Module skytools.em_law
Module containing emission law functions. These produce the frequency scaling of different emission components relevant to the CMB observations.
Expand source code
#######################################################################
# This file is a part of SkyTools
#
# Sky Tools
# Copyright (C) 2023 Shamik Ghosh
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
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# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
#
# For more information about SkyTools please visit
# <https://github.com/1cosmologist/skytools> or contact Shamik Ghosh
# at shamik@lbl.gov
#
#########################################################################
"""
Module containing emission law functions. These produce the frequency scaling
of different emission components relevant to the CMB observations.
"""
import numpy as np
import scipy.constants as con
T_CMB = 2.72548 # K
def B_nu_T(nu_in_GHz, T_planck=T_CMB):
"""
B_nu_T is the Planck distribution function, defined as:
.. math::
B(\\nu, T) = \\frac{2 h \\nu^3}{c^2} \\frac{1}{e^{x} - 1},
with
.. math::
x = \\frac{h \\nu}{k_B T}.
It returns the Planck function values for nu (can be vectorized)
for a given blackbody temperature.
Parameters
----------
nu_in_GHz : float or numpy ndarray
Frequency in GHz at which we want the value of the Planck function.
T_planck : float, optional
Temperature of the Planck distribution in Kelvin.
Default value set to the CMB monopole temperature of 2.7255 K.
Returns
-------
float or numpy ndarray
A float value (or ndarray) for the value(s) of the Planck distribution.
"""
nu_in_Hz = nu_in_GHz * con.giga
x = con.h * nu_in_Hz / con.k / T_planck
prefactor = 2. * con.h * nu_in_Hz**3. / con.c**2.
return prefactor / (np.exp(x) - 1.)
def B_prime_nu_T(nu_in_GHz, T_planck=T_CMB):
"""
B_prime_nu_T is the derivative Planck distribution function defined as:
.. math::
\\frac{dB(\\nu, T)}{dT} = \\frac{h \\nu}{k_B T^2} \\frac{e^{x}}{e^{x} - 1} B(\\nu, T),
with
.. math::
x = \\frac{h \\nu}{k_B T}.
It returns the first derivative of the Planck function with respect to T,
for nu (can be vectorized) at a given blackbody temperature.
Parameters
----------
nu_in_GHz : float or numpy ndarray
Frequency in GHz at which we want the value of the Planck function
derivative.
T_planck : float, optional
Temperature of the Planck distribution in Kelvin.
Default value set to the CMB monopole temperature of 2.7255 K.
Returns
-------
float or numpy ndarray
A float value (or ndarray) for the value(s) of the differential Planck distribution.
"""
nu_in_Hz = nu_in_GHz * con.giga
x = con.h * nu_in_Hz / con.k / T_planck
prefactor = con.h * nu_in_Hz / con.k / T_planck**2.
return prefactor * np.exp(x) / (np.exp(x) - 1.) * B_nu_T(nu_in_GHz)
def ysz_spectral_law(nu_in_GHz):
"""
ysz_spectral_law is the SED function for Compton y parameter, defined as:
.. math::
y_{SZ} = \\frac{dB(\\nu, T)}{dT} \\left(\\frac{x}{\\tanh(\\frac{x}{2})} - 4\\right) T,
with
.. math::
x = \\frac{h \\nu}{k_B T}.
It returns the frequency scaling of y_SZ for nu (can be vectorized).
Parameters
----------
nu_in_GHz : float or numpy ndarray
Frequency in GHz at which we want the value of the y_SZ SED.
Notes
-----
Temperature is set to the CMB monopole temperature of 2.7255 K.
Returns
-------
float or numpy 1D array
A float value (or ndarray) for the value(s) of the y_SZ SED.
"""
nu_in_Hz = nu_in_GHz * con.giga
x = con.h * nu_in_Hz / con.k / T_CMB
g_nu_part = x / np.tanh(x/2.) - 4.
return B_prime_nu_T(nu_in_GHz) * g_nu_part * T_CMB
def greybody(nu_in_GHz, nu_ref_in_GHz, spec_ind, T_grey, flux_ref=1.):
"""
greybody is the SED function for a greybody distribution, defined as:
.. math::
I_\\nu = A \\left(\\frac{\\nu}{\\nu_0}\\right)^{\\beta} \\frac{B(\\nu, T_{grey})}{B(\\nu_0, T_{grey})}.
This function allows to set flux at reference frequency. If not used
the output is just the frequency scaling of the greybody.
