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Julian Krolik

Publications and source records attributed to Julian Krolik.

At least 19 recordsLinked to original sources
arXiv2026

The Origin of the Magnetic Flux Driving TDE Jets

Tidal Disruption Events (TDEs) occur when a star approaches a black hole closely enough to be torn apart by tidal forces, after which the stellar debris begins to orbit the black hole. Over the past decade, hundreds of TDEs have been identified, with many more expected from upcoming surveys. A small subset of these events launch transient, highly luminous relativistic jets (10^47-10^48 erg/s isotropic-equivalent). These are generally interpreted in terms of the Blandford--Znajek mechanism, implying the presence of substantial magnetic flux near the black hole horizon. The question arises: What is the origin of this flux? In this paper, we investigate three candidate sources: stellar magnetic fields, magnetic flux from large radii around the black hole (the Lasso mechanism), and magnetic flux stored in the inner portion of a pre-existing accretion disk. We find that: observed stellar magnetic fields are insufficient to power these jets; the Lasso mechanism requires shallow radial magnetic field profiles and a mechanism to trap the magnetic flux brought by the debris in the vicinity of the black hole; pre-existing magnetic flux near the black hole in an accretion disk is the most plausible source.

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arXiv2026

Unexpectedly Weak General Relativistic Effects in Strongly Relativistic Tidal Disruption Events

Tidal disruption events (TDEs) occur when stars are destroyed by supermassive black holes and are among the brightest nuclear transients. It has been thought that strong relativistic effects rapidly dissipate orbital energy and produce prompt disk formation when the stellar pericenter is smaller than $\sim 10$ gravitational radii. Using a general relativistic hydrodynamic simulation of a strongly relativistic TDE involving a Sun-like star and a $10^{6}\,M_{\odot}$ non-spinning black hole, we find instead that the overall evolution is similar to weakly relativistic TDEs: the debris remains highly eccentric, with most of the returned mass residing near the orbital apocenter ($\sim 250\times$ the initial pericenter distance), and shocks, rather than accretion, power the event. The simulation starts from the initial stellar approach and follows the debris evolution up to $35$\,days after the peak mass-return time ($\simeq$ $23$\,days). Although early shocks driven by strong relativistic apsidal precession and pericenter nozzle compression dissipate orbital energy efficiently, they last only about a week ($\sim 0.3$ of the peak mass-return time). Stream self-interactions increase the incoming stream's angular momentum, thereby expanding its pericenter distance, weakening precession and shocks, and reducing dissipation. These results suggest that circularization in TDEs may proceed slowly regardless of the strength of apsidal precession, with the flow remaining highly eccentric and extended during the peak optical/UV luminosity.

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arXiv2026

The TDE Population from First-Principles Models of Stellar Disruption and Debris Dynamics

We present a physically-grounded population model for optical tidal disruption events (TDEs) that combines first-principles hydrodynamic simulations of stellar disruption with statistical inference of the underlying stellar and black hole populations. The model's prediction of peak luminosity is based directly on recent global simulations that follow the disruption self-consistently and contains no tunable parameters related to the emission physics. We construct the predicted joint distribution of peak luminosity and black hole mass, including both full and partial disruptions, and compare it to a sample of observed TDEs using Bayesian inference and Markov chain Monte Carlo sampling. We find that the model reproduces the distribution in the ($M_{BH},L_{peak}$) plane for the bulk of the observed TDE population with good statistical consistency. The data strongly favor an old stellar population, with a sharp suppression of stars above $M_* \simeq 1.5 - 2 M_\odot$. They also indicate that, at fixed stellar mass, the volumetric TDE rate is nearly independent of black hole mass. Partial disruptions contribute a substantial fraction ($\sim 30\%$) of detected events in flux-limited samples and are essential for reproducing the observed distribution. The inferred population properties are robust to different approximations to the stellar mass-radius relation, although the event rate at high luminosity is sensitive to the form of this relation for massive stars. We predict a large population of difficult to detect low luminosity TDEs, implying that the true volumetric TDE rate may exceed that inferred from present samples by up to an order of magnitude.

