arXiv subjects

Eliot Quataert

Publications and source records attributed to Eliot Quataert.

At least 55 records · Page 3Linked to original sources

arXiv2025

The Type Ia Supernova and Asymptotic Giant Branch Stellar Ejecta-regulated Interstellar Medium of Massive Galaxies

Observations and theory suggest that Type Ia supernovae (SNIa) heating and mass loss from asymptotic giant branch (AGB) stars play a crucial role in the interstellar medium (ISM) of massive galaxies. We perform 3D hydrodynamic simulations of the central few kiloparsecs of massive galaxies, including radiative cooling and mass and energy injection from AGB winds and SNIa (resolving each SNIa remnant, a few $\times10~\mathrm{pc}$ in size), excluding black hole feedback. We study systems with different initial core thermodynamic profiles, focusing on NGC 1399. Our simulations reproduce its observed density and entropy profiles well. Over $100~\mathrm{Myr}$, two steady-state profiles emerge, depending on the inner circumgalactic medium (CGM) pressure and the ratio of Ia heating to cooling: (i) if SNIa heating is less than cooling, a cooling flow develops; (ii) if SNIa heating is comparable to or exceeds cooling, SNIa heating drives a slow subsonic outflow of AGB ejecta, with black hole accretion at small radii. This outflow, pressure-confined by the CGM, adapts the ISM to the CGM properties: a low entropy CGM results in a dense, low entropy ISM with higher black hole accretion, while a high entropy CGM leads to a less dense, high entropy ISM with lower accretion. This suggests that the AGB-SNIa regulated ISM connects CGM and galaxy scales, potentially influencing black hole feedback in massive halos. Approximate methods of modeling Ia heating, such as clustered SNIa and smoothly distributed heating, produce unrealistic ISM profiles over $100~\mathrm{Myr}$, highlighting the importance of resolving SNIa in simulations.

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arXiv2025

Super-Eddington accretion onto black holes and its application to fallback accretion

We study the problem of steady-state spherical accretion onto a black hole, in which the internal energy of the flow is governed by radiation and photon diffusion dominates the energy flux at large radii. In the free-fall limit, the fluid equations can admit two types of solutions for a given accretion rate: (1) accretion flows that become isothermal at large radii and (2) solutions in which the temperature at infinity vanishes as a power law of the radius. Using boundary layer theory, we obtain analytic solutions for the two cases and apply our results to fallback accretion onto a black hole following a failed supernova explosion. We give predictions for the observational signature of fallback accretion using realistic progenitor properties from MESA, both for a fully ionized inflow and for the more realistic case in which recombination/ionization take place due to low photospheric temperatures. The observed fading sources coincident with the failed-supernova candidates in NGC 6946 and M31 are too luminous to be powered by spherical accretion onto newly formed black holes; the observed sources are instead likely due to accretion of the turbulent, convective envelope of the supergiant progenitor.

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arXiv2025

Self-Similar Cosmic-Ray Transport in High-Resolution Magnetohydrodynamic Turbulence

We study the propagation of cosmic rays (CRs) through a simulation of magnetohydrodynamic (MHD) turbulence at unprecedented resolution of $10{,}240^3$. We drive turbulence that is subsonic and super-Alfvénic, characterized by $δB_{\rm rms}/B_0=2$. The high resolution enables an extended inertial range such that the Alfvén scale $l_A$, where $δB (l_A)\approx B_0$, is well resolved. This allows us to properly capture how the cascade transitions from large amplitudes on large scales to small amplitudes on small scales. We find that sharp bends in the magnetic field are key mediators of particle transport even on small scales via resonant curvature scattering. We further find that particle scattering in the turbulence shows strong hints of self-similarity: (1) the diffusion has weak energy dependence over almost two decades in particle energy and (2) the particles' random walk exhibits a broad power-law distribution of collision times such that the diffusion is dominated by the rarest, long-distance excursions. Our results suggest that large-amplitude MHD turbulence can provide efficient scattering over a wide range of CR energies and may help explain many CR observations above a $\sim$TeV: the flattening of the B/C spectrum, the hardening of CR primary spectra and the weak dependence of arrival anisotropy on CR energy.

