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# Copyright (c) 2017-2018 Weitian LI <weitian@aaronly.me>
# MIT license
"""
Calculate the synchrotron emission for a given relativistic electron
spectrum, e.g., derived for the 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
from functools import lru_cache
import numpy as np
import scipy.special
from scipy import integrate, 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
@lru_cache()
def B_gauss(self):
"""
Magnetic field in unit of [G] (i.e., Gauss)
"""
return self.B * 1e-6 # [uG] -> [G]
@property
@lru_cache()
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,
the pitch angle.
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 using interpolation to improve speed.
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))
The pitch angles of electrons w.r.t. the magnetic field are assumed
to be ``pi/2``, which maybe a good simplification.
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
nu_c = self.frequency_crit(self.gamma)
frequencies = np.array(frequencies, ndmin=1)
syncem = np.zeros(shape=frequencies.shape)
for i, freq in enumerate(frequencies):
logger.debug("Calculating emissivity at %.2f [MHz]" % freq)
kernel = self.F(freq / nu_c)
# Integrate over energy ``gamma`` in logarithmic grid
syncem[i] = j_coef * integrate.simps(
self.n_e*kernel*self.gamma, x=np.log(self.gamma))
if len(syncem) == 1:
return syncem[0]
else:
return syncem
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