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hydrogen.c
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hydrogen.c
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/********************************************************************************************************/
/* HYREC-2: Hydrogen and Helium Recombination Code */
/* Written by Yacine Ali-Haimoud and Chris Hirata (C2010-17) */
/* with contributions from Nanoom Lee (2020) */
/* */
/* hydrogen.c: all functions related to Hydrogen recombination */
/* */
/* Units used: cgs + eV (all temperatures in eV) */
/* */
/* Revision history: */
/* - January 2020: - Added new mode, SWIFT */
/* - December 2014: Added effect of non-standard energy injection */
/* - May 2012: - Now solve for the photon distortion instead of absolute value */
/* of radiation field (less prone to numerical errors) */
/* - Improved the post-Saha expansion to properly account for */
/* non-thermal distortions */
/* - Added explicit dependence on fine-structure constant */
/* (fsR = alpha/alpha0) and electron mass (meR = me/me0 ~ mue/mue0) */
/* - Included dependence on xe and xHII in case HeII still present */
/* - January 2011: - changed the post-Saha expansion to use the full derivative */
/* (including two-photon processes and diffusion) rather than Peebles'ODE */
/* - post-Saha expansion can now pass the difference from Saha value */
/* to external routines */
/* - differential 2s--1s rate is now systematically normalized to */
/* total 2s--1s rate that can be set by user in hydrogen.h */
/* - Written November 2010 */
/********************************************************************************************************/
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <string.h>
#include "hyrectools.h"
#include "hydrogen.h"
/***********************************************************************************************************
Some constants appropriately rescaled for different values of the fine-structure constant and electron mass
***********************************************************************************************************/
double SAHA_FACT(double fsR, double meR) {
return 3.016103031869581e21 * cube(fsR*meR); /* (2 pi mu_e * EI/EI0)^(3/2)/h^3 in eV^(-3/2) cm^(-3), used for Saha equilibrium and detailed balance */
}
double LYA_FACT(double fsR, double meR) {
return 4.662899067555897e15 * cube(fsR*fsR*meR); /* 8pi/(3 lambda_Lya^3) in cm^(-3), used as prefactor for Lyman-alpha Sobolev escape probability */
}
double L2s_rescaled(double fsR, double meR) { /* Rescaled two-photon decay rate */
return L2s1s * square(fsR*fsR*fsR*fsR) * meR;
}
/*****************************************************************************************************
Temperature rescaling given fine-structure constant and electron mass, so we can use today's EI value
*****************************************************************************************************/
void rescale_T(double *T, double fsR, double meR) {
*T /= fsR*fsR*meR;
}
/**************************************************************************************************
Case-B recombination coefficient and photoionization rate, fit of Pequignot et al 1991, in cm^3 s^{-1}
INPUT TEMPERATURE ASSUMED TO BE ALREADY RESCALED FOR VALUES OF alpha_fs and me
***************************************************************************************************/
double alphaB_PPB(double TM, double fsR, double meR) {
double t4;
t4 = TM/kBoltz/1e4;
return square(fsR/meR) * 4.309e-13*pow(t4,-0.6166)/(1.+ 0.6703*pow(t4,0.5300));
}
/**************************************************************************************************
Effective three-level atom model with adjustable fudge factor.
Fudge = 1 is Peebles' model. Fudge = 1.14 is similar to RecFast (Seager et al. 1999, 2000).
Changes (May 2012):
- Correction: detailed balance implies that the photoionization rate is proportional to alpha_B(Tr) rather than alpha_B(Tm)
- Analytically subtract nearly-cancelling terms at high-z
- Explicit dependence on alpha_fs and m_e now accounted for
- Added explicit dependence on xHII, which is not necessarily equal to xe if Helium has not entirely recombined
December 2014:
- Added additional ionization and excitations due to additional energy injection.
dEdtdV is the rate of energy deposition rate per unit volume (in eV/s/cm^3)
***************************************************************************************************/
double rec_TLA_dxHIIdlna(REC_COSMOPARAMS *cosmo, double xe, double xHII, double nH, double H, double TM, double TR, double Fudge) {
double RLya, alphaB_TM, alphaB_TR, four_betaB, C, s, Dxe2, DalphaB;
double fsR = cosmo->fsR, meR = cosmo->meR;
rescale_T(&TM, fsR, meR);
rescale_T(&TR, fsR, meR);
RLya = LYA_FACT(fsR, meR) * H / nH / (1.-xHII);
alphaB_TM = Fudge * alphaB_PPB(TM, fsR, meR);
alphaB_TR = Fudge * alphaB_PPB(TR, fsR, meR);
four_betaB = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-0.25*EI/TR) * alphaB_TR;
C = (3.*RLya + L2s_rescaled(fsR, meR))/(3.*RLya + L2s_rescaled(fsR, meR) + four_betaB); /* Peebles' C-factor */
s = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-EI/TR)/nH;
Dxe2 = xe*xHII - s*(1.-xHII); /* xe xp - xe xp[Saha eq with 1s] -- gives more compact expressions */
DalphaB = alphaB_TM - alphaB_TR;
return -nH*(s*(1.-xHII)*DalphaB + Dxe2*alphaB_TM)*C/H + (cosmo->inj_params->ion + (1.-C)*cosmo->inj_params->exclya)/H;
}
/**********************************************************************************************
Allocates memory for the structure HYREC_ATOMIC, and reads and stores
effective-few-level rates and two-photon rates.
This function is a merger of two previous functions for effective rates and two-photon rates.
