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\begin{document}
\title{Equations developed in the programing formalism for the angular
momentum tranport following Mathis \& Zahn 2004} 
\author{} 
\maketitle

Following Mathis \& Zahn 2004, the complete set of differential
equations to be solved are (I always imposse the approximation $\Omega_2=0$):

\begin{enumerate}

\item Eq. B.1.1 (B.1), mean rotation rate: This equation does not
  change.

\item Eq. B.1.2 (B.2), mean chemical composition: This equation does
  not change.

\item Eq. B.2.1 (B.3), meridional circulation:

In this case

\begin{equation}
0={\cal B}_2-{M\bar g\bar\rho C_p\bar T(\nabla_{ad}-\nabla)\over LP}U_2
\end{equation}

with

\begin{eqnarray}
{\cal B}_2=-2\Big[1-{1\over 6\pi G\bar\rho R^2}{\partial
    R^3\bar\Omega^2\over\partial
    R}-{\bar\epsilon+\bar\epsilon_g\over\epsilon_m}\Big]\Big[{R^2\over\bar
    g^2}{d\bar g\over dR}{1\over 3}\bar\Omega^2-{R\over\bar g}{2\over
    3}\bar\Omega^2-{d\over dR}\Big({\hat\phi_2\over g_0}\Big)\Big]
    \nonumber\\ 
    -{1\over\pi G\bar\rho}{R\over
    3}\bar\Omega{\partial\bar\Omega\over\partial
    R}+{\rho_m\over\bar\rho}\Big[{R\over
    3}{\partial{\cal A}_2\over\partial R}-{2H_t\over R}\Big(1+{D_h\over
    K}\Big)\Psi_2\Big]
    \nonumber\\
    +{\bar\epsilon+\bar\epsilon_g\over\epsilon_m}
    \Big[{\cal A}_2+(f_\epsilon(\epsilon_T-\delta)+\delta)\Psi_2+
    (f_\epsilon(\epsilon_\mu+\varphi)-\varphi)\Lambda_2\Big]-{M\over
    L}C_p\bar T\Big({d\Psi_2\over dt}+\Phi{d\ln\bar\mu\over
    dt}\Lambda_2\Big)
\end{eqnarray}

From now on, I will use the approximation and notation that any
quantity can be written in the form (unless confusion)

\begin{equation}
\bar X\approx X_0\equiv X
\end{equation}

This equation can be divided in 16 terms

\begin{enumerate}

\item
\begin{eqnarray}
  -{2R^2\over 3g^2}{dg\over dR}\Omega^2=-{2R^2_\odot r^2\over
  3g^2}{8\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\sqrt{\nu}}{\partial g\over \partial\nu}\Omega^2
  \nonumber\\
  =-{16\pi\over
  9}{R^4_\odot\over M_\odot}{r^4\rho\over g\sqrt{\nu}}{\partial\ln g\over
  \partial\nu}\Omega^2
\end{eqnarray}

\item
\begin{equation}
{4\over 3}{R\over g}\Omega^2={4\over 3}R_\odot{r\over g}\Omega^2
\end{equation}

\item
\begin{eqnarray}
  2{d\over dR}\Big({\hat\phi_2\over g}\Big)=2{8\pi\over
  3}{R^2_\odot\over M_\odot}{r^2\rho\over\sqrt{\nu}}\Big({1\over
  g}{\partial\hat\phi_2\over\partial\nu}-{1\over
  g^2}\hat\phi_2{\partial g\over\partial\nu}\Big)
  \nonumber\\
  ={16\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
  g\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}-
  {16\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\hat\phi_2
\end{eqnarray}

\item
\begin{eqnarray}
  {1\over 3\pi G\rho R^2}{\partial R^3\Omega^2\over\partial
  R}{R^2\over g^2}{dg\over dR}{1\over 3}\Omega^2={1\over 9\pi G\rho
  g}{d \ln g\over
  dR}\Omega^2\Big(3R^2\Omega^2+2R^3\Omega{\partial\Omega\over\partial
  R}\Big)
  \nonumber\\
  ={1\over 9\pi G\rho}{8\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Omega^2\Big(3R^2_\odot
  r^2\Omega^2+2R^3_\odot r^3\Omega{8\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\sqrt{\nu}}{\partial\Omega\over\partial\nu}\Big)
  \nonumber\\
  ={8\over 27}{R^2_\odot\over GM_\odot}{r^2\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Omega^2\Big(3R^2_\odot
  r^2\Omega^2+{16\pi\over 3}{R^5_\odot\over
  M_\odot}{r^5\rho\over\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}\Big)
  \nonumber\\
  ={8\over 9}{R^4_\odot\over GM_\odot}{r^4\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Omega^4+{128\pi\over
  81}{R^7_\odot\over GM^2_\odot}{r^7\rho\over g\nu}{\partial\ln
  g\over\partial\nu}\Omega^3{\partial\Omega\over\partial\nu}
\end{eqnarray}

