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plot_fig3.py
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plot_fig3.py
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#import:
import matplotlib
matplotlib.use('Agg')
from matplotlib.pyplot import *
import numpy as np
from numpy import linspace, logspace, empty, zeros, ones, array, fromfile
from numpy import pi, exp, log, sqrt, sin, cos, arccos, arctan2
from numpy import absolute, sign, floor, ceil,ma
# from numpy.polynomial.laguerre import laggauss
# from numpy.polynomial.legendre import leggauss
# from scipy.interpolate import interp1d
# from scipy.special import kn
# from bisect import bisect
colors=[
# 'xkcd:brownish red',
'xkcd:red',
# 'xkcd:orange',
# 'xkcd:dark yellow',
# 'xkcd:dark yellow green',
'xkcd:deep green',
# 'xkcd:dark cyan',
'xkcd:blue',
'xkcd:purple'
]
#physical constants:
evere=.5109989e6 # electron volts in elecron rest energy
G=13275412528e1 # G*M_sol in km^3/s^2
c=299792458e-3 # speed of light in km/s
NPhi = 550#500#150#128*2#120#128#120 # Number of equidistant phase points
phi, phi_weight=linspace(0,360,num=NPhi,endpoint=True,retstep=True) #Size of spacing between samples = phi_weight
philag = ma.array(zeros(NPhi))
sphilag = ma.array(zeros(NPhi))
chi = ma.array(zeros(NPhi))
chi_sph = ma.array(zeros(NPhi))
chi_old = ma.array(zeros(NPhi))
chi_pri = ma.array(zeros(NPhi))
chi_one = ma.array(zeros(NPhi))
chi_pri_sph = ma.array(zeros(NPhi))
chi_nul = ma.array(zeros(NPhi))
chi_nul_sph = ma.array(zeros(NPhi))
diffchi_old = ma.array(zeros(NPhi))
diffchi = ma.array(zeros(NPhi))
mu_0 = ma.array(zeros(NPhi))
nu=[600]*4 # star rotation frequency in Hz
# nu=[600]*4# [1,200,400,600] # star rotation frequency in Hz
M=1.4#1.6#1.4 # star mass in solar masses
# R_g=M*2.95 # gravitational Schwarzschild radius
R_0=12.0 # equatorial radius of the star in kilometers
# incl = [40*pi/180]*4 # [20*pi/180,40*pi/180,60*pi/180,80*pi/180] #pi*4/18#1.5 # line of sight colatitude #in radians
# incl = [50*pi/180]*4 # [20*pi/180,40*pi/180,60*pi/180,80*pi/180] #pi*4/18#1.5 # line of sight colatitude #in radians
incl = [15*pi/180,30*pi/180,45*pi/180,60*pi/180] #pi*4/18#1.5 # line of sight colatitude #in radians
# theta = [15*pi/180,30*pi/180,45*pi/180,60*pi/180]# [pi/4]
# theta = [3*pi/18]*4 #[15*pi/180,30*pi/180,45*pi/180,60*pi/180]# [pi/4]
theta = [35*pi/180]*4 #[15*pi/180,30*pi/180,45*pi/180,60*pi/180]# [pi/4]
#,3*pi/18,6*pi/18,pi/18]
NSpots = len(theta)
#rc("text", usetex=True)
figA = figure(figsize=(14,12), dpi=300) #8,6
# dads78
#rc("font", family="serif")
