MICRO-TIDAL DISRUPTION EVENTS BY STELLAR COMPACT OBJECTS AND THE PRODUCTION OF ULTRA-LONG GRBs

In the following we consider two cases: disruption with a low mass ratio with a high mass ratio. In the former, the mass of the disrupted star/planet is assumed to be negligible compared with the CO mass, and the CO can be assumed to be stationary at the center of mass of the system. Such a case corresponds to the tidal disruptions of planets or low-mass stars by stellar COs. However, when the mass of the disrupted object is relatively large (a tenth to a few tenths of the CO mass), the CO can no longer be assumed to be stationary or to reside at the center of mass of the system. The first scenario, with a low mass ratio, has been discussed extensively in the literature in the context of widely studied TDEs by MBHs; although different in scale the results should also apply to the μTDE case of low mass ratio; we briefly review the results obtained for that case. The scenario with a high mass ratio has been little studied and we therefore run hydrodynamical simulations of such tidal disruptions to characterize some of their basic properties.

Tidal disruptions with low mass ratio. In the first stage following the disruption, the bound fraction of the debris, f_fall, falls back and returns to the pericenter. The first bound material returns after a time

$$\begin{eqnarray}{t}_{\mathrm{min}} & = & \displaystyle \frac{2\pi {R}_{p}^{3}}{{({{GM}}_{\bullet })}^{1/2}{(2{R}_{\star })}^{3/2}}\\ & \approx & 3.52\times {10}^{5}{\left(\displaystyle \frac{{R}_{p}}{2.15{R}_{\odot }}\right)}^{3}{\left(\displaystyle \frac{{R}_{\star }}{0.1{R}_{\odot }}\right)}^{-3/2}{\left(\displaystyle \frac{{M}_{\bullet }}{10{M}_{\odot }}\right)}^{-1/2}\;{\rm{s}},\end{eqnarray} \tag{ 1 }$$

where the normalization was done for a Jupiter-like planet with radius ${R}_{\star }=0.1{R}_{\odot }$ disrupted by a BH with a typical mass of $10\;{M}_{\odot }$ at a closest approach of ${R}_{p}={R}_{t}=2.15\;{R}_{\odot }$. Assuming a flat distribution of debris energies, the return rate of the bound material to pericenter at late times (Rees 1988; Phinney 1989; Ulmer 1999) is

$$\begin{eqnarray}&&\dot{M}\sim \displaystyle \frac{1}{3}\displaystyle \frac{{M}_{\star }}{{t}_{\mathrm{min}}}{\left(\displaystyle \frac{t}{{t}_{\mathrm{min}}}\right)}^{-5/3},\end{eqnarray} \tag{ 2 }$$

where the peak return rate occurs at about $t\sim 1.5{t}_{\mathrm{min}}$ (Evans & Kochanek 1989) and half of the mass of the fallback debris returns by about $t\sim 6{t}_{\mathrm{min}}$. Indeed, such behavior is seen in our hydrodynamical simulations of a BH tidally disrupting a Jupiter-mass planet (see Figure 1). Note, however, that simulations by Ayal et al. (2000) show that a large fraction of the returned debris later becomes unbound. They find the total accreted mass of debris to be four times smaller than found earlier, with an approximately constant accretion rate. Nevertheless, this does not make a significant change to the overall derived timescale. We also mention the work by Coughlin & Begelman (2014), who take a somewhat different approach and suggest the formation of an extended jet-producing envelope.

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Partial tidal disruptions with high mass ratio. The ${t}^{-5/3}$ infall rate back to the CO discussed above corresponds to the case of complete or nearly complete tidal disruption of a low-mass object (compared with the disrupting CO). If the object is completely disrupted then there is nothing special about the debris field near gas that is marginally bound to the black hole, and the classic ${t}^{-5/3}$ behavior follows. The results change somewhat when the object is only partially disrupted, as can happen more frequently in cases of high mass ratio or when the object has a dense core. The surviving remnant affects the distribution of gas marginally bound to the black hole, and a fallback rate steeper than ${t}^{-5/3}$ results; see, e.g., MacLeod et al. (2012) for simulations of partial disruptions of giant stars by a supermassive black hole, and discussion of these issues by Guillochon & Ramirez-Ruiz (2013) and Hayasaki et al. (2013).

To study partial tidal disruptions in cases of high mass ratio, we have carried a set of hydrodynamical simulations using the StarSmasher code. StarSmasher is a smoothed particle hydrodynamics (SPH) code that evolved from StarCrash (Faber et al. 2010), which itself has its origins in the SPH code of Rasio (1991). The primary enhancement of StarSmasher over its predecessors is that it incorporates equations of motion derived from a variational principle to ensure accurate evolution of energy and entropy (Gaburov et al. 2010). In addition, gravitational forces between particles are calculated by direct summation on NVIDIA graphics cards.

We consider the tidal disruption of a solar-mass star or a Jupiter-like planet, modeled with 395,000 particles, by a 10 ${M}_{\odot }$ BH point particle. A wider range of parameter space for tidal disruptions will be discussed in more detail elsewhere. The solar-mass star is evolved with the TWIN stellar evolution code to an age of 4590 Myr to give a radius $\sim 1{R}_{\odot };$ the hydrodynamical simulations employ an equation of state (EOS) consisting of ideal-gas and radiation-pressure components. The Jupiter-like planet is modeled as an n = 1 polytrope of mass ${10}^{-3}{M}_{\odot }$ and radius $0.1{R}_{\odot }$, with a ${\rm{\Gamma }}=2$ EOS. We consider both parabolic and hyperbolic encounters (but we discuss only the former cases here, more relevant for the expected relative velocities), and study cases of different closest approach ${R}_{p}=0.6,0.9$, and $1\;{R}_{t}$ (that is, ${R}_{p}=1.29\;{R}_{\odot }$, $1.94\;{R}_{\odot }$, and 2.15 ${R}_{\odot }$). Runs with larger closest approach (${R}_{p}\sim 4\;{R}_{\odot }$) resulted in only a negligible fraction of the stellar (or planetary) mass falling back onto the BH. None of our runs implements radiative cooling. As can be seen in Table 1, our simulations results in a partial disruption, leaving behind the denser core of the star, which is not disrupted. We did not follow the long-term stellar evolution of such disruption remnants, which could be of interest in itself.

In order to estimate the fallback rate, we use the so-called freezing model as in MacLeod et al. (2012):

$$\begin{eqnarray*}&&\displaystyle \frac{{dM}}{{dt}}=\displaystyle \frac{{dM}}{{dE}}\displaystyle \frac{{dE}}{{dt}}=\displaystyle \frac{1}{3}{(2\pi {{GM}}_{\bullet })}^{2/3}\displaystyle \frac{{dM}}{{dE}}{t}^{-5/3}.\end{eqnarray*}$$

Here Kepler's third law has been used to relate the specific binding energy $E={{GM}}_{\bullet }/(2a)$, where a is the semimajor axis of a parcel of debris, to the time t for that material to return to periapsis. This approximation neglects the influence of the remnant star, if it exists, on the fallback time. Note that the canonical ${t}^{-5/3}$ behavior of dM/dt follows from having a flat distribution of dM/dE versus E. More generally, dM/dE depends on the strength of the tidal interaction as well as the density profile of objects being disrupted. Objects that are completely or nearly disrupted tend to give relatively flat dM/dE at small E and therefore ${dM}/{dt}\sim {t}^{-5/3}$ at late times. In contrast, objects that are only partially disrupted yield a fallback rate dM/dt that drops off more quickly with time.

The actual distribution of dM/dE from our simulations is determined by sorting the mass of unbound particles into bins of E. For each unbound SPH particle, we calculate E as the sum of its specific kinetic energy (relative to the black hole) and its specific gravitational potential energy due to all mass in the system. For our Jupiter-like planetary disruptions (with ${R}_{\ast }\ll {R}_{p}\sim {R}_{t}$), unbound gas forms a long stream with one end that reaches back to the black hole only once the planet (or what remains of it) has retreated well outside the tidal radius. Since the tidal disruption plays out fully before the return of the gas to the black hole, it is sufficient to determine the distribution of dM/dE from a single snapshot of the simulation, taken shortly before the first stripped material returns.

In the disruption of our Sun-like stars (with ${R}_{\ast }\sim {R}_{p}\sim {R}_{t}$), stripped material from the star encircles the black hole and collides with other infalling material before the tidal disruption has fully completed, and it is consequently not possible to use a single snapshot to determine the fallback mass rate over all times. We therefore use a sequence of snapshots when binning particles by their specific energy in these cases. The first snapshot is taken about 10³ s after the periapse passage: all particles that can be identified as being bound to the black hole are included in this binning, although these particles may have their bin locations adjusted while we step through about an additional five future snapshots spaced over about five dynamical timescales. As new particle are identified as being bound to the black hole, they are included in the binning. If in a later snapshot a previously binned particle has not yet reached 20% of its expected fallback time t, then it is instead rebinned according to its binding energy E calculated from that later snapshot. This procedure avoids issues associated with particles that have encircled the black hole colliding with infalling material.

As can be seen in Figure 1 the return rate of the most bound material happens on timescales comparable with scaling for the case of low mass ratio (Equation (1)), but the profile of the return rate is steeper, approximately $\sim {t}^{-7/3}$, with only a weak dependence on the pericenter separation ${R}_{p}$ for the cases considered. We also note that in these simulations the fraction of material bound to the black hole after the disruption is of the order of 0.04 for a closest approach at the tidal radius (${R}_{p}=2.15{R}_{\odot }$), whereas impacts at deeper penetrations result in higher fractions (0.06 and 0.2 for ${R}_{p}=1.94\;{R}_{\odot }$ and ${R}_{p}=1.29\;{R}_{\odot }$, respectively); for the following we adopt ${f}_{\mathrm{fall}}=0.1$ as a typical value.

After circularization the bound debris from the tidal disruption forms a torus or disk around the CO (for example, see Ruffert 1992) at a radius of ${r}_{{\rm{c}}}\simeq 2{R}_{p}$. The accretion timescale for this disk, assuming it is thick (with the ratio of disk scale height to its radius, h, of the order of 1), is of the order of the viscous time

$$\begin{eqnarray}{t}_{{\rm{acc}}} & \approx & \displaystyle \frac{{t}_{{\rm{kep}}}(2{R}_{p})}{\alpha 2\pi {h}^{2}}\\ & \approx & 4.5\times {10}^{4}\;{h}^{-2}{\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1}{\left(\displaystyle \frac{{R}_{p}}{2.15{R}_{\odot }}\right)}^{3/2}{\left(\displaystyle \frac{{M}_{\bullet }}{10{M}_{\odot }}\right)}^{-1/2}\;{\rm{s}}\end{eqnarray} \tag{ 3 }$$

where ${t}_{{\rm{kep}}}(r)$ is the Keplerian orbital period at orbital radius r and α is the viscosity constant. The value of α is unknown, and we normalize it with the commonly used value $\alpha =0.1$.

We therefore expect the formation of an accretion disk during the few thousand seconds after the tidal disruption (Equation (1)), for the assumed parameters. This might be observable in the timescale for the rise of the flare. Later on the debris material will be accreted on timescales of a few 10⁴–10⁵ s onto the CO (Equation (3)), whereas additional fallback material is expected to accrete continuously onto the CO at a rate that decays as a power law (Equation (2) or somewhat steeper for disruption with a high mass ratio). The exact timescale for the formation of the accretion disk is therefore less important as the evolution is dominated by the longer accretion timescales. We do note, however, that for the longer-term evolution as well as in the case of the disruption of larger, evolved star (e.g., red giant), occurring at a much larger radius, the accretion rate will be dominated by the fallback time rather than the accretion timescale. It was suggested that such a scenario may explain the origin of some X-ray binaries (J. Steiner & J. Guillochon 2015, private communication; see also Hills 1976a and Krolik 1984 for long-lived X-ray sources formed by tidal capture).

The encounter rates between stars have been studied by many (see, e.g., Di Stefano & Rappaport 1992 for very similar calculations). Here we give the rates of encounters leading to a TDE, i.e., where the closest approach of a star/planet to a given CO is smaller that the tidal radius. Such encounter rates are dominated by the contribution from the densest stellar systems, such as globular clusters (GCs) and galactic nuclei.

Consider a CO of mass M_co in the core of a cluster containing ${N}_{\ast }$ single stars (for a discussion of binaries, see below). A tidal disruption will take place if the distance of closest approach between this CO and a star is less than the tidal radius ${R}_{t}$. If each ordinary star had a mass ${M}_{\ast }$, the tidal disruption rate from the one CO would be

$$\begin{eqnarray*}&&{\dot{p}}_{\mathrm{co}}({R}_{t},\sigma )=2\pi G({M}_{\mathrm{co}}+{M}_{\ast }){N}_{\ast }{R}_{t}{\sigma }^{-1}{V}_{{\rm{c}}}^{-1}\end{eqnarray*}$$

where σ is the relative velocity dispersion and the core volume ${V}_{{\rm{c}}}$ can be written in terms of the core radius ${R}_{{\rm{c}}}$ of the GC as ${V}_{{\rm{c}}}=4\pi {R}_{{\rm{c}}}^{3}/3$. Here we are assuming that the collision cross section is dominated by gravitational focusing. The total disruption rate in a single GC is then

$$\begin{eqnarray*}&&{{\rm{\Gamma }}}_{\mathrm{co}}\approx {\dot{p}}_{\mathrm{co}}{N}_{\mathrm{co}}\approx {\dot{p}}_{\mathrm{co}}{n}_{0}{f}_{\mathrm{co}}{V}_{{\rm{c}}},\end{eqnarray*}$$

where N_co is the total number of COs in the GC core, n₀ is the number density of stars in the core, and f_co is the fraction of COs in the stellar population (see Di Stefano & Rappaport 1992).

