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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.units import (Units as AU, Constants as AC)
logger = logging.getLogger(__name__)
def _interp_sync_kernel(xmin=1e-5, xmax=20.0, xsample=128):
"""
Sample the synchrotron kernel function at the specified X
positions and make an interpolation, to optimize the speed
when invoked to calculate the synchrotron emissivity.
Parameters
----------
xmin, xmax : float, optional
The lower and upper cuts for the kernel function.
Default: [1e-5, 20.0]
xsample : int, optional
Number of samples within [xmin, xmax] used to do interpolation.
NOTE: The kernel function is quiet smooth and slow-varying.
Returns
-------
F_interp : function
The interpolated kernel function ``F(x)``.
"""
xx = np.logspace(np.log10(xmin), np.log10(xmax), num=xsample)
Fxx = [xp * integrate.quad(lambda t: scipy.special.kv(5/3, t),
a=xp, b=np.inf)[0]
for xp in xx]
F_interp = interpolate.interp1d(
xx, Fxx, kind="quadratic", bounds_error=False,
fill_value=(Fxx[0], Fxx[-1]), assume_sorted=True)
return F_interp
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]
B : float
The assumed uniform magnetic field within the cluster ICM.
Unit: [uG]
"""
# The interpolated synchrotron kernel function ``F(x)``.
F_interp = _interp_sync_kernel()
def __init__(self, gamma, n_e, B):
self.gamma = np.asarray(gamma)
self.n_e = np.asarray(n_e)
self.B = B # [uG]
@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
@classmethod
def F(cls, 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.
"""
return cls.F_interp(x)
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
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