Radiation-hydrodynamics of star-disc collisions: From system parameters to outflows and lightcurves

Abstract: Quasi-periodic eruptions (QPEs) are nuclear transients producing bright, repeating soft X-ray flares superimposed on quiescent emission. A promising interpretation is that they are powered by star-disc collisions, in which a star crosses an accretion disc around a supermassive black hole, drives shocks, and launches dense outflows from which radiation emerges. We present a systematic study of star-disc collisions, linking the physical parameters of the collision to the resulting outflows and emerging bolometric luminosities. We perform three-dimensional local radiation-hydrodynamics simulations, varying the disc surface density and vertical density profile, stellar velocity and radius, and local collision angle. We focus on the regime where the star remains unperturbed by the collision. We find that the variations of stellar velocity and disc surface density leave the bow shock and outflow morphology largely unchanged. However, faster stars produce brighter flares, while denser discs mainly increase the flare duration. Increasing the stellar radius increases the momentum of the forward outflow and produces brighter and longer flares. More centrally concentrated discs yield brighter and shorter flares because radiation escapes more efficiently through outer low-density layers. More oblique crossings reduce the momentum and luminosity asymmetry of two outflows and lengthen the flares. We provide empirical scalings of the peak luminosity and flare duration with the individual system parameters and apply them to GSN 069. The best candidate solutions favour a star with a radius $\sim R_\odot$ on a retrograde orbit, colliding with a dense post-TDE disc with a vertically concentrated density profile. Our findings suggest that specific combinations of system parameters can reproduce characteristic flare amplitudes, durations, duty cycles, and strong-weak flare patterns observed in QPE sources.

• The paper establishes empirical scaling relations linking star–disc collision parameters to observed QPE flare properties.
• It employs 3D SPH simulations with flux-limited diffusion to quantify shock heating, outflow asymmetries, and lightcurve evolution.
• The results support the star–disc collision scenario as a viable engine for QPEs and call for more comprehensive global modeling.

Systematic Radiation-Hydrodynamic Study of Star–Disc Collisions: Parameter Dependence of Outflows and Lightcurves

Introduction

This work presents a comprehensive parameter study of star–disc collisions using three-dimensional local radiation-hydrodynamics simulations. The context is the origin of quasi-periodic eruptions (QPEs), a class of soft X-ray nuclear transients associated with accreting supermassive black holes (SMBHs). QPEs display recurrent, hour-to-day flares with peak luminosities $L_{\rm peak} \sim 10^{41}$–$10^{43}$ erg s$^{-1}$ and alternate long–short, strong–weak patterns. One leading model posits that intermittent star–disc collisions are the energy source for these transients, requiring robust, predictive models to connect local hydrodynamics with observed flare properties. This paper focuses on systematically varying the local disc and stellar parameters to quantify their effects on the hydrodynamics, outflow characteristics, and observable signatures.

Simulation Setup and Methodology

Simulations are performed with the Phantom SPH code, adopting a local Cartesian domain centered on the disc midplane. Fiducial parameters correspond to a $10^6\, M_\odot$ SMBH and a $4$ h QPE period, with a solar-radius star crossing a radiation pressure-dominated, optically thick $\alpha$-disc ($\alpha=0.1$, $\dot{M}/\dot{M}_{\rm Edd}=0.1$). The study independently varies:

• Stellar velocity $v_\star$
• Disc surface density $\Sigma_{\rm d}$
• Stellar radius $10^{43}$0
• Disc vertical density profile
• Local collision angle $10^{43}$1

Radiation transport employs the flux-limited diffusion approximation in LTE, and the star is modelled as a rigid sphere. Effects such as disc self-gravity, rotation, coronal layers, and detailed stellar response are neglected to focus on the local, short-timescale hydrodynamics dominant during a crossing.

Dynamical Evolution of Star–Disc Collisions

The collision process robustly proceeds through four key stages: bow-shock formation, heating and shock propagation, lateral and wake outflow formation, and breakout-driven radiative losses. The shocked gas and associated radiation energy densities are initially trapped, subsequently being liberated as the outflow expands and photon diffusion timescales decrease.

Figure 1: Gas density slices in the $10^{43}$2-plane at $10^{43}$3 illustrate outflow structures for fiducial and parameter-varied simulations at successive stages.

Figure 2: Radiation energy density slices at matching epochs highlight the redistribution of photon energy as a function of system parameters.

Qualitatively, the outflow morphology retains its generic structure under variations of $10^{43}$4 and $10^{43}$5, but becomes increasingly asymmetric for larger $10^{43}$6 and for more perpendicular ($10^{43}$7) crossings. More grazing impact geometries ($10^{43}$8) produce elongated cavities and reduce forward–backward outflow asymmetries.

