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Stellar disruption by a supermassive black hole: is the light curve really proportional to $t^{-5/3}$?
G. Lodato, A. R. King, J. E. Pringle
TL;DR
The paper asks whether stellar tidal-disruption flares generally follow the standard t^-5/3 light curve. Using an analytic model tied to stellar density profiles and numerical simulations of polytropic stars, it finds that this scaling is reached only at late times, especially for centrally concentrated stars.
Problem
The paper revisits the assumption that tidal-disruption flare light curves generally follow L(t) ∝ t^-5/3, since that scaling requires a uniform debris energy distribution.
Method
The authors derive an analytic light-curve model from stellar density structure and test it with SPH simulations of polytropic stellar models.
Results
For centrally concentrated, solar-type stars, the light curve is shallower near peak and reaches t^-5/3 only after luminosity falls by at least two magnitudes.
Takeaways & Limitations
The t^-5/3 profile should be expected primarily at late stages, while early light-curve shape carries information about stellar structure.
Takeaways & Limitations
The study fixes the mass ratio and one set of orbital parameters, although the results are expected to depend on additional parameters such as pericentre ratio and orbital eccentricity.
Abstract
from arXiv · showhide
In this paper we revisit the arguments for the basis of the time evolution of the flares expected to arise when a star is disrupted by a supermassive black hole. We present a simple analytic model relating the lightcurve to the internal density structure of the star. We thus show that the standard lightcurve proportional to $t^{-5/3}$ only holds at late times. Close to the peak luminosity the lightcurve is shallower, deviating more strongly from $t^{-5/3}$ for more centrally concentrated (e.g. solar--type) stars. We test our model numerically by simulating the tidal disruption of several stellar models, described by simple polytropic spheres with index $γ$. The simulations agree with the analytical model given two considerations. First, the stars are somewhat inflated on reaching pericentre because of the effective reduction of gravity in the tidal field of the black hole. This is well described by a homologous expansion by a factor which becomes smaller as the polytropic index becomes larger. Second, for large polytropic indices wings appear in the tails of the energy distribution, indicating that some material is pushed further away from parabolic orbits by shocks in the tidal tails. In all our simulations, the $t^{-5/3}$ lightcurve is achieved only at late stages. In particular we predict that for solar type stars, this happens only after the luminosity has dropped by at least two magnitudes from the peak. We discuss our results in the light of recent observations of flares in otherwise quiescent galaxies and note the dependence of these results on further parameters, such as the star/hole mass ratio and the stellar orbit.
1 INTRODUCTION
The paper revisits the standard L(t) ∝ t^-5/3 prediction for stellar tidal-disruption flares, showing that it depends on the disrupted star’s internal density structure. It combines an analytic density-based model with SPH simulations to examine this dependence.
- 1 INTRODUCTION: The paper develops a simple model relating the disrupted debris energy distribution and resulting light curve to the star’s density profile.The model starts from the stellar density structure and is supplemented by numerical SPH calculations.
- 1 INTRODUCTION: The study places its analysis within prior theoretical and numerical work that commonly fitted t^-5/3 declines to candidate stellar-disruption events.It notes that SPH has been widely used because it can follow systems across broad physical scales.
- 1 INTRODUCTION: More centrally concentrated stars tend to produce shallower light curves than the standard t^-5/3 form.This predicts that the light-curve shape near the flare peak depends strongly on the stellar structure.
- 1 INTRODUCTION: After disruption, the debris energy distribution determines orbital return times, while rapid energy and angular-momentum loss at pericentre makes the return-time distribution effectively the accretion rate.This provides the paper’s physical link from debris dynamics to the observed flare luminosity.
- 1 INTRODUCTION: The standard L(t) ∝ t^-5/3 light curve requires a uniform distribution of debris specific energy, an assumption not established by the original theory.The paper argues that the energy distribution instead depends on stellar properties, particularly internal structure.
2 THE PROCESS OF TIDAL DISRUPTION OF A STAR BY A SMBH
The analytic model maps a star’s internal density profile to its debris-energy distribution and resulting fallback light curve, showing why t^-5/3 is generally a late-time behavior.
- Analytic model: The impulse approximation derives the debris energy distribution from the stellar density profile during the brief pericentre interaction.It is expected to be qualitatively reliable for parabolic encounters, although encounter details may differ because the interaction is not instantaneous.
- Analytic model: The model uses dimensionless energy, radius, mass, time, density, penetration factor, and mass ratio to describe the disruption.For q = 10^6, β = 1 corresponds to Rp = 100R⋆.
- Predicted light curves: More centrally concentrated stars produce shallower flares because they contain less mass at large energies and more mass returning at low energies.Their light curves begin with a relatively longer delay and approach t^-5/3 only after the energy distribution flattens at low energies.
- Predicted light curves: Polytropic models with γ = 5/3, 1.4, and 4/3 predict different energy distributions and accretion-rate evolutions.The right panel tracks the evolving power-law index against the reference t^-5/3 profile.
