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# Copyright (c) 2017 Weitian LI <liweitianux@live.com>
# MIT license
"""
Calculate the synchrotron emission for simulated radio halos.
"""
import logging
import numpy as np
import scipy.integrate
import scipy.special
from ...utils.units import (Units as AU, Constants as AC)
logger = logging.getLogger(__name__)
class SynchrotronEmission:
"""
Calculate the synchrotron emission from a given population
of electrons.
References
----------
[1] Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
Appendix.C
"""
def __init__(self, B):
# Uniform magnetic field strength
self.B = B # [uG]
@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
def F(self, 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, p, n_e):
"""
Calculate the synchrotron emissivity (power emitted per volume
and per frequency) at the specified frequency from the given
electron number & energy distribution
Parameters
----------
nu : float
Frequency (unit: MHz) where to calculate the emissivity.
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]
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(p.shape)
for i in range(len(p)):
# Integrate over ``theta``
func_p[i] = scipy.integrate.quad(
lambda t: func(t, p[i], n_e[i]),
a=0, b=np.pi/2)[0]
# Integrate over ``p``
j_nu = coef * scipy.integrate.trapz(func_p, p)
return j_nu
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