Abstract
After a star has been tidally disrupted by a black hole, the debris forms an elongated stream. We start by studying the evolution of this gas before its bound part returns to the original stellar pericenter. While the axial motion is entirely ballistic, the transverse directions of the stream are usually thinner due to the confining effects of self-gravity. This basic picture may also be influenced by additional physical effects such as clump formation, hydrogen recombination, magnetic fields and the interaction with the ambient medium. We then examine the fate of this stream when it comes back to the vicinity of the black hole to form an accretion flow. Despite recent progress, the hydrodynamics of this phase remains uncertain due to computational limitations that have so far prevented us from performing a fully self-consistent simulation. Most of the initial energy dissipation appears to be provided by a self-crossing shock that results from an intersection of the stream with itself. The debris evolution during this collision depends on relativistic apsidal precession, expansion of the stream from pericenter, and nodal precession induced by the black hole spin. Although the combined influence of these effects is not fully understood, current works suggest that this interaction is typically too weak to significantly circularize the trajectories, with its main consequence being an expansion of the shocked gas. Global simulations of disc formation performed for simplified initial conditions find that the debris experiences additional collisions that cause its orbits to become more circular until eventually settling into a thick and extended structure. These works suggest that this process completes faster for more relativistic encounters due to the stronger shocks involved. It is instead significantly delayed if weaker shocks take place, allowing the gas to retain large eccentricities during multiple orbits. Radiation produced as the matter gets heated by circularizing shocks may leave the system through photon diffusion and participate in the emerging luminosity. This current picture of accretion flow formation results from recent theoretical works synthesizing the interplay between different aspects of physics. In comparison, early analytical works correctly identified the essential processes involved in disc formation, but had difficulty developing analytic frameworks that accurately combined non-linear hydrodynamical processes with the underlying relativistic dynamics. However, important aspects still remain to be understood at the time of writing, due to numerical challenges and the complexity of this process.
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Notes
Despite its fast outward motion, the unbound extremity of the stream remains close to the semi-major axis of the most bound gas at \(t\approx t_{\mathrm{min}}\) since its velocity near this location can be approximated by \(v_{\mathrm{unb}} \approx\sqrt{G M_{\mathrm{h}}/ a_{\mathrm {min}}}\) such that the distance reached is \(v_{\mathrm{unb}} t \approx a_{\mathrm{min}}\).
Remarkably, the width evolution becomes homologous with \(H\propto R\) near the black hole due to the increased tidal force that corresponds to an up-turn in the density profile seen at \(R \lesssim200 R_{\mathrm{t}}\) in Fig. 1.
Note that the rate of tidal disruptions in such systems may be increased due to the modification of the gravitational potential by the mass of the disc (Karas and Šubr 2007), as discussed in more details in the Rates Chapter.
If all the debris have the same specific angular momentum as that of the star on its original parabolic orbit, \(l_{\star} = \sqrt{2 G M_{\mathrm{h}}R_{\mathrm{p}}}\), the lowest energy state is a circular orbit at a distance \(R_{\mathrm{circ}} = l^{2}_{\star}/G M_{\mathrm{h}}= 2 R_{\mathrm {p}}\) from the black hole. Although the most bound part of the stream is on an elliptical orbit, this estimate remains valid since its typical eccentricity is \(e_{\mathrm{min}}= 0.98 \approx1\) as shown in equation (5).
Although the most bound part of the stream comes back to pericenter with an already negative energy given by equation (2), it is much larger than that required to reach complete circularization, since \(\Delta\varepsilon/\Delta\varepsilon_{\mathrm{circ}} \approx R_{ \mathrm{p}}/a_{\mathrm{min}}\ll1\), and therefore \(\Delta\varepsilon\) is irrelevant in the computation of the energy loss necessary to reach this configuration.
There is also observational evidence supporting the notion that disc formation occurs differently from what early works assumed. For example, a compact accretion disc cannot reproduced the high level of optical emission detected from many events (Lodato and Price 2010; Miller 2015). Additionally, most TDEs have an integrated energy lower than \(M_{\star}\Delta\varepsilon_{\mathrm{circ}} \approx10^{52}~\text{erg}\), by at least an order of magnitude. This “inverse energy crisis” is discussed in Sect. 3.5.2 along with possible solutions.
