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astro.py
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"""
This is a module for the basics of the astrophysical expressions
"""
from __future__ import division
import numpy as np
from numpy import pi, sqrt, exp, power, log, log10
from scipy.integrate import trapz
from scipy.special import erf, lambertw
from inspect import getargspec
import os
# related to the map handling
import healpy as hp
from astropy_healpix import HEALPix
from astropy.coordinates import SkyCoord
# from astropy.io import fits
from astropy.coordinates import Galactic
# from astropy import units as u
import tools as tl
import constants as ct
import particle as pt
import ska as sk
# Dark matter and galactic functions:
def rho_NFW(r, rs, rhos):
"""
The density profile [GeV/cm**3] of an NFW halo.
Parameters
----------
r : the distance from the center [kpc]
rs : the NFW r_s parameter [kpc]
rhos : the NFW rho_s parameter [GeV/cm**3]
"""
res = rhos/(r/rs)/(1.+r/rs)**2
return res
def rho_MW(r, DM_profile="NFW"):
"""
The density [GeV/cm**3] of the Milky Way.
Parameters
----------
x : distance from the center [kpc]
"""
if DM_profile == "NFW":
rs = ct._rs_of_MW_
rho_at_sun = ct._rho_local_DM_
rsun = ct._Sun_to_gal_center_
rhos = rho_at_sun * (rsun/rs) * (1. + rsun/rs)**2
res = rho_NFW(r, rs, rhos)
try:
res = res.astype(float)
except:
pass
elif DM_profile == "Burkert":
x = r / ct._r_H_
res = ct._rho_H_ / (1. + x) / (1. + x**2)
else:
raise Exception(
"The DM profile can only be NFW or Burkert. Please check the input.")
return res
def theta_gal_ctr(l, b, output_radians=True):
"""
Computes the theta angle [radians/degrees] that a point of galactic coordinates (l, b) makes with the galactic center.
Parameters
----------
l : galactic longitude [deg]
b : galactic latitude [deg]
output_radians : whether the result is given in radians (default: True)
"""
theta = np.arccos(np.cos(l*pi/180.)*np.cos(b*pi/180.))
if output_radians:
return theta
else:
return theta*180./pi
def r_to_gal(x, th, r0=ct._Sun_to_gal_center_):
"""
The radius [kpc] of a point to the galactic center.
Parameters
----------
x : distance of a point on the line of sight [kpc]
th : angle between galaxy center and light source [radian]
r0 : distance from the Sun to the galaxy center [kpc] (default: 8.1)
"""
r2 = x**2 + r0**2 - 2.*x*r0*np.cos(th)
r = sqrt(r2)
try:
r = r.astype(float)
except:
pass
return r
#########################################
# Light-related functions:
def flux_density_to_psd(nu, Snu, Omega):
"""
Convert flux density S_nu to phase space density f. Returns Energy [eV] and f = S_nu/Omega * 2*pi^2/E^3.
Parameters
----------
nu : array of the frequency [GHz]
Snu : array of S_nu [Jy]
Omega : the solid angle the source subtends [sr]
"""
Snu_in_eV3 = Snu * ct._Jy_over_eV3_
E = ct._h_ * nu * ct._GHz_over_eV_
res = Snu_in_eV3 / Omega * 2. * np.pi**2 / E**3
return (E, res)
def psd_to_flux_density(E, f, Omega):
"""
Converting phase space density f to flux density S_nu. Returns frequency nu [GHz] and spectral irradiance S_nu = E^3/2./pi^2*f*Omega [Jy].
Parameters
----------
E : array of energy [eV]
f : p.s.d
Omega : the angle the source subtends [sr]
"""
Snu = E**3 / 2. / np.pi**2 * f * Omega / ct._Jy_over_eV3_
nu = E / ct._GHz_over_eV_ / ct._h_
return (nu, Snu)
def flux(nu, Snu):
"""
Integrate S_nu w.r.t. nu to get S [eV^4]. Returns the flux (irradiance) [eV^4].