Shock Heating and Outflow Kinematics

Shock heating, quantified via the energy injection rate $10^{43}$9, strongly scales with $^{-1}$0 and $^{-1}$1, but is largely invariant in dimensionless form across changes in $^{-1}$2 and $^{-1}$3 at fixed geometry.

Figure 3: Shock heating rate $^{-1}$4 over time for varying parameters demonstrates near-invariant temporal profiles for $^{-1}$5 and $^{-1}$6.

Vertical momentum analysis reveals systematic forward–backward outflow asymmetries that are controlled by $^{-1}$7 and $^{-1}$8; specifically, $^{-1}$9 and $10^6\, M_\odot$0, with asymmetry vanishing for $10^6\, M_\odot$1.

Figure 4: Time evolution and parameter-dependence of outflow momentum, emphasizing reduced symmetry for small $10^6\, M_\odot$2 or large $10^6\, M_\odot$3.

Mass outflow rates $10^6\, M_\odot$4 for both forward and backward directions follow analogous trends, with late-time constancy linked to the lateral expansion geometry and buildup of shocked gas.

Figure 5: Mass outflow rates for different parameter regimes, demonstrating consistent trends for $10^6\, M_\odot$5, $10^6\, M_\odot$6, and distinct evolution for $10^6\, M_\odot$7, $10^6\, M_\odot$8, and disc density profile.

Outflow Emission and Synthetic Lightcurves

The simulated bolometric lightcurves are obtained by integrating diffusive and advective fluxes escaping through the photosphere of the outflow, resolved independently for forward and backward directions.

Figure 6: Temporal evolution of emergent luminosity for forward and backward outflows, and impact of parameter variation on peak, duration, and asymmetry.

Key scaling relations are derived and fit empirically:

• $10^6\, M_\odot$9; $4$0
• $4$1 (forward), $4$2
• $4$3 (backward), $4$4
• Disc vertical concentration (smaller $4$5) produces brighter and shorter flares

These trends are captured in two-dimensional heatmap illustrations:

Figure 7: Heatmaps of $4$6 and $4$7 reveal monotonic or weak dependencies for each varied parameter, clarifying the governing physical processes.

Application to Observed QPEs: Parameter Constraints

Utilizing these empirical relations, the parameter space compatible with well-studied QPE sources (GSN 069) is mapped by matching observed $4$8, flare duration, duty cycles, and strong–weak patterns.

Figure 8: Corner plot of parameter space regions reproducing GSN 069 flare properties within a factor of three, showing preference for retrograde, post-TDE disc solutions or, alternatively, very grazing low-density discs.

Candidate solutions divide into two families:

1. High surface density ($4$9 fiducial), vertically concentrated discs ($\alpha$0), mildly sub-solar $\alpha$1, and retrograde collisions;
2. Lower $\alpha$2, more standard disc thickness, but requiring extremely grazing ($\alpha$3) geometries.

The first branch is most compatible with TDE-formed discs, with geometrically thin, high-density structures inferred.

Implications and Theoretical Consequences

The results provide robust quantitative support for the star–disc collision scenario as a plausible engine for QPEs, particularly when disc structure, collision geometry, and stellar properties conspire to produce the range of observed flare behavior. The sensitivity of $\alpha$4 and $\alpha$5 to $\alpha$6, $\alpha$7, $\alpha$8, and $\alpha$9 reinforces prior semianalytic and lower-dimensional findings, while the 3D modeling confirms that overall collision morphology is resilient to most parameter variations but that radiative and kinematic asymmetries are highly parameter-dependent.

The modeling also exposes limitations—most notably, the need for future simulations incorporating global disc rotation, magnetic fields, non-Gaussian disc profiles, evolving stellar structure, and full spectral transport, rather than grey LTE. The neglect of viewing angle, shearing, and post-collision disc evolution restricts direct comparison to detailed X-ray lightcurves and spectra.

Conclusion

This work establishes empirical scaling relations linking local star–disc collision parameters to outflow dynamics and observable flare properties using 3D radiation-hydrodynamics simulations. It demonstrates that QPE phenomenology, including the strong–weak alternation and parameter-dependent durations, can be reproduced within the star–disc collision framework for specific ranges of disc and impact geometry parameters. The findings strongly motivate further global, long-duration modeling and spectroscopic transport studies for definitive theoretical–observational synthesis, and they provide a critical foundation for interpreting forthcoming QPE data, both photometric and spectroscopic, from current and future high-cadence X-ray missions.