Preshock Region Research Articles

Cyclotron line formation in the radiative shock of an accreting magnetized neutron star

Context. Magnetic neutron stars (NSs) often exhibit a cyclotron resonant scattering feature (CRSF) in their X-ray spectra. Accretion onto their magnetic poles is responsible for the emergence of X-rays, but the site of the CRSF formation is still a puzzle. A promising candidate for high-luminosity sources has always been the radiative shock in the accretion column. Nevertheless, no quantitative calculations of spectral formation at the radiative shock have been performed so far. Aims. It is well accepted that, in the accretion column of a high-luminosity, accreting magnetic NS, a radiative shock is formed. Here, we aim to explore the scenario where the shock is the site of the cyclotron-line formation. We studied spectral formation at the radiative shock and the emergent spectral shape across a wide range of the parameter space and determined which parameters hold an important role in shaping a prominent CRSF. Methods. We developed a Monte Carlo (MC) code based on the forced first collision numerical scheme to conduct radiation transfer simulations at the radiative shock. The seed photons were due to bremsstrahlung and were emitted in the post-shock region. We properly treated bulk-motion Comptonization in the pre-shock region, thermal Comptonization in the post-shock region, and resonant Compton scattering in both regions. We adopted a fully relativistic scheme for the interaction between radiation and electrons, employing an appropriate polarization-averaged differential cross-section. As a result, we calculated the angle- and energy-dependent emergent X-ray spectrum from the radiative shock, focusing on both the CRSF and the X-ray continuum, under diverse conditions. The accretion column was characterized by cylindrical symmetry, and the radiative shock was treated as a mathematical discontinuity. Results. We find that a power law, hard X-ray continuum, and a CRSF are naturally produced by the first-order Fermi energization as the photons criss-cross the shock. The depth and the width of the CRSF depend mainly on the transverse optical depth and the post-shock temperature. We show that the cyclotron-line energy centroid is shifted by ∼(20 − 30)% to lower energies compared to the classical cyclotron energy; this is due to the Doppler boosting between the shock reference frame and the bulk-motion frame. We demonstrate that a “bump” feature arises in the right wing of the CRSF due to the up-scattering of photons by the accreting plasma and extends to higher energies for larger optical depths and post-shock temperatures. Conclusions. We conclude that resonant Compton scattering of photons by electrons in a radiative shock is efficient in producing a power-law X-ray continuum with a high-energy cutoff accompanied by a prominent CRSF. The implications of the Doppler effect on the centroid of the emergent absorption feature must be considered if an accurate determination of the magnetic field strength is desired.

Constraining the White-dwarf Mass and Magnetic Field Strength of a New Intermediate Polar through X-Ray Observations

We report timing and broadband spectral analysis of a Galactic X-ray source, CXOGBS J174517.0−321356 (J1745), with a 614 s periodicity. Chandra discovered the source in the direction of the Galactic Bulge. Gong proposed that J1745 was either an intermediate polar (IP) with a mass of ∼1 M ⊙, or an ultracompact X-ray binary (UCXB). To confirm J1745's nature, we jointly fit XMM-Newton and NuSTAR spectra, ruling out a UCXB origin. We have developed a physically realistic model that considers a finite magnetosphere radius, X-ray absorption from the preshock region, and reflection from the white-dwarf (WD) surface to properly determine the IP properties, especially its WD mass. To assess systematic errors on the WD mass measurement, we consider a broad range of specific accretion rates ( g cm−2 s−1) based on the uncertain source distance (d = 3–8 kpc) and fractional accretion area (f = 0.001–0.025). Our model properly implements the fitted accretion column height in the X-ray reflection model and accounts for the underestimated mass accretion rate due to the (unobserved) soft X-ray blackbody and cyclotron cooling emissions. We found that the lowest accretion rate of = 0.6 g cm−2 s−1, which corresponds to the nearest source distance and maximum f value, yields a WD mass of (0.92 ± 0.08)M ⊙. On the other hand, as long as the accretion rate is g cm−2 s−1, the WD mass is robustly measured to be (0.81 ± 0.06)M ⊙, nearly independent of . The derived WD mass range is consistent with the mean WD mass of nearby IPs. Assuming spin equilibrium between the WD and accretion disk, we constrained the WD magnetic field to B ≳ 7 MG, indicating that it could be a highly magnetized IP. Our analysis presents the most comprehensive methodology for constraining the WD mass and B field of an IP by consolidating the effects of cyclotron cooling, finite magnetospheric radius, and accretion column height.

