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sco_model_LOG_dskb.f90
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SUBROUTINE sco_MODEL_LOGdskb(disk_size, corona_size, Tcorona, Tdisk, tau, QPO_frequency, DHext, eta_frac, Nsss, Ssss,Tsss, &
Nreal, Sreal, Treal, Nimag,Simag, Timag, dTe_mod, dTs_mod, dTe_arg, dTs_arg, Hexo0_out, eta_int)
USE iso_fortran_env, ONLY : WP => REAL64
USE sco_global
USE sco_arrays
IMPLICIT NONE
! scalar arguments
REAL(WP), INTENT(INOUT) :: disk_size, corona_size, Tcorona, Tdisk, tau, QPO_frequency, DHext, eta_frac
REAL(WP) Emin_adim, Emax_adim, Emin, Emax
! REAL :: param(5), photar(meshlog-1), EAR(0:meshlog-1), photer(meshlog-1)
REAL :: Tdisk2, h_T, Tdisk3, Tdskb
REAL :: photardskb(meshlog-2), eardskb(0:meshlog-2), photerdskb(meshlog-2), photardskb2(meshlog-2), photerdskb2(meshlog-2)
REAL :: photardskb3(meshlog-2), photerdskb3(meshlog-2)
! integer :: near
INTEGER :: ifl
! CHARACTER(4) method
! array arguments
REAL(WP) :: x2(meshlog) , x_use(meshlog-2), xlog(meshlog), xlog_use(meshlog-2), ddskb_t(meshlog-2), bbdisk(meshlog-2)
! parameters steady state solution
REAL(WP) :: c2(meshlog-2), nc, c5, c6, c11, Nesc(meshlog-2), Vc, omega, dxlog, xtot_low, xtot_up, dx
REAL(WP) :: KN_corr_interpol(meshlog-2), ccsub(meshlog-2), bbdisk2(meshlog-2), bbdisk3(meshlog-2)
INTEGER Ntri, columns_CC, INFO, nestsol, Nsss, Nreal, Nimag
REAL(WP) :: L(meshlog-3), U(meshlog-3), D(meshlog-2), n0(meshlog)
! !parameters perturbative solution
REAL(WP) :: Nescp(meshlog), c2p(meshlog), dn0log(meshlog), dn02log(meshlog), factor1
REAL(WP) :: KNp_int(meshlog), Hexo0, eta_max, eta, transf, xlog_square(meshlog), xlog_use_square(meshlog-2)
REAL(WP) :: x_use_square(meshlog-2), x2square(meshlog)
REAL(WP) :: Q1(meshlog-2), Q2(meshlog-2), Q3(meshlog-2), stau_kn(meshlog-2)
REAL(WP) :: A1(meshlog-2), A2(meshlog-2), p1, x_withunit(meshlog), rad_sphere, surf, corona_simps
REAL(WP) :: corona_Lum, corona_Lum_out, vect4(meshlog), Iex01, Iex02, Iex03, Nescp_use(meshlog-2)
REAL(WP) :: area, tc, to_phys, vect1(meshlog-2), vect2(meshlog-2), vect3(meshlog-2), xlog_trans(meshlog)
REAL(WP) :: SOLsss(meshlog), SOLreal(meshlog), SOLimag(meshlog), Tsss(meshlog+4), Ssss(meshlog+4)
REAL(WP) :: Treal(meshlog+4), Sreal(meshlog+4), Timag(meshlog+4), Simag(meshlog+4)
COMPLEX(WP), DIMENSION(meshlog-3) :: Lp, Up
COMPLEX(WP), DIMENSION(meshlog-2) :: Dp, sol_ongrid, k0n
COMPLEX(WP), DIMENSION(meshlog) :: solution, auxiliar, auxiliar2, auxiliar3
COMPLEX(WP) :: dTe, dTs, dTesimps1, dTesimps2, dTssimps, denom, k0, k1, k2
REAL(WP) :: dTesimps1_real, dTesimps2_real, dTesimps1_imag, dTesimps2_imag, dTssimps_real, dTssimps_imag
REAL(WP) :: dTe_mod, dTs_mod, dTe_arg, dTs_arg, Hexo0_out, eta_int
call sco_constants(dist, mass, time, energy_norm, eV2J, keV2J, MeV2J, J2keV, Etrans, kbol, hplanck, c, cc2, me, sigma, stau)
!---We define the energy regime for the BVP solution
Emin_adim = 1.e-6
Emax_adim = 40.
