# Copyright (c) 2017 Weitian LI # MIT license """ Simulate (giant) radio halo originating from the last/recent cluster-cluster major merger event, following the "statistical magneto-turbulent model" proposed by [cassano2005]_, but with many modifications and simplifications. References ---------- .. [brunetti2011] Brunetti & Lazarian 2011, MNRAS, 410, 127 http://adsabs.harvard.edu/abs/2011MNRAS.410..127B .. [brunetti2016] Brunetti 2016, PPCF, 58, 014011 http://adsabs.harvard.edu/abs/2016PPCF...58a4011B .. [cassano2005] Cassano & Brunetti 2005, MNRAS, 357, 1313 http://adsabs.harvard.edu/abs/2005MNRAS.357.1313C .. [cassano2006] Cassano, Brunetti & Setti, 2006, MNRAS, 369, 1577 http://adsabs.harvard.edu/abs/2006MNRAS.369.1577C .. [cassano2012] Cassano et al. 2012, A&A, 548, A100 http://adsabs.harvard.edu/abs/2012A%26A...548A.100C .. [donnert2013] Donnert 2013, AN, 334, 615 http://adsabs.harvard.edu/abs/2013AN....334..515D .. [donnert2014] Donnert & Brunetti 2014, MNRAS, 443, 3564 http://adsabs.harvard.edu/abs/2014MNRAS.443.3564D .. [hogg1999] Hogg 1999, arXiv:astro-ph/9905116 http://adsabs.harvard.edu/abs/1999astro.ph..5116H .. [miniati2015] Miniati & Beresnyak 2015, Nature, 523, 59 http://adsabs.harvard.edu/abs/2015Natur.523...59M .. [sarazin1999] Sarazin 1999, ApJ, 520, 529 http://adsabs.harvard.edu/abs/1999ApJ...520..529S """ import logging from functools import lru_cache import numpy as np from . import helper from .solver import FokkerPlanckSolver from .emission import SynchrotronEmission from ...share import CONFIGS, COSMO from ...utils.units import (Units as AU, UnitConversions as AUC, Constants as AC) from ...utils.convert import Fnu_to_Tb logger = logging.getLogger(__name__) class RadioHalo: """ Simulate the extended radio halo emission from the galaxy cluster experiencing on-going/recent merger. Description ----------- 1. Calculate the merger crossing time (t_cross; ~1 Gyr); 2. Calculate the diffusion coefficient (Dpp) from the systematic acceleration timescale (tau_acc; ~0.1 Gyr). The acceleration diffusion is assumed to have an action time ~ t_cross (i.e., only during merger crossing), and then been disabled (i.e., only radiation and ionization losses later); 3. Assume the electrons are constantly injected and has a power-law energy spectrum, determine the injection rate by further assuming that the total injected electrons has energy of a fraction (eta_e) of the ICM total thermal energy; 4. Set the initial electron density/spectrum be the total injected electrons during t_merger time; 5. Calculate the magnetic field from the cluster total mass (which is assumed to be growth linearly from M_main+M_sub to M_obs); 6. Calculate the energy losses for the coefficients of Fokker-Planck equation; 7. Solve the Fokker-Planck equation to derive the relativistic electron spectrum at t_obs (i.e., z_obs); 8. Calculate the synchrotron emissivity from the derived electron spectrum. Parameters ---------- M_obs : float Cluster virial mass at the current observation (simulation end) time. Unit: [Msun] z_obs : float Redshift of the current observation (simulation end) time. M_main, M_sub : float The main and sub cluster masses before the (major) merger. Unit: [Msun] z_merger : float The redshift when the (major) merger begins. Attributes ---------- fpsolver : `~FokkerPlanckSolver` The solver instance to calculate the electron spectrum evolution. radius : float The halo radius Unit: [kpc] gamma : 1D float `~numpy.ndarray` The Lorentz factors of the adopted logarithmic grid to solve the equation. electron_spec : 1D