\item
\begin{eqnarray}
  -{1\over 3\pi G\rho R^2}{\partial R^3\Omega^2\over\partial R}{R\over
  g}{2\over 3}\Omega^2=-{2\over 9\pi G\rho
  Rg}\Omega^2\Big(3R^2\Omega^2+2R^3\Omega{\partial\Omega\over\partial
  R}\Big)
  \nonumber\\
  =-{2\over 3\pi G}R_\odot{r\over\rho g}\Omega^4-{4\over 9\pi
  G\rho g}R^2_\odot r^2{8\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\sqrt{\nu}}\Omega^3{\partial\Omega\over\partial\nu}
  \nonumber\\
  =-{2\over 3\pi G}R_\odot{r\over\rho g}\Omega^4-{32\over
  27}{R^4_\odot\over GM_\odot}{r^4\over
  g\sqrt{\nu}}\Omega^3{\partial\Omega\over\partial\nu}
\end{eqnarray}

\item
\begin{eqnarray}
  -{1\over 3\pi G\rho R^2}{\partial R^3\Omega^2\over\partial R}{d\over
   dR}\Big({\hat\phi_2\over g}\Big)=
  \nonumber\\
  -{1\over 3\pi G\rho
   R^2}\Big(3R^2\Omega^2+2R^3\Omega{\partial\Omega\over\partial
   R}\Big)\Big({1\over g}{d\hat\phi_2\over dR}-{1\over
   g}\hat\phi_2{d\ln g\over dR}\Big)
  \nonumber\\
  =-{1\over\pi G}{1\over\rho
   g}\Omega^2{d\hat\phi_2\over dR}+{1\over\pi G}{1\over\rho
   g}{d\ln g\over dR}\Omega^2\hat\phi_2-{2\over 3\pi
   G}{R\over\rho g}\Omega{d\Omega\over
   dR}{\partial\hat\phi_2\over\partial R}+
  \nonumber\\
   {2\over 3\pi
   G}{R\over\rho g}\Omega{d\Omega\over
   dR}{\partial\ln g\over\partial R}\hat\phi_2
  \nonumber\\
  =-{1\over\pi G}{1\over\rho g}{8\pi\over 3}{R^2_\odot\over M_\odot}
  {r^2\rho\over\sqrt{\nu}}\Omega^2{\partial\hat\phi_2\over\partial\nu}+
  {1\over\pi G}{1\over\rho g}{8\pi\over 3}{R^2_\odot\over M_\odot}
  {r^2\rho\over\sqrt{\nu}}\Omega^2{\partial\ln
   g\over\partial\nu}\hat\phi_2-
  \nonumber\\
  {2\over 3\pi G}{R\over\rho
    g}\Omega{64\pi^2\over 9}{R^4_\odot\over
   M^2_\odot}{r^4\rho^2\over\nu}{\partial\Omega\over\partial\nu}
  {\partial\hat\phi_2\over\partial\nu}+
      {2\over 3\pi G}{R\over\rho
    g}\Omega{64\pi^2\over 9}{R^4_\odot\over
   M^2_\odot}{r^4\rho^2\over\nu}{\partial\ln g\over\partial\nu}
  {\partial\Omega\over\partial\nu}\hat\phi_2
  \nonumber\\
  =-{8\over 3}{R^2_\odot\over GM_\odot}{r^2\over
    g\sqrt{\nu}}\Omega^2{\partial\hat\phi_2\over\partial\nu}+
  {8\over 3}{R^2_\odot\over GM_\odot}{r^2\over
    g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Omega^2\hat\phi_2-
  \nonumber\\
  {128\pi\over 27}{R^5_\odot\over GM^2_\odot}{r^5\rho\over g\nu}\Omega
  {\partial\Omega\over\partial\nu}{\partial\hat\phi_2\over\partial\nu}+
  {128\pi\over 27}{R^5_\odot\over GM^2_\odot}{r^5\rho\over g\nu}
  {\partial\ln
   g\over\partial\nu}\Omega{\partial\Omega\over\partial\nu}\hat\phi_2
\end{eqnarray}