Parameters
----------
nu : the range of the frequency to be integrated over [GHz]
Snu : the flux density [Jy]
"""
# fix unit
nu_in_eV = nu * ct._GHz_over_eV_
Snu_in_eV3 = Snu * ct._Jy_over_eV3_
res = trapz(Snu_in_eV3, nu_in_eV)
return res
def irrad(distance, lumin):
"""
Returns the flux density (spectral irradiance) [Jy] of a source.
Parameters
----------
distance : distance to source [kpc]
lumin : spectral luminosity of source [erg * s^-1 * Hz^-1]
"""
area = 4.*pi * \
(distance*ct._kpc_over_cm_)**2. # [cm^2] sphere covering the source
irrad = (lumin/area) / ct._Jy_over_cgs_irrad_ # [Jy]
return irrad
def lumin(distance, irrad):
"""
Returns the spectral luminosity [erg * s^-1 * Hz^-1] of a source.
Parameters
----------
distance : distance to source [kpc]
irrad : flux density (spectral irradiance) of source [Jy]
"""
area = 4.*pi * \
(distance*ct._kpc_over_cm_)**2. # [cm^2] sphere covering the source
lumin = (irrad*ct._Jy_over_cgs_irrad_)*area # [erg * s^-1 * Hz^-1]
return lumin
def S_cygA(nu):
"""
The flux density [Jy] S_nu of Cygnus A.
Parameters
----------
nu : frequency, can be scalar or array [GHz]
"""
log_res = []
nu_lst, flg_scalar = tl.treat_as_arr(nu) # scalar --> array trick
nu_lst *= 1.e3 # converting to MHz
for nui in nu_lst:
if nui < 2000.:
a = 4.695
b = 0.085
c = -0.178
else:
a = 7.161
b = -1.244
c = 0.
log_res.append(a + b * np.log10(nui) + c * np.log10(nui)**2)
log_res = np.asarray(log_res)
res = 10.**(log_res)
if flg_scalar:
res = np.squeeze(res)
return res
#########################################
# Source model functions:
def FreeErrorMssg(model):
"""
A custom error message for different free-expansion evolutions.
"""
if model in pars_early.keys():
return "For model=={}, we need the arguments: {}.".format(model, ['t']+pars_early[model])
else:
return "Currently, this function can only support model=='eff' (Bietenholz et al., arXiv:2011.11737) or model=='thy'/'thy-alt' (Weiler et al., 1986ApJ...301..790W). model={} is not supported.".format(str(model))
def AdiabaticErrorMssg():
"""
A custom error message for the adiabatic evolution.
"""
return "L_adiab needs the arguments: ['t_ref', 'L_ref', 'gamma']."
# adiabatic and spectral indices
def gamma_from_alpha(alpha):
"""
Adiabatic index from spectral index, Sedov-Taylor expansion.
Parameters
----------
alpha : spectral index
"""
gamma = (4./5)*(2.*alpha + 1.)
return gamma
def alpha_from_gamma(gamma):
"""
Spectral index from adiabatic index, Sedov-Taylor expansion.
Parameters
----------
gamma : adiabatic index
"""
alpha = ((5./4)*gamma - 1.)/2.
return alpha
def nu_factor(nu, nu_pivot, alpha):
"""
Frequency rescaling factor, depending on the spectral index.
Parameters
----------
nu : frequency [GHz]
nu_pivot : pivot (i.e. reference) frequency [GHz]
alpha : spectral index
"""
return (nu/nu_pivot)**-alpha
# SNR light-curves
def L_eff(t, L_peak, t_peak):
"""
Returns the early evolution (i.e. during free expansion) of the SNR radio spectral luminosity [erg * s^-1 * Hz^-1] as a function of time after explosion [years] and frequency [GHz], according to the effective model from Bietenholz et al., arXiv:2011.11737. NOTE: frequency dependence prefactor, usually (nu/nu_pivot)^-alpha, is factorized out.
Parameters
----------
t : time after explosion [years]
L_peak : peak spectral luminosity [erg * s^-1 * Hz^-1]
t_peak : peak time [days]
"""
t_days = t*365. # [years] -> [days]
return L_peak * exp(-1.5 * (t_peak/t_days - 1.)) * (t_days/t_peak)**-1.5
def L_thy(t, L_norm, beta, K2, delta, tau_factor=1.):
"""
Returns the early evolution (i.e. during free expansion) of the SNR radio spectral luminosity [erg * s^-1 * Hz^-1] as a function of time after explosion [years] and frequency [GHz], according to the theoretical model from Weiler et al., 1986ApJ...301..790W. NOTE: frequency dependence prefactor, usually (nu/nu_pivot)^-alpha, is factorized out.