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arXiv2025

The merger of spinning, accreting supermassive black hole binaries

Because they are likely to accrete substantial amounts of interstellar gas, merging supermassive binary black holes are expected to be strong multimessenger sources, radiating gravitational waves, photons from thermal gas, and photons from relativistic electrons energized by relativistic jets. Here we report on a numerical simulation that covers the late inspiral, merger, and initial postmerger phase of such a system where both black holes have the same mass and spin, and both spin axes are parallel to the orbital angular momentum. The simulation incorporates both 3D general relativistic magnetohydrodynamics and numerical relativity. The thermal photon power during the late inspiral, merger, and immediate postmerger phases is drawn from strong shocks rather than dissipation of turbulence inside a smoothly structured accretion disk as typically found around accreting single black holes. We find that the thermal photon and jet Poynting flux outputs are closely related in time, and we posit a mechanism that enforces this relation. The power radiated in both photons and jets diminishes gradually as merger is approached, but jumps sharply at merger to a noisy plateau. Such a distinct lightcurve should aid efforts to identify supermassive black hole mergers, with or without accompanying gravitational wave detections.

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arXiv2025

Relativistic gas accretion onto supermassive black Hole binaries from inspiral through merger

Accreting supermassive black hole binaries are powerful multimessenger sources emitting both gravitational and EM radiation. Understanding the accretion dynamics of these systems and predicting their distinctive EM signals is crucial to informing and guiding upcoming efforts aimed at detecting gravitational waves produced by these binaries. To this end, accurate numerical modeling is required to describe both the spacetime and the magnetized gas around the black holes. In this paper, we present two key advances in this field of research. First, we have developed a novel 3D GRMHD framework that combines multiple numerical codes to simulate the inspiral and merger of supermassive black hole binaries starting from realistic initial data and running all the way through merger. Throughout the evolution, we adopt a simple but functional prescription to account for gas cooling through photon emission. Next, we have applied our new computational method to follow the time evolution of a circular, equal-mass, nonspinning black hole binary for ~200 orbits, starting from a separation of 20r_g and reaching the postmerger evolutionary stage of the system. We have shown how mass continues to flow toward the binary even after the binary "decouples" from its surrounding disk, but the accretion rate onto the black holes diminishes. We have identified how the minidisks orbiting each black hole are slowly drained and eventually dissolve as the binary compresses. We confirm previous findings that the system's luminosity decreases by a factor of a few during inspiral; however, we observe an abrupt increase by ~50% in this quantity at the time of merger, likely accompanied by an equally abrupt change in spectrum. Finally, we have demonstrated that during the inspiral, fluid ram pressure regulates the fraction of the magnetic flux transported to the binary that attaches to the black holes' horizons.

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arXiv2025

Follow the Mass -- A Concordance Picture of Tidal Disruption Events

Three recent global simulations of tidal disruption events (TDEs) have produced, using different numerical techniques and parameters, very similar pictures of their dynamics. In typical TDEs, after the star is disrupted by a supermassive black hole, the bound portion of the stellar debris follows highly eccentric trajectories, reaching apocenters of several thousand gravitational radii. Only a very small fraction is captured upon returning to the vicinity of the supermassive black hole. Nearly all the debris returns to the apocenter, where shocks produce a thick irregular cloud on this radial scale and power the optical/UV flare. These simulation results imply that over a few years, the thick cloud settles into an accretion flow responsible for the long term emission. Despite not being designed to match observations, and without adjusting any parameters, the dynamical picture given by the three simulations aligns well with observations of typical events, correctly predicting the flares' typical total radiated energy, luminosity, temperature and emission line width. On the basis of these predictions, we provide an updated method (TDEmass) to infer the stellar and black hole masses from a flare's peak luminosity and temperature. This picture also correctly predicts that the luminosity observed years after the flare should be nearly constant. In addition, we show that in a magnitude-limited survey, if the intrinsic rate of TDEs is independent of black hole mass, the detected events will preferentially have black hole masses $\sim 10^{6.3 \pm 0.3} M_\odot$ and stellar masses $\sim 1 M_\odot$, with the width of the mass distribution for disrupted stars sensitive to the stellar mass function in the host galaxy's center.

