Tidal disruption rate of stars by spinning supermassive black holes

0.46 MB·English

by  Michael Kesden

Tidal disruption rate of stars by spinning supermassive black holes
Michael Kesden1,∗
1Center for Cosmology and Particle Physics, Department of Physics,
New York University, New York, New York 10003
(Dated: August 2011)
A supermassive black hole can disrupt a star when its tidal field exceeds the star’s self-gravity,
andcandirectlycapturestarsthatcrossitseventhorizon. ForblackholeswithmassM (cid:38)107M ,
(cid:12)
tidal disruption of main-sequence stars occurs close enough to the event horizon that a Newtonian
treatment of the tidal field is no longer valid. The fraction of stars that are directly captured is
also no longer negligible. We calculate generically oriented stellar orbits in the Kerr metric, and
evaluate the relativistic tidal tensor at the pericenter for those stars not directly captured by the
black hole. We combine this relativistic analysis with previous calculations of how these orbits are
2
populated to determine tidal-disruption rates for spinning black holes. We find, consistent with
1
previous results, that black-hole spin increases the upper limit on the mass of a black hole capable
0 oftidallydisruptingsolarlikestarsto∼7×108M . Morequantitatively,wefindthatdirectstellar
2 (cid:12)
capture reduces tidal-disruption rates by a factor ∼ 2/3 (1/10) at M (cid:39) 107(108)M . The strong
(cid:12)
n dependenceoftidal-disruptionratesonblack-holespinforM (cid:38)108M impliesthatfuturesurveys
(cid:12)
a like the Large Synoptic Survey Telescope that discover thousands of tidal-disruption events can
J constrain supermassive black-hole spin demographics.

5
2
I. INTRODUCTION within the loss cone have velocities lying in a cone about

] the radial direction. TDE rates are set by the rate at
O
whichstellardiffusionfromotherportionsofphasespace
In 1943, active galactic nuclei (AGN) were discovered
C withemissionlinesDopplerbroadenedtowidths(cid:38)1,000 refills this loss cone. Frank and Rees evaluated stellar
. fluxes into the loss cone at a critical radius r at which
h km/s [1]. Twenty years later, theorists proposed that crit
p these AGN were powered by accretion onto compact ob- stellardiffusionoperatingonareferencetimescaletR[20]
- jects of masses 105 −108M [2]. Such massive objects could refill the loss cone within tdyn. Cohn and Kulsrud
o (cid:12)
[21] provided a more sophisticated treatment of stellar
r cannot support themselves against gravitational collapse
st into supermassive black holes (SBHs) [3]. SBH masses diffusion into the loss cone by numerically integrating
the Fokker-Planck equation in energy-angular momen-
a are tightly correlated with the luminosity [4], mass [5],
[ tum space. More recently, Wang and Merritt [22] have
and velocity dispersion [6] of the spheroidal component

revisedpredictedTDEratesusingmorerealisticgalactic
2 of their host galaxies.
density profiles and the observed relation between SBH
v SBHs primarily grow by accreting gas driven into
mass and host-galaxy velocity dispersion [6].
9 galactic centers by tidal torques during major mergers
2 [7, 8]. However, SBHs can also grow by directly captur- These analyses focused on smaller SBHs for which a
3 ing stars that cross their event horizons or by accreting Newtoniantreatmentoftidalforcesisvalidandthenum-
6
debris from stars passing close enough to be tidally dis- ber of directly captured stars is negligible compared to
.
9 rupted [9]. Such tidal-disruption events (TDEs) could thenumberthataretidallydisrupted. ManasseandMis-
0 also power bright flares of radiation as the stellar de- ner [23] introduced Fermi normal coordinates that are
1
bris is shock heated and subsequently accreted [10, 11]. ideal for a relativistic treatment of the tidal tensor, and
1
: Several potential TDEs have been found in x-ray [12], calculated this tidal tensor for radial geodesics of the
v UV [13], and optical [14] surveys, and tidal debris may Schwarzschild metric for nonspinning SBHs. Marck [24]
Xi also fuel recent blazar activity seen by the Swift satellite generalized this calculation to generically oriented time-
[15–18]. The handful of TDEs found in current surveys like geodesics of the Kerr metric [25] for spinning SBHs.
r
a implies that thousands more may be found each year in Beloborodov et al. [26] used this tidal tensor to calcu-
future surveys by the Large Synoptic Survey Telescope late the relativistic cross sections for tidal disruption for
[14, 19]. A detailed comparison between predicted and a range of initial orbital inclinations with respect to the
observed TDE rates will provide important constraints SBH spin. Ivanov and Chernyakova [27] used a numeri-
on SBHs and the galactic centers in which they reside. callyfastLagrangianmodelofatidallydisruptedstarto
investigate how stellar hydrodynamics affects these rela-
Frank and Rees [9] were among the first to estimate
tivistic cross sections. In this paper, we combine a sim-
TDErates. Theyintroducedtheconceptofa”losscone”
ilar relativistic treatment of tidal disruption and direct
in the stellar phase space that would be depopulated
capture with existing calculations of loss-cone physics to
by tidal disruption within a dynamical time t . Stars
dyn
deriveimprovedpredictionsofTDEratesformassiveand
highly spinning SBHs.