Parameters
----------
nu_in_GHz : float or numpy 1D array
Frequency in GHz at which we want the value of the greybody function.
nu_ref_in_GHz : float
Greybody reference frequency in GHz.
spec_ind : float
Spectral index of the greybody.
T_grey : float
Greybody temperature in Kelvin.
flux_ref : float, optional
Amplitude A of greybody emissions at reference frequency in arbitrary units.
Default value is 1 (returns frequency scaling).
Returns
-------
float or numpy ndarray
A float value (or ndarray) for the value(s) of the greybody SED.
"""
# flux_ref sets the flux at reference frequency. If not set then takes default value of 1.
# This would give just the frequency scaling of the greybody.
powlaw = (nu_in_GHz / nu_ref_in_GHz)**(spec_ind)
return flux_ref * powlaw * B_nu_T(nu_in_GHz, T_planck=T_grey) / B_nu_T(nu_ref_in_GHz, T_planck=T_grey)
def powerlaw(nu_in_GHz, nu_ref_in_GHz, spec_ind=1.):
"""
powerlaw is the SED function for a powerlaw distribution, defined as:
.. math::
\\left(\\frac{\\nu}{\\nu_0}\\right)^\\beta.
This function outputs the frequency scaling of a powerlaw distribution.
Parameters
----------
nu_in_GHz : float or numpy 1D array
Frequency in GHz at which we want the value of the powerlaw function.
nu_ref_in_GHz : float
Powerlaw reference frequency in GHz.
spec_ind : float, optional
Spectral index of the powerlaw. Default value set to 1.
Returns
-------
float or numpy ndarray
A float value (or ndarray) for the value(s) of the powerlaw freqency
scaling.
"""
return (nu_in_GHz / nu_ref_in_GHz)**spec_ind
def modified_blackbody(nu_in_GHz, spec_ind, T_bb):
"""
modified_blackbody is the frequency scaling function for a modified blackbody
(MBB) distribution, defined as:
.. math::
\\nu^\\beta B(\\nu, T_{bb}).
Comparing with greybody function, this provides modified blackbody frequency scaling
without a reference frequency. In principle it return only the numerator part of
the greybody function.
Parameters
----------
nu_in_GHz : float or numpy ndarray
Frequency in GHz at which we want the value of the MBB function.
spec_ind : float
Spectral index of the MBB.
T_bb : float
MBB temperature in Kelvin.
Returns
-------
float or numpy ndarray
A float value (or ndarray) for the value(s) of the MBB frequency scaling.
"""
return (nu_in_GHz * con.giga)**spec_ind * B_nu_T(nu_in_GHz, T_planck=T_bb)Functions
def B_nu_T(nu_in_GHz, T_planck=2.72548)-
B_nu_T is the Planck distribution function, defined as: B(\nu, T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{x} - 1}, with x = \frac{h \nu}{k_B T}. It returns the Planck function values for nu (can be vectorized) for a given blackbody temperature.
Parameters
nu_in_GHz:floatornumpy ndarray- Frequency in GHz at which we want the value of the Planck function.
T_planck:float, optional- Temperature of the Planck distribution in Kelvin. Default value set to the CMB monopole temperature of 2.7255 K.
Returns
floatornumpy ndarray- A float value (or ndarray) for the value(s) of the Planck distribution.
Expand source code
def B_nu_T(nu_in_GHz, T_planck=T_CMB): """ B_nu_T is the Planck distribution function, defined as: .. math:: B(\\nu, T) = \\frac{2 h \\nu^3}{c^2} \\frac{1}{e^{x} - 1}, with .. math:: x = \\frac{h \\nu}{k_B T}. It returns the Planck function values for nu (can be vectorized) for a given blackbody temperature. Parameters ---------- nu_in_GHz : float or numpy ndarray Frequency in GHz at which we want the value of the Planck function. T_planck : float, optional Temperature of the Planck distribution in Kelvin. Default value set to the CMB monopole temperature of 2.7255 K. Returns ------- float or numpy ndarray A float value (or ndarray) for the value(s) of the Planck distribution. """ nu_in_Hz = nu_in_GHz * con.giga x = con.h * nu_in_Hz / con.k / T_planck prefactor = 2. * con.h * nu_in_Hz**3. / con.c**2. return prefactor / (np.exp(x) - 1.) def B_prime_nu_T(nu_in_GHz, T_planck=2.72548)-
B_prime_nu_T is the derivative Planck distribution function defined as: \frac{dB(\nu, T)}{dT} = \frac{h \nu}{k_B T^2} \frac{e^{x}}{e^{x} - 1} B(\nu, T), with x = \frac{h \nu}{k_B T}. It returns the first derivative of the Planck function with respect to T, for nu (can be vectorized) at a given blackbody temperature.