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arXiv2024

The Dynamics of Debris Disk Creation in Neutron Star Mergers

The detection of GW170817/AT2017gfo inaugurated an era of multimessenger astrophysics, in which gravitational wave and multiwavelength photon observations complement one another to provide unique insight on astrophysical systems. A broad theoretical consensus exists in which the photon phenomenology of neutron star mergers largely rests upon the evolution of the small amount of matter left on bound orbits around the black hole or massive neutron star remaining after the merger. Because this accretion disk is far from inflow equilibrium, its subsequent evolution depends very strongly on its initial state, yet very little is known about how this state is determined. Using both snapshot and tracer particle data from a numerical relativity/MHD simulation of an equal-mass neutron star merger that collapses to a black hole, we show how gravitational forces arising in a non-axisymmetric, dynamical spacetime supplement hydrodynamical effects in shaping the initial structure of the bound debris disk. The work done by hydrodynamical forces is ${\sim}10$ times greater than that due to time-dependent gravity. Although gravitational torques prior to remnant relaxation are an order of magnitude larger than hydrodynamical torques, their intrinsic sign symmetry leads to strong cancellation; as a result, hydrodynamical and gravitational torques have comparable effect. We also show that the debris disk's initial specific angular momentum distribution is sharply peaked at roughly the specific angular momentum of the merged neutron star's outer layers, a few $r_g c$, and identify the regulating mechanism.

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arXiv2023

Shocks Power Tidal Disruption Events

Accretion of debris seems to be the natural mechanism to power the radiation emitted during a tidal disruption event (TDE), in which a supermassive black hole tears apart a star. However, this requires the prompt formation of a compact accretion disk. Here, using a fully relativistic global simulation for the long-term evolution of debris in a TDE with realistic initial conditions, we show that at most a tiny fraction of the bound mass enters such a disk on the timescale of observed flares. To "circularize" most of the bound mass entails an increase in the binding energy of that mass by a factor $\sim 30$; we find at most an order unity change. Our simulation suggests it would take a time scale comparable to a few tens of the characteristic mass fallback time to dissipate enough energy for "circularization". Instead, the bound debris forms an extended eccentric accretion flow with eccentricity $\simeq 0.4-0.5$ by $\sim 2$ fallback times. Although the energy dissipated in shocks in this large-scale flow is much smaller than the "circularization" energy, it matches the observed radiated energy very well. Nonetheless, the impact of shocks is not strong enough to unbind initially bound debris into an outflow.

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arXiv2023

Extremely Relativistic Tidal Disruption Events

Extreme tidal disruption events (eTDEs), which occur when a star passes very close to a supermassive black hole, may provide a way to observe a long-sought general relativistic effect: orbits that wind several times around a black hole and then leave. Through general relativistic hydrodynamics simulations, we show that such eTDEs are easily distinguished from most tidal disruptions, in which stars come close, but not so close, to the black hole. Following the stellar orbit, the debris in eTDEs is initially distributed in a crescent that quickly turns into tight spirals, from which some mass later falls back toward the black hole, while the remainder is ejected. Internal shocks within the infalling debris power the observed emission. The resulting light-curve rises rapidly to roughly the Eddington luminosity, maintains this level for between a few weeks and a year (depending on both the stellar mass and the black hole mass), and then drops. Most of its power is in thermal X-rays at a temperature $\sim (1-2)\times 10^{6}$ K ($\sim 100-200$ eV). The debris evolution and observational features of eTDEs are qualitatively different from ordinary TDEs, making eTDEs a new type of TDE. Although eTDEs are relatively rare for lower-mass black holes, most tidal disruptions around higher-mass black holes are extreme. Their detection offers a view of an exotic relativistic phenomenon previously inaccessible.

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arXiv2021

The Impact of Shocks on the Vertical Structure of Eccentric Disks

Accretion disks whose matter follows eccentric orbits can arise in multiple astrophysical situations. Unlike circular orbit disks, the vertical gravity in eccentric disks varies around the orbit. In this paper, we investigate some of the dynamical effects of this varying gravity on the vertical structure using $1D$ hydrodynamics simulations of individual gas columns assumed to be mutually non-interacting. We find that time-dependent gravitational pumping generically creates shocks near pericenter; the energy dissipated in the shocks is taken from the orbital energy. Because the kinetic energy per unit mass in vertical motion near pericenter can be large compared to the net orbital energy, the shocked gas can be heated to nearly the virial temperature, and some of it becomes unbound. These shocks affect larger fractions of the disk mass for larger eccentricity and/or disk aspect ratio. If the orbit can be maintained despite orbital energy loss, diverse initial structures evolve in only a few orbits so that they follow a limit-cycle characterized by a low-entropy midplane and a much higher entropy outer layer. In favorable cases (such as the tidal disruption of stars by supermassive black holes), these effects could be a potentially important energy dissipation and mass loss mechanism.