#rc("font",serif="Times")
matplotlib.pyplot.figure(1)
lbfontsz = 25
lwidth= 2.5
rc("xtick", labelsize=lbfontsz)
rc("ytick", labelsize=lbfontsz)
rc("axes", linewidth=lwidth)
#figA.clear()
matplotlib.pyplot.rcParams.update({'axes.titlesize': lbfontsz})
matplotlib.pyplot.rcParams.update({'font.size': lbfontsz})
matplotlib.pyplot.rcParams.update({'lines.linewidth': lwidth})
matplotlib.pyplot.rcParams.update({'ytick.major.width': lwidth})
matplotlib.pyplot.rcParams.update({'xtick.major.width': lwidth})
matplotlib.pyplot.rcParams.update({'ytick.major.size': 10.0})
matplotlib.pyplot.rcParams.update({'xtick.major.size': 10.0})
matplotlib.pyplot.rcParams.update({'font.family': 'serif'})
#matplotlib.pyplot.rcParams.update({'font.serif': 'Times'})
matplotlib.pyplot.subplots_adjust(wspace=0, hspace=0)
plotAc=figA.add_subplot(2,1,1,yscale='linear')
plotAd=figA.add_subplot(2,1,2,) #
def Beloborodov(cos_psi):
"""Beloborodov's approximation for cos_alpha(cos_psi) light bending function
takes the cos psi
returns the cos alpha and its derivative
"""
return 1. + (cos_psi - 1.)/redshift**2 ,1./redshift**2
def Poutanen(u,y):
return ( 1 - u )*y*( 1 + u*u*y*y/112 - np.e/1e2*u*y*( np.log( 1 - y/2 ) + y/2 ) )
def Schwarzschild(R,alpha):
"""Schwarzschild exact relation between the \psi and \\alpha angles, where
\\alpha is the angle between radius vector of the spot and the direction of the outgoing photon near the surface
and \psi is the angle between normal and light propagation at the limit of infinite distance.
For given distance from the mass center and the emission angle \\alpha
this function returns two numbers:
the corresponding angle \psi
and the time lag over against the fotons emited with zero impact parameter at the radius.
"""
kx,wx=IntBend
eps=(1+kx[0])/4e2
u=R_g/R
b=sin(alpha)/sqrt(1-u)*R # impact parameter
if 2*alpha>pi+eps:
cos_3eta=sqrt(27)*R_g/2/b
if cos_3eta > 1:
return pi+2*eps,0 # the timelag
closest_approach=-2*b/sqrt(3)*cos(arccos(cos_3eta)/3 + 2*pi/3)
psi_max, lag_max= Schwarzschild(closest_approach,pi/2.)
psi_min, lag_min= Schwarzschild(R,pi-alpha)
psi=2*psi_max - psi_min
lag=2*lag_max - lag_min # + 2*(R - closest_approach + R_g*log((R - R_g)/(closest_approach - R_g)))/c
if psi>pi:
return pi+eps,lag
else:
psi=0
lag=(R_e - R + R_g*log( (R_e - R_g)/(R - R_g) ) )/c
for i in range(NBend):
ex=(kx[i]+1)/2
q=(2. - ex*ex - u*(1 - ex*ex)**2/(1 - u))*sin(alpha)**2
sr=sqrt(cos(alpha)**2+ex*ex*q)
if 2*alpha>pi-eps:
dpsi=b/R/sqrt(q)*wx[i] #*2/2
else:
dpsi=ex*b/R/sr*wx[i] #*2/2
dlag=dpsi*b/c/(1+sr) #*2/2