Typical GC cores have densities of ${n}_{0}\simeq {10}^{5}\ {{\rm{pc}}}^{-3}$, and a typical core radius ${R}_{{\rm{c}}}=1\ {\rm{pc}}$ (Pryor & Meylan 1993; Gnedin & Ostriker 1997). Typical velocity dispersions in GC cores are roughly $\sigma \simeq 20\ \mathrm{km}\ {{\rm{s}}}^{-1}$ (see, e.g., Di Stefano & Rappaport 1992). The fraction of COs in the stellar population is taken to be ${f}_{{\rm{NS}}}=0.017$ and ${f}_{{\rm{BH}}}=0.012$ (taking a Salpeter mass function between $0.6\quad \mathrm{and}\quad 120\;{M}_{\odot }$; NSs are assumed to originate from stars of mass between 8 and 15 ${M}_{\odot }$ and BHs are assumed to originate from stars more massive than $15\;{M}_{\odot }$); however, due to NS natal kicks and binary heating of BHs in a cluster, only a fraction of these COs are retained in the cluster. We assume a retention fraction of 0.05, following Pfahl et al. (2002), and we take typical masses of a neutron star and black hole to be $1.4\;{M}_{\odot }$ and $10\;{M}_{\odot }$, respectively. For COs of 1–10 M_⊙, the tidal radius falls approximately in the range ${R}_{\odot }\lesssim {r}_{t}\lesssim 2\;{R}_{\odot }$ for stars (and also for Jupiter-like planets, whereas terrestrial planets are disrupted only at distances about half of this because of their higher average densities). Using these typical values, we calculate the rates of disruption of stars by COs. We find ${{\rm{\Gamma }}}_{{\rm{NS}}}=3.2\times {10}^{-9}\;{{\rm{yr}}}^{-1}$ and ${{\rm{\Gamma }}}_{{\rm{BH}}}=1.8\times {10}^{-8}\;{{\rm{yr}}}^{-1}$. These rates are consistent with more detailed cluster simulations where physical collisions were considered (e.g., Ivanova et al. 2008). Although the population of BHs is smaller than that of NSs, their larger mass makes the encounter cross section much larger, thus enhancing the rate of close encounters. For the ∼150 GCs observed in the Milky Way (MW) galaxy we finally get the tidal disruption rates per MW-like galaxy to be ${{\rm{\Gamma }}}_{{\rm{NS}}}^{{\rm{gal}}}\;=4.8\times {10}^{-7}({f}_{\mathrm{NS}}/0.017)({f}_{\mathrm{ret}}/0.05)\;{{\rm{yr}}}^{-1}$ and ${{\rm{\Gamma }}}_{{\rm{BH}}}^{{\rm{gal}}}=2.8\times {10}^{-6}({f}_{\mathrm{BH}}/0.012)({f}_{\mathrm{ret}}/0.05)\;{{\rm{yr}}}^{-1}$. We point out that many galaxies contain much larger number of GCs, and thus these estimates are only lower limits.

The rates of μTDEs of planets depend on the unknown fraction of free-floating planets in GCs; for a ratio of free-floating planets to stars, ${f}_{\mathrm{ffp} \mbox{-} \mathrm{star}}$, of one (one free-floating planet per star) the rate should be slightly lower (due to the smaller combined mass), but comparable to that of stellar disruptions, and the rate should scale linearly with ${f}_{\mathrm{ffp} \mbox{-} \mathrm{star}}$.

Interactions between three and four (or more) bodies are much more complicated, as they may involve resonant encounters in which the stars can pass by each other several times. These could show chaotic behavior in which the stars may pass each other at almost any arbitrary distance (see, e.g., Valtonen & Karttunen 2006, and references within). Such behavior can much enhance the possibility of very close passages followed by tidal disruptions. Calculation of these rates requires better knowledge of the characteristics and distributions of binaries and compact binaries in clusters, and a detailed treatment that is beyond the scope of this work. However, from comparison with the somewhat similar scenario of stellar collisions and tidal captures (Krolik et al. 1984; Fregeau et al. 2004; Leigh & Sills 2011), in which such encounters were found to make an important contribution, we note that our results thus give only a lower limit on the tidal disruption rates, which could be higher by a factor of a few as a result of these few-body encounters.