Emin = Emin_adim * Tcorona
Emax = Emax_adim * Tcorona
!---The output is an array, whose components are evenly spaced numbers between log(Emin/Tcorona) and log(Emax/Tcorona
CALL sco_linspace(log(Emin_adim), log(Emax_adim), meshlog, xlog)
!---We define the grid of energy spaced evenly on a log scale
x2 = exp(xlog)
!---We transform input parameters to the internal units
Tcorona = Tcorona * Etrans
Tdisk = Tdisk * Etrans
disk_size = disk_size * 1000. * (1. / dist)
corona_size = corona_size * 1000. * (1. / dist)
QPO_frequency = QPO_frequency * time
!---We define the energy step size for the numerical integration
dxlog = xlog(4) - xlog(3)
dx = x2(4) - x2(3)
!---We define the integration limits for the energy averaged rms
xtot_low = 2. * Etrans / Tcorona
xtot_up = 60. * Etrans / Tcorona
DO I = 1, meshlog-2
xlog_use(i) = xlog(i+1)
x_use(i) = x2(i+1)
ENDDO
!--------BEGIN OF: construction of the steady state solution----------
Ntri=meshlog-2 !dimension of the x_use and then it will be the dimension of the tridiagonal matrix &
!the constant vector of the system that we need to solve
CALL sco_par(disk_size, corona_size, Tcorona, Tdisk, tau, QPO_frequency, Ntri, x_use, c2, nc, c5, c6, c11, Nesc, &
Vc, KN_corr_interpol)
omega = 2.0 * PI * QPO_frequency
!---Preparing to solve the steady state Kompaneets equation (SS) after discretization
DO I=2, Ntri
L(I-1) = 1. / (dxlog **2) - (x_use(i)-1.) / (2. * dxlog) ! sub-diagonal elements
U(I-1) = 1. / (dxlog **2) + (x_use(i-1)-1.) / (2. * dxlog) ! super-diagonal elements
ENDDO
D = -2. + 2. * x_use - c2 - 2. / (dxlog **2) ! diagonal elements: x-dependent
!---Array that has the boundaries of the energy channels
eardskb(0) = real(x_use(1)*(Tcorona/Etrans) / sqrt(exp(dxlog)))
DO I = 1, Ntri
eardskb(i) = real(eardskb(i-1) * exp(dxlog))
ENDDO
Tdskb = real(Tdisk/Etrans)
ifl = 0
CALL xsdskb(eardskb, Ntri, Tdskb, ifl, photardskb, photerdskb)
DO i = 1, Ntri
bbdisk(i) = photardskb(i) / ( (Tcorona/Etrans)**2 * 1.0344E-3 * (eardskb(i)-eardskb(i-1))) ! the subroutine gives as dskb(E) * dE * 1.0344E-3
ccsub (i) = -bbdisk(i)
ENDDO
!---Derivative of dskb with respect to Tdisk
ifl = 0
h_T = real(Tdisk/Etrans) * 1.e-2
Tdisk2 = real(Tdisk/Etrans) + h_T
Tdisk3 = real(Tdisk/Etrans)-h_T
CALL xsdskb(eardskb, Ntri, Tdisk2, ifl, photardskb2, photerdskb2)
DO i = 1, Ntri
bbdisk2(i) = photardskb2(i) / ( (Tcorona/Etrans)**2 * 1.0344E-3 * (eardskb(i)-eardskb(i-1))) ! the subroutine gives as dskb(E) * dE * 1.0344E-3
ENDDO
CALL xsdskb(eardskb, Ntri, Tdisk3, ifl, photardskb3, photerdskb3)
DO i = 1, Ntri
bbdisk3(i) = photardskb3(i) / ( (Tcorona/Etrans)**2 * 1.0344E-3 * (eardskb(i)-eardskb(i-1))) ! the subroutine gives as dskb(E) * dE * 1.0344E-3