float `~numpy.ndarray` The derived electron (number density) distribution/spectrum at the final time (``zend``), which is set by the methods ``self.calc_electron_spectrum()`` or ``self.set_electron_spectrum()``. Unit: [cm^-3] """ def __init__(self, M_obs, z_obs, M_main, M_sub, z_merger, configs=CONFIGS): self.M_obs = M_obs self.z_obs = z_obs self.M_main = M_main self.M_sub = M_sub self.z_merger = z_merger self.configs = configs self._set_configs() self._set_solver() def _set_configs(self): comp = "extragalactic/halos" self.f_lturb = self.configs.getn(comp+"/f_lturb") self.f_acc = self.configs.getn(comp+"/f_acc") self.eta_turb = self.configs.getn(comp+"/eta_turb") self.eta_e = self.configs.getn(comp+"/eta_e") self.gamma_min = self.configs.getn(comp+"/gamma_min") self.gamma_max = self.configs.getn(comp+"/gamma_max") self.gamma_np = self.configs.getn(comp+"/gamma_np") self.buffer_np = self.configs.getn(comp+"/buffer_np") self.time_step = self.configs.getn(comp+"/time_step") self.injection_index = self.configs.getn(comp+"/injection_index") def _set_solver(self): self.fpsolver = FokkerPlanckSolver( xmin=self.gamma_min, xmax=self.gamma_max, x_np=self.gamma_np, tstep=self.time_step, f_advection=self.fp_advection, f_diffusion=self.fp_diffusion, f_injection=self.fp_injection, buffer_np=self.buffer_np, ) @property @lru_cache() def gamma(self): """ The logarithmic grid adopted for solving the equation. """ return self.fpsolver.x @property def age_obs(self): return COSMO.age(self.z_obs) @property def age_merger(self): return COSMO.age(self.z_merger) @property def tback_merger(self): """ The time from the observation (``z_obs``) back to the merger (``z_merger``). """ return (self.age_obs - self.age_merger) # [Gyr] @property @lru_cache() def time_crossing(self): """ The time duration of the sub-cluster crossing the main cluster, which is also used to approximate the merging time, during which the turbulence acceleration is regarded as effective. Unit: [Gyr] """ return helper.time_crossing(self.M_main, self.M_sub, z=self.z_merger) @property def radius_virial_obs(self): """ The virial radius of the "current" cluster (``M_obs``) at ``z_obs``. Unit: [kpc] """ return helper.radius_virial(mass=self.M_obs, z=self.z_obs) @property def radius_virial_main(self): """ The virial radius of the main cluster at ``z_merger``. """ return helper.radius_virial(mass=self.M_main, z=self.z_merger) @property def radius_virial_sub(self): return helper.radius_virial(mass=self.M_sub, z=self.z_merger) @property @lru_cache() def radius(self): """ The estimated radius for the simulated radio halo. NOTE ---- The halo radius is assumed to be the virial radius of the falling sub-cluster. See ``helper.radius_halo()`` for more details. Unit: [kpc] """ r_halo = helper.radius_halo(self.M_main, self.M_sub, self.z_merger) return r_halo @property def angular_radius(self): """ The angular radius of the radio halo. Unit: [arcsec] """ DA = COSMO.DA(self.z_obs) * 1e3 # [Mpc] -> [kpc] theta = self.radius / DA # [rad] return theta * AUC.rad2arcsec @property def volume(self): """ The halo volume, calculated from the above radius. Unit: [kpc^3] """ return (4*np.pi/3) * self.radius**3 @property @lru_cache() def B_obs(self): """ The magnetic field strength at the simulated observation time (i.e., cluster mass of ``self.M_obs``), will be used to calculate the synchrotron emissions. Unit: [uG] """ return helper.magnetic_field(mass=self.M_obs, z=self.z_obs) @property @lru_cache() def kT_main(self): """ The mean temperature of the main cluster ICM at ``z_merger`` when