# Copyright (c) 2017-2019 Weitian LI # MIT License """ Simulate (giant) radio halos originating from the recent merger events, which generate cluster-wide turbulence and accelerate the primary (i.e., fossil) relativistic electrons to high energies to be synchrotron-bright. This *turbulence re-acceleration* model is currently most widely accepted to explain the (giant) radio halos. The simulation method is somewhat based on the statistical (Monte Carlo) method proposed by [cassano2005]_, but with extensive modifications and improvements. References ---------- .. [brunetti2011] Brunetti & Lazarian 2011, MNRAS, 410, 127 http://adsabs.harvard.edu/abs/2011MNRAS.410..127B .. [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 .. [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 .. [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 2015, ApJ, 800, 60 http://adsabs.harvard.edu/abs/2015ApJ...800...60M .. [pinzke2017] Pinzke, Oh & Pfrommer 2017, MNRAS, 465, 4800 http://adsabs.harvard.edu/abs/2017MNRAS.465.4800P .. [sarazin1999] Sarazin 1999, ApJ, 520, 529 http://adsabs.harvard.edu/abs/1999ApJ...520..529S .. [vazza2011] Vazza et al. 2011, A&A, 529, A17 http://adsabs.harvard.edu/abs/2011A%26A...529A..17V """ import logging from functools import lru_cache import numpy as np from scipy import integrate from . import helper from .solver import FokkerPlanckSolver from .emission import HaloEmission from ...share import CONFIGS, COSMO from ...utils.units import (Units as AU, UnitConversions as AUC, Constants as AC) logger = logging.getLogger(__name__) class RadioHalo1M: """ Simulate the radio halo properties for a galaxy cluster that is experiencing an on-going merger or had a merger recently. Description ----------- 1. Calculate the turbulence persistence time (tau_turb; ~<1 Gyr); 2. Calculate the diffusion coefficient (D_pp) from the systematic acceleration timescale (tau_acc; ~0.1 Gyr). The acceleration diffusion is assumed to have an action time ~ tau_turb (i.e., only during turbulence persistence), and then is 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 electron density/spectrum be the accumulated electrons injected during t_merger time, then evolve it for time_init period considering only losses and constant injection, in order to derive an approximately steady electron spectrum for following use; 5. Calculate the magnetic field from the cluster total mass (which is assumed to be growth linearly from M_main 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. _acceleration_disabled : bool Whether the turbulence acceleration is intentionally disabled? """ compID = "extragalactic/halos" name = "giant radio halos" 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.t_obs = COSMO.age(z_obs) self.M_main = M_main self.M_sub = M_sub self.z_merger = z_merger self.t_merger = COSMO.age(z_merger) self._acceleration_disabled = False self._set_configs(configs) self._set_solver() def _set_configs(self, configs): sec = self.compID self.configs = configs self.f_acc = configs.getn(sec+"/f_acc") self.f_radius = configs.getn(sec+"/f_radius") self.x_turb = configs.getn(sec+"/x_turb") self.eta_turb = configs.getn(sec+"/eta_turb") self.eta_e = configs.getn(sec+"/eta_e") self.x_cr = configs.getn(sec+"/x_cr") self.eta_b = self.x_cr # Equipartition between magnetic field and CR self.mass_index = configs.getn(sec+"/mass_index") self.radius_index = configs.getn(sec+"/radius_index") self.gamma_min = configs.getn(sec+"/gamma_min") self.gamma_max = configs.getn(sec+"/gamma_max") self.gamma_np = configs.getn(sec+"/gamma_np") self.buffer_np = configs.getn(sec+"/buffer_np") if self.buffer_np == 0: self.buffer_np = None self.time_step = configs.getn(sec+"/time_step") self.time_init = configs.getn(sec+"/time_init") self.injection_index = configs.getn(sec+"/injection_index") self.f_rc = configs.getn(sec+"/f_rc") self.beta = configs.getn(sec+"/beta") self.bolo_freq_min = configs.getn(sec+"/bolo_freq_min") self.bolo_freq_max = configs.getn(sec+"/bolo_freq_max") self.bolo_freq_num = configs.getn(sec+"/bolo_freq_num") self.kT_out = configs.getn("extragalactic/clusters/kT_out") 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 t_begin(self): """ The cosmic time when the merger begins. Unit: [Gyr] """ return self.t_merger @property @lru_cache() def t_merger_end(self): """ The time when the merger (i.e., turbulence) ends. Unit: [Gyr] """ return self.t_merger + self.duration_turb(self.t_merger) @property def radius(self): """ The estimated radius of the simulated radio halo. It is known that the halo radius scales non-linearly as the hosting cluster, breaking the self-similiarity, which may be caused by the magnetic field and the releativistic electron distributions. Reference: Ref.[cassano2007],Sec.4 Unit: [kpc] """ r_turb = self.radius_turb(self.t_merger) r_cl = helper.radius_cluster(mass=self.M_obs, z=self.z_obs) r0_cl = helper.radius_cluster(mass=1e15, z=0) return r_turb * self.f_radius * (r_cl/r0_cl)**self.radius_index @lru_cache() def radius_strip(self, t_merger): """ The stripping radius of the in-falling sub-cluster at time t. Unit: [kpc] """ self._validate_time(t_merger) z = COSMO.redshift(t_merger) M_main = self.mass_main(t_merger) M_sub = self.mass_sub(t_merger) return helper.radius_stripping(M_main, M_sub, z, f_rc=self.f_rc, beta=self.beta) @lru_cache() def radius_turb(self, t_merger): """ The radius of the turbulence region, which is estimated as the sum of stripping radius ``r_s`` of the sub-cluster and the core radius ``r_c`` of the main cluster. Unit: [kpc] """ self._validate_time(t_merger) z = COSMO.redshift(t_merger) M_main = self.mass_main(t_merger) r_s = self.radius_strip(t_merger) r_c = self.f_rc * helper.radius_cluster(M_main, z) return r_s + r_c @lru_cache() def duration_turb(self, t_merger): """ The duration that the turbulence persists strong enough to be able to effectively accelerate the electrons, which is estimated as: τ_turb ~ d / v_i ~ 2*R_turb / v_i, where ``v_i`` is the relative impact velocity. Reference: [miniati2015],Sec.5 Unit: [Gyr] """ self._validate_time(t_merger, include_end=False) z_merger = COSMO.redshift(t_merger) M_main = self.mass_main(t_merger) M_sub = self.mass_sub(t_merger) d = 2 * self.radius_turb(t_merger) v_i = helper.velocity_impact(M_main, M_sub, z_merger) uconv = AUC.kpc2km * AUC.s2Gyr # [kpc]/[km/s] => [Gyr] return uconv * d / v_i # [Gyr] @lru_cache() def velocity_turb_base(self, t): """ Calculate the velocity dispersion of the base turbulence. Without injection by mergers, the ICM can has some turbulence, which can amount ~< 5% of the thermal energy in relaxed systems (Ref.[vazza2011]). ε_turb = (1/2) * ρ_gas * = x_turb * ε_th ε_th = (3/2) * n_th * kT ρ_gas = μ * m_u * n_th c_s^2 = γ_gas * kT / (μ * m_u) => v_turb = c_s * sqrt(3 * x_turb / γ_gas) Parameters ---------- t : float The beginning or ending time of the merger. Unit: [Gyr] Returns ------- v_turb : float The velocity dispersion of the base turbulence. Unit: [km/s] """ self._validate_time(t) if np.isclose(self.x_turb, 0): return 0 c_s = helper.speed_sound(self.kT(t)) # [km/s] return c_s * np.sqrt(3*self.x_turb / AC.gamma) @lru_cache() def velocity_turb(self, t): """ Calculate the turbulence velocity dispersion. NOTE ---- During the merger, a fraction of the merger kinetic energy is transferred into the turbulence within the region of radius R_turb. Then estimate the turbulence velocity dispersion from its energy. Merger energy: E_merger ≅ <ρ_gas> * v_i^2 * V_turb V_turb = ᴨ * r_s^2 * (R_vir+r_s) Turbulence