**********************************************************************************************/
void allocate_and_read_atomic(HYREC_ATOMIC *atomic, int *error, char *path_to_hyrec, char error_message[SIZE_ErrorM]){
/*********** Effective rates *************/
char *alpha_file, *rr_file, *twog_file;
char sub_message[128];
alpha_file = malloc(SIZE_InputFile);
alpha_file[0] = 0;
strcat(alpha_file, path_to_hyrec);
strcat(alpha_file, ALPHA_FILE);
FILE *fA = fopen(alpha_file, "r");
if (fA == NULL) {
sprintf(sub_message, "in allocate_and_read_atomic: could not open file %s \n", alpha_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
rr_file = malloc(SIZE_InputFile);
rr_file[0] = 0;
strcat(rr_file, path_to_hyrec);
strcat(rr_file, RR_FILE);
FILE *fR = fopen(rr_file, "r");
if (fR == NULL) {
sprintf(sub_message, "in allocate_and_read_atomic: could not open file %s \n", rr_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
unsigned i, j, l;
/* Allocate memory */
atomic->logAlpha_tab[0] = create_2D_array(NTM, NTR, error, error_message);
atomic->logAlpha_tab[1] = create_2D_array(NTM, NTR, error, error_message);
atomic->logAlpha_tab[2] = create_2D_array(NTM, NTR, error, error_message);
atomic->logAlpha_tab[3] = create_2D_array(NTM, NTR, error, error_message);
maketab(log(TR_MIN), log(TR_MAX), NTR, atomic->logTR_tab);
maketab(T_RATIO_MIN, T_RATIO_MAX, NTM, atomic->T_RATIO_tab);
atomic->DlogTR = atomic->logTR_tab[1] - atomic->logTR_tab[0];
atomic->DT_RATIO = atomic->T_RATIO_tab[1] - atomic->T_RATIO_tab[0];
for (i = 0; i < NTR; i++) {
for (j = 0; j < NTM; j++) for (l = 0; l <= 3; l++) {
if( fscanf(fA, "%le", &(atomic->logAlpha_tab[l][j][i])) != 1){
sprintf(sub_message, "in allocate_and_read_atomic: could not read file %s completely -- The file might be corrupted\n", alpha_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
atomic->logAlpha_tab[l][j][i] = log(atomic->logAlpha_tab[l][j][i]);
}
if ( fscanf(fR, "%le", &(atomic->logR2p2s_tab[i])) != 1){
sprintf(sub_message, "in allocate_and_read_atomic: could not read file %s completely -- The file might be corrupted\n", rr_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
atomic->logR2p2s_tab[i] = log(atomic->logR2p2s_tab[i]);
}
fclose(fA);
fclose(fR);
/************ Two-photon rates ************/
FILE *f2g;
unsigned b;
double L2s1s_current;
int fscanf_counter;
twog_file = malloc(SIZE_InputFile);
twog_file[0] = 0;
strcat(twog_file, path_to_hyrec);
strcat(twog_file, TWOG_FILE);
f2g = fopen(twog_file, "r");
if (f2g == NULL) {
sprintf(sub_message, "in allocate_and_read_atomic: could not open file %s \n", twog_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
for (b = 0; b < NVIRT; b++) {
fscanf_counter = 0;
fscanf_counter+= fscanf(f2g, "%le", &(atomic->Eb_tab[b]));
fscanf_counter+= fscanf(f2g, "%le", &(atomic->A1s_tab[b]));
fscanf_counter+= fscanf(f2g, "%le", &(atomic->A2s_tab[b]));
fscanf_counter+= fscanf(f2g, "%le", &(atomic->A3s3d_tab[b]));
fscanf_counter+= fscanf(f2g, "%le", &(atomic->A4s4d_tab[b]));
if(fscanf_counter!=5){
sprintf(sub_message, "in allocate_and_read_atomic: could not read file %s completely -- The file might be corrupted\n", twog_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
}
fclose(f2g);
/* Normalize 2s--1s differential decay rate to L2s1s (can be set by user in hydrogen.h) */
L2s1s_current = 0.;
for (b = 0; b < NSUBLYA; b++) L2s1s_current += atomic->A2s_tab[b];
for (b = 0; b < NSUBLYA; b++) atomic->A2s_tab[b] *= L2s1s/L2s1s_current;
/* Switches for the various effects considered in Hirata (2008) and diffusion:
Effect A: correct 2s-->1s rate, with stimulated decays and absorptions of non-thermal photons
Effect B: Sub-Lyman-alpha two-photon decays
Effect C: Super-Lyman-alpha two-photon decays
Effect D: Raman scattering */
#if (EFFECT_A == 0)
for (b = 0; b < NSUBLYA; b++) atomic->A2s_tab[b] = 0;
#endif
#if (EFFECT_B == 0)
for (b = 0; b < NSUBLYA; b++) atomic->A3s3d_tab[b] = atomic->A4s4d_tab[b] = 0;
#endif
#if (EFFECT_C == 0)
for (b = NSUBLYA; b < NVIRT; b++) atomic->A3s3d_tab[b] = atomic->A4s4d_tab[b] = 0;
#endif
#if (EFFECT_D == 0)
for (b = NSUBLYA; b < NVIRT; b++) atomic->A2s_tab[b] = 0;
for (b = NSUBLYB; b < NVIRT; b++) atomic->A3s3d_tab[b] = 0;
#endif
#if (DIFFUSION == 0)
for (b = 0; b < NVIRT; b++) atomic->A1s_tab[b] = 0;
#endif
free(alpha_file);
free(rr_file);
free(twog_file);
}
/***********************************************************************************************
Free the memory for rate tables.
***********************************************************************************************/
void free_atomic(HYREC_ATOMIC *atomic){
free_2D_array(atomic->logAlpha_tab[0], NTM);
free_2D_array(atomic->logAlpha_tab[1], NTM);
free_2D_array(atomic->logAlpha_tab[2], NTM);
free_2D_array(atomic->logAlpha_tab[3], NTM);
}
/**********************************************************************************************
Allocates memory for the structure FIT_FUNC, and reads and stores
correction function for SWIFT mode
**********************************************************************************************/
void allocate_and_read_fit(FIT_FUNC *fit, int *error, char *path_to_hyrec, char error_message[SIZE_ErrorM]){
char sub_message[128];
/*********** Effective rates *************/
char *fit_file;
fit_file = malloc(SIZE_InputFile);
fit_file[0] = 0;
strcat(fit_file, path_to_hyrec);
strcat(fit_file, FIT_FILE);
FILE *fA = fopen(fit_file, "r");
if (fA == NULL) {
sprintf(sub_message, "in allocate_and_read_fit: could not open file %s \n", fit_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
unsigned i, j;
/* Allocate memory */
fit->swift_func[0] = create_1D_array(DKK_SIZE, error, error_message);
fit->swift_func[1] = create_1D_array(DKK_SIZE, error, error_message);
fit->swift_func[2] = create_1D_array(DKK_SIZE, error, error_message);
fit->swift_func[3] = create_1D_array(DKK_SIZE, error, error_message);
fit->swift_func[4] = create_1D_array(DKK_SIZE, error, error_message);
for (i = 0; i < DKK_SIZE; i++) {
for (j = 0; j < 5; j++) {
if( fscanf(fA,"%le", &(fit->swift_func[j][i])) != 1){
sprintf(sub_message, "in allocate_and_read_atomic: could not read file %s completely -- The file might be corrupted\n", fit_file);
strcat(error_message, sub_message);
*error = 1;
return;
}
}
}
fclose(fA);
free(fit_file);
}
/***********************************************************************************************
Free the memory for rate tables.