psi+= dpsi
lag+= dlag
return psi,lag
# flattening=0
def foldchi(c):
for i in range(len(c)):
c[i] = (c[i]*180/pi+90)%180-90
if abs(c[i]-c[i-1])>90:
c[i] = ma.masked
return c
if True:
# sphere case
for p in range(NSpots):
sin_i=sin(incl[p])
cos_i=cos(incl[p])
sin_theta=sin(theta[p])
cos_theta=cos(theta[p])
R=R_0
dR=0.0
R_g=M*2.95
u = R_g/R
redshift=1.0/sqrt(1.0 - u) # 1/sqrt(1-R_g/R) = 1+ z = redshift
f=0
sin_gamma=0.0
cos_gamma=1.0
beta=2*pi*nu[p]*R*redshift*sin_theta/c
Gamma=1.0/sqrt(1.0 - beta**2)
Gamma1= (1.0-sqrt(1.0 - beta**2) )/ beta
print('theta: ',theta[p],'gamma: ',arctan2(f,1.0)*180/pi)
for t in range(NPhi):
# if True: # find mu
phi0 = phi[t]*pi/180+pi
sin_phi=sin(phi0)
cos_phi=cos(phi0)
cos_psi=cos_i*cos_theta + sin_i*sin_theta*cos_phi
sin_psi=sqrt(1. - cos_psi**2)
cos_alpha = 1.0 - Poutanen(u, 1.0 - cos_psi) # insert exact formula here
sin_alpha = sqrt(1. - cos_alpha**2)
sin_alpha_over_sin_psi= sin_alpha/sin_psi if sin_psi > 1e-4 else 1./redshift
cos_xi = - sin_alpha_over_sin_psi*sin_i*sin_phi
delta = 1./Gamma/(1.-beta*cos_xi)
cos_sigma = cos_gamma*cos_alpha + sin_alpha_over_sin_psi*sin_gamma*(cos_i*sin_theta - sin_i*cos_theta*cos_phi)
sin_sigma = sqrt(1. - cos_sigma**2)
mu0=delta*cos_sigma # cos(sigma')
# Omega=dS[p]*mu0*redshift**2*dcos_alpha #*Gamma*R*R/cos_gamma #
sin_chi_0= - sin_theta*sin_phi # times sin psi
cos_chi_0=sin_i*cos_theta - sin_theta*cos_i*cos_phi # times sin psi
chi_0=arctan2(sin_chi_0,cos_chi_0)
# chi_0 = 0
# sin_chi_1=sin_gamma*sin_i*sin_phi*sin_alpha_over_sin_psi #times sin alpha sin sigma
# cos_chi_1=cos_gamma - cos_alpha*cos_sigma #times sin alpha sin sigma
# chi_1=arctan2(sin_chi_1,cos_chi_1)
# sin_lambda=sin_theta*cos_gamma - sin_gamma*cos_theta
# cos_lambda=cos_theta*cos_gamma + sin_theta*sin_gamma
# cos_eps = sin_alpha_over_sin_psi*(cos_i*sin_lambda - sin_i*cos_lambda*cos_phi + cos_psi*sin_gamma) - cos_alpha*sin_gamma
# # this line is the longest one
# # alt_cos_eps=(cos_sigma*cos_gamma - cos_alpha)/sin_gamma # legit! thanks God I checked it!
# # sin_chi_prime=cos_eps*mu0*Gamma*beta # times something
# sin_chi_prime=cos_eps*mu0*delta*Gamma*beta*(1-Gamma1*cos_xi)# times something
# # cos_chi_prime=1. - cos_sigma**2 /(1. - beta*cos_xi) # times the samething
# cos_chi_prime=sin_sigma**2 - Gamma*mu0**2*beta*cos_xi*(1 - Gamma1*cos_xi) # times the samething
# chi_prime=arctan2(sin_chi_prime,cos_chi_prime)
# chi[t] = foldchi(chi_0+chi_prime+ chi_1)
cos_eps_sph = sin_alpha_over_sin_psi*(cos_i*sin_theta - sin_i*cos_theta*cos_phi)
