1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
# 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 emission 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 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]
"""
coef = AC.e / (2*np.pi * AU.mec) # [Hz/G]
coef *= 1e-12 # [Hz/G] -> [MHz/uG]
nu = coef * self.B # [MHz]
return nu
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
@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.
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
----------
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]
"""
# 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)
kernel = self.F(nu / nu_c)
# 2D samples over width to do the integration
s2d = kernel * np.outer(self.n_e, np.sin(theta)**2)
# Integrate over ``theta`` (the last axis)
s1d = integrate.simps(s2d, x=theta)
# Integrate over energy ``gamma`` in logarithmic grid
j_nu = integrate.simps(s1d*self.gamma, np.log(self.gamma))
coef = np.sqrt(3) * AC.e**3 * self.B / AC.c
j_nu *= coef
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
|