As will be described more precisely in Sect. 3.3, it is for example possible that this disc remains globally eccentric, or inclined with respect to the black hole spin.
This relation can also be understood from the fact that the stream has a width similar to the original stellar radius when it comes back near pericenter. Because the compression takes place near the tidal radius, it occurs on the dynamical timescale of the star, which leads to \(v_{\mathrm{z,s}} \approx\sqrt{GM_{\star}/R_{\star}}\).
Another source of interactions for the stream involves the sequential tidal disruption of the two components of a binary star by a black hole (Bonnerot and Rossi 2019), during which the two debris streams can collide with each other due to the difference in their trajectory induced by the previous binary separation.
Note that this treatment is approximate due to the difference in orbital energy between the two gas components involved in the collision. In reality, the part of the stream moving away from the black hole is more bound than the approaching one with a lower apocenter distance.
Although for simplicity this description involves a single width \(H_{2}\) for the outgoing stream component, it is not guaranteed that the gas distribution remains cylindrical following the expansion.
As explained in the Rates Chapter, victim stars sourced by two-body relaxation approach the black hole with a quasi-isotropic distribution of orientations, although alternative scenarios (Wernke and Madigan 2019) are possible. As a result, the stellar angular momentum is in general not aligned with the black hole spin: \(i\neq0\).
Due to the time-dependence of the fallback rate, the inflow rate through the collision point may differ for the two stream components, especially if the intersection happens at large distances with \(R_{\mathrm{int}}\gtrsim a_{\mathrm{min}}\).
The self-crossing shock leads to an increase of the gas temperature to \(T_{\mathrm{g}} \approx m_{\mathrm{p}} v^{2}_{\mathrm{r}} / k_{\mathrm{B}} \approx10^{9}~\mathrm{K}\) using equation (15) and a typical radial velocity \(v_{\mathrm{r}} \approx0.01 \, c\) for the colliding gas. This shocked matter rapidly cools by emitting photons that results in an increase of radiation energy while that of the gas diminishes. At the end of this process, most of the energy is in the form of photons with an equilibrium temperature of \(T_{\mathrm{eq}} \approx(\rho_{\mathrm{s}} v^{2}_{\mathrm {r}}/a)^{1/4} \approx10^{6}~\mathrm{{K}}\), where in the lattermost equality we have adopted a density inside the stream of \(\rho_{\mathrm{s}} \approx10^{-7}~\text{g}\,\text{cm}^{-3}\) as derived in Sect. 2 (see Fig. 1). The ratio of gas to radiation pressure is then \(P_{\mathrm{g}} / P_{\mathrm{r}} \approx(\rho k_{\mathrm{B}} T_{\mathrm{eq}}/m_{\mathrm{p}}) / (a T^{4}_{\mathrm{eq}}/3) \approx10^{-3}\).
For the strong self-crossing shock they consider, the radiation-hydrodynamics simulation performed by Jiang et al. (2016b) finds that about \(16\%\) of the debris gets unbound, which is lower than the value of \(50\%\) obtained from the adiabatic study of Lu and Bonnerot (2020). A possible origin for this discrepancy is that radiative losses reduce the impact of radiation pressure on the gas that therefore gets accelerated to a lower terminal speed, although it may also result from differences in the numerical setup used or the initial properties of the colliding streams.
Note that the self-crossing shock found in most global simulations of disc formation presented in Sect. 3.3 actually differ from the ideal situation of identical stream components, but this difference may, at least partially, originate from the simplified initial conditions used in these studies.
Analytically, this unbinding of mass at pericenter is disfavoured by the fact that the energy dissipated by the nozzle shock is much less than the binding energy of the most bound debris with a ratio \(\Delta\varepsilon_{\mathrm{no}} / \Delta\varepsilon= \beta^{2} (M_{ \mathrm{h}}/ M_{\star})^{-1/3} \approx0.01 \ll1\) using equations (2) and (10). This effect has also not been found in any of the simulations performed at higher resolution, where the stream instead expands around its original trajectory (e.g. Guillochon et al. 2014). Its numerical origin in the low resolution simulations by Ayal et al. (2000) likely results from an inaccurate computation of pressure forces at pericenter. Because this matter moves on near-parabolic trajectories, the resulting variations in velocities may cause the artificial unbinding.