Parameters
----------
t : time after explosion [years]
L_norm : normalization spectral luminosity [erg * s^-1 * Hz^-1]
beta : post-peak free expansion evolution [defined in the exponent with a minus sign]
K2 : optical depth normalization
delta : optical depth time evolution [defined in the exponent with a minus sign]
tau_factor : frequency-dependent power for opacity tau; in Weiler's it's (nu/5.)**-2.1 (default: 1.)
"""
# In Weiler's, factor = (nu/5)^-alpha, while tau_factor = (nu/5.)**-2.1
t_days = t*365. # [years] -> [days]
tau = tau_factor * K2 * t_days**(-delta)
return L_norm * t_days**(-beta) * exp(-tau)
def L_free(t, model='eff', **kwargs):
"""
Returns the free expansion evolution of the SNR radio spectral luminosity [erg * s^-1 * Hz^-1] as a function of time after explosion [years] and frequency [GHz]. NOTE: frequency dependence prefactor, usually (nu/nu_pivot)^-alpha, is factorized out.
Parameters
----------
t : time after explosion [years]
model : model for the free expansion evolution, either 'eff' or 'thy' (default: 'eff')
"""
# Checking kwargs:
# subset of kwargs that is in pars_early[model]
pars_inter = set(pars_early[model]).intersection(kwargs.keys())
if pars_inter != set(pars_early[model]):
raise KeyError("The kwargs passed do not contain all the arguments for the model requested. For model={} these are: {}".format(
model, pars_early[model]))
# reading off the arguments of the early evolution functions
if model == 'eff':
Learly_pars = getargspec(L_eff)[0]
elif model == 'thy':
Learly_pars = getargspec(L_thy)[0]
# building the kwargs for the free expansion lightcurve
free_kwargs = {}
for par in Learly_pars:
if par in kwargs.keys():
free_kwargs[par] = kwargs[par]
try:
if model == 'eff':
Lf = L_eff(t, **free_kwargs)
elif model == 'thy':
# in principle, L_norm could be absent from free_kwargs; instead there being 'K1' and 'distance'. An old version of this code allowed for that. We have decided to restrict it.
Lf = L_thy(t, **free_kwargs)
except TypeError:
raise TypeError(FreeErrorMssg(model=model))
return Lf
def L_adiab(t, t_ref, L_ref, gamma):
"""
Returns the adiabatic (Sedov-Taylor) evolution of the SNR radio spectral luminosity [erg * s^-1 * Hz^-1] as a function of time after explosion [years] and frequency [GHz]. NOTE: frequency dependence prefactor, usually (nu/nu_pivot)^-alpha, is factorized out.
Parameters
----------
t : time after explosion [years]
t_ref : reference time after explosion [years]
L_ref : spectral luminosity at reference time [erg * s^-1 * Hz^-1]
gamma : adiabatic expansion evolution [defined in the exponent with a minus sign]
"""
return L_ref * (t/t_ref)**(-gamma)
def sandwich_logeqn(Lpk, tpk, L0, t0, gamma, tt):
"""
Returns the equation to minimize in order to find the transition time between the effective free expansion and the adiabatic expansion.
Parameters
----------
Lpk : peak spectral luminosity [erg * s^-1 * Hz^-1]
tpk : peak time [days]
L0 : SNR spectral luminosity today [erg * s^-1 * Hz^-1]
t0 : SNR age [years]
gamma : adiabatic index
tt : free-adiabatic phases transition time [years]
"""
tpk_yr = tpk/365.
eqn = log10(Lpk/L0) + (1.5)*(1. - tpk_yr/tt)*log10(np.e) + \
(gamma-1.5)*log10(tt) + 1.5*log10(tpk_yr) - gamma*log10(t0)
return eqn
def tage_compute(Lpk, tpk, t_trans, L_today, gamma):
"""
Returns the age [years] of a SNR given Lpk, tpk, t_trans, L_today, and gamma.