***********************************************************************************************/
void free_fit(FIT_FUNC *fit){
unsigned j;
for (j = 0; j < 5; j++) free(fit->swift_func[j]);
}
/************************************************************************************************
Interpolation of tabulated effective rates
To be (slightly) more efficient, not using the external interpolation routine.
Gets the correct rates for given fine-structure constant and electron mass.
- Modified May 2012: - Accounts for different alpha_fs and m_e
- Also returns DAlpha[2], table of ALpha(Tm, Tr) - ALpha(Tr, Tr)
INPUT TEMPERATURE ASSUMED TO BE ALREADY RESCALED FOR VALUES OF alpha_fs and me
- Modified December 2014: additional heating sometimes brings Tm slightly larger than Tr.
If this is the case, use Tm = Tr in the recombination rates.
Will eventually re-tabulate the effective rates for Tm/Tr > 1 to fix this.
************************************************************************************************/
void interpolate_rates(double Alpha[2], double DAlpha[2], double Beta[2], double *R2p2s, double TR, double TM_TR,
HYREC_ATOMIC *atomic, double fsR, double meR, int *error, char error_message[SIZE_ErrorM]) {
unsigned l, k, i;
long iTM, iTR;
double frac1, frac2;
double logTR, T_RATIO;
double coeff1[4], coeff2[4], temp[4];
double Alpha_eq[2];
char sub_message[128];
/* Check that TM/TR is in range */
if (TM_TR < T_RATIO_MIN) {
sprintf(sub_message, "in interpolate_rates: TM/TR = %f is out of range.\n", TM_TR);
strcat(error_message, sub_message);
*error = 1;
return;
}
/* T_RATIO is defined to be min(TM_TR, TR_TM) */
if (TM_TR > 1.) {
T_RATIO = 1./TM_TR; i = 2;
}
else {
T_RATIO = TM_TR; i = 0;
}
/* Check if log(TR) is in the range tabulated */
if (TR < TR_MIN || TR > TR_MAX) {
sprintf(sub_message, "in interpolate_rates: TR = %f is out of range.\n", TR);
strcat(error_message, sub_message);
*error = 1;
return;
}
/**** TR-only-dependent functions ****/
/* Identify location to interpolate in log(TR) */
logTR = log(TR);
iTR = (long)floor((logTR - log(TR_MIN))/atomic->DlogTR);
if (iTR < 1) iTR = 1;
if (iTR > NTR-3) iTR = NTR-3;
frac2 = (logTR - log(TR_MIN))/atomic->DlogTR - iTR;
coeff2[0] = frac2*(frac2-1.)*(2.-frac2)/6.;
coeff2[1] = (1.+frac2)*(1.-frac2)*(2.-frac2)/2.;
coeff2[2] = (1.+frac2)*frac2*(2.-frac2)/2.;
coeff2[3] = (1.+frac2)*frac2*(frac2-1.)/6.;
for (l = 0; l <= 1; l++) {
/* Alpha evaluated at Tm = Tr */
Alpha_eq[l] = square(fsR/meR)* exp(atomic->logAlpha_tab[l][NTM-1][iTR-1]*coeff2[0]
+atomic->logAlpha_tab[l][NTM-1][iTR]*coeff2[1]
+atomic->logAlpha_tab[l][NTM-1][iTR+1]*coeff2[2]
+atomic->logAlpha_tab[l][NTM-1][iTR+2]*coeff2[3]);
/* Beta obtained by detailed balance from Alpha(Tr, Tr) */
/* prefactor = pow(2.0 * M_PI * mue *TR / hPc / hPc, 1.5)) * exp(-0.25*EI/TR) */
Beta[l] = Alpha_eq[l] * SAHA_FACT(fsR, meR) * TR*sqrt(TR) *exp(-0.25*EI/TR)/(2.*l+1.);
}
/* Effective 2p->2s rate */
*R2p2s = fsR*fsR*fsR*fsR*fsR*meR *
exp(atomic->logR2p2s_tab[iTR-1]*coeff2[0]
+atomic->logR2p2s_tab[iTR]*coeff2[1]
+atomic->logR2p2s_tab[iTR+1]*coeff2[2]
+atomic->logR2p2s_tab[iTR+2]*coeff2[3]);
/**** Effective recombination coefficients Alpha(Tm, Tr) ****/
/* Identify location to interpolate in T_RATIO */
iTM = (long)floor((T_RATIO - T_RATIO_MIN)/atomic->DT_RATIO);
if (iTM < 1) iTM = 1;
if (iTM > NTM-3) iTM = NTM-3;
frac1 = (T_RATIO - T_RATIO_MIN)/atomic->DT_RATIO - iTM;
coeff1[0] = frac1*(frac1-1.)*(2.-frac1)/6.;
coeff1[1] = (1.+frac1)*(1.-frac1)*(2.-frac1)/2.;
coeff1[2] = (1.+frac1)*frac1*(2.-frac1)/2.;
