Notes on ${\beta }_0$

The goal of ${\beta }_0$ is to capture the expansion of a displaced fluid element due to the stratification of the atmosphere. MAESTROeX computes ${\beta }_0$ as:

(360)$${\beta }_0(r,t)={\rho }_0\exp \left({\int }_0^r\frac{1}{{\bar{\mathrm{\Gamma }}}_1{p}_0}\frac{\partial p_0}{\partial r^{\prime }}d{r}^{\prime }\right)$$

Constant Composition

Consider an isentropically stratified atmosphere, with a constant composition as a function of $r$. If you displace a parcel of fluid upwards, it will expand adabatically and continue to rise until its density matches the background density. Even if ${\bar{\mathrm{\Gamma }}}_1$ is not constant in $r$, following from the definition of ${\beta }_0$,

(361)$$\frac{1}{{\beta }_0}\frac{d{\beta }_0}{dr}=\frac{1}{{\bar{\mathrm{\Gamma }}}_1{p}_0}\frac{d{p}_0}{dr}$$

and the definition of ${\mathrm{\Gamma }}_1$,

(362)$${\mathrm{\Gamma }}_1={\frac{d\log p}{d\log \rho }|}_s$$

So, at constant entropy, from the definition of ${\mathrm{\Gamma }}_1$, it must hold that

(363)$$\frac{1}{\rho }\frac{d\rho }{dr}=\frac{1}{{\mathrm{\Gamma }}_{1}p}\frac{dp}{dz}.$$

Comparing to the definition of ${\beta }_0$ then

(364)$$\frac{1}{{\beta }_0}\frac{{\beta }_0}{dr}=\frac{1}{{\bar{\mathrm{\Gamma }}}_1{p}_0}\frac{d{p}_0}{dr}=\frac{1}{{\rho }_0}\frac{d{\rho }_0}{dr}.$$

Therefore, ${\beta }_0={\rho }_0$.

This means that if we have a constant composition and an isentropically stratified atmosphere, as we displace a fluid element, it will always remain neutrally buoyant.

Composition Gradient

If there is a change in composition with $r$, the situation is more complicated. Consider again an isentropically stratified atmosphere, now with a composition gradient. If you displace a parcel of fluid upwards, it will rise. If there are no processes that change the composition (e.g. reactions), then the composition in the fluid element will remain fixed. As it rises, it will the ambient medium will have a different composition that it has. In this case, what is the path to equilibrium?

On the Effect of Chemical Potential

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)$$\alpha \equiv -\left(\frac{(1-\rho h_p)p_T-\rho c_p}{{\rho }^2{c}_p{p}_{\rho }}\right)$$

where

(366)$$h_p\equiv {\left(\frac{\partial h}{\partial p}\right)}_{T,X},c_p\equiv {\left(\frac{\partial h}{\partial T}\right)}_{p,X},p_T\equiv {\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X},p_{\rho }\equiv {\left(\frac{\partial p}{\partial \rho }\right)}_{T,X}$$

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)$$\alpha =\frac{1}{{\mathrm{\Gamma }}_1{p}_0}$$

used CG’s discussion of the various adiabatic $\mathrm{\Gamma }$’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)$$\alpha =-\frac{1}{p{\chi }_{\rho }{c}_p}[(\frac{1}{\rho {\chi }_{\rho }}-\frac{\rho e_{\rho }}{p{\chi }_{\rho }})\frac{p{\chi }_T}{T}-c_p],$$

where

(369)$$\begin{array}{r}\begin{array}{rl}{\chi }_{\rho }& \equiv {\left(\frac{\partial \ln p}{\partial \ln \rho }\right)}_{T,X}\\ {\chi }_T& \equiv {\left(\frac{\partial \ln p}{\partial \ln T}\right)}_{\rho ,X}.\end{array}\end{array}$$

Following the results of paper I, we want to find a relation between $p{\chi }_{\rho }$ and ${\mathrm{\Gamma }}_1$.