sin_chi_prime_sph=cos_eps_sph*mu0*delta*Gamma*beta*(1-Gamma1*cos_xi)# times something
cos_chi_prime_sph=sin_alpha**2 - Gamma*mu0**2*beta*cos_xi*(1 - Gamma1*cos_xi) # times the samething
chi_prime_sph= arctan2(sin_chi_prime_sph,cos_chi_prime_sph)
chi_sph[t] = (chi_0+ chi_prime_sph)
nup = nu[p]
M_bar = M/1.4
nu_cr = 1278*sqrt(M_bar)*(10/R)**1.5
nu_bar = nup/nu_cr
R_e = R_0*(0.9766 + 0.025/(1.07- nu_bar)+0.07*M_bar**1.5*nu_bar**2)
a1 = 0.001*M_bar**1.5
a0 = 1.0 - a1/1.1
a2 = 10*a1
M_prime = M*(a0 + a1/(1.1-nu_bar)+ a2*nu_bar**2)
print(M_prime/M)
print(R_e/R_0)
R_g=M_prime*2.95
sin_i=sin(incl[p])
cos_i=cos(incl[p])
Omega_bar=2*pi*nu[p]*sqrt(2*R_e**3/R_g)/c
# print('_O_^2',Omega_bar**2,'_O_',Omega_bar)
flattening=(0.788-0.515*R_g/R_e)*Omega_bar**2
# print(R_e*(1-flattening))
sin_theta=sin(theta[p])
cos_theta=cos(theta[p])
R=R_e*(1 - flattening*cos_theta**2)
dR=2*R_e*flattening*cos_theta*sin_theta # dR / d\theta
u = R_g/R
redshift=1.0/sqrt(1.0 - u) # 1/sqrt(1-R_g/R) = 1+ z = redshift
f=redshift/R*dR
sin_gamma=f/sqrt(1 + f**2) # angle gamma is positive towards the north pole
cos_gamma=1.0/sqrt(1 + f**2)
beta=2*pi*nu[p]*R*redshift*sin_theta/c
Gamma=1.0/sqrt(1.0 - beta**2)
Gamma1= (1.0-sqrt(1.0 - beta**2) )/ beta
print('theta: ',theta[p],'gamma: ',arctan2(f,1.0)*180/pi)
for t in range(NPhi):
# if True: # find mu
phi0 = phi[t]*pi/180+pi
sin_phi=sin(phi0)
cos_phi=cos(phi0)
cos_psi=cos_i*cos_theta + sin_i*sin_theta*cos_phi
sin_psi=sqrt(1. - cos_psi**2)
cos_alpha = 1.0 - Poutanen(u, 1.0 - cos_psi) # insert exact formula here
sin_alpha = sqrt(1. - cos_alpha**2)
sin_alpha_over_sin_psi= sin_alpha/sin_psi if sin_psi > 1e-4 else 1./redshift
cos_xi = - sin_alpha_over_sin_psi*sin_i*sin_phi
delta = 1./Gamma/(1.-beta*cos_xi)
cos_sigma = cos_gamma*cos_alpha + sin_alpha_over_sin_psi*sin_gamma*(cos_i*sin_theta - sin_i*cos_theta*cos_phi)
sin_sigma = sqrt(1. - cos_sigma**2)
mu0=delta*cos_sigma # cos(sigma')
mu_0[t]=(1-mu0)/(1+3.582*mu0)*117
# Omega=dS[p]*mu0*redshift**2*dcos_alpha #*Gamma*R*R/cos_gamma #
sin_chi_0= - sin_theta*sin_phi # times sin psi
cos_chi_0=sin_i*cos_theta - sin_theta*cos_i*cos_phi # times sin psi
chi_0=arctan2(sin_chi_0,cos_chi_0)
# chi_0 = 0
sin_chi_1=sin_gamma*sin_i*sin_phi*sin_alpha_over_sin_psi #times sin alpha sin sigma
cos_chi_1=cos_gamma - cos_alpha*cos_sigma #times sin alpha sin sigma
chi_1=arctan2(sin_chi_1,cos_chi_1)
sin_lambda=sin_theta*cos_gamma - sin_gamma*cos_theta
cos_lambda=cos_theta*cos_gamma + sin_theta*sin_gamma
cos_eps = sin_alpha_over_sin_psi*(cos_i*sin_lambda - sin_i*cos_lambda*cos_phi + cos_psi*sin_gamma) - cos_alpha*sin_gamma