Fast particles overtaking shock front in two-dimensional Yukawa solids

High-speed particles overtaking the shock front during the propagation of compressional shocks in two-dimensional (2D) Yukawa solids are investigated using molecular dynamical simulations. When the compressional speed is lower, all particles around the shock front are almost accelerated synchronously. However, when the compressional speed is much higher, some particles penetrate the shock front to enter the preshock region. Around the shock front, it is found that the particle velocity profile at the first peak of the dispersive shock wave (DSW) is able to be described using the Gaussian distribution, so that the amplitudes of the DSW can be well characterized. As the compressional speed increases, the particle velocity corresponding to these DSW's amplitudes increase more substantially than the shock front speed. These amplitudes of the DSW are found to be able to predict the occurrence of the fast particles. Combined with the previous study of the DSW's period, it is demonstrated that the properties of the DSW are nearly not affected by the conditions of the 2D Yukawa systems, but only related to the compressional speed.

PSP Observations of a Slow Shock Pair Bounding a Large‐Scale Plasmoid/Macro Magnetic Hole

AbstractSlow shocks are introduced to be the main dissipation sites in Petschek reconnection model, but they are seldom observed in interplanetary space. We report a slow shock pair bounding a plasmoid/macro magnetic hole observed by Parker Solar Probe. The jump conditions across the shocks are examined and confirmed to satisfy the Rankine‐Hugoniot relations. The flow speed in the preshock and postshock regions of both shocks match up with the characteristics of slow shocks. The slow shock pair is suggested to be a part of a curved slow shock bounding the plasmoid produced by magnetic reconnection. Our findings append a new dissipation site to the picture of energy conversion through solar wind magnetic reconnection. In addition, the results suggest that plasmoids emitted by local magnetic reconnections could be a possible origin of macro magnetic holes in the solar wind.

Electron Preacceleration at Weak Quasi-perpendicular Intracluster Shocks: Effects of Preexisting Nonthermal Electrons

Abstract Radio relics in the outskirts of galaxy clusters imply the diffusive shock acceleration (DSA) of electrons at merger-driven shocks with Mach number M s ≲ 3–4 in the intracluster medium (ICM). Recent studies have suggested that electron preacceleration and injection, prerequisite steps for DSA, could occur at supercritical shocks with M s ≳ 2.3 in the ICM, thanks to the generation of multiscale waves by microinstabilities such as the Alfvén ion cyclotron (AIC) instability, the electron firehose instability (EFI), and the whistler instability (WI). On the other hand, some relics are observed to have subcritical shocks with M s ≲ 2.3, leaving DSA at such weak shocks as an outstanding problem. Reacceleration of preexisting nonthermal electrons has been contemplated as one of possible solutions for that puzzle. To explore this idea, we perform particle-in-cell simulations for weak quasi-perpendicular shocks in high-β (β = P gas/P B ) plasmas with power-law suprathermal electrons in addition to Maxwellian thermal electrons. We find that suprathermal electrons enhance the excitation of electron-scale waves via the EFI and WI. However, they do not affect the ion reflection and the ensuing generation of ion-scale waves via the AIC instability. The presence of ion-scale waves is the key for the preacceleration of electrons up to the injection momentum; thus, the shock criticality condition for electron injection to DSA is preserved. Based on the results, we conclude that preexisting nonthermal electrons in the preshock region alone would not resolve the issue of electron preacceleration at subcritical ICM shocks.

Nonlinear evolution of fast neutrino flavor conversion in the preshock region of core-collapse supernovae

In environments with high dense neutrino gases, such as in core-collapse supernovae, the neutrinos can experience collective neutrino oscillation due to their self-interactions. In particular, fast flavor conversion driven by the crossings in the neutrino angular distribution can affect explosion mechanism, nucleosynthesis, and neutrino observation. We perform the numerical computation of nonlinear flavor evolution on the neutrino angular distribution with tiny crossings expected to be generated in the preshock region. We demonstrate that the fast instability is triggered and a cascade develops under a realistic three-flavor model considering muon production and weak magnetism in the supernova dynamics. The tiny crossing excites specific spatial modes, and then the flavor instability propagates into other modes which otherwise remain stable due to the nonlinear effects. Our results indicate that fast flavor conversion can rise in the preshock region and have a sufficient impact on the flavor contents.

