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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
.. [you1998]
You 1998
The Radiation Mechanisms in Astrophysics, 2nd Edition, Beijing
Sec.4.2.3, p.187
"""
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-3, xmax=10.0, xsample=256):
"""
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.
WARNING
-------
Do NOT simply bound the synchrotron kernel within the specified
[xmin, xmax] range, since it decreases as a power law of index
1/3 at the left end, and decreases exponentially at the right end.
Bounding it with interpolation will cause the synchrotron emissivity
been *overestimated* on the higher frequencies.
Parameters
----------
xmin, xmax : float, optional
The lower and upper cuts for the kernel function.
Default: [1e-3, 10.0]
xsample : int, optional
Number of samples within [xmin, xmax] used to do interpolation.
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=True, 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)`` within
# the specified range.
# NOTE: See the *WARNING* above.
F_xmin = 1e-3
F_xmax = 10.0
F_xsample = 256
F_interp = _interp_sync_kernel(F_xmin, F_xmax, F_xsample)
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.
References: Ref.[you1998]
"""
x = np.array(x, ndmin=1)
y = np.zeros(x.shape)
idx = (x >= cls.F_xmin) & (x <= cls.F_xmax)
y[idx] = cls.F_interp(x[idx])
# Left end: power law of index 1/3
idx = (x < cls.F_xmin)
A = cls.F_interp(cls.F_xmin)
y[idx] = A * (x[idx] / cls.F_xmin)**(1/3)
# Right end: exponentially decrease
idx = (x > cls.F_xmax)
y[idx] = (0.5*np.pi * x[idx])**0.5 * np.exp(-x[idx])
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
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