Parameters
----------
Lpk : peak spectral luminosity [erg * s^-1 * Hz^-1]
tpk : peak time [days]
t_trans : free-adiabatic phases transition time [years]
L_today : SNR spectral luminosity today [erg * s^-1 * Hz^-1]
gamma : adiabatic index
"""
def Lf_fn(tt): return L_eff(tt, Lpk, tpk)
L_trans = Lf_fn(t_trans) # L_trans
t_age = (L_today/L_trans)**(-1/gamma) * t_trans # t_age
return t_age
def L_source(t, model='eff', output_pars=False, **kwargs):
"""
Returns the SNR radio luminosity [erg * s^-1 * Hz^-1] as a function of time after explosion [years] and frequency [GHz]. NOTE: frequency dependence prefactor, usually (nu/nu_pivot)^-alpha, is factorized out.
Parameters
----------
t : time after explosion [years]
model : model for the free expansion evolution, either 'eff' or 'thy' (default: 'eff')
output_pars : whether to output a dictionary with the parameters used
"""
if not 'gamma' in kwargs.keys():
raise KeyError(
"'gamma' is not in kwargs. Please pass 'gamma' as an argument.")
gamma = kwargs['gamma']
# list of lightcurve parameters that are known (i.e. present in kwargs)
known = {}
# list of lightcurve parameters that are to be deduced (i.e. are missing in kwargs)
to_deduce = []
for par in pars_lightcurve[model]:
if par in kwargs.keys():
known[par] = kwargs[par]
else:
to_deduce.append(par)
if (model == 'thy') and ('tau_factor' in kwargs.keys()): # special case
known['tau_factor'] = kwargs['tau_factor']
if len(to_deduce) > 1: # too many missing parameters
raise ValueError(
"to_deduce={} is too large. Please include more parameters in kwargs.".format(to_deduce))
try:
to_deduce = to_deduce[0] # extracting the only parameter to be deduced
# the parameters required in kwargs
required = pars_required[(model, to_deduce)]
if not set(required).issubset(set(known.keys())):
raise ValueError("known={} is too short. It should be a subset of the required parameters, which are {}. Please include more parameters in kwargs.".format(
known.keys(), required))
except IndexError:
to_deduce = None # no parameter to deduce!
if model == 'eff':
if (to_deduce == 'L_today') or (to_deduce == None): # based on peak info, deduce L_today
t_today = known['t_age'] # age [years]
t_trans = known['t_trans'] # transition time [years]
# computing transition luminosity
L_trans = L_free(t_trans, model=model, **known)
adiab_kwargs = {'L_ref': L_trans, 't_ref': t_trans, 'gamma': gamma}
L_today = L_adiab(t_today, **adiab_kwargs)
known.update({'L_today': L_today})
elif to_deduce == 'L_peak': # based on t_peak and today's info, deduce L_peak
t_age = known['t_age'] # age [years]
t_trans = known['t_trans'] # transition time [years]
L_today = known['L_today'] # spectral luminosity [today]
adiab_kwargs = {'L_ref': L_today, 't_ref': t_age, 'gamma': gamma}
eff_kwargs = {}
eff_pars = getargspec(L_eff)[0]
for par, val in known.items():
if par in eff_pars:
eff_kwargs[par] = val
# computing transition luminosity
L_trans = L_adiab(t_trans, **adiab_kwargs)
def LogLeff_fn(LogLpk): return log10(
L_eff(t_trans, L_peak=10**LogLpk, **eff_kwargs)) - log10(L_trans)
L_peak = tl.zeros(LogLeff_fn, log10(L_arr_default))
L_peak = np.squeeze(10.**L_peak)
known.update({'L_peak': L_peak})
elif to_deduce == 't_trans': # based on t_age and both today's and peak info, deduce t_trans
t_peak = known['t_peak']
t_age = known['t_age'] # age
L_peak = known['L_peak']
L_today = known['L_today'] # spectral luminosity today
adiab_kwargs = {'L_ref': L_today, 't_ref': t_age, 'gamma': gamma}
def fn(Ltt): return sandwich_logeqn(L_peak, t_peak, L_today,