ENDDO
ddskb_T = (bbdisk2-bbdisk3)/(2*h_T)
columns_CC=1
CALL dgtsv(Ntri, columns_CC, L, D, U, CCsub, Ntri, INFO) ! on exit, CC has the solution of (LDU)*X=CC
n0(1) = 0.0
n0(meshlog) = 0.0
DO I =1, Ntri
n0(I+1) = CCsub(I)
ENDDO
!--- n0 is the steady state solution
!---BEGIN OF: construction of the linearized equation
CALL sco_par(disk_size, corona_size, Tcorona, Tdisk, tau, QPO_frequency, meshlog, x2, c2p, &
nc, c5, c6, c11, Nescp, Vc, KNp_int)
!---We define the first and second order derivative of n0
dn0log(1) = (-n0(3) + 4. * n0(2) - 3. * n0(1)) / (2. * dxlog)
dn0log(meshlog) = (3. * n0(meshlog) - 4. * n0(meshlog-1) + n0(meshlog-2)) / (2. * dxlog)
dn02log(1) = (2. * n0(1) - 5. * n0(2) + 4. * n0(3) - n0(4))
dn02log(meshlog) = (2. * n0(meshlog) + 5. * n0(meshlog-1) + 4. * n0(meshlog-2) + n0(meshlog-3))
DO I = 1, Ntri
dn0log(i+1) = (n0(i+2) - n0(i)) / (2.0 * dxlog)
dn02log(i+1) = (n0(i+2) - 2.0 * n0(i+1) + n0(i)) / (dxlog * dxlog)
x_use_square(i) = (x_use(i))**2
xlog_use_square(i) = (xlog_use(i))**2
Nescp_use(i) = Nescp(i+1)
ENDDO
DO I = 1, meshlog
x2square(i) = (x2(i))**2
xlog_square(i) = (xlog(i))**2
ENDDO
Dp= dcmplx(2. + (dxlog**2) * bbdisk / CCsub , -c5 * dxlog**2 )
DO I=1, Ntri -1
Lp (i) = dcmplx(-1. + (x_use(i+1)-1) * dxlog/2. + (dn0log(i+2) * dxlog)/ CCsub(i+1) , 0)
Up (i)= dcmplx(-1. - (x_use(i)-1) * dxlog/2. - (dn0log(i+1) * dxlog)/ CCsub(I) , 0)
ENDDO
stau_kn = (3. / 4.) * stau * KN_corr_interpol ! klein Nishina correction
Q2 = x_use_square * x_use * CCsub * stau_kn
Q1 = CCsub * x_use_square * stau_kn
Q3 = CCsub * x_use_square / Nescp_use
vect1 = CCsub * x_use_square * stau_kn
CALL sco_SIMPSON(Ntri,vect1,xlog_use,Iex01)
vect2 = CCsub * x_use * x_use_square * stau_kn
CALL sco_SIMPSON(Ntri,vect2,xlog_use,Iex02)
vect3 = CCsub * x_use_square/ Nescp_use
CALL sco_SIMPSON(Ntri,vect3,xlog_use,Iex03)
factor1 = ((Tcorona ** 3) * nc) / (me * c)
Hexo0 = factor1 * (4. * Iex01 - Iex02)
eta_max = c11 / Iex03
eta = eta_frac * eta_max
denom = dcmplx(4. * factor1 * Iex01 , - (3. / 2.) * omega * Tcorona) ! denominator in eq (A6) of Karpouzas et al 2019
eta_int = eta ! we record eta_int (\tilde\eta)
k0 = DHext * Hexo0 / denom ! first term in eq (A6) of Karpouzas et al 2019
k1 = -4. * factor1 / denom ! multiplicative factor of the 3rd term in eq (A6) of Karpouzas et al 2019
k2 = factor1 / denom ! multiplicative factor of the 2nd term in eq (A6) of Karpouzas et al 2019
DO I = 1, Ntri
A1(I) = (dxlog**2) * (-2. -dn0log(i+1)/ccsub(i) + dn02log(I+1) / CCsub(I))
ENDDO
A2 = (dxlog**2) *(Tdisk/Etrans) * (ddskb_T / CCsub)
p1 = c6 * eta ! multiplicative factor in eq (A7) of Karpouzas et al 2019
