School of Natural Sciences and Mathematics
Unified Treatment of Tidal Disruption by Schwarzschild Black Holes
UT Dallas Author(s): Juan Servin Michael Kesden
Rights: ©2017 American Physical Society.
Citation: Servin, J., and M. Kesden. 2017. "Unified treatment of tidal disruption by Schwarzschild black holes." 95(8), doi:10.1103/PhysRevD.95.083001
This document is being made freely available by the Eugene McDermott Library of the University of Texas at Dallas with permission of the copyright owner. All rights are reserved under United States copyright law unless specified otherwise. PHYSICAL REVIEW D 95, 083001 (2017) Unified treatment of tidal disruption by Schwarzschild black holes
† Juan Servin* and Michael Kesden Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA (Received 9 November 2016; published 3 April 2017) Stars on orbits with pericenters sufficiently close to the supermassive black hole at the center of their host galaxy can be ripped apart by tidal stresses. Some of the resulting stellar debris becomes more tightly bound to the hole and can potentially produce an observable flare called a tidal-disruption event (TDE). We provide a self-consistent, unified treatment of TDEs by nonspinning (Schwarzschild) black holes, investigating several effects of general relativity including changes to the boundary in phase space that defines the loss-cone orbits on which stars are tidally disrupted or captured. TDE rates decrease rapidly at large black hole masses due to direct stellar capture, but this effect is slightly countered by the widening of the loss cone due to the stronger tidal fields in general relativity. We provide a new mapping procedure that translates between Newtonian gravity and general relativity, allowing us to better compare predictions in both gravitational theories. Partial tidal disruptions in relativity will strip more material from the star and produce more tightly bound debris than in Newtonian gravity for a stellar orbit with the same angular momentum. However, for deep encounters leading to full disruption in both theories, the stronger tidal forces in relativity imply that the star is disrupted further from the black hole and that the debris is therefore less tightly bound, leading to a smaller peak fallback accretion rate. We also examine the capture of tidal debris by the horizon and the relativistic pericenter precession of tidal debris, finding that black holes of 106 solar masses and above generate tidal debris precessing by 10° or more per orbit.
DOI: 10.1103/PhysRevD.95.083001
I. INTRODUCTION consistent with those previously predicted by early models, resulting in an increased interest in TDEs in recent years. Supermassive black holes (SBHs) with masses in the 6 10 Analysis of TDEs has been conducted under both range from 10 M⊙ ≤ M• ≤ 10 M⊙ are found at the Newtonian gravity [5,27–32] and general relativity centers of most large galaxies [1,2]. Observations of [33–40]. In this paper, we provide a unified, self-consistent quasars [3] and active galactic nuclei (AGN) have aided treatment of several aspects of the tidal-disruption process our understanding of these massive celestial objects, but that are sensitive to general relativity. We explore how these constitute only a small fraction [4] of all SBHs in the relativity changes the boundaries of the loss cone in local Universe, making it difficult to take a fully represen- phase space that determines TDE rates and the distribution tative census of galactic nuclei. Studying quiescent SBHs, of orbital elements for tidally disrupted stars. We then visible via their interactions with nearby stellar objects examine how relativity affects the distributions of the through the phenomena of tidal disruption events (TDEs) specific energy and angular momentum of the tidal debris [5], would allow one to fill in this population gap, and combine these results with recent hydrodynamical particularly for lower mass SBHs. Quiescent SBHs also simulations to predict the rate at which this debris falls have dynamical effects on neighboring stars and gas, but back onto the SBH to fuel a tidal flare [41]. these can only be observed locally where the influence We restrict ourselves in this paper to the Schwarzschild radius can be resolved. spacetime [42] corresponding to nonspinning SBHs. ’ TDEs occur when tidal forces due to a SBH s gravita- The spherical symmetry of the Schwarzschild metric ’ tional field overcome an orbiting star s self gravity, causing implies that only the magnitudes of the orbital energy roughly half of the resulting debris to become more tightly and angular momentum are needed to define the boun- bound to the SBH [5]. This bound debris quickly forms a daries of the loss cone. However, in the Kerr spacetime disk that is accreted by the SBH, releasing large amounts [43] of spinning SBHs, the orientation of the tidally of energy in what is known as a tidal flare. Flares can be disrupted star’s orbit with respect to the SBH’s equatorial – – found in the optical [6 12], ultraviolet [13 16], soft x-ray plane affects the magnitude of the tidal stresses [37]. – – [17 22], and hard x-ray [23 25] bands. A recent survey This creates a higher-dimensional parameter space, the [26] discovered many candidate TDEs, showing rates analysis of which is beyond the scope of this paper. We will, however, provide some comments on SBH spin when relevant throughout the paper and lay the *[email protected] foundation for future work that will focus on incorpo- † [email protected] rating spin dependence into our procedure.
2470-0010=2017=95(8)=083001(18) 083001-1 © 2017 American Physical Society JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017) The rest of this paper is structured as follows. Section II in phase space because the velocity vector lies in a cone provides a brief overview of how loss-cone theory is used about the radial direction. We assume that the stellar to calculate TDE rates in Newtonian gravity. Section III distributions in galactic centers are described by isothermal details how relativity changes the boundaries of the loss spheres with density profile cone, allowing us to compare the TDE rates predicted in 2 Newtonian gravity and general relativity. We then focus on σ ρðrÞ¼ ; ð4Þ trying to compare individual TDEs in the two gravitational 2πGr2 theories, which introduces the somewhat subtle issue σ − σ of which parameter to hold constant in these comparisons. where is the velocity dispersion given by the M• We introduce a new procedure to map between Newtonian relation [44,45]. This profile is slightly steeper than the −7 4 orbits and relativistic geodesics in Sec. IV that allows us to ρ ∝ r = Bahcall-Wolf equilibrium solution [46] towards estimate how relativity would modify simulations per- which stellar distributions are expected to relax. formed in Newtonian gravity. In Sec. V, we use this Observationally, galaxies can be classified as having either mapping and loss-cone theory to predict distributions of “cored” or “power-law” stellar density profiles [47]; physical quantities like the peak fallback accretion rate isothermal spheres are reasonable approximations for the and amount of apsidal precession that affect observed TDE power-law galaxies that dominate the global TDE rate properties. A summary of our key results, their implications [27,28]. Recent observations suggest that poststarburst for TDE observations, and our plans to generalize to the galaxies with even steeper stellar density profiles than Kerr spacetime are provided in Sec. VI. isothermal spheres are heavily overrepresented among TDE host galaxies [48,49]. II. NEWTONIAN LOSS-CONE THEORY We assume that loss-cone orbits are refilled by two-body nonresonant relaxation [27–30], in which case the differ- We begin by providing a brief summary of Newtonian ential flux F of stars into the loss cone per unit dimension- loss-cone theory [27–30]. We define the tidal radius as less specific binding energy ε ≡ ε=σ2, following the 1=3 approach and notation of Merritt [50],isgivenby ≡ M• rt R⋆; ð1Þ Z M⋆ y ε 4π2 ε 2 d ε Fð Þ¼ qð ÞLd