Breaking the Degeneracy in Magnetic Cataclysmic Variable X-Ray Spectral Modeling Using X-Ray Light Curves

Abstract We present an analysis of mock X-ray spectra and light curves of magnetic cataclysmic variables using an upgraded version of the 3D cyclops code. This 3D representation of the accretion flow allows us to properly model total and partial occultation of the postshock region by the white dwarf as well as the modulation of the X-ray light curves due to the phase-dependent extinction of the preshock region. We carried out detailed postshock region modeling in a four-dimensional parameter space by varying the white dwarf mass and magnetic field strength as well as the magnetosphere radius and the specific accretion rate. To calculate the postshock region temperature and density profiles, we assumed equipartition between ions and electrons; took into account the white dwarf gravitational potential, the finite size of the magnetosphere, and a dipole-like magnetic field geometry; and considered cooling by both bremsstrahlung and cyclotron radiative processes. By investigating the impact of the parameters on the resulting X-ray continuum spectra, we show that there is an inevitable degeneracy in the four-dimensional parameter space investigated here, which compromises X-ray continuum spectral fitting strategies and can lead to incorrect parameter estimates. However, the inclusion of X-ray light curves in different energy ranges can break this degeneracy, and it therefore remains, in principle, possible to use X-ray data to derive fundamental parameters of magnetic cataclysmic variables, which represents an essential step toward understanding their formation and evolution.

The Role of Complex Ionized Absorbers in the Soft X-Ray Spectra of Intermediate Polars

Abstract In magnetic cataclysmic variables (mCVs), X-ray radiation originates from a shock-heated, multi-temperature plasma in the post-shock region near the white dwarf’s surface. These X-rays are modified by a complex distribution of absorbers in the pre-shock region. The presence of photoionized lines and warm absorber features in the soft X-ray spectra of these mCVs suggests that these absorbers are ionized. We developed the ionized complex absorber model zxipab, which is represented by a power-law distribution of ionized absorbers in the pre-shock flow. Using the ionized absorber model zxipab along with a cooling flow model with a reflection component, we model the broadband Chandra/High-Energy Transmission Grating (HETG) and Nuclear Spectroscopic Telescope Array (NuSTAR) spectra of two intermediate polars (IPs): NY Lup and V1223 Sgr. We find that this model describes well many of the H- and He-like emission lines from medium-Z elements, which arise from a collisionally excited plasma. However the model fails to account for some of the He-like triplets from the medium-Z elements, which points toward their photoionized origin. We do not find compelling evidence for a blackbody component to model the soft excess seen in the residuals of the Chandra/HETG spectra, which could be due to uncertainties in the estimation of the interstellar absorption of these sources using the Chandra/HETG data and/or excess fluxes seen in some photoionized emission lines which are not accounted for by the cooling flow model. We describe the implications of this model with respect to the geometry of the pre-shock region for these two IPs.

Small-scale turbulent dynamo in astrophysical environments: nonlinear dynamo and dynamo in a partially ionized plasma

Small-scale turbulent dynamo is responsible for the amplification of magnetic fields on scales smaller than the driving scale of turbulence in diverse astrophysical media. Most earlier dynamo theories concern the kinematic regime and small-scale magnetic field amplification. Here we review our recent progress in developing the theories for the nonlinear dynamo and the dynamo regime in a partially ionized plasma. The importance of reconnection diffusion of magnetic fields is identified for both the nonlinear dynamo and magnetic field amplification during gravitational contraction. For the dynamo in a partially ionized plasma, the coupling state between neutrals and ions and the ion-neutral collisional damping can significantly affect the dynamo behavior and the resulting magnetic field structure. We present both our analytical predictions and numerical tests with a two-fluid dynamo simulation on the dynamo features in this regime. In addition, to illustrate the astrophysical implications, we discuss several examples for the applications of the dynamo theory to studying magnetic field evolution in both preshock and postshock regions of supernova remnants, in weakly magnetized molecular clouds, during the (primordial) star formation, and during the first galaxy formation.