k0n = k0 ! the mppinv needs a vector
!---Calculation of the solution
CALL sco_MPPINV(Lp, Dp, Up, p1, A1, A2, k0n, k1, k2, Q1, Q2, Q3, dxlog, Ntri, sol_ongrid)
solution(1) = (0,0) ! boundary conditions
solution(meshlog) = (0,0)
DO I = 1, Ntri
solution(i+1) = sol_ongrid(i)
ENDDO
SOLreal = REALPART(solution)
SOLimag = IMAGPART(solution)
x_withunit = x2 * Tcorona ! energy grid in internal units
rad_sphere = disk_size + corona_size
surf = 4. * pi * rad_sphere ** 2
vect4 = x2 * n0/ Nescp
CALL sco_SIMPSON(meshlog,vect4,xlog,corona_simps)
!corona_Lum = surf * (1. - eta) * c * nc * Tcorona ** 2 * corona_simps ! this is probably wrong
!corona_Lum_out = corona_Lum * keV2J / (time * Etrans)
!---Vector to go from grid units to physical units in ph cm^-2 s^-1 keV^-1 @ 1kpc
area = 4. * pi * (3e19/dist) **2
tc = disk_size / (c * tau)
to_phys = (1.-eta)*Vc*nc/area/(tau+tau**2/3)/tc
to_phys = to_phys* 1e-4 / dist**2 /time * Etrans
transf = Etrans/Tcorona ! parameter used to transform the input grid from keV to internal units
xlog_trans = x2/transf
nestsol= meshlog + 4
SOLsss = n0*transf*to_phys
!---Calculation of the fractional amplitude of the corona temperature oscillation
auxiliar = x2square * n0 * solution * (3. / 4.) * stau * KNp_int
CALL sco_SIMPSON(size(xlog),realpart(auxiliar),xlog,dTesimps1_real)
CALL sco_SIMPSON(size(xlog),imagpart(auxiliar),xlog,dTesimps1_imag)
dTesimps1 = complex(dTesimps1_real,dTesimps1_imag)
auxiliar2 = x2 * auxiliar
CALL sco_SIMPSON(size(xlog),realpart(auxiliar2),xlog,dTesimps2_real)
CALL sco_SIMPSON(size(xlog),imagpart(auxiliar2),xlog,dTesimps2_imag)
dTesimps2 = complex(dTesimps2_real,dTesimps2_imag)
dTe = k0 + k1 * dTesimps1 + k2 * dTesimps2
dTe_mod = cdabs(dTe)
dTe_arg = atan2(imagpart(dTe),realpart(dTe))
!---Calculation of the fractional amplitude of the seed source temperature oscillation
auxiliar3 = x2square * n0 * solution / Nescp
CALL sco_SIMPSON(size(xlog),realpart(auxiliar3),xlog,dTssimps_real)
CALL sco_SIMPSON(size(xlog),imagpart(auxiliar3),xlog,dTssimps_imag)
dTssimps=complex(dTssimps_real,dTssimps_imag)
dTs = p1 * dTssimps
dTs_mod = cdabs(dTs)
dTs_arg = atan2(imagpart(dTs),realpart(dTs))
!---Calculation of the external heating rate at thermal equilibrium
Hexo0_out = Hexo0 * keV2J / (time * Etrans) ! units : Joules per second
!---Interpolating splines
CALL sco_InterpolatedUnivariateSpline(meshlog,xlog_trans,SOLsss,nestsol,Nsss,Tsss,Ssss)
CALL sco_InterpolatedUnivariateSpline(meshlog,xlog_trans,SOLreal,nestsol,Nreal,Treal,Sreal)
CALL sco_InterpolatedUnivariateSpline(meshlog,xlog_trans,SOLimag,nestsol,Nimag,Timag,Simag)
ENDSUBROUTINE