fð ;yÞdy : ð5Þ 0 where M• is the mass of the SBH, M⋆ is the mass of the tidally disrupted star, and R⋆ is the stellar radius. Here qðε Þ is the orbital period divided by the time a star This definition provides an order-of-magnitude estimate ≡ 2 2 of the distance from the SBH at which tidal disruption takes to diffuse across the loss cone, y L =qLd is a ≡ 1 occurs. Stars on parabolic orbits with pericenters equal to r dimensionless angular-momentum variable, yd =q, and t ε will have orbital angular momentum fð ;yÞ is the stellar phase-space density as a function of our dimensionless variables. For the stellar density profile 1=2 of Eq. (4), Lt ≡ ð2GM•rtÞ : ð2Þ 2 −2 ≲ 32π M⋆ hðε Þ rd Any star on an orbit with angular momentum L Lt will qðε Þ¼ pffiffiffi ln Λ ; ð6Þ 3 2 ψ − ε become tidally disrupted once it reaches a distance r ∼ rt M• ðrdÞ rh from the SBH. We can quantify the threshold for full tidal disruption by introducing the parameter α to account for where ln Λ is the Coulomb logarithm [51–53], ψ ðrÞ ≡ 2 stellar structure, allowing us to define −ΦðrÞ=σ is the negative dimensionless potential energy (including contributions from both the isothermal stellar 2 σ2 rd ≡ α rt; ð3aÞ density profile and the SBH), rh ¼ GM•= is the influence radius [54], and hðε Þ [Eq. (6.78) of Merritt [50]] is a function Ld ≡ αLt; ð3bÞ derived from the isotropic distribution function consistent with our isothermal stellar density profile ρðrÞ.Weset where rd is the radial distance and Ld is the maximum Λ ¼ 0.4M•=M⊙, appropriate for stars whose velocity dis- orbital angular momentum at which a star described by tribution is consistent with the virial theorem [28,55]. parameter α is fully disrupted. Throughout this paper, we The ratio q helps provide intuition about how two-body use the value α ¼ 1.9−1=2 ≃ 0.725 consistent with the relaxation feeds stars into SBHs. Portions of phase space threshold for full tidal disruption in Guillochon and with q ≪ 1 are said to belong to the empty loss-cone Ramirez-Ruiz [41] for a solar-type star with polytropic (ELC) regime because diffusion is not efficient enough to index γ ¼ 4=3. repopulate loss-cone orbits on their dynamical time scale. The TDE rate thus depends on the rate at which stars are Conversely, q ≫ 1 corresponds to the full loss-cone (FLC) driven onto such orbits, which are said to lie in the loss cone regime where diffusion is effective enough to keep orbits
083001-2 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017) 7 in these parts of phase space filled. SBHs with M• ≳ 10 M⊙ the Schwarzschild metric is both time independent and have q ≪ 1 for ε > 1 and thus belong primarily to the spherically symmetric. Note that the specific energy ELC regime, implying lower TDE rates despite their larger ν tidal radii [29]. Approximately 30% of TDEs occur in the μ ∂ 2M• dt E ≡ −gμνU ¼ 1 − ; ð11Þ FLC regime for the galaxy sample considered in Stone and ∂t r dτ Metzger [56], where ∼50% result in full rather than partial disruptions. where The integrand in Eq. (5) for the isotropic distribution μ function consistent with the isothermal density profile of μ dx dt dr dθ dϕ U ¼ ¼ ; ; ; ð12Þ Eq. (4) is given by dτ dτ dτ dτ dτ ffiffiffi X∞ −γ2 =4 p 2 e m J0ðγ yÞ is the 4-velocity, asymptotes to unity for particles at rest 1 − ffiffiffiffiffi m ffiffiffiffiffi fðyÞ¼fðydÞ p γ γ p ; ð7Þ far from the SBH since it contains the rest-mass energy. yd 1 m J1ð m ydÞ m¼ We approximate E ¼ 1 for the orbits of tidally disrupted stars because the velocity dispersion σ of their host galaxies where J0 and J1 are Bessel functions of the first kind and pffiffiffiffiffi is much less than the speed of light. This 4-velocity satisfies γm is defined such that γm yd is the mth zero of J0 [29]. The flux of stars into the loss cone is then, the geodesic equation μ 2 2 dU μ ν α Fðε Þ¼4π L fðy ÞξðqÞ; ð8Þ þ ΓναU U ¼ 0; ð13Þ d d dτ where ξ q is defined to be q times the integral in Eq. (5) μ ð Þ where Γνα are the Christoffel symbols for the ξ ≈ 2 4 1=4 and is well approximated by ðqÞ q=ðq þ q Þ . Schwarzschild metric. Note the resemblance to Newton’s ξ ε Note that ðqÞ is implicitly dependent on through second law dv=dt ¼ 0 for force-free motion for vanishing Eq. (6). The total TDE rate is obtained by integrating over Christoffel symbols as can be chosen for flat space. all binding energies, In general relativity, tidal forces arise because of the Z tendency of parallel geodesics to deviate from each other in N_ ¼ Fðε Þdε ; ð9Þ the presence of spacetime curvature. If two neighboring particles with 4-velocity Uμ are separated by an infinitesi- mal spacelike deviation 4-vector Xβ, this deviation will and depends implicitly on the SBH mass M•. evolve with proper time τ according to the geodesic deviation equation III. RELATIVISTIC TIDAL FORCES 2 β In Newtonian gravity, the tidal force acting on a fluid d X μ α β β μ α ν ¼ U ∇μðU ∇αX Þ¼−R μανU X U element of a star is simply the gravitational force in a frame dτ2 ’ β α freely falling with the star s center of mass. In general ¼ −C αX ; ð14Þ relativity however, gravity affects particle motion through α the curvature of spacetime. In this section, we review how where ∇μ are covariant derivatives, d=dτ ≡ U ∇α is the tides arise in general relativity so that we can determine the β derivative with respect to the proper time, R μαν is the value of the angular momentum L that sets the boundary d Riemann curvature tensor, and of the loss cone in this gravitational theory. We restrict our analysis in this paper to the Schwarzschild metric [42] β β μ ν C α ≡ R μανU U ð15Þ describing nonspinning SBHs, which in Boyer-Lindquist coordinates [57] is is the tidal tensor. Although we restrict ourselves to this 2 lowest-order result in this paper, the geodesic deviation 2M• r ds2 − 1 − dt2 dr2 equation in both the Newtonian [58] and relativistic [59] ¼ þ 2 − 2 r r M•r regimes is corrected by octupole and higher tides at higher α þ r2ðdθ2 þ sin2θdϕ2Þ: ð10Þ order in the deviation vector X . In Newtonian gravity, a fluid element is tidally stripped from the surface of a star Massive test particles experiencing no nongravitational when the tidal force exerted by the SBH exceeds the force forces travel on timelike geodesics xμðτÞ of this metric, exerted on this fluid element by that star’s self gravity. If the the generalization of straight lines in flat space. We can escape velocity of the star is not relativistic, we can choose to parametrize these geodesics by the proper time τ. approximate its self gravity as Newtonian even if general Particles traveling on these geodesics have conserved relativity is needed to describe the geodesics of the SBH. specific energy E and specific angular momentum L since In this approximation, the fluid element is tidally stripped
083001-3 JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017) when the acceleration due to this self gravity is exceeded by the proper acceleration d2X=dτ2 given by the geodesic deviation equation (14) [60]. We can solve this equation more easily by transforming from the global Boyer-Lindquist coordinates given by Eq. (10) to local Fermi normal coordinates (τ;XðiÞ) valid in a neighborhood of spacetime about the center of mass of the star at an arbitrarily chosen proper time τ ¼ 0 [61]. These coordinates define an orthonormal tetrad: the star’s λμ 4-velocity provides the timelike 4-vector ð0Þ, and three λμ spacelike 4-vectors ðiÞ are chosen that are parallel trans- ported along the geodesic. Spatial indices that run from 1 to 3 are denoted by Latin indices, unlike Greek indices that run from 0 to 3. All of the tensors appearing in the geodesic deviation equation can be projected into this basis,
μ λμ U ¼ ð0Þ ð16aÞ
