In MAESTRO, we do things in an operator split fashion — the hydro is
de-coupled from the burning. This means that during the hydro parts
of the algorithm (where \(\beta_0\) is used), the system is fixed in
chemical equilibrium. For completeness, however, here we describe the
effects of the species’ chemical potentials, which were neglected in
the original derivation of \(\beta_0\). Note that similar terms appear
in the calculation of things such as specific heats,
which may be important in the burning step — there appears to
be very little about this in the literature, but everyone seems to
assume it makes little difference.
Derivation of \(\alpha\)
In paper I, \(\alpha\) is defined as
(365)\[\alpha\equiv -\left(
\frac{(1-\rho h_p)p_T-\rho c_p}{\rho^2c_pp_\rho}\right)\]
where
(366)\[h_p \equiv \left(\frac{\partial h}{\partial p}\right)_{T,X}, \quad
c_p \equiv \left(\frac{\partial h}{\partial T}\right)_{p,X}, \quad
p_T \equiv \left(\frac{\partial p}{\partial T}\right)_{\rho,X}, \quad
p_\rho \equiv \left(\frac{\partial p}{\partial \rho}\right)_{T,X}\]
where the subscript \(X\) means holding all \(X_i\) constant. In the
absence of reactions, the \(X\) subscript can be dropped from all
derivatives and with the use of the equation of state \(p=p(\rho,T)\),
\(\alpha\) can be written as \(\alpha=\alpha(\rho,T)\). Such a system
without reactions and in thermal equilibrium could be either a pure
system of one species, or a system of many species in chemical (and
therefore thermodynamic) equilibrium. Cox and Giuli (hereafter
CG) call the former type of system a “simple system” and therefore
the latter a “non-simple system” in chemical equilibrium. The
analysis in paper I that reduced Eq.365 to
(367)\[\alpha = \frac{1}{\Gamma_1p_0}\]
used CG’s discussion of the various adiabatic \(\Gamma\)’s. However,
their discussion only pertains to “simple systems” or “non-simple
systems” in chemical equilibrium. In general, nuclear reactions will
be important and therefore this analysis needs to be reformed.
Even in the presence of reactions, Eq.365 can be rewritten
as was done in paper I:
(368)\[\alpha = -\frac{1}{p\chi_\rho c_p}\left[\left(\frac{1}{\rho\chi_\rho}
- \frac{\rho e_\rho}{p\chi_\rho}\right)\frac{p\chi_T}{T} - c_p\right],\]
where
(369)\[\begin{split}\begin{aligned}
\chi_{\rho} &\equiv \left(\frac{\partial\ln p}{\partial\ln\rho}
\right)_{T,X} \\
\chi_{T} &\equiv \left(\frac{\partial\ln p}{\partial\ln T}
\right)_{\rho,X}.\end{aligned}\end{split}\]
Following the results of paper I, we want to find a relation
between \(p\chi_\rho\) and \(\Gamma_1\).
For an equation of state \(p=p(\rho,T,X)\) we have
(370)\[d\ln p = \left(\frac{\partial\ln p}{\partial\ln\rho}\right)_{T,X}d\ln\rho +
\left(\frac{\partial\ln p}{\partial\ln T}\right)_{\rho,X}d\ln T +
\sum_i\left(\frac{\partial\ln p}{\partial\ln X_i}\right)_{\rho,T,(X_j,j
\neq i)} d\ln X_i.\]
We define another logarithmic derivative
(371)\[\begin{aligned}
\chi_{X_{i}} &\equiv \left(\frac{\partial\ln p}{\partial\ln X_i}
\right)_{\rho,T,(X_j,j\neq i)}\end{aligned}\]
and therefore
(372)\[d\ln p = \chi_\rho \ d\ln\rho + \chi_T \ d\ln T + \sum_i \chi_{X_i}\
d\ln X_i.\]
From here we get the general statement
(373)\[\frac{\partial\ln p}{\partial \ln \rho} = \chi_\rho +
\chi_T\frac{\partial \ln T}{\partial\ln \rho} +
\sum_i\chi_{X_i}\frac{\partial\ln X_i}{\partial\ln \rho}\]
which must hold for an adiabatic process as well, and therefore we have
(374)\[\Gamma_1 = \chi_\rho + \chi_T\left(\Gamma_3-1\right)
+ \sum_i\chi_{X_i}\Gamma_{4,i}\]
where we use CG’s definition of \(\Gamma_1\) and \(\Gamma_3\) and introduce a
fourth gamma function:
(375)\[\Gamma_1 \equiv \left(
\frac{\partial \ln p}{\partial \ln \rho}\right)_{\text{AD}},\quad
\Gamma_3-1\equiv \left(
\frac{\partial \ln T}{\partial \ln \rho}\right)_{\text{AD}},\quad
\Gamma_{4,i}\equiv \left(
\frac{\partial\ln X_i}{\partial\ln\rho}\right)_{\text{AD}},\]
where the subscript AD means along an adiabat. We now derive an expression
for \(\Gamma_3\).
