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

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

Problem and Context
Quasi-periodic eruptions (QPEs) are nuclear transients associated with supermassive black holes (SMBHs), characterized by repeating soft X-ray flares. While several models exist, the star–disc collision scenario—where a star on a tight, eccentric, and inclined orbit repeatedly intersects an accretion disc—is a leading candidate. This model naturally explains the recurrence times, the alternating "long–short" intervals, and the "strong–weak" flare patterns observed in sources like GSN 069. However, connecting the local physical parameters of these collisions to the observable flare properties (luminosity, duration, asymmetry) requires understanding the coupled evolution of shocks, outflow geometry, and radiative transport in an expanding, optically thick medium. Previous studies have explored parts of this problem using 1D, 2D, or non-radiative approaches, but a dedicated systematic 3D radiation-hydrodynamics study linking system parameters to lightcurves has been lacking.

Methodology
The authors performed a suite of three-dimensional local radiation-hydrodynamics simulations using the smoothed-particle hydrodynamics (SPH) code Phantom. The simulations were conducted in a local Cartesian frame centered on the disc midplane, neglecting global orbital dynamics, general relativity, and the self-gravity of the gas, as the collision timescales are much shorter than orbital timescales.

The study utilized a fiducial model based on a main-sequence star ($R_\star \approx R_\odot$) colliding with an optically thick, radiation-pressure-dominated $\alpha$-disc around a $10^6 M_\odot$ SMBH. The star was modeled as a rigid sphere moving at $v_\star = 0.1c$. The authors systematically varied five key parameters relative to the fiducial model, varying one at a time:

1. Stellar velocity ($v_\star$): Ranging from $0.5$ to $1.5$ times the fiducial value.
2. Disc surface density ($\Sigma_d$): Ranging from $1$ to $100$ times the fiducial value, spanning from standard thin discs to dense post-tidal disruption event (TDE) discs.
3. Stellar radius ($R_\star$): Ranging from $0.5$ to $4$ times the fiducial value, effectively varying the ratio of disc scale height to stellar radius ($H/R_\star$).
4. Vertical disc density profile: Modeled as a Gaussian distribution $\rho(z) \propto e^{-z^2/2\sigma^2}$, varying the concentration of mass toward the midplane ($\sigma/R_\odot$).
5. Local collision angle ($i$): The angle between the relative velocity vector and the disc midplane, ranging from $90^\circ$ (perpendicular) to $30^\circ$ (oblique).

Radiation transport was treated using the flux-limited diffusion approximation with electron-scattering opacity. The emerging bolometric luminosity was calculated by integrating diffusion and advective fluxes through a dynamically determined photosphere.

Key Contributions and Results
The study provides empirical scalings for peak luminosity ($L_{\text{peak}}$) and flare duration ($\Delta t$) as functions of the system parameters. The main findings are:

• Collision Dynamics: The star drives a bow shock, creating two outflows (forward and backward) on opposite sides of the disc. The forward outflow is generally more massive and energetic. The qualitative morphology of the bow shock and outflows remains largely unchanged across variations in $v_\star$ and $\Sigma_d$, though the thermodynamic normalization changes.
• Stellar Velocity ($v_\star$): Faster stars produce brighter flares ($L_{\text{peak}} \propto v_\star^2$) due to higher shock heating rates. They also produce longer flares ($\Delta t \propto v_\star^{1.4-2.2}$) because more energy is injected, though the luminosity decays faster due to stronger adiabatic losses.
• Disc Surface Density ($\Sigma_d$): Denser discs increase the mass and momentum of the outflows but have a weak effect on peak luminosity ($L_{\text{peak}} \propto \Sigma_d^{-0.1}$ for forward, $\propto \Sigma_d^{0.2}$ for backward). This is due to a competition between increased shock heating and stronger adiabatic losses in optically thicker ejecta. However, denser discs significantly prolong the flare duration ($\Delta t \propto \Sigma_d^{0.5-0.6}$) due to longer photon diffusion times.
• Stellar Radius ($R_\star$): Larger stars interact with more gas, producing brighter ($L_{\text{peak}} \propto R_\star^{1.5}$ forward, $\propto R_\star^{0.7}$ backward) and longer ($\Delta t \propto R_\star^{1.5}$) flares. Increasing $R_\star$ enhances the momentum asymmetry between forward and backward outflows.
• Vertical Density Profile ($\sigma$): More centrally concentrated discs (smaller $\sigma$) yield brighter and shorter flares. The dense central layers absorb more shock energy, while the low-density outer layers allow radiation to escape more efficiently with reduced adiabatic losses.
• Collision Angle ($i$): More oblique crossings (smaller $i$) reduce the momentum and luminosity asymmetry between the two outflows. The backward outflow becomes significantly brighter ($L_{\text{peak}} \propto (\sin i)^{-0.8}$), and flares become longer ($\Delta t \propto (\sin i)^{-0.9 \text{ to } -1}$) due to the longer interaction time and more symmetric redistribution of shocked gas.

Application to GSN 069
The authors applied their empirical scalings to the well-characterized source GSN 069. By sampling the parameter space to match observed peak luminosities ($\sim 3 \times 10^{42}$ erg s$^{-1}$), durations ($\sim 1$ h), and the strong-weak amplitude ratio, they identified two viable regions:

1. Preferred Solution: A retrograde orbit colliding with a dense, vertically concentrated post-TDE disc ($\Sigma_d \sim 45 \times$ fiducial, $\sigma/R_\odot \sim 0.3$). This model reproduces the observed flare properties well.
2. Alternative Solution: A retrograde orbit with a standard thin disc density but a very grazing collision geometry ($i \sim 6^\circ$). This solution is less accurate and requires extreme parameters.

The authors note that the preference for a dense, vertically concentrated disc is consistent with observational evidence of TDE-like activity in GSN 069.

Significance and Limitations
The paper claims to provide the first systematic 3D radiation-hydrodynamics study linking local collision parameters to QPE lightcurves. The derived empirical scalings serve as a bridge between local simulations and global QPE models, allowing for the prediction of flare properties based on system parameters. The authors emphasize that their findings suggest specific combinations of parameters can reproduce the characteristic amplitudes, durations, and duty cycles of QPEs.

The study modestly acknowledges limitations: the models assume separable one-parameter scalings, neglect disc rotation and shear, treat the star as a rigid sphere, and do not account for viewing angle effects or spectral evolution (e.g., Comptonization). Consequently, the results are presented as physically motivated regions of parameter space rather than unique, definitive best-fitting models. The authors suggest that future work should incorporate realistic stellar structures, disc dynamics, and viewing-angle-dependent luminosities to fully quantify the model's ability to reproduce the diversity of observed QPE sources.