the merger begins. Unit: [keV] """ return helper.kT_cluster(mass=self.M_main, z=self.z_merger, configs=self.configs) @property @lru_cache() def kT_sub(self): return helper.kT_cluster(mass=self.M_sub, z=self.z_merger, configs=self.configs) @property @lru_cache() def kT_obs(self): """ The "current" cluster ICM mean temperature at ``z_obs``. """ return helper.kT_cluster(self.M_obs, z=self.z_obs, configs=self.configs) # [keV] @property @lru_cache() def Mach_turbulence(self): """ The Mach number of the merger-induced turbulence. The turbulence Mach number: Mach_turb = sqrt(<δv>^2) / c_s ≅ sqrt(sqrt(3)/α) * sqrt(η_turb/0.37) where: c_s is the sound speed, α is a parameter ranges about 1.5-3, and we take it as: α = 3^(3/2) / 2 ≅ 2.6 η_turb describes the fraction of thermal energy originating from turbulent dissipation, ~0.3. Reference: Ref.[miniati2015],Eq.(1) """ alpha = 3**1.5 / 2 mach = np.sqrt(3**0.5 * self.eta_turb / alpha / 0.37) return mach @property @lru_cache() def tau_acceleration(self): """ Calculate the electron acceleration timescale due to turbulent waves at the given (cosmic) time, which describes the turbulent acceleration efficiency. Unit: [Gyr] NOTE ---- Generally, the turbulent acceleration timescale is about 0.1 Gyr. It is shown that this acceleration timescale depends weakly on cluster mass and redshift, therefore, its value is derived at the beginning of the merger and assumed to be constant throughout the merging period. Reference: Ref.[brunetti2016],Eq.(8,9) """ Mach = self.Mach_turbulence Rvir = helper.radius_virial(mass=self.M_main, z=self.z_merger) cs = helper.speed_sound(self.kT_main) # [km/s] # Turbulence injection scale L0 = self.f_lturb * Rvir # [kpc] x = cs*AUC.km2cm / AC.c fx = x * (x**4/4 + x*x - (1+2*x*x) * np.log(x) - 5/4) term1 = self.f_acc * 2.5 / fx / (Mach/0.5)**4 term2 = (L0/300) / (cs/1500) tau = term1 * term2 / 1000 # [Gyr] return tau @property @lru_cache() def injection_rate(self): """ The constant electron injection rate assumed. Unit: [cm^-3 Gyr^-1] The injection rate is parametrized by assuming that the total energy injected in the relativistic electrons during the cluster life (e.g., ``age_obs`` here) is a fraction (``self.eta_e``) of the total thermal energy of the cluster. The electrons are assumed to be injected throughout the cluster ICM/volume, i.e., do not restricted inside the halo volume. Qe(γ) = Ke * γ^(-s), int[ Qe(γ) γ me c^2 ]dγ * t_cluster = eta_e * e_th => Ke = [(s-2) * eta_e * e_th * γ_min^(s-2) / (me * c^2 * t_cluster)] References ---------- Ref.[cassano2005],Eqs.(31,32,33) """ s = self.injection_index e_thermal = helper.density_energy_thermal(self.M_obs, self.z_obs) term1 = (s-2) * self.eta_e * e_thermal # [erg cm^-3] term2 = self.gamma_min**(s-2) term3 = AU.mec2 * self.age_obs # [erg Gyr] Ke = term1 * term2 / term3 # [cm^-3 Gyr^-1] return Ke def calc_electron_spectrum(self, tstart=None, tstop=None, n0_e=None): """ Calculate the relativistic electron spectrum by solving the Fokker-Planck equation. Parameters ---------- tstart : float, optional The (cosmic) time from when to solve the Fokker-Planck equation for relativistic electrons evolution. Default: ``self.age_merger``. Unit: [Gyr] tstop : float, optional The (cosmic) time when to derive final relativistic electrons spectrum for synchrotron emission calculations. Default: ``self.age_obs``. Unit: [Gyr] n0_e : 1D `~numpy.ndarray`, optional The initial electron spectrum (number distribution). Default: accumulated constantly injected electrons until ``tstart``. Unit: [cm^-3] Returns ------- electron_spec : float 1D `~numpy.ndarray` The solved electron spectrum at ``tstop``. Unit: [cm^-3] """ if tstart is None: tstart = self.age_merger if tstop is None: tstop = self.age_obs if n0_e is None: # Accumulated constantly injected electrons until ``tstart``. n_inj = self.fp_injection(self.gamma) n0_e = n_inj * tstart # When the evolution time is too short, decrease the time step # to improve the results. # XXX: is this necessary??? nstep_min = 20 if (tstop - tstart) / self.time_step < nstep_min: tstep = (tstop - tstart) / nstep_min logger.debug("Decreased time step: %g -> %g [Gyr]" % (self.time_step, self.fpsolver.tstep)) self.fpsolver.tstep = tstep self.electron_spec = self.fpsolver.solve(u0=n0_e, tstart=tstart, tstop=tstop) return self.electron_spec def set_electron_spectrum(self, n_e): """ Check the given electron spectrum and set it to the ``self.electron_spec``. Parameters ---------- n_e : float 1D `~numpy.ndarray` The solved electron spectrum at ``zend``. Unit: [cm^-3] """ n_e = np.array(n_e) # make a copy if n_e.shape == self.gamma.shape: self.electron_spec = n_e else: raise ValueError("given electron spectrum has wrong shape!") def calc_emissivity(self, frequencies, n_e=None, gamma=None, B=None): """ Calculate the synchrotron emissivity for the derived electron spectrum. Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the synchrotron emissivity. Unit: [MHz] n_e : 1D `~numpy.ndarray`, optional The electron spectrum (w.r.t. Lorentz factors γ). If not provided, then use the cached ``self.electron_spec`` that was solved at above. Unit: [cm^-3] gamma : 1D `~numpy.ndarray`, optional The Lorentz factors γ of the electron spectrum. If not provided, then use ``self.gamma``. B : float, optional The magnetic field strength. If not provided, then use ``self.B_obs``. Unit: [uG] Returns ------- emissivity : float, or 1D `~numpy.ndarray` The calculated synchrotron emissivity at each specified frequency. Unit: [erg/s/cm^3/Hz] """ if n_e is None: n_e = self.electron_spec if gamma is None: gamma = self.gamma if B is None: B = self.B_obs syncem = SynchrotronEmission(gamma=gamma, n_e=n_e, B=B) emissivity = syncem.emissivity(frequencies) return emissivity def calc_power(self, frequencies, emissivity=None, **kwargs): """ Calculate the halo synchrotron power (i.e., power *emitted* per unit frequency) by assuming the emissivity is uniform throughout the halo volume. NOTE ---- The calculated power (a.k.a. spectral luminosity) is in units of [W/Hz] which is common in radio astronomy, instead of [erg/s/Hz]. 1 [W] = 1e7 [erg/s] Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the synchrotron power. Unit: [MHz] emissivity : float, or 1D `~numpy.ndarray`, optional The synchrotron emissivity at the input frequencies. If not provided, then invoke above ``calc_emissivity()`` method to calculate them. Unit: [erg/s/cm^3/Hz] **kwargs : optional arguments, i.e., ``n_e``, ``gamma``, and ``B``. Returns ------- power : float, or 1D `~numpy.ndarray` The calculated synchrotron power at each input frequency. Unit: [W/Hz] """ frequencies = np.asarray(frequencies) if emissivity is None: emissivity = self.calc_emissivity(frequencies=frequencies, **kwargs) else: emissivity = np.asarray(emissivity) if emissivity.shape != frequencies.shape: raise ValueError("input 'frequencies' and 'emissivity' " "do not match") power = emissivity * (self.volume * AUC.kpc2cm**3) # [erg/s/Hz] power *= 1e-7 # [erg/s/Hz] -> [W/Hz] return power def calc_flux(self, frequencies, **kwargs): """ Calculate the synchrotron flux density (i.e., power *observed* per