The first law of thermodynamics can be written as
(376)\[dQ = dE + pdV - \sum_i\mu_idN_i\]
where \(\mu_i=\left(
\frac{\partial E}{\partial N_i}\right)_{\text{AD},\rho,(N_j,j\neq i)}\) is
the chemical potential; or per unit mass we have
(377)\[\begin{split}\begin{aligned}
dq &= de - \frac{p}{\rho^2}d\rho - \sum_i\mu_id
\left(\frac{n_i}{\rho}\right)\\
&= de - \frac{p}{\rho^2}d\rho - \sum_i
\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}dX_i\end{aligned}\end{split}\]
where we have used \(X_i \equiv \rho_i/\rho = A_in_i/\rho N_\text{A}\)
and the chemical potential has been replaced with
\(\mu_i = \frac{A_i}{N_\text{A}}\left(\frac{\partial e}{\partial X_i} \right)_{\rho,\text{AD},(X_j,j\neq i)}\).
Using this and expressing the specific internal energy as \(e=e(\rho,T,X)\)
we then have
(378)\[dq = c_vdT +
\left[\left(\frac{\partial e}{\partial \rho}\right)_{T,X} -\frac{p}{\rho^2}
\right]d\rho +
\sum_i\left[
\left(\frac{\partial e}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)} -
\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right]dX_i\]
and
(379)\[\begin{split}\begin{aligned}
\left(\frac{d\ln T}{d\ln\rho}\right)_\text{AD} \equiv \Gamma_3-1
&= \frac{1}{c_vT}\left[
\frac{p}{\rho} - \left(\frac{\partial e}{\partial\ln\rho}\right)_{T,X} +
\right.{}\nonumber\\
&\qquad\qquad \left.\sum_i \left[
\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
-
\left(\frac{\partial e}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)}
\right]X_i\Gamma_{4,i}\right]
\end{aligned}\end{split}\]
Now we need to evaluate \(\left(\partial e/\partial \ln\rho\right)_{T,X}\).
Again using the first law and the fact that \(ds=dq/T\) is an exact
differential (i.e. mixed derivatives are equal) we have
(380)\[\begin{split}\begin{aligned}
\label{eq:dedlnrho}
\left(
\frac{\partial}{\partial\rho}\left[\frac{c_v}{T}\right]\right)_{T,X} &=
\left(\frac{\partial}{\partial T}\left[\frac{1}{T}
\left(\frac{\partial e}{\partial\rho}\right)_{T,X} - \frac{p}{T\rho^2}
\right]\right)_{\rho,X}{}\nonumber\\
\frac{1}{T}\left(\frac{\partial}{\partial\rho}\left(
\frac{\partial e}{\partial T}\right)_{\rho,X}\right)_{T,X} &=
-\frac{1}{T^2}\left(\frac{\partial e}{\partial\rho}\right)_{T,X} +
\frac{1}{T}\left(\frac{\partial}{\partial T}\left(
\frac{\partial e}{\partial\rho}\right)_{T,X}\right)_{\rho,X}
+\frac{p}{T^2\rho^2} -
\frac{1}{T\rho^2}\left(\frac{\partial p}{\partial T}\right)_{\rho,X}
{}\nonumber\\
\therefore\quad \left(\frac{\partial e}{\partial\ln \rho}\right)_{T,X} &=
\frac{p}{\rho}\left(1-\chi_T\right),\end{aligned}\end{split}\]
exactly the same result if we were to exclude species information.