Parameters
nu_in_GHz:floatornumpy ndarray- Frequency in GHz at which we want the value of the Planck function derivative.
T_planck:float, optional- Temperature of the Planck distribution in Kelvin. Default value set to the CMB monopole temperature of 2.7255 K.
Returns
floatornumpy ndarray- A float value (or ndarray) for the value(s) of the differential Planck distribution.
Expand source code
def B_prime_nu_T(nu_in_GHz, T_planck=T_CMB): """ B_prime_nu_T is the derivative Planck distribution function defined as: .. math:: \\frac{dB(\\nu, T)}{dT} = \\frac{h \\nu}{k_B T^2} \\frac{e^{x}}{e^{x} - 1} B(\\nu, T), with .. math:: x = \\frac{h \\nu}{k_B T}. It returns the first derivative of the Planck function with respect to T, for nu (can be vectorized) at a given blackbody temperature. Parameters ---------- nu_in_GHz : float or numpy ndarray Frequency in GHz at which we want the value of the Planck function derivative. T_planck : float, optional Temperature of the Planck distribution in Kelvin. Default value set to the CMB monopole temperature of 2.7255 K. Returns ------- float or numpy ndarray A float value (or ndarray) for the value(s) of the differential Planck distribution. """ nu_in_Hz = nu_in_GHz * con.giga x = con.h * nu_in_Hz / con.k / T_planck prefactor = con.h * nu_in_Hz / con.k / T_planck**2. return prefactor * np.exp(x) / (np.exp(x) - 1.) * B_nu_T(nu_in_GHz) def greybody(nu_in_GHz, nu_ref_in_GHz, spec_ind, T_grey, flux_ref=1.0)-
greybody is the SED function for a greybody distribution, defined as: I_\nu = A \left(\frac{\nu}{\nu_0}\right)^{\beta} \frac{B(\nu, T_{grey})}{B(\nu_0, T_{grey})}. This function allows to set flux at reference frequency. If not used the output is just the frequency scaling of the greybody.
Parameters
nu_in_GHz:floatornumpy 1D array- Frequency in GHz at which we want the value of the greybody function.
nu_ref_in_GHz:float- Greybody reference frequency in GHz.
spec_ind:float- Spectral index of the greybody.
T_grey:float- Greybody temperature in Kelvin.
flux_ref:float, optional- Amplitude A of greybody emissions at reference frequency in arbitrary units. Default value is 1 (returns frequency scaling).
Returns
floatornumpy ndarray- A float value (or ndarray) for the value(s) of the greybody SED.
Expand source code
def greybody(nu_in_GHz, nu_ref_in_GHz, spec_ind, T_grey, flux_ref=1.): """ greybody is the SED function for a greybody distribution, defined as: .. math:: I_\\nu = A \\left(\\frac{\\nu}{\\nu_0}\\right)^{\\beta} \\frac{B(\\nu, T_{grey})}{B(\\nu_0, T_{grey})}. This function allows to set flux at reference frequency. If not used the output is just the frequency scaling of the greybody. Parameters ---------- nu_in_GHz : float or numpy 1D array Frequency in GHz at which we want the value of the greybody function. nu_ref_in_GHz : float Greybody reference frequency in GHz. spec_ind : float Spectral index of the greybody. T_grey : float Greybody temperature in Kelvin. flux_ref : float, optional Amplitude A of greybody emissions at reference frequency in arbitrary units. Default value is 1 (returns frequency scaling). Returns ------- float or numpy ndarray A float value (or ndarray) for the value(s) of the greybody SED. """ # flux_ref sets the flux at reference frequency. If not set then takes default value of 1. # This would give just the frequency scaling of the greybody. powlaw = (nu_in_GHz / nu_ref_in_GHz)**(spec_ind) return flux_ref * powlaw * B_nu_T(nu_in_GHz, T_planck=T_grey) / B_nu_T(nu_ref_in_GHz, T_planck=T_grey) def modified_blackbody(nu_in_GHz, spec_ind, T_bb)-
modified_blackbody is the frequency scaling function for a modified blackbody (MBB) distribution, defined as: \nu^\beta B(\nu, T_{bb}). Comparing with greybody function, this provides modified blackbody frequency scaling without a reference frequency. In principle it return only the numerator part of the greybody function.
