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# Copyright (c) 2017 Weitian LI <weitian@aaronly.me>
# MIT license
"""
Helper functions
References
----------
.. [arnaud2005]
Arnaud, Pointecouteau & Pratt 2005, A&A, 441, 893;
http://adsabs.harvard.edu/abs/2005A%26A...441..893
.. [cassano2005]
Cassano & Brunetti 2005, MNRAS, 357, 1313
http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
.. [cassano2007]
Cassano et al. 2007, MNRAS, 378, 1565;
http://adsabs.harvard.edu/abs/2007MNRAS.378.1565C
.. [cassano2012]
Cassano et al. 2012, A&A, 548, A100
http://adsabs.harvard.edu/abs/2012A%26A...548A.100C
.. [zandanel2014]
Zandanel, Pfrommer & Prada 2014, MNRAS, 438, 124
http://adsabs.harvard.edu/abs/2014MNRAS.438..124Z
"""
import numpy as np
from scipy import integrate
from ...configs import CONFIGS
from ...utils import COSMO
from ...utils.units import (Units as AU,
Constants as AC,
UnitConversions as AUC)
def radius_virial(mass, z=0.0):
"""
Calculate the virial radius of a cluster at a given redshift.
Parameters
----------
mass : float
Total (virial) mass of the cluster
Unit: [Msun]
z : float, optional
Redshift
Default: 0.0 (i.e., present day)
Returns
-------
R_vir : float
Virial radius of the cluster
Unit: [kpc]
"""
Dc = COSMO.overdensity_virial(z)
rho = COSMO.rho_crit(z) # [g/cm^3]
R_vir = (3*mass*AUC.Msun2g / (4*np.pi * Dc * rho))**(1/3) # [cm]
R_vir *= AUC.cm2kpc # [kpc]
return R_vir
def radius_halo(mass, z=0.0):
"""
Calculate the radius of (giant) radio halo for a cluster.
The halo radius is assumed to linearly scale with the virial radius,
and is estimated by:
R_halo = R_vir / 4
* halo radius is ~3-6 times smaller than the virial radius;
Ref.[cassano2007],Sec.(1)
* halo half radius is ~R500/4, therefore, R_halo ~ R_vir/4;
Ref.[zandanel2014],Sec.(6.2)
Parameters
----------
mass : float
Total (virial) mass of the cluster
Unit: [Msun]
z : float, optional
Redshift
Default: 0.0 (i.e., present day)
Returns
-------
R_halo : float
Radius of the (expected) giant radio halo
Unit: [kpc]
"""
R_vir = radius_virial(mass=mass, z=z) # [kpc]
R_halo = R_vir / 4.0 # [kpc]
return R_halo
def mass_to_kT(mass, z=0.0):
"""
Calculate the cluster ICM temperature from its mass using the
mass-temperature scaling relation (its inversion used here)
derived from observations.
The following M-T scaling relation from Ref.[arnaud2005],Tab.2:
M200 * E(z) = A200 * (kT / 5 keV)^α ,
where:
A200 = (5.34 +/- 0.22) [1e14 Msun]
α = (1.72 +/- 0.10)
Its inversion:
kT = (5 keV) * [M200 * E(z) / A200]^(1/α).
NOTE: M200 (i.e., Δ=200) is used to approximate the virial mass.
Parameters
----------
mass : float
Total (virial) mass of the cluster.
Unit: [Msun]
z : float, optional
Redshift of the cluster
Returns
-------
kT : float
The ICM mean temperature.
Unit: [keV]
"""
# A = (5.34 + np.random.normal(scale=0.22)) * 1e14 # [Msun]
A = 5.34 * 1e14 # [Msun]
# alpha = 1.72 + np.random.normal(scale=0.10)
alpha = 1.72
Ez = COSMO.E(z)
kT = 5.0 * (mass * Ez / A) ** (1/alpha)
return kT
def density_number_thermal(mass, z=0.0):
"""
Calculate the number density of the ICM thermal plasma.