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arXiv2025

Cyclic Zoom: Multiscale GRMHD Modeling of Black Hole Accretion and Feedback

We present a ``cyclic zoom'' method to capture the dynamics of accretion flows onto black holes across a vast range of spatial and temporal scales in general relativistic magnetohydrodynamic (GRMHD) simulations. In this method, we cyclically zoom out (derefine) and zoom in (refine) the simulation domain while using a central mask region containing a careful treatment of the coarsened fluid variables to preserve the small-scale physics, in particular the magnetic field dynamics. The method can accelerate GRMHD simulations by $\gtrsim 10^5$ times for problems with large-scale separation. We demonstrate the validity of the technique using a series of tests, including spherically symmetric Bondi accretion, the Blandford-Znajek monopole, magnetized turbulent Bondi accretion, accretion of a magnetized rotating torus, and the long-term evolution of an accreting torus about both Schwarzschild and Kerr black holes. As applications, we simulate Bondi and rotating torus accretion onto black holes from galactic scales, covering an extremely large dynamic range. In Bondi accretion, the accretion rate is suppressed relative to the Bondi rate by $\sim(10r_\mathrm{g}/r_\mathrm{B})^{1/2}$ with a feedback power of $\sim 0.01 \dot{M} c^2$ for vanishing spin, and $\sim 0.1 \dot{M} c^2$ for spin $a\approx0.9$. In the long-term evolution of a rotating torus, the accretion rate decreases with time as $\dot{M}\propto t^{-2}$ on timescales much longer than the viscous timescale, demonstrating that our method can capture not only quasi-steady problems but also secular evolution. Our new method likewise holds significant promise for applications to many other problems that need to cover vast spatial and temporal scales.

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arXiv2025

Modeling Cosmic Ray Electron Spectra and Synchrotron Emission in the Multiphase ISM

We model the transport and spectral evolution of 1-100 GeV cosmic ray (CR) electrons (CREs) in TIGRESS MHD simulations of the magnetized, multiphase interstellar medium. We post-process a kpc-sized galactic disk patch representative of the solar neighborhood using a two-moment method for CR transport that includes advection, streaming, and diffusion. The diffusion coefficient is set by balancing wave growth via the CR streaming instability against wave damping (nonlinear Landau and ion-neutral collisions), depending on local gas and CR properties. Implemented energy loss mechanisms include synchrotron, inverse Compton, ionization, and bremsstrahlung. We evaluate CRE losses by different mechanisms as a function of energy and distance from the midplane, and compare loss timescales to transport and diffusion timescales. This comparison shows that CRE spectral steepening above p = 1 GeV/c is due to a combination of energy-dependent transport and losses. Our evolved CRE spectra are consistent with direct observations in the solar neighborhood, with a spectral index that steepens from an injected value of -2.3 to an energy dependent value between -2.7 and -3.3. We also show that the steepening is independent of the injection spectrum. Finally, we present potential applications of our models, including to the production of synthetic synchrotron emission. Our simulations demonstrate that the CRE spectral slope can be accurately recovered from pairs of radio observations in the range 1.5-45 GHz.

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arXiv2025

Cosmic Rays Masquerading as Hot CGM Gas: An Inverse-Compton Origin for Diffuse X-ray Emission in the Circumgalactic Medium

Observations have argued that Milky Way (MW), Andromeda, and lower-mass galaxies exhibit extended soft X-ray diffuse halos to radii $R\gtrsim100\,$kpc in the circumgalactic medium (CGM). If interpreted as thermal emission, the shallow surface brightness profiles $S_{X}\propto R^{-1}$ are difficult to explain and contradict other observations. We show that such halos instead arise from inverse Compton (IC) scattering of CMB photons with GeV cosmic ray (CR) electrons. GeV electrons have ~Gyr lifetimes and escape the galaxy, forming a shallow extended profile out to $\gtrsim100\,$kpc, where IC off the CMB should produce soft, thermal-like X-ray spectra peaked at ~keV. The observed keV halo luminosities and brightness profiles agree well with those expected for CRs observed in the local interstellar medium (LISM) escaping the galaxy, with energetics consistent with known CRs from SNe and/or AGN, around galaxies with stellar masses $M_{\ast}\lesssim2\times 10^{11}\,M_{\odot}$. At higher masses observed X-ray luminosities are larger than predicted from IC and should be dominated by hot gas. In the MW+M31, the same models of escaping CRs reproduce gamma-ray observations if we assume an LISM-like proton-to-electron ratio and CR-pressure-dominated halo. In all other halos, the radio and $γ$-ray brightness is below detectable limits. If true, the observations provide qualitatively new constraints on CGM and CR physics: X-ray brightness directly traces the CR lepton energy density in the CGM. This agrees with LISM values within 10 kpc, which following the profile expected for escaping CRs in the CGM. The inferred CR pressure is a major part of the MW CGM pressure budget. X-ray surface brightness and luminosity allows one to further determine the CGM diffusivity at radii $\sim10-1000\,$kpc. These also agree with LISM values at small radii but increase in the CGM.