α ðiÞλα FIG. 1. The angular-momentum threshold L for tidal disrup- X ¼ X ðiÞ ð16bÞ d tion as a function of SBH mass M• in both Newtonian gravity (solid black) and general relativity (solid green). As the mass of β ðγÞ λβ λðδÞ λðκÞ λðξÞ R μαν ¼ R ðδÞðκÞðξÞ ðγÞ μ α ν ð16cÞ the SBH increases, Ld falls less steeply in relativity than in Newtonian gravity because of the stronger tidal forces. The horizontal red dashed line at L ¼ 4M• indicates that direct β ðiÞ λβ λðjÞ ðiÞ λβ λðjÞ cap C α ¼ C ðjÞ ðiÞ α ¼ R ð0ÞðjÞð0Þ ðiÞ α: ð16dÞ capture by the event horizon occurs below this value, implying ≃ 107.39 that SBHs with M• >Mmax M⊙ cannot fully disrupt The Boyer-Lindquist indices are raised and lowered by the solar-type stars. Schwarzschild metric (10) while the Fermi normal coordinate indices in brackets are raised and lowered 2 2 2M•r by the Lorentz metric η μ ν . Note that the symmetry of ð Þð Þ L ¼ − 2 ; ð19Þ the Riemann tensor implies that the tidal tensor is sym- r M• metric and only has spatial components in Fermi normal coordinates. Inserting Eq. (16) into Eq. (14) yields the between the orbital angular momentum L and Boyer- geodesic deviation equation in Fermi normal coordinates Lindquist coordinate r at pericenter provides two equations that can be solved for the relativistic tidal radius rd and angular-momentum threshold L for tidal disruption. Note d2XðiÞ d −CðiÞ XðjÞ: that in the limit r ≫ 2M• these reduce to the Newtonian τ2 ¼ ðjÞ ð17Þ d expressions of Eqs. (2) and (3). Figure 1 shows how the angular-momentum threshold Because CðiÞ is a real, symmetric 3 × 3 tensor, it has three ðjÞ Ld depends on SBH mass M• in both Newtonian gravity real eigenvalues which in Boyer-Lindquist coordinates are and general relativity. All orbits with angular momentum L 3 2 2 3 2 2 3 M•=r , ð1 þ 3L =r ÞM•=r , and −2ð1 þ 3L =2r ÞM•=r . below this threshold are considered to be inside the loss The negative sign in Eq. (17) implies that the eigenvectors 1=3 cone. The Newtonian tidal radius rt ∝ M• according to associated with the positive eigenvalues correspond to 2 Eq. (1), while the gravitational radius r ≡ GM•=c scales directions along which the star is compressed, while the g linearly with SBH mass. This implies that relativistic eigenvector associated with the negative eigenvalue corre- effects become negligible for small SBH masses, and the sponds to the direction in which the star is stretched. two curves in Fig. 1 converge towards the left edge of the Equating the magnitude of this negative eigenvalue times 3 2 plot. The fact that the term in brackets on the left-hand side the stellar radius R⋆ with the acceleration 2M⋆=ðα R⋆Þ ’ of Eq. (18) is greater than unity implies that tidal forces are due to the star s self gravity yields stronger in general relativity than in Newtonian gravity for orbits with the same angular momentum. This explains why 3 2 2 2 1 L M•R⋆ M⋆ the relativistic curve in Fig. 1 is above the Newtonian curve; þ 2 3 ¼ 3 2 : ð18Þ 2r r ðα R⋆Þ stronger tides allow the SBH to disrupt stars at larger angular momentum Ld in relativity. This result of stronger Combining this result with the relation tides in relativity is confirmed by comparisons between
083001-4 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017) polytropic index γ ¼ 4=3 can be partially disrupted on −1=2 orbits with L ≤ Lpd ¼ αpdLt, where αpd ¼ 0.6 ≃ 1.29. Equation (9) gives the rate of full TDEs, while the rate of partial TDEs is determined by summing the contributions from orbits with Ld ≤ L ≤ Lpd as described in greater detail in Sec. V. Relativity modifies these Newtonian predictions in two ways. It changes the thresholds rd and Ld in Eqs. (6) and (7) to the maximum of the thresholds set by full disruption and direct capture, and modifies the limits of the integral in Eq. (5) to account for the three outcomes of partial disruption, full disruption, and direct capture. For SBH masses M•
083001-5 JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017) metric in Boyer-Lindquist coordinates given in Eq. (10), the physical significance of this mapping is that it identifies parabolic orbits such that the circular orbits in the two gravitational theories with the same pericenter r would have the same circumference 2πr. It is unclear why this choice of mapping would be particularly useful for ana- lyzing the tidal disruption of stars on noncircular orbits. Mapping (2) identifies orbits in the two theories with the same values of the gauge-invariant orbital angular momen- tum, defined for the Schwarzschild metric as ν μ ∂ 2 dϕ L ≡ gμνU ¼ r ð20Þ ∂ϕ dτ for equatorial orbits (θ ¼ π=2). This mapping seems like a useful choice, as TDE properties do depend on the orbital angular momentum as seen in the recent simulations of Guillochon and Ramirez-Ruiz [41]. However, tides are FIG. 3. The mapping between Schwarzschild geodesics with penetration factor β and orbits in Newtonian gravity with penetra- stronger on Schwarzschild geodesics than on Keplerian β orbits with the same value of L as established in Sec. III,so tion factor N on which stars experience the same tidal forces at pericenter. The solid blue, dashed magenta, dot-dashed red, and it seems unlikely that TDEs on orbits in the two theories solid green curves show this mapping for SBHs with masses identified in this manner would have the same properties. 5 6 7 7.39 M•=M⊙ of 10 , 10 , 10 ,and10 , respectively. The solid black We conjecture that TDEs resulting from stars initially on diagonal βN ¼ β shows the limit of these curves as M• → 0, while β β orbits identified by mapping (3) will have similar properties the dotted black curve shows the relation Nð capÞ beyond which because the stars were subjected to similar tidal forces. This these mappings are undefined. The horizontal black dashed line “ ” conjecture is supported by the freezing model of Lodato, βN;d ¼ 1.9 corresponds to full disruption in the hydrodynamical King, and Pringle [32] which proposed that the orbital simulations of Guillochon and Ramirez-Ruiz [41]. energy distribution of tidal debris is frozen in following an instantaneous tidal disruption of an unperturbed star. This Instead of using the angular momentum L to identify model was used with reasonable success to describe the orbits, the TDE literature often uses the penetration factor light curve of the TDE PS1-10jh [7]. We will use this freezing assumption later in Sec. V to determine the L2 β ≡ t ; ð22Þ appropriate relativistic correction to the energy distribution, L2 but we will apply the correction at the disruption radius rd rather than pericenter. This is consistent with the analytic where Lt is defined in Eq. (2). This choice is convenient β ≥ β ≡ 2 2 model of Stone, Sari, and Loeb [63] and some hydrody- because stars on orbits with d Lt =Ld are tidally β namical simulations of TDEs [41,64] which showed that disrupted, where d is of order unity. Our mapping LNðLÞ β β the spread in debris energy was largely independent of the implies an equivalent mapping Nð Þ, where orbital angular momentum L for full disrupted stars. 2 1=3 β ≡ Lt r þ M• rt To implement mapping (3), we must find the angular N 2 ¼ − 2 ; ð23Þ LN r M• r momentum LN of the orbit in Newtonian gravity that has the same peak tidal force as that experienced by a star and r is an implicit function of L and hence β through on a Schwarzschild geodesic with angular momentum L Eq. (19). We show this mapping for several values of the and pericenter r. We accomplish this by setting the SBH mass M• in Fig. 3. These mappings are undefined for β β ≡ 2 2 magnitude of the negative eigenvalue of the tidal tensor > cap Lt =Lcap since such geodesics plunge directly ðiÞ C ðjÞ equal to its Newtonian limit for an orbit with into the event horizon and therefore do not have pericenters angular momentum LN, at which one can calculate the tidal force. The mappings β β Nð Þ are above the diagonal and have positive curvature 2 3 3L 1 2M• because tides are stronger in general relativity, requiring 1 : 21 þ 2 2 3 ¼ 2 ð Þ Newtonian orbits to have deeper penetration factors β to r r LN N match the tides on Schwarzschild geodesics at pericenter. ≃ 107.39 This equation, combined with Eq. (19) relating the The mapping for SBH mass Mmax M⊙ terminates β β β 1 9 angular momentum L to the pericenter r in Boyer- at Nð capÞ¼ N;d ¼ . , precisely the threshold for the Lindquist coordinates, can be solved to determine full disruption of Solar-type stars [41] seen in Fig. 1 LNðLÞ for mapping (3). for Ld ¼ Lcap.
