Tidal disruption flares from stars on eccentric orbits
aa r X i v : . [ a s t r o - ph . H E ] O c t Tidal disruption flares from stars on eccentric orbits
Kimitake Hayasaki , , , a , Nicholas Stone , and Abraham Loeb Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA, 02138, USA Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Ko-rea
Abstract.
We study tidal disruption and subsequent mass fallback for stars approachingsupermassive black holes on bound orbits, by performing three dimensional SmoothedParticle Hydrodynamics simulations with a pseudo-Newtonian potential. We find that themass fallback rate decays with the expected -5 / ff ers from the -5 / There is substantial evidence that galactic nuclei harbor supermassive black holes (SMBHs), the ma-jority of which are quiescent and not active galactic nuclei. The tidal disruption of a star by a SMBH,and subsequent flaring activity, provides a rare observational diagnosis for the large population of qui-escent SMBHs. These powerful flares are expected to have a luminosity at least comparable to theEddington luminosity [1,2].The standard picture of a tidal disruption event (TDE) involves a star at large separation fallinginto a massive black hole on an almost parabolic orbit. After the star is tidally disrupted by the SMBH,half the stellar debris becomes gravitationally bound to the SMBH as it loses orbital energy inside thetidal radius. The bound debris finally falls back and accretes onto the black hole. Kepler’s third lawimplies that the accretion rate decays with the -5 / ffi ciently small to make detailed test-ing of theoretical models di ffi cult. Observations suggest that the TDE rate is ∼ − yr − per galaxy [5].This observed rate is in rough agreement with theoretical rate estimates based on two-body scatteringat ∼ pc scales, which motivates the assumption of nearly parabolic orbits [6].However, recent theoretical studies on rates of tidal separation of binary stars by SMBHs suggestthat a significant fraction of tidal disruption flares may occur from stars approaching the black hole onsomewhat eccentric orbits, significantly less parabolic than in the standard picture [7]. Other sourcesof TDEs from stars on more eccentric orbits include binary SMBH systems and recoiling SMBHs [8].These latter two sources are capable of producing TDEs with even lower values of orbital eccentricitythan in the binary separation scenario, and motivate our work here. In this paper, we explore throughhydrodynamical simulations how mass fallback rates in TDEs vary between the canonical, paraboliccase and the underexplored eccentric scenario. a e-mail: [email protected]; [email protected] PJ Web of Conferences
We describe here procedures for numerically modeling the tidal disruption of stars on bound orbits.The simulations presented below were performed with a three-dimensional (3D) Smoothed ParticleHydrodynamics (SPH) code, which is based on a version originally developed by Ref. [9]. We modelthe initial star as a polytropic gas sphere in hydrostatic equilibrium. The tidal disruption process isthen simulated by setting the star in motion through the gravitational field of an SMBH.A star is tidally disrupted when the tidal force of the black hole acting on the star is stronger thanthe star’s self-gravity. The radius where these two forces balance is defined as the tidal disruptionradius r t = M BH m ∗ ! / r ∗ , (1)where M BH is the black hole mass and m ∗ is the stellar mass. The star-black hole system is put on the x - y plane, where both axes are normalized by r t and the black hole is put at the origin of the system. Theinitial position of the star is given by r = ( r cos φ , r sin φ , r = r t is the radial distancefrom the black hole and φ shows the angle between x -axis and r . In our simulations, the black holeis represented by a sink particle with the appropriate gravitational mass M BH . All gas particles that fallwithin a specified accretion radius are accreted by the sink particle. We set the accretion radius of theblack hole as equal to the Schwarzshild radius r S = GM BH / c , with c being the speed of light.In order to treat approximately the relativistic precession of a test particle in the Schwarzschildmetric, we incorporate into our SPH code the following pseudo-Newtonian potential [10]: U ( r ) = − GM BH r " c + − c − c ( r S / r ) + c r S r , (2)where we adopt c = ( − / + √ c = (4 √ − c = ( − / √ − c = c = c = ff ects such as the black hole spin or gravitational wave emission.We have performed five simulations of tidal disruption events with di ff erent parameters. The com-mon parameters through all of simulations are following: m ∗ = M ⊙ , r ∗ = R ⊙ , M BH = M ⊙ , φ = − . π , and γ = /
3. The total number of SPH particles used in each simulation is 10 , and the termi-nation time of each simulation is 4 Ω − ∗ , where Ω − ∗ ≡ p r ∗ / Gm ∗ ≃ . × − ( r ∗ / R ⊙ ) / ( M ⊙ / m ∗ ) / yr.We also adopt standard SPH artificial viscosity parameters α SPH = β SPH =
2. Table 1 summa-rizes each model, where the penetration factor β represents the ratio of the tidal disruption radius topericenter distance, r p . As an approaching star enters into the tidal disruption radius, its fluid elements become dominated bythe tidal force of the black hole, while their own self-gravity and pressure forces become relativelynegligible. The tidal force then produces a spread in specific energy of the stellar debris ∆ǫ ≈ GM BH r ∗ r . (3)The total mass of the stellar debris is defined with the di ff erential mass distribution m ( ǫ ) ≡ dM ( ǫ ) / d ǫ ,where M ( ǫ ) ≡ R ∞−∞ m ( ǫ ′ ) d ǫ ′ . When a star is disrupted from a parabolic orbit, m ( ǫ ) will be centered onzero and distributed over − ∆ǫ ≤ ǫ ≤ ∆ǫ .Since the stellar debris with negative specific energy is bound to the SMBH, it returns to pericenterand will eventually accrete onto the black hole. If we define its binding energy, ǫ = − GM BH / a (the semi-major axis of the stellar debris is a ), then the mass fallback rate is given by [3] dM / dt = ( dM ( ǫ ) / d ǫ ) | d ǫ/ dt | ( ǫ < d ǫ/ dt = − (1 / π GM BH ) / t − / . This is derived from Kepler’sthird law.idal Disruption events and AGN outbursts workshop Table 1.
