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diff --git a/fg21sim/extragalactic/clusters/helper.py b/fg21sim/extragalactic/clusters/helper.py
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+# Copyright (c) 2017 Weitian LI <weitian@aaronly.me>
+# MIT license
+
+"""
+Helper functions
+
+References
+----------
+.. [cassano2007]
+   Cassano et al. 2007, MNRAS, 378, 1565;
+   http://adsabs.harvard.edu/abs/2007MNRAS.378.1565C
+.. [arnaud2005]
+   Arnaud, Pointecouteau & Pratt 2005, A&A, 441, 893;
+   http://adsabs.harvard.edu/abs/2005A%26A...441..893
+"""
+
+
+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