Relativistic apsidal precession can be important for TDEs involving intermediate-mass black holes if \(\beta\gg1\) but this type of deeply-penetrating encounter is uncommon and poses its own computational challenges related to accurately resolving mid-plane compression in the disrupting star and the returning debris streams.
Note that several investigations (Guillochon et al. 2014; Bonnerot et al. 2016b) nevertheless adopt an entirely Keplerian description. This approach is legitimate when the precession angle is small but can also be useful to identify whether some features of the gas evolution occur independently of general relativistic effects.
Because of the bound stellar trajectory, the difference in period between the different parts of the stream is negligible compared to the period of the star. For this reason, when discussing idealized simulations with significantly sub-parabolic stellar eccentricities, we express the circularization timescale in terms of the stellar period rather than that of the most bound debris.
A more extreme situation is that of an ultra-deep encounter where the stellar pericenter is similar to the gravitational radius of the black hole. In this case, the disruption of the star is accompanied by a stretching of the debris into a elongated structure. Due to the very large values of the apsidal precession angle with \(\Delta\phi\gtrsim\pi\), it is possible that this gas collides with itself during the first passage of the star near pericenter. Numerical investigations of this process (Haas et al. 2012; Evans et al. 2015; Darbha et al. 2019) find that this early self-crossing shock results in the fast formation of an accretion flow around the black hole. We note however that such relativistic pericenters are rare for black holes substantially smaller than the Hills mass.
This difference could originate from the non-zero black hole spin, that may significantly increase the energy dissipated during the nozzle shock if nodal precession causes the formation of oblique collisions, as proposed in Sect. 3.1.1.
Numerical studies considering intermediate-mass black holes were carried out earlier as well (Rosswog et al. 2008, 2009; Ramirez-Ruiz and Rosswog 2009; Guillochon et al. 2014). However, they do not run for long enough to capture the completion of disc formation, and we therefore do not present them in detail here.
For the parameters used by Shiokawa et al. (2015), the relativistic apsidal angle is the same as for the disruption of a solar-type star by a black hole of mass \(M_{\mathrm{h}}= 3 \times10^{5}~\mathrm{M}_{\odot}\) for \(\beta=1\).
Note that even though the disc could form as very eccentric, it is unclear whether this configuration can be retained due to the various dissipation processes impacting the gas evolution, such as those associated with accretion and the interaction with the stream of loosely bound debris that continues returning to interact with the disc.
Earlier on, Lodato (2012) also estimated the energy that must be radiated for the gas to completely circularize, proposing that this could be an observable power source without further development.
This situation is for example expected if early interactions take place near pericenter either as a result of strong relativistic apsidal precession or due to gas deflection induced by a fast expansion at the nozzle shock (see Sect. 3.1.2).
We note, however, that over much of the computational domain of Sadowski et al. (2016), the magnetorotational instability is marginally or under-resolved.
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Acknowledgements
We gratefully acknowledge conversations with, and detailed comments from T. Piran, as well as his edits on an earlier version of this manuscript. We are also grateful to E.M. Rossi for insightful discussions during the writing of this chapter. The research of CB was funded by the Gordon and Betty Moore Foundation through Grant GBMF5076. NCS was supported by the NASA Astrophysics Theory Research program (grant NNX17AK43G; PI B. Metzger), and from the Israel Science Foundation (Individual Research Grant 2565/19).
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The Tidal Disruption of Stars by Massive Black Holes
Edited by Peter G. Jonker, Sterl Phinney, Elena Maria Rossi, Sjoert van Velzen, Iair Arcavi and Maurizio Falanga
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Bonnerot, C., Stone, N.C. Formation of an Accretion Flow. Space Sci Rev 217, 16 (2021). https://doi.org/10.1007/s11214-020-00789-1
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DOI: https://doi.org/10.1007/s11214-020-00789-1