# this line is the longest one
# alt_cos_eps=(cos_sigma*cos_gamma - cos_alpha)/sin_gamma # legit! thanks God I checked it!
# sin_chi_prime=cos_eps*mu0*Gamma*beta # times something
sin_chi_prime=cos_eps*mu0*delta*Gamma*beta*(1-Gamma1*cos_xi)# times something
# cos_chi_prime=1. - cos_sigma**2 /(1. - beta*cos_xi) # times the samething
cos_chi_prime=sin_sigma**2 - Gamma*mu0**2*beta*cos_xi*(1 - Gamma1*cos_xi) # times the samething
chi_prime=arctan2(sin_chi_prime,cos_chi_prime)
chi[t] = (chi_0+chi_prime+ chi_1)
nup = nu[p]
M_bar = M/1.4
nu_cr = 1278*sqrt(M_bar)*(10/R)**1.5
nu_bar = nup/nu_cr
R_e = R_0#*(0.9766 + 0.025/(1.07- nu_bar)+0.07*M_bar**1.5*nu_bar**2)
a1 = 0.001*M_bar**1.5
a0 = 1.0 - a1/1.1
a2 = 10*a1
M_prime = M#*(a0 + a1/(1.1-nu_bar)+ a2*nu_bar**2)
print(M_prime/M)
print(R_e/R_0)
R_g=M_prime*2.95
sin_i=sin(incl[p])
cos_i=cos(incl[p])
Omega_bar=2*pi*nu[p]*sqrt(2*R_e**3/R_g)/c
# print('_O_^2',Omega_bar**2,'_O_',Omega_bar)
flattening=(0.788-0.515*R_g/R_e)*Omega_bar**2
# print(R_e*(1-flattening))
sin_theta=sin(theta[p])
cos_theta=cos(theta[p])
R=R_e*(1 - flattening*cos_theta**2)
dR=2*R_e*flattening*cos_theta*sin_theta # dR / d\theta
u = R_g/R
redshift=1.0/sqrt(1.0 - u) # 1/sqrt(1-R_g/R) = 1+ z = redshift
f=redshift/R*dR
sin_gamma=f/sqrt(1 + f**2) # angle gamma is positive towards the north pole
cos_gamma=1.0/sqrt(1 + f**2)
beta=2*pi*nu[p]*R*redshift*sin_theta/c
Gamma=1.0/sqrt(1.0 - beta**2)
Gamma1= (1.0-sqrt(1.0 - beta**2) )/ beta
print('theta: ',theta[p],'gamma: ',arctan2(f,1.0)*180/pi)
for t in range(NPhi):
# if True: # find mu
phi0 = phi[t]*pi/180+pi
sin_phi=sin(phi0)
cos_phi=cos(phi0)
cos_psi=cos_i*cos_theta + sin_i*sin_theta*cos_phi
sin_psi=sqrt(1. - cos_psi**2)
cos_alpha = 1.0 - Poutanen(u, 1.0 - cos_psi) # insert exact formula here
sin_alpha = sqrt(1. - cos_alpha**2)
sin_alpha_over_sin_psi= sin_alpha/sin_psi if sin_psi > 1e-4 else 1./redshift
cos_xi = - sin_alpha_over_sin_psi*sin_i*sin_phi
delta = 1./Gamma/(1.-beta*cos_xi)
cos_sigma = cos_gamma*cos_alpha + sin_alpha_over_sin_psi*sin_gamma*(cos_i*sin_theta - sin_i*cos_theta*cos_phi)
sin_sigma = sqrt(1. - cos_sigma**2)
mu0=delta*cos_sigma # cos(sigma')
# Omega=dS[p]*mu0*redshift**2*dcos_alpha #*Gamma*R*R/cos_gamma #
sin_chi_0= - sin_theta*sin_phi # times sin psi
cos_chi_0=sin_i*cos_theta - sin_theta*cos_i*cos_phi # times sin psi
chi_0=arctan2(sin_chi_0,cos_chi_0)
# chi_0 = 0
sin_chi_1=sin_gamma*sin_i*sin_phi*sin_alpha_over_sin_psi #times sin alpha sin sigma
cos_chi_1=cos_gamma - cos_alpha*cos_sigma #times sin alpha sin sigma
chi_1=arctan2(sin_chi_1,cos_chi_1)
sin_lambda=sin_theta*cos_gamma - sin_gamma*cos_theta
cos_lambda=cos_theta*cos_gamma + sin_theta*sin_gamma
cos_eps = sin_alpha_over_sin_psi*(cos_i*sin_lambda - sin_i*cos_lambda*cos_phi + cos_psi*sin_gamma) - cos_alpha*sin_gamma