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arXiv2025

Cosmic Ray Feedback in Massive Halos: Implications for the Distribution of Baryons

We use order of magnitude estimates and observational constraints to argue that feedback from relativistic cosmic rays (CRs) produced by massive black holes is likely to have a particularly large effect at radii of order the virial radius and larger in group-mass halos. We show that for a range of plausible (but uncertain) CR transport parameters and energetics, the pressure produced by CRs generated by the central massive black hole over its lifetime can be of order the thermal gas pressure in the outskirts of $\sim 10^{13-14} M_\odot$ halos (but not in more massive clusters). The properties of this CR feedback at low redshift are not well predicted by the radiative cooling rate of hot gas at smaller radii, which is often used as a proxy for `current' black hole feedback. This is because most black hole growth happens early in massive halos, and CR transport timescales in halo outskirts are Gyr or more; the accumulated CR energy thus depends on the full history of black hole activity in the halo. The large CR pressure in group-mass systems likely leads to CR-driven outflows that move gas from large halo radii to outside the virial radius. Such feedback would not be captured by current cosmological simulations that focus on mechanical black hole feedback; in particular, CR feedback remains active even long after the mechanical feedback sourcing the CRs has turned off. We speculate that this CR feedback may be important for explaining the weak lensing $S_8$ tension and the evidence for strong feedback at large halo radii from kinetic Sunyaev-Zeldovich measurements. Prospects for testing this mechanism observationally and implementing the necessary physics in cosmological simulations are discussed.

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arXiv2025

The Double Tidal Disruption Event AT 2022dbl Implies That at Least Some "Standard" Optical TDEs are Partial Disruptions

Flares produced following the tidal disruption of stars by supermassive black holes can reveal the properties of the otherwise dormant majority of black holes and the physics of accretion. In the past decade, a class of optical-ultraviolet tidal disruption flares has been discovered whose emission properties do not match theoretical predictions. This has led to extensive efforts to model the dynamics and emission mechanisms of optical-ultraviolet tidal disruptions in order to establish them as probes of supermassive black holes. Here we present the optical-ultraviolet tidal disruption event AT 2022dbl, which showed a nearly identical repetition 700 days after the first flare. Ruling out gravitational lensing and two chance unrelated disruptions, we conclude that at least the first flare represents the partial disruption of a star, possibly captured through the Hills mechanism. Since both flares are typical of the optical-ultraviolet class of tidal disruptions in terms of their radiated energy, temperature, luminosity, and spectral features, it follows that either the entire class are partial rather than full stellar disruptions, contrary to the prevalent assumption, or that some members of the class are partial disruptions, having nearly the same observational characteristics as full disruptions. Whichever option is true, these findings could require revised models for the emission mechanisms of optical-ultraviolet tidal disruption flares and a reassessment of their expected rates.