t_age, gamma, 10.**Ltt) # function to minimize
try:
# NOTE: added 'np.squeeze'
Lt_cross = np.squeeze(tl.zeros(fn, log10(t_arr_default)))
t_cross = 10.**Lt_cross
try:
t_trans = max(t_cross)
except:
t_trans = t_cross
except ValueError:
t_trans = 1.e11 # made-up, very large value
known.update({'t_trans': t_trans})
elif to_deduce == 't_age': # based on t_trans and both today's and peak info, deduce t_age
t_peak = known['t_peak']
t_trans = known['t_trans'] # transition time
L_peak = known['L_peak']
L_today = known['L_today'] # spectral luminosity today
# simplified free expansion function
def Lf_fn(tt): return L_free(tt, model=model, **known)
L_trans = Lf_fn(t_trans) # L_trans
adiab_kwargs = {'L_ref': L_trans, 't_ref': t_trans, 'gamma': gamma}
t_age = tage_compute(L_peak, t_peak, t_trans,
L_today, gamma) # age
known.update({'t_age': t_age})
else:
raise ValueError(
"to_deduce={}, known={} is currently not supported.".format(to_deduce, known))
elif model == 'thy':
if (to_deduce == 'L_today') or (to_deduce == None):
t_today = known['t_age'] # age [years]
t_trans = known['t_trans'] # transition time [years]
# computing transition luminosity
L_trans = L_free(t_trans, model=model, **known)
adiab_kwargs = {'L_ref': L_trans, 't_ref': t_trans, 'gamma': gamma}
L_today = L_adiab(t_today, **adiab_kwargs)
known.update({'L_today': L_today})
elif to_deduce == 'L_norm':
t_age = known['t_age'] # age [years]
t_trans = known['t_trans'] # transition time [years]
L_today = known['L_today'] # spectral luminosity [today]
adiab_kwargs = {'L_ref': L_today, 't_ref': t_age, 'gamma': gamma}
thy_kwargs = {}
thy_pars = getargspec(L_thy)[0]
for par, val in known.items():
if par in thy_pars:
thy_kwargs[par] = val
# computing transition luminosity
L_trans = L_adiab(t_trans, **adiab_kwargs)
def LogLthy_fn(LogLnorm): return log10(
L_thy(t_trans, L_norm=10**LogLnorm, **thy_kwargs)) - log10(L_trans)
L_norm = tl.zeros(LogLthy_fn, log(L_arr_default))
L_norm = np.squeeze(10.**L_norm)
known.update({'L_norm': L_norm})
if output_pars:
return_pars = {}
return_pars.update(known)
return_pars.update(adiab_kwargs)
Lf = L_free(t, model=model, **known) # computing early-times luminosity
La = L_adiab(t, **adiab_kwargs) # computing adiabatic luminosity
if not 'use_free_expansion' in kwargs.keys():
use_free_expansion = 1.
else:
if kwargs['use_free_expansion'] is False:
use_free_expansion = 0.
else:
use_free_expansion = 1.
Lum = Lf*np.heaviside(t_trans - t, 1.) * \
use_free_expansion + La*np.heaviside(t - t_trans, 0.)
if output_pars:
return Lum, return_pars
else:
return Lum
# SNR light-curve analytic formulas, assuming the Bietenholz effective model.
def Snu_supp(gamma, frac_tpk, frac_tt):
"""
Fractional suppression of spectral irradiance (i.e. S0/S_peak = L0/L_peak).
Parameters
----------
gamma : adiabatic index
frac_tpk : ratio of peak day to SNR age
frac_tt : ratio of transition time to SNR age
"""
first = exp(1.5 - 1.5*frac_tpk/frac_tt)
second = (frac_tt/frac_tpk)**-1.5
third = (1./frac_tt)**-gamma
return first*second*third
def ftt(gamma, frac_tpk, sup):
"""
Fractional transition time (i.e. t_trans/t_age).
Parameters
----------
gamma : adiabatic index
frac_tpk : ratio of peak day to SNR age
sup : spectral irradiance suppression
"""
const = 1./(2.*gamma/3. - 1.)
arg = const * exp(const) * frac_tpk**(const+1.) * sup**(-2.*const/3.)
num = const*frac_tpk
den = lambertw(arg)
return np.real(num/den)
def dimless_free(frac_tpk, tau):
"""
Dimensionless free expansion lightcurve.
