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\documentstyle[11pt]{article}
\topmargin=-.5cm
\textheight=22cm
\oddsidemargin=.0cm
\textwidth=15.8cm
\baselineskip=18pt
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\<{\noindent }
\begin{document}
\title{\bf Users Manual for RNS: version 2.0}
\author{Sharon Morsink}
\date{\today}
\maketitle
\section{Introduction}
RNS version 2.0 is a set of routines which will integrate the
Einstein field equations for a rapidly rotating neutron star
given a perfect fluid equation of state. This version of the code
will be most useful for people who need to use the metric in another
application. This manual will explain how to use the routines.
\section{Metric and Coordinates}
The neutron star models which we compute are assumed to be stationary,
axisymmetric, uniformly rotating perfect fluid solutions of the Einstein
field equations. The assumptions of stationarity and axisymmetry allow the
introduction of two coordinates $\phi$ and $t$ on which the
space-time metric doesn't depend.
The metric, $g_{\alpha \beta}$ can be written
as
\begin{equation}
ds^2 = - e^{\gamma + \rho} dt^2
+ e^{\gamma - \rho} \bar{r}^2 \sin ^2 \theta \left(
d\phi - \omega dt \right)^2
+ e^{2 \alpha} \left( d\bar{r}^2 + \bar{r}^2 d\theta^2\right),
\label{metric}
\end{equation}
where the metric potentials $\rho, \gamma, \alpha$ and $\omega$ depend only
on the coordinates $\bar{r}$ and $\theta$.
The function $\frac{1}{2}(\gamma + \rho)$
is the relativistic
generalization of the Newtonian gravitational potential; the time dilation
factor between an observer moving with angular velocity $\omega$ and an
observer at infinity is $e^{\frac{1}{2}(\gamma + \rho)}$.
The coordinate $\bar{r}$ is not the same as the Schwarzschild
coordinate $r$. In the limit of spherical symmetry,
$\bar{r}$ corresponds to the
isotropic Schwarzschild coordinate. Circles centred about the axis of
symmetry have circumference $2 \pi r$ where $r$ is related to our
coordinates $\bar{r},\theta$ by
\be
r = e^{\frac{1}{2}(\gamma - \rho)} \bar{r} \sin \theta .
\label{radius}
\ee
The metric potential $\omega$ is the angular velocity about the symmetry
axis of
zero angular momentum observers
(ZAMOs) and is responsible for the Lense-Thirring effect.
The fourth metric potential, $\alpha$ specifies the geometry of the
two-surfaces of constant $t$ and $\phi$. When the star is non-rotating, the
exterior geometry is that of the isotropic Schwarzschild metric, with
\be
e^{\frac{1}{2}(\gamma + \rho)} = \frac{1-M/2\bar{r}}{1+M/2\bar{r}}, \;
e^{\frac{1}{2}(\gamma - \rho)} = e^\alpha = \left( 1 + M/2\bar{r} \right)^2,
\; \omega=0.
\label{isotropic}
\ee
The program uses a compactified coordinate $s$ which is related to
$\bar{r}$ by
\be
\bar{r} = \bar{r}_e \left( \frac{s}{1-s} \right),
\ee
where $\bar{r}_e$ is the value of $\bar{r}$ at the star's equator.
This definition of $s$ gives
\begin{eqnarray}
s = 0.5 && \Leftrightarrow \bar{r} = \bar{r}_e \\
s = 1.0 && \Leftrightarrow \bar{r} \rightarrow \infty
\end{eqnarray}
The angular variable $\mu$, defined by
\[
\mu = \cos \theta
\]
is used by the program.
\section{Creating the Numerical Grid}
The user specifies the numerical grid size in the makefile.
The parameter {\tt MDIV} ({\em divisions of $\mu$})
specifies the number of spokes in the
angular direction, while {\tt SDIV} ({\em divisions of $s$})
specifies the number of spokes in the radial direction.