Spatiotemporal linear instability analysis of collective neutrino flavor conversion in four-dimensional spacetime

In an environment with high-density neutrinos formed in a core-collapse supernova (CCSN), the neutrinos exhibit nonlinear and complex oscillation behaviors due to their self-interactions. The onset of this nonlinear oscillation can be investigated by linearizing the evolution equation for small perturbations around the flavor eigenstates. While the condition under which the flavor eigenstates are unstable has been investigated in many studies, how the perturbations evolve in spacetime has yet to be elucidated. In this paper, we analytically and correctly derive the asymptotic behaviors of the linear perturbations in 4-dimensional spacetime in the linear regime for a 2-beam neutrino model using the recently proposed Lefschetz thimble formulation. The result suggests that the perturbations grow in the directions between the two neutrino beams. We also briefly discuss the possible effects of neutrino flavor conversion on the explosion mechanism of a CCSN. In particular, the result implies that the flavor instability in the preshock region may propagate into the postshock region, contrary to the previous study focusing on the group velocity in 1-dimensional space. How to treat the case of a more realistic continuous spectrum is also discussed.

On the characteristics of fast neutrino flavor instabilities in three-dimensional core-collapse supernova models

We assess the occurrence of fast neutrino flavor instabilities in two three-dimensional state-of-the-art core-collapse supernova simulations performed using a two-moment three-species neutrino transport scheme: one with an exploding 9$\mathrm{M_{\odot}}$ and one with a non-exploding 20$\mathrm{M_{\odot}}$ model. Apart from confirming the presence of fast instabilities occurring within the neutrino decoupling and the supernova pre-shock regions, we detect flavor instabilities in the post-shock region for the exploding model. These instabilities are likely to be scattering-induced. In addition, the failure in achieving a successful explosion in the heavier supernova model seems to seriously hinder the occurrence of fast instabilities in the post-shock region. This is a consequence of the large matter densities behind the stalled or retreating shock, which implies high neutrino scattering rates and thus more isotropic distributions of neutrinos and antineutrinos. Our findings suggest that the supernova model properties and the fate of the explosion can remarkably affect the occurrence of fast instabilities. Hence, a larger set of realistic hydrodynamical simulations of the stellar collapse is needed in order to make reliable predictions on the flavor conversion physics.

Fast neutrino-flavor conversion in the preshock region of core-collapse supernovae

We make a strong case that the fast neutrino-flavor conversion, one of the collective flavor oscillation modes, commonly occurs in core-collapse supernovae (CCSNe). It is confirmed in the numerical data obtained in realistic simulations of CCSNe but the argument is much more generic and applicable universally: the coherent neutrino-nucleus scattering makes the electron lepton number (ELN) change signs at some inward direction and trigger the flavor conversion in the outward direction in the pre-shock region. Although the ELN crossing is tiny and that is why it has eluded recognition so far, it is still large enough to induce the flavor conversion. Our findings will have an important observational consequences for CCSNe neutrinos.

Radiative shocks in spherical accretion

ABSTRACT In order to explore various aspects of radiative shocks, we examine standing radiative shock waves in spherical accretion flows onto a central gravitating body under the equilibrium diffusion approximation. In contrast to the usual one-dimensional shock, in radiative shocks a radiative precursor appears in the pre-shock region before the shock front, due to the radiative diffusion effect. Furthermore, in spherical flows around a central object the gravitational potential varies in this radiative precursor, and a curvature effect also exists. We first formulate such radiative shocks in spherical flows, derive the overall jump conditions, and solve the structure of the radiative precursor for both the gas and radiation pressure dominated cases. Since the jump conditions contain the coordinates of both ends of the radiative precursor, we must obtain both the solution and the endpoints of the precursor simultaneously. We find that the gravitational effect is not significant, although it cannot be ignored. The curvature effect exerts a strong influence on the structure and width of the precursor. The precursor starting point x1 normalized by the shock radius is roughly expressed by $x_1={\cal M}_1^{1/7}$ for a radiation pressure dominated shock, while $x_1=1.21^{({\cal M}_1-1)}$ for a gas pressure one, where ${\cal M}_1$ is the pre-shock Mach number.

Compound Chondrule Formation in Optically Thin Shock Waves

Abstract Shock-wave heating within the solar nebula is one of the leading candidates for the source of chondrule-forming events. Here we examine the possibility of compound chondrule formation via optically thin shock waves. Several features of compound chondrules indicate that they are formed via the collisions of supercooled precursors. We evaluate whether compound chondrules can be formed via the collision of supercooled chondrule precursors in the framework of the shock-wave heating model by using semi-analytical methods and discuss whether most of the crystallized chondrules can avoid destruction upon collision in the post-shock region. We find that chondrule precursors immediately turn into supercooled droplets when the shock waves are optically thin, and they can maintain supercooling until the condensation of evaporated fine dust grains. Owing to the large viscosity of supercooled melts, supercooled chondrule precursors can survive high-speed collisions on the order of 1 km s−1 when the temperature is below ∼1400 K. From the perspective of the survivability of crystallized chondrules, shock waves with a spatial scale of ∼104 km may be potent candidates for the chondrule formation mechanism. Based on our results from one-dimensional calculations, a fraction of compound chondrules can be reproduced when the chondrule-to-gas mass ratio in the pre-shock region is ∼2 × 10−3, which is approximately half of the solar metallicity.