- Predicted light curves: After 1 year, the power-law index is n ≈ −1.5 for γ = 5/3 but remains n ≈ −0.8 for the more solar-like γ = 4/3 model.The γ = 5/3 result is reasonably close to −5/3, whereas the γ = 4/3 result is substantially shallower.
3 NUMERICAL SIMULATIONS
The simulations model parabolic stellar encounters with SPH polytropes spanning several structural indices, then compare their evolving debris with the analytic predictions.
- Simulation design: The analytic treatment is approximate because it assumes the stellar structure remains unchanged until pericentre and treats the encounter as instantaneous.The simulations are used to test the resulting energy distribution and light curve.
- Simulation design: The simulations use a non-relativistic SPH code for parabolic encounters with β = 1 and q = 10^6, placing the pericentre at 100 stellar radii.The setup follows the dimensionless variables introduced for the analytic model.
- Simulation design: Four polytropic stars are simulated: γ = 1.4, 1.5, 5/3, and 1.8, initialized by constructing and relaxing their target density profiles.The particle placement uses close sphere packing followed by differential radial stretching.
- Encounter evolution: Figure 4 compares projected stellar densities at pericentre and after the encounter at roughly twice the pericentre distance for all four γ values.The black hole lies outside the image at the coordinate-system origin.
- Stellar models: The γ = 1.4 model is the most centrally concentrated and γ = 1.8 the least, while larger γ models are less compressible.The analytic density profiles and relaxed SPH realizations are compared in Figure 3.
4 RESULTS
The simulations show that stellar inflation near pericentre and shocks in tidal tails are needed to explain the disrupted debris’ energy distributions. Increasing γ reduces the required homologous expansion but increases shocked mass and energy-distribution wings, while the return-time profile approaches t^-5/3 more rapidly.
- 4.1 The γ = 5/3 case: At pericentre, the γ = 5/3 star is already distorted and expanded because the black hole’s tidal field reduces its effective gravity.The energy distribution broadens as the star approaches the black hole and reaches approximately the width predicted by the analytical model.
- 4.1 The γ = 5/3 case: At roughly 20 pericentre distances, the simulated γ = 5/3 energy distribution approximately matches the analytical model near its peak after rescaling the model by ≈1.6.The comparison uses a distribution averaged over 10 time units and normalized profiles.
- 4.2 Varying the polytropic index: As γ increases, the stellar density becomes more uniform, the high-density region more extended, and the tidal-tail edge more sharply defined.The elongated core is more aligned with the black-hole direction for smaller γ.
- 4.2 Varying the polytropic index: For γ = 1.4, 1.5, 5/3 and 1.8, the simulations match analytically predicted distributions using homologous expansion factors ξ = 2.5, 2.1, 1.63 and 1.6, respectively.The expansion factor decreases as γ increases, reflecting the reduced response of higher-γ stars.
- 4.2 Varying the polytropic index: Higher γ produces more shocks in the tidal tails and progressively stronger wings in the energy distribution, with shocks concentrated near the tail edges.For γ = 5/3, shocks occur where |q| > 1; the shocked-mass curves show increasing shocked mass with γ and a common peak near t ≈ 4.
- 4.2 Varying the polytropic index: Flatter energy distributions at larger γ produce steeper return-time distributions that approach the t^-5/3 profile more rapidly.The corresponding accretion rates are shown for γ = 1.4, 1.5, 5/3 and 1.8.
5 DISCUSSION AND CONCLUSIONS
The paper finds that the canonical t^-5/3 flare profile generally emerges only at late times, with the early lightcurve shaped by stellar density structure and modified by specific simulation effects. For solar-type stars, the profile is reached only after the luminosity falls by at least two magnitudes.
- 5 DISCUSSION AND CONCLUSIONS: Observed flares with only a small luminosity decline can appear shallower than t^-5/3, whereas longer-baseline observations can approach the standard fall-off.Reported optical fits include n ≈ -1.1 and n ≈ -0.82, consistent with the predicted early-time behavior.
- 5 DISCUSSION AND CONCLUSIONS: The simulations agree with the analytical model after accounting for stellar inflation near pericentre and energy-distribution wings produced by shocks in tidal tails.The inflation is represented by homologous expansion, while the wings occur for large polytropic indices.
- 5 DISCUSSION AND CONCLUSIONS: The t^-5/3 profile generally appears only at late times, rather than throughout the flare evolution.The result holds across the simulations and reflects the non-flat energy distribution of disrupted debris.
- 5 DISCUSSION AND CONCLUSIONS: At early times, more centrally concentrated stars produce shallower lightcurves than the standard t^-5/3 profile.The lightcurve is especially sensitive to stellar structure near peak luminosity.
- 5 DISCUSSION AND CONCLUSIONS: For solar-type stars, the t^-5/3 profile is reached only after the luminosity has dropped by at least two magnitudes from peak.Stars with relatively flat density profiles reach the profile earlier.
- 5 DISCUSSION AND CONCLUSIONS: The study uses a simple setup with fixed star–black-hole mass ratio and one set of orbital parameters, so those parameters may further affect the results.The authors specifically identify the tidal-radius-to-pericentre ratio and orbital eccentricity as additional dependencies.