where we have defined the mass fraction,
We write the average molar mass and average proton number as:
Our algorithm requires terms involving the derivative thermodynamic variables
(
From Eq.327 and Eq.328 we have
where after differentiation we have used Eq.326 to write
We therefore have
Before it was brought to our attention by Frank Timmes, we were missing the second term in Eq.330. The only place where such terms appear in our algorithm is in a sum over all species, such as:
The second term in Eq.334 is identically zero because
This second term arises from what was added to Eq.330 by Frank’s correction. Therefore, although important for individual derivatives with respect to composition, this correction term has no effect on our solution.
Convective stability criterion
Here we look at the criterion for convective stability in the case of non-uniform chemical composition. This section follows Cox & Giuli :raw-latex:`\cite{cg-ed2}` closely (see chapter 13).
Consider a fluid parcel that gets displaced upwards (against gravity) from
a radial location
Since the parcel originates at r,
Since the total pressure,
Using the equation of state
For convenience we introduce
Then we can rearrange Eq.339 to get
Then the general stability criterion is
Here’s where various assumptions/simplifications get used.
If no assumptions are made, you can’t get any further than Eq.342. Even in view of an infinitesimally small initial perturbation, you can’t, in general, assume the
’s in parcel are the same as the ’s in the background. This applies in the case where nuclear reactions and/or ionization change the composition of the parcel. This case tends not to be of much interest for two reasons. Either composition effects get incorporated implicitly through assuming chemical equilibrium. Or both of these terms can be neglected in the rising parcel. This would be justified if the timescale for reactions is long compared to the convective timescale, and either the same is true for ionization or the fluid is fully ionized.If we assume that
remains constant in the parcel, then drops out for the parcel. In this case, we can assume, in view of the arbitrarily small initial perturbation of the parcel, that and to have the same values in the parcel as in the background. Then the stability criterion becomes(343)The Ledoux stability criterion is obtained by assuming that the parcel moves adiabatically.
If we assume that the background is in chemical equilibrium and the parcel achieves instantaneous chemical equilibrium, then
for the background and the parcel. (Note that we aren’t requiring constant composition in the parcel here.) The effect of variable composition are then absorbed into and . Again, we can take and to have the same values in the parcel as in the background. The criterion then is(344)We obtain the Schwarzchild criterion for stability if we also assume the parcel moves adiabatically.
The Scharwzchild criterion can be recast in terms of entropy if the EOS is taken as
instead of . Then, in place of Eq.339 we have(345)We can substitute this into Eq.337 for stability, and assuming the parcel moves adiabatically, we get
(346)One of Maxwell’s relations is
(347)All thermodynamically stable substances have temperatures that increase upon adiabatic compression, i.e.
. So Maxwell’s relation implies that . The stability criterion then becomes(348)
Determining which stability criterion we want to enforce in creating the initial model is complicated by the phenomenon of semiconvection, which occurs when the Ledoux criterion is satisfied but the Schwarzchild is not, i.e.
(Note that
When we set up an initial model, we want to minimize any initial tendency towards convective motions, as we want these to be driven by the heating due to nuclear reactions, not the initial configuration we supply. Thus I think we want to guard against semiconvection as well as “traditional” convection by using the stability criterion
Although this looks like the Schwarschild criterion (and, because I’m not
entirely sure on vocabulary, it might even be called the Schwarzchild
criterion), this does not simplify to Eq.348
because we need to keep the explicit
The question of whether we’re in chemical equilibrium or not might be a moot
point since our EOS (or any other part of the code) doesn’t enforce chemical
equilibrium. Thus, even
in the case of chemical equilbrium, we can’t in general
drop the explicit
Adiabatic Excess
The adiabatic excess,
where the subscript “ad” means along an adiabat. We can combine the exponents to get the following relation
The adiabatic excess is defined as
where
is the thermal gradient. It is important to note that these thermal
gradients are only along the radial direction. The “actual”
gradient can be found from finite differencing the data whereas the
adiabatic term,
The Schwarzschild criterion does not care about changes in composition
and we therefore write
where
Dividing Eq.355 by
Using the
Combining Eq.352 and Eq.358
to eliminate
which uses only terms which are easily returned from an EOS call.