For an equation of state $p=p(\rho ,T,X)$ we have

(370)$$d\ln p={\left(\frac{\partial \ln p}{\partial \ln \rho }\right)}_{T,X}d\ln \rho +{\left(\frac{\partial \ln p}{\partial \ln T}\right)}_{\rho ,X}d\ln T+\sum _i{\left(\frac{\partial \ln p}{\partial \ln X_i}\right)}_{\rho ,T,(X_j,j\ne i)}d\ln X_i.$$

We define another logarithmic derivative

(371)$$\begin{array}{rl}{\chi }_{X_i}& \equiv {\left(\frac{\partial \ln p}{\partial \ln X_i}\right)}_{\rho ,T,(X_j,j\ne i)}\end{array}$$

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)$${\mathrm{\Gamma }}_1={\chi }_{\rho }+{\chi }_T({\mathrm{\Gamma }}_3-1)+\sum _i{\chi }_{X_i}{\mathrm{\Gamma }}_{4,i}$$

where we use CG’s definition of ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_3$ and introduce a fourth gamma function:

(375)$${\mathrm{\Gamma }}_1\equiv {\left(\frac{\partial \ln p}{\partial \ln \rho }\right)}_{\text{AD}},{\mathrm{\Gamma }}_3-1\equiv {\left(\frac{\partial \ln T}{\partial \ln \rho }\right)}_{\text{AD}},{\mathrm{\Gamma }}_{4,i}\equiv {\left(\frac{\partial \ln X_i}{\partial \ln \rho }\right)}_{\text{AD}},$$

where the subscript AD means along an adiabat. We now derive an expression for ${\mathrm{\Gamma }}_3$.

The first law of thermodynamics can be written as

(376)$$dQ=dE+pdV-\sum _i{\mu }_{i}d{N}_i$$

where ${\mu }_i={\left(\frac{\partial E}{\partial N_i}\right)}_{\text{AD},\rho ,(N_j,j\ne i)}$ is the chemical potential; or per unit mass we have

(377)$$\begin{array}{r}\begin{array}{rl}dq& =de-\frac{p}{{\rho }^2}d\rho -\sum _i{\mu }_{i}d\left(\frac{n_i}{\rho }\right)\\ & =de-\frac{p}{{\rho }^2}d\rho -\sum _i{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}d{X}_i\end{array}\end{array}$$

where we have used $X_i\equiv {\rho }_i/\rho =A_i{n}_i/\rho N_{\text{A}}$ and the chemical potential has been replaced with ${\mu }_i=\frac{A_i}{N_{\text{A}}}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}$. Using this and expressing the specific internal energy as $e=e(\rho ,T,X)$ we then have

(378)$$dq=c_{v}dT+[{\left(\frac{\partial e}{\partial \rho }\right)}_{T,X}-\frac{p}{{\rho }^2}]d\rho +\sum _i[{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}-{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}]d{X}_i$$

and

(379)$$\begin{array}{r}\begin{array}{rl}{\left(\frac{d\ln T}{d\ln \rho }\right)}_{\text{AD}}\equiv {\mathrm{\Gamma }}_3-1& =\frac{1}{c_{v}T}[\frac{p}{\rho }-{\left(\frac{\partial e}{\partial \ln \rho }\right)}_{T,X}+\\ & \sum _i[{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}-{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}]X_i{\mathrm{\Gamma }}_{4,i}]\end{array}\end{array}$$

Now we need to evaluate ${(\partial e/\partial \ln \rho )}_{T,X}$. 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)$$\begin{array}{r}\begin{array}{rl}{\left(\frac{\partial }{\partial \rho }\left[\frac{c_v}{T}\right]\right)}_{T,X}& ={\left(\frac{\partial }{\partial T}[\frac{1}{T}{\left(\frac{\partial e}{\partial \rho }\right)}_{T,X}-\frac{p}{T{\rho }^2}]\right)}_{\rho ,X}\\ \frac{1}{T}{\left(\frac{\partial }{\partial \rho }{\left(\frac{\partial e}{\partial T}\right)}_{\rho ,X}\right)}_{T,X}& =-\frac{1}{T^2}{\left(\frac{\partial e}{\partial \rho }\right)}_{T,X}+\frac{1}{T}{\left(\frac{\partial }{\partial T}{\left(\frac{\partial e}{\partial \rho }\right)}_{T,X}\right)}_{\rho ,X}+\frac{p}{T^2{\rho }^2}-\frac{1}{T{\rho }^2}{\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X}\\ \therefore {\left(\frac{\partial e}{\partial \ln \rho }\right)}_{T,X}& =\frac{p}{\rho }(1-{\chi }_T),\end{array}\end{array}$$

