# Copyright (c) 2017 Weitian LI # MIT license """ Calculate the synchrotron emission and inverse Compton emission for simulated radio halos. References ---------- .. [cassano2005] Cassano & Brunetti 2005, MNRAS, 357, 1313 http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C Appendix.C .. [era2016] Condon & Ransom 2016 Essential Radio Astronomy https://science.nrao.edu/opportunities/courses/era/ Chapter.5 """ import logging import numpy as np import scipy.special from scipy import integrate from scipy import interpolate from ...utils import COSMO from ...utils.units import (Units as AU, UnitConversions as AUC, Constants as AC) from ...utils.convert import Fnu_to_Tb_fast logger = logging.getLogger(__name__) class SynchrotronEmission: """ Calculate the synchrotron emissivity from a given population of electrons. Parameters ---------- gamma : `~numpy.ndarray` The Lorentz factors of electrons. n_e : `~numpy.ndarray` Electron number density spectrum. Unit: [cm^-3] z : float Redshift of the cluster/halo been observed/simulated. B : float The assumed uniform magnetic field within the cluster ICM. Unit: [uG] radius : float The radius of the halo, within which the uniform magnetic field and electron distribution are assumed. Unit: [kpc] """ def __init__(self, gamma, n_e, z, B, radius): self.gamma = np.asarray(gamma) self.n_e = np.asarray(n_e) self.z = z self.B = B # [uG] self.radius = radius # [kpc] @property def B_gauss(self): """ Magnetic field in unit of [G] (i.e., gauss) """ return self.B * 1e-6 # [uG] -> [G] @property def frequency_larmor(self): """ Electron Larmor frequency (a.k.a. gyro frequency): ν_L = e * B / (2*π * m0 * c) = e * B / (2*π * mec) => ν_L [MHz] = 2.8 * B [G] Unit: [MHz] """ nu_larmor = AC.e * self.B_gauss / (2*np.pi * AU.mec) # [Hz] return nu_larmor * 1e-6 # [Hz] -> [MHz] def frequency_crit(self, gamma, theta=np.pi/2): """ Synchrotron critical frequency. Critical frequency: ν_c = (3/2) * γ^2 * sin(θ) * ν_L Parameters ---------- gamma : `~numpy.ndarray` Electron Lorentz factors γ theta : `~numpy.ndarray`, optional The angles between the electron velocity and the magnetic field. Unit: [rad] Returns ------- nu_c : `~numpy.ndarray` Critical frequencies Unit: [MHz] """ nu_c = 1.5 * gamma**2 * np.sin(theta) * self.frequency_larmor return nu_c def F(self, x): """ Synchrotron kernel function. NOTE ---- * Use interpolation to optimize the speed, also avoid instabilities near the lower end (e.g., x < 1e-5). * Interpolation also helps vectorize this function for easier calling. * Cache the interpolation results, since this function will be called multiple times for each frequency. Parameters ---------- x : `~numpy.ndarray` Points where to calculate the kernel function values. NOTE: X values will be bounded, e.g., within [1e-5, 20] Returns ------- y : `~numpy.ndarray` Calculated kernel function values. """ if not hasattr(self, "_F_interp"): # Interpolate the kernel function and cache the results # # The lower and upper cuts xmin = 1e-5 xmax = 20.0 # Number of samples within [xmin, xmax] # NOTE: this kernel function is quiet smooth and slow-varying. nsamples = 128 # Make an interpolation xx = np.logspace(np.log10(xmin), np.log10(xmax), num=nsamples) Fxx = [xp * integrate.quad(lambda t: scipy.special.kv(5/3, t), a=xp, b=np.inf)[0] for xp in xx] self._F_interp = interpolate.interp1d(xx, Fxx, kind="quadratic") x = np.array(x) # Make a copy! x[x < xmin] = xmin x[x > xmax] = xmax y = self._F_interp(x) return y def emissivity(self, frequencies): """ Calculate the synchrotron emissivity (power emitted per volume and per frequency) at the requested frequency. NOTE ---- Since ``self.gamma`` and ``self.n_e`` are sampled on a logarithmic grid, we integrate over ``ln(gamma)`` instead of ``gamma`` directly: I = int_gmin^gmax f(g) d(g) = int_ln(gmin)^ln(gmax) f(g) g d(ln(g)) Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the synchrotron emissivity. Unit: [MHz] Returns ------- syncem : float, or 1D `~numpy.ndarray` The calculated synchrotron emissivity at each specified frequency. Unit: [erg/s/cm^3/Hz] """ j_coef = np.sqrt(3) * AC.e**3 * self.B_gauss / AU.mec2 # Ignore the zero angle theta = np.linspace(0, np.pi/2, num=len(self.gamma))[1:] theta_grid, gamma_grid = np.meshgrid(theta, self.gamma) nu_c = self.frequency_crit(gamma_grid, theta_grid) # 2D grid of ``n_e(gamma) * sin^2(theta)`` nsin2 = np.outer(self.n_e, np.sin(theta)**2) frequencies = np.array(frequencies, ndmin=1) syncem = np.zeros(shape=frequencies.shape) for i, freq in zip(range(len(frequencies)), frequencies): logger.debug("Calc synchrotron emissivity at %.2f [MHz]" % freq) kernel = self.F(freq / nu_c) # 2D samples over width to do the integration s2d = kernel * nsin2 # Integrate over ``theta`` (the last axis) s1d = integrate.simps(s2d, x=theta) # Integrate over energy ``gamma`` in logarithmic grid syncem[i] = j_coef * integrate.simps(s1d*self.gamma, np.log(self.gamma)) if len(syncem) == 1: return syncem[0] else: return syncem def power(self, nu): """ Calculate the synchrotron power (power emitted per frequency) at the requested frequency. Returns ------- P_nu : float Synchrotron power at frequency ``nu``. Unit: [erg/s/Hz] """ r_cm = self.radius * AUC.kpc2cm volume = (4.0/3.0) * np.pi * r_cm**3 P_nu = self.emissivity(nu) * volume return P_nu def flux(self, nu): """ Calculate the synchrotron flux (power observed per frequency) at the requested frequency. Returns ------- F_nu : float Synchrotron flux at frequency ``nu``. Unit: [Jy] = 1e-23 [erg/s/cm^2/Hz] """ DL = COSMO.DL(self.z) * AUC.Mpc2cm # [cm] P_nu = self.power(nu) F_nu = 1e23 * P_nu / (4*np.pi * DL*DL) # [Jy] return F_nu def brightness(self, nu, pixelsize): """ Calculate the synchrotron surface brightness (power observed per frequency and per solid angle) at the specified frequency. NOTE ---- If the radio halo has solid angle less than the pixel area, then it is assumed to have solid angle of 1 pixel. Parameters ---------- pixelsize : float The pixel size of the output simulated sky image Unit: [arcsec] Returns ------- Tb : float Synchrotron surface brightness at frequency ``nu``. Unit: [K] <-> [Jy/pixel] """ DA = COSMO.DL(self.z) * AUC.Mpc2cm # [cm] radius = self.radius * AUC.kpc2cm # [cm] omega = (np.pi * radius**2 / DA**2) * AUC.rad2deg**2 # [deg^2] pixelarea = (pixelsize * AUC.arcsec2deg) ** 2 # [deg^2] if omega < pixelarea: omega = pixelarea F_nu = self.flux(nu) Tb = Fnu_to_Tb_fast(F_nu, omega, nu) return Tb