\item
\begin{eqnarray}
  2{\bar\epsilon+\bar\epsilon_g\over\epsilon_m}{R^2\over g^2}{dg\over
  dR}{1\over 3}\Omega^2={2\over 3}{\partial\ln l\over\partial\ln
  m}R^2_\odot{r^2\over g^2}{8\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\sqrt{\nu}}{\partial g\over\partial\nu}\Omega^2
  \nonumber\\
  ={16\pi\over 9}{R^4_\odot\over M_\odot}{r^4\rho\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}{\partial\ln
  l\over\partial\ln m}\Omega^2
\end{eqnarray}

\item
\begin{equation}
  -2{\bar\epsilon+\bar\epsilon_g\over\epsilon_m}{R\over g}{2\over
   3}\Omega^2=-{4\over 3}R_\odot{r\over g}{\partial\ln
   l\over\partial\ln m}\Omega^2
\end{equation}

\item
\begin{eqnarray}
  -2{\bar\epsilon+\bar\epsilon_g\over\epsilon_m}{d\over
   dR}\Big({\hat\phi_2\over g}\Big)=-2{\partial\ln l\over\partial\ln
   m}\Big({1\over g}{d\hat\phi_2\over dR}-{1\over
   g^2}\hat\phi_2{dg\over dR}\Big)
  \nonumber\\
  =-{16\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
   g\sqrt{\nu}}{\partial\ln l\over\partial\ln
   m}{\partial\hat\phi_2\over\partial\nu}+{16\pi\over 3}{R^2_\odot\over
   M_\odot}{r^2\rho\over g\sqrt{\nu}}{\partial\ln l\over\partial\ln
   m}{\partial\ln g\over\partial\nu}\hat\phi_2
\end{eqnarray}

\item
\begin{equation}
  -{1\over \pi G\rho}{R\over 3}\Omega{\partial\Omega\over\partial R}=
  -{8\over 9}{R^3_\odot\over
  GM_\odot}{r^3\over\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}
\end{equation}

\item
\begin{equation}
  {\rho_m\over\rho}{R\over 3}{\partial{\cal A}_2\over\partial R}=
  {3M_\odot\nu^{3/2}\over 4\pi R^3_\odot r^3\rho}{R_\odot r\over
  3}{8\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over\sqrt{\nu}}
  {\partial{\cal A}_2\over\partial\nu}={2\over 3}\nu{\partial{\cal
  A}_2\over\partial\nu}
\end{equation}

\item
\begin{eqnarray}
  -{\rho_m\over\rho}2{H_T\over R}\Big(1+{D_h\over K}\Big)\Psi_2=
  -{3M_\odot\nu^{3/2}\over 4\pi R^3_\odot r^3\rho}2{H_T\over R}
  \Big(1+{D_h\over K}\Big)\Psi_2
  \nonumber\\
  =-{3\over 2\pi}{M_\odot\over R^4_\odot}{\nu^{3/2}\over r^4\rho}H_T
  \Big(1+{D_h\over K}\Big)\Psi_2
\end{eqnarray}

\item
\begin{equation}
  {\bar\epsilon+\bar\epsilon_g\over\epsilon_m}{\cal A}_2=
  {\partial\ln l\over\partial\ln m}{\cal A}_2
\end{equation}

\item
\begin{eqnarray}
  {\bar\epsilon+\bar\epsilon_g\over\epsilon_m}(f_\epsilon(\epsilon_T-\delta)+
  \delta)\Psi_2={\partial\ln l\over\partial\ln
  m}\Big({M_\odot\nu^{3/2}\over L_\odot l}{\partial\ln
  m\over\partial\ln l}\epsilon_N(\epsilon_T-\delta)+\delta\Big)\Psi_2
  \nonumber\\
  =\Big({M_\odot\nu^{3/2}\over L_\odot
  l}\epsilon_N(\epsilon_T-\delta)+{\partial\ln l\over\partial\ln
  m}\delta\Big)\Psi_2
\end{eqnarray}

\item
\begin{eqnarray}
  {\bar\epsilon+\bar\epsilon_g\over\epsilon_m}(f_\epsilon(\epsilon_\mu+\varphi)
  -\varphi)\Lambda_2={\partial\ln l\over\partial\ln
  m}\Big({M_\odot\nu^{3/2}\over L_\odot l}{\partial\ln
  m\over\partial\ln l}\epsilon_N(\epsilon_\mu+\psi)-\psi\Big)\Lambda_2
  \nonumber\\
  =\Big({M_\odot\nu^{3/2}\over L_\odot
  l}\epsilon_N(\epsilon_\mu+\varphi)-{\partial\ln l\over\partial\ln
  m}\varphi\Big)\Lambda_2
\end{eqnarray}