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arXiv2020

Tidal disruptions of main sequence stars -- IV. Relativistic effects and dependence on black hole mass

Using a suite of fully relativistic hydrodynamic simulations applied to main-sequence stars with realistic internal density profiles, we examine full and partial tidal disruptions across a wide range of black hole mass ($10^{5}\leq M_{\rm BH}/\mathrm{M}_{\odot}\leq 5\times 10^{7}$) and stellar mass ($0.3 \leq M_{\star} /\mathrm{M}_{\odot}\leq 3$) as larger $M_{\rm BH}$ leads to stronger relativistic effects. For fixed $M_{\star}$, as $M_{\rm BH}$ increases, the ratio of the maximum pericenter distance yielding full disruptions ($\mathcal{R}_{\rm t}$) to its Newtonian prediction rises rapidly, becoming triple the Newtonian value for $M_{\rm BH} = 5\times10^{7}~{\rm M}_\odot$, while the ratio of the energy width of the stellar debris for full disruptions to the Newtonian prediction decreases steeply, resulting in a factor of two correction at $M_{\rm BH} = 5 \times 10^7~{\rm M}_\odot$. We find that for partial disruptions, the fractional remnant mass for a given ratio of the pericenter to $\mathcal{R}_{\rm t}$ is higher for larger $M_{\rm BH}$. These results have several implications. As $M_{\rm BH}$ increases above $\sim 10^7~{\rm M}_\odot$, the cross section for complete disruptions is suppressed by competition with direct capture. However, the cross section ratio for partial to complete disruptions depends only weakly on $M_{\rm BH}$. The relativistic correction to the debris energy width delays the time of peak mass-return rate and diminishes the magnitude of the peak return rate. For $M_{\rm BH} \gtrsim 10^7~{\rm M}_\odot$, the $M_{\rm BH}$-dependence of the full disruption cross section and the peak mass-return rate and time is influenced more by relativistic effects than by Newtonian dynamics.

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arXiv2020

Tidal disruptions of main sequence stars -- II. Simulation methodology and stellar mass dependence of the character of full tidal disruptions

This is the second in a series of papers presenting the results of fully general relativistic simulations of stellar tidal disruptions in which the stars' initial states are realistic main-sequence models. In the first paper (Paper I), we gave an overview of this program and discussed the principal observational implications of our work. Here we describe our calculational method and provide details about the outcomes of full disruptions, focusing on the stellar mass dependence of the outcomes for a black hole of mass $10^{6}\rm{M}_{\odot}$. We consider eight different stellar masses, from $0.15~{\rm M}_\odot$ to $10~{\rm M}_\odot$. We find that, relative to the traditional order-of-magnitude estimate $r_{\rm t}$, the physical tidal radius of low-mass stars ($M_{\star} \lesssim 0.7~ {\rm M}_\odot$) is larger by tens of percent, while for high-mass stars ($M_{\star} \gtrsim1~ {\rm M}_\odot$) it is smaller by a factor 2--2.5. The traditional estimate of the range of energies found in the debris is $\approx 1.4\times$ too large for low-mass stars, but is a factor $\sim 2$ too small for high-mass stars; in addition, the energy distribution for high-mass stars has significant wings. For all stars undergoing tidal encounters, we find that mass-loss continues for many stellar vibration times because the black hole's tidal gravity competes with the instantaneous stellar gravity at the star's surface until the star has reached a distance from the black hole $\sim O(10)r_{\rm t}$.

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arXiv2020

Measuring stellar and black hole masses of tidal disruption events

The flare produced when a star is tidally disrupted by a supermassive black hole holds potential as a diagnostic of both the black hole mass and the star mass. We propose a new method to realize this potential based upon a physical model of optical/UV light production in which shocks near the apocenters of debris orbits dissipate orbital energy, which is then radiated from that region. Measurement of the optical/UV luminosity and color temperature at the peak of the flare leads directly to the two masses. The black hole mass depends mostly on the temperature observed at peak luminosity, while the mass of the disrupted star depends mostly on the peak luminosity. We introduce {\sc TDEmass}, a method to infer the black hole and stellar masses given these two input quantities. Using {\sc TDEmass}, we find, for 21 well-measured events, black hole masses between $5\times 10^5$ and $10^7 M_\odot$ and disrupted stars with initial masses between 0.6 and $13M_\odot$. An open-source {\sc python}-based tool for {\sc TDEmass} is available at https://github.com/taehoryu/TDEmass.git.