For example the following line in the makefile:
\vspace{2mm}
\begin{verbatim}
#STANDARD
SIZE=-DMDIV=65 -DSDIV=129
\end{verbatim}
\vspace{2mm}
sets the number of angular spokes to 65 and the number of
radial spokes to 129.
The function make$\underline{\;\;}$grid is called at the
beginning of the application in order to set up the numerical
grid.
\subsection{make$\underline{\;\;}$grid}
\begin{verbatim}
/*******************************************************************/
/* Create computational grid. */
/* Points in the mu-direction are stored in the array mu[i]. */
/* Points in the s-direction are stored in the array s_gp[j]. */
/*******************************************************************/
void make_grid(double s_gp[SDIV+1],
double mu[MDIV+1])
{
int m, s; /* counters */
for(s=1;s<=SDIV;s++)
s_gp[s] = SMAX*(s-1.0)/(SDIV-1.0);
/* s_gp[1] = 0.0 corresponds to the center of the star
s_gp[SDIV] = SMAX corresponds to infinity */
/* SMAX is defined in the file consts.h */
for(m=1;m<=MDIV;m++)
mu[m] = (m-1.0)/(MDIV-1.0);
/* mu[1] = 0.0 corresponds to the plane of the equator
mu[MDIV] = 1.0 corresponds to the axis of symmetry */
/* s_gp[0] and mu[0] are not used by the program */
}
\end{verbatim}
\section{Loading the Equation of State}
RNS needs a tabulated, zero-temperature equation of state (EOS), as an input,
in order to run.
You will have to format the EOS file in a specific way, as described here:
The first line in the EOS file should contain the number of tabulated points.
The remaining lines should consist of four columns - energy
density (in ${\rm gr/cm^3}$), pressure (in ${\rm dynes/cm^2}$), enthalpy (in
${\rm cm^2/s^2}$), and baryon number density (in ${\rm cm^{-3}}$). The
{\it enthalpy} is defined as
\be H(P)=\int_0^P { c^2 dP \over (\epsilon +P)}, \ee
\<where $\epsilon$ is the energy density, $P$ is the pressure and $c$ is the
speed of light.
The number of points should be limited to 200. Example files
(e.g. {\tt eosC}) are supplied with the source code.
The function load$\underline{\;\;}$eos is called at the beginning
of the application to load the specified equation of state file.
\subsection{load$\underline{\;\;}$eos}
\begin{verbatim}
/*************************************************************************/
/* Load EOS file. */
/*************************************************************************/
void load_eos( char eos_file[],
double log_e_tab[201],
double log_p_tab[201],
double log_h_tab[201],
double log_n0_tab[201],
int *n_tab)
{
int i; /* counter */
double p, /* pressure */
rho, /* density */
h, /* enthalpy */
n0; /* number density */
FILE *f_eos; /* pointer to eos_file */
/* OPEN FILE TO READ */
if((f_eos=fopen(eos_file,"r")) == NULL ) {
printf("cannot open file: %s\n",eos_file);
exit(0);
}
/* READ NUMBER OF TABULATED POINTS */
fscanf(f_eos,"%d\n",n_tab);
/* READ EOS, H, N0 AND MAKE THEM DIMENSIONLESS */
for(i=1;i<=(*n_tab);i++) {
fscanf(f_eos,"%lf %lf %lf %lf\n",&rho,&p,&h,&n0) ;
log_e_tab[i]=log10(rho*C*C*KSCALE); /* multiply by C^2 to get */
log_p_tab[i]=log10(p*KSCALE); /* energy density. */
log_h_tab[i]=log10(h/(C*C));
log_n0_tab[i]=log10(n0);
}
}
\end{verbatim}
\section{Setting Program Defaults}
There are some parameters used in the program whose default values
need to be set. I will probably build their values into the routines
at some unspecified later date. Until then you must include the
following lines of code in your application:
\begin{verbatim}
double
cf, /* convergence factor */
accuracy;
double
e_surface, /* value of energy density at star's surface */
p_surface, /* value of pressure at star's surface */
enthalpy_min; /* minimum value of enthalpy */
/* set program defaults */
cf=1.0;
accuracy=1e-5;
if(strcmp(eos_type,"tab")==0) {
e_surface=7.8*C*C*KSCALE;
p_surface=1.01e8*KSCALE;
enthalpy_min=1.0/(C*C);
}
else{
e_surface=0.0;
p_surface=0.0;
enthalpy_min=0.0;
}
\end{verbatim}
\section{Compute Pressure and Enthalpy at Star's Center}
The user specifies the value of the star's energy density
at the star's center. The application must then compute the
values of pressure and enthalpy at the center of the star.