unit frequency) of the halo, with k-correction considered. NOTE ---- The *k-correction* must be applied to the flux density (Sν) or specific luminosity (Lν) because the redshifted object is emitting flux in a different band than that in which you are observing. And the k-correction depends on the spectrum of the object in question. For any other spectrum (i.e., vLv != const.), the flux density Sv is related to the specific luminosity Lv by: Sv = (1+z) L_v(1+z) / (4π DL^2), where * L_v(1+z) is the specific luminosity emitting at frequency v(1+z), * DL is the luminosity distance to the object at redshift z. Reference: Ref.[hogg1999],Eq.(22) Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the flux density. Unit: [MHz] **kwargs : optional arguments, i.e., ``n_e``, ``gamma``, and ``B``. Returns ------- flux : float, or 1D `~numpy.ndarray` The calculated flux density w.r.t. each input frequency. Unit: [Jy] = 1e-23 [erg/s/cm^2/Hz] = 1e-26 [W/m^2/Hz] """ z = self.z_obs freqz = np.asarray(frequencies) * (1+z) power = self.calc_power(freqz, **kwargs) # [W/Hz] DL = COSMO.DL(self.z_obs) * AUC.Mpc2m # [m] flux = 1e26 * (1+z) * power / (4*np.pi * DL*DL) # [Jy] return flux def calc_brightness_mean(self, frequencies, flux=None, pixelsize=None, **kwargs): """ Calculate the mean surface brightness (power observed per unit frequency and per unit solid angle) expressed in *brightness temperature* at the specified frequencies. NOTE ---- If the solid angle that the object extends is smaller than the specified pixel area, then is is assumed to have size of 1 pixel. Parameters ---------- frequencies : float, or 1D `~numpy.ndarray` The frequencies where to calculate the mean brightness temperature Unit: [MHz] flux : float, or 1D `~numpy.ndarray`, optional The flux density w.r.t. each input frequency. Unit: [Jy] pixelsize : float, optional The pixel size of the output simulated sky image. If not provided, then invoke above ``calc_flux()`` method to calculate them. Unit: [arcsec] **kwargs : optional arguments, i.e., ``n_e``, ``gamma``, and ``B``. Returns ------- Tb : float, or 1D `~numpy.ndarray` The mean brightness temperature at each frequency. Unit: [K] <-> [Jy/pixel] """ frequencies = np.asarray(frequencies) if flux is None: flux = self.calc_flux(frequencies=frequencies, **kwargs) # [Jy] else: flux = np.asarray(flux) if flux.shape != frequencies.shape: raise ValueError("input 'frequencies' and 'flux' do not match") omega = np.pi * self.angular_radius**2 # [arcsec^2] if pixelsize and (omega < pixelsize**2): omega = pixelsize ** 2 # [arcsec^2] logger.warning("Object size < 1 pixel; force to be 1 pixel!") Tb = Fnu_to_Tb(flux, omega, frequencies) # [K] return Tb def fp_injection(self, gamma, t=None): """ Electron injection (rate) term for the Fokker-Planck equation. NOTE ---- The injected electrons are assumed to have a power-law spectrum and a constant injection rate. Qe(γ) = Ke * γ^(-s), Ke: constant injection rate Parameters ---------- gamma : float, or float 1D `~numpy.ndarray` Lorentz factors of electrons t : None Currently a constant injection rate is assumed, therefore this parameter is not used. Keep it for the consistency with other functions. Returns ------- Qe : float, or float 1D `~numpy.ndarray` Current electron injection rate at specified energies (gamma). Unit: [cm^-3 Gyr^-1] References ---------- Ref.[cassano2005],Eqs.