Abstract
Tidal disruption events occur rarely in any individual galaxy. Over the last decade, however, time-domain surveys have begun to accumulate statistical samples of these flares. What dynamical processes are responsible for feeding stars to supermassive black holes? At what rate are stars tidally disrupted in realistic galactic nuclei? What may we learn about supermassive black holes and broader astrophysical questions by estimating tidal disruption event rates from observational samples of flares? These are the questions we aim to address in this Chapter, which summarizes current theoretical knowledge about rates of stellar tidal disruption, and compares theoretical predictions to the current state of observations.
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Notes
As we see later in Sect. 3.4, factor (iii) is generally unimportant for determining TDE rates.
In reality, the exact criterion for tidal disruption of a main sequence star is that \(R_{\mathrm{p}} < R_{\mathrm{t}}/\beta _{\mathrm{crit}}\), where \(\beta _{\mathrm{crit}}\approx 0.95\mbox{--}1.85\) is a dimensionless constant dependent on the central concentration of the star, and can be measured precisely with numerical hydrodynamics simulations (Guillochon and Ramirez-Ruiz 2013; Mainetti et al. 2017; see also the Disruption Chapter).
If a broad spectrum of stellar masses exist, it is generally the heaviest species that relaxes to the \(n(r) \propto r^{-7/4}\) profile, while lighter species will achieve shallower, \(n(r) \propto r^{-3/2}\) distributions (Bahcall and Wolf 1977). However, strong mass-segregation can give rise to steeper distributions, as in Alexander and Hopman (2009), Preto and Amaro-Seoane (2010), Aharon and Perets (2016).
For a less rigorous—but in some respects more intuitive—approach operating entirely in coordinate space, see the work of Syer and Ulmer (1999), which obtains qualitatively similar results to those presented here.
With this definition, the flux is positive if the stars diffuse towards the loss cone, as they usually do. Note that the sign convention is the opposite in some studies.
In general, closed-form expressions for the variables in this section do not exist. However, for the special case of a singular isothermal sphere density profile, with \(n(r) \propto r^{-2}\), Wang and Merritt (2004) provide analytic expressions for \(q(E)\), \(\mathcal{F}(E)\), and \(\dot{N}\).
An interesting caveat to this discussion concerns extremely steep stellar cusps, i.e. those with density profiles falling off faster than \(n(r) \propto r^{-9/4}\). Such density profiles are not self-consistent, because they predict that, as \(E\to -\infty \), \(\mathcal{F}_{\mathrm{empty}}\to \infty \) (Syer and Ulmer 1999). Density profiles of this steepness are rarely seen in nature, with the possible exception of post-starburst galaxies, which we discuss later Sect. 5.2.1.
But see also Arca-Sedda and Capuzzo-Dolcetta (2017) for counterexample simulations, where star cluster infall leads to a tangential bias. Ultimately, the final \(b(E)\) profile is likely sensitive to the orbital properties of infalling star clusters.
The drain region is often called the “loss wedge” in the axisymmetric case—e.g. Magorrian and Tremaine (1999).
The right-hand side of Eq. (23) should be multiplied by \(\beta _{\mathrm{crit}}^{3}\) to account for stellar structure.
This calculation assumes the full loss-cone limit, in computing the relativistic correction factor, although the per-galaxy TDE rates shown in Fig. 3 are based on loss-cone calculations using contributions from both the full and empty regions.
Although we note that a power-law tail of high-\(\beta \) TDEs will occur, even when \(q \ll 1\), due to the effects of strong scattering (Weissbein and Sari 2017).
We note that since the counterparts of HVSs can be captured around the MBH, the distribution of such stars could also reflect the processes leading to, and the rates of, TDEs (Perets and Gualandris 2010).
The assumptions of spherical symmetry and quasi-isotropy are relaxed in Magorrian and Tremaine (1999), but for brevity we focus primarily on the simplest case.