This is done by calling the function make$\underline{\;\;}$center().
\subsection{make$\underline{\;\;}$center()}
\begin{verbatim}
void make_center(
char eos_file[],
double log_e_tab[201],
double log_p_tab[201],
double log_h_tab[201],
double log_n0_tab[201],
int n_tab,
char eos_type[],
double Gamma_P,
double e_center,
double *p_center,
double *h_center)
{
int n_nearest;
double rho0_center;
n_nearest=n_tab/2;
if(strcmp(eos_type,"tab")==0) {
/* If the EOS is tabulated, interpolate to find
p_center and h_center given the value of e_center */
(*p_center) = p_at_e( e_center, log_p_tab, log_e_tab, n_tab, &n_nearest);
(*h_center) = h_at_p( (*p_center), log_h_tab, log_p_tab, n_tab, &n_nearest);
}
else {
/* If the EOS is polytropic, use standard formulae to compute
p_center and h_center */
rho0_center = rtsec_G( e_of_rho0, Gamma_P, 0.0,e_center,DBL_EPSILON,
e_center );
(*p_center) = pow(rho0_center,Gamma_P);
(*h_center) = log((e_center+(*p_center))/rho0_center);
}
}
\end{verbatim}
\section{Allocate Memory}
Memory for the metric functions, energy density, pressure, etc. must
be allocated at the beginning of the application. Each of these
functions is two-dimensional and is represented by a
{\tt SDIV} $\times$ {\tt MDIV} matrix. The metric functions
are represented by the following matrices:
\begin{eqnarray}
\rho &\Leftrightarrow& \hbox{rho} \\
\gamma &\Leftrightarrow& \hbox{gama} \quad \hbox{Note the spelling!} \\
\omega &\Leftrightarrow& \hbox{omega} \quad \hbox{Note the lower case o}\\
\alpha &\Leftrightarrow& \hbox{alpha}
\end{eqnarray}
Declare and allocate memory for these variables by including the
following lines of code in your application:
\begin{verbatim}
double
**rho,
**gama,
**omega,
**alpha,
**energy, /* The star's energy-density */
**pressure, /* The star's pressure */
**enthalpy, /* The star's enthalpy */
**velocity_sq, /* The square of the fluid's velocity */
*v_plus, /* vel. of co-rot. particle wrt ZAMO */
*v_minus; /* vel. of counter-rot. ... */
/* ALLLOCATE MEMORY (Numerical Recipes routines)*/
rho = dmatrix(1,SDIV,1,MDIV);
gama = dmatrix(1,SDIV,1,MDIV);
alpha = dmatrix(1,SDIV,1,MDIV);
omega = dmatrix(1,SDIV,1,MDIV);
energy = dmatrix(1,SDIV,1,MDIV);
pressure = dmatrix(1,SDIV,1,MDIV);
enthalpy = dmatrix(1,SDIV,1,MDIV);
velocity_sq = dmatrix(1,SDIV,1,MDIV);
v_plus = dvector(1,SDIV);
v_minus = dvector(1,SDIV);
\end{verbatim}
\section{Compute a Spherical Star}
Before computing a rotating neutron star, you will need to compute
the form of a spherical neutron star. The spherical star will be
used as a first approximation for the rotating neutron star.
The function sphere() is called by the application to compute the
metric of a spherical star. The function sphere is given information
about the equation of state and the values of energy density,
pressure and enthalpy at the center and surface of the star. The
function computes the coordinate radius of the star $r_e$
as well as the metric functions $\rho, \gamma, \omega$ and $\alpha$.