Electron Preacceleration in Weak Quasi-perpendicular Shocks in High-beta Intracluster Medium

Abstract Giant radio relics in the outskirts of galaxy clusters are known to be lit up by the relativistic electrons produced via diffusive shock acceleration (DSA) in shocks with low sonic Mach numbers, M s ≲ 3. The particle acceleration at these collisionless shocks critically depends on the kinetic plasma processes that govern the injection to DSA. Here, we study the preacceleration of suprathermal electrons in weak, quasi-perpendicular (Q ⊥) shocks in the hot, high-β (β = P gas/P B) intracluster medium (ICM) through two-dimensional particle-in-cell simulations. Guo et al. showed that, in high-β Q ⊥-shocks, some of the incoming electrons could be reflected upstream and gain energy via shock drift acceleration (SDA). The temperature anisotropy due to the SDA-energized electrons then induces the electron firehose instability (EFI), and oblique waves are generated, leading to a Fermi-like process and multiple cycles of SDA in the preshock region. We find that such electron preacceleration is effective only in shocks above a critical Mach number . This means that, in ICM plasmas, Q ⊥-shocks with M s ≲ 2.3 may not efficiently accelerate electrons. We also find that, even in Q ⊥-shocks with M s ≳ 2.3, electrons may not reach high enough energies to be injected to the full Fermi-I process of DSA, because long-wavelength waves are not developed via the EFI alone. Our results indicate that additional electron preaccelerations are required for DSA in ICM shocks, and the presence of fossil relativistic electrons in the shock upstream region may be necessary to explain observed radio relics.

Radiative shocks in disk accretion

Abstract Radiative shock waves standing in disk accretion flows are examined under the equilibrium diffusion approximation (1T limit) in the optically thick case, taking into account the hydrostatic equilibrium in the vertical direction. In contrast to the usual one-dimensional shock, where the gas density of the post-shock region increases due to the shock compression, if the shock is sufficiently strong, the gas density in the post-shock region often decreases due to the vertical expansion behind the shock front. However, the surface density behaves like the gas density in the usual shocks, and increases up to 7 in the radiation pressure dominated shock. Hence, the vertical optical depth in the post-shock region rises, in spite of the reduction of the gas density. In addition, similar to the usual radiative shock, there appears a radiative precursor in the pre-shock region before the shock front, due to the radiative diffusion effect. We derive the overall jump conditions for the radiative shock in disk accretion flows, and solve the structure of the radiative precursor for both the gas and radiation pressure dominated cases. The solutions are quite fundamental in disk-accretion shock problems, and should be developed in various aspects.

Bow Shock in Merging Cluster A520: The Edge of the Radio Halo and the Electron–Proton Equilibration Timescale

Abstract We studied the prominent bow shock in the merging galaxy cluster A520 using a deep Chandra X-ray observation and archival VLA radio data. This shock is a useful diagnostic tool, owing to its clear geometry and relatively high Mach number. At the “nose” of the shock, we measure a Mach number of . The shock becomes oblique away from the merger axis, with the Mach number falling to ≃1.6 around 30° from the nose. The electron temperature immediately behind the shock nose is consistent with that from the Rankine–Hugoniot adiabat, and is higher (at a 95% confidence) than expected for adiabatic compression of electrons followed by Coulomb electron–proton equilibration, indicating the presence of equilibration mechanisms faster than Coulomb collisions. This is similar to an earlier finding for the Bullet cluster. We also combined four archival VLA data sets to obtain a better image of the cluster’s giant radio halo at 1.4 GHz. An abrupt edge of the radio halo traces the shock front, and no emission is detected in the pre-shock region. If the radio edge were due only to adiabatic compression of relativistic electrons in pre-shock plasma, we would expect a pre-shock radio emission detectable in this radio data set; however, an interferometric artifact dominates the uncertainty, so we cannot rule this model out. Other interesting features of the radio halo include a peak at the remnant of the cool core, suggesting that the core used to have a radio minihalo, and a peak marking a possible region of high turbulence.