The first step in calculating TDE rates is to estab-
∗Electronicaddress: [email protected] lish criteria for determining when a star is tidally dis-
2
rupted. Most tidally disrupted stars are initially on totidaldisruption. Wethenusethesesimulationstocal-
highly eccentric or unbound orbits characterized by the culate expected TDE rates in Sec. IV. The implications
distance r of their pericenters from the SBH. An order- of our findings are discussed in Sec. V.
of-magnitude estimate of the maximum value of r for
which tidal disruption occurs can be obtained by equat-
ing the differential acceleration GMR /r3 experienced II. TIDAL FIELDS ALONG KERR GEODESICS
∗
by a star of mass m and radius R in the tidal field
∗ ∗
of a black hole of mass M to the star’s self-gravity In Boyer-Lindquist coordinates [34] and units where
Gm∗/R∗2. This implies that a star will be tidally dis- G=c=1, the Kerr metric takes the form
rupted when r < r (cid:39) (M/m )1/3R . A star will be
TD ∗ ∗
directly captured by the SBH when r is less than the ra- (cid:18) 2Mr(cid:19) 4Marsin2θ Σ
ds2 = − 1− dt2− dtdφ+ dr2
dius of the event horizon, which for a nonspinning SBH Σ Σ ∆
isequaltotheSchwarzschildradiusrS =2GM/c2. Since (cid:18) 2Ma2rsin2θ(cid:19)
rTD ∝M1/3 while rS ∝M, the ratio of tidally disrupted +Σdθ2+ r2+a2+ sin2θdφ2(2)
Σ
to directly captured stars will decrease with increasing
SBH mass. Equating these two distances, we find that a
where Σ ≡ r2+a2cos2θ and ∆ ≡ r2−2Mr+a2. This
SBH with mass M greater than
metricis bothstationary(independentof t)and axisym-
c3 (cid:18)R (cid:19)3/2 metric (independent of φ). Massive test particles travel
Mmax (cid:39) m1/2 2G∗ ontimelikegeodesicsoftheKerrmetric. Individualstars
∗ have masses m ∼ M much less than those of SBHs
∗ (cid:12)
(cid:18)m (cid:19)−1/2(cid:18)R (cid:19)3/2 (106M (cid:46)M (cid:46)1010M ),andradiiR ∼R (cid:39)7×1010
= 1.1×108M ∗ ∗ (1) (cid:12) (cid:12) ∗ (cid:12)
(cid:12) M R cm less than the Schwarzschild radius
(cid:12) (cid:12)
(cid:18) (cid:19)
shoulddirectlycapturestarsinsteadoftidallydisrupting r = 2GM =2.95×1011 cm M . (3)
them. S c2 106M
(cid:12)
Our estimate of r assumed that the gravitational
TD
We can therefore consider them to be test particles for
field of the SBH was that of a Newtonian point par-
the purpose of determining their orbits. The position
ticle, which should only be valid for r (cid:29) r . One
TD S
(r,θ,φ) of a star as a function of proper time τ evolves
should be very suspicious of using this estimate at the
according to the equations [35]
event horizon, as we did when deriving M above. In
max
a proper general-relativistic treatment, the spacetime of
(cid:18)dr(cid:19)2
a spinning SBH is described by the Kerr metric [25]. Σ2 = [E(r2+a2)−aL ]2
The Kerr metric is a two-parameter family of solutions dτ z
to Einstein’s equation fully specified by the SBH mass −∆[r2+(L −aE)2+Q] (4a)
z
M and dimensionless spin a/M < 1. Theoretical esti-
(cid:18)dθ(cid:19)2
mates of SBH spins depend sensitively on the extent to Σ2 = Q−L2cot2θ−a2(1−E2)cos2θ(4b)
which SBHs grow by accretion or mergers. SBHs accret- dτ z
ing from a standard thin disk [28] can attain a limiting (cid:18)dφ(cid:19) 2MarE a2L
Σ = L csc2θ+ − z , (4c)
spin a/M√(cid:39) 0.998 [29] after increasing their mass by a dτ z ∆ ∆
factor ∼ 6 [30]. The spins of SBHs formed in mergers
vary greatly depending on whether the spins of the ini- wherethespecificenergyE,angularmomentumL ,and
z
tial binary black holes become aligned with their orbital Carter constant Q are conserved along geodesics.