NOTE
----
This number density is independent of cluster (virial) mass,
but (mostly) increases with redshifts.
Parameters
----------
mass : float
Mass of the cluster
Unit: [Msun]
z : float, optional
Redshift
Returns
-------
n_th : float
Number density of the ICM thermal plasma
Unit: [cm^-3]
"""
N = mass * AUC.Msun2g * COSMO.baryon_fraction / (AC.mu * AC.u)
R_vir = radius_virial(mass, z) * AUC.kpc2cm # [cm]
volume = (4*np.pi / 3) * R_vir**3 # [cm^3]
n_th = N / volume # [cm^-3]
return n_th
def density_energy_thermal(mass, z=0.0):
"""
Calculate the thermal energy density of the ICM.
Returns
-------
e_th : float
Energy density of the ICM
Unit: [erg cm^-3]
"""
n_th = density_number_thermal(mass, z) # [cm^-3]
kT = mass_to_kT(mass, z) * AUC.keV2erg # [erg]
e_th = (3.0/2) * kT * n_th
return e_th
def density_energy_electron(spectrum, gamma):
"""
Calculate the energy density of relativistic electrons.
Parameters
----------
spectrum : 1D float `~numpy.ndarray`
The number density of the electrons w.r.t. Lorentz factors
Unit: [cm^-3]
gamma : 1D float `~numpy.ndarray`
The Lorentz factors of electrons
Returns
-------
e_re : float
The energy density of the relativistic electrons.
Unit: [erg cm^-3]
"""
e_re = integrate.trapz(spectrum*gamma*AU.mec2, gamma)
return e_re
def velocity_impact(M_main, M_sub, z=0.0):
"""
Estimate the relative impact velocity between the two merging
clusters when they are at a distance of the virial radius.
Parameters
----------
M_main, M_sub : float
Total (virial) masses of the main and sub clusters
Unit: [Msun]
z : float, optional
Redshift
Returns
-------
vi : float
Relative impact velocity
Unit: [km/s]
References
----------
Ref.[cassano2005],Eq.(9)
"""
eta_v = 4 * (1 + M_main/M_sub) ** (1/3)
R_vir = radius_virial(M_main, z) * AUC.kpc2cm # [cm]
vi = np.sqrt(2*AC.G * (1-1/eta_v) *
(M_main+M_sub)*AUC.Msun2g / R_vir) # [cm/s]
vi /= AUC.km2cm # [km/s]
return vi
def time_crossing(M_main, M_sub, z=0.0):
"""
Estimate the crossing time of the sub cluster during a merger.
NOTE: The crossing time is estimated to be τ ~ R_vir / v_impact.
Parameters
----------
M_main, M_sub : float
Total (virial) masses of the main and sub clusters
Unit: [Msun]
z : float, optional
Redshift
Returns
-------
time : float
Crossing time
Unit: [Gyr]
References
----------
Ref.[cassano2005],Sec.(4.1)
"""
R_vir = radius_virial(M_main, z) # [kpc]
vi = velocity_impact(M_main, M_sub, z) # [km/s]
# Unit conversion coefficient: [s kpc/km] => [Gyr]
uconv = AUC.kpc2km * AUC.s2Gyr
time = uconv * R_vir / vi # [Gyr]
return time
def magnetic_field(mass):
"""
Calculate the mean magnetic field strength according to the
scaling relation between magnetic field and cluster mass.
Parameters
----------
mass : float
Cluster mass
Unit: [Msun]
Returns
-------
B : float
The mean magnetic field strength
Unit: [uG]
References
----------
Ref.[cassano2012],Eq.(1)
"""
comp = "extragalactic/clusters"
b_mean = CONFIGS.getn(comp+"/b_mean")
b_index = CONFIGS.getn(comp+"/b_index")
M_mean = 1.6e15 # [Msun]
B = b_mean * (mass/M_mean) ** b_index
return B
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