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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 scipy import integrate
from scipy import interpolate
from ...utils import COSMO
from ...utils.units import (Units as AU,
UnitConversions as AUC,
Constants as AC)
from ...utils.convert import Fnu_to_Tb_fast
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]
@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.
NOTE
----
Use interpolation to optimize the speed, also avoid instabilities
near the lower end (e.g., x < 1e-5).
Interpolation also helps vectorize this function for easier calling.
Parameters
----------
x : `~numpy.ndarray`
Points where to calculate the kernel function values.
NOTE: X values will be bounded, e.g., within [1e-5, 20]
Returns
-------
y : `~numpy.ndarray`
Calculated kernel function values.
"""
# The lower and upper cuts
xmin = 1e-5
xmax = 20.0
# Number of samples within [xmin, xmax]
# NOTE: this kernel function is quiet smooth and slow-varying.
nsamples = 128
# Make an interpolation
x_interp = np.logspace(np.log10(xmin), np.log10(xmax),
num=nsamples)
F_interp = [
xp * integrate.quad(lambda t: scipy.special.kv(5/3, t),
a=xp, b=np.inf)[0]
for xp in x_interp
]
func_interp = interpolate.interp1d(x_interp, F_interp,
kind="quadratic")
x = np.array(x) # Make a copy!
x[x < xmin] = xmin
x[x > xmax] = xmax
y = func_interp(x)
return y
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 = 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 = 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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