083001-6 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017) affect the observed properties of TDEs. These properties _ include the peak accretion rate Mpeak at which tidal debris falls back onto the SBH, the time delay tpeak between tidal disruption and when this peak fallback accretion occurs, a parameter fL measuring the vulnerability of the tidal debris to direct capture by the event horizon of the SBH, and the pericenter precession Δω of the disrupted star. The peak _ accretion rate Mpeak may be proportional to the peak bolometric luminosity if accretion remains below the Eddington rate [32]. The time delay tpeak could in principle be measured if in addition to the peak of the TDE light curve one could also observe a prompt electromagnetic signature associated with disruption like an x-ray shock breakout [66,67]. The direct capture of tidal debris, occur- ring for fL < 0, could suppress emission responsible for the early portion of the TDE light curve [39]. Relativistic pericenter precession has been shown in theoretical work to FIG. 4. The minimum penetration factors βd and βpd for full and partial tidal disruption in general relativity as functions of affect the time needed to circularize the orbits of the tidal SBH mass M•. The horizontal dot-dashed green and blue lines debris and produce emission from the newly formed disk – show the thresholds βN;d ¼ 1.9 and βN;pd ¼ 0.6 for full and [64,68 71]. partial disruption in Newtonian gravity [41]. The solid green and Guillochon and Ramirez-Ruiz [41] performed Newtonian blue curves show the values of β for Schwarzschild geodesics on hydrodynamical simulations to determine the peak fallback d _ which stars experience the same tidal forces at pericenter. The accretion rate Mpeak and time delay tpeak. They then provided intersections between these curves and the dashed red curve analytic fits to these quantities as functions of the Newtonian β ðM•Þ, marked by the vertical dashed lines, occur at the cap penetration factor βN. Under the approximation that these ≃ 107.39 ≃ maximum SBH masses Mmax;FD M⊙ and Mmax;PD quantities depend only on the peak tidal forces along each 8.15 10 M⊙ capable of full and partial tidal disruption. orbit, we use the inverse of the mapping βNðβÞ shown in _ Fig. 3 to derive fits to Mpeak and tpeak as functions of the To further illustrate how relativity affects the minimum β β relativistic penetration factor in the Schwarzschild space- penetration factor d for tidal disruption, we show its time. We then use the loss-cone theory discussed in Sec. II, dependence on SBH mass M• in Fig. 4 for both full and modified by the relativistic corrections to the boundaries partial tidal disruptions. Newtonian hydrodynamical simu- of the loss cone shown in Fig. 1, to predict the distributions of lations [41] indicate that solar-type stars are fully disrupted for β β 1 9 these quantities for different SBH masses. N > N;d ¼ . , while partial disruptions occur for In Sec. VA, we determine the differential TDE rate β β 0 6 β N > N;pd ¼ . .AsM• increases and d approaches _ β β β β dN=d per unit relativistic penetration factor .In cap,the d curves fall increasingly below these Newtonian _ β Sec. VB, we use this rate and our corrected fits Mpeakð Þ thresholds because of the stronger tides in general relativity. β β and tpeakð Þ to derive initial estimates for the differential The values of M• at which these curves intersect the cap _ _ _ ≃ 107.39 TDE rates dN=dMpeak and dN=dtpeak per unit peak fallback curve are the maximum masses Mmax;FD M⊙ and 8.15 accretion rate and time delay, respectively. These initial M ≃ 10 M⊙ capable of full and partial tidal max;PD estimates do not account for an additional relativistic effect disruption. These limits are consistent with recent work discussed in Kesden [39], that the gradient of the potential suggesting that observable TDEs by SBHs with masses 8 well that determines the width of debris energy distribution above 10 M⊙ result exclusively from giant stars [65]. differs in general relativity and Newtonian gravity. We Determinations of Mmax that failed to account for the stronger β β review this result and then derive revised estimates for relativistic tides, effectively using mapping (2) d ¼ N;d _ _ _ shown by the horizontal lines in Fig. 4, would underestimate dN=dMpeak and dN=dtpeak in Sec. VC. The stellar debris M by a factor of ∼4.5 for both full and partial disruptions. produced following tidal disruption will also have a max distribution of orbital angular momentum, but this often Future work will explore how SBH spin can push Mmax to even higher values for stars on prograde orbits. receives less attention in Newtonian gravity where the orbital periods that determine the fallback accretion rate depend on energy but not angular momentum. However, V. DISTRIBUTIONS OF RELATIVISTIC TDE the distribution of orbital angular momentum can be very PROPERTIES important when considering highly relativistic TDEs for In this section, we use concepts developed earlier in the which much of the debris can lose enough specific angular paper to predict distributions of several quantities that may momentum to fall below the threshold for capture Lcap even
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FIG. 5. The differential TDE rate dN=d_ β per unit penetration factor for various SBH masses under Newtonian gravity (left) and relativity (right). The dashed black line in both panels shows the Newtonian threshold βN;d ¼ 1.9 for full disruption, while the blue, magenta, and red dashed lines in the right panel show the thresholds βd in relativity defined through the mapping βNðβdÞ¼βN;d between Newtonian orbits and Schwarzschild geodesics with the same tidal forces at pericenter. if the initial star was above this threshold. We discuss this M• consistent with the TDE rates seen in Fig. 2. The blue regime in Sec. VDand show that this effect sharply limits the curve in the left panel of Fig. 5 extends smoothly beyond relativistic correction to the peak fallback accretion rate. βN;d because much of the phase space beyond this value is Finally, in Sec. VEwe use our relativistically corrected loss- still in the full loss-cone regime q ≫ 1 for which dN=d_ β ∝ _ −1 7 cone theory to determine the differential TDE rate dN=dΔω β [56].However,asM• increases towards 10 M⊙, the per unit relativistic shift in argument of pericenter, an differential TDE rate becomes exponentially suppressed for important effect responsible for tidal stream crossings and βN > βN;d reflecting that much of the phase space beyond the prompt circularization of debris [64,68–71]. this point now lies in the empty loss-cone regime [56]. Relativity introduces two changes into the calculation of A. Distribution of penetration factor β the differential TDE rate dN=d_ β: (1) the stronger tides _ We begin by calculating the differential TDE rate dN=dβ increase the numerical value of Ld from the Newtonian in Newtonian gravity. The angular-momentum variable y result given by the black curve in Fig. 1 to the relativistic can be expressed in terms of the penetration factor β as result given by the green curve in this figure, and (2) capture by the event horizon cause the differential rate to fall L2 discontinuously to zero for β ≥ β . As most stars diffusing y ¼ t : ð24Þ cap qL2β into the loss cone have apocenters near the influence radius d 2 2 rh ¼ GM•=σ ≫ rg ¼ GM•=c [72], the Newtonian treat- This relation allows us to differentiate Eq. (5) with respect ment of stellar diffusion encapsulated in the ratio q remains β to , which when combined with the distribution function an accurate approximation. Both of the changes listed in Eq. (7) yields above leave signatures in the relativistic differential TDE _ β dF 2πL 2 rate dN=d seen in the right panel of Fig. 5. The bends in ¼ t fðy Þ these curves marking the threshold for full disruption in the dβ β d empty loss-cone regime migrate to lower values of β ∞ 2 pffiffiffiffiffiffi X −γm=4 pffiffiffi e J0ðγmLt=Ld qβÞ consistent with the SBH mass dependence of the mapping × 1 − 2 q pffiffiffi : ð25Þ γ γ βNðβÞ shown in Fig. 3. The curves also end abruptly at β 1 m J1ð m= qÞ cap m¼ as can be seen in the red curve corresponding to SBH 107 β ≃ 1 27 Integrating this expression with respect to the dimension- mass M• ¼ M⊙ for which cap . . Integrating the less specific binding energy ε gives us the Newtonian curves in Fig. 5 within the appropriate β ranges _ differential TDE rate dN=dβ shown in the left panel of (0.6 < βN < 1.9 for partial disruptions and 1.9 < βN < β Fig. 5 above. The total TDE rate is the area under this curve cap for full disruptions) would reproduce the full and for βN > βN;d which decreases with increasing SBH mass partial TDE rates shown in Fig. 2.
083001-8 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017) 7 B. Distributions of peak fallback accretion More quantitatively, for a 10 M⊙ SBH, relativity increases _ rate Mpeak and time delay tpeak _ the mean value of the (scaled) peak accretion rate Mpeak for Guillochon and Ramirez-Ruiz [41] performed a series of full TDEs to 2.965 M⊙=yr from its Newtonian value of Newtonian hydrodynamical simulations at different pen- 2.757 M⊙=yr and decreases the mean value of the (scaled) etration factors βN and used them to derive analytic fits to time delay tpeak to 0.078 yr from its Newtonian value of _ the peak fallback accretion rate dMpeak=dt and time delay 0.0792 yr. These changes occur because direct capture by γ 4 3 tpeak for solar-type stars with polytropic index ¼ = : the SBH event horizon prevents stars on deeply penetrating orbits from producing observable TDEs, and simulations −1 2 _ M• = [41] show that such deeply penetrating TDEs yield lower Mpeak ¼ A4=3 6 ; ð26aÞ peak accretion and longer delays. 10 M⊙ 1 2 M• = C. Relativistic correction to the debris energy tpeak ¼ B4=3 6 ; ð26bÞ 10 M⊙ distribution _ _ The estimated differential TDE rates dN=dMpeak and where, _ dN=dtpeak in the previous section accounted for relativistic corrections to the boundaries of the loss cone, but in the 27 3 − 27 5β 3 87β2 . . N þ . N A4 3 ; interest of a systematic exploration of relativistic effects we = ¼ exp 1 − 3 26β − 1 39β2 ð27aÞ . N . N did not simultaneously include relativistic corrections to the ffiffiffiffiffiffi energy distribution of the tidal debris. We turn our attention −0 387 0 573pβ − 0 312β . þ . ffiffiffiffiffiffi N . N to this effect in this section. B4=3 ¼ p ; ð27bÞ 1 − 1.27 βN − 0.9βN A key approximation often used in the analysis of TDEs is that the tidal debris travels on ballistic trajectories 0 6 ≤ β ≤ 4 0 β β and . N . . Given these fits, the mapping Nð Þ following disruption, and that orbit circularization followed _ derived in Sec. IV, and the differential TDE rate dN=dβ by viscous accretion occurs promptly after the bound debris obtained in the previous section, we can calculate differ- falls back to pericenter [5,31,73]. The fallback accretion _ ential rates rate onto the SBH, and thus its peak value Mpeak and the time t between disruption and when this peak occurs, is dN_ dN_ dβ dX −1 peak N determined by the distribution of orbital periods of the tidal ¼ β β β ð28Þ dX d d d N debris, which is in turn set by the energy distribution ’ ∈ _ through Kepler s third law for X fMpeak;tpeakg. We show these differential TDE rates in Fig. 6 above. A 3 1=2 a −3=2 complication arises from the fact that the dependence on τ ¼ 2π ¼ 2πM•ð2EÞ : ð29Þ _ M• penetration factor in Eq. (27) is not monotonic; Mpeak (tpeak) increases (decreases) with βN until the threshold for full This Newtonian relation remains an excellent approxima- disruption βN;d is reached, then decreases (increases) for TDEs that penetrate more deeply into the SBH’s potential tion even for relativistic TDEs, because most of the debris is on highly eccentric orbits prior to circularization and well. This βN dependence contradicts the naive predictions of freezing models which assume that the energy distri- spends most of its time near apocenter where Eq. (29) bution of the tidal debris is frozen in at the disruption radius holds. However, if the tidal disruption itself occurs near pericenter where relativistic effects are strongest, rd, but does not fatally compromise our analysis. We address this complication by plotting the TDEs from partial Newtonian predictions for the specific binding energy E entering into Eq. (29) may not be accurate. TDE simu- (βN < βN;d) and full (βN > βN;d) disruptions separately in the left and right panels of Fig. 6. The differential rates lations in general relativity [59,62,64,71,74] naturally yield proper relativistic energy distributions, but our goal in this diverge at the extremum βN ¼ βN;d shown by the vertical black dashed lines in these plots, but the total rates (areas section is to study how relativistic corrections might alter under the curves) remain finite. We see that the stronger the predictions of Newtonian simulations. tides of relativity suppress the rates of both full and partial We addressed this issue in Kesden [39], where we found 7 that for an undistorted star of radius R⋆, the width of the TDEs, particularly for SBH masses M• ≥ 10 M⊙ for which portions of phase space with β > β lie primarily potential-energy distribution before disruption and thus the N N;d width of the debris energy distribution after disruption is in the empty loss-cone regime. This emptiness even reduces given in general relativity by the expression the rate of partial TDEs because of the sharper gradients driving stronger diffusion across the boundary as seen by σ λα ∇ λβ λα Γγ the dip near the threshold in the top left panel of Fig. 6. E;GR ¼j ðiÞ αEjR⋆ ¼jgβγ ð0Þ ðiÞ αtjR⋆; ð30Þ
083001-9 JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017)
_ _ _ FIG. 6. The differential TDE rates dN=dMpeak (top panels) and dN=dtpeak (bottom panels) as functions of the peak fallback accretion _ rate Mpeak and time delay tpeak for partial (left panels) and full (right panels) disruptions. To clarify the presentation of relativistic effects, we have scaled out the explicit dependence on SBH mass M• given on the right-hand side of Eq. (26). The solid (dot-dashed) curves _ show the differential TDE rates in Newtonian gravity (general relativity). The vertical dashed black lines indicate the values of Mpeak and time delay tpeak and the threshold βN;d ¼ 1.9 of full disrpution. Γγ 1=3 1=3 where gβγ is the metric tensor, αt are the Christoffel symbols, ≡ M•R⋆ M• M⋆ M• λβ λα Et 2 ¼ ¼ E⋆ ð32Þ and ð0Þ and ðiÞ are the orthonormal tetrad of basis rt M⋆ R⋆ M⋆ 4-vectors determined by our choice of Fermi normal coor- dinates in Sec. III.Forβ < βd corresponding to partial is an order-of-magnitude estimate for the width of this energy disruptions, tidal debris is assumed to be liberated at peri- distribution, and E⋆ ≡ M⋆=R⋆ is a similar estimate for the center where the tidal forces are strongest. Evaluating specific self-binding energy of the star. The mass hierarchy Eq. (30) at pericenter for such orbits yields M⋆ ≪ M• implies that E⋆ ≪ Et, supporting the assumption of freezing models that the relative velocities of fluid −1 2 M•R⋆ 2M• = σ ¼ 1 − elements at the time of disruption can be neglected when E;GR;PD r2 r determining the debris energy distribution. 2 2 −1=2 In trying to compare this relativistic result to Newtonian rt 1 − M• ¼ Et : ð31Þ predictions, we encounter the same issue of choosing a r r mapping between the two gravitational theories that we The pericenter r in Boyer-Lindquist coordinates can be addressed in Sec. IV. Mapping (1), which was used in expressed in terms of the angular momentum L using Kesden [39], compared Eq. (31) to Newtonian orbits with Eq. (19), the same pericenter coordinate and found