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)$$\begin{array}{r}\begin{array}{rl}{\left(\frac{\partial }{\partial X_i}\left[\frac{c_v}{T}\right]\right)}_{\rho ,T,(X_j,j\ne i)}& ={\left(\frac{\partial }{\partial T}[\frac{1}{T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}-\frac{1}{T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}]\right)}_{\rho ,X}\\ \frac{1}{T}{\left(\frac{\partial }{\partial X_i}{\left(\frac{\partial e}{\partial T}\right)}_{\rho ,X}\right)}_{\rho ,T,(X_j,j\ne i)}& =\frac{1}{T^2}[{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}-{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}]+\\ & \frac{1}{T}[{\left(\frac{\partial }{\partial T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}\right)}_{\rho ,X}-{\left(\frac{\partial }{\partial T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}]\\ \therefore {\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}& ={\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}-{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}.\end{array}\end{array}$$

Plugging these back into Eq.379 we have

(382)$${\mathrm{\Gamma }}_3-1=\frac{1}{c_{v}T}[\frac{p}{\rho }{\chi }_T+\sum _i{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i{\mathrm{\Gamma }}_{4,i}],$$

or

(383)$$c_v=\frac{1}{T({\mathrm{\Gamma }}_3-1)}[\frac{p}{\rho }{\chi }_T+\sum _i{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i{\mathrm{\Gamma }}_{4,i}].$$

We can obtain an expression for the specific heat at constant pressure from the enthalpy

(384)$$\begin{array}{r}\begin{array}{rl}{c}_p\equiv {\left(\frac{\partial h}{\partial T}\right)}_{p,X}& ={\left(\frac{\partial e}{\partial T}\right)}_{p,X}-\frac{p}{{\rho }^2}{\left(\frac{\partial \rho }{\partial T}\right)}_{p,X}\\ & ={\left(\frac{\partial e}{\partial T}\right)}_{p,X}+\frac{p}{{\rho }^2}{\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X}{\left(\frac{\partial \rho }{\partial p}\right)}_{T,X}\\ & ={\left(\frac{\partial e}{\partial T}\right)}_{p,X}+\frac{p}{\rho T}\frac{{\chi }_t}{{\chi }_{\rho }}.\end{array}\end{array}$$

The first term on the rhs can be obtained from writing $e=e(p,T,X)$ and $p=p(\rho ,T,X)$:

(385)$$\begin{array}{r}\begin{array}{rl}de& ={\left(\frac{\partial e}{\partial p}\right)}_{T,X}dp+{\left(\frac{\partial e}{\partial T}\right)}_{p,X}dT+\sum _i{\left(\frac{\partial e}{\partial X_i}\right)}_{p,T,(X_j,j\ne i)}d{X}_i\\ dp& ={\left(\frac{\partial p}{\partial \rho }\right)}_{T,X}d\rho +{\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X}dT+\sum _i{\left(\frac{\partial p}{\partial X_i}\right)}_{\rho ,T,(X_j,j\ne i)}d{X}_i\\ \therefore {\left(\frac{\partial e}{\partial T}\right)}_{\rho ,X}& ={\left(\frac{\partial e}{\partial p}\right)}_{T,X}{\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X}+{\left(\frac{\partial e}{\partial T}\right)}_{p,X}\\ \Rightarrow {\left(\frac{\partial e}{\partial T}\right)}_{p,X}& =c_v-{\left(\frac{\partial e}{\partial \rho }\right)}_{T,X}{\left(\frac{\partial \rho }{\partial p}\right)}_{T,X}{\left(\frac{\partial p}{\partial T}\right)}_{\rho ,X}\\ & =c_v-\frac{p{\chi }_T}{\rho T{\chi }_{\rho }}(1-{\chi }_T)\end{array}\end{array}$$

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 $\mathrm{\Gamma }$ ’s, Eq.374, we then have

