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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
----------
.. [cassano2005]
Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
Appendix.C
.. [era2016]
Condon & Ransom 2016
Essential Radio Astronomy
https://science.nrao.edu/opportunities/courses/era/
Chapter.5
"""
import logging
import numpy as np
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 emissivity from a given population
of electrons.
Parameters
----------
gamma : `~numpy.ndarray`
The Lorentz factors of electrons.
n_e : `~numpy.ndarray`
Electron number density spectrum.
Unit: [cm^-3]
z : float
Redshift of the cluster/halo been observed/simulated.
B : float
The assumed uniform magnetic field within the cluster ICM.
Unit: [uG]
radius : float
The radius of the halo, within which the uniform magnetic field
and electron distribution are assumed.
Unit: [kpc]
"""
def __init__(self, gamma, n_e, z, B, radius):
self.gamma = np.asarray(gamma)
self.n_e = np.asarray(n_e)
self.z = z
self.B = B # [uG]
self.radius = radius # [kpc]
@property
def B_gauss(self):
"""
Magnetic field in unit of [G] (i.e., gauss)
"""
return self.B * 1e-6 # [uG] -> [G]
@property
def frequency_larmor(self):
"""
Electron Larmor frequency (a.k.a. gyro frequency):
ν_L = e * B / (2*π * m0 * c) = e * B / (2*π * mec)
=> ν_L [MHz] = 2.8 * B [G]
Unit: [MHz]
"""
nu_larmor = AC.e * self.B_gauss / (2*np.pi * AU.mec) # [Hz]
return nu_larmor * 1e-6 # [Hz] -> [MHz]
def frequency_crit(self, gamma, theta=np.pi/2):
"""
Synchrotron critical frequency.
Critical frequency:
ν_c = (3/2) * γ^2 * sin(θ) * ν_L
Parameters
----------
gamma : `~numpy.ndarray`
Electron Lorentz factors γ
theta : `~numpy.ndarray`, optional
The angles between the electron velocity and the magnetic field.
Unit: [rad]
Returns
-------
nu_c : `~numpy.ndarray`
Critical frequencies
Unit: [MHz]
"""
nu_c = 1.5 * gamma**2 * np.sin(theta) * self.frequency_larmor
return nu_c
def F(self, 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.
* Cache the interpolation results, since this function will be called
multiple times for each frequency.
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.
"""
if not hasattr(self, "_F_interp"):
# Interpolate the kernel function and cache the results
#
# 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
xx = np.logspace(np.log10(xmin), np.log10(xmax), num=nsamples)
Fxx = [xp * integrate.quad(lambda t: scipy.special.kv(5/3, t),
a=xp, b=np.inf)[0]
for xp in xx]
self._F_interp = interpolate.interp1d(xx, Fxx, kind="quadratic")
x = np.array(x) # Make a copy!
x[x < xmin] = xmin
x[x > xmax] = xmax
y = self._F_interp(x)
return y
def emissivity(self, frequencies):
"""
Calculate the synchrotron emissivity (power emitted per volume
and per frequency) at the requested frequency.
NOTE
----
Since ``self.gamma`` and ``self.n_e`` are sampled on a logarithmic
grid, we integrate over ``ln(gamma)`` instead of ``gamma`` directly:
I = int_gmin^gmax f(g) d(g)
= int_ln(gmin)^ln(gmax) f(g) g d(ln(g))
Parameters
----------
frequencies : float, or 1D `~numpy.ndarray`
The frequencies where to calculate the synchrotron emissivity.
Unit: [MHz]
Returns
-------
syncem : float, or 1D `~numpy.ndarray`
The calculated synchrotron emissivity at each specified
frequency.
Unit: [erg/s/cm^3/Hz]
"""
j_coef = np.sqrt(3) * AC.e**3 * self.B_gauss / AU.mec2
# Ignore the zero angle
theta = np.linspace(0, np.pi/2, num=len(self.gamma))[1:]
theta_grid, gamma_grid = np.meshgrid(theta, self.gamma)
nu_c = self.frequency_crit(gamma_grid, theta_grid)
# 2D grid of ``n_e(gamma) * sin^2(theta)``
nsin2 = np.outer(self.n_e, np.sin(theta)**2)
frequencies = np.array(frequencies, ndmin=1)
syncem = np.zeros(shape=frequencies.shape)
for i, freq in zip(range(len(frequencies)), frequencies):
logger.debug("Calc synchrotron emissivity at %.2f [MHz]" % freq)
kernel = self.F(freq / nu_c)
# 2D samples over width to do the integration
s2d = kernel * nsin2
# Integrate over ``theta`` (the last axis)
s1d = integrate.simps(s2d, x=theta)
# Integrate over energy ``gamma`` in logarithmic grid
syncem[i] = j_coef * integrate.simps(s1d*self.gamma,
np.log(self.gamma))
if len(syncem) == 1:
return syncem[0]
else:
return syncem
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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