angular momentum prior to merger [31]. SBH spins can Although Boyer-Lindquist coordinates reduce to flat-
be inferred from observations of iron Kα lines in AGN space spherical coordinates in the limit r → ∞, the
x-rayspectra[32]. Largespinshavebeenmeasured,such nonzero off-diagonal elements of the Kerr metric (2) im-
asa/M =0.989+0.009 intheSeyfert1.2galaxyMCG-06- ply that these coordinate vectors do not constitute an
−0.002
30-15 [33], although reliable estimates are only available orthogonaltetradatfiniter. Thegravitational-fieldgra-
for a small number of systems. dients experienced by freely falling observers are more
Given the large sample of observed TDEs expected in conveniently expressed by projecting them onto an or-
the near future and the wide range of predicted SBH thonormal tetrad λ like that provided by Fermi normal
µ
spins, it is important to determine the extent to which coordinates [23]. This coordinate system can be used
TDEratesdependonSBHspin. Thisistheprimarygoal to specify points in the neighborhood of a central time-
ofthispaper. Thegreaterthespindependence,themore like geodesic, such as that traversed by a star orbiting
tightlyobservedTDEswillbeabletoconstrainthedistri- a Kerr SBH. The timelike member of this tetrad λ is
0
bution of SBH spins. In Sec. II, we review how the tidal the tangent vector along the central geodesic, while the
field is calculated along timelike geodesics of the Kerr spacelikevectorsλ (i=1,2,3)spantheplaneinthetan-
i
metric. In Sec. III, we describe the Monte Carlo simula- gent space othogonal to λ . The point (τ,xi) in Fermi
0
tionsweperformedtodeterminewhichstellarorbitslead normalcoordinatesisreachedbystartingatthelocation
3
of the star at proper time τ and moving a proper dis- where
tance R=(cid:112)(cid:80) (xi)2 along the spacelike geodesic whose
i
tangent vector is (cid:80) xiλ . K ≡ (L −aE)2+Q (8a)
i i z
S ≡ r2+K (8b)
InFerminormalcoordinates,thetime-timecomponent
of the metric can be Taylor expanded as T ≡ K−a2+cos2θ (8c)
Mr
I ≡ (r2−3a2cos2θ) (8d)
1 Σ3
g λµλν =−1−R xixj +... , (5)
µν 0 0 0i0j Macosθ
I ≡ (3r2−a2cos2θ) . (8e)
2 Σ3
where R is the Riemann curvature tensor projected
αβγδ TheangleΨevolvesalongthegeodesictoensurethatλ
1
onto the orthonormal tetrad λ . This implies that the
µ and λ are parallel propagated.
3
tidal potential Φ experienced by a star is
tidal ThefullygeneraltidaltensorofEq.(7)isintimidating,
but we can gain insight by considering the tidal tensor
for equatorial geodesics (θ =π/2,Q=0) whose nonzero
1 1
Φ =− (g λµλν +1)= C xixj +... , (6) elements are
tidal 2 µν 0 0 2 ij
(cid:18) r2+K (cid:19)M
C = 1−3 cos2Ψ (9a)
where C ≡ R is the tidal tensor. Although the 11 r2 r3
ij 0i0j
higher-order corrections to the tidal potential denoted r2+K
C = −3 McosΨsinΨ (9b)
by the ellipsis can sometimes be significant [36], in this 13 r5
paper we consider only the term quadratic in xi. The (cid:18) (cid:19)
K M
tidal tensor C is a symmetric, traceless 3×3 matrix C = 1+3 (9c)
ij 22 r2 r3
whoseeigenvectorsdenotetheprincipalaxesalongwhich
the star is stretched or squeezed, and whose eigenvalues C = (cid:18)1−3r2+K sin2Ψ(cid:19)M . (9d)
denote the extent of this stretching and squeezing. 33 r2 r3
The problem of calculating the tidal potential Φ
tidal The eigenvalues of this tensor are M/r3, (1 +
thus reduces to choosing an appropriate orthonormal
3K/r2)M/r3, and −2(1+3K/2r2)M/r3. Since the tidal
tetrad λ for generic Kerr geodesics and projecting the
µ force is
Riemann tensor onto this tetrad. This has already been
accomplished for us by Marck [24], who found
F =−∇ Φ =−C xj , (10)
i i tidal ij
(cid:18) ST(r2−a2cos2θ) (cid:19) thepositiveeigenvaluescorrespondtodirectionsinwhich
C11 = 1−3 KΣ2 cos2Ψ I1 the star is squeezed while the negative eigenvalues cor-
respond to the direction in which it is streched. In the
+6arcosθ ST cos2ΨI (7a) Newtonian limit K/r2 → 0, the eigenvalues reduce to
KΣ2 2 −2M/r3 and the doubly degenerate eigenvalue M/r3.
C12 = [−arcosθ(S+T)I1 This degeneracy reflects the restoration of symmetry be-
√
ST tween the θ and φ directions at large r, where the effects
+(a2cos2θS−r2T)I2]3KΣ2 cosΨ (7b) oftheSBH’sspinarenegligible. Stretchingoccursinthe
radialdirectioncorrespondingtotheeigenvalue−2M/r3.
