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
|
# Copyright (c) 2016-2017 Weitian LI <weitian@aaronly.me>
# MIT license
"""
Utilities for conversion among common astronomical quantities.
"""
import numpy as np
import astropy.units as au
import numba
def Fnu_to_Tb(Fnu, omega, freq):
"""
Convert flux density to brightness temperature, using the
Rayleigh-Jeans law, in the Rayleigh-Jeans limit.
Parameters
----------
Fnu : `~astropy.units.Quantity`
Input flux density, e.g., `1.0*au.Jy`
omega : `~astropy.units.Quantity`
Source angular size/area, e.g., `100*au.arcsec**2`
freq : `~astropy.units.Quantity`
Frequency where the flux density measured, e.g., `120*au.MHz`
Returns
-------
Tb : `~astropy.units.Quantity`
Brightness temperature, with default unit `au.K`
References
----------
- Brightness and Flux
http://www.cv.nrao.edu/course/astr534/Brightness.html
- Wikipedia: Brightness Temperature
https://en.wikipedia.org/wiki/Brightness_temperature
- NJIT: Physics 728: Introduction to Radio Astronomy: Lecture #1
https://web.njit.edu/~gary/728/Lecture1.html
- Astropy: Equivalencies: Brightness Temperature / Flux Density
http://docs.astropy.org/en/stable/units/equivalencies.html
"""
equiv = au.brightness_temperature(omega, freq)
Tb = Fnu.to(au.K, equivalencies=equiv)
return Tb
def Sb_to_Tb(Sb, freq):
"""
Convert surface brightness to brightness temperature, using the
Rayleigh-Jeans law, in the Rayleigh-Jeans limit.
Parameters
----------
Sb : `~astropy.units.Quantity`
Input surface brightness, e.g., `1.0*(au.Jy/au.arcsec**2)`
freq : `~astropy.units.Quantity`
Frequency where the flux density measured, e.g., `120*au.MHz`
Returns
-------
Tb : `~astropy.units.Quantity`
Brightness temperature, with default unit `au.K`
"""
omega = 1.0 * au.arcsec**2
Fnu = (Sb * omega).to(au.Jy) # [Jy]
return Fnu_to_Tb(Fnu, omega, freq)
@numba.jit(nopython=True)
def Sb_to_Tb_fast(Sb, freq):
"""
Convert surface brightness to brightness temperature, using the
Rayleigh-Jeans law, in the Rayleigh-Jeans limit.
This function does the calculations explicitly, and does NOT rely
on the `astropy.units`, therefore it is much faster. However, the
input parameters must be in right units.
Tb = Sb * c^2 / (2 * k_B * nu^2)
where `Sb` is the surface brightness density measured at a certain
frequency (unit: [ Jy/rad^2 ] = [ erg/s/cm^2/Hz/rad^2 ]).
Parameters
----------
Sb : float
Input surface brightness
Unit: [Jy/deg^2]
freq : float
Frequency where the flux density measured
Unit: [MHz]
Returns
-------
Tb : float
Calculated brightness temperature
Unit: [K]
"""
# NOTE: `radian` is dimensionless
rad2_to_deg2 = np.rad2deg(1.0) * np.rad2deg(1.0)
Sb_rad2 = Sb * rad2_to_deg2 # unit: [ Jy/rad^2 ] -> [ Jy ]
c = 29979245800.0 # speed of light, [ cm/s ]
k_B = 1.3806488e-16 # Boltzmann constant, [ erg/K ]
coef = 1e-35 # take care the unit conversions
Tb = coef * (Sb_rad2 * c*c) / (2 * k_B * freq*freq) # unit: [ K ]
return Tb
@numba.jit(nopython=True)
def Fnu_to_Tb_fast(Fnu, omega, freq):
"""
Convert flux density to brightness temperature, using the
Rayleigh-Jeans law, in the Rayleigh-Jeans limit.
This function does NOT invoke the `astropy.units`, therefore it is
much faster.
Parameters
----------
Fnu : float
Input flux density
Unit: [Jy] = 1e-23 [erg/s/cm^2/Hz] = 1e-26 [W/m^2/Hz]
omega : float
Source angular size/area
Unit: [deg^2]
freq : float
Frequency where the flux density measured
Unit: [MHz]
Returns
-------
Tb : float
Calculated brightness temperature
Unit: [K]
"""
Sb = Fnu / omega # [Jy/deg^2]
return Sb_to_Tb_fast(Sb, freq)
|