Parameters
----------
frac_tpk : ratio of peak day to SNR age
tau : ratio of time to SNR age
"""
return exp(1.5 - 1.5*frac_tpk/tau) * (tau/frac_tpk)**-1.5
def dimless_adiab(gamma, sup, tau):
"""
Dimensionless adiabatic expansion lightcurve.
Parameters
----------
gamma : adiabatic index
sup : spectral irradiance suppression
tau : ratio of time to SNR age
"""
return sup*(tau)**-gamma
def dimless_lum(gamma, frac_tpk, sup, tau):
"""
Dimensionless full lightcurve.
Parameters
----------
gamma : adiabatic index
frac_tpk : ratio of peak day to SNR age
sup : spectral irradiance suppression
tau : ratio of time to SNR age
"""
frac_tt = ftt(gamma, frac_tpk, sup)
free = dimless_free(frac_tpk, tau)
adiab = dimless_adiab(gamma, sup, tau)
return free*np.heaviside(frac_tt-tau, 0.) + adiab*np.heaviside(tau-frac_tt, 1.)
###############################
# SNR evolution analytic models
###############################
# ---------------------------------
# Phenomenological evolution model
def R_pheno(t, Rst=3.8, tst=360., eta1=1., eta2=0.4):
"""
SNR radius [pc] as a simple phenomenological broken power law. The default values are taken from Tables 2 & 3 of the Truelove-McKee '99 paper for n=0 ejecta, with M_ej = 1 M_sun, E_sn = 1.e51 erg, and n0 = 0.2 cm^-3.
Parameters
----------
t : SNR age [years]
Rst : radius [pc] at the start of the Sedov-Taylor (adiabatic) expansion phase (default: 3.8)
tst : age [years] at the start of the Sedov-Taylor (adiabatic) expansion phase (default 360.)
eta1 : the power scaling R~t^eta1 during the Ejecta-Dominated phase (default: 1.)
eta2 : the power scaling R`t^eta2 during the Sedov-Taylor expansion phase (default: 2/5 = 0.4)
"""
ed = Rst * (t/tst)**eta1 * np.heaviside(tst-t, 0.)
ad = Rst * (t/tst)**eta2 * np.heaviside(t-tst, 1.)
return ed+ad
def pheno_age(R, Rst=3.8, tst=360., eta1=1., eta2=0.4):
"""
SNR age [years] as deduced from a phenomenological broken power-law function of the SNR blast radius [pc]. The default values are taken from Tables 2 & 3 of the Truelove-McKee '99 paper for n=0 ejecta, with M_ej = 1 M_sun, E_sn = 1.e51 erg, and n0 = 0.2 cm^-3.
Parameters
----------
R : SNR radius [pc]
Rst : radius [pc] at the start of the Sedov-Taylor (adiabatic) expansion phase (default: 3.8)
tst : age [years] at the start of the Sedov-Taylor (adiabatic) expansion phase (default 360.)
eta1 : the power scaling R~t^eta1 during the Ejecta-Dominated phase (default: 1.)
eta2 : the power scaling R`t^eta2 during the Sedov-Taylor expansion phase (default: 2/5 = 0.4)
"""
def logDelRt(t, r): return log10(
R_pheno(t, Rst, tst, eta1=eta1, eta2=eta2)) - log10(r)
R_arr, _ = tl.treat_as_arr(R)
age = np.array([tl.zeros(logDelRt, t_arr_default, r) for r in R_arr])
age = np.squeeze(age)
return age
# -------------------------
# Physical evolution model
def ED_fn(t, t_bench, R_bench, model):
"""
Blast radius [pc] as a function of time for the Ejecta-Dominated era. Based on Truelove & McKee 1999 (TM99).
Parameters
----------
t : time [years]
t_bench : benchmark time [years]
R_bench : benchmark radius [pc]
model : model of the expansion ('TM99-simple'/'TM99-0' for a simplified version of TM99 or for the case with n=0 ejecta.)