coeff1[3] = (1.+frac1)*frac1*(frac1-1.)/6.;
for (l = 0; l <= 1; l++) {
/* effective recombination coefficient to each level */
for (k = 0; k < 4; k++) {
temp[k] = atomic->logAlpha_tab[l+i][iTM-1+k][iTR-1]*coeff2[0]
+ atomic->logAlpha_tab[l+i][iTM-1+k][iTR]*coeff2[1]
+ atomic->logAlpha_tab[l+i][iTM-1+k][iTR+1]*coeff2[2]
+ atomic->logAlpha_tab[l+i][iTM-1+k][iTR+2]*coeff2[3];
}
Alpha[l] = square(fsR/meR)* exp(temp[0]*coeff1[0]+temp[1]*coeff1[1]
+temp[2]*coeff1[2]+temp[3]*coeff1[3]);
DAlpha[l] = Alpha[l] - Alpha_eq[l];
}
}
double rec_swift_hyrec_dxHIIdlna(HYREC_DATA *data, double xe, double xHII, double nH, double H, double TM, double TR, double z){
REC_COSMOPARAMS *cosmo = data-> cosmo;
HYREC_ATOMIC *atomic = data->atomic;
FIT_FUNC *fit = data->fit;
int *error = &data->error;
double fsR = cosmo->fsR, meR = cosmo->meR;
double Alpha[2], DAlpha[2], Beta[2], R2p2s, RLya;
double DK_K_fid=0., DK_K, fitted_RLya;
double C_2s, C_2p, gamma_2s, gamma_2p, s, Dxe2;
static double diff[3];
unsigned i;
double ratio;
char sub_message[128];
double T0fid_T0;
if (*error == 1) return 0.;
ratio = TM/TR;
rescale_T(&TR, fsR, meR);
TM = ratio * TR; // This way ensure that TM<=TR is preserved
T0fid_T0 = 2.7255 / (TR/kBoltz/(1.+z));
/* The numbers in the following lines are fiducial parameters for correction function (Do not change) */
diff[0] = (cosmo->ocbh2 - 0.14175)*pow(T0fid_T0,3);
diff[1] = (cosmo->obh2*(1-cosmo->YHe) - 0.02242*(1-0.246738546372))*pow(T0fid_T0,3);
diff[2] = cosmo->Neff-3.046;
interpolate_rates(Alpha, DAlpha, Beta, &R2p2s, TR, TM/TR, atomic, fsR, meR, error, data->error_message);
RLya = LYA_FACT(fsR, meR) *H/nH/(1.-xHII); // 8 PI H/(3 nH x1s lambda_Lya^3)
if (TR/kBoltz > fit->swift_func[0][DKK_SIZE-1]) DK_K = 0.;
else {
DK_K_fid = rec_interp1d(fit->swift_func[0][0], 10., fit->swift_func[1], DKK_SIZE, TR/kBoltz, error, data->error_message);
for (i = 0; i < 3; i++) {
DK_K_fid = DK_K_fid + diff[i]*rec_interp1d(fit->swift_func[0][0], 10., fit->swift_func[i+2],
DKK_SIZE, TR/kBoltz, error, data->error_message);
}
}
DK_K = DK_K_fid;
fitted_RLya = RLya / (1.+DK_K);
gamma_2s = Beta[0] + 3.*R2p2s + L2s_rescaled(fsR, meR);
gamma_2p = Beta[1] + R2p2s + fitted_RLya;
C_2s = (L2s_rescaled(fsR, meR)+3.*R2p2s*fitted_RLya/gamma_2p)/(gamma_2s-3.*R2p2s*R2p2s/gamma_2p);
C_2p = (fitted_RLya+R2p2s*L2s_rescaled(fsR, meR)/gamma_2s)/(gamma_2p-R2p2s*3.*R2p2s/gamma_2s);
s = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-EI/TR)/nH;
Dxe2 = xe*xHII - s*(1.-xHII); // xe^2 - xe^2[Saha eq with 1s] -- gives more compact expressions
if (*error == 1) {
sprintf(sub_message, " called from rec_swift_hyrec_dxHIIdlna\n");
strcat(data->error_message, sub_message);
return 0.;
}
return -nH/H *( (s*(1.-xHII)*DAlpha[0] + Alpha[0]*Dxe2)*C_2s + (s*(1.-xHII)*DAlpha[1] + Alpha[1]*Dxe2)*C_2p)
+ (cosmo->inj_params->ion + (0.25*(1.-C_2s) + 0.75*(1.-C_2p))*cosmo->inj_params->exclya)/H ;
}
/************************************************************************************************
Solves for the populations of the 2s and 2p states in steady-state, and returns dxe/dlna.
Uses standard rate for 2s-->1s decay and Sobolev for Lyman alpha (no feedback),
and fully accounts for virtually all transitions among excited states through effective rates.
Inputs: xe, nH in cm^{-3}, H in s^{-1}, TM, TR in eV. Output: dxe/dlna
Changes May 2012:
- Now use the analytic expressions using the generalized Peebles' C factors
[see companion paper, Eqs. (43)-(46)].
- Also explicitly subtract nearly cancelling terms at early times.