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arXiv2020

Tidal Disruptions of Main Sequence Stars -- V. The Varieties of Disruptions

Tidal disruption events (TDEs), events in which a star passes very close to a supermassive black hole, are generally imagined as leading either to the star's complete disruption or to its passage directly into the black hole. In the former case it is widely believed that in all cases the bound portion of the debris quickly "circularizes" due to relativistic apsidal precession, i.e., forms a compact accretion disk, and emits a flare of standardized lightcurve and spectrum. We show here that TDEs are more diverse and can be grouped into several distinct categories on the basis of stellar pericenter distance $r_p$; we calculate the relative frequency of these categories. In particular, because rapid circularization requires $r_p \lesssim 10r_g$ ($r_g \equiv GM_{\rm BH}/c^2$), it can happen in only a minority of total disruptions, $\lesssim 1/4$ when the black hole has mass $M_{\rm BH} = 10^6 M_\odot$. For larger pericenter distances, $10 < r_p/r_g < 27$ (for $M_{\rm BH}=10^6M_\odot$), main sequence stars are completely disrupted, but the bound debris orbits are highly eccentric and possess semimajor axes $\sim 100\times$ the scale of the expected compact disk. Partial disruptions with fractional mass-loss $\gtrsim 10\%$ should occur with a rate similar to that of total disruptions; for fractional mass-loss $\gtrsim 50\%$, the rate is $\approx 1/3$ as large. Partial disruptions -- which must precede total disruptions when the stars' angular momenta evolve in the "empty loss-cone" regime -- change the orbital energy by factors $\gtrsim O(1)$. Remnants of partial disruptions are in general far from thermal equilibrium. Depending on the orbital energy of the remnant and conditions within the stellar cluster surrounding the SMBH, it may return after hundreds or thousands of years and be fully disrupted, or it may rejoin the stellar cluster.

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arXiv2020

Tidal disruptions of main sequence stars -- III. Stellar mass dependence of the character of partial disruptions

In this paper, the third in this series, we continue our study of tidal disruption events of main-sequence stars by a non-spinning $10^{6}~\rm{M}_\odot$ supermassive black hole. Here we focus on the stellar mass dependence of the outcomes of partial disruptions. As the encounter becomes weaker, the debris mass is increasingly concentrated near the outer edges of the energy distribution. As a result, the mass fallback rate can deviate substantially from a $t^{-5/3}$ power-law, becoming more like a single peak with a tail declining as $t^{-p}$ with $p\simeq2-5$. Surviving remnants are spun-up in the prograde direction and are hotter than main sequence stars of the same mass. Their specific orbital energy is $\simeq10^{-3}\times$ that of the debris, but of either sign with respect to the black hole potential, while their specific angular momentum is close to that of the original star. Even for strong encounters, remnants have speeds at infinity relative to the black hole potential $\lesssim 300$ km s$^{-1}$, so they are unable to travel far out into the galactic bulge. The remnants most deeply bound to the black hole go through a second tidal disruption event upon their first return to pericenter; if they have not thermally relaxed, they will be completely disrupted.

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arXiv2020

Tidal Disruptions of Main Sequence Stars -- I. Observable Quantities and their Dependence on Stellar and Black Hole Mass

This paper introduces a series of papers presenting a quantitative theory for the tidal disruption of main sequence stars by supermassive black holes. Using fully general relativistic hydrodynamics simulations and MESA-model initial conditions, we explore the pericenter-dependence of tidal disruption properties for eight stellar masses ($0.15 \leq M_*/M_\odot \leq 10$) and six black hole masses ($10^5 \leq M_{BH}/M_\odot \leq 5 \times 10^7$). We present here the results most relevant to observations. The effects of internal stellar structure and relativity decouple for both the disruption cross section and the characteristic energy width of the debris. Moreover, the full disruption cross section is almost independent of $M_*$ for $M_*/M_\odot \lesssim 3$. Independent of $M_*$, relativistic effects increase the critical pericenter distance for full disruptions by up to a factor $\sim 3$ relative to the Newtonian prediction. The probability of a direct capture is also independent of $M_*$; at $M_{BH}/M_\odot \simeq 5 \times 10^6$ this probability is equal to that of a complete disruption. The width of the debris energy distribution $ΔE$ can differ from the standard estimate by factors from 0.35 to 2, depending on $M_*$ and $M_{BH}$, implying a corresponding change in the characteristic mass-return timescale. The "frozen-in approximation" is inconsistent with $ΔE$, and mass-loss continues over a long span of time. We provide analytic forms, suitable for use in both event rate estimates and parameter inference, to describe all these trends. For partial disruptions, we find a nearly-universal relation between the star's angular momentum and the fraction of $M_*$ remaining. Within the "empty loss-cone" regime, partial disruptions must precede full disruptions. These partial disruptions can drastically affect the rate and appearance of subsequent total disruptions.