C = [(a2cos2θ−r2)I
13 1 Notethatdespiteone’spossibleintuitiontothecontrary,
ST
+2arcosθI ]3 cosΨsinΨ (7c) thetidalforceremainsfiniteatboththeinnermoststable
2 KΣ2 circular orbit and even the event horizon itself.
(cid:18) r2T2−a2cos2θS2(cid:19) To determine whether a star on a given orbit is
C = 1+3 I
22 KΣ2 1 tidallydisrupted,wecheckatthepericenterofthatorbit
ST whether the outward tidal force in the direction corre-
−6arcosθ I (7d)
KΣ2 2 sponding to the negative eigenvalue of the tidal tensor
exceeds the inwards Newtonian self-gravity of the star.
C = [−arcosθ(S+T)I
23 1
√ We assume that the tidal field is maximized at the peri-
ST
+(a2cos2θS−r2T)I ]3 sinΨ (7e) center as in the Newtonian limit. If β− denotes the nu-
2 KΣ2 merical value of this eigenvalue, tidal disruption occurs
(cid:18) ST(r2−a2cos2θ) (cid:19) if
C = 1−3 sin2Ψ I
33 KΣ2 1
+6arcosθKSΣT2 sin2ΨI2 , (7f) r <rTD =(cid:20)(cid:18)M|β/−r|3(cid:19)(cid:18)mM∗(cid:19)(cid:21)1/3R∗ . (11)
4
In the Newtonian limit β = −2M/r3 discussed above,
−
this condition is equivalent to the more familiar expres-
sion
(cid:18)2M(cid:19)1/3
r <r = R . (12)
TD m ∗
∗
Althoughourcondition(11)fortidaldisruptionisonly
approximate, we expect it to be conservative for several
reasons. Itneglectsthatthetidalforcehasalreadyraised
bulges on the star’s surface before the star reaches the
pericenter, so the stellar radius R appearing in Eq. (11)
∗
should exceed its value in hydrostatic equilibrium far
from the SBH. It also assumes that the star is nonro-
tating, while in reality the torques exerted on the tidally
distorted star will cause it to partially corotate with its
orbit. These torques are likely to be complicated for
a generic nonequatorial Kerr geodesic, but we can gain
some insight by again considering the Newtonian limit.
Stars rotating with angular velocity Ω will be disrupted
at a radius
(cid:20)(cid:18) Ω2r3(cid:19)(cid:18)M (cid:19)(cid:21)1/3
r <r (Ω)= 2+ R (13)
TD GM m ∗
∗
FIG. 1: The mass M of the heaviest SBH capable of dis-
inthislimit. Forcorotatingstarsoncircularorbits(Ω2 = rupting a star of solamramxass and radius as a function of SBH
GM/r3),thefirstfactorinparenthesesontheright-hand spin a/M. The red squares show the values listed in Table
sideofEq.(13)equals3asinthedefinitionoftheradius 2 of [27] derived using a simple hydrodynamical model. The
of the Hill’s sphere [37]. For a star corotating at the solid blue curve shows our prediction according to the rela-
pericenter of a parabolic orbit like that expected for a tivisticcriterion(11),whilethedashedblackcurveshowsthe
starapproachingaSBHfromalargedistance,thisfactor Newtonian prediction (12).
equals 4. Our assumption that the star is nonrotating is
conservativebecausethecondition(13)ismostrestrictive
for Ω = 0, although r (Ω) only varies by the modest spin,starsonprograde,equatorial,marginallyboundor-
TD
factor 21/3. bits[39]arethemostlikelytobetidallydisrupted. Using
Ourcriterion(11)mightoverestimatetherateatwhich a simple but computationally inexpensive hydrodynam-
starsarefullydisruptedsincetheymightlosetheirouter ical model, they calculated the mass Mmax of the heav-
layers while maintaining their dense cores. In the New- iest SBH capable of tidally disrupting stars without di-
tonian limit, Phinney [38] showed that stars will not be rectlycapturingthem. AsintheNewtonianpredictionof
fully disrupted until Eq.(1),M ∝m−1/2R3/2. InFig.1,wecomparetheir
max ∗ ∗
predictions(redsquares)toourownusingtherelativistic
(cid:18)k(cid:19)1/6(cid:18)M (cid:19)1/3 criterion (11) (solid blue curve) and Newtonian criterion
r <r = R , (14)
TD f m∗ ∗ (12) (dashed black curve) for stellar mass m∗ =M(cid:12) and
radius R = R . We see that the relativistic correc-
∗ (cid:12)
where k is the constant of apsidal motion and fGm2/R tion to the Newtonian prediction is significant, and that
∗ ∗
is the star’s binding energy. The factor k/f = 0.3(0.02) our simple criterion (11) does a reasonable job given the
forstarswithconvective(radiative)atmospheres,butthe ∼50% uncertainty in the simulations [27].