- Account for explicit dependence on alpha_fs and m_e
- Added explicit dependence on xHII, which is not necessarily equal to xe if Helium has not entirely recombined
December 2014:
- Added dependence on extra energy deposited in the plasma, dEdtdV in eV/s/cm^3
************************************************************************************************/
double rec_HMLA_dxHIIdlna(HYREC_DATA *data, double xe, double xHII, double nH, double H, double TM, double TR) {
REC_COSMOPARAMS *cosmo = data->cosmo;
HYREC_ATOMIC *atomic = data->atomic;
int *error = &data->error;
double fsR = cosmo->fsR, meR = cosmo->meR;
double Alpha[2], DAlpha[2], Beta[2], R2p2s, RLya;
double Gamma_2s, Gamma_2p, C2s, C2p, s, Dxe2;
double ratio;
char sub_message[128];
if (*error == 1) return 0.;
ratio = TM/TR;
rescale_T(&TR, fsR, meR);
TM = ratio * TR; /* This way ensure that TM<=TR is preserved */
interpolate_rates(Alpha, DAlpha, Beta, &R2p2s, TR, TM/TR, atomic, fsR, meR, error, data->error_message);
if (*error == 1) {
sprintf(sub_message, " called from rec_HMLA_dxHIIdlna\n");
strcat(data->error_message, sub_message);
return 0.;
}
RLya = LYA_FACT(fsR, meR) *H/nH/(1.-xHII); /* 8 PI H/(3 nH x1s lambda_Lya^3) */
/* Effective inverse lifetimes of 2s and 2p states */
Gamma_2s = Beta[0] + 3.*R2p2s + L2s_rescaled(fsR, meR);
Gamma_2p = Beta[1] + R2p2s + RLya;
/* Generalization of Peebles' C factor */
C2s = (L2s_rescaled(fsR, meR) + 3.*R2p2s * RLya/Gamma_2p)/(Gamma_2s - 3.*R2p2s*R2p2s/Gamma_2p);
C2p = (RLya + R2p2s * L2s_rescaled(fsR, meR)/Gamma_2s)/(Gamma_2p - 3.*R2p2s*R2p2s/Gamma_2s);
s = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-EI/TR)/nH;
Dxe2 = xe*xHII - s*(1.-xHII); /* xe^2 - xe^2[Saha eq with 1s] -- gives more compact expressions */
return -nH/H *( (s*(1.-xHII)*DAlpha[0] + Alpha[0]*Dxe2)*C2s + (s*(1.-xHII)*DAlpha[1] + Alpha[1]*Dxe2)*C2p )
+ (cosmo->inj_params->ion + (0.25*(1.-C2s) + 0.75*(1.-C2p))*cosmo->inj_params->exclya)/H ;
}
/********************************************************************************************************
Compute the A_{b,b+/-1} "Einstein A-"coefficients between virtual states, due to diffusion
(In the notation of the paper, A_{b,b\pm1} = R_{b,b\pm1})
Aup[b] = A_{b, b+1} Adn[b] = A_{b, b-1}
********************************************************************************************************/
void populate_Diffusion(double *Aup, double *Adn, double *A2p_up, double *A2p_dn,
double TM, double Eb_tab[NVIRT], double A1s_tab[NVIRT]) {
unsigned b;
double DE2;
DE2 = E21*E21*2.*TM/mH;
/****** RED WING ******/
b = NSUBLYA - NDIFF/2;
Aup[b] = DE2 / square(Eb_tab[b+1] - Eb_tab[b]) * A1s_tab[b]; /* A{0,1}. Assume A{0,-1} = 0 */
for (b = NSUBLYA - NDIFF/2 + 1; b < NSUBLYA-1; b++) {
Adn[b] = exp((Eb_tab[b] - Eb_tab[b-1])/TM) * Aup[b-1]; /* Detailed balance */
Aup[b] = (DE2 * A1s_tab[b] - square(Eb_tab[b] - Eb_tab[b-1]) * Adn[b])
/square(Eb_tab[b+1] - Eb_tab[b]); /* Aup[b] , Adn[b] must add up to correct diffusion rate */
}
/* Last bin below Lyman alpha */
b = NSUBLYA - 1;
Adn[b] = exp((Eb_tab[b] - Eb_tab[b-1])/TM) * Aup[b-1];
Aup[b] = (DE2 * A1s_tab[b] - square(Eb_tab[b] - Eb_tab[b-1]) * Adn[b])
/square(E21 - Eb_tab[b]);
*A2p_dn = exp((E21 - Eb_tab[b])/TM)/3.* Aup[b]; /* 2p -> NSUBLYA-1 rate obtained by detailed balance */
/****** BLUE WING ******/
b = NSUBLYA + NDIFF/2 - 1;
Adn[b] = DE2 / square(Eb_tab[b] - Eb_tab[b-1]) * A1s_tab[b];
for (b = NSUBLYA + NDIFF/2 - 2; b > NSUBLYA; b--) {
Aup[b] = exp((Eb_tab[b] - Eb_tab[b+1])/TM) * Adn[b+1];
Adn[b] = (DE2 * A1s_tab[b] - square(Eb_tab[b+1] - Eb_tab[b]) * Aup[b])
/square(Eb_tab[b] - Eb_tab[b-1]);
}
/* First bin above Lyman alpha */
b = NSUBLYA;
Aup[b] = exp((Eb_tab[b] - Eb_tab[b+1])/TM) * Adn[b+1];
Adn[b] = (DE2 * A1s_tab[b] - square(Eb_tab[b+1] - Eb_tab[b]) * Aup[b])
/square(Eb_tab[b] - E21);
*A2p_up = exp((E21 - Eb_tab[b])/TM)/3. * Adn[b]; /* 2p -> NSUBLYA rate obtained by detailed balance */
}
/*********************************************************************************************************
Populate the real-real, real-virtual, virtual-real and virtual-virtual T-matrices,
as well as the source vectors sr, sv, given Dxe = xe - xe[Saha] as an input.
WITH DIFFUSION. Tvv[0][b] is the diagonal element Tbb, Tvv[1][b] = T{b,b-1} and Tvv[2][b] = T{b,b+1}
Also, computes and stores the optical depths Delta tau_b for future use
Modified May 2012: - now uses the photon distortion
instead of absolute photon occupation number.
- Accounts for explicit dependence on alpha_fs and m_e
INPUT TEMPERATURE ASSUMED TO BE ALREADY RESCALED FOR VALUES OF alpha_fs and me
- Added explicit dependence on xHII, which is not necessarily equal to xe if Helium has not entirely recombined
December 2014: Added dependence on additional energy deposition dEdtdV in eV/s/cm^3.