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FIG. 7. Dimensionless width σE=Et of the tidal debris energy distribution as a function of penetration factor β. The solid curves show the Newtonian predictions σE;N=Et, while the dashed blue, magenta, red, and green curves show the relativistic predictions σE;GR=Et for 5 6 7 7.39 M•=M⊙ ¼ 10 , 10 , 10 , and 10 . The last value is Mmax, the most massive SBH capable of fully disrupting a solar-type star. The β β vertical dotted lines show the thresholds d for full disruption; each of the relativistic curves end at cap. The single black curve in the left panel shows the Newtonian prediction in mapping (2) which is independent of SBH mass, while the three solid curves in the right panel show the Newtonian predictions for mapping (3). 2 σ M•R⋆ rt However, the weaker tides in Newtonian gravity imply E;N;PDð1Þ ¼ 2 ¼ Et : ð33Þ β β r r that stars can reach larger penetration factors N;d > d before being fully disrupted. SBHs with masses M• ≃ 106 β β Dividingffiffiffi Eq. (31) by Eq. (33) yields a peak correction of M⊙ can have deeply penetrating encounters d < < p β 2 for r ¼ 4M•, the minimum pericenter for an orbit that cap where the star still manages to avoid direct capture. For avoids direct capture by the horizon. As argued in Sec. IV such penetration factors, the relativistic prediction can fall however, this mapping is a somewhat unnatural way to below that in Newtonian gravity as can be seen near the identify orbits in the two theories. right edge of the left panel of Fig. 7. It is interesting to note Mapping (2) compared Schwarzschild geodesics to that the relativistic predictions retain some mild β depend- Newtonian orbits with the same angular momentum L ence for full disruptions β > βd, unlike in Newtonian β 2 2 ðiÞ (and hence penetration factor ¼ Lt =L ). For this choice, freezing models. This is because the tidal tensor C ðjÞ σ the width of the Newtonian energy distribution is and energy width E;GR are velocity-dependent in relativity, even if evaluated at the fixed tidal acceleration 2 2 3 2 σ M•R⋆ M• β2 2M⋆=ðα R⋆Þ set by the star’s self gravity. E;N;PDð2Þ ¼ 2 ¼ Et 2 ¼ Et; ð34Þ rL L Mapping (3) solves the problem of stars being fully disrupted at different values of β in the two theories by where rL is the pericenter of a Newtonian orbit with angular identifying the Schwarzschild geodesic with penetration momentum L. Freezing models posit that the energy factor β with the Newtonian orbit with penetration factor β β distribution is frozen in for β > βN;d, implying that Nð Þ on which a star experiences the same peak tidal force σ β2 at pericenter as shown in Fig. 3. In this case, the width of E;N;FDð2Þ ¼ N;dEt for full disruptions in mapping (2). Although there is no nice analytic expression for the widths the energy distribution for partial disruptions is σ for full disruption in general relativity, they can be E;GR;FD 2=3 calculated numerically using Eq. (30). We show the widths σ M•R⋆ β2 β M•R⋆ r þ M• E;N;PDð3Þ ¼ 2 ¼ Nð ÞEt ¼ 2 − 2 ; of these distributions in mapping (2) for both partial and rF r r M• full disruptions in the left panel of Fig. 7. We see that for ð35Þ partial disruptions β < βd, the energy distribution is broader in relativity than Newtonian gravity. The ratio pffiffiffi where rF is the pericenter of a Newtonian orbit with σ σ 4 2 E;GR= E;Nð2Þ reaches a maximum value of at Mmax, penetration factor βNðβÞ and r as previously is the peri- even larger than the peak correction in mapping (1). center in Boyer-Lindquist coordinates of a Schwarzschild
083001-11 JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017) geodesic with penetration factor β. This width again freezes σ β2 out at E;N;FDð3Þ ¼ N;dEt for full disruptions as in mapping (2), but these full disruptions now occur for β > βd as in σ σ general relativity. We show E;GR and E;Nð3Þ as functions of penetration factor β in the right panel of Fig. 7. We see that full disruption occurs at βd in both theories with this mapping, but that for a given penetration factor β, the σ relativistic predictions E;GR are now below the Newtonian σ predictions E;Nð3Þ. The deeper penetration needed for tidal disruption in Newtonian gravity leads to steeper potential gradients and thus broader tidal debris energy distributions. σ σ The ratio E;GR= E;Nð3Þ reaches a minimum value of 128 625 1=6 ≃ 0 768 ð = Þ . at Mmax. The Newtonian orbit 4 mapped to the Schwarzschild geodesic with Lcap ¼ M• 1024 5 1=6 ≃ 2 43 has LN ¼ð = Þ M• . M•
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_ _ _ FIG. 9. The differential TDE rates dN=dMpeak (left panel) and dN=dtpeak (right panel) as functions of the peak fallback accretion rate _ Mpeak and time delay tpeak between disruption and when this peak is reached. The blue, magenta, and red curves show SBH masses 5 6 7 M•=M⊙ ¼ 10 , 10 , and 10 . The solid curves show partial disruptions while the dashed curves show full disruptions. The vertical black dotted lines show the values in Newtonian gravity at the threshold βN;d for full disruption. debris will have a distribution of specific angular momen- specific angular momentum threshold for direct capture by tum L with a width given by the horizon. An element of the tidal debris with fL < 0 will be captured by the SBH, and fL ¼ −1 is the minimum value σ λα ∇ λβ λα Γγ L;GR ¼j ðiÞ αLjR⋆ ¼jgβγ ð0Þ ðiÞ αϕjR⋆ ð37Þ of this parameter for a star that is not originally on a capture orbit (L>Lcap). analogous to Eq. (30) giving the width of the specific We need an estimate for ΔL to evaluate our new energy distribution. This width is set at pericenter r for parameter fL. In the freezing model, an element of the partial disruptions, allowing us to evaluate Eq. (37) as tidal debris located at XðiÞ in Fermi normal coordinates at tidal disruption has its specific energy and angular momen- 2 1=2 1=3 1=2 tum changed by an amount σ M• M⋆ rt L;GR;PD ¼ R⋆ ¼ Lt : ð38Þ r M• r Δ ≡ ðiÞ ðiÞλα ∇ E X rEðiÞ ¼ X ðiÞ αE; ð40aÞ The mass hierarchy M⋆ ≪ M• between the star and SBH ðiÞ ðiÞ α σ ≪ ΔL ≡ X r X λ ∇αL; 40b implies that L;GR Lt and thus that it is usually a good LðiÞ ¼ ðiÞ ð Þ approximation to assume that the specific angular momen- tum of the tidal debris is equal to that of the initial star. in the tidal-disruption process. The element of the star This contrasts with the debris energy distribution for which that will become the most tightly bound element of the ðiÞ E ≫ σ2 implying that the tidal debris is much more tightly tidal debris will have X of magnitude R⋆ antialigned t σ bound to the SBH than the initial star. However, for highly with rEðiÞ, leading to the specific binding energy E;GR relativistic TDEs, the small amount of specific angular givenbyEq.(30). The element of the star that loses the momentum lost in the disruption process may be enough most angular momentum in the tidal-disruption process ðiÞ for some of the tidal debris to be captured by the horizon. will have X of magnitude R⋆ antialigned with rLðiÞ and σ We examine this possibility by defining the dimensionless have its angular momentum reduced by an amount L;GR parameter givenbyEq.(37).IfrEðiÞ and rLðiÞ are parallel to each other (as at pericenter in Newtonian gravity, where both L þ ΔL − L L − 4M• f ≡ cap ¼ − 1; ð39Þ point in the radial direction), the same element of the tidal L jΔLj jΔLj debris will be both the mostly tightly bound (the first to fall back onto the SBH) and have lost the most angular where L is the specific angular momentum of the initial star, momentum (and thus be at greatest risk for direct ΔL<0 is the change in specific angular momentum of a capture). This is also true in general relativity when 4 β ≤ β fluid element in the disruption process, and Lcap ¼ M• is the freezing occurs at pericenter, i.e. d. A consequence
