jnu_mixed.c

/***********************************************************************************
    Copyright 2013 Joshua C. Dolence, Charles F. Gammie, Monika Mo\'scibrodzka,
                   and Po Kin Leung

                        GRMONTY  version 1.0   (released February 1, 2013)

    This file is part of GRMONTY.  GRMONTY v1.0 is a program that calculates the
    emergent spectrum from a model using a Monte Carlo technique.

    This version of GRMONTY is configured to use input files from the HARM code
    available on the same site.   It assumes that the source is a plasma near a
    black hole described by Kerr-Schild coordinates that radiates via thermal 
    synchrotron and inverse compton scattering.
    
    You are morally obligated to cite the following paper in any
    scientific literature that results from use of any part of GRMONTY:

    Dolence, J.C., Gammie, C.F., Mo\'scibrodzka, M., \& Leung, P.-K. 2009,
        Astrophysical Journal Supplement, 184, 387

    Further, we strongly encourage you to obtain the latest version of 
    GRMONTY directly from our distribution website:
    http://rainman.astro.illinois.edu/codelib/

    GRMONTY is free software; you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation; either version 2 of the License, or
    (at your option) any later version.

    GRMONTY is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with GRMONTY; if not, write to the Free Software
    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA

***********************************************************************************/


#include "decs.h"
#pragma omp threadprivate(r)
/* 

"mixed" emissivity formula 

interpolates between Petrosian limit and
classical thermal synchrotron limit

good for Thetae > 1

*/

#define CST 1.88774862536	/* 2^{11/12} */
double jnu_synch(double nu, double Ne, double Thetae, double B,
		 double theta)
{
	double K2, nuc, nus, x, f, j, sth, xp1, xx;
	double K2_eval(double Thetae);

	if (Thetae < THETAE_MIN)
		return 0.;

	K2 = K2_eval(Thetae);

	nuc = EE * B / (2. * M_PI * ME * CL);
	sth = sin(theta);
	nus = (2. / 9.) * nuc * Thetae * Thetae * sth;
	if (nu > 1.e12 * nus)
		return (0.);
	x = nu / nus;
	xp1 = pow(x, 1. / 3.);
	xx = sqrt(x) + CST * sqrt(xp1);
	f = xx * xx;
	j = (M_SQRT2 * M_PI * EE * EE * Ne * nus / (3. * CL * K2)) * f *
	    exp(-xp1);

	return (j);
}

#undef CST

#define JCST	(M_SQRT2*EE*EE*EE/(27*ME*CL*CL))
double int_jnu(double Ne, double Thetae, double Bmag, double nu)
{
/* Returns energy per unit time at							*
 * frequency nu in cgs										*/

	double j_fac, K2;
	double F_eval(double Thetae, double B, double nu);
	double K2_eval(double Thetae);


	if (Thetae < THETAE_MIN)
		return 0.;

	K2 = K2_eval(Thetae);
	if (K2 == 0.)
		return 0.;

	j_fac = Ne * Bmag * Thetae * Thetae / K2;

	return JCST * j_fac * F_eval(Thetae, Bmag, nu);
}

#undef JCST

#define CST 1.88774862536	/* 2^{11/12} */
double jnu_integrand(double th, void *params)
{

	double K = *(double *) params;
	double sth = sin(th);
	double x = K / sth;

	if (sth < 1.e-150 || x > 2.e8)
		return 0.;

	return sth * sth * pow(sqrt(x) + CST * pow(x, 1. / 6.),
			       2.) * exp(-pow(x, 1. / 3.));
}

#undef CST

/* Tables */
double F[N_ESAMP + 1], K2[N_ESAMP + 1];
double lK_min, dlK;
double lT_min, dlT;

#define EPSABS 0.
#define EPSREL 1.e-6
#define KMIN (0.002)
#define KMAX (1.e7)
#define TMIN (THETAE_MIN)
#define TMAX (1.e2)
void init_emiss_tables(void)
{

	int k;
	double result, err, K, T;
	gsl_function func;
	gsl_integration_workspace *w;

	func.function = &jnu_integrand;
	func.params = &K;

	lK_min = log(KMIN);
	dlK = log(KMAX / KMIN) / (N_ESAMP);

	lT_min = log(TMIN);
	dlT = log(TMAX / TMIN) / (N_ESAMP);

	/*  build table for F(K) where F(K) is given by
	   \int_0^\pi ( (K/\sin\theta)^{1/2} + 2^{11/12}(K/\sin\theta)^{1/6})^2 \exp[-(K/\sin\theta)^{1/3}]
	   so that J_{\nu} = const.*F(K)
	 */
	w = gsl_integration_workspace_alloc(1000);
	for (k = 0; k <= N_ESAMP; k++) {
		K = exp(k * dlK + lK_min);
		gsl_integration_qag(&func, 0., M_PI / 2., EPSABS, EPSREL,
				    1000, GSL_INTEG_GAUSS61, w, &result,
				    &err);
		F[k] = log(4 * M_PI * result);
	}
	gsl_integration_workspace_free(w);

	/*  build table for quick evaluation of the bessel function K2 for emissivity */
	for (k = 0; k <= N_ESAMP; k++) {
		T = exp(k * dlT + lT_min);
		K2[k] = log(gsl_sf_bessel_Kn(2, 1. / T));

	}

	/* Avoid doing divisions later */
	dlK = 1. / dlK;
	dlT = 1. / dlT;

	fprintf(stderr, "done.\n\n");

	return;
}

/* rapid evaluation of K_2(1/\Thetae) */

double K2_eval(double Thetae)
{

	double linear_interp_K2(double);

	if (Thetae < THETAE_MIN)
		return 0.;
	if (Thetae > TMAX)
		return 2. * Thetae * Thetae;

	return linear_interp_K2(Thetae);
}

#define KFAC	(9*M_PI*ME*CL/EE)
double F_eval(double Thetae, double Bmag, double nu)
{

	double K, x;
	double linear_interp_F(double);

	K = KFAC * nu / (Bmag * Thetae * Thetae);

	if (K > KMAX) {
		return 0.;
	} else if (K < KMIN) {
		/* use a good approximation */
		x = pow(K, 0.333333333333333333);
		return (x * (37.67503800178 + 2.240274341836 * x));
	} else {
		return linear_interp_F(K);
	}
}

#undef KFAC
#undef KMIN
#undef KMAX
#undef EPSABS
#undef EPSREL

double linear_interp_K2(double Thetae)
{

	int i;
	double di, lT;

	lT = log(Thetae);

	di = (lT - lT_min) * dlT;
	i = (int) di;
	di = di - i;

	return exp((1. - di) * K2[i] + di * K2[i + 1]);
}

double linear_interp_F(double K)
{

	int i;
	double di, lK;

	lK = log(K);

	di = (lK - lK_min) * dlK;
	i = (int) di;
	di = di - i;

	return exp((1. - di) * F[i] + di * F[i + 1]);
}