"""
tstar = t/t_bench
if model == 'TM99-simple':
# power law R~t
Rstar = tstar
elif model == 'TM99-0':
# TM99, Table 5
Rstar = 2.01*tstar*(1. + 1.72 * tstar**(3./2.))**(-2./3.)
return R_bench*Rstar
def ST_fn(t, t_bench, R_bench, model):
"""
Blast radius [pc] as a function of time for the Sedov-Taylor era. Based on Truelove & McKee 1999 (TM99).
Parameters
----------
t : time [years]
t_bench : benchmark time [years]
R_bench : benchmark radius [pc]
model : model of the expansion ('TM99-simple'/'TM99-0' for a simplified version of TM99 or for the case with n=0 ejecta.)
"""
tstar = t/t_bench
if model == 'TM99-simple':
# power law R~t^(2/5)
Rstar = tstar**(2./5.)
elif model == 'TM99-0':
# TM99, Table 5
arg = (1.42*tstar - 0.254)
# regularizing:
arg = np.where(arg <= 0., 1.e-100, arg)
Rstar = arg**(2./5.)
return R_bench*Rstar
def Rb_TM99(t, t_bench, R_bench, model):
"""
Blast radius [pc] as a function of SNR age [years]. Based on Truelove & McKee 1999 (TM99).
Parameters
----------
t : time [years]
t_bench : benchmark time [years]
R_bench : benchmark radius [pc]
model : model of the expansion ('TM99-simple'/'TM99-0' for a simplified version of TM99 or for the case with n=0 ejecta.)
"""
# broken function:
if model == 'TM99-simple':
two_phase = np.minimum.reduce(
[ED_fn(t, t_bench, R_bench, model), ST_fn(t, t_bench, R_bench, model)])
elif model == 'TM99-0':
two_phase = np.maximum.reduce(
[ED_fn(t, t_bench, R_bench, model), ST_fn(t, t_bench, R_bench, model)])
return two_phase
def physics_age(R, model='estimate', M_ej=1., E_sn=1., rho0=1.):
"""
SNR age [years] as deduced from a physically-motivated a function of the SNR blast radius [pc]. Using either a simple model, or formulas by Truelove & McKee 1999 (TM99).
Parameters
----------
R : SNR radius today [pc]
model : 'estimate'/'TM99-simple'/TM99-0'': whether the simple one-phase 'estimate' model is used, or instead the two-phase Truelove-McKee model (ED-ST, or Ejecta-Dominated -- Sedov-Taylor), either in a simplified form, or for n=0 (uniform) ejecta profile.
M_ej : Mass of the ejecta [M_sun] (default: 1.)
E_sn : Energy of the SNR [1.e51 ergs] (default: 1.)
rho0 : Mass density of surrounding medium [m_proton/cm^3] (default: 1.)
"""
if not model in ['estimate', 'TM99-simple', 'TM99-0']:
raise ValueError("model={} is currently unsupported.".format(model))
if model == 'estimate':
# https://chandra.harvard.edu/edu/formal/age_snr/pencil_paper.pdf
R_m = R*(ct._kpc_over_m_/1000.) # [m]
vol_m3 = (4./3.) * pi * R_m**3. # [m^3]
m_proton = 0.93827208816*ct._GeV_over_g_ * 1.e-3 # [kg]
density = (m_proton*rho0)*1.e6 # [kg/m^3]
mass = density*vol_m3 # [kg]
E_sn_J = (E_sn*1.e51)*(1.e-7) # [J]
ke = 0.25*E_sn_J # [J]
vel = sqrt(2.*ke / mass) # [m/s]
age = (R_m/vel)/ct._year_over_s_ # [years]
return age
elif model in ['TM99-simple', 'TM99-0']:
# Article: https://iopscience.iop.org/article/10.1086/313176
# Erratum: https://iopscience.iop.org/article/10.1086/313385
# Revisited: arXiv:2109.03612
n0 = rho0/1. # number density [cm^-3]
# Characteristic scales: Table 2
R_ch = 3.07 * M_ej**(1./3.) * n0**(-1./3.) # [pc]
t_ch = 423. * E_sn**(-1./2.) * M_ej**(5./6.) * n0**(-1./3.) # [years]
# Sedov-Taylor scales: Table 3 (rounding up, for various ejecta power-law indices)
if model == 'TM99-simple':
tstar_ST = 0.4
Rstar_ST = 0.7
elif model == 'TM99-0':
tstar_ST = 0.4950
Rstar_ST = 0.7276
# Sedov-Taylor radius and age
t_ST = tstar_ST*t_ch
R_ST = Rstar_ST*R_ch
if model == 'TM99-simple':
t_bench = t_ST
R_bench = R_ST
elif model == 'TM99-0':
t_bench = t_ch
R_bench = R_ch
# defining the evolution in the two phases
def LogDelRb(t, r):
"""
Function whose zeros we need to find in order to solve for the time [years] as a function of the radius [pc].
