From 93dc96a0a6a336419b2e80b221257a8a7532019a Mon Sep 17 00:00:00 2001
From: Aaron LI <aaronly.me@outlook.com>
Date: Mon, 16 Jan 2017 21:10:30 +0800
Subject: Add clusters/emission.py: Calculate synchrotron emissivity

Calculate the synchrotron emissivity from the given electron spectrum at
specified frequency.

Add the electron charge to units.Constants
---
 fg21sim/extragalactic/clusters/emission.py | 117 +++++++++++++++++++++++++++++
 fg21sim/extragalactic/clusters/units.py    |   6 ++
 2 files changed, 123 insertions(+)
 create mode 100644 fg21sim/extragalactic/clusters/emission.py

diff --git a/fg21sim/extragalactic/clusters/emission.py b/fg21sim/extragalactic/clusters/emission.py
new file mode 100644
index 0000000..e71632b
--- /dev/null
+++ b/fg21sim/extragalactic/clusters/emission.py
@@ -0,0 +1,117 @@
+# Copyright (c) 2017 Weitian LI <liweitianux@live.com>
+# MIT license
+
+"""
+Calculate the synchrotron emission for simulated radio halos.
+"""
+
+import logging
+
+import numpy as np
+import scipy.integrate
+import scipy.special
+
+from .units import (Units as AU, Constants as AC)
+
+
+logger = logging.getLogger(__name__)
+
+
+class SynchrotronEmission:
+    """
+    Calculate the synchrotron emission from a given population
+    of electrons.
+
+    References
+    ----------
+    [1] Cassano & Brunetti 2005, MNRAS, 357, 1313
+        http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C
+        Appendix.C
+    """
+    def __init__(self, B):
+        # Uniform magnetic field strength
+        self.B = B  # [uG]
+
+    @property
+    def frequency_larmor(self):
+        """
+        Electron Larmor frequency:
+            ν_L = e * B / (2*π * m0 * c) = e * B / (2*π * mec)
+
+        Unit: MHz
+        """
+        coef = AC.e / (2*np.pi * AU.mec)  # [Hz/G]
+        coef *= 1e-12  # [MHz/uG]
+        nu = coef * self.B  # [MHz]
+        return nu
+
+    def frequency_crit(self, p, theta=np.pi/2):
+        """
+        Synchrotron critical frequency.
+
+        Critical frequency:
+            ν_c = (3/2) * γ^2 * sin(θ) * ν_L
+
+        Parameters
+        ----------
+        p : float
+            Electron momentum (unit: mec), i.e., Lorentz factor γ
+        theta : float, optional
+            The angle between the electron velocity and the magnetic field.
+            (unit: radian)
+
+        Returns
+        -------
+        nu : float
+            Critical frequency, unit: MHz
+        """
+        nu_L = self.frequency_larmor
+        nu = (3/2) * p**2 * np.sin(theta) * nu_L
+        return nu
+
+    def F(self, x):
+        """
+        Synchrotron kernel function.
+        """
+        val = x * scipy.integrate.quad(lambda t: scipy.special.kv(5/3, t),
+                                       a=x, b=np.inf)[0]
+        return val
+
+    def emissivity(self, nu, p, n_e):
+        """
+        Calculate the synchrotron emissivity (power emitted per volume
+        and per frequency) at the specified frequency from the given
+        electron number & energy distribution
+
+        Parameters
+        ----------
+        nu : float
+            Frequency (unit: MHz) where to calculate the emissivity.
+        p : `~numpy.ndarray`
+            The momentum grid adopted when solving the Fokker-Planck equation.
+            Unit: [mec]
+        n_e : `~numpy.ndarray`
+            Electron spectrum by solving the Fokker-Planck equation.
+            Unit: [cm^-3 mec^-1]
+
+        Returns
+        -------
+        j_nu : float
+            Synchrotron emissivity at frequency ``nu``.
+            Unit: [erg/s/cm^3/Hz]
+        """
+        def func(theta, _p, _n_e):
+            nu_c = self.frequency_crit(_p, theta)
+            x = nu / nu_c
+            return (np.sin(theta)**2 * _n_e * self.F(x))
+
+        coef = np.sqrt(3) * AC.e**3 * self.B / AC.c  # multiplied a [mec]
+        func_p = np.zeros(p.shape)
+        for i in range(len(p)):
+            # Integrate over ``theta``
+            func_p[i] = scipy.integrate.quad(
+                lambda t: func(t, p[i], n_e[i]),
+                a=0, b=np.pi/2)[0]
+        # Integrate over ``p``
+        j_nu = coef * scipy.integrate.trapz(func_p, p)
+        return j_nu
diff --git a/fg21sim/extragalactic/clusters/units.py b/fg21sim/extragalactic/clusters/units.py
index 01891fa..a7e4ef5 100644
--- a/fg21sim/extragalactic/clusters/units.py
+++ b/fg21sim/extragalactic/clusters/units.py
@@ -30,10 +30,13 @@ class UnitConversions:
     Hold the conversion relations directly to avoid repeated/unnecessary
     calculations.
     """
+    # Mass
     Msun2g = au.solMass.to(au.g)
     g2Msun = au.g.to(au.solMass)
+    # Time
     Gyr2s = au.Gyr.to(au.s)
     s2Gyr = au.s.to(au.Gyr)
+    # Length
     kpc2cm = au.kpc.to(au.cm)
     cm2kpc = au.cm.to(au.kpc)
     Mpc2cm = au.Mpc.to(au.cm)
@@ -43,6 +46,7 @@ class UnitConversions:
     kpc2km = au.kpc.to(au.km)
     km2kpc = au.km.to(au.kpc)
     km2cm = au.km.to(au.cm)
+    # Energy
     keV2erg = au.keV.to(au.erg)
 
 
@@ -60,6 +64,8 @@ class Constants:
     u = ac.u.cgs.value  # [g]
     # Gravitational constant
     G = ac.G.cgs.value  # [cm^3/g/s^2]
+    # Electron charge
+    e = ac.e.gauss.value  # [Fr] = [esu]
 
     # Mean molecular weight
     # Ref.: Ettori et al, 2013, Space Science Review, 177, 119-154, Eq.(6)
-- 
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