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FIG. 10. The parameter fL ≡ ðL þ ΔL − LcapÞ=jΔLj, a measure of whether tidal debris is at risk of direct capture by the event horizon, as a function of SBH mass M•. The green (black) curves correspond to stars at the thresholds β ¼ βdðβpdÞ for full (partial) disruptions. The solid curves correspond to elements of the tidal debris that fall back onto the SBH when fallback accretion peaks, while the dashed curves represent the most tightly bound (mtb) elements (first to fall back). The red (blue) horizontal dashed lines show the reference values fL ¼ 0ð−1Þ.
of rEðiÞ and rLðiÞ being parallel is a one-to-one relationship to the thresholds for full disruption and nonzero partial between the change in specific orbital energy and angular disruption. For the most tightly bound elements with Δ −σ momentum, symmetric for gains and losses. Numerical L ¼ L;GR,Eq.(39) yields simulations of full disruptions in general relativity show 1=3 1=2 1 1=3pffiffiffiffiffiffiffiffi that this symmetric one-to-one relationship begins to M• r ffiffiffi M• fL ¼ p − 8E⋆ − 1: break down for relativistic TDEs only mildly beyond M⋆ rt β M⋆ the threshold for full disruption [62]. ≤ 0 Δ −σ ð42Þ TDEs for which fL for L ¼ L;GR in Eq. (39) will have suppressed fallback accretion rates and greater For M⋆ ≪ M•, the term in the square brackets goes to delays between disruption and the beginning of emission β−1=2 ∝ 1=3 because direct capture will have removed the most tightly and fL M• ,howeverasM• increases this term decreases and ultimately reaches zero (fL ¼ −1)forL ¼ bound debris. Furthermore, if rEðiÞ and rLðiÞ are parallel, there is a one-to-one relationship Lcap when the entire star is directly captured. This increase then subsequent decrease in fL as a function of M• is seen in the left panel of Fig. 10 indicating that tidal debris is jrLðiÞj ΔL ¼ E ð41Þ safe from direct capture for all but the most massive SBHs. jrEðiÞj ∼20% Only when M• is within of Mmax does the direct capture of tidal debris become significant as seen in the between the specific binding energy E of an element of right panel of Fig. 10. Although such events constitute tidal debris and the specific angular momentum ΔL it loses only a small fraction of the total TDE rate, they are the during tidal disruption. If we use Kepler’s third law (29) to TDEs subject to the most extreme relativistic effects such relate the time delay t to the specific binding energy peak as the pericenter precession that will be considered in the Epeak of debris accreted at that time, we can use Eq. (41) to next section. estimate ΔL and thus fL for such debris. When this estimate of fL becomes negative, all of the debris accreted before the fallback accretion rate reaches its peak will be E. Relativistic pericenter precession captured by the horizon. Early work on TDEs assumed that tidal debris would We show fL for both peak and most tightly bound fluid promptly circularize after falling back to pericenter [5], elements as a function of SBH mass M• in Fig. 10,for but Newtonian hydrodynamical simulations demonstrated stars with penetration factors βd and βpd corresponding that energy dissipation at pericenter was inefficient for
083001-14 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017)
M⋆ ≪ M• [69]. Early analytic work [73] suggested that relativistic pericenter precession might promote orbit cir- cularization, because this precession would lead to steam crossings at which the inelastic collisions of tidal elements would transform orbital kinetic energy into heat that could be subsequently radiated away. This suggestion was later supported by relativistic TDE simulations in which such precession was indeed shown to generate tidal stream crossings [62,64,68,70,71]. The angular coordinate ω specifying the location of pericenter with respect to a reference axis in the orbital plane is known as the argument of pericenter. In Newtonian gravity, the argument of pericenter is a constant of motion, but this is not true for Schwarzschild geodesics of the metric (10) in Boyer-Lindquist coordinates. At lowest post- Newtonian (PN) order, the argument of pericenter changes by an amount FIG. 11. Precession of argument of pericenter Δω (in units of 2 2 3 radians) as a function of the specific angular momentum L. The M• M• = Δω 6π 3πβ solid magenta and dot-dashed blue curves show the exact 1PN ¼ ¼ E⋆: ð43Þ L M⋆ precession and its 1PN approximation, respectively. The vertical dashed red line show the capture threshold L ¼ 4M• and the ≃ cap This PN approximation breaks down for L Lcap for horizontal dotted line shows the 1PN prediction for precession at which pericenter precession must be integrated numerically this threshold. along Schwarzschild geodesics, Z _ Δω ∞ ϕ_ dN=d . We show this rate in Fig. 12 for several SBH Δω 2 0 − 2π M• S ¼ _ dr ; ð44Þ masses . As expected from Eq. (43),TDEsbymore r jrj massive SBHs have greater amounts of pericenter precession suggesting that relativistic orbit circularization might be where more effective in such events. Greater pericenter precession 2 2 1 2 may also lead to hotter accretion disks, accounting for the 2M• L L = r_ ¼ 1 þ − ð45aÞ r r r
L ϕ_ ¼ ð45bÞ r2 are the first-order derivatives of the Boyer-Lindquist coordinates r and ϕ with respect to proper time τ. This breakdown is also seen in Fig. 5 of Cheng and Bogdanović [62], where larger deviations from the PN approximation occur for deeply penetrating orbits. The lower limit of the integral in Eq. (44) is the pericenter r which depends implicitly on L and hence β through Eq. (19). We can set the upper limit of this integral to ∞ since we assume the tidally disrupted stars are initially on nearly parabolic orbits and the tidal debris remains highly eccentric. Equation (38) also indicates that we can approximate the specific angular momentum of the tidal debris as equal to that of the initial Δω Δω star. We plot 1PN and S in Fig. 11, which shows that pericenter precession diverges at L ¼ L guaranteeing a cap FIG. 12. The differential TDE rate dN=d_ Δω as a function of the tidal stream crossing and perhaps subsequent orbit circu- precession of the argument of pericenter Δω. The solid blue, larization of the tidal debris. magenta, and red curves show this distribution for SBH masses Δω β Given that S is a function of ,wecanusethe 105 106 107 _ M•=M⊙ ¼ , , and , respectively. The dashed black differential TDE rate dN=dβ derived in Sec. VA and shown curve connects the lower limit of each rate curve corresponding to in the right panel of Fig. 12 to derive the differential TDE rate the threshold βpd of partial disruption.