"""
return log10(Rb_TM99(t, t_bench, R_bench, model)) - log10(r)
R_arr, _ = tl.treat_as_arr(R)
age = np.array([tl.zeros(LogDelRb, t_arr_default, r) for r in R_arr])
age = np.squeeze(age)
return age
#########################################
# Noise and signal
def bg_408_temp(l, b, size=None, average=False, verbose=True, load_on_the_fly=False):
"""
Reads the background brightness temperature at 408 MHz at galactic coordinates l and b.
Parameters
----------
l : longitude [deg]
b : latitude [deg]
size : angular size [sr] of the region of interest (default: None)
average : whether the brightness temperature is averaged over the size of the region of interest (default: False)
verbose: control of warning output. Switch to False only if you know what you are doing.
load_on_the_fly: flag to load the map on the fly. It's for debugging purpose. Switch to True only if you know what you are doing.
"""
if load_on_the_fly:
if verbose:
print('You requested to load Haslem every time bg_408_temp is called. This is highly time-consuming. You should use this only for debugging purpose.')
# initialize the haslem map
try:
path = os.path.dirname(os.path.abspath(__file__))+'/data/'
path = os.path.join(path, 'lambda_haslam408_dsds.fits')
global map_allsky_408
map_allsky_408 = hp.read_map(path)
except FileNotFoundError:
raise Exception(
'Haslam map (lambda_haslam408_dsds.fits) is not found.')
healp = HEALPix(nside=512, order='ring', frame=Galactic())
pos_coords = SkyCoord(frame="galactic", l=l, b=b, unit='deg')
pos_pix = healp.skycoord_to_healpix(pos_coords)
vec = hp.pix2vec(nside=512, ipix=pos_pix)
if size is not None:
size, is_scalar = tl.treat_as_arr(size) # scalar --> array trick
bg_T408 = []
if average:
for size_val in size:
radial_angle = ct.solid_angle_to_angle(
size_val)/2. # need to divide by 2 to get radius
new_pos_pix = hp.query_disc(
nside=512, vec=vec, radius=radial_angle)
bg_T408.append(np.average(map_allsky_408[new_pos_pix]))
else:
# this should be used only for debugging
# i.e. manually switch average == False
print('Warning: you are setting average flag to False')
for size_val in size:
new_pos_pix = pos_pix
bg_T408.append(np.average(map_allsky_408[new_pos_pix]))
if is_scalar:
bg_T408 = np.squeeze(bg_T408)
else:
bg_T408 = np.array(bg_T408)
else:
new_pos_pix = pos_pix
bg_T408 = np.average(map_allsky_408[new_pos_pix])
# query the background temperature for the echo
return bg_T408
def T_atm(nu):
"""Compute the atmospheric temperature [K].
NOTE: There is an unphysical break at the boundary.
:param nu: Frequency [GHz]
"""
nu, is_scalar = tl.treat_as_arr(nu)
nu_low = nu[nu <= ct._nu_min_ska_mid_]
nu_mid = nu[nu > ct._nu_min_ska_mid_]
res_low = Tatm_low_fn(nu_low)
res_mid = Tatm_mid_fn(nu_mid)
res = np.concatenate((res_low, res_mid))
if is_scalar:
res = np.squeeze(res)
return res
def Tatm_mid_fn(nu):
"""The atmospheric temperature for SKA1-Mid frequency
:param nu: frequency [GHz]
"""