**********************************************************************************************************/
void populateTS_2photon(double Trr[2][2], double *Trv[2], double *Tvr[2], double *Tvv[3],
double sr[2], double sv[NVIRT], double Dtau[NVIRT],
double xe, double xHII, double TM, double TR, double nH, double H, HYREC_ATOMIC *atomic,
double Dfplus[NVIRT], double Dfplus_Ly[],
double Alpha[2], double DAlpha[2], double Beta[2], double fsR, double meR, double exclya,
int *error, char error_message[SIZE_ErrorM]) {
unsigned b;
double R2p2s, RLya, Gammab, one_minus_Pib, dbfact, x1s, s, Dxe2;
double A2p_up, A2p_dn, rescale2g, rescalediff;
double *Aup, *Adn;
char sub_message[128];
/*** Added May 2012: rescalings for dependence on alpha and me ***/
rescale2g = square(fsR*fsR*fsR*fsR)*meR; /* for two-photon rates */
rescalediff = rescale2g * fsR*fsR*meR;
/* diffusion rate ~ two-photon rate * TM ~ two-photon rate * alpha^2 me * rescaled(TM) [which is assumed as an input] */
Aup = create_1D_array(NVIRT, error, error_message);
Adn = create_1D_array(NVIRT, error, error_message);
x1s = 1.-xHII;
s = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-EI/TR)/nH;
Dxe2 = xe*xHII - s*x1s;
RLya = LYA_FACT(fsR, meR) *H /nH/x1s; /*8 PI H/(3 nH x1s lambda_Lya^3) */
interpolate_rates(Alpha, DAlpha, Beta, &R2p2s, TR, TM / TR, atomic, fsR, meR, error, error_message);
if (*error == 1) {
sprintf(sub_message, " called from populateTS_2photon\n");
strcat(error_message, sub_message);
return;
}
/****** 2s row and column ******/
Trr[0][0] = Beta[0] + 3.*R2p2s
+ 3.* RLya * (1.664786871919931 *exp(-E32/TR) /* Ly-beta escape */
+ 1.953125 *exp(-E42/TR)); /* Ly-gamma escape */
Trr[0][1] = -R2p2s;
sr[0] = nH * (s*x1s*DAlpha[0] + Alpha[0]*Dxe2) + 3.* RLya * x1s * 1.664786871919931 *Dfplus_Ly[1];
sr[0] += 0.25 *exclya;
#if (EFFECT_A == 0) /* Standard treatment of 2s-->1s two-photon decays */
Trr[0][0] += L2s1s*rescale2g; /* rescaled for alpha, me */
#endif
/****** 2p row and column ******/
Trr[1][1] = Beta[1] + R2p2s + RLya;
Trr[1][0] = -3.*R2p2s;
sr[1] = nH * (s*x1s*DAlpha[1] + Alpha[1]*Dxe2) + 3.*RLya * x1s * Dfplus_Ly[0];
sr[1] += 0.75 *exclya;
/***** Two-photon transitions: populating Trv, Tvr and updating Trr ******/
for (b = 0; b < NVIRT; b++) {
dbfact = exp((atomic->Eb_tab[b] - E21)/TR);
Trr[0][0] -= Tvr[0][b] = -rescale2g*atomic->A2s_tab[b]/fabs(exp((atomic->Eb_tab[b] - E21)/TR)-1.);
Trv[0][b] = Tvr[0][b] *dbfact;
Trr[1][1] -= Tvr[1][b] = -exp(-E32/TR)/3. * rescale2g*atomic->A3s3d_tab[b]/fabs(exp((atomic->Eb_tab[b] - E31)/TR)-1.)
-exp(-E42/TR)/3. * rescale2g*atomic->A4s4d_tab[b]/fabs(exp((atomic->Eb_tab[b] - E41)/TR)-1.);
Trv[1][b] = Tvr[1][b] *3.*dbfact;
}
/****** Tvv and sv. Accounting for DIFFUSION ******/
populate_Diffusion(Aup, Adn, &A2p_up, &A2p_dn, TM, atomic->Eb_tab, atomic->A1s_tab);
/*** Added May 2012: rescale for dependence on alpha and me ***/
A2p_up *= rescalediff;
A2p_dn *= rescalediff;
for (b = 0; b < NVIRT; b++) {
Aup[b] *= rescalediff;
Adn[b] *= rescalediff;
}
/* Updating Tvr, Trv, Trr for diffusion between line center ("2p state") and two neighboring bins */
Trr[1][1] += (A2p_dn + A2p_up);
for (b = 0; b < NVIRT; b++) {
Gammab = -(Trv[0][b] + Trv[1][b]) + Aup[b] + Adn[b]; /* Inverse lifetime of virtual state b */
/*** Diffusion region ***/
if ( (b >= NSUBLYA - NDIFF/2 && b < NSUBLYA - 1) ||(b > NSUBLYA && b < NSUBLYA + NDIFF/2)) {
Tvv[1][b] = -Aup[b-1];
Tvv[2][b] = -Adn[b+1];
}
/* Bins neighboring Lyman alpha. */
if (b == NSUBLYA-1) {
Tvv[1][b] = -Aup[b-1];
Tvv[2][b] = 0.;
Tvr[1][b] -= A2p_dn;
Trv[1][b] -= Aup[b];
}
if (b == NSUBLYA) {
Tvv[1][b] = 0.;
Tvv[2][b] = -Adn[b+1];
Tvr[1][b] -= A2p_up;
Trv[1][b] -= Adn[b];
}
/*********************/
Dtau[b] = Gammab * x1s * cube(hPc/atomic->Eb_tab[b]/fsR/fsR/meR) * nH /8. /M_PI /H;
/* Rescaled for alpha, me*/
one_minus_Pib = Dtau[b] > 1e-6 ? 1.- (1.-exp(-Dtau[b]))/Dtau[b] : Dtau[b]/2. - square(Dtau[b])/6.;
Tvv[0][b] = Dtau[b] > 0.? Gammab/one_minus_Pib : 2./(x1s * cube(hPc/atomic->Eb_tab[b]/fsR/fsR/meR) * nH /8. /M_PI /H); /* Added May 2012: proper limit Dtau->0 */
sv[b] = Tvv[0][b] * x1s * Dfplus[b] * (1.-one_minus_Pib);
}
free(Aup);
free(Adn);
}
/*********************************************************************
Solves the linear system T*X = B, where T is a DIAGONALLY DOMINANT
tridiagonal matrix, and X and B are both one-column vectors of N elements.
diag[i] = T_{ii}, updiag[i] = T_{i,i+1}, dndiag[i] = T_{i,i-1}
IMPORTANT: This is NOT THE MOST GENERAL ALGORITHM. Only adapted for the
case we will consider, i.e. |T_{ii}| > |T_{i,i+1}| + |T_{i,i-1}|.