\subsection{sphere()}
\begin{verbatim}
void sphere(double s_gp[SDIV+1],
double log_e_tab[201],
double log_p_tab[201],
double log_h_tab[201],
double log_n0_tab[201],
int n_tab,
char eos_type[],
double Gamma_P,
double e_center,
double p_center,
double h_center,
double p_surface,
double e_surface,
double **rho,
double **gama,
double **alpha,
double **omega,
double *r_e)
{
int s,
m,
n_nearest;
double r_is_s,
r_is_final,
r_final,
m_final,
lambda_s,
nu_s,
r_is_gp[RDIV+1],
lambda_gp[RDIV+1],
nu_gp[RDIV+1],
gama_mu_0[SDIV+1],
rho_mu_0[SDIV+1],
gama_eq,
rho_eq,
s_e=0.5;
/* The function TOV integrates the TOV equations. The function
can be found in the file equil.c */
TOV(1, eos_type, e_center, p_center, p_surface, e_surface, Gamma_P,
log_e_tab, log_p_tab, log_h_tab, n_tab, r_is_gp, lambda_gp,
nu_gp, &r_is_final, &r_final, &m_final);
TOV(2, eos_type, e_center, p_center, p_surface, e_surface, Gamma_P,
log_e_tab, log_p_tab, log_h_tab, n_tab, r_is_gp, lambda_gp,
nu_gp, &r_is_final, &r_final, &m_final);
TOV(3, eos_type, e_center, p_center, p_surface, e_surface, Gamma_P,
log_e_tab, log_p_tab, log_h_tab, n_tab, r_is_gp, lambda_gp,
nu_gp, &r_is_final, &r_final, &m_final);
n_nearest=RDIV/2;
for(s=1;s<=SDIV;s++) {
r_is_s=r_is_final*(s_gp[s]/(1.0-s_gp[s]));
if(r_is_s<r_is_final) {
lambda_s=interp(r_is_gp,lambda_gp,RDIV,r_is_s,&n_nearest);
nu_s=interp(r_is_gp,nu_gp,RDIV,r_is_s,&n_nearest);
}
else {
lambda_s=2.0*log(1.0+m_final/(2.0*r_is_s));
nu_s=log((1.0-m_final/(2.0*r_is_s))/(1.0+m_final/(2*r_is_s)));
}
gama[s][1]=nu_s+lambda_s;
rho[s][1]=nu_s-lambda_s;
for(m=1;m<=MDIV;m++) {
gama[s][m]=gama[s][1];
rho[s][m]=rho[s][1];
alpha[s][m]=(gama[s][1]-rho[s][1])/2.0;
omega[s][m]=0.0;
}
gama_mu_0[s]=gama[s][1]; /* gama at \mu=0 */
rho_mu_0[s]=rho[s][1]; /* rho at \mu=0 */
}
n_nearest=SDIV/2;
gama_eq = interp(s_gp,gama_mu_0,SDIV,s_e,&n_nearest); /* gama at equator */
rho_eq = interp(s_gp,rho_mu_0,SDIV,s_e,&n_nearest); /* rho at equator */
(*r_e)= r_final*exp(0.5*(rho_eq-gama_eq));
}
\end{verbatim}
\section{Integrate a Rotating Star}
Once the spherical star's metric has been computed the application
is ready to integrate the Einstein field equations for a rotating
neutron star. Before doing so, the application must choose
a value of the ratio of the star's polar radius to the equatorial radius.
This ratio is denoted
\begin{verbatim}
r_ratio
\end{verbatim}
Typically, one has to try many values of the ratio before the
desired value of angular velocity, mass, etc is found. Once the value
of the ratio has been set, call the function spin().
