Resolving the (Debate About) Nozzle Shocks in Tidal Disruption Events

TL;DR

This work resolves the nozzle-shock debate in tidal disruption events by building a computationally efficient pipeline that couples 3D SPH disruption, an affine-model evolution of debris slices, and 1D hydrodynamics near pericenter to resolve the nozzle shock with a realistic EOS. Hydrogen recombination and H$_2$ formation broaden the debris stream prior to the nozzle, increasing dissipation but not fully circularizing in-plane motion; instead, they enhance the likelihood and characteristics of self-intersection on the second orbit. The study shows that nozzle-induced dissipation, coupled with differential apsidal and nodal precession, can significantly influence the geometry and timing of self-intersections, which in turn shapes the observed luminosity and emission mechanisms. The results provide a foundation for more realistic circularization and emission models in TDEs, with clear implications for interpreting current and future observations from LSST and related surveys.

Abstract

When a star passes within the Roche limit of a supermassive black hole (SMBH), it is pulled apart by the BH's tidal field in a tidal disruption event (TDE). The resulting flare is powered by the circularization and accretion of bound stellar debris, which initially returns to the BH on eccentric orbits in a thin debris stream. The returning fluid elements follow inclined orbits that converge near pericenter, resulting in extreme vertical compression to scales $10^{-4}~R_\odot$ and the formation of a nozzle shock. Dissipation at the nozzle shock may affect circularization by altering the properties of the debris stream, but its role is the subject of ongoing debate. We develop an idealized model for the debris stream evolution combining 3D smoothed-particle hydrodynamics simulations, the semi-analytic affine model, and 1D finite-volume hydrodynamic simulations. Because our model is computationally cheap, we can unambiguously resolve the nozzle shock, use a realistic equation of state, and follow the debris stream evolution at many different times. Near peak fallback, Hydrogen recombination and molecular Hydrogen formation broaden the stream by a factor $\sim 5$, enhancing dissipation at the nozzle. However, the dissipation is still insufficient to directly circularize the debris by in-plane pressure gradients. Instead, the thicker stream substantially increases the likelihood that the stream self-intersects on the second orbit, despite relativistic nodal precession. The stream properties at self-intersection are sensitive to dissipation at the nozzle and the timing of focal points where the ballistic trajectories of the debris converge. Our results clarify the nozzle shock's role in circularization in TDEs, providing a foundation for more realistic circularization and emission models.

Figures (17)

• Figure 1: A schematic diagram of the TDE problem, highlighting key steps in the evolution of the stellar debris and our modeling approach: (i) we model the initial $\beta=2$ disruption of a $1~M_\odot$ star by a $10^6~M_\odot$ SMBH using the 3D SPH code phantom; (ii) the star gets tidally disrupted near pericenter; (iii) once the stellar debris forms a stream, we use a snapshot of the phantom simulation to initialize the affine model for the evolution of a stream slice; (iv) the release of latent heat associated with Hydrogen recombination significantly expands the stream slice; (v) as the stream slice approaches apocenter, it enters the ballistic regime where its dynamics are dominated by the tidal field of the SMBH; (vi) once the stream slice falls back to $10~r_{\rm p}$, we use the affine model to initialize a 1D Athena++ simulation; (vii) during the second pericenter passage, the stream slice undergoes a nozzle shock and the bulk of the relativistic apsidal precession occurs; (viii) when the stream returns to $10~r_{\rm p}$, we map the Athena++ simulation back into the affine model; (ix) due to the previous apsidal precession, outgoing and incoming stream slices collide at an angle $\Theta_{\rm si}$, dissipating orbital energy and driving circularization. In the bottom of the diagram, we show our coordinate system with origin $\vb{r}_{\rm c}$ and stream slice orthogonal to $\vu{x}'$. On the left side, we show an edge-on view of the stream slice from the $+\vu{z}$-direction. On the right side, we show a face-on view of the stream slice from the $+\vu{x}'$-direction. The unit vector $\vu{x}'$ forms angles $\alpha$ and $\delta$ relative to $\vu{x}$ and $\vu{r}_{\rm c}$ respectively. $H$, $\Delta$, and $L$ describe the dimensions of the stream slice in the $\vu{x}'$, $\vu{y}'$, and $\vu{z}$ directions respectively.
• Figure 2: Contour plots of mean molecular weight (left panel) and effective adiabatic index $\Gamma_1 \equiv \partial \ln p / \partial \ln \rho |_s$ (right panel) as a function of density and temperature using the Tomida+2013 EOS. In the left panel, we label different regions based on the dominant chemical species: (i), ${\rm H}^+$ and ${\rm He}^{2+}$ (dark blue, $\mu \simeq 0.62$); (ii), ${\rm H}^+$ and ${\rm He}^{+}$ (light blue, $\mu \simeq 0.65$); (iii), ${\rm H}^+$ and ${\rm He}$ (green, $\mu \simeq 0.68$); (iv), ${\rm H}$ and ${\rm He}$ (orange, $\mu \simeq 1.29$); and (v), ${\rm H_2}$ and ${\rm He}$ (red, $\mu \simeq 2.35$). In regions of partial ionization or dissociation, the effective adiabatic index approaches unity because $p\dd V$ work goes into the chemical reaction rather than changing the temperature and pressure.
• Figure 3: The logarithmic density in the $xy$-plane averaged over the $z$-direction and weighted by mass from our simulation of the initial disruption in phantom. In the main panel, we superimpose snapshots at 2.7, 11, 33, and 168 hours. In the inset panels, we show a zoom-in of each snapshot centered on the marginally bound debris. Each inset panel has a side length $2~r_{\rm p}$. We indicate the SMBH using a white dot and an annotation. After the initial disruption, the tidal field elongates the stellar debris into a quasi-cylindrical structure which is well-described by the affine model. We use the snapshot at $2.7~{\rm hr}$ to initialize our affine model calculations.
• Figure 4: The mass fallback rate as a function of time (bottom $x$-axis) and specific orbital energy (top $x$-axis) in units of $M_\odot/{\rm yr}$ (left $y$-axis) and Eddington accretion rate (right $y$-axis) from our phantom simulation. Each dot corresponds to a stream slice we simulate using the affine model and Athena++. The peak fallback rate $160~\dot{M}_{\rm Edd}$ occurs $27~{\rm day}$ after the star passes pericenter, corresponding to orbital energy $-0.0067~G M_\bullet / r_{\rm p}$.
• Figure 5: The vertical ($H$; solid lines) and in-plane ($\Delta$; dashed lines) stream widths as a function of time for the fiducial stream slice on the first orbit. We show results with $\gamma$-law (blue lines), Saha ionization (orange lines), and Tomida+2013 EOSs (green lines). We also show the orbital radius (grey line). Chemical processes augment pressure gradients and promote stream expansion by keeping the debris stream hot as it expands. Including H recombination increases the maximum pre-nozzle stream widths by a factor $\sim 5$. Including other chemical processes adds an additional order-unity factor.