**********************************************************************/
void solveTXeqB(double *diag, double *updiag, double *dndiag, double *X, double *B,
unsigned N, int *error, char error_message[SIZE_ErrorM]){
int i;
double denom;
double *alpha = create_1D_array(N, error, error_message);
double *gamma = create_1D_array(N, error, error_message); /* X[i] = gamma[i] - alpha[i] * X[i+1] */
alpha[0] = updiag[0] / diag[0];
gamma[0] = B[0] / diag[0];
for (i = 1; i < N; i++) {
denom = diag[i] - dndiag[i] * alpha[i-1];
alpha[i] = updiag[i] / denom;
gamma[i] = (B[i] - dndiag[i] * gamma[i-1]) / denom;
}
X[N-1] = gamma[N-1];
for (i = N-2; i >= 0; i--) X[i] = gamma[i] - alpha[i] * X[i+1];
free(alpha);
free(gamma);
}
/**************************************************************************************************************
Solves for the populations of the real (2s, 2p) and virtual states
***************************************************************************************************************/
void solve_real_virt(double xr[2], double xv[NVIRT], double Trr[2][2], double *Trv[2], double *Tvr[2], double *Tvv[3],
double sr[2], double sv[NVIRT], int *error, char error_message[SIZE_ErrorM]){
double *Tvv_inv_Tvr[2];
double *Tvv_inv_sv;
double Trr_new[2][2];
double sr_new[2];
unsigned i, j, b;
double det;
unsigned NSUBDIFF;
NSUBDIFF = NSUBLYA - NDIFF/2; /* lowest bin of the diffusion region */
/** Allocate memory **/
for (i = 0; i < 2; i++) Tvv_inv_Tvr[i] = create_1D_array(NVIRT, error, error_message);
Tvv_inv_sv = create_1D_array(NVIRT, error, error_message);
/*** Computing Tvv^{-1}.Tvr ***/
for (i = 0; i < 2; i++) {
for (b = 0; b < NSUBDIFF; b++) Tvv_inv_Tvr[i][b] = Tvr[i][b]/Tvv[0][b];
for (b = NSUBLYA + NDIFF/2; b < NVIRT; b++) Tvv_inv_Tvr[i][b] = Tvr[i][b]/Tvv[0][b];
solveTXeqB(Tvv[0]+NSUBDIFF, Tvv[2]+NSUBDIFF, Tvv[1]+NSUBDIFF, Tvv_inv_Tvr[i]+NSUBDIFF, Tvr[i]+NSUBDIFF, NDIFF, error, error_message);
}
/*** Trr_new = Trr - Trv.Tvv^{-1}.Tvr ***/
for (i = 0; i < 2; i++) for (j = 0; j < 2; j++) {
Trr_new[i][j] = Trr[i][j];
for (b = 0; b < NVIRT; b++) Trr_new[i][j] -= Trv[i][b]*Tvv_inv_Tvr[j][b];
}
/*** Computing Tvv^{-1}.sv ***/
for (b = 0; b < NSUBDIFF; b++) Tvv_inv_sv[b] = sv[b]/Tvv[0][b];
for (b = NSUBLYA + NDIFF/2; b < NVIRT; b++) Tvv_inv_sv[b] = sv[b]/Tvv[0][b];
solveTXeqB(Tvv[0]+NSUBDIFF, Tvv[2]+NSUBDIFF, Tvv[1]+NSUBDIFF, Tvv_inv_sv+NSUBDIFF, sv+NSUBDIFF, NDIFF, error, error_message);
/*** sr_new = sr - Trv.Tvv^{-1}sv ***/
for (i = 0; i < 2; i++) {
sr_new[i] = sr[i];
for (b = 0; b < NVIRT; b++) sr_new[i] -= Trv[i][b]*Tvv_inv_sv[b];
}
/*** Solve 2 by 2 system Trr_new.xr = sr_new ***/
det = Trr_new[0][0] * Trr_new[1][1] - Trr_new[0][1] * Trr_new[1][0];
xr[0] = (Trr_new[1][1] * sr_new[0] - Trr_new[0][1] * sr_new[1])/det;
xr[1] = (Trr_new[0][0] * sr_new[1] - Trr_new[1][0] * sr_new[0])/det;
/*** xv = Tvv^{-1}(sv - Tvr.xr) ***/
for (b = 0; b < NVIRT; b++) xv[b] = Tvv_inv_sv[b] - Tvv_inv_Tvr[0][b]*xr[0] - Tvv_inv_Tvr[1][b]*xr[1];
/** Free memory **/
for (i = 0; i < 2; i++) free(Tvv_inv_Tvr[i]);
free(Tvv_inv_sv);
}
/*********************************************************************************************
Interpolation of the photon distortion used to get f+ from f- at a higher frequency bin and earlier time.
Use a simple linear interpolation so the spectrum is always positive.
Added May 2012
*********************************************************************************************/
double interp_Dfnu(double lna_start, double dlna, double *ytab, unsigned int iz, double lna){
long ind;
double frac;
/* If iz = 0 or 1, radiation field at earlier times is still thermal.
Also thermal if iz > 1 and lna < lna_start. */
if (iz == 0 || iz == 1 || lna < lna_start) return 0.;
/* Check if in range */
if (lna >= lna_start + dlna*(iz-1)) {
fprintf(stderr, "Error in interp_Dfnu: lna-value out of range in interpolation routine\n");
fprintf(stderr, "The time-step used is probably too large\n");
exit(1);
}
/* If iz >= 2, do a linear interpolation so the spectrum is always positive */
ind = (long) floor((lna-lna_start)/dlna);
frac = (lna-lna_start)/dlna - ind;
return (1.-frac)*ytab[ind] + frac*ytab[ind+1];
}
/*************************************************************************************************************
Obtain fplus at each bin, given the history of fminus (simple free-streaming). iz is the current time step.
fminus[0..iz-1] is known.