energy: E_turb ≅ 0.5 * M_turb * ≅ 0.5 * M_turb * + η_turb * E_merger => Velocity dispersion: ≅ + 2*η_turb * <ρ_gas> * v_i^2 * V_turb / M_turb M_turb = int_0^R_turb[ ρ_gas(r)*4ᴨ*r^2 ]dr where: <ρ_gas>: mean gas density of the main cluster R_vir: virial radius of the main cluster R_turb: radius of turbulence region v_i: impact velocity r_s: stripping radius of the in-falling sub-cluster Returns ------- v_turb : float The turbulence velocity dispersion Unit: [km/s] """ self._validate_time(t) z = COSMO.redshift(t) M_main = self.mass_main(t) M_sub = self.mass_sub(t) r_s = self.radius_strip(t) # [kpc] R_turb = self.radius_turb(t) # [kpc] v_turb_base = self.velocity_turb_base(t) # [km/s] rho_gas_f = helper.calc_gas_density_profile( M_main+M_sub, z, f_rc=self.f_rc, beta=self.beta) M_turb = 4*np.pi * integrate.quad( lambda r: rho_gas_f(r) * r**2, a=0, b=R_turb)[0] # [Msun] v_i = helper.velocity_impact(M_main, M_sub, z) # [km/s] rho_main = helper.density_number_thermal(M_main, z) # [cm^-3] rho_main *= AC.mu*AC.u * AUC.g2Msun * AUC.kpc2cm**3 # [Msun/kpc^3] R_vir = helper.radius_cluster(M_main, z) # [kpc] volume = np.pi * r_s**2 * R_vir # [kpc^3] E_add = self.eta_turb * rho_main * v_i**2 * volume v2_turb = v_turb_base**2 + 2 * E_add / M_turb # [km/s]^2 return np.sqrt(v2_turb) @lru_cache() def tau_acceleration(self, t, is_end=False): """ Calculate the electron acceleration timescale due to turbulent waves, which describes the turbulent acceleration efficiency. Here we consider the turbulence cascade mode through scattering in the high-β ICM mediated by plasma instabilities (firehose, mirror) rather than Coulomb scattering. Therefore, the fast modes damp by TTD (transit time damping) on relativistic rather than thermal particles, and the diffusion coefficient is given by: D'_γγ = 2 * γ^2 * ζ * k_L * v_t^4 / (c_s^3 * X_cr) where: ζ: factor describing the effectiveness of plasma instabilities X_cr: relative energy density of cosmic rays k_L (= 2π/L): turbulence injection scale v_t: turbulence velocity dispersion c_s: sound speed Hence, the acceleration timescale is: τ'_acc = γ^2 / (4 * D'_γγ) = X_cr * c_s^3 / (8 * ζ * k_L * v_t^4) Previous studies show that more massive clusters are more efficient to accelerate electrons to be radio bright. To further account for this scaling relation: D_γγ = D'_γγ * f_m * (M_main / 1e15)^m where: m: scaling index f_m: normalization Therefore, the final acceleration timescale is: τ_acc = τ'_acc * (M_main / 1e15)^(-m) / f_m = X_cr * c_s^3 * (M_main/1e15)^(-m) / (8*f_acc * k_L * v_t^4) with: f_acc = f_m * ζ References ---------- * Ref.[pinzke2017],Eq.(37) * Ref.[miniati2015],Eq.(29) Parameters ---------- t : float The beginning time or ending time (if ``is_end=True``) of the merger. Unit: [Gyr] is_end : bool, optional Whether the given time ``t`` is the ending time of the merger. Default: False Returns ------- tau : float The acceleration timescale of the merger. Unit: [Gyr] """ self._validate_time(t, include_end=is_end) if is_end: v_t = self.velocity_turb_base(t) # [km/s] else: v_t = self.velocity_turb(t) # [km/s] if np.isclose(v_t, 0): return np.inf k_L = 2 * np.pi / self.radius_turb(t) # [kpc^-1] c_s = helper.speed_sound(self.kT(t)) # [km/s] tau = self.x_cr * c_s**3 / (8*k_L * v_t**4) tau *= AUC.s2Gyr * AUC.kpc2km # [s kpc/km] -> [Gyr] # Scale by cluster mass M_main = self.mass_main(t) f_mass = (M_main / 1e15) ** (-self.mass_index) tau *= f_mass tau /= self.f_acc # tune factor (folded with ζ "zeta_ins") 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., ``t_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 = η_e * e_th => Ke = [(s-2) * η_e * e_th * γ_min^(s-2) / (me * c^2 * t_cluster)] References ---------- Ref.[cassano2005],Eqs.