Similarly, we can find an expression for the derivative of energy with
respect to composition
(381)\[\begin{split}\begin{aligned}
\left(\frac{\partial}{\partial X_i}\left[
\frac{c_v}{T}\right]\right)_{\rho,T,(X_j,j\neq i)} &=
\left(\frac{\partial}{\partial T}\left[
\frac{1}{T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,T,(X_j,j\neq i)} - \frac{1}{T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right]\right)_{\rho,X}\\
\frac{1}{T}\left(\frac{\partial }{\partial X_i}\left(
\frac{\partial e}{\partial T}\right)_{\rho,X}\right)_{\rho,T,(X_j,j\neq i)}
&= \frac{1}{T^2}\left[\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)} -
\left(\frac{\partial e}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)}
\right] + \\
&\ \ \ \ \ \frac{1}{T}\left[
\left(\frac{\partial}{\partial T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)}
\right)_{\rho,X} -
\left(\frac{\partial }{\partial T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}\right]\\
\therefore\quad
\left(\frac{\partial e}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)} &=
\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)} - \left(
\frac{\partial}{\partial\ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}.\end{aligned}\end{split}\]
Plugging these back into Eq.379 we have
(382)\[\Gamma_3-1 = \frac{1}{c_vT}\left[\frac{p}{\rho}\chi_T +\sum_i
\left(\frac{\partial}{\partial\ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}X_i
\Gamma_{4,i}\right],\]
or
(383)\[c_v = \frac{1}{T(\Gamma_3-1)}\left[\frac{p}{\rho}\chi_T +\sum_i
\left(\frac{\partial}{\partial\ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}X_i
\Gamma_{4,i}\right].\]
We can obtain an expression for the specific heat at constant pressure
from the enthalpy
(384)\[\begin{split}\begin{aligned}
c_p \equiv \left(\frac{\partial h}{\partial T}\right)_{p,X} &=
\left(\frac{\partial e}{\partial T}\right)_{p,X} - \frac{p}{\rho^2}
\left(\frac{\partial \rho}{\partial T}\right)_{p,X}\\
&= \left(\frac{\partial e}{\partial T}\right)_{p,X} + \frac{p}{\rho^2}
\left(\frac{\partial p}{\partial T}\right)_{\rho,X}
\left(\frac{\partial \rho}{\partial p}\right)_{T,X}\\
&=\left(\frac{\partial e}{\partial T}\right)_{p,X} + \frac{p}{\rho T}
\frac{\chi_t}{\chi_\rho}.\end{aligned}\end{split}\]
The first term on the rhs can be obtained from writing \(e=e(p,T,X)\) and
\(p=p(\rho,T,X)\):
(385)\[\begin{split}\begin{aligned}
de &= \left(\frac{\partial e}{\partial p}\right)_{T,X}dp
+ \left(\frac{\partial e}{\partial T}\right)_{p,X}dT +
\sum_i \left(\frac{\partial e}{\partial X_i}\right)_{p,T,(X_j,j\neq i)}
dX_i\\
dp &= \left(\frac{\partial p}{\partial \rho}\right)_{T,X}d\rho +
\left(\frac{\partial p}{\partial T}\right)_{\rho,X}dT + \sum_i
\left(\frac{\partial p}{\partial X_i}\right)_{\rho,T,(X_j,j\neq i)}dX_i\\
\therefore \ \left(\frac{\partial e}{\partial T}\right)_{\rho,X} &=
\left(\frac{\partial e}{\partial p}\right)_{T,X}
\left(\frac{\partial p}{\partial T}\right)_{\rho,X} +
\left(\frac{\partial e}{\partial T}\right)_{p,X}\\
\Rightarrow \ \left(\frac{\partial e}{\partial T}\right)_{p,X} &= c_v -
\left(\frac{\partial e}{\partial \rho}\right)_{T,X}