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arXiv2020

Tidal radii of main sequence stars -- I. Physical tidal radius, semi-analytic model and their implications

A star is tidally disrupted by a supermassive black hole when their separation is shorter than the "tidal radius". This quantity is often estimated on an order-of-magnitude basis without reference to the star's internal structure. Using MESA models for main sequence stars and fully general relativistic dynamics, we find the physical tidal radius for complete disruption $\cal{R}_t$ for a $10^6M_\odot$ black hole (BH). We find that across a factor $\sim20$ in stellar mass $M_*$, i.e., $0.15M_{\odot}\leq M_*\leq3M_\odot$, $\cal{R}_t\sim27\times$(BH's gravitational radius). When comparing $\cal{R}_t$ with the commonly used order-of-magnitude estimate $r_t$, we find that $\cal{R}_t\sim1.05-1.45r_t$ for $0.15M_\odot\leq M_*\leq0.5M_\odot$, but between $0.5 M_\odot$ and $1 M_\odot$, $\cal{R}_t$ drops to $\sim 0.45r_t$, and it remains at this value up to $10 M_\odot$. The near-constancy of $\cal{R}_t$ implies a weaker dependence of the full disruption rate on $M_*$ than when predicted with $r_t$. The characteristic energy width of the debris $ΔE$ ranges from $\sim1.2Δ\cal{E}$ for low-mass stars to $\sim 0.35Δ\cal{E}$ for higher-mass stars, where $Δ\cal{E}=GM_{\rm BH}R_*/\cal{R}_t^{2}$. We present analytic fits for the $M_*$ dependence of $\cal{R}_t$ and $ΔE$; these fits lead to analytic expressions for the time of peak mass fallback rate and the maximal mass fallback rate. Our results also bear on the fraction of events leading to fast or slow circularization, as well as on the character of the tidal event occurring when the remnant of a partial disruption returns to the black hole. Using a semi-analytic model, we show that $\cal{R}_t$ is primarily determined by the star's central density rather than its mean density. For high-mass stars, the full disruption rate is roughly 1/4 the partial disruption rate, while this ratio is close to unity for low-mass stars.

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arXiv2020

Tidal radii of main sequence stars -- II. Simulation methodology and the character of full tidal disruptions

This is the second in a series of papers presenting the results of fully general relativistic simulations of stellar tidal disruptions in which the stars' initial states are realistic main-sequence models. We consider eight different stellar masses, from $0.15~{\rm M}_\odot$ to $10~{\rm M}_\odot$. In the first paper (Ryu et al. 2019a), we gave an overview of this program and discussed the principal observational implications of our work. Here we describe our calculational method and provide details about the outcomes of full disruptions. We find that, relative to the traditional order-of-magnitude estimate $r_{\rm t}$, the physical tidal radius of low-mass stars is larger by tens of percent, while for high-mass stars ($M_{\star} \gtrsim1~ {\rm M}_\odot$) it is smaller by a factor $2-2.5$. The traditional estimate of the range of energies found in the debris is approximately accurate for low-mass stars, but is a factor $\sim 2$ too small for high-mass stars; in addition, the energy distribution for high-mass stars has significant wings. For all stars undergoing tidal encounters, we find that mass-loss continues for a long time because the ${\it instantaneous}$ tidal radius, the distance out to which the black hole's tidal gravity competes with the instantaneous stellar gravity at the star's surface, stays comparable to the distance to the black hole until the star has reached $O(10)~r_{\rm t}$. These findings indicate significant failings in the popular "frozen-in" approximation.

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