(31,32,33) """ kT_out = self.configs.getn("extragalactic/clusters/kT_out") s = self.injection_index e_th = helper.density_energy_thermal(self.M_obs, self.z_obs, kT_out=kT_out) term1 = (s-2) * self.eta_e * e_th # [erg cm^-3] term2 = self.gamma_min**(s-2) term3 = AU.mec2 * self.t_obs # [erg Gyr] Ke = term1 * term2 / term3 # [cm^-3 Gyr^-1] return Ke @property def electron_spec_init(self): """ The electron spectrum at ``t_begin`` to be used as the initial condition for the Fokker-Planck equation. This initial electron spectrum is derived from the accumulated electron spectrum injected throughout the ``t_begin`` period, by solving the same Fokker-Planck equation, but only considering energy losses and constant injection, evolving for a period of ``time_init`` in order to obtain an approximately steady electron spectrum. Units: [cm^-3] """ # Accumulated electrons constantly injected until ``t_begin`` n_inj = self.fp_injection(self.gamma) n0_e = n_inj * (self.t_begin - self.time_init) logger.debug("Deriving the initial electron spectrum ...") self._acceleration_disabled = True tstart = self.t_begin tstop = self.t_begin + self.time_init self.fpsolver.tstep = self.time_step * 3 # To save time n_e = self.fpsolver.solve(u0=n0_e, tstart=tstart, tstop=tstop) self._acceleration_disabled = False self.fpsolver.tstep = self.time_step return n_e def calc_electron_spectrum(self, tstart=None, tstop=None, n0_e=None, fiducial=False): """ 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.t_begin``. Unit: [Gyr] tstop : float, optional The (cosmic) time when to derive final relativistic electrons spectrum for synchrotron emission calculations. Default: ``self.t_obs``. Unit: [Gyr] n0_e : 1D `~numpy.ndarray`, optional The initial electron spectrum (number distribution). Default: ``self.electron_spec_init`` Unit: [cm^-3] fiducial : bool Whether to disable the turbulent acceleration and derive the fiducial electron spectrum? Default: ``False`` Returns ------- n_e : float 1D `~numpy.ndarray` The solved electron spectrum. Unit: [cm^-3] """ if tstart is None: tstart = self.t_begin if tstop is None: tstop = self.t_obs if n0_e is None: n0_e = self.electron_spec_init if fiducial: self._acceleration_disabled = True self.fpsolver.tstep = self.time_step * 2 # To save time logger.debug("Calculating the %s electron spectrum ..." % ("[fiducial]" if fiducial else "")) n_e = self.fpsolver.solve(u0=n0_e, tstart=tstart, tstop=tstop) self._acceleration_disabled = False self.fpsolver.tstep = self.time_step return n_e def calc_acc_factor(self, n_e, n_e_fiducial=None): """ Calculate the turbulence acceleration factor, which is estimated as the ratio of the bolometric emissivity between the accelerated electron spectrum and the fiducial electron spectrum derived with turbulent acceleration turned off. Parameters ---------- n_e : float 1D `~numpy.ndarray` The derived (accelerated) electron spectrum. Unit: [cm^-3] n_e_fiducial : float 1D `~numpy.ndarray`, optional The fiducial electron spectrum. Unit: [cm^-3] Returns ------- factor : float Acceleration factor of the bolometric emissivity. """ haloem = HaloEmission(gamma=self.gamma, n_e=n_e, B=1) em = haloem.calc_emissivity_bolo(freq_min=self.bolo_freq_min, freq_max=self.bolo_freq_max, freq_num=self.bolo_freq_num) if n_e_fiducial is None: n_e_fiducial = self.calc_electron_spectrum(fiducial=True) haloem.n_e = n_e_fiducial em_fiducial = haloem.calc_emissivity_bolo( freq_min=self.bolo_freq_min, freq_max=self.bolo_freq_max, freq_num=self.bolo_freq_num) return em / em_fiducial 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 and calculated from the acceleration timescale ``tau_acc``. WARNING ------- A zero diffusion coefficient may lead to unstable/wrong results, since it is not properly taken care of by the solver. By carrying out some tests, the