\subsection{spin()}
\begin{verbatim}
/*************************************************************************/
/* Main iteration cycle for computation of the rotating star's metric */
/*************************************************************************/
void spin(double s_gp[SDIV+1],
double mu[MDIV+1],
double log_e_tab[201],
double log_p_tab[201],
double log_h_tab[201],
double log_n0_tab[201],
int n_tab,
char eos_type[],
double Gamma_P,
double h_center,
double enthalpy_min,
double **rho,
double **gama,
double **alpha,
double **omega,
double **energy,
double **pressure,
double **enthalpy,
double **velocity_sq,
int a_check,
double accuracy,
double cf,
double r_ratio,
double *r_e_new,
double *Omega)
{
int m, /* counter */
s, /* counter */
n, /* counter */
k, /* counter */
n_of_it=0, /* number of iterations */
n_nearest,
print_dif = 0,
i,
j;
double **D2_rho,
**D2_gama,
**D2_omega;
float ***f_rho,
***f_gama;
double sum_rho=0.0, /* intermediate sum in eqn for rho */
sum_gama=0.0, /* intermediate sum in eqn for gama */
sum_omega=0.0, /* intermediate sum in eqn for omega */
r_e_old, /* equatorial radius in previus cycle */
dif=1.0, /* difference | r_e_old - r_e | */
d_gama_s, /* derivative of gama w.r.t. s */
d_gama_m, /* derivative of gama w.r.t. m */
d_rho_s, /* derivative of rho w.r.t. s */
d_rho_m, /* derivative of rho w.r.t. m */
d_omega_s, /* derivative of omega w.r.t. s */
d_omega_m, /* derivative of omega w.r.t. m */
d_gama_ss, /* 2nd derivative of gama w.r.t. s */
d_gama_mm, /* 2nd derivative of gama w.r.t. m */
d_gama_sm, /* derivative of gama w.r.t. m and s */
temp1, /* temporary term in da_dm */
temp2,
temp3,
temp4,
temp5,
temp6,
temp7,
temp8,
m1,
s1,
s2,
ea,
rsm,
gsm,
omsm,
esm,
psm,
v2sm,
mum,
sgp,
s_1,
e_gsm,
e_rsm,
rho0sm,
term_in_Omega_h,
r_p,
s_p,
gama_pole_h, /* gama^hat at pole */
gama_center_h, /* gama^hat at center */
gama_equator_h, /* gama^hat at equator */
rho_pole_h, /* rho^hat at pole */
rho_center_h, /* rho^hat at center */
rho_equator_h, /* rho^hat at equator */
omega_equator_h, /* omega^hat at equator */
gama_mu_1[SDIV+1], /* gama at \mu=1 */
gama_mu_0[SDIV+1], /* gama at \mu=0 */
rho_mu_1[SDIV+1], /* rho at \mu=1 */
rho_mu_0[SDIV+1], /* rho at \mu=0 */
omega_mu_0[SDIV+1], /* omega at \mu=0 */
s_e=0.5,
**da_dm,
**dgds,
**dgdm,
**D1_rho,
**D1_gama,
**D1_omega,
**S_gama,
**S_rho,
**S_omega,
**f2n,
**P_2n,
**P1_2n_1,
Omega_h,
sin_theta[MDIV+1],
theta[MDIV+1],
sk,
sj,
sk1,
sj1,
r_e;
f2n = dmatrix(1,LMAX+1,1,SDIV);
f_rho = f3tensor(1,SDIV,1,LMAX+1,1,SDIV);
f_gama = f3tensor(1,SDIV,1,LMAX+1,1,SDIV);
P_2n = dmatrix(1,MDIV,1,LMAX+1);
P1_2n_1 = dmatrix(1,MDIV,1,LMAX+1);
for(n=0;n<=LMAX;n++)
for(i=2;i<=SDIV;i++) f2n[n+1][i] = pow((1.0-s_gp[i])/s_gp[i],2.0*n);
if(SMAX!=1.0) {
for(j=2;j<=SDIV;j++)
