path: root/fg21sim/extragalactic/clusters/helper.py

# 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
"""


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 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)
    slope = 2.63
    # intercept = 2.3 + np.random.normal(scale=0.05)
    intercept = 2.3
    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)
    A = 5.34
    # 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