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# Copyright (c) 2017 Weitian LI <weitian@aaronly.me>
# MIT license
"""
Calculate the synchrotron emission and inverse Compton emission
for simulated radio halos.
References
----------
[1] Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
Appendix.C
"""
import logging
import numpy as np
import scipy.integrate
import scipy.special
from ...utils.units import (Units as AU,
UnitConversions as AUC,
Constants as AC)
from ...utils.convert import Fnu_to_Tb_fast
from ...utils.cosmology import Cosmology
logger = logging.getLogger(__name__)
class SynchrotronEmission:
"""
Calculate the synchrotron emission spectrum from a given population
of electrons.
Parameters
----------
B : float
The assumed uniform magnetic field of the galaxy cluster.
Unit: [uG]
p : `~numpy.ndarray`
The momentum grid adopted when solving the Fokker-Planck equation.
Unit: [mec]
n_e : `~numpy.ndarray`
Electron spectrum by solving the Fokker-Planck equation.
Unit: [cm^-3 mec^-1]
radius, float
The radius of the galaxy cluster/halo, within which the uniform
magnetic field and electron distribution are assumed.
Unit: [kpc]
z : float
Redshift of the galaxy cluster/halo
"""
def __init__(self, B, p, n_e, radius, z):
self.B = B # [uG]
self.p = p
self.n_e = n_e
self.z = z
self.radius = radius # [kpc]
self.cosmo = Cosmology()
@property
def frequency_larmor(self):
"""
Electron Larmor frequency:
ν_L = e * B / (2*π * m0 * c) = e * B / (2*π * mec)
Unit: MHz
"""
coef = AC.e / (2*np.pi * AU.mec) # [Hz/G]
coef *= 1e-12 # [MHz/uG]
nu = coef * self.B # [MHz]
return nu
def frequency_crit(self, p, theta=np.pi/2):
"""
Synchrotron critical frequency.
Critical frequency:
ν_c = (3/2) * γ^2 * sin(θ) * ν_L
Parameters
----------
p : float
Electron momentum (unit: mec), i.e., Lorentz factor γ
theta : float, optional
The angle between the electron velocity and the magnetic field.
(unit: radian)
Returns
-------
nu : float
Critical frequency, unit: MHz
"""
nu_L = self.frequency_larmor
nu = (3/2) * p**2 * np.sin(theta) * nu_L
return nu
@staticmethod
def F(x):
"""
Synchrotron kernel function.
"""
val = x * scipy.integrate.quad(lambda t: scipy.special.kv(5/3, t),
a=x, b=np.inf)[0]
return val
def emissivity(self, nu):
"""
Calculate the synchrotron emissivity (power emitted per volume
and per frequency) at the requested frequency.
Parameters
----------
nu : float
Frequency where to calculate the emissivity.
Unit: [MHz]
Returns
-------
j_nu : float
Synchrotron emissivity at frequency ``nu``.
Unit: [erg/s/cm^3/Hz]
"""
def func(theta, _p, _n_e):
nu_c = self.frequency_crit(_p, theta)
x = nu / nu_c
return (np.sin(theta)**2 * _n_e * self.F(x))
coef = np.sqrt(3) * AC.e**3 * self.B / AC.c # multiplied a [mec]
func_p = np.zeros(self.p.shape)
for i in range(len(self.p)):
# Integrate over ``theta``
func_p[i] = scipy.integrate.quad(
lambda t: func(t, self.p[i], self.n_e[i]),
a=0, b=np.pi/2)[0]
# Integrate over ``p``
j_nu = coef * scipy.integrate.trapz(func_p, self.p)
return j_nu
def power(self, nu):
"""
Calculate the synchrotron power (power emitted per frequency)
at the requested frequency.
Returns
-------
P_nu : float
Synchrotron power at frequency ``nu``.
Unit: [erg/s/Hz]
"""
r_cm = self.radius * AUC.kpc2cm
volume = (4.0/3.0) * np.pi * r_cm**3
P_nu = self.emissivity(nu) * volume
return P_nu
def flux(self, nu):
"""
Calculate the synchrotron flux (power observed per frequency)
at the requested frequency.
Returns
-------
F_nu : float
Synchrotron flux at frequency ``nu``.
Unit: [Jy] = 1e-23 [erg/s/cm^2/Hz]
"""
DL = self.cosmo.DL(self.z) * AUC.Mpc2cm # [cm]
P_nu = self.power(nu)
F_nu = 1e23 * P_nu / (4*np.pi * DL*DL) # [Jy]
return F_nu
def brightness(self, nu, pixelsize):
"""
Calculate the synchrotron surface brightness (power observed
per frequency and per solid angle) at the specified frequency.
NOTE
----
If the radio halo has solid angle less than the pixel area, then
it is assumed to have solid angle of 1 pixel.
Parameters
----------
pixelsize : float
The pixel size of the output simulated sky image
Unit: [arcsec]
Returns
-------
Tb : float
Synchrotron surface brightness at frequency ``nu``.
Unit: [K] <-> [Jy/pixel]
"""
DA = self.cosmo.DL(self.z) * AUC.Mpc2cm # [cm]
radius = self.radius * AUC.kpc2cm # [cm]
omega = (np.pi * radius**2 / DA**2) * AUC.rad2deg**2 # [deg^2]
pixelarea = (pixelsize * AUC.arcsec2deg) ** 2 # [deg^2]
if omega < pixelarea:
omega = pixelarea
F_nu = self.flux(nu)
Tb = Fnu_to_Tb_fast(F_nu, omega, nu)
return Tb
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