for(n=1;n<=LMAX;n++)
for(k=2;k<=SDIV;k++) {
sk=s_gp[k];
sj=s_gp[j];
sk1=1.0-sk;
sj1=1.0-sj;
if(k<j) {
f_rho[j][n+1][k] = f2n[n+1][j]*sj1/(sj*
f2n[n+1][k]*sk1*sk1);
f_gama[j][n+1][k] = f2n[n+1][j]/(f2n[n+1][k]*sk*sk1);
}else {
f_rho[j][n+1][k] = f2n[n+1][k]/(f2n[n+1][j]*sk*sk1);
f_gama[j][n+1][k] = f2n[n+1][k]*sj1*sj1*sk/(sj*sj
*f2n[n+1][j]*sk1*sk1*sk1);
}
}
j=1;
n=0;
for(k=2;k<=SDIV;k++) {
sk=s_gp[k];
f_rho[j][n+1][k]=1.0/(sk*(1.0-sk));
}
n=1;
for(k=2;k<=SDIV;k++) {
sk=s_gp[k];
sk1=1.0-sk;
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=1.0/(sk*sk1);
}
for(n=2;n<=LMAX;n++)
for(k=1;k<=SDIV;k++) {
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=0.0;
}
k=1;
n=0;
for(j=1;j<=SDIV;j++)
f_rho[j][n+1][k]=0.0;
for(j=1;j<=SDIV;j++)
for(n=1;n<=LMAX;n++) {
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=0.0;
}
n=0;
for(j=2;j<=SDIV;j++)
for(k=2;k<=SDIV;k++) {
sk=s_gp[k];
sj=s_gp[j];
sk1=1.0-sk;
sj1=1.0-sj;
if(k<j)
f_rho[j][n+1][k] = sj1/(sj*sk1*sk1);
else
f_rho[j][n+1][k] = 1.0/(sk*sk1);
}
}
else{
for(j=2;j<=SDIV-1;j++)
for(n=1;n<=LMAX;n++)
for(k=2;k<=SDIV-1;k++) {
sk=s_gp[k];
sj=s_gp[j];
sk1=1.0-sk;
sj1=1.0-sj;
if(k<j) {
f_rho[j][n+1][k] = f2n[n+1][j]*sj1/(sj*
f2n[n+1][k]*sk1*sk1);
f_gama[j][n+1][k] = f2n[n+1][j]/(f2n[n+1][k]*sk*sk1);
}else {
f_rho[j][n+1][k] = f2n[n+1][k]/(f2n[n+1][j]*sk*sk1);
f_gama[j][n+1][k] = f2n[n+1][k]*sj1*sj1*sk/(sj*sj
*f2n[n+1][j]*sk1*sk1*sk1);
}
}
j=1;
n=0;
for(k=2;k<=SDIV-1;k++) {
sk=s_gp[k];
f_rho[j][n+1][k]=1.0/(sk*(1.0-sk));
}
n=1;
for(k=2;k<=SDIV-1;k++) {
sk=s_gp[k];
sk1=1.0-sk;
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=1.0/(sk*sk1);
}
for(n=2;n<=LMAX;n++)
for(k=1;k<=SDIV-1;k++) {
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=0.0;
}
k=1;
n=0;
for(j=1;j<=SDIV-1;j++)
f_rho[j][n+1][k]=0.0;
for(j=1;j<=SDIV-1;j++)
for(n=1;n<=LMAX;n++) {
f_rho[j][n+1][k]=0.0;
f_gama[j][n+1][k]=0.0;
}
n=0;
for(j=2;j<=SDIV-1;j++)
for(k=2;k<=SDIV-1;k++) {
sk=s_gp[k];
sj=s_gp[j];
sk1=1.0-sk;
sj1=1.0-sj;
if(k<j)
f_rho[j][n+1][k] = sj1/(sj*sk1*sk1);
else
f_rho[j][n+1][k] = 1.0/(sk*sk1);
}
j=SDIV;
for(n=1;n<=LMAX;n++)
for(k=1;k<=SDIV;k++) {
f_rho[j][n+1][k] = 0.0;
f_gama[j][n+1][k] = 0.0;
}
k=SDIV;
for(j=1;j<=SDIV;j++)
for(n=1;n<=LMAX;n++) {
f_rho[j][n+1][k] = 0.0;
f_gama[j][n+1][k] = 0.0;
}
}
n=0;
for(i=1;i<=MDIV;i++)
P_2n[i][n+1]=legendre(2*n,mu[i]);
for(i=1;i<=MDIV;i++)
for(n=1;n<=LMAX;n++) {
P_2n[i][n+1]=legendre(2*n,mu[i]);
P1_2n_1[i][n+1] = plgndr(2*n-1 ,1,mu[i]);
}
free_dmatrix(f2n,1,LMAX+1,1,SDIV);
for(m=1;m<=MDIV;m++) {
sin_theta[m] = sqrt(1.0-mu[m]*mu[m]);
theta[m] = asin(sin_theta[m]);
}
r_e = (*r_e_new);
while(dif> accuracy || n_of_it<2) {
if(print_dif!=0)
printf("%4.3e\n",dif);
/* Rescale potentials and construct arrays with the potentials along
| the equatorial and polar directions.
*/
for(s=1;s<=SDIV;s++) {
for(m=1;m<=MDIV;m++) {
rho[s][m] /= SQ(r_e);
gama[s][m] /= SQ(r_e);
alpha[s][m] /= SQ(r_e);
omega[s][m] *= r_e;
}
rho_mu_0[s]=rho[s][1];
gama_mu_0[s]=gama[s][1];
omega_mu_0[s]=omega[s][1];
rho_mu_1[s]=rho[s][MDIV];
gama_mu_1[s]=gama[s][MDIV];
}
/* Compute new r_e. */
r_e_old=r_e;
r_p=r_ratio*r_e;
s_p=r_p/(r_p+r_e);
n_nearest= SDIV/2;
gama_pole_h=interp(s_gp,gama_mu_1,SDIV,s_p,&n_nearest);
gama_equator_h=interp(s_gp,gama_mu_0,SDIV,s_e,&n_nearest);
gama_center_h=gama[1][1];
rho_pole_h=interp(s_gp,rho_mu_1,SDIV,s_p,&n_nearest);
rho_equator_h=interp(s_gp,rho_mu_0,SDIV,s_e,&n_nearest);
rho_center_h=rho[1][1];
r_e=sqrt(2*h_center/(gama_pole_h+rho_pole_h-gama_center_h-rho_center_h));
/* Compute angular velocity Omega. */
if(r_ratio==1.0) {
Omega_h=0.0;
omega_equator_h=0.0;
}
else {
omega_equator_h=interp(s_gp,omega_mu_0,SDIV,s_e, &n_nearest);
term_in_Omega_h=1.0-exp(SQ(r_e)*(gama_pole_h+rho_pole_h
-gama_equator_h-rho_equator_h));
if(term_in_Omega_h>=0.0)
Omega_h = omega_equator_h + exp(SQ(r_e)*rho_equator_h)
*sqrt(term_in_Omega_h);
else {
Omega_h=0.0;
}
}
/* Compute velocity, energy density and pressure. */
n_nearest=n_tab/2;
for(s=1;s<=SDIV;s++) {
sgp=s_gp[s];
for(m=1;m<=MDIV;m++) {
rsm=rho[s][m];
if(r_ratio==1.0)
velocity_sq[s][m]=0.0;
else
velocity_sq[s][m]=SQ((Omega_h-omega[s][m])*(sgp/(1.0-sgp))
*sin_theta[m]*exp(-rsm*SQ(r_e)));
if(velocity_sq[s][m]>=1.0)
velocity_sq[s][m]=0.0;
enthalpy[s][m]=enthalpy_min + 0.5*(SQ(r_e)*(gama_pole_h+rho_pole_h
-gama[s][m]-rsm)-log(1.0-velocity_sq[s][m]));
if((enthalpy[s][m]<=enthalpy_min) || (sgp>s_e)) {
pressure[s][m]=0.0;
energy[s][m]=0.0;
}
else { if(strcmp(eos_type,"tab")==0) {
pressure[s][m]=p_at_h(enthalpy[s][m], log_p_tab,
log_h_tab, n_tab, &n_nearest);
energy[s][m]=e_at_p(pressure[s][m], log_e_tab,
log_p_tab, n_tab, &n_nearest, eos_type,
Gamma_P);
}
else {
rho0sm=pow(((Gamma_P-1.0)/Gamma_P)
*(exp(enthalpy[s][m])-1.0),1.0/(Gamma_P-1.0));
pressure[s][m]=pow(rho0sm,Gamma_P);
energy[s][m]=pressure[s][m]/(Gamma_P-1.0)+rho0sm;
}
}
/* Rescale back metric potentials (except omega) */
rho[s][m] *= SQ(r_e);
gama[s][m] *= SQ(r_e);
alpha[s][m] *= SQ(r_e);
}
}
/* Compute metric potentials */
S_gama = dmatrix(1,SDIV,1,MDIV);
S_rho = dmatrix(1,SDIV,1,MDIV);
S_omega = dmatrix(1,SDIV,1,MDIV);
for(s=1;s<=SDIV;s++)
for(m=1;m<=MDIV;m++) {
rsm=rho[s][m];
gsm=gama[s][m];
omsm=omega[s][m];
esm=energy[s][m];
psm=pressure[s][m];
e_gsm=exp(0.5*gsm);
e_rsm=exp(-rsm);
v2sm=velocity_sq[s][m];
mum=mu[m];
m1=1.0-SQ(mum);
sgp=s_gp[s];
s_1=1.0-sgp;
s1=sgp*s_1;
s2=SQ(sgp/s_1);
ea=16.0*PI*exp(2.0*alpha[s][m])*SQ(r_e);
if(s==1) {
d_gama_s=0.0;
d_gama_m=0.0;
d_rho_s=0.0;
d_rho_m=0.0;
d_omega_s=0.0;
d_omega_m=0.0;
}else{
d_gama_s=deriv_s(gama,s,m);
d_gama_m=deriv_m(gama,s,m);
d_rho_s=deriv_s(rho,s,m);
d_rho_m=deriv_m(rho,s,m);
d_omega_s=deriv_s(omega,s,m);
d_omega_m=deriv_m(omega,s,m);
}
S_rho[s][m] = e_gsm*(0.5*ea*(esm + psm)*s2*(1.0+v2sm)/(1.0-v2sm)
+ s2*m1*SQ(e_rsm)*(SQ(s1*d_omega_s)
+ m1*SQ(d_omega_m))
+ s1*d_gama_s - mum*d_gama_m + 0.5*rsm*(ea*psm*s2
- s1*d_gama_s*(0.5*s1*d_gama_s+1.0)
- d_gama_m*(0.5*m1*d_gama_m-mum)));
S_gama[s][m] = e_gsm*(ea*psm*s2 + 0.5*gsm*(ea*psm*s2 - 0.5*SQ(s1
*d_gama_s) - 0.5*m1*SQ(d_gama_m)));
S_omega[s][m]=e_gsm*e_rsm*( -ea*(Omega_h-omsm)*(esm+psm)
*s2/(1.0-v2sm) + omsm*( -0.5*ea*(((1.0+v2sm)*esm
+ 2.0*v2sm*psm)/(1.0-v2sm))*s2
- s1*(2*d_rho_s+0.5*d_gama_s)
+ mum*(2*d_rho_m+0.5*d_gama_m) + 0.25*SQ(s1)*(4
*SQ(d_rho_s)-SQ(d_gama_s)) + 0.25*m1*(4*SQ(d_rho_m)
- SQ(d_gama_m)) - m1*SQ(e_rsm)*(SQ(SQ(sgp)*d_omega_s)
+ s2*m1*SQ(d_omega_m))));
}
/* ANGULAR INTEGRATION */
D1_rho = dmatrix(1,LMAX+1,1,SDIV);
D1_gama = dmatrix(1,LMAX+1,1,SDIV);
D1_omega = dmatrix(1,LMAX+1,1,SDIV);
n=0;