\left(\frac{\partial \rho}{\partial p}\right)_{T,X}
\left(\frac{\partial p}{\partial T}\right)_{\rho,X}\\
&= c_v - \frac{p\chi_T}{\rho T\chi_\rho}\left(1-\chi_T\right)\end{aligned}\end{split}\]
and
(386)\[c_p = \frac{p}{\rho T}\frac{\chi_T^2}{\chi_\rho} + c_v\]
Dividing this by Eq.383 and using the relation between the
\(\Gamma\) ’s, Eq.374, we then have
(387)\[\begin{split}\begin{aligned}
\label{eq:pchirho}
\gamma \equiv \frac{c_p}{c_v} &= 1 + \frac{p(\Gamma_3-1)}{\rho }
\frac{\chi_T^2}{\chi_\rho}\left[\frac{p}{\rho}\chi_T +\sum_i
\left(\frac{\partial}{\partial\ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}X_i
\Gamma_{4,i}\right]^{-1}{}\nonumber\\
&= 1 + \frac{p\chi_T\left(\Gamma_1 - \chi_\rho -
\sum_i \chi_{X_i}\Gamma_{4,i}\right)}{p\chi_\rho\chi_T + \rho
\chi_\rho\sum_i \left(
\frac{\partial}{\partial \ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}X_i\Gamma_{4,i}}{}\nonumber\\
&= \frac{p\chi_T\Gamma_1 + \sum_i \left[\rho\chi_\rho\left(
\frac{\partial}{\partial \ln T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right)_{\rho,X}X_i - p\chi_T
\chi_{X_i}\right]\Gamma_{4,i}}{p\chi_\rho\chi_T + \rho
\chi_\rho\sum_i \left(
\frac{\partial}{\partial \ln T}\left(
\frac{\partial e}{\partial X_i}\right)_{\rho,\text{AD},(X_j,j\neq i)}
\right)_{\rho,X}X_i\Gamma_{4,i}}{}\nonumber\\
\Rightarrow p\chi_\rho &= \frac{1}{\chi_T\gamma}\left[p\chi_T\Gamma_1 +
\sum_i \left[\rho\chi_\rho\left(1-\gamma\right)\left(
\frac{\partial}{\partial \ln T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right)_{\rho,X}X_i - p\chi_T
\chi_{X_i}\right]\Gamma_{4,i}\right].\end{aligned}\end{split}\]
Plugging [eq:pchirho] into Eq.368 and rewriting the
partial derivative of \(e\) with the help of [eq:dedlnrho] we have
(388)\[\begin{split}\begin{aligned}
\alpha &= -\frac{1}{p\chi_\rho c_p}\left[\left(\frac{1}{\rho\chi_\rho}
- \frac{\rho e_\rho}{p\chi_\rho}\right)\frac{p\chi_T}{T} - c_p\right] \\
&=\frac{\gamma}{c_p}\frac{c_p\chi_T + \left(\rho
\left(\frac{\partial e}{\partial\ln\rho}\right)_{T,X}-p\right)
\frac{\chi_T^2}{T\rho\chi_\rho}}
{p\chi_T\Gamma_1 +
\sum_i \left[\rho\chi_\rho\left(1-\gamma\right)\left(
\frac{\partial}{\partial \ln T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right)_{\rho,X}X_i - p\chi_T
\chi_{X_i}\right]\Gamma_{4,i}}\\
&=\frac{\gamma}{\Gamma_1 p c_p}\left[\frac{c_p - \frac{p\chi_T^2}
{T\rho\chi_\rho}}
{1 + \sum_i \left[\frac{\rho\chi_\rho}{p\chi_T}
\left(1-\gamma\right)\left(
\frac{\partial}{\partial \ln T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right)_{\rho,X}X_i -
\chi_{X_i}\right]\frac{\Gamma_{4,i}}{\Gamma_1}}\right]\\
&=\left(\frac{1}{\Gamma_1p}\right)
\left[1 + \sum_i \left[\frac{\rho\chi_\rho}{p\chi_T}
\left(1-\gamma\right)\left(
\frac{\partial}{\partial \ln T}\left(\frac{\partial e}{\partial X_i}
\right)_{\rho,\text{AD},(X_j,j\neq i)}\right)_{\rho,X}X_i -
\chi_{X_i}\right]\frac{\Gamma_{4,i}}{\Gamma_1}\right]^{-1}\\\end{aligned}\end{split}\]
(389)\[\boxed{
\alpha = \frac{1}{\Gamma_1p}\left[1 + \sum_i\left[\frac{\rho^2p_\rho}
{pp_T}(1-\gamma)\frac{N_\text{A}}{A_i}
\left(\frac{\partial\mu_i}{\partial T}\right)_{\rho,X}X_i -
\chi_{X_i}\right]\frac{\Gamma_{4,i}}{\Gamma_1}\right]^{-1}
}\]