083001-15 JUAN SERVIN and MICHAEL KESDEN PHYSICAL REVIEW D 95, 083001 (2017) discrepancy in color temperatures between TDEs discovered respectively. These limits would have been under- in the optical and x-ray [75]. Further investigation of this estimated by a factor of ∼4.5 without accounting possibility requires a criterion for orbit circularization that for the stronger tides in relativity. we hope to explore in future work. We also note that the (4) Event horizons can capture a portion of the tidal differential TDE rate dN=d_ β, and thus all the distributions debris in general relativity, even if the tidally shown in this section including dN=d_ Δω, depends on the disrupted star is not initially on a capture orbit. stellar density profile ρðrÞ. A profile yielding a larger Tidal debris loses both energy and angular momen- fraction of TDEs in the full loss-cone (pinhole) regime tum during tidal disruption. If the specific angular would provide more stars on deeply penetrating orbits momentum of the initial star was already close to the and thus a fattening of the large Δω tails of the curves capture threshold, this additional loss can cause in Fig. 12. some of the debris to plunge directly into the horizon. This captured debris is the most tightly β ≲ β VI. DISCUSSION bound part of the tidal stream for d, so its capture suppresses the early portions of the TDE In this paper, we have systematically compared tidal light curve assuming the luminosity traces the fall- disruption by Schwarzschild black holes in general rela- back accretion rate. tivity and point masses in Newtonian gravity. Differences (5) Tidal streams precess in general relativity, poten- between the two theories have potentially observable tially leading to inelastic collisions between parts of consequences for both TDE rates and the properties of the stream that allow energy to be dissipated and individual events: debris orbits to circularize. At lowest PN order, this (1) Tidal forces are stronger in general relativity than 2=3 precession scales as M• at the threshold βd for full Newtonian gravity for orbits with the same penetra- 2 2 disruption, but it increases more steeply with SBH tion factor β ≡ Lt =L . This implies that the loss mass as βd approaches the threshold for direct cone within which stars are tidally disrupted is larger capture β . in general relativity: L >L ⇒ β < β . Par- cap d d;N d d;N In future work, we plan to explore how SBH spin affects tially disrupted stars with β < β will lose more d all five of these relativistic effects. By breaking the material in relativity than Newtonian gravity, and spherical symmetry of the spacetime, SBH spin increases stars with β < β < β will be fully disrupted in d d;N the parameter space to include spin magnitude, orbital relativity but not Newtonian gravity. inclination, and argument of pericenter. As the tidal forces (2) The width σE of the energy distribution of tidal debris is larger in general relativity than Newtonian along geodesics depend on all these parameters, the gravity for orbits with the same penetration factor β. diffusion equation determining the rate at which stars enter This implies that stars will not only lose more the loss cone will likely need to be solved numerically to material in partial disruptions in relativity, but this account for the higher-dimensional boundary conditions. material will become more tightly bound to the SBH Qualitatively, we expect TDE rates to be biased in favor of and fall back to pericenter more quickly, leading to a stars on retrograde orbits for moderate SBH masses, but higher peak luminosity if the radiative efficiency is this bias should switch towards stars on prograde orbits as fixed. However, stars that are fully disrupted in both the retrograde threshold for disruption falls below that for theories (β > βd;N) will be disrupted higher in the direct capture. Both of these biases will tend to spin down potential well in relativity than Newtonian gravity the SBH, as a larger fraction of the stellar material will be because of the stronger tides. This implies that the accreted in each case from retrograde orbits. TDEs from tidal debris will be less tightly bound despite the stars on prograde orbits however will be more luminous relativistic correction and therefore that the peak because of the more tightly bound prograde innermost fallback accretion rate is lower in relativity than stable circular orbits, leading to a potential observational Newtonian gravity. bias in favor of prograde TDEs. (3) Black holes have event horizons in general rela- SBH spin also affects the relativistic precession of tidal tivity that allow them to directly capture stars. This debris, as has been found in numerical simulations [76]. reduces the TDE rate in relativity compared to At 1.5PN order, spin-dependent pericenter precession Newtonian gravity, since tidal debris captured by opposes (supports) the 1PN pericenter precession con- the horizon cannot emit photons detectable by sidered in this paper on prograde (retrograde) orbits [50], observers. As the threshold Lcap for direct capture imposing a further bias in favor of retrograde TDEs if increases more steeply with SBH mass than the such precession is required for orbit circularization. SBH thresholds Ld and Lpd for full and partial disrup- spin induces precession of the longitude of ascending tions, SBHs with masses greater than Mmax;d ¼ node which can cause tidal streams to precess out of their 2 5 107 1 4 108 . × M⊙ and Mmax;pd ¼ . × M⊙ will no initial orbital planes, inhibiting stream crossings and longer be capable of full and partial disruption, delaying orbit circularization [70,73,77]. Spin-dependent
083001-16 UNIFIED TREATMENT OF TIDAL DISRUPTION BY … PHYSICAL REVIEW D 95, 083001 (2017) solutions of stellar diffusion into the loss cone will allow ACKNOWLEDGMENTS us to calculate the fraction of TDEs experiencing such We would like to thank Roseanne Cheng, Chris delays between disruption and peak fallback accretion. Kochanek, David Merritt, and Nicholas Stone for useful By providing a unified treatment of TDEs in the conversations, and both Tamara Bogdanović and an anony- Schwarzschild spacetime, this paper performs an impor- mous second referee for valuable suggestions. M. K. is tant service in its own right and establishes a beachhead supported by the Alfred P. Sloan Foundation Grant No. FG- for attacking the more ambitious problem of tidal dis- 2015-65299 and NSF Grant No. PHY-1607031. ruption by spinning Kerr black holes.
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