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\title{Actual opacity, or its derivative?}
\author{}
\date{\today,~ $ $Revision: 1.5 $ $}
\begin{document}
\maketitle

Concerning Regner's remark ``It is the actual opacity, and not its
derivatives, that determines convective stability,'' we examine the
solution for
\EQ
\kappa(\rho,T)=\rho f(T),
\EN
where $b(T)\equiv\dd\ln f/\dd\ln T$ quantifies the local slope of
$\kappa(T)$.
If $b(T)$ were a constant, the solution is a polytrope with polytropic
index $n=(3-b)/(1+a)$, where $a\equiv\dd\ln\kappa/\dd\ln\rho=1$; see
Eq.~(8) of \cite{BB14}.
Consider now the following form for $\ln f(\ln T)$:
\EQ
\ln f=\ln f_1+\half\Delta\ln f\erfc(-\sqrt{\pi} b_{\max} \Delta\ln T/\Delta\ln f),
\EN
where $\Delta\ln f=\ln f_2-\ln f_1$, and
$f_1$ and $f_2$ are the values of $f$ for $T\to0$ and $T\to\infty$, respectively.
This yields the following expression for the function $b(T)$:
\EQ
b(T)=b_{\max} \exp\left[-\pi b_{\max}^2 (\Delta\ln T/\Delta\ln f)^2\right].
\EN
These profiles for $f(T)$ are plotted in \Fig{kappa_erfc} for
$f_1=10^6$ and $f_2=10^8$ in units where $T$ is in K,
length in units of $\Mm$, velocity in units of $\km\s^{-1}$
and density in units of $\g\cm^{-3}$.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{kappa_erfc}
\end{center}\caption[]{
Plots of $\kappa(T)$ and $b(T)$
for $b_{\rm max}=0$ (black), 1 (red), and 10 (blue).
}\label{kappa_erfc}\end{figure}

To obtain the hydrostatic (nonconvecting) solution, we first integrate \citep{Bra16}
\EQ
\ln T=\frac{F c_P\nabla_{\rm ad}}{g}\int \frac{\dd\ln P}{K(T,P)}
\EN
where $K=16\sigma_{\rm SB} T^3/3\kappa\rho$ is the radiative conductivity,
$\rho=P/(c_P\nabla_{\rm ad} T)$ is the perfect gas density,
$F=\sigma_{\rm SB} T_{\rm eff}^4$ is the radiative flux,
$T_{\rm eff}=2^{1/4}T_{\rm top}$ is the effective temperature in
terms of the constant temperature at the top of the layer,
$c_P=0.0346\km^2\s^{-2}\K^{-1}$ is the specific heat at constant pressure,
$\nabla_{\rm ad}=1-1/\gamma=0.4$ for $\gamma=5/3$,
and $g=274\km^2\s^{-2}\Mm^{-1}$ is the solar value of the gravitational acceleration.
We choose $T_0=14,000\K$ for the temperature where the bump occurs.

\begin{figure}[h!]\begin{center}
\includegraphics[width=.49\columnwidth]{p_erfc}
\includegraphics[width=.49\columnwidth]{p_erfc_SS}
\end{center}\caption[]{
Plots of $T(z)$ and $S(T)\equiv T/P^{2/5}$
for $b_{\rm max}=0$ (black), 1 (red), and 10 (blue).
}\label{p_erfc}\end{figure}

We choose three values of $b_{\rm max}$: 0, 1, and 10.
For $b_{\rm max}=0$, we choose $T_{\rm top}=5300\K$, while
for the other values, we adjust $T_{\rm top}$ such that the
temperature profiles $T(P)$ overlap for large temperature;
see \Tab{Tres} for a summary.

\begin{table}[h]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{rccccccc}
$b_{\rm max}$ & $T_{\rm top}$ & $\rho_{\rm bot}$\\
\hline
 0 & $5300\K$ & $1.30\times10^{-7}$ \\
 1 & $3660\K$ & $1.84\times10^{-7}$ \\
10 & $3000\K$ & $1.45\times10^{-7}$ \\
\label{Tres}\end{tabular}}\end{table}

The blue lines in \Fig{p_erfc} are always Schwarzschild unstable
in the deeper parts.
To recover stability at the bottom, we can put $a=0$, so that
$\kappa$ is independent of $\rho$, i.e., we put $\kappa(T)=f(T)$.
The result is shown in \Fig{p_erfc_a0}.

\begin{figure}[h!]\begin{center}
\includegraphics[width=.49\columnwidth]{p_erfc_a0}
\includegraphics[width=.49\columnwidth]{p_erfc_a0_SS}
\end{center}\caption[]{
Similar to \Fig{p_erfc}, but with 
$\kappa(T)=f(T)$, being independent of $\rho$
and for $b_{\rm max}=3$ (black), 5 (red), and 10 (blue).
}\label{p_erfc_a0}\end{figure}

%Regner: ``We need the exponents of the Kramers' expression because
%they are needed to counter the explicit exponents of $T$ and $p$ in
%the expression for $\nabla_{\rm rad}$, in order for it to stay below
%(or above) the (assumed) constant $\nabla_{\rm ad}$. This can be done
%since we know the forms of $T$ and $p$ in polytropes (but not for
%actual stars!). It is, however, still the actual opacity, and not its
%derivatives, that determines convective stability.''


%\begin{enumerate}\itemsep-0.2mm
%\item
%\end{enumerate}

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%$R$ & $T$ & $\kf$ & $\omf$ & $\EEGW$ & $\hrms$ & $k_{\rm peak}$ \\
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

%r e f
\begin{thebibliography}{}

\bibitem[Barekat \& Brandenburg(2014)]{BB14}
Barekat, A., \& Brandenburg, A.\yanaN{2014}{571}{A68}
{Near-polytropic stellar simulations with a radiative surface}

\bibitem[Brandenburg(2016)]{Bra16}
Brandenburg A.\yapjN{2016}{832}{6}
{Stellar mixing length theory with entropy rain}

\end{thebibliography}

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