The first column shows each simulated scenario. The second, third, and fourth columns are the penetra-tion factor β = r p / r T , the initial orbital eccentricity of star-black hole system e ∗ , and its initial semi-major axis a ∗ ,respectively. The last column describes the remark for each model.Model β = r t / r p e ∗ a ∗ [ r t ] Remarks1 1 1 . − Newtonian2 1 1 . − Pseudo-Newtonian3 5 1 . − Pseudo-Newtonian4 1 0 .
98 50 . .
98 10 . Fig. 1. Di ff erential mass distributions over specific energy of stellar debris and their corresponding mass-fall backrates. While the di ff erential mass distribution is shown in panel (a), the mass fallback rate is shown in panel (b).In both panels, the dot-dashed line (green), dotted line (blue), dashed line (red), dashed-three-dotted line (black),and solid line (black) represent the mass distributions and corresponding mass fallback rates of Model 1, Model 2,Model 3, Model 4 and Model 5, respectively. The energy is measured in units of ∆ǫ given by equation (3). The specific orbital energy of a star on an eccentric orbit is given by ǫ orb ≈ − GM BH a ∗ = − GM BH r t β (1 − e ∗ ) , (4)where a ∗ and e ∗ are the initial semi-major axis of the star-black hole system and its initial orbitaleccentricity, respectively. This quantity is less than zero because of the finite value of a ∗ , in contrast tothe standard, parabolic orbit of a star. If ǫ orb is less than ∆ǫ , all the stellar debris should be bounded bythe black hole, even after the tidal disruption. The condition ǫ orb = ∆ǫ therefore gives a critical valueof orbital eccentricity of the star e crit ≈ − β m ∗ M BH ! / , (5)below which all the stellar debris should remain gravitationally bound to the black hole. The criticaleccentricity is evaluated to be e crit = .
98 for Model 4, whereas e crit = .
996 for Model 5. For the ec-centric TDEs, the orbital period of the most tightly bound orbit, t min , and the orbital period of the mostloosely bound orbit, t max , are obtained by using Kepler’s third law with ǫ = ∆ǫ ± ǫ orb and equation (4).The duration time of mass fallback for eccentric TDEs with e ∗ < e crit is thus predicted to be finite andcan be written by ∆ t = t max − t min = ( π/ √ Ω − ∗ / [ β (1 − e ∗ )] / ) (cid:18)h / − (1 /β (1 − e ∗ )( m ∗ / M BH ) / i − / − (cid:19) . Evaluating this gives ∆ t ≈ Ω − ∗ for Model 5, whereas ∆ t → ∞ for Model 4 in spite of smaller t min than that of Models 1-3.Figure 1 show di ff erential mass distributions and their corresponding mass fallback rates in Mod-els 1-5. While the di ff erential mass distribution is shown in panel (a), the mass fallback rate is shown inPJ Web of Conferencespanel (b). In panel (b), the horizontal solid line denotes the Eddington rate: ˙ M Edd = (1 /η )( L Edd / c ) ≃ . × − ( η/ . − ( M BH / M ⊙ ) M ⊙ yr − , where L Edd = π GM BH m p c /σ T is the Eddington luminos-ity with m p and σ T denoting the proton mass and Thomson scattering cross section, respectively, and η is the mass-to-energy conversion e ffi ciency, which is set to 0 . ∆ǫ before and after the tidal disruption. Thecorresponding mass fallback rates are proportional to t − / . The slight deviation from time to the − / ∆ǫ and the central peak rising from 0 . ∆ǫ to 0 (see also[11]). Simulations of Models 2 and 3 have performed with the pseudo-Newtonian potential given byequation (2). Model 3 has the same simulation parameters as Model 2 except for β =
5. Since thepotential is deeper as β is higher, the re-congregation of the mass due to the self-gravity of the stellardebris is prevented. This leads to the mildly-sloped mass distribution, and therefore the peak of themass fallback rate also smooths.The mass is not distributed around zero but around − ∆ǫ in Model 4, and around − ∆ǫ in Model 5.This is because the specific energy of initial stellar orbit is originally negative (see equation 4). Clearly,most of mass in Model 4 is bounded by the negative shift of the center. The resultant energy spread isslightly larger than we analytically expected. This suggests that the critical eccentricity is smaller thanthe value in equation (5). In Model 5, all of mass is bounded and falls back to the black hole in a muchshorter time than that of Models 1-3. As shown in panel (b), the mass fallback rate of Model 5 is fourorders of magnitude greater than the Eddington rate. We have performed 3D SPH simulations of tidal disruption processes for stars on bound orbits. Ourmain conclusions are summarized as follows:1. There is a critical orbital eccentricity below which all stellar debris falls back to the black hole.The simulated critical eccentricity is slightly lower than expected from our analytical prediction.2. In an eccentric TDE with orbital eccentricity below the critical eccentricity, all the stellar debrisfalls back to the black hole in a much shorter time than that of the standard TDE. The resultantmass fallback rate substantially exceeds the Eddington rate and di ff ers from the -5 / References
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