\item
\begin{equation}
  -{M\over L}C_PT{d\Psi_2\over dt}=-{M_\odot\nu^{3/2}\over L_\odot
   l}C_PT{\Psi_2-\Psi_2(t)\over\Delta t}
\end{equation}
\end{enumerate}

These are all the terms evolved in ${\cal B}_2$. I must admit I do not
know how to translate to this formalism the term

\begin{equation}
  \Phi{d\ln\hat\mu\over dt}\Lambda_2
\end{equation}

Therefore I have left it, but its addition is trivial.

Therefore, the complete equation for the meridional circulation can be
written as (groupping by variables)

\begin{eqnarray}
  \Big({8\over 9}{R^4_\odot\over GM_\odot}{r^4\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}-{2\over 3\pi
  G}R_\odot{r\over\rho g}\Big)\Omega^4
  \nonumber\\
  +\Big({128\pi\over 81}{R^7_\odot\over GM^2_\odot}{r^7\rho\over
  g\nu}{\partial\ln g\over\partial\nu}-{32\over 27}{R^4_\odot\over
  GM_\odot}{r^4\over
  g\sqrt{\nu}}\Big)\Omega^3{\partial\Omega\over\partial\nu}
  \nonumber\\
  +\Big({4\over 3}R_\odot{r\over g}-{16\pi\over 9}{R^4_\odot\over
  M_\odot}{r^4\rho\over g\sqrt{\nu}}{\partial\ln
  g\over\partial\nu}\Big)\Big(1-{\partial\ln l\over\partial\ln
  m}\Big)\Omega^2
  \nonumber\\
  -{8\over 9}{R^3_\odot\over
  GM_\odot}{r^3\over\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}
  \nonumber\\
  -{8\over 3}{R^2_\odot\over
  GM_\odot}{r^2\over
  g\sqrt{\nu}}\Omega^2{\partial\hat\phi_2\over\partial\nu}
  \nonumber\\
  +{8\over 3}{R^2_\odot\over
  GM_\odot}{r^2\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Omega^2\hat\phi_2
  \nonumber\\
  -{128\pi\over 27}{R^5_\odot\over GM^2_\odot}{r^5\rho\over
  g\nu}\Omega{\partial\Omega\over\partial\nu}
  {\partial\hat\phi_2\over\partial\nu}
  \nonumber\\
  +{128\pi\over 27}{R^5_\odot\over GM^2_\odot}{r^5\rho\over
  g\nu}{\partial\ln g\over\partial\nu}\Omega\hat\phi_2
  {\partial\Omega\over\partial\nu}
  \nonumber\\
  +{16\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
  g\sqrt{\nu}}{\partial\ln g\over\partial\nu}\Big({\partial\ln
  l\over\partial\ln m}-1\Big)\hat\phi_2
  \nonumber\\
  -{16\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over
  g\sqrt{\nu}}\Big({\partial\ln l\over\partial\ln m}-1\Big)
  {\partial\hat\phi_2\over\partial\nu}
  \nonumber\\
  +{2\over 3}\nu{\partial{\cal A}_2\over\partial\nu}+{\partial\ln
  l\over\partial\ln m}{\cal A}_2
  \nonumber\\
  +\Big({M_\odot\over L_\odot}{\nu^{3/2}\over
  l}\epsilon_N(\epsilon_T-\delta)+{\partial\ln l\over\partial\ln
  m}\delta-{3\over 2\pi}{M_\odot\over R^4_\odot}{\nu^{3/2}\over\rho
  r^4}H_T\Big(1+{D_h\over K}\Big)-{M_\odot\over
  L_\odot}{\nu^{3/2}C_PT\over l}{1\over\Delta t}\Big)\Psi_2
  \nonumber\\
  +\Big({M_\odot\over L_\odot}{\nu^{3/2}\over
  l}\epsilon_N(\epsilon_\mu+\varphi)-{\partial\ln l\over\partial\ln
  m}\varphi-{M_\odot\over L_\odot}{\nu^{3/2}C_PT\over
  l}\Phi{d\ln\bar\mu\over dt}\Big)\Lambda_2
  \nonumber\\
  +{M_\odot\over L_\odot}{\nu^{3/2}C_PT\over l}{1\over\Delta
  t}\Psi_2(t)
  \nonumber\\
  -{M_\odot\over L_\odot}{\nu^{3/2}g\rho C_PT(\nabla_{ad}-\nabla)\over
  lP}U_2=0
\end{eqnarray}