maximum acceleration timescale ``tau_acc`` is assumed to be 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] """ tau_acc = tau_max = 10.0 # [Gyr] if self._is_turb_active(t): tt = self._merger_time(t) is_end = False else: tt = self._merger_end_time(t) is_end = True tau_acc = self.tau_acceleration(tt, is_end) if tau_acc > tau_max: tau_acc = tau_max return np.square(gamma) / (4 * tau_acc) # [Gyr^-1] def fp_advection(self, gamma, t): """ Advection term/coefficient for the Fokker-Planck equation, which describes a systematic tendency for upward or downward 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 coefficient. Unit: [Gyr^-1] """ advection = (abs(self._energy_loss(gamma, t)) - (self.fp_diffusion(gamma, t) * 2 / gamma)) return advection def _energy_loss(self, gamma, t): """ Energy loss mechanisms: * inverse Compton scattering off the CMB photons * synchrotron radiation * Coulomb collisions Reference: Ref.[sarazin1999],Eqs.(6,7,9) Unit: [Gyr^-1] """ gamma = np.asarray(gamma) z = COSMO.redshift(t) B = self.magnetic_field(t) # [uG] mass = self.mass_main(t) n_th = helper.density_number_thermal(mass, z) # [cm^-3] loss_ic = -4.32e-4 * gamma**2 * (1+z)**4 loss_syn = -4.10e-5 * gamma**2 * B**2 loss_coul = -3.79e4 * n_th * (1 + np.log(gamma/n_th) / 75) return loss_ic + loss_syn + loss_coul def _merger_time(self, t=None): """ The (cosmic) time when the merger begins. Unit: [Gyr] """ return self.t_merger def _merger_end_time(self, t=None): """ The (cosmic) time when the merger ends. Unit: [Gyr] """ return self.t_merger_end def _validate_time(self, t, include_end=True): """ Validate that the given time ``t`` is the time when a merger begins or ends (if ``include_end`` is ``True``), otherwise raise an error. """ if np.any(np.isclose(t, self.t_merger)): return if include_end and np.any(np.isclose(t, self.t_merger_end)): return raise ValueError("Not a merger begin/end time: %f" % t) def _is_turb_active(self, t): """ Is the turbulence acceleration active at the given time? """ t_begin = self._merger_time(t) t_end = t_begin + self.duration_turb(t_begin) return (t >= t_begin) and (t <= t_end) def mass_merged(self, t=None): """ The mass of the merged cluster. Unit: [Msun] """ return self.M_main + self.M_sub def mass_sub(self, t=None): """ The mass of the sub cluster. Unit: [Msun] """ return self.M_sub def mass_main(self, t): """ Calculate the main cluster mass at the given (cosmic) time. The main cluster is assumed to grow linearly in time from (M_main, z_merger) to (M_obs, z_obs). Unit: [Msun] """ t0 = self.t_begin rate = (self.M_obs - self.M_main) / (self.t_obs - t0) mass = rate * (t - t0) + self.M_main # [Msun] return mass def kT(self, t): """ The ICM mean temperature of the main cluster. Unit: [keV] """ kT_out = self.configs.getn("extragalactic/clusters/kT_out") M_main = self.mass_main(t) z = COSMO.redshift(t) return helper.kT_cluster(mass=M_main, z=z, kT_out=kT_out) def magnetic_field(self, t): """ Calculate the mean magnetic field strength of the main cluster mass at the given (cosmic) time. Unit: [uG] """ z = COSMO.redshift(t) mass = self.mass_main(t) # [Msun] return helper.magnetic_field(mass=mass, z=z, eta_b=self.eta_b, kT_out=self.kT_out) class RadioHaloAM(RadioHalo1M): """ Simulate the radio halo properties for a galaxy cluster with all its on-going merger and past merger events taken into account. Parameters ---------- M_obs : float Cluster virial mass at the observation (simulation end) time. Unit: [Msun] z_obs : float Redshift of the observation (simulation end) time. M_main, M_sub : list[float] List of main and sub cluster masses at each merger event, from current to earlier time. Unit: [Msun] z_merger : list[float] The redshifts at each merger event, from small to large. merger_num : int Number of merger events traced for the cluster. """ def __init__(self, M_obs, z_obs, M_main, M_sub, z_merger, merger_num, configs=CONFIGS): M_main = np.asarray(M_main[:merger_num]) M_sub = np.asarray(M_sub[:merger_num]) z_merger = np.asarray(z_merger[:merger_num]) # increasing super().