exponent of 1/6 ensures that r is only weakly depen- We see that in the maximally spinning limit (a/M →
TD
dent on this factor. We ignore this factor and keep our 1), a SBH as massive as ∼ 109M is capable of tidally
(cid:12)
criterion (11) for the remainder of this paper, but the disruptingmain-sequencestars. Thispredictionisconsis-
Monte Carlo simulations described in the next section tent with earlier simulations [40, 41] that demonstrated
couldeasilybeevaluatedwithanewcriterionthatincor- this possibility. The above scaling of M with stel-
max
porates this factor or a different choice of stellar proper- lar mass and radius suggests that a white dwarf with
ties than m =M , R =R . m (cid:39) M ,R (cid:39) 0.01R could be tidally disrupted by
∗ (cid:12) ∗ (cid:12) ∗ (cid:12) ∗ (cid:12)
A detailed study of the fraction of mass loss as a func- a maximally spinning SBH as massive as 106M . This
(cid:12)
tion of SBH and orbital parameters is beyond the scope conclusionhelpsalleviatetensionbetweenthesmallSBH
of this paper, but such a study for selected orbital incli- massrequiredfortheinterpretationofSwiftJ1644+57as
nations has been performed by Ivanov and Chernyakova a white-dwarf tidal disruption [42] and the larger value
[27]. They recognized that for a given SBH mass and of M inferred from the relation between SBH mass and
5
the equations of motion (4) with the star located at an
initial position (r,θ,φ) in Boyer-Lindquist coordinates.
Sincetheseequationsareindependentofφasisthetidal
tensorC ,wedonotactuallyneedtointegrateEq.(4c).
ij
We choose an initial radius r =2000M, where relativisic
corrections are small, and check that our results are in-
sensitive to this choice. In this limit, the constants of
motion are given by
L = rvsinθsinΘsinΦ (15a)
z
Q = L2 +L2
x y
= (rvsinΘ)2(cos2Φ+cos2θsin2Φ) , (15b)
wherev =(2M/r)1/2 isthemagnitudeoftheinitialstel-
lar velocity and Θ and Φ are the angles described in
Fig. 2. Since the stars at r (cid:29) r do not know about
TD
the direction of the SBH spin, the stellar distribution
is axisymmetric about ˆr and there is a uniform distri-
bution in Φ. Although astrophysical spheroids do not
necessarily have isotropic velocity dispersions at large r,
thestarsapproachingthisclosetotheSBHbelongtothe
tinyfractionofthedistributionwherethevelocityliesin
a loss cone centered about the radial direction [9]. Since
there is no reason to expect the distribution function to
FIG. 2: Our choice of coordinates for determining the ini-
be varying strongly in this small portion of phase space,
tialconditionsforintegratingtheequationsofmotion(4)for
thereisauniformdistributionin−1≤cosΘ≤1aswell.
stellar orbits. The SBH is located at the origin, and the star
is located at Boyer-Lindquist coordinates (r,θ,φ). Θ is the However,sincetherateatwhichstarsofvelocityv enter
angle between the stellar velocity v and the inwards radial the sphere of radius r is proportional to v·ˆr, we weight
direction −ˆr, while Φ is the angle between the component of our distribution by cosΘ during our Monte Carlo sim-
vperpendiculartoˆrandtheunitvectorˆe intheθdirection. ulations of stellar orbits. We choose a maximum value
θ
Θ toavoidwastingcomputationaltimeonorbitsthat
max
do not closely approach the SBH.
host-galaxy velocity dispersion [6]. With this choice of initial conditions, we integrate the
equationsofmotion(4)untilthestarreachesthepericen-
ter. We then calculate and tabulate the negative eigen-
III. MONTE CARLO SIMULATIONS value β of the tidal tensor C (7). We also tabulate
− ij
which stars are directly captured by the SBH when their
Unlike the Newtonian two-body problem, there is no orbitsencountertheSBH’seventhorizon. Weintegrated
general analytic solution to the relativistic equations of 250,000 stellar orbits for each of several SBH spins, with
motion(4). Wemustintegratetheseequationsexplicitly an additional 250,000 with a smaller choice of Θmax to
for every orbit we consider. Stars that will eventually be increase our sampling of the small number of orbits that
tidally disrupted are scattered onto their final orbits at lead to tidal disruption when M →Mmax.
radii r (cid:29) r . These orbits may or may not be grav-
TD
itationally bound to the SBH, but their Newtonian or-
bital energies ∼ m σ2, where σ is a typical velocity at IV. TDE RATES
∗
r (cid:29)r , are much less than the rest-mass energy m c2.
