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Eric R. Coughlin

Publications and source records attributed to Eric R. Coughlin.

At least 19 recordsLinked to original sources
arXiv2026

Outflows in steep density gradients: diversity of behavior and implications for tidal disruption events and luminous fast blue optical transients

Powerful explosions may undergo sustained energy injection as a central engine launches a wind into the surrounding gas, generating a forward and a reverse shock separated by a contact discontinuity. During the adiabatic phase, the dynamics depend strongly on the wind-to-ambient density ratio $f \equiv ρ_{\rm w} / ρ_{\rm a}$. For $f << 1$, the reverse shock lies well inside the contact discontinuity, and the mechanical energy deposited by the wind is retained in a radially extended, approximately isobaric shocked-wind region whose pressure drives the swept-up ambient shell. For $f \gg 1$, the reverse shock remains close to the contact, and the expansion is governed by the ram-pressure interaction between the freely expanding wind and the swept-up ambient gas. We use analytic scalings and one-dimensional shock-capturing hydrodynamic simulations to determine how outflows in these two limits evolve in ambient density profiles $ρ_{\rm a} \propto r^{-n}$, where $2 \leq n \leq 3$, and whether their shock structures accelerate or coast at constant velocity. For $n > 2$, initially underdense outflows produce accelerating forward shocks whose radii evolve as $R_{\rm s} \propto t^{3/(5-n)}$. Because $ρ_{\rm w} \propto r^{-2}$, f increases with radius, causing the reverse-shocked wind region to contract relative to the contact position as the forward shock transitions toward constant-velocity expansion. This occurs when $f \sim$ a few at $t_{\rm dec} \propto f_0^{1/(2-n)}$, where $f_0$ is the initial wind-to-ambient density ratio. By contrast, outflows initialized with $f_0 \gg 1$ do not develop an extended accelerating phase and remain approximately coasting throughout their adiabatic evolution. We discuss applications to tidal disruption event outflows and luminous fast blue optical transients, whose environments are often inferred to have steep density profiles with $n > 2$.

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arXiv2026

Systematic Error in Approximate Models of the GRB Early Afterglow

Gamma-ray burst (GRB) afterglows are thought to arise when relativistic ejecta launched by a compact central engine drive a blast wave into the surrounding circumburst medium, producing broadband synchrotron emission. We present a rigorous assessment, based on high-resolution special relativistic hydrodynamics simulations, of a widely adopted `two-zone model' for approximating the dynamics of the early afterglow phase. Before the onset of the Blandford-McKee (BMK) self-similar solution, the outflow generally produces two emission components, associated with the forward-shocked circumburst medium and the reverse-shocked ejecta. The subsequent evolution depends on whether the reverse shock significantly decelerates the ejecta as it crosses the shell, separating the so-called relativistic and Newtonian reverse shock regimes. We show that when the reverse shock is Newtonian, it crosses the ejecta shell long before BMK self-similarity is established, leaving a prolonged interval that can span $\sim$ hours in observer time in which the true hydrodynamic evolution is not captured by standard semi-analytic prescriptions. We demonstrate that this mismatch can, for representative afterglow parameters, substantially overpredict the reverse-shock emission from radio through ultraviolet frequencies, or overpredict the forward-shock emission at X-ray frequencies, depending on how the transition away from the two-zone model is prescribed.

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arXiv2026

The Role of Stellar Spin in Repeating Partial Tidal Disruption Events

The repeated tidal stripping of a star by a supermassive black hole, known as a repeating partial tidal disruption event (rpTDE), can give rise to a transient that rebrightens months to years after the first outburst. Among the rpTDE candidates so far observed, some exhibit dimmer peak luminosities during each successive outburst, which is a trend that has not been reproduced from theoretical models when the star survives more than one encounter with the black hole. Here we suggest that this trend can be recovered if the partially disrupted star is initially (i.e., prior to its first mass-stripping event) rapidly rotating, which is expected if the star was placed on its orbit through the Hills breakup of a tidally locked and tight binary. We test this hypothesis with hydrodynamical simulations of high-mass ($\geqslant 1 M_{\odot}$) main sequence stars repeatedly partially disrupted by a $10^6 M_{\odot}$ black hole, and demonstrate that successively dimmer outbursts are indeed recovered for high (tens of percent breakup) and prograde (i.e., aligned with the orbital angular momentum) stellar spins. Our results provide strong indirect evidence for the operation of the Hills mechanism in seeding the stars in rpTDEs.