Assume the Lyman lines are optically thick
Dfminus_hist is a NVIRT by nz array of previous Delta f(nu_b - epsilon)(z)
Dfminus_Ly_hist is a 3 by nz array of previous Deltaf(nu - epsilon) redward of Ly alpha, beta and gamma lines
Changed May 2012: Now using the interpolation function interp_Dfnu, which only interpolates
over 2 nearest neighbors, which ensures that the distortion is always positive
*************************************************************************************************************/
void fplus_from_fminus(double Dfplus[NVIRT], double Dfplus_Ly[], double **Dfminus_hist, double **Dfminus_Ly_hist,
double TR, double zstart, unsigned iz, double z, double Eb_tab[NVIRT]) {
unsigned b;
double ainv, lna_start, zp1;
zp1 = 1.+z;
lna_start = -log(1.+zstart);
/*** Bins below Lyman alpha ***/
for (b = 0; b < NSUBLYA-1; b++) {
ainv = zp1*Eb_tab[b+1]/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_hist[b+1], iz, -log(ainv));
}
/*** highest bin below Ly-alpha: feedback from optically thick Ly-alpha ***/
b = NSUBLYA-1;
ainv = zp1*E21/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_Ly_hist[0], iz, -log(ainv));
/*** incoming photon occupation number at Lyman alpha ***/
b = NSUBLYA; /* next highest bin */
ainv = zp1*Eb_tab[b]/E21;
Dfplus_Ly[0] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_hist[b], iz, -log(ainv));
/*** Bins between Lyman alpha and beta ***/
for (b = NSUBLYA; b < NSUBLYB-1; b++) {
ainv = zp1*Eb_tab[b+1]/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_hist[b+1], iz, -log(ainv));
}
/*** highest bin below Ly-beta: feedback from Ly-beta ***/
b = NSUBLYB-1;
ainv = zp1*E31/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_Ly_hist[1], iz, -log(ainv));
/*** incoming photon occupation number at Lyman beta ***/
b = NSUBLYB; /* next highest bin */
ainv = zp1*Eb_tab[b]/E31;
Dfplus_Ly[1] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_hist[b], iz, -log(ainv));
/*** Bins between Lyman beta and gamma ***/
for (b = NSUBLYB; b < NVIRT-1; b++) {
ainv = zp1*Eb_tab[b+1]/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_hist[b+1], iz, -log(ainv));
}
/*** highest energy bin: feedback from Ly-gamma ***/
b = NVIRT-1;
ainv = zp1*E41/Eb_tab[b];
Dfplus[b] = interp_Dfnu(lna_start, DLNA_HYREC, Dfminus_Ly_hist[2], iz, -log(ainv));
}
/******************************************************************************************************************
dxe/dlna when including two-photon processes.
Assume fminus[0..iz-1] is known. Update fminus[iz]
Modified May 2012:
- now use the photon distortion instead of absolute photon occupation number
- Accounts for explicit dependence on alpha_fs and m_e
- Added Dfnu_hist as a variable. Will contain the *average* distortion within each bin
December 2014: added dependence on additional energy injection dEdtdV in eV/s/cm^3.
The fractions that goes into ionizations, excitations and heat are assumed to be those of Chen & Kamionkowski 2004.
In the next version I'll make them potentialy changeable.
******************************************************************************************************************/
double rec_HMLA_2photon_dxHIIdlna(HYREC_DATA *data, double xe, double xHII, double nH, double H, double TM, double TR,
unsigned iz, double z) {
REC_COSMOPARAMS *cosmo = data->cosmo;
HYREC_ATOMIC *atomic = data->atomic;
int *error = &data->error;
double **Dfminus_hist = data->rad->Dfminus_hist;
double **Dfminus_Ly_hist = data->rad->Dfminus_Ly_hist;
double **Dfnu_hist = data->rad->Dfnu_hist;
double zstart = data->rad->z0;
double fsR = cosmo->fsR, meR = cosmo->meR;
double xr[2], xv[NVIRT], Dfplus[NVIRT], Dfplus_Ly[2]; /* Assume incoming radiation blueward of Ly-gamma is Blackbody */
double dxedlna, one_minus_Pib, one_minus_exptau, Dfeq, s, x1s, Dxe2;
unsigned b, i;
double Trr[2][2];
double *Trv[2];
double *Tvr[2];
double *Tvv[3];
double sr[2];
double sv[NVIRT];
double Dtau[NVIRT];
double Alpha[2], DAlpha[2], Beta[2];
double ratio;
char sub_message[128];
if (*error == 1) return 0.;
ratio = TM/TR;
rescale_T(&TR, fsR, meR);
TM = ratio * TR; /* This way ensure that TM<=TR is preserved */
for (i = 0; i < 2; i++) Trv[i] = create_1D_array(NVIRT, error, data->error_message);
for (i = 0; i < 2; i++) Tvr[i] = create_1D_array(NVIRT, error, data->error_message);
for (i = 0; i < 3; i++) Tvv[i] = create_1D_array(NVIRT, error, data->error_message);
/* Redshift photon occupation number from previous times and higher energy bins */
fplus_from_fminus(Dfplus, Dfplus_Ly, Dfminus_hist, Dfminus_Ly_hist, TR, zstart, iz, z, atomic->Eb_tab);
/* Compute real-real, real-virtual and virtual-virtual transition rates */
populateTS_2photon(Trr, Trv, Tvr, Tvv, sr, sv, Dtau, xe, xHII, TM, TR, nH, H, atomic,
Dfplus, Dfplus_Ly, Alpha, DAlpha, Beta, fsR, meR, cosmo->inj_params->exclya, error, data->error_message);
if (*error == 1) {
sprintf(sub_message, " called from rec_HMLA_2photon_dxHIIdlna\n");
strcat(data->error_message, sub_message);
return 0.;
}
/* Solve for the population of the real and virtual states
(in fact, for the difference xi - xi[eq with 1s]) */
solve_real_virt(xr, xv, Trr, Trv, Tvr, Tvv, sr, sv, error, data->error_message);
/* Obtain xe_dot */
x1s = 1.-xHII;
s = SAHA_FACT(fsR, meR) *TR*sqrt(TR) *exp(-EI/TR)/nH;
Dxe2 = xe*xHII - s*x1s;
dxedlna = -(nH *(s*x1s*DAlpha[0] + Alpha[0]*Dxe2) - xr[0]*Beta[0]
+nH *(s*x1s*DAlpha[1] + Alpha[1]*Dxe2) - xr[1]*Beta[1])/H
+ cosmo->inj_params->ion/H; /* First term automatically includes the additional excitations
since x2s, x2p are computed accounting for them */
/* Update fminuses */
for (b = 0; b < NVIRT; b++) {
if (Dtau[b] > 1e-30) {
one_minus_Pib = Dtau[b] > 1e-6 ? 1.- (1.-exp(-Dtau[b]))/Dtau[b] : Dtau[b]/2. - square(Dtau[b])/6.;
Dfeq = -xr[0]*Tvr[0][b] - xr[1]*Tvr[1][b];