Parameters
nu_in_GHz:floatornumpy ndarray- Frequency in GHz at which we want the value of the MBB function.
spec_ind:float- Spectral index of the MBB.
T_bb:float- MBB temperature in Kelvin.
Returns
floatornumpy ndarray- A float value (or ndarray) for the value(s) of the MBB frequency scaling.
Expand source code
def modified_blackbody(nu_in_GHz, spec_ind, T_bb): """ modified_blackbody is the frequency scaling function for a modified blackbody (MBB) distribution, defined as: .. math:: \\nu^\\beta B(\\nu, T_{bb}). Comparing with greybody function, this provides modified blackbody frequency scaling without a reference frequency. In principle it return only the numerator part of the greybody function. Parameters ---------- nu_in_GHz : float or numpy ndarray Frequency in GHz at which we want the value of the MBB function. spec_ind : float Spectral index of the MBB. T_bb : float MBB temperature in Kelvin. Returns ------- float or numpy ndarray A float value (or ndarray) for the value(s) of the MBB frequency scaling. """ return (nu_in_GHz * con.giga)**spec_ind * B_nu_T(nu_in_GHz, T_planck=T_bb) def powerlaw(nu_in_GHz, nu_ref_in_GHz, spec_ind=1.0)-
powerlaw is the SED function for a powerlaw distribution, defined as: \left(\frac{\nu}{\nu_0}\right)^\beta. This function outputs the frequency scaling of a powerlaw distribution.
Parameters
nu_in_GHz:floatornumpy 1D array- Frequency in GHz at which we want the value of the powerlaw function.
nu_ref_in_GHz:float- Powerlaw reference frequency in GHz.
spec_ind:float, optional- Spectral index of the powerlaw. Default value set to 1.
Returns
floatornumpy ndarray- A float value (or ndarray) for the value(s) of the powerlaw freqency scaling.
Expand source code
def powerlaw(nu_in_GHz, nu_ref_in_GHz, spec_ind=1.): """ powerlaw is the SED function for a powerlaw distribution, defined as: .. math:: \\left(\\frac{\\nu}{\\nu_0}\\right)^\\beta. This function outputs the frequency scaling of a powerlaw distribution. Parameters ---------- nu_in_GHz : float or numpy 1D array Frequency in GHz at which we want the value of the powerlaw function. nu_ref_in_GHz : float Powerlaw reference frequency in GHz. spec_ind : float, optional Spectral index of the powerlaw. Default value set to 1. Returns ------- float or numpy ndarray A float value (or ndarray) for the value(s) of the powerlaw freqency scaling. """ return (nu_in_GHz / nu_ref_in_GHz)**spec_ind def ysz_spectral_law(nu_in_GHz)-
ysz_spectral_law is the SED function for Compton y parameter, defined as: y_{SZ} = \frac{dB(\nu, T)}{dT} \left(\frac{x}{\tanh(\frac{x}{2})} - 4\right) T, with x = \frac{h \nu}{k_B T}. It returns the frequency scaling of y_SZ for nu (can be vectorized).
Parameters
nu_in_GHz:floatornumpy ndarray- Frequency in GHz at which we want the value of the y_SZ SED.
Notes
Temperature is set to the CMB monopole temperature of 2.7255 K.
Returns
floatornumpy 1D array- A float value (or ndarray) for the value(s) of the y_SZ SED.
Expand source code
def ysz_spectral_law(nu_in_GHz): """ ysz_spectral_law is the SED function for Compton y parameter, defined as: .. math:: y_{SZ} = \\frac{dB(\\nu, T)}{dT} \\left(\\frac{x}{\\tanh(\\frac{x}{2})} - 4\\right) T, with .. math:: x = \\frac{h \\nu}{k_B T}. It returns the frequency scaling of y_SZ for nu (can be vectorized). Parameters ---------- nu_in_GHz : float or numpy ndarray Frequency in GHz at which we want the value of the y_SZ SED. Notes ----- Temperature is set to the CMB monopole temperature of 2.7255 K. Returns ------- float or numpy 1D array A float value (or ndarray) for the value(s) of the y_SZ SED. """ nu_in_Hz = nu_in_GHz * con.giga x = con.h * nu_in_Hz / con.k / T_CMB g_nu_part = x / np.tanh(x/2.) - 4. return B_prime_nu_T(nu_in_GHz) * g_nu_part * T_CMB