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arXiv2026

The interaction phase of engine-driven explosions and high-energy winds

Wide-angle outflows, or winds, are associated with a broad range of astrophysical systems, including protostars, massive stars, X-ray binaries, tidal disruption events (TDEs), luminous fast blue optical transients (LFBOTs), and starburst galaxies. When these winds first ``turn on," they inflate a ``bubble" into their surroundings, bounded by two shocks and a contact discontinuity, and evolve through distinct adiabatic phases prior to the onset of significant radiative cooling. For sufficiently overdense ejecta, the flow quickly relaxes into an interaction-dominated similarity state at early times and later enters an energy-conserving regime. We present a systematic study of these phases for adiabatic winds expanding into power-law density profiles $ρ\propto r^{-n}$ with $0 \leq n \leq 2$. Using analytic scalings together with one-dimensional shock-capturing hydrodynamic simulations, we quantify both the relaxation timescales and the accuracy with which the corresponding similarity solutions reproduce the fluid velocity, density, and pressure throughout the shocked bubble. We show that the interaction solutions are attained within only a few dynamical times and remain valid until the reverse-shocked shell is no longer thin relative to the forward-shocked shell, corresponding in practice to an instantaneous overdensity of order unity. For $n < 2$, the flow subsequently converges to the generalized energy-conserving scaling $R_s \propto t^{3/(5-n)}$, while the special case $n=2$ exhibits a single persistent similarity state. We discuss the durations and implications of these phases for stellar and galactic outflows, TDEs, and LFBOTs.

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arXiv2026

Resolving the (Debate About) Nozzle Shocks in Tidal Disruption Events

When a star passes within the Roche limit of a supermassive black hole (SMBH), it is pulled apart by the BH's tidal field in a tidal disruption event (TDE). The resulting flare is powered by the circularization and accretion of bound stellar debris, which initially returns to the BH on eccentric orbits in a thin debris stream. The returning fluid elements follow inclined orbits that converge near pericenter, resulting in extreme vertical compression to scales $10^{-4}~R_\odot$ and the formation of a nozzle shock. Dissipation at the nozzle shock may affect circularization by altering the properties of the debris stream, but its role is the subject of ongoing debate. We develop an idealized model for the debris stream evolution combining 3D smoothed-particle hydrodynamics simulations, the semi-analytic affine model, and 1D finite-volume hydrodynamic simulations. Because our model is computationally cheap, we can unambiguously resolve the nozzle shock, use a realistic equation of state, and follow the debris stream evolution at many different times. Near peak fallback, Hydrogen recombination and molecular Hydrogen formation broaden the stream by a factor $\sim 5$, enhancing dissipation at the nozzle. However, the dissipation is still insufficient to directly circularize the debris by in-plane pressure gradients. Instead, the thicker stream substantially increases the likelihood that the stream self-intersects on the second orbit, despite relativistic nodal precession. The stream properties at self-intersection are sensitive to dissipation at the nozzle and the timing of focal points where the ballistic trajectories of the debris converge. Our results clarify the nozzle shock's role in circularization in TDEs, providing a foundation for more realistic circularization and emission models.

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arXiv2026

On the origin of anomalous dissipation in simulations of tidal disruption events

In a tidal disruption event (TDE), a star is destroyed by the tidal field of a supermassive black hole. The stellar debris is initially placed on highly elliptical orbits, and a longstanding question in TDE theory is: How does the stellar debris circularize into a disc and accrete? The originally proposed answer to this question is self-intersection shocks, where relativistic apsidal precession results in a strong collision between the incoming and outgoing material. However, global simulations of TDEs tend to find enhanced hydrodynamical dissipation prior to any intersections of the debris orbits, with the material ``fanning out'' into a wide-angle and partially-unbound outflow upon passing through pericenter. We show that this dissipation is numerical in origin and arises from a combination of 1) the change in the kinematics of the debris as it passes through pericenter, with its velocity profile along the stream transitioning from strongly diverging pre-pericenter to strongly converging post-pericenter, and 2) the dependence of numerical algorithms (viscosity switches for particle-based methods and Riemann solvers for Godunov-based schemes) on the diverging vs. converging nature of the fluid. We support this conclusion with analytical and numerical modeling. We discuss possible resolutions to these issues as well as the implications of our findings in the context of observations.