__init__(M_obs=M_obs, z_obs=z_obs, M_main=M_main, M_sub=M_sub, z_merger=z_merger, configs=configs) self.merger_num = merger_num @property def t_begin(self): """ The cosmic time when the first merger begins. Unit: [Gyr] """ return self.t_merger[-1] @property @lru_cache() def t_merger_end(self): """ The times when the mergers (i.e., turbulence) end. (NOTE: decreasing order) Unit: [Gyr] """ return np.array([t+self.duration_turb(t) for t in self.t_merger]) @property def radius(self): """ The halo radius estimated by using the maximum turbulence radius. Unit: [kpc] """ r_turb = self.radius_turb_max r_cl = helper.radius_cluster(mass=self.M_obs, z=self.z_obs) r0_cl = helper.radius_cluster(mass=1e15, z=0) return r_turb * self.f_radius * (r_cl/r0_cl)**self.radius_index @property @lru_cache() def radius_turb_max(self): """ The maximum turbulence radius. Unit: [kpc] """ return max([self.radius_turb(tm) for tm in self.t_merger]) def radius_turb_eff(self, t, use_last=True): """ Get the effective turbulence radius, i.e., the largest one if multiple mergers are active at the given time. Parameters ---------- use_last : bool If ``True``, return the turbulence radius of the last merger event when there is no active turbulence at the given time. Otherwise, return 0. Unit: [kpc] """ mergers = [(t, t+self.duration_turb(t), self.radius_turb(t)) for t in self.t_merger] # time decreasing try: r_eff = max([r for t1, t2, r in mergers if t >= t1 and t < t2]) except ValueError: # No active turbulence at this time if use_last: r_eff = next(r for __, t2, r in mergers if t >= t2) else: r_eff = 0 return r_eff def _merger_event(self, t): """ Return the most recent merger event happend before the given time, i.e., the merger event that the given time locates in. """ idx = (self.t_merger > t).sum() # 't_merger' in decreasing order return { "idx": idx, "M_main": self.M_main[idx], "M_sub": self.M_sub[idx], "z": self.z_merger[idx], "t": self.t_merger[idx], } def mass_merged(self, t): """ The mass of merged cluster at the given (cosmic) time. Unit: [Msun] """ if t >= self.t_obs: return self.M_obs else: merger = self._merger_event(t) return (merger["M_main"] + merger["M_sub"]) def mass_sub(self, t): """ The mass of the sub cluster at the given (cosmic) time. Unit: [Msun] """ merger = self._merger_event(t) return merger["M_sub"] def mass_main(self, t): """ Calculate the main cluster mass, which is assumed to grow along the merger/accretion processes, at the given (cosmic) time. Unit: [Msun] """ merger1 = self._merger_event(t) idx1 = merger1["idx"] mass1 = merger1["M_main"] t1 = merger1["t"] if idx1 == 0: mass0 = self.M_obs t0 = self.t_obs else: idx0 = idx1 - 1 mass0 = self.M_main[idx0] t0 = self.t_merger[idx0] rate = (mass0 - mass1) / (t0 - t1) return (mass1 + rate * (t - t1)) def _merger_time(self, t): """ Determine the beginning time of the merger event that is doing effective acceleration at the given time. At a certain time, there may be multiple past merger events with different turbulence durations (``tau_turb``) and acceleration efficiencies (``tau_acc``). Therefore, multiple mergers can cover the given time. The one with the largest acceleration efficiency (i.e., smallest ``tau_acc``) is chosen and its beginning time is returned. Otherwise, the most recent merger event happened before the given time is chosen. Unit: [Gyr] """ mergers = [(tm, tm+self.duration_turb(tm), self.tau_acceleration(tm)) for tm in self.t_merger] m_active = [(tm, tend, tau) for (tm, tend, tau) in mergers if t >= tm and t < tend] if m_active: m_eff = min(m_active, key=lambda item: item[2]) return m_eff[0] else: m = self._merger_event(t) return m["t"] def _merger_end_time(self, t): """ Determine the ending time of the most recent merger happened before the given time. Unit: [Gyr] """ t_end = np.sort(self.t_merger_end) # increasing order idx = (t_end <= t).sum() - 1 return t_end[idx] @property @lru_cache() def _time_adjust(self): """ Determine the time points when spectrum adjustment is needed. Different mergers generate turbulence in regions of different radius, therefore, the accelerated spectrum needs appropriate adjustment. Returns ------- t_adj : list[float] List of (cosmic) times when the adjustment is needed. NOTE: May be empty, e.g., only one merger event. Unit: [Gyr] """ mergers = [(t, t+self.duration_turb(t)) for t in self.t_merger] t_begin = [t for t, __ in mergers] t_end = [t for __, t in mergers] tps = sorted(t_begin + t_end) radii = [self.radius_turb_eff(t, use_last=False) for t in tps] tinfo = {t: {"begin": t in t_begin, "end": t in t_end, "radius": r} for t, r in zip(tps, radii)} t_adj = [] for r1, r2, t in zip(radii, radii[1:], tps[1:]): if np.isclose(r1, r2): continue ti = tinfo[t] if ti["end"] and np.isclose(ti["radius"], 0): continue t_adj.append(t) return t_adj def _adjust_spectrum(self, spec_in, t, spec_ref): """ Adjust the electron spectrum to take into account the change of turbulence region size. If the current turbulence radius is smaller than the maximum turbulence radius, then dilute the accelerated part of the spectrum according to the volume ratio. Parameters ---------- spec_in : 1D `~numpy.ndarray` The spectrum at the ending of the given acceleration period. t : float The corresponding time of the given spectrum ``spec_in``. Unit: [Gyr] spec_ref : 1D `~numpy.ndarray` The spectrum at the beginning of the given acceleration period. Returns ------- spec : Adjusted spectrum. """ r = self.radius_turb_eff(t, use_last=True) r_max = self.radius_turb_max if np.isclose(r, r_max): return spec_in logger.debug("Adjusting the accelerated spectrum ...") spec_diff = spec_in - spec_ref idx = spec_diff > 0 spec = np.array(spec_ref) spec[idx] += spec_diff[idx] * (r/r_max)**3 return spec def calc_electron_spectrum(self, tstart=None, tstop=None, n0_e=None, fiducial=False): """ Calculate the relativistic electron spectrum by solving the Fokker-Planck equation. Given that different mergers have different turbulence radii, the spectrum needs appropriate adjustments to take this into account. At the beginning of each merger, the accelerated part of the spectrum (i.e., where the electron density increases compared to last adjustment) is scaled according to the ratio of the previous turbulence volume to the maximum turbulence volume, i.e., dilute the accelerated spectrum to the maximum turbulence volume. """ if tstart is None: tstart = self.t_begin if tstop is None: tstop = self.t_obs if n0_e is None: n0_e = self.electron_spec_init if fiducial: self._acceleration_disabled = True self.fpsolver.tstep = self.time_step * 2 # To save time logger.debug("Calculating the [fiducial] electron spectrum ...") n_e = self.fpsolver.solve(u0=n0_e, tstart=tstart, tstop=tstop) self._acceleration_disabled = False self.fpsolver.tstep = self.time_step return n_e logger.debug("Calculating the electron spectrum ...") tps = [self.t_begin] + self._time_adjust + [self.t_obs] n1_e = n0_e for t1, t2 in zip(tps, tps[1:]): if tstart >= t2 or tstop < t1: continue if tstart > t1: t1 = tstart if tstop < t2: t2 = tstop logger.debug("Time period: [%.2f, %.2f] [Gyr] ..." % (t1, t2)) n2_e = self.fpsolver.solve(u0=n1_e, tstart=t1, tstop=t2) n2_e = self._adjust_spectrum(n2_e, t2, spec_ref=n1_e) n1_e = n2_e return n2_e