(387)$$\begin{array}{r}\begin{array}{rl}\gamma \equiv \frac{c_p}{c_v}& =1+\frac{p({\mathrm{\Gamma }}_3-1)}{\rho }\frac{{\chi }_T^2}{{\chi }_{\rho }}{[\frac{p}{\rho }{\chi }_T+\sum _i{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i{\mathrm{\Gamma }}_{4,i}]}^{-1}\\ & =1+\frac{p{\chi }_T({\mathrm{\Gamma }}_1-{\chi }_{\rho }-\sum _i{\chi }_{X_i}{\mathrm{\Gamma }}_{4,i})}{p{\chi }_{\rho }{\chi }_T+\rho {\chi }_{\rho }\sum _i{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i{\mathrm{\Gamma }}_{4,i}}\\ & =\frac{p{\chi }_T{\mathrm{\Gamma }}_1+\sum _i[\rho {\chi }_{\rho }{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i-p{\chi }_T{\chi }_{X_i}]{\mathrm{\Gamma }}_{4,i}}{p{\chi }_{\rho }{\chi }_T+\rho {\chi }_{\rho }\sum _i{\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i{\mathrm{\Gamma }}_{4,i}}\\ \Rightarrow p{\chi }_{\rho }& =\frac{1}{{\chi }_T\gamma }[p{\chi }_T{\mathrm{\Gamma }}_1+\sum _i[\rho {\chi }_{\rho }(1-\gamma ){\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i-p{\chi }_T{\chi }_{X_i}]{\mathrm{\Gamma }}_{4,i}].\end{array}\end{array}$$

Plugging [eq:pchirho] into Eq.368 and rewriting the partial derivative of $e$ with the help of [eq:dedlnrho] we have

(388)$$\begin{array}{r}\begin{array}{rl}\alpha & =-\frac{1}{p{\chi }_{\rho }{c}_p}[(\frac{1}{\rho {\chi }_{\rho }}-\frac{\rho e_{\rho }}{p{\chi }_{\rho }})\frac{p{\chi }_T}{T}-c_p]\\ & =\frac{\gamma }{c_p}\frac{c_p{\chi }_T+(\rho {\left(\frac{\partial e}{\partial \ln \rho }\right)}_{T,X}-p)\frac{{\chi }_T^2}{T\rho {\chi }_{\rho }}}{p{\chi }_T{\mathrm{\Gamma }}_1+\sum _i[\rho {\chi }_{\rho }(1-\gamma ){\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i-p{\chi }_T{\chi }_{X_i}]{\mathrm{\Gamma }}_{4,i}}\\ & =\frac{\gamma }{{\mathrm{\Gamma }}_{1}p{c}_p}\left[\frac{c_p-\frac{p{\chi }_T^2}{T\rho {\chi }_{\rho }}}{1+\sum _i[\frac{\rho {\chi }_{\rho }}{p{\chi }_T}(1-\gamma ){\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i-{\chi }_{X_i}]\frac{{\mathrm{\Gamma }}_{4,i}}{{\mathrm{\Gamma }}_1}}\right]\\ & =\left(\frac{1}{{\mathrm{\Gamma }}_{1}p}\right){[1+\sum _i[\frac{\rho {\chi }_{\rho }}{p{\chi }_T}(1-\gamma ){\left(\frac{\partial }{\partial \ln T}{\left(\frac{\partial e}{\partial X_i}\right)}_{\rho ,\text{AD},(X_j,j\ne i)}\right)}_{\rho ,X}{X}_i-{\chi }_{X_i}]\frac{{\mathrm{\Gamma }}_{4,i}}{{\mathrm{\Gamma }}_1}]}^{-1}\end{array}\end{array}$$

(389)$$\boxed{{\displaystyle \alpha =\frac{1}{{\mathrm{\Gamma }}_{1}p}{[1+\sum _i[\frac{{\rho }^2{p}_{\rho }}{p{p}_T}(1-\gamma )\frac{N_{\text{A}}}{A_i}{\left(\frac{\partial {\mu }_i}{\partial T}\right)}_{\rho ,X}{X}_i-{\chi }_{X_i}]\frac{{\mathrm{\Gamma }}_{4,i}}{{\mathrm{\Gamma }}_1}]}^{-1}}}$$

Recalling Derivation of ${\beta }_0$

Recall from paper I that ${\beta }_0$ was derived from the equation

(390)$$\mathrm{\nabla }\cdot \mathbf{U}+\alpha \mathbf{U}\cdot \mathrm{\nabla }{p}_0=\tilde{S}$$

in such a fashion that we ended up with an equation of the form

(391)$$\mathrm{\nabla }\cdot ({\beta }_0(r)\mathbf{U})={\beta }_0\tilde{S}.$$

The derivation in Appendix B of paper I for a ${\beta }_0$ that satisfies [eq:beta constraint] automatically assumed $\alpha ={\left({\mathrm{\Gamma }}_{1_0}{p}_0\right)}^{-1}$. This would have to be modified with the above derivation of $\alpha$ to be correct in a non-operator split fashion.