(31,32,33) """ Ke = self.injection_rate # [cm^-3 Gyr^-1] Qe = Ke * gamma**(-self.injection_index) return Qe def fp_diffusion(self, gamma, t): """ Diffusion term/coefficient for the Fokker-Planck equation. The diffusion is directly related to the electron acceleration which is described by the ``tau_acc`` acceleration timescale parameter. NOTE ---- Considering that the turbulence acceleration is a 2nd-order Fermi process, it has only an effective acceleration time of several 1e8 years. Therefore, the turbulence is assumed to only accelerate the electrons during the merging period, i.e., the acceleration timescale is set to be infinite after "t_merger + time_cross". However, a zero diffusion coefficient may lead to unstable/wrong results, so constrain the acceleration timescale to be a large enough but finite number (e.g., 10 Gyr). Parameters ---------- gamma : float, or float 1D `~numpy.ndarray` The Lorentz factors of electrons t : float Current (cosmic) time when solving the equation Unit: [Gyr] Returns ------- diffusion : float, or float 1D `~numpy.ndarray` Diffusion coefficients Unit: [Gyr^-1] References ---------- Ref.[donnert2013],Eq.(15) """ if t < (self.age_merger + self.time_crossing): tau_acc = self.tau_acceleration # [Gyr] else: # The large enough timescale to avoid unstable results tau_acc = 10.0 # [Gyr] diffusion = gamma**2 / 4 / tau_acc return diffusion def fp_advection(self, gamma, t): """ Advection term/coefficient for the Fokker-Planck equation, which describes a systematic tendency for upward or downard drift of particles. This term is also called the "generalized cooling function" by [donnert2014], which includes all relevant energy loss functions and the energy gain function due to turbulence. Returns ------- advection : float, or float 1D `~numpy.ndarray` Advection coefficients, describing the energy loss/gain rates. Unit: [Gyr^-1] """ advection = (abs(self._loss_ion(gamma, t)) + abs(self._loss_rad(gamma, t)) - (self.fp_diffusion(gamma, t) * 2 / gamma)) return advection def _mass(self, t): """ Calculate the main cluster mass at the given (cosmic) time. NOTE ---- We assume that the main cluster grows (i.e., gains mass) linearly in time from (M_main, z_merge) to (M_obs, z_obs). Parameters ---------- t : float The (cosmic) time/age. Unit: [Gyr] Returns ------- mass : float The mass of the main cluster. Unit: [Msun] """ t_merger = self.age_merger rate = (self.M_obs - self.M_main) / (self.age_obs - t_merger) mass = rate * (t - t_merger) + self.M_main return mass def _magnetic_field(self, t): """ Calculate the mean magnetic field strength of the main cluster mass at the given (cosmic) time. Parameters ---------- t : float The (cosmic) time/age. Unit: [Gyr] Returns ------- B : float The mean magnetic field strength of the main cluster. Unit: [uG] """ z = COSMO.redshift(t) mass = self._mass(t) # [Msun] B = helper.magnetic_field(mass=mass, z=z) # [uG] return B def _loss_ion(self, gamma, t): """ Energy loss through ionization and Coulomb collisions. Parameters ---------- gamma : float, or float 1D `~numpy.ndarray` The Lorentz factors of electrons t : float The cosmic time/age Unit: [Gyr] Returns ------- loss : float, or float 1D `~numpy.ndarray` The energy loss rates Unit: [Gyr^-1] References ---------- Ref.[sarazin1999],Eq.(9) """ z = COSMO.redshift(t) mass = self._mass(t) n_th = helper.density_number_thermal(mass, z) # [cm^-3] loss = -3.79e4 * n_th * (1 + np.log(gamma/n_th) / 75) return loss def _loss_rad(self, gamma, t): """ Energy loss via synchrotron emission and inverse Compton scattering off the CMB photons. References ---------- Ref.[sarazin1999],Eq.(6,7) """ B = self._magnetic_field(t) # [uG] z = COSMO.redshift(t) loss = -4.32e-4 * gamma**2 * ((B/3.25)**2 + (1+z)**4) return loss