With

\begin{equation}
  {8\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\sqrt{\mu}}H_T{\partial\Psi_2\over\partial\nu}-
  (1-\delta+\chi_T)\Psi_2-(\varphi+\chi_\mu)\Lambda_2-{\cal A}_2=0
\end{equation}

\item Eq. B.2.2 (B.6), Baroclinic relation:
\begin{equation}
  \Theta=\varphi\Lambda_2-\delta\Psi_2={R\over g}{2\over
  3}R\Omega{\partial\Omega\over\partial R}
\end{equation}

therefore

\begin{equation}
  {16\pi\over 9}{R^4_\odot\over M_\odot}{r^4\rho\over
  g\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}-
  \varphi\Lambda_2+\delta\Psi_2=0
\end{equation}

\item Eq. B.2.3 (B.7), Horizontal fluctuation of the molecular weight:

\begin{equation}
  {d\Lambda_2\over dt}-{d\ln\bar\mu\over dt}\Lambda_2={U_2\over
  H_P}\nabla_\mu-{6\over R^2}D_h\Lambda_2
\end{equation}

This gives

\begin{equation}
  {\nabla_\mu\over H_P}U_2+\Big({d\ln\bar\mu\over dt}-{1\over\Delta
  t}-6{1\over R^2_\odot}{1\over
  r^2}D_h\Big)\Lambda_2+{\Lambda_2(t)\over\Delta t}=0
\end{equation}

\item Eq. B.2.4 (B.8), Poisson equation (for $\ell=2$):

\begin{equation}
  {1\over R}{d^2(R\hat\phi_2)\over dR^2}-{\ell(\ell+1)\over
  R^2}\hat\phi_2-{4\pi G\over g}{d\rho\over dr}\hat\phi_2={4\pi G\over
  g}\Big(-\rho{2\over 3}R\Omega^2+{1\over 3}{dR^2\rho\Omega^2\over dR}\Big)
\end{equation}

This equation has five terms:

\begin{enumerate}

\item
\begin{eqnarray}
  {1\over R}{d^2(R\hat\phi_2)\over dR^2}={1\over R}{d\over
  dR}\Big(R{d\hat\phi_2\over dR}+\hat\phi_2\Big)={1\over R}
  \Big({d\hat\phi_2\over dR}+R{d^2\hat\phi_2\over
  dR^2}+{d\hat\phi_2\over dR}\Big)
  \nonumber\\
  ={2\over R}{d\hat\phi_2\over dR}+{d\over
  dR}{d\hat\phi_2\over dR}
  ={2\over R}{d\hat\phi_2\over dR}+{d\over dR}\Big({8\pi\over
  3}{1\over
  M_\odot}{R^2\rho\over\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}\Big)
  \nonumber\\
  ={2\over R}{d\hat\phi_2\over dR}+{8\pi\over 3}{1\over
  M_\odot}\Big({2R\rho\over\sqrt{\nu}}+{R^2\over\sqrt{\nu}}{d\rho\over
  dR}-{1\over 2}{R^2\rho\over\nu^{3/2}}{d\nu\over
  dR}+{R^2\rho\over\sqrt{\nu}}{d\over
  dR}\Big){\partial\hat\phi_2\over\partial\nu}
  \nonumber\\
  ={16\pi\over 3}{R_\odot\over
  M_\odot}{r\rho\over\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}+
  {16\pi\over 3}{R_\odot\over
  M_\odot}{r\rho\over\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}+
  {8\pi\over 3}{R^2_\odot\over M_\odot}{r^2\over\sqrt{\nu}}{8\pi\over
  3}{R^2_\odot\over M_\odot}{r^2\rho\over\sqrt{\nu}}
  {\partial\rho\over\partial\nu}{\partial\hat\phi_2\over\partial\nu}-
  \nonumber\\
  {4\pi\over 3}{R^2_\odot\over
  M_\odot}{r^2\rho\over\nu^{3/2}}{8\pi\over 3}{R^2\over
  M_\odot}{r^2\rho\over\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}
  +{8\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over\sqrt{\nu}}
  {8\pi\over 3}{R^2_\odot\over M_\odot}{r^2\rho\over\sqrt{\nu}}
  {\partial^2\hat\phi_2\over\partial\nu^2}
  \nonumber\\
  ={32\pi\over 3}{R_\odot\over
  M_\odot}{r\rho\over\sqrt{\nu}}{\partial\hat\phi_2\over\partial\nu}+
  {64\pi^2\over 9}{R^4_\odot\over
  M^2_\odot}{r^4\rho^2\over\nu}{\partial\ln\rho\over\partial\nu}
  {\partial\hat\phi_2\over\partial\nu}-
  \nonumber\\
  {32\pi^2\over 9}{R^4_\odot\over M^2_\odot}{r^4\rho^2\over\nu^2}
  {\partial\hat\phi_2\over\partial\nu}+
  {64\pi^2\over 9}{R^4_\odot\over M^2_\odot}{r^4\rho^2\over\nu}
  {\partial^2\hat\phi_2\over\partial\nu^2}
  \nonumber\\
  ={32\pi\over 3}{R_\odot\over M_\odot}{r\rho\over\sqrt{\nu}}
  \Big(1+{2\pi\over 3}{R^3_\odot\over
  M_\odot}{r^3\rho\over\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}-{\pi\over
  3}{R^3\over
  M_\odot}{r^3\rho\over\nu^{3/2}}\Big){\partial\hat\phi_2\over\partial\nu}+
  {64\pi^2\over 9}{R^4_\odot\over M^2_\odot}{r^4\rho^2\over\nu}
  {\partial^2\hat\phi_2\over\partial\nu^2}
\end{eqnarray}

