# 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
"""
import numpy as np
from ...utils import cosmo
from ...utils.units import (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 derived from the virial radius using the scaling
relation in [cassano2007]_.
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]
References
----------
Ref.[cassano2007],Fig.(11)
"""
slope = 2.63 + np.random.normal(scale=0.5)
intercept = 2.3 + np.random.normal(scale=0.05)
R_vir = radius_virial(mass=mass, z=z) # [kpc]
R_halo = 10 ** (slope * np.log10(R_vir) + intercept)
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)
alpha = 1.72 + np.random.normal(scale=0.10)
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.
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 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