If the stellar density profile \(n(r)\) is too shallow, the DF \(f(E)\) obtained from Eq. (31) will have negative values, which is an unphysical outcome. In the limit of a Kepler potential, the shallowest self-consistently isotropic power-law density profile is \(n(r) \propto r^{-1/2}\); shallower density profiles require some degree of tangential anisotropy to remain positive-definite in \(f(E,\mathcal{R})\).
This assumption is unlikely to be generically true. The clearest caveat here concerns the strong preference among observed TDE flares to reside in rare E+A and, more generally, post-starburst galaxies (see discussion in Sect. 5.2.1). Because E+A and post-starburst galaxies make up very small fractions of the low-\(z\) galaxy population (\(\sim 0.2\%\) and \(2.3\%\), respectively; French et al. 2016), this implies the presence of unusual stellar dynamics enhancing TDE rates in these galaxies by at least an order of magnitude. However, the fraction of all galaxies that have a post-starburst nature increases steeply as a function of redshift. For example, going from \(z\approx 0.5\) to \(z\approx 2\) increases the fraction of post-starburst galaxies by a factor of \(\approx 5\) (Wild et al. 2016), suggesting that at high \(z\), the decline in \(\dot{n}\) due to the decreasing volume density of SMBHs may be overwhelmed by the growing abundance of this rare galaxy type.
Note that in the results of van Velzen and Farrar (2014), statistical uncertainties are denoted in superscript/subscript error ranges, while systematic uncertainties (associated with the uncertain choice of model light curve used to back out true rates from flux-limited samples) are denoted in prefactor error ranges.
The situation becomes more complicated if galactic nuclei are significantly triaxial, in which case larger galaxies may have larger individual TDE rates \(\dot{N}\). From an observational point of view, the prevalence of nuclear triaxiality remains uncertain.
This seems to be indicated by resolved color gradients in nearby E+A galaxies, see e.g. Pracy et al. (2012).
Such large anisotropies may be vulnerable to the radial orbit instability (Polyachenko and Shukhman 1981).
A notable exception to this trend is the overdensity scenario; ultrasteep density cusps will produce almost all their TDEs in the empty loss cone regime.
Although it is worth noting that some observational rate inferences, such as Esquej et al. (2008), would not be in tension with conservative theoretical rate estimates.
In this calculation, we have assumed that half of the disrupted star accretes onto the SMBH. For nearly-parabolic stellar orbits, precisely half of the disrupted star is dynamically bound to the SMBH, although we caution that hydrodynamic shocks and radiation pressure in super-Eddington accretion may unbind a portion of this dynamically bound half (see the Formation of the Accretion Flow Chapter and the Accretion Disc Chapter for more discussion of these uncertainties).
In principle, if TDE rates are dominated by mechanisms (such as nuclear triaxiality, or eccentric stellar discs) that preferentially supply stars from a specific orbital orientation, TDEs may act to spin up SMBHs.
The marginally bound radius \(R_{\mathrm{mb}}\) is the minimum pericenter that avoids capture by the event horizon and is of order \(R_{\mathrm{g}}\) (Bardeen et al. 1972).
The online “Open TDE Catalog” https://tde.space/ is a useful resource for the observationally-curious reader.
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Acknowledgements
N.C.S. received financial support from the NASA Astrophysics Theory Research Program (Grant NNX17AK43G; PI B. Metzger). M. K. acknowledges support from NSF Grant No. PHY-1607031 and NASA Award No. 80NSSC18K0639. E.M.R. acknowledges support from NWO TOP grant Module 2, project number 614.001.401. P.A.S. acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, the COST Action GWverse CA16104, and the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303).
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The Tidal Disruption of Stars by Massive Black Holes
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Stone, N.C., Vasiliev, E., Kesden, M. et al. Rates of Stellar Tidal Disruption. Space Sci Rev 216, 35 (2020). https://doi.org/10.1007/s11214-020-00651-4
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DOI: https://doi.org/10.1007/s11214-020-00651-4