Modeling Tidal Disruptions with Dynamical Tides

TL;DR

The paper tackles efficient modeling of tidal disruption events (TDEs) by coupling dynamical tides to a post-disruption debris phase. It introduces a two-stage framework in which linear perturbation theory computes the star's tidal deformation (via driven quadrupole moments) up to a disruption criterion $E_Q/|U_{bind}|=\gamma$, with $\gamma$ calibrated against hydrodynamical simulations through the critical $β$; once disrupted, debris is treated as free particles to obtain the energy distribution $dM/dE$ and fallback rate $dM/dT$. Applied to MESA middle-age main-sequence stars on parabolic orbits, the model reproduces broad $dM/dE$ and $dM/dT$ shapes, with the energy partition among $g$-, $f$-, and $p$-modes depending on stellar structure. The approach yields rapid computations (≈1 minute) and provides physical insight while highlighting limitations from neglecting self-gravity and relativistic effects, suggesting straightforward extensions to Kerr gravity and nonlinearities.

Abstract

Tidal disruption events (TDEs) occur when stars pass close enough to supermassive black holes to be torn apart by tidal forces. Traditionally, these events are studied with computationally intensive hydrodynamical simulations. In this paper, we present a fast, physically motivated two-stage model for TDEs. In the first stage, we model the star's tidal deformation using linear stellar perturbation theory, treating the star as a collection of driven harmonic oscillators. When the tidal energy exceeds a fraction $γ$ of the star's gravitational binding energy (with $γ\sim \mathcal O(1)$), we transition to the second stage, where we model the disrupted material as free particles. The parameter $γ$ is determined with a one-time calibration to the critical impact parameter obtained in hydrodynamical simulations. This method enables fast computation of the energy distribution ${\rm d} M/{\rm d}E$ and fallback rate ${\rm d} M/{\rm d} T$, while offering physical insight into the disruption process. We apply our model to MESA-generated profiles of middle-age main-sequence stars. Our code is available on GitHub.

Figures (5)

• Figure 1: Illustration of our two-stage model of TDEs. In the first stage, we compute the excitation of the stellar oscillation modes due to the tidal field of the BH. When the tidal energy $E_Q$ becomes a fraction $\gamma$ of the gravitational binding energy $U_{\rm bind}$, we transition into the second stage, where each fluid element is modeled as a free particle orbiting the BH. The value of $\gamma$ can be pre-calibrated by matching to numerical simulations. The figure depicts a tidal disruption of a $1M_\odot$ MESA MAMS star, on an orbit with $\beta\approx1.76\beta_{\rm crit}\approx4.44$ (corresponding to a pericenter distance $R_p=11GM_{\rm BH}$) by a BH with mass $M_{\rm BH}=10^6M_\odot$. All elements in the figure are drawn to scale, including the BH size, the density profile of the star and the randomly sampled free particle trajectories after the disruption.
• Figure 2: Calibration factor $\gamma$, as defined in Eq. \ref{['eq:disruptioncriteria']}, as a function of the stellar mass for MESA stars. The calibration factor is computed as in Eq. \ref{['eqn:gamma-def']}, for an impact parameter equal to $\beta_{\rm crit}$, given in Eq. \ref{['eqn:beta-crit']}.
• Figure 3: Left: time evolution of the components of the quadruple moment $Q_{ij}$ and the tidal energy $E_Q$, defined in Eq. \ref{['eqn:E_Q']}. The pericenter is at $t=0$ and the orbital parameters, as well as the star's and BH's masses, are the same as in Fig. \ref{['fig:money-figure']}. We highlight the point where $E_Q/|U_{\rm bind}|=\gamma$, which corresponds to the disruption time $t_{\rm TDE}$. The first stage of our model ends at $t=t_{\rm TDE}$. The perturbative calculation at $t>t_{\rm TDE}$ is no longer applicable, therefore we draw all lines as dotted. Right: fractional contribution (as defined in Eq. \ref{['eqn:E_Q-decomposed']}) of $g$-, $f$- and $p$-modes to the total tidal energy $E_Q$ at $t=t_{\rm TDE}$, as a function of $M_\star$. The solid lines assume $\beta=1.76\beta_{\rm crit}$, while the dashed line assume $r_p=17GM_{\rm BH}$. It is apparent that the result is largely insensitive to the pericenter distance.
• Figure 4: Specific energy distribution (left panel) and mass fallback rate (right panel) for MESA MAMS stars with masses $0.5M_\odot$, $1M_\odot$, $2M_\odot$, and $8M_\odot$. All cases assume $M_{\rm BH}=10^6M_\odot$ and $\beta=1.76\beta_{\rm crit}$, which equals $2.61$, $4.44$, $5.60$, $4.37$ in the four cases respectively. The left panel is normalized according to Eq. \ref{['eqn:deltaEdeltaT']}, while the right panel displays physical units.
• Figure 5: Equatorial density slices at $t=t_{\rm TDE}$ for different values of $M_\star$ and $\beta$. The top row corresponds to $\beta=1.05\beta_{\rm crit}$, the middle row to $\beta=1.76\beta_{\rm crit}$ (equivalent to the four cases described in Fig. \ref{['fig:dMdE-dMdT']}), the bottom row to $\beta=5\beta_{\rm crit}$. All cases assume $M_{\rm BH}=10^6M_\odot$. In the top row, the disruption time $t_{\rm TDE}$ is after pericenter; in the bottow two rows, it is before pericenter. Note that the orientation of this figure is rotated counterclockwise by $90°$ with respect to Fig. \ref{['fig:money-figure']}. We denote the direction pointing to the BH with an arrow. At pericenter, the arrow points left. Each panel has a width of $4R_\star$ and a height of $6R_\star$, while the logarithmic color scale is normalized in each panel to the maximum star density.