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arXiv2026

On the Origin of Mass Ejection in Failed Supernovae

Some high-mass stars likely end their lives in underluminous implosions that leave behind a black hole, known as failed supernovae (FSNe). However, neutrinos radiated during proto-neutron star formation generate a weak (Mach $\gtrsim 1$) shockwave in the outer layers of the star, which produces a unique transient as it breaks out of the dying star and signals its imminent disappearance. It was recently shown that there are two self-similar solutions that describe the propagation of this weak shockwave, and these solutions simultaneously contain outward-moving ejecta and fallback accretion onto the black hole. Here we show that the larger Mach number solutions are unstable, such that the Mach number of the shock grows with time $t$ and deviates from the self-similar prediction as $\propto t^α$, with $α\lesssim 0.1$, whereas the smaller Mach number solutions are stable. We also show that, above a critical mass loss that is readily achievable in core-collapse supernovae, the shock asymptotically strengthens and approaches the strong limit. Our results imply that it is the mass lost to neutrinos \textit{relative} to the mass enclosed by the shockwave, as well as the stellar density gradient where the shock forms, that primarily dictate its strength and the amount of material it ejects. These criteria explain why red supergiants, which have relative mass losses well in excess of the critical value at the time of shock formation, more readily eject material and create more luminous explosions compared to more compact progenitors.

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arXiv2026

AT2024wpp: An Extremely Luminous Fast Ultraviolet Transient Powered by Accretion onto a Black Hole

We present the discovery of AT 2024wpp ("Whippet"), a fast and luminous 18cow-like transient. At a redshift of z=0.0868, revealed by Keck Cosmic Web Imager spectroscopy of its faint star-forming host, it is the fourth-nearest example of its class to date. Rapid identification of the source in the Zwicky Transient Facility data stream permitted ultraviolet-through-optical observations to be obtained prior to peak, allowing the first determination of the peak bolometric luminosity (2x10^45 erg/s), maximum photospheric radius (10^15 cm), and total radiated energy (10^51 erg) of an 18cow-like object. We present results from a comprehensive multiwavelength observing campaign, including a far-UV spectrum from the Cosmic Origins Spectrograph on the Hubble Space Telescope and deep imaging extending >100 days post-explosion from the Very Large Telescope, Hubble Space Telescope, Very Large Array, and Atacama Large Millimetre Array. We interpret the observations under a model in which a rapidly-accreting central engine blows a fast (~0.2c) wind into the surrounding medium and irradiates it with X-rays. The high Doppler velocities and intense ionization within this wind prevent identifiable spectroscopic features from appearing in the ejecta or in the surrounding circumstellar material. Weak H and He signatures do emerge in the spectra after 35 days in the form of double-peaked narrow lines. Each peak is individually narrow (full width ~3000 km/s) but the two components are separated by ~6600 km/s, indicating stable structures of denser material, possibly representing streams of tidal ejecta or an ablated companion star.

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arXiv2026

Repeating Nuclear Transients from Repeating Partial Tidal Disruption Events

Extragalactic nuclear transients that exhibit repeating outbursts can be modeled as the repeated dynamical interaction between bound stars and supermassive black holes (SMBHs). A subset of these transients, with recurrence timescales of months-to-years, have been explained as accretion flares from the repeated tidal stripping of a star by an SMBH, in a repeating partial tidal disruption event (rpTDE). We outline the scope of the rpTDE model and discuss hydrodynamical simulations and analytical predictions for the stability of stars undergoing repeated mass loss, and the long-term evolution of these flares as a function of stellar type and orbital parameters. Our findings demonstrate that high-mass and centrally concentrated stars undergo negligible changes in structure in response to small amounts ($\sim 1-10\% M_\star$) of mass loss, and can survive many mass-stripping encounters with an SMBH. Contrarily, low-mass and less evolved stars are unstable to mass loss, and would be destroyed within a few orbits. We discuss the implications of these results for constraining the stellar type and orbital parameters of observed sources, such as ASASSN-14ko, for which $\gtrsim 20$ flares have been observed, and AT2020vdq, which exhibits a second flare that is brighter than its primary outburst.

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arXiv2026

AT2024lhc and AT2024kmq in the landscape of featureless tidal disruption events

We study AT2024kmq and AT2024lhc, two tidal disruption events (TDEs) with blue featureless spectra associated with high-mass black holes ($M_{\rm BH}\sim 10^8\,M_\odot$). Both events show optical precursors consistent with shock dissipation from stream self-intersection. Their X-ray emission is luminous ($L_{\rm X}\sim 10^{44}\,{\rm erg\,s^{-1}}$), highly variable (with minimum observed variability timescales of 1.3\,hr and 4.8\,hr for factor of $\sim3$ flux changes), long-lasting ($>1\,\rm yr$), emerging no later than the optical peak, and well characterized by power-laws with $1.7<Γ<3$ (where $f_ν\propto ν^{1-Γ}$). The X-ray properties and radio non-detections support a compact corona ($\lesssim 10 r_{\rm g}$) producing Comptonized X-ray emission. Using all published featureless TDEs, we find statistically significant bimodality in the distribution of their peak UV/optical blackbody luminosities and radii. We assemble a comparison TDE sample with early-time X-ray observations with eROSITA, in which we find different $M_{\rm BH}$ distributions in TDEs with different X-ray spectral evolution properties: low-mass black holes ($M_{\rm BH} \sim 10^6 M_\odot$) remain soft ($Γ>4$) within $t\lesssim 2$\,yr, intermediate masses ($\sim 10^7 M_\odot$) transition from soft to hard at $\sim$1 yr, while high masses ($\sim 10^8 M_\odot$) are hard ($1.5<Γ\lesssim 3$) from the outset. We interpret this result as evidence that the soft-to-hard state transition in TDEs occurs at the critical threshold of $\dot{M}_{\rm acc} \sim 0.03 \dot M_{\rm Edd}$ (similar to X-ray binaries), using the fact that the transition timescale predicted by simple disk theory scales with black hole mass as $t_{\rm tr}\propto M_{\rm BH}^{-3/4}$.

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arXiv2026

The Maximum Gravity Model for partial Tidal Disruption Events: Mass Loss, Peak Fallback Rate and Dependence on Stellar Properties

A star entering the tidal sphere of a supermassive black hole (SMBH) can be partially stripped of mass, resulting in a partial tidal disruption event (TDE). Here we develop an analytical model for properties of these events, including the peak fallback rate, $\dot{M}_{\rm peak}$, the time at which the peak occurs, $t_{\rm peak}$, and the amount of mass removed from the star, $ΔM$, for any star and any pericenter distance associated with the stellar orbit about the black hole. We compare the model predictions to 1276 hydrodynamical simulations of partial TDEs of main-sequence stars by a $10^6 M_\odot$ SMBH. The model yields $t_{\rm peak}$ predictions that are in good agreement (to within tens of percent) with the numerical simulations for any stellar mass and age. The agreement for $\dot{M}_{\rm peak}$ is weaker due to the influence of self-gravity on the debris stream dynamics, which remains dynamically important for partial TDEs; the agreement for $\dot{M}_{\rm peak}$ is, however, to within a factor of $\sim 2-3$ in the majority of cases considered, with larger differences for low-mass stars ($M_\star \lesssim 0.5 M_\odot$) on grazing orbits with small mass loss. We show that partial TDE lightcurves for disruptions caused by $\sim 10^6M_\odot$ SMBHs can span $\sim 20-100$ day peak timescales, whereas grazing encounters of high-mass stars with high-mass SMBHs can yield longer peak timescales ($t\gtrsim 1000$ days), associated with some observed transients. Our model provides a significant step toward an analytical prescription for TDE lightcurves and luminosity functions.

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arXiv2025

A Wind-Driven Origin for the Firework Morphology of the Supernova Remnant Pa 30

Pa 30 -- the likely remnant of the Galactic type Iax supernova of 1181 AD -- displays an unusual, firework-like morphology, consisting of radial filaments extending from a common center, where a white dwarf (WD) currently drives a very fast wind (speed $\gtrsim 10^{4}$ km s$^{-1}$). We propose the filaments arose from the Rayleigh-Taylor-unstable nature of the interface between the circumstellar medium (CSM) and the shocked wind launched by the natal WD; the filaments then elongated intact due to the Kelvin-Helmholtz-stable nature of the large initial density contrast between the wind and CSM, supplemented by the slowly declining wind density profile (relative to homologously expanding ejecta). To support this interpretation, we present two-dimensional hydrodynamical simulations and derive the filament properties, including their speed, density, and temperature, all of which are consistent with observations. We suggest the filaments elongate until the wind and CSM densities become comparable at the contact discontinuity, which occurs within 1--10 years, and then truncate because the RTI halts. The subsequent KHI growth timescale across the current width of the filaments is longer than the age of Pa 30, so they remain intact. The filament-less central region in Pa 30 is therefore more likely a consequence of the finite timescale over which the RTI operates, rather than a wind termination shock. In general, firework-like filaments may form in other systems, provided there is a sufficiently large density contrast between the ejecta and its surroundings.

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arXiv2025

Tidal Disruption Events

Stars that orbit too close to a black hole can be ripped apart by strong tides, producing a type of luminous transient event called a ``tidal disruption event" (TDE). Tidal disruption events of stars by supermassive black holes (SMBHs) provide windows into the nuclei of galaxies at size scales that are difficult to observe directly outside our own galactic neighborhood. They provide a unique opportunity to study these supermassive black holes under feeding conditions that change dramatically over ~week-month timescales, and that regularly reach super-Eddington mass inflow rates. Their light curves are dependent on the properties of the disrupting black hole, and can be used to help constrain the lower mass end of the SMBH mass function -- a region of parameter space that is difficult to access with classic dynamical mass measurements.

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arXiv2025

Optical/infrared observations of the extraordinary GRB 250702B: a highly obscured afterglow in a massive galaxy consistent with multiple possible progenitors

GRB 250702B was the longest gamma-ray burst ever detected, with a duration that challenges standard collapsar models and suggests an exotic progenitor. We collected a rich set of optical and infrared follow-up observations of its rapidly fading afterglow using a suite of telescopes including the W. M. Keck Observatory, the Gemini telescopes, the Magellan Baade Telescope, the Victor M. Blanco 4-meter telescope, and the Fraunhofer Telescope at Wendelstein Observatory. Our analysis reveals that the afterglow emission is well described by forward shock emission from a highly obscured relativistic jet. Deep photometric observations of the host galaxy reveal a massive (10^10.66 solar masses), dusty, and extremely asymmetric system that is consistent with two galaxies undergoing a major merger. The galactocentric offset, host galaxy properties, and jet characteristics disfavor a jetted TDE around a supermassive black hole but do not definitively distinguish between competing progenitor scenarios. We find that the afterglow and host are consistent with a range of progenitors including an atypical collapsar, a merger between a helium star and a stellar mass black hole, the disruption of a star by a stellar mass compact object (micro-TDE), and the tidal disruption of a star by an off-nuclear intermediate mass black hole.

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arXiv2025

Tidal disruption events with SPH-EXA: resolving the return of the stream

In a tidal disruption event (TDE), a star is disrupted by the tidal field of a massive black hole, creating a debris stream that returns to the black hole, forms an accretion flow, and powers a luminous flare. Over the last few decades, several numerical studies have concluded that shock-induced dissipation occurs as the stream returns to pericentre (i.e., pre-self-intersection), resulting in efficient circularisation of the debris. However, the efficacy of these shocks is the subject of intense debate. We present high-resolution simulations (up to 10^10 particles) of the disruption of a solar-like star by a 10^6M_sun black hole with the new, GPU-based, smoothed-particle hydrodynamics code SPH-EXA, including the relativistic apsidal precession of the stellar debris orbits; our simulations run from initial disruption to the moment of stream self-intersection. With 10^8 particles - corresponding to the highest-resolution SPH simulations of TDEs in the pre-existing literature - we find significant, in-plane spreading of the debris as the stream returns through pericenter, in line with previous works that suggested this is a significant source of dissipation and luminous emission. However, with increasing resolution this effect is dramatically diminished, and with 10^10 particles there is effectively no change between the incoming and the outgoing stream widths. Our results demonstrate that the paradigm of significant dissipation of kinetic energy during pericentre passage is incorrect, and instead it is likely that debris circularisation is mediated by the originally proposed, stream-stream collision scenario.

