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

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

Problem Statement
Nuclear star clusters (NSCs), such as the one surrounding the supermassive black hole (SMBH) in the Milky Way, represent extreme collisional environments. The SMBH accelerates stars to high velocities ($\gtrsim 100\text{--}1000$ km/s), leading to frequent close encounters and direct collisions. While the rate of these collisions is significant enough to influence stellar populations and cluster evolution over gigayear timescales, modeling them in large-scale cluster simulations is computationally prohibitive. Running full hydrodynamic simulations for every potential collision over 10 Gyr is infeasible. Consequently, existing models rely on physically motivated fitting formulae derived from Smoothed-Particle Hydrodynamics (SPH) datasets. However, previous studies have largely focused on mass loss and merger outcomes, leaving the effects of high-speed collisions on stellar trajectories (velocity changes and deflection angles) less well-constrained. Furthermore, as the parameter space of initial conditions (mass ratios, stellar ages, structures) expands, analytic fitting formulae may become increasingly cumbersome to formulate and implement.

Methodology
The authors present a suite of 236 high-resolution SPH simulations using the StarSmasher code to model hyperbolic collisions between two equal-mass ($1 M_\odot$) main-sequence stars. The stellar structure is derived from a 2.5 Gyr MESA profile, and each star is resolved with $N = 10^5$ SPH particles. The simulations explore a two-dimensional parameter space defined by:

1. Relative speed at infinity ($v_\infty$): Ranging from 100 to 5100 km/s, representative of velocities within the sphere of influence of an SMBH.
2. Periapsis distance ($r_p$): Ranging from 0 (head-on) to 1.8 $R_\odot$ (grazing), with four additional simulations at $r_p = 2.5 R_\odot$ to test tidal capture expectations.

The study employs a three-pronged approach:

1. Qualitative Analysis: Categorizing outcomes into three regimes: stellar mergers ("one star"), hit-and-run collisions ("two stars"), and complete destruction ("zero stars").
2. Analytic Fitting: Developing physically motivated fitting formulae for capture boundaries, fractional mass loss ($f_{ML}$), deflection angles ($\Delta\theta$), and fractional speed changes ($\Delta v/v_\infty$).
3. Machine Learning (ML) Benchmarking: Testing two ML algorithms—k-Nearest Neighbors (kNN) and Neural Networks (NN)—to predict collision outcomes (classification) and continuous variables (regression). The dataset was split into training (60%), validation (20%), and testing (20%) sets.

Key Contributions and Results

• Collision Regimes and Capture Boundaries:
  The simulations confirm three distinct outcomes based on $v_\infty$ and $r_p$. The authors validate the capture radius fitting formula from Lai et al. (1993) for $1 M_\odot$ stars by scaling the dimensionless constant $A_{cap}$. They find that all physical collisions resulting in a bound system ultimately lead to a merger, even if a temporary binary forms initially.

• Fractional Mass Loss ($f_{ML}$):
  The authors derive separate fitting formulae for mergers (Eq. 2) and hit-and-run collisions (Eq. 3). The merger formula reproduces the parabolic collision limit ($v_\infty \ll v_{esc}$) from Lombardi et al. (2002). Both formulae are accurate to within $\sim 4\%$ across the dataset. The transition to total destruction ($f_{ML} = 1$) is captured by the hit-and-run formula for $r_p > 0$.

• Orbital Effects (Deflection and Speed):

  • Deflection Angle: The authors demonstrate that for periapses $r_p \gtrsim 0.3 \times (R_1 + R_2)$, the deflection angle is well-approximated by point-particle dynamics (Eq. 4), despite the physical collision. A new fitting formula (Eq. 5) bridges the gap between the point-particle limit and the physical collision regime, achieving accuracy within 4 degrees for all data. Negative deflection angles (recoil) are observed in high-speed, near-disruption encounters.
  • Speed Change: The fractional change in speed is modeled as a sum of tidal dissipation and shock/collision contributions (Eq. 10). The tidal term dominates for grazing encounters, while the collision term dominates for head-on impacts. The formula is generally accurate to within a few percent, with a single exception near the capture boundary.

• Machine Learning Performance:

  • Classification: The Neural Network (NN) achieved 100% balanced accuracy on the test set for predicting collision outcomes (merger, hit-and-run, destruction), significantly outperforming the kNN model (86.7% balanced accuracy). The kNN model struggled near decision boundaries.
  • Regression: For predicting $f_{ML}$, $\Delta\theta$, and $\Delta v/v_\infty$, the NN consistently yielded lower Root Mean Squared Errors (RMSE) and Maximum Absolute Errors (MAE) than kNN.
  • Comparison with Analytic Fits: The NN performance was on par with, and in some cases (specifically for deflection angles at low $v_\infty$), exceeded the accuracy of the custom-tailored analytic fitting formulae. Both ML models and fitting formulae exhibited the largest errors near the boundaries between collision outcome classes.

Significance and Claims
The paper claims that while physically motivated fitting formulae remain valuable for providing physical insight and handling limiting cases, Machine Learning—specifically Neural Networks—offers a robust alternative for modeling stellar collisions in dense environments. The authors argue that as the parameter space expands to include diverse stellar masses, ages, and structures, the complexity of formulating analytic fits may render them less practical. In contrast, ML models can effectively learn the underlying dynamics from high-resolution simulation data, delivering accuracy comparable to or better than tailored analytic expressions. This suggests that ML is a promising tool for incorporating realistic collision physics into large-scale N-body simulations of galactic nuclei, where computational efficiency and accuracy are paramount.