\item
\begin{equation}
-{\ell(\ell+1)\over R^2}\hat\phi_2=-{\ell(\ell+1)\over
 R^2_\odot}{1\over r^2}\hat\phi_2
\end{equation}

\item
\begin{equation}
  -{4\pi G\over g}{d\rho\over dR}\hat\phi_2=-{32\pi^2G\over
   3}{R^2_\odot\over
   M_\odot}{r^2\rho^2\over g\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}
  \hat\phi_2
\end{equation}

\item
\begin{equation}
  -{4\pi G\rho\over g}{2\over 3}R\Omega^2=-{8\pi G\over
   3}R_\odot{r\rho\over g}\Omega^2
\end{equation}

\item
\begin{eqnarray}
  {4\pi G\over 3g}{dR^2(\rho\Omega^2)\over dR}={4\pi G\over 3g}
  \Big(2R\rho\Omega^2+R^2\Omega^2{d\rho\over
  dR}+2R^2\rho\Omega{d\Omega\over dR}\Big)
  \nonumber\\
  ={8\pi G\over 3}R_\odot{r\rho\over g}\Omega^2+
  {32\pi^2G\over 9}{R^4_\odot\over M_\odot}{r^4\rho^2\over
  g\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}\Omega^2+
  {64\pi^2G\over 9}{R^4_\odot\over M_\odot}{r^4\rho^2\over
  g\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}
\end{eqnarray}
\end{enumerate}

With this, the equation finally is

\begin{eqnarray}
  {64\pi^2\over 9}{R^4_\odot\over M^2_\odot}{r^2\rho^2\over\nu}
  {\partial^2\hat\phi_2\over\partial\nu^2}
  \nonumber\\
  +{32\pi\over 3}{R_\odot\over M_\odot}{r\rho\over\sqrt{\nu}}
  \Big(1+{2\pi\over 3}{R^3_\odot\over
  M_\odot}{r^3\rho\over\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}-{\pi\over
  3}{R^3\over
  M_\odot}{r^3\rho\over\nu^{3/2}}\Big){\partial\hat\phi_2\over\partial\nu}
  \nonumber\\
  -\Big({\ell(\ell+1)\over R^2_\odot}{1\over r^2}+
  {32\pi^2G\over 3}{R^2_\odot\over
   M_\odot}{r^2\rho^2\over\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}
  \hat\phi_2\Big)
  \nonumber\\
  -{64\pi^2G\over 9}{R^4_\odot\over M_\odot}{r^4\rho^2\over
  g\sqrt{\nu}}\Omega{\partial\Omega\over\partial\nu}
  \nonumber\\
  -{32\pi^2G\over 9}{R^4_\odot\over M_\odot}{r^4\rho^2\over
  g\sqrt{\nu}}{\partial\ln\rho\over\partial\nu}\Omega^2=0
\end{eqnarray}
\end{enumerate}
\end{document}

