Modeling Stellar Collisions in Galactic Nuclei Using Hydrodynamic Simulations and Machine Learning

TL;DR

This work tackles stellar collisions in galactic nuclei by combining high-resolution SPH simulations of equal-mass $1\,M_\odot$ stars across a broad range of speeds $v_{\infty}$ and closest approaches $r_p$ with physically motivated fitting formulae for capture, mass loss, and kinematic changes. It also benchmarks two ML approaches, kNN and neural networks, on predicting collision outcomes and properties, finding that neural networks often match or surpass the accuracy of the analytic fits while offering scalable modeling as the initial-condition space grows. The 236-run SPH dataset provides insights into tidal dissipation, deflection, and energy deposition, and demonstrates that ML, particularly NN, can efficiently interpolate complex hydrodynamic outcomes beyond simple parametric fits. Together, these methods enhance the modeling of collisions in dense stellar environments and support improved predictions for the dynamical evolution of nuclear star clusters and related transients.

Abstract

Nuclear star clusters represent some of the most extreme collisional environments in the Universe. A typical nuclear star cluster harbors a supermassive black hole at its center, which accelerates stars to high speeds ($\gtrsim 100$-$1000$ km/s) in a region where millions of other stars reside. Direct collisions occur frequently in such high-density environments, where they can shape the stellar populations and drive the evolution of the cluster. We present a suite of a couple hundred high-resolution smoothed-particle hydrodynamics (SPH) simulations of collisions between $1$ M$_\odot$ stars, at impact speeds representative of galactic nuclei. We use our SPH dataset to develop physically-motivated fitting formulae for predicting collision outcomes. While collision-driven mass loss has been examined in detail in the literature, we present a new framework for understanding the effects of "hit-and-run" collisions on a star's trajectory. We demonstrate that the change in stellar velocity follows the tidal-dissipation limit for grazing encounters, while the deflection angle is well-approximated by point-particle dynamics for periapses $\gtrsim0.3$ times the stellar radii. We use our SPH dataset to test two machine learning (ML) algorithms, k-Nearest Neighbors and neural networks, for predicting collision outcomes and properties. We find that the neural network out-performs k-Nearest Neighbors and delivers results on par with and in some cases exceeding the accuracy of our fitting formulae. We conclude that both fitting formulae and ML have merits for modeling collisions in dense stellar environments, however ML may prove more effective as the parameter space of initial conditions expands.

Figures (8)

• Figure 1: Various collisions of two identical $1\,M_{\odot}$ stars. (A) The top row illustrates the case with $r_{\rm p}=0.2 R_\odot$, $v_{\infty}=900$ km s$^{-1}$, resulting in a stellar merger. (B) The second row of snapshots shows $r_{\rm p}=0.2 R_\odot$, $v_{\infty}=3700$ km s$^{-1}$, in which both stars remain, having been stripped by the collisions. (C) The third row is for $r_{\rm p}=0.2 R_\odot$, $v_{\infty}=3700$ km s$^{-1}$: although two separate overdense regions are seen in the final panel, both dissipate as time progresses, leaving no final stars. Color bars show column density on a logarithmic scale in g cm$^{-2}$.
• Figure 2: Parameters of interest from our grid of SPH simulations. Upper left: The general collision outcome for each pair of initial relative speed ($v_{\infty}$) and distance of closest approach between the stars ($r_{p}$) sampled. Blue circles represent cases where the collision culminated in a merger, i.e., resulted a single star. Green crosses indicate where the collision destroyed both stars. Orange dots represent cases where the speed was too high or the impact parameter too large for the stars to capture one another and merge. We show the fit for the boundary between mergers and higher-speed "hit-and-run" type collisions in the black dashed line (Eq. (\ref{['eq:capture_fit_ourunits']}), adapted from Lai+93's equation 4). Lower left: The fractional mass loss from the stars ($f_{ML}$). A fractional mass loss of $1$ indicates a complete disruption of the stars. Fractional mass loss is maximized for small impact parameter and high speed. Right column: The effects of high-speed collisions ("two star" cases) on the velocity vector of the stars. The upper panel shows the fractional decrease in speed ($\Delta v / v_{\infty}$) and the bottom panel, the deflection angle ($\Delta \theta$). The deflection angle is maximized for low-speed collisions, while the fractional change in speed is maximized for smaller $r_{p}$, corresponding to more physical overlap between the stars.
• Figure 3: Fractional mass loss ($f_{ML}$) from the SPH simulations versus $v_\infty$ for different values of $r_p$. We divide the data up into mergers, or "one star", cases and hit-and-run type collisions, or "two star" cases. However, we also include relevant "zero star" cases -- total destruction, $f_{ML} = 1$ -- to show the transition between the other two classes and fully destructive collisions. The orange lines represent our fits from Eq. \ref{['eq:fML_mergers']} (left) and Eq. \ref{['eq:fML_highspeed']} (right), which reproduce the transitions to $f_{ML} = 1$.
• Figure 4: Deflection angle following a hit-and-run (two-star) collision versus $v_\infty$ for all of our values of periapsis with more than $5$ data points, which are the SPH data used to tailor the fitting formula given by Eq. \ref{['eq:theta_fit']}. Dots represent the SPH results, while the grey dashed curve is the deflection angle for a hyperbolic encounter between point particles (Eq. \ref{['eq:theta']}). The orange solid curves represent the predictions of our fitting formula (Eq. \ref{['eq:theta_fit']}), which is generally accurate to within $4$ degrees for all of our data. We note that for $r_p>0.6$ R$_\odot$, the equation for a purely hyperbolic encounter predicts the deflection angle to within $0.1$ radians, or $6$ degrees.
• Figure 5: On the left, we show our $T_{2,\mathrm{fit}}$ and $T_{3,\mathrm{fit}}$ for Eq. \ref{['eq:deltaE_tides_onlyT2T3']}, which are dimensionless functions that quantify a star's internal response to a tidal field. For $r_p>2 R_{\odot}$, our fits begin to converge with $T_2$ and $T_3$ as calculated for tidal capture using GYRE. On the right, we compare our fitting formula for the change in speed with the SPH data for hit-and-run collisions. The orange curves show our fitting formula, while the open circles represent the SPH data.