The structure and spectrum of the accretion shock in the atmospheres of young stars

The structure and spectrum of the accretion shock have been self-consistently simulated for a wide range of parameters typical for Classical T Tauri Stars (CTTS). Radiative cooling of the shocked gas was calculated, taking into account the self-absorption and non-equilibrium (time-dependent) effects in the level populations. These effects modify the standard cooling curve for an optically thin plasma in coronal equilibrium, however the shape of high-temperature ($T>3\times10^5$ K) part of the curve remains unchanged. The applied methods allow us to smoothly describe the transition from the cooling flow to the hydrostatic stellar atmosphere. Thanks to this approach, it has been found that the narrow component of HeII lines is formed predominantly in the irradiated stationary atmosphere (hotspot), i.e. at velocities of the settling gas $<2$ km s$^{-1}$. The structure of the pre-shock region is calculated simultaneously with the heated atmosphere. The simulation shows that the pre-shock gas produces a noticeable emission component in HeII lines and practically does not manifest itself in HeI lines ($\lambda\lambda$ 5876, 10830 {\AA}). The UV spectrum of the hotspot is distorted by the pre-shock gas, namely numerous red-shifted emission and absorption lines overlap each other forming a pseudo-continuum. The spectrum of the accretion region at high pre-shock densities $\sim10^{14}$ cm$^{-3}$ is fully formed in the in-falling gas and can be qualitatively described as a spectrum of a star with an effective temperature derived from the Stefan-Boltzmann law via the full energy flux.

SHOCK ACCELERATION MODEL WITH POSTSHOCK TURBULENCE FOR GIANT RADIO RELICS

We explore the shock acceleration model for giant radio relics, in which relativistic electrons are accelerated via diffusive shock acceleration (DSA) by merger-driven shocks in the outskirts of galaxy clusters. In addition to DSA, turbulent acceleration by compressive MHD mode downstream of the shock is included as well as energy losses of postshock electrons by Coulomb scattering, synchrotron emission, and inverse Compton scattering off the cosmic background radiation. Considering that only a small fraction of merging clusters host radio relics, we favor the reacceleration scenario in which radio relics are generated preferentially when shocks encounter the regions containing low-energy ($\gamma_{\rm e} \lesssim 300$) cosmic ray electrons (CRe). We perform time-dependent DSA simulations of spherically expanding shocks with physical parameters relevant for the Sausage radio relic, and calculate the radio synchrotron emission from the accelerated CRe. We find that significant level of postshock turbulent acceleration is required in order to reproduce broad profiles of the observed radio flux densities of the Sausage relic. Moreover, the spectral curvature in the observed integrated radio spectrum can be explained, if the putative shock should have swept up and exited out of the preshock region of fossil CRe about 10~Myr ago.

Shock Acceleration Model for the Toothbrush Radio Relic

Abstract Although many of the observed properties of giant radio relics detected in the outskirts of galaxy clusters can be explained by relativistic electrons accelerated at merger-driven shocks, significant puzzles remain. In the case of the so-called Toothbrush relic, the shock Mach number estimated from X-ray observations ( ) is substantially weaker than that inferred from the radio spectral index ( ). Toward understanding such a discrepancy, we here consider the following diffusive shock acceleration (DSA) models: (1) weak-shock models with and a preexisting population of cosmic-ray electrons (CRe) with a flat energy spectrum, and (2) strong-shock models with and either shock-generated suprathermal electrons or preexisting fossil CRe. We calculate the synchrotron emission from the accelerated CRe, following the time evolution of the electron DSA, and the subsequent radiative cooling and postshock turbulent acceleration (TA). We find that both models could reproduce reasonably well the observed integrated radio spectrum of the Toothbrush relic, but the observed broad transverse profile requires the stochastic acceleration by downstream turbulence, which we label “turbulent acceleration” or TA to distinguish it from DSA. Moreover, to account for the almost uniform radio spectral index profile along the length of the relic, the weak-shock models require a preshock region over 400 kpc with a uniform population of preexisting CRe with a high cutoff energy ( ). Due to the short cooling time, it is challenging to explain the origin of such energetic electrons. Therefore, we suggest the strong-shock models with low-energy seed CRe ( ) are preferred for the radio observations of this relic.