# this line is the longest one
# alt_cos_eps=(cos_sigma*cos_gamma - cos_alpha)/sin_gamma # legit! thanks God I checked it!
# sin_chi_prime=cos_eps*mu0*Gamma*beta # times something
sin_chi_prime=cos_eps*mu0*delta*Gamma*beta*(1-Gamma1*cos_xi)# times something
# cos_chi_prime=1. - cos_sigma**2 /(1. - beta*cos_xi) # times the samething
cos_chi_prime=sin_sigma**2 - Gamma*mu0**2*beta*cos_xi*(1 - Gamma1*cos_xi) # times the samething
chi_prime=arctan2(sin_chi_prime,cos_chi_prime)
chi_old[t] = (chi_0+chi_prime+ chi_1)
diffchi = chi - chi_sph
diffchi_old = chi_old - chi_sph
# print(diffchi*180/pi)
plotAc.plot(phi,foldchi(chi),"-",linewidth=3,color=colors[p] )
plotAc.plot(phi,foldchi(chi_sph),"--",linewidth=2,color=colors[p])
plotAd.plot(phi,foldchi(diffchi),"-",linewidth=3,color=colors[p])
if p==3:
# plotAc.plot(phi,mu_0,"--",linewidth=1.5,color='orange')
# plotAd.plot(phi,mu_0,"--",linewidth=1.5,color='orange')
# plotAc.plot(phi,foldchi(chi_old),"--",linewidth=1.5,color='black')
# plotAd.plot(phi,diffchi_old*180/pi,"--",linewidth=1.5,color='black')
# plotAd.plot(phi,(chi - chi_old)*180/pi,"--",linewidth=1.5,color='purple')
# plotAc.plot(phi,[0]*NPhi,"--",linewidth=1,color="brown")
# plotAd.plot(phi,[0]*NPhi,"--",linewidth=1,color="brown")
# print((chi - chi_old)*180/pi)
pass
fontsize = 30
# plotAc.xaxis.set_major_formatter(matplotlib.pyplot.NullFormatter())
# plotAd.xaxis.set_major_formatter(matplotlib.pyplot.NullFormatter())
plotAd.set_xticks([0,60,120,180,240,300,360])
plotAc.set_xticks([36,108,180,252,324])
plotAd.set_xticks([36,108,180,252,324])
plotAd.set_xticklabels(["-0.4","-0.2","0","0.2","0.4"])
plotAc.set_yticks([0,-45,-90,45,90])
# plotAd.set_yticks([0,-10,-20,20,10])
plotAd.set_yticks([0,-10,-20,20,10])
plotAd.margins(x =0)
plotAd.set_ylim((-27,27))
plotAc.margins(x =0,y=0)
plotAc.tick_params(axis="both", which="both", pad=10,top=True,right = True)#
plotAd.tick_params(axis="both", which="both", pad=10,top=True,right = True)#
plotAc.tick_params(axis='x', which='major', bottom = True, labelbottom=False)
plotAc.tick_params(axis='both', which='major', labelsize=fontsize,direction='in')
plotAd.tick_params(axis='both', which='major', labelsize=fontsize,direction='in')
# plotAd.set_ylabel(r'$\chi_{\mathrm{obl}}-\chi_{\mathrm{sph}}$',fontsize=fontsize)
plotAd.set_xlabel(r'$\phi/(2\pi)}$',fontsize=fontsize)
plotAd.set_ylabel(r'$\chi_{\mathrm{obl}}-\chi_{\mathrm{sph}}\,\mathrm{[deg]}$',fontsize=fontsize)
plotAc.set_ylabel(r"$\chi\,\mathrm{[deg]}$",fontsize=fontsize)
# figA.tight_layout()
figA.subplots_adjust(left=0.15)
figA.savefig('fig3.pdf',bbox_inches='tight')#.format(e))
figA.clf()