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

TL;DR

This work extends the Maximum Gravity (MG) model to partial tidal disruption events by linking the tidal encounter strength, quantified by the penetration factor $β$, to the mass stripped $\Delta M$, the peak fallback time $t_{\rm peak}$, and the peak fallback rate $\dot{M}_{\rm peak}$ for a wide range of stellar masses and ages. The authors derive analytic expressions that tie $t_{\rm peak}$ and $\dot{M}_{\rm peak}$ to a critical radius inside the star where the tidal field matches the star’s maximum self-gravity, and they validate these predictions against 1276 hydrodynamical SPH simulations of MS stars disrupted by a $10^6\,M_\odot$ black hole. The comparisons show robust agreement for $t_{\rm peak}$ (within tens of percent) and reasonable agreement for $\dot{M}_{\rm peak}$ (within a factor of $\sim2$–$3$), with larger deviations for high-$β$ grazing encounters and for certain evolved stars where self-gravity and core dynamics play a larger role. The model thus offers a practical analytical prescription for TDE lightcurves and luminosity functions, highlighting how $t_{\rm peak}$ encodes SMBH and stellar properties and how long-duration transients can arise from grazing disruptions of high-mass stars around massive SMBHs.

Abstract

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.

Figures (14)

• Figure 1: Schematic showing a star of mass $M_\star$, radius $R_\star$, being partially tidally stripped by an SMBH. The tidal field of the SMBH exceeds the self-gravitational field of the mass contained within some radius $R\leqslant R_\star$ at a pericenter distance $r_{\rm p}=r_{\rm t,c}(R)$, such that the mass exterior to this radius, $\Delta M$, is tidally stripped. When the tidal field of the SMBH exceeds the maximum self gravitational field within the star, it leads to complete disruption of the star, i.e., $\Delta M = M_\star$.
• Figure 2: The numerically obtained fallback rates for a $1.0M_\odot$ ZAMS star (shown by the discrete data points) along with the fits to the functional form given by Equation \ref{['eq:pade-approx']} (solid curves).
• Figure 3: Fallback rates for a $0.2M_\odot$ ZAMS star (top left), a $0.5 M_\odot$ ZAMS star (top right), a $0.8 M_\odot$ ZAMS star (bottom left), and a $0.8M_\odot$ star at $14$Gyr (botom right), with the penetration factor $\beta$ ranging from $0.6$ to $\beta_{\rm c}$ (critical $\beta$ for complete disruption). The fallback rates for the partial disruptions scale as $\propto t^{-9/4}$ at late times. The peak of each fallback rate is indicated with a circle.
• Figure 4: The peak timescale $t_{\rm peak}$ and peak fallback rate $\dot{M}_{\rm peak}$ for a $0.2M_\odot$ ZAMS star (top left), a $0.5 M_\odot$ ZAMS star (top right), a $0.8 M_\odot$ ZAMS star (bottom left), and a $0.8M_\odot$ star at $14$Gyr (botom right), with the penetration factor $\beta$ ranging from $0.6$ to $\beta_{\rm c}$ (the MG model prediction for the critical penetration factor), as indicated in the legend. With an increase in $\beta$, the peak timescale $t_{\rm peak}$ shifts to earlier times, and the peak fallback rate $\dot{M}_{\rm peak}$ increases. The error bars show the duration for which $\dot{M}>95\%\dot{M}_{\rm peak}$.
• Figure 5: The relative error in $t_{\rm peak}$, defined as $\sigma_{\rm t_{\rm peak}} \equiv (t_{\rm peak,hydro}-t_{\rm peak,MG})/t_{\rm peak,hydro}$ (where $t_{\rm peak,hydro}$ is the peak timescale measured from the phantom simulation and $t_{\rm peak,MG}$ is the model prediction), as a function of $\beta$ for a $0.2M_\odot$ ZAMS star (top left), a $0.5 M_\odot$ ZAMS star (top right), a $0.8 M_\odot$ ZAMS star (bottom left), and a $0.8M_\odot$ star at $14$Gyr (botom right).