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arXiv2025

Comprehensive X-ray Observations of the Exceptional Ultra-long X-ray and Gamma-ray Transient GRB 250702B with Swift, NuSTAR, and Chandra: Insights from the X-ray Afterglow Properties

GRB 250702B is an exceptional transient that produced multiple episodes of luminous gamma-ray radiation lasting for $>25$ ks, placing it among the class of ultra-long gamma-ray bursts (GRBs). However, unlike any known GRB, the \textit{Einstein Probe} detected soft X-ray emission up to 24 hours before the gamma-ray triggers. We present comprehensive X-ray observations of the transient's afterglow obtained with the Neil Gehrels Swift Observatory, the Nuclear Spectroscopic Telescope Array, and the Chandra X-ray Observatory between 0.5 to 65 days (observer frame) after the initial high-energy trigger. The X-ray emission decays steeply as $\sim t^{-1.9}$, and shows short timescale X-ray variability ($ΔT/T < 0.03$) in both Swift and NuSTAR, consistent with flares superposed on an external shock continuum. Serendipitous detections by the Swift Burst Alert Telescope (BAT) out to $\sim$0.3 days and continued NuSTAR variability to $\sim$2 days imply sustained central engine activity; including the precursor, the required engine duration is $\gtrsim 3$ days. Afterglow modeling favors the combination of forward and reverse shock emission in a wind-like ($k \approx 2$) environment. These properties, especially the long-lived engine and early soft X-ray emission, are difficult to reconcile with a collapsar origin, and GRB 250702B does not fit neatly with canonical ultra-long GRBs or relativistic tidal disruption events (TDEs). A hybrid scenario in which a star is disrupted by a stellar-mass black hole (a micro-TDE) provides a plausible explanation, although a relativistic TDE from an intermediate-mass black hole remains viable.

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arXiv2025

Predicting the Properties of the Fallback Rate from Tidal Disruption Events: Investigating the Maximum Gravity Model

A star destroyed by the tidal field of a supermassive black hole (SMBH) in a tidal disruption event (TDE) gives rise to a luminous flare. TDEs are being detected at an ever-increasing rate, motivating the need for accurate models of their lightcurves. The ``maximum gravity'' (MG) model posits that a star is completely destroyed when the tidal field of the SMBH exceeds the maximum self-gravitational field within the star, $g_{\rm max}$, and predicts the peak fallback rate $\dot{M}_{\rm peak}$ and the time to peak $t_{\rm peak}$. Here we perform hydrodynamical simulations of the complete disruption of 24 stars with masses ranging from $0.2-5.0 M_\odot$, at different stages of their main sequence evolution, to test the predictions of this model. We find excellent agreement between the MG model predictions and our simulations for stars near the zero-age main sequence, while the predictions are less accurate (but still within $\sim 35-50\%$ of the simulation results) for highly evolved stars. We also generalize the MG model to incorporate the Paczy{ń}ski-Wiita potential to assess the impact of strong-gravity effects -- which are especially important for deep encounters that are required to completely destroy evolved and centrally concentrated stars -- and find good agreement with recent works that include relativistic gravity. Our results demonstrate that this model provides accurate constraints on the peak timescale of TDE lightcurves and their correlation with black hole mass.

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arXiv2025

What's kickin' in partial tidal disruption events?

Stars partially destroyed by a supermassive black hole (SMBH) in a partial tidal disruption event (TDE) can be ejected from the SMBH. Previous investigations attributed this positive-energy/velocity kick to asymmetries in the mass lost by the star near pericenter. We propose that asymmetric mass loss is not predominantly responsible for "kicking" the star, and that these kicks instead arise from the combination of a) the reformation of the core following an initial phase of quasi-ballistic motion, and b) the differential shear between the unbound and marginally bound (to the SMBH) material during this phase. We predict that the kick speed $v_{\rm kick}$ is weakly dependent on the stellar properties, and for SMBH masses $M_{\bullet} \gtrsim 10^{3} M_{\odot}$, $v_{\rm kick}$ is independent of SMBH mass, is not limited to the stellar escape speed $v_{\rm esc}$, and is related to the surviving core mass $M_{\rm c}$ approximately as $v_{\rm kick} \simeq 0.45 \left(M_{\rm c}/M_{\star}\right)^{-1/3}$, where $M_{\star}$ is the original stellar mass. For $M_{\bullet} \lesssim 10^{3} M_{\odot}$, we find that the maximum-attainable kick speed depends on SMBH mass, satisfies $v_{\rm kick, max} \simeq 0.4 \, v_{\rm esc}\left(M_{\bullet}/M_{\star}\right)^{1/6}$, and is reached for core masses that satisfy $M_{\rm c}/M_{\star} \lesssim 1.7\left(M_{\bullet}/M_{\star}\right)^{-1/2}$. This model predicts that massive stars with $M_{\star}\gtrsim few\times 10 M_{\odot}$ could be ejected at speeds $\gtrsim (1-2)\times 10^3$ km s$^{-1}$ if stripped of $\gtrsim 50\%$ of their mass.

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