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arXiv2025

Rapid, strongly magnetized accretion in the zero-net-vertical-flux shearing box

We show that there exist two qualitatively distinct turbulent states of the zero-net-vertical-flux shearing box. The first, which has been studied in detail previously, is characterized by a weakly magnetized ($β\sim50$) midplane with slow periodic reversals of the mean azimuthal field (dynamo cycles). The second, the 'low-$β$ state,' which is the main subject of this paper, is characterized by a strongly magnetized $β\sim 1$ midplane dominated by a coherent azimuthal field with much stronger turbulence and much larger accretion stress ($α\sim 1$). The low-$β$ state emerges in simulations initialized with sufficiently strong azimuthal magnetic fields. The mean azimuthal field in the low-$β$ state is quasi steady (no cycles) and is sustained by a dynamo mechanism that compensates for the continued loss of magnetic flux through the vertical boundaries; we attribute the dynamo to the combination of differential rotation and the Parker instability, although many of its details remain unclear. Vertical force balance in the low-$β$ state is dominated by the mean magnetic pressure except at the midplane, where thermal pressure support is always important (this holds true even when simulations are initialized at $β\ll 1$, provided the thermal scale height of the disk is well resolved). The efficient angular momentum transport in the low-$β$ state may resolve long-standing tension between predictions of magnetorotational turbulence (at high $β$) and observations; likewise, the bifurcation in accretion states we identify may be important for understanding the state transitions observed in dwarf novae, X-ray binaries, and changing-look AGN. We discuss directions for future work, including the implications of our results for global accretion disk models and simulations.

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arXiv2025

Limb-Brightened Jet in M87 from Anisotropic Nonthermal Electrons

Very long baseline interferometry observations reveal that relativistic jets like the one in M87 have a limb-brightened, double-edged structure. Analytic and numerical models struggle to reproduce this limb-brightening. We propose a model in which we invoke anisotropy in the distribution function of synchrotron-emitting nonthermal electrons such that electron velocities are preferentially directed parallel to magnetic field lines, as suggested by recent particle-in-cell simulations of electron acceleration and the effects of synchrotron cooling. We assume that the energy injected into nonthermal electrons is proportional to the jet Poynting flux, and we account for synchrotron cooling via a broken power-law energy distribution. We implement our emission model in both general relativistic magnetohydrodynamic (GRMHD) simulations and axisymmetric force-free electrodynamic (GRFFE) jet models and produce simulated jet images at multiple scales and frequencies using polarized general relativistic radiative transfer. We find that the synchrotron emission is concentrated parallel to the local helical magnetic field and that this feature produces limb-brightened jet images on scales ranging from tens of microarcseconds to hundreds of milliarcseconds in M87. We present theoretical predictions for horizon-scale M87 jet images at 230 and 345 GHz that can be tested with next generation instruments. Due to the scale-invariance of the GRMHD and GRFFE models, our emission prescription can be applied to other targets and serve as a foundation for a unified description of limb-brightened synchrotron images of extragalactic jets.

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arXiv2025

Signatures of Black Hole Spin and Plasma Acceleration in Jet Polarimetry

We study the polarization of black hole jets on scales of $10-10^3\,GM/c^2$ and show that large spatial swings in the polarization occur at three characteristic distances from the black hole: the radius where the counter-jet dims, the radius where the magnetic field becomes azimuthally dominated (the light cylinder), and the radius where the plasma reaches its terminal Lorentz factor. To demonstrate the existence of these swings, we derive a correspondence between axisymmetric magnetohydrodynamic outflows and their force-free limits, which allows us to analytically compute the plasma kinematics and magnetic field structure of collimated, general relativistic jets. We then use this method to ray trace polarized images of black hole jets with a wide range of physical parameters, focusing on roughly face-on jets like that of M87. We show that the location of the polarization swings is strongly tied to the location of the light cylinder and thus to the black hole's spin, illustrating a new method of measuring spin from polarized images of the jet. This signature of black hole spin should be observable by future interferometric arrays like the (Next Generation) Event Horizon Telescope, which will be able to resolve the polarized emission of the jet down to the near-horizon region at high dynamic range.

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arXiv2025

Evolution of SMBHs in light of PTA measurements: implications for growth by mergers and accretion

We study the growth of supermassive black holes accounting for both accretion and mergers. The former is informed by observations of the quasar luminosity function (QLF) and the latter by the gravitational wave-background (GWB) recently detected by PTAs, while estimates of the present-day black hole mass function provide a boundary condition. The GWB is dominated by the most massive black holes ($\gtrsim10^{9}M_{\odot}$). We show that their evolution can be simplified into a two-step process: mergers dominate at $z\leq1$, while accretion peaks at $1.4\leq z\leq2$. The large amplitude of the observed GWB suggests a significant number of mergers. We show that this generically implies a higher average Eddington ratio for quasars relative to a scenario in which mergers are negligible. In the absence of mergers, matching local estimates of BH abundance to the QLF implies a radiative efficiency $ε_r=0.12$ and Eddington ratio $λ=0.2$. With mergers, a progenitor of mass $M_i$ is boosted to a final total mass $M_f$ and there is a direct relation between the mass gained in mergers and the average Eddington ratio of the quasar population, given by $M_f/M_i\simλ/0.2$. There is thus a tension between the observed GWB, quasar properties, and the BH mass function: estimates of the mass function consistent with Eddington ratios inferred in quasars and $ε_r\sim0.1$ underpredict the GWB; multiple/equal mass mergers can boost the GWB, but lead to a high Eddington ratio. If the local mass function is on the high end of current estimates, the GWB is more readily explained, but requires low efficiencies $ε_r\sim10^{-2}$ not expected in standard luminous accretion models. The significant merger rate implied by the GWB also strongly suggests that the most massive BHs in the local universe have significant spin due to the orbital angular momentum from mergers, perhaps $a\sim0.5$.

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arXiv2025

Strongly magnetized accretion with low angular momentum produces a weak jet

We study the spherical accretion of magnetized plasma with low angular momentum onto a supermassive black hole, utilizing global general relativistic magnetohydrodynamic simulations. Black hole-driven feedback in the form of magnetic eruptions and jets triggers magnetized turbulence in the surrounding medium. We find that when the Bondi radius exceeds a certain value relative to the black hole's gravitational radius, this turbulence restricts the subsequent inflow of magnetic flux, strongly suppressing the strength of the jet. Consequently, magnetically arrested disks and powerful jets are not a generic outcome of the accretion of magnetized plasma, even if there is an abundance of magnetic flux available in the system. However, if there is significant angular momentum in the inflowing gas, the eruption-driven turbulence is suppressed (sheared out), allowing for the presence of a powerful jet. Both the initially rotating and nonrotating flows go through periods of low and high gas angular momentum, showing that the angular momentum content of the inflowing gas is not just a feature of the ambient medium, but is strongly modified by the eruption and jet-driven black hole feedback. In the lower-angular-momentum states, our results predict that there should be dynamically strong magnetic fields on horizon scales, but no powerful jet; this state may be consistent with Sgr A* in the Galactic center.

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arXiv2024

Star-Disk Collisions: Implications for Quasi-periodic Eruptions and Other Transients Near Supermassive Black Holes

We use Athena++ to study the hydrodynamics of repeated star-accretion disk collisions close to supermassive black holes, and discuss their implications for the origin of quasi-periodic eruptions (QPEs) and other repeating nuclear transients. We quantify the impact of the collisions on the stellar structure, the amount of stripped stellar debris, and the debris' orbital properties. We provide simple fitting functions for the stellar mass-loss per collision; the mass-loss is much larger after repeated collisions due to the dilute stellar atmosphere shock-heated in earlier collisions. The lifetime of the QPE-emitting phase set by stellar mass-loss in star-disk collision models for QPEs is thus at most ~1000 years; it is shortest for eRO-QPE2, of order a few decades. The mass of the stripped stellar debris per collision and its orbital properties imply that currently observed QPEs are not powered by direct star-disk collisions but rather by collisions between the stellar debris liberated in previous collisions and the accretion disk (`circularization shocks'). We discuss how the hydrodynamics of this interaction can explain the diverse timing properties of QPEs including the regular timing of GSN 069 and eRO-QPE2 and the large flare-to-flare timing variations observed in eRO-QPE1. QPEs with recurrence times of many days, if observed, may have more regular timing.

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arXiv2024

Elevated UV luminosity density at Cosmic Dawn explained by non-evolving, weakly mass-dependent star formation efficiency

Recent observations with the James Webb Space Telescope (JWST) have uncovered unexpectedly high cosmic star formation activity in the early Universe, mere hundreds of millions of years after the Big Bang. These observations are often understood to reflect an evolutionary shift in star formation efficiency (SFE) caused by changing galactic conditions during these early epochs. We present FIREbox-HR, a high-resolution, cosmological hydrodynamical simulation from the Feedback in Realistic Environments project, which offers insights into the SFE of galaxies during the first billion years of cosmic time. FIREbox-HR re-simulates the cosmic volume (L = 22.1 cMpc) of the original FIREbox run with eight times higher mass resolution (m_b ~ 7800 M_sun), but with identical physics, down to z ~ 6. FIREbox-HR predicts ultraviolet (UV) luminosity functions in good agreement with available observational data. The simulation also successfully reproduces the observed cosmic UV luminosity density at z ~ 6 - 14, demonstrating that relatively high star formation activity in the early Universe is a natural outcome of the baryonic processes encoded in the FIRE-2 model. According to FIREbox-HR, the SFE - halo mass relation for intermediate mass halos (M_halo ~ 10^9 - 10^11 M_sun) does not significantly evolve with redshift and is only weakly mass-dependent. These properties of the SFE - halo mass relation lead to a larger contribution from lower mass halos at higher z, driving the gradual evolution of the observed cosmic UV luminosity density. A theoretical model based on the SFE - halo mass relation inferred from FIREbox-HR allows us to explore implications for galaxy evolution. Future observations of UV faint galaxies at z > 12 will provide an opportunity to further test these predictions and deepen our understanding of star formation during Cosmic Dawn.

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arXiv2024

A Unified Model of Cosmic Ray Propagation and Radio Extreme Scattering Events from Intermittent Interstellar Structures

Intermittent magnetic structures are a plausible candidate for explaining cosmic-ray (CR) diffusion rates derived from observed CR energy spectra. Independently, studies of extreme scattering events (ESEs) of radio quasars and pulsar scintillation have hinted that very straight, large-aspect-ratio, magnetic current sheets may be responsible for the localized large scattering of radio waves. The required shortest axis of the typical structures producing ESEs is of the same scale ($\sim$AU) as the gyroradii of $\sim$GeV CRs. In this paper, we propose that the same magnetic/density sheets can produce large scattering of both CRs and radio waves. We demonstrate that the geometry and volume filling factor of the sheets derived from quasar ESEs can explain the observed mean free path of GeV CRs without introducing free parameters. The model places constraints on the sheet geometry, such as straightness and large aspect ratio, and assumes the statistics of the sheets are similar throughout the Galactic volume. We, therefore, discuss observational tests of the sheet model, which includes observations of echoes in pulsars and fast radio bursts, gravitationally lensed quasars, the distribution of ESE durations, and spatial correlations between ESE events and rotation-measure fluctuations. Such tests will be enabled by upcoming wide-field radio instruments, including Canadian Hydrogen Observatory and Radio-transient Detector (CHORD) and Deep Synoptic Array 2000 Antennas (DSA-2000).

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arXiv2024

The Peak Frequency and Luminosity of Synchrotron Emitting Shocks: from Non-Relativistic to Ultra-Relativistic Explosions

Synchrotron emission is ubiquitous in explosive astrophysical events -- it is a natural byproduct of shocks formed when matter expelled by the explosion collides with ambient material. This emission is well-observed in various classes of transients, and is often interpreted within a canonical `equipartition' framework that allows physical properties of the shock to be inferred from the frequency and luminosity at which the observed spectral energy distribution (SED) peaks. This framework has been remarkably successful in explaining observations of radio supernovae. It has also been used for trans-relativistic explosions, where the shock velocities approach the speed of light. However, the conventional framework does not incorporate relativistic effects. Neither does it account for thermal electrons, which have been shown to be important for high-velocity shocks. In this paper we describe a revised framework that accounts for these two effects, and is applicable to non-relativistic, trans-relativistic, and ultra-relativistic explosions. We show that accounting for these effects can dramatically change the inferred parameters of high-velocity shocks, and in particular -- that the shock velocity, ambient density, and total energy are overestimated by the conventional non-relativistic framework. We delineate the phase-space where such modifications are important in terms of observationally measurable parameters. We also find a novel upper limit on the peak synchrotron luminosity of shock-powered transients, which is remarkably consistent with existing observations. Finally, we discuss a prediction of the model -- that the SED will qualitatively change as a function of shock velocity -- and show that this is broadly consistent with data for representative events (e.g., SN1998bw, AT2018cow, CSS161010, AT2020xnd).

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arXiv2024

Nonlinear perturbations and weak shock waves in isentropic atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave-steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham's approximation can be used to solve for the shock's dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate ($\sim$ few--15) Mach numbers, where the accuracy increases with the adiabatic index.

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