TD ∗
It is therefore an excellent approximation to set the spe- Given a stellar phase-space distribution function, it is
cific energy E appearing in Eqs. (4) equal to unity in reasonablystraightforwardtocalculatetherateofTDEs
units where c=1. The Kerr metric (2) is axisymmetric, using the Monte Carlo simulations described in the pre-
soourresultsareindependentoftheinitialvalueofφ. We vious section. In Sec. IVA below, we calculate the TDE
mustperformMonteCarlosimulationswithanappropri- rate as a function of SBH mass assuming that the stars
ate distribution of the remaining variables {r,θ,L ,Q} approach a Maxwellian distribution with fixed number
z
todeterminewhatfractionoforbitsaretidallydisrupted density and velocity dispersion far from the SBH. This
accordingtoourrelativisticcriterion(11),wheretheneg- calculation illustrates the dependence of TDE rates on
ativeeigenvalueβ ofthetidaltensorC dependsonall SBH spin. However, astrophysical SBHs reside in galaxy
− ij
these variables. spheroids whose properties are tightly correlated with
We illustrate the geometry of the problem and our SBH mass [4–6]. In addition, the very luminous early-
choice of coordinates in Fig. 2. We begin integrating typegalaxiesthathostthemostmassiveSBHshavecored
6
profilesattheircentersunlikethepower-lawprofilesthat the “Newtonian” prediction would be that a star is di-
characterize less luminous early-type galaxies and late- rectly captured by the SBH if the pericenter of its or-
type bulges [43]. Predicted TDE rates are sensitive to bit is less than the Schwarzschild radius (3). The peri-
whethergalacticcentersaredescribedbycoredorpower- center of a parabolic (E = 0) orbit with specific an-
N
law profiles [22]. Recent observations [44] suggest that gular momentum L is L2 /2GM. Equating this to
N N
even the nuclear star cluster at our own Galactic center, the Schwarzschild radius, a star is directly captured if
long believed to have a cuspy profile (ρ ∝ r−7/4) [45], L ≤ L ≡ 2GM/c, which according to Eq. (20) im-
N cap
may in fact have a core of radius r (cid:39) 0.5 pc [46]. plies a capture rate
core
Given these uncertainties, it is difficult to make precise
estimates of astrophysical TDE rates. Despite this, in (cid:90) Lcap ∂Γ (32π)1/2n(GM)2
Γ = dL =
Sec. IVB we calculate the TDE rate assuming galaxies cap ∂L N σc2
0 N
have isothermal (ρ ∝ r−2) profiles at their centers and (cid:18) M (cid:19)2(cid:18) n (cid:19)
hostSBHswithmassescorrelatedwiththeirvelocitydis- = 2.1×10−6yr−1
106M 105pc−3
persions. The results of this calculation shown in Fig. 4 (cid:12)
illustrate how SBH spin affects TDE rates. (cid:18) σ (cid:19)−1
× (21)
100 km/s
A. Maxwellian distribution A star will be tidally disrupted if the pericenter of its
orbit is less than the tidal-disruption radius r (12),
TD
which implies an angular momentum
AssumethatstarsfarfromtheSBHhaveaMaxwellian
distribution function
(cid:18)(2M)4/3GR (cid:19)1/2
f(r,v)= (2πσn2)3/2e−v2/2σ2 (16) LN ≤LTD ≡ m1∗/3 ∗ . (22)
This implies a TDE rate
with number density n and velocity dispersion σ. The
differential rate at which stars with Newtonian specific
(cid:90) LTD ∂Γ (8π)1/2nGMR (cid:18)2M(cid:19)1/3
energyEN andangularmomentumLN enterasphereof Γ(cid:48)TD = ∂L dLN = σ ∗ m
radius r is given by 0 N ∗
(cid:18) M (cid:19)4/3(cid:18) n (cid:19)
∂2Γ (cid:90) = 6.3×10−5yr−1
=4πr2 d3vv δ(E(cid:48)−E )δ(L(cid:48)−L )f(r,v), 106M 105pc−3
∂E ∂L z N N (cid:12)
N N (17) (cid:18) σ (cid:19)−1
× (23)
wherethevolumeintegralextendsovertheregionv >0 100 km/s
z
and E(cid:48) and L(cid:48) are given by
ThisrateagreeswiththatinEq.(16b)ofFrankandRees
1 [9] which applies when the critical radius at which the
E(cid:48) = v2 , (18a)
2 loss cone refills on a dynamical time exceeds the SBH’s
L(cid:48) = |r×v| . (18b) radius of influence. If TDEs can only be observed when
the tidal debris is not directly captured by the SBH, the
We can use Eq. (16) and the delta functions to evaluate observed TDE rate will be Γ = Γ(cid:48) − Γ . Since
TD TD cap
the integral to find Γ(cid:48) ∝M4/3 while Γ ∝M2, the TDE rate will vanish
TD cap
for M ≥M (1) at which r =r .
max TD s
∂2Γ = (8π)1/2nLNe−EN/σ2 . (19) We can use this same differential rate ∂Γ/∂LN (20) to
∂E ∂L σ3 calculate the relativistic direct-capture and TDE rates.
N N
However, we must now rely on the Monte Carlo simula-
If σ (cid:28) c, orbits near the SBH will be insensitive to the tions of Sec. III to determine whether stars are directly
value of EN and we can integrate over this variable to capturedortidallydisrupted,insteadofthesimpleNew-
yield a differential rate tonian expressions for L and L given above. The
cap TD
simulated orbits have a maximum angular momentum
∂Γ =(cid:90) ∞ ∂2Γ dE = (8π)1/2nLN . (20) Lmax ≡ (2GMr)1/2sinΘmax. The total rate at which
∂L ∂E ∂L N σ stars on these orbits enter a sphere of radius r =2000M
N 0 N N
is
The divergence of this rate as σ → 0 results from gravi-
tational focusing, which would channel all stars into the (cid:90) Lmax ∂Γ (8π)1/2nGMr
Γ = dL = sin2Θ .
SBH in the absence of tangential velocities. tot ∂L N σ max
0 N
Before proceeding to the relativistic calculation, let (24)
us review the Newtonian predictions. Although the The rate Γ at which stars are directly captured by
cap
event horizon is fundamentally a relativistic concept, the SBH is found by multiplying this total rate Γ by
tot
7
mildly dependent on spin, but we do not see any obvi-
ous reason for this to be the case. Young et al. [48]
calculated the ratio of the capture rate Γ (a) for Kerr
cap
SBHs of spin a to the capture rate Γ (0) for nonspin-
cap
ningSBHs. Equation(B2)oftheirpapershowsthatthis
ratio is approximately given by
Γ (a) (cid:16) a (cid:17)2 (cid:16) a (cid:17)4 (cid:16) a (cid:17)6
cap =1−0.0820 +0.0717 −0.0864 .
Γ (0) M M M
cap
(25)
The small numerical values of the coefficients in this
expression indicate the weak dependence of the direct-
capture rate on SBH spin; the ratio is between 0.9 and
unity over the full range of spins 0≤a/M ≤1.
The TDE rate Γ exhibits a much stronger depen-
TD
dence on SBH spin, as illustrated by the strongly vary-
ing solid colored curves in Fig. 3. At small SBH masses,
r (cid:29)M andtheTDErateforallspinsconvergestothe
TD
Newtonian result as expected. However, as M increases,
tidaldisruptionoccursclosertotheSBHwheretheNew-
tonianapproximationisincreasinglyinvalid. Thisismost
glaringly apparent for masses M (cid:38) M of Eq. (1) for
max
whichtidaldisruptionwouldnotbepossibleintheNew-
tonian limit. The true maximum mass, where the solid
FIG.3: Theratesatwhichstarsaredirectlycaptured(dotted colored curves in Fig. 3 intersect ΓTD = 0, is given as
lines)andtidallydisrupted(solidanddashedcurves)bySBHs a function of spin in Fig. 1. Since Γ(cid:48) ∝ M4/3, these
TD
of mass M in constant-density cores with n = 105 pc−3 and massiveSBHsarecapableoftidallydisruptingevenmore
σ=100km/s. TheblackcurvesshowtheNewtonianratesof stars than their less massive counterparts. Although the
Eqs.(21)and(23),whilethecoloredcurvesshowtherelativis- spins a/M = 0.99 and 0.999 depicted by the blue and
ticratesforSBHswithspinsa/M =0(red),0.5(orange),0.9
purplecurvesinFig.3mayseemextreme,thesimplesce-
(green), 0.99 (blue), and 0.999 (purple). The capture rates
nario of growing a SBH from a standard thin accretion
mildly decrease with SBH spin, while for M ≥ 108M the
(cid:12) disk leads to a limiting spin a/m (cid:39) 0.998 quite close to
TDE rates greatly increase with SBH spin.
the purple curve [29]. Although uncertain, cosmological
predictionsforSBHspindistributionscanalsobepeaked
near these large values [31]. The primary conclusions to
the fraction F of simulated geodesics that cross the
cap drawfromouranalysisarethatrelativisticcorrectionsto
event horizon. The TDE rate Γ is similarly found by
TD the TDE rate can alter predictions by a factor of several
multiplying Γtot by the fraction FTD of orbits that vi- forM (cid:38)107M , andthattheycanallowTDEstooccur
(cid:12)
olate the relativistic criterion (11) for tidal disruption.
for SBHs as large as ∼109M .
(cid:12)
If r and Θ are chosen large enough, these fractions
max
F ∝ (rsin2Θ )−1 so that the physical rates are inde-
max
pendent of our choice of initial conditions.
B. Real galaxies
In Fig. 3, we show the direct-capture rate Γ and
cap
TDE rate Γ as functions of SBH mass M for our fidu-
TD Following Frank and Rees [9], the rates we calculated
cial choices n = 105 pc−3 and σ = 100 km/s. The New-
in the preceding subsection assumed that galaxies had
tonian prediction for Γ underestimates the true rela-
cap constant-density cores outside the SBH’s radius of influ-
tivistic capture rate by about a factor of 4. If we had
ence,
used the true specific angular momentum L = 4GM/c
z
for marginally bound geodesics of a Schwarzschild SBH GM (cid:18) M (cid:19)(cid:18) σ (cid:19)−2
as the upper limit of the integral in our Newtonian pre- rh ≡ σ2 =0.43 pc 106M 100 km/s . (26)
(cid:12)
diction(21),wecouldhavenearlyreproducedthecorrect
relativisticresult. Thecapturerateisnearlyindependent Real galaxies with either power-law or core profiles have
of SBH spin as indicated by the colored dotted lines ly- mass-density profiles ρ(r) that monotonically decrease
ing almost on top of each other. This is surprising, since with radius. This raises the question of what is the ap-
the specific angular momentum L for prograde (retro- propriate number density n to insert in our expressions
z
grade) marginally bound equatorial orbits varies from for direct-capture and TDE rates. Frank and Rees [9]
4M (−4M) to 2M (−4.828M) as a/M increases from argued that the appropriate density to use is that at the
0 to 1. Near perfect cancellation over orbital orientation criticalradiusr atwhichstellardiffusioncanrefillthe
crit
(θ,Θ,Φ) must occur for the capture rate Γ to be so lossconeoftidallydisruptedorbitsonadynamicaltime.
cap
8
A very crude estimate of this density can be made by
assuming that r (cid:39)r , an assumption roughly true for
crit h
real galaxies as indicated by Fig. 6 of Wang and Mer-
ritt [22]. If we further assume that the density profile of
galactic centers is that of a single isothermal sphere,
σ2
ρ(r)= , (27)
2πGr2
then inserting n=ρ(r )/m into Eq. (23) implies
h ∗
(cid:18) M (cid:19)−2/3(cid:18) σ (cid:19)5
Γ(cid:48) (cid:39)1.3×10−3yr−1 .
TD 106M 100 km/s
(cid:12)
(28)
WangandMerritt[22]usetheisotropicdistributionfunc-
tion appropriate for a single isothermal sphere to calcu-
late the true rate at which the loss cone is refilled by
stellar diffusion. They find that their results are well
approximated by the fit
(cid:18) M (cid:19)−1(cid:18) σ (cid:19)7/2
Γ(cid:48) (cid:39)2.5×10−3yr−1 .
TD 106M 100 km/s
(cid:12)
(29)
If we combine this estimate with a recent determination
oftherelationbetweenSBHmassandhost-galaxyveloc-
FIG. 4: The rates at which stars are tidally disrupted by
ity dispersion [47],
SBHs of mass M in power-law galaxies obeying the M −σ
M (cid:18) σ (cid:19)4.32 relation. ThedashedblacklineisthepredictionofWangand
=7.58 , (30) Merritt [22] for Γ(cid:48)TD with an updated M −σ relation. The
106M(cid:12) 100 km/s colored curves show our relativistic corrections ΓTD to this
prediction. The TDE rate increases with SBH spin, with the
we arrive at a final TDE rate of
given curves corresponding to a/M = 0 (red), 0.5 (orange),
(cid:18) M (cid:19)−0.19 0.9 (green), 0.99 (blue), and 0.999 (purple).
Γ(cid:48) (cid:39)4.8×10−4yr−1 (31)
TD 106M
(cid:12)
in the Newtonian limit. This estimate should be reason- V. DISCUSSION
able for the power-law galaxies that dominate the total
TDE rate; the core galaxies that host the most massive Astronomers have sought to observe the electromag-
SBHs have TDE rates Γ(cid:48) (cid:39)10−5 yr−1 about an order netic flares associated with TDEs ever since this possi-
TD
of magnitude below that of comparable-mass power-law bilitywasproposedbyRees[10]. SeveralpotentialTDEs
galaxies [22]. were discovered over the past 15 years by the Roentgen
In Fig. 4, we show how the direct capture of stars by Satellite (ROSAT) [12] and the Galaxy Evolution Ex-
spinning SBHs changes this prediction. This figure was plorer [13], and the recent discovery of additional TDEs
prepared with the same set of Monte Carlo simulations byboththeSloanDigitalSkySurvey[14]andSwift[15–
describedinSec.III.Althoughthereareconsiderabledif- 18] has renewed interest in this phenomenon. While in-
ferences between the Newtonian predictions of Eqs. (23) dividualTDEsmayprovidenewinsightsintoSBHaccre-
and (31), these differences result from different treat- tion physics, the large samples that may soon be avail-
ments of the stellar populations far from the SBH. We able[14]willuniquelyprobethewholepopulationofboth
may therefore simply renormalize our relativistic predic- active and quiescent SBHs. While overall TDE rates de-
tions Γ = F Γ of the previous subsection by di- pendonstellarpopulationsatgalacticcenters,theupper
TD TD tot
viding by Eq. (23) and multiplying by Eq. (31) at each boundonthemassM ofSBHscapableoftidaldisruption
SBHmassM. DirectcapturereducesthepredictedTDE is a sensitive measure of SBH spins. For M (cid:38) 107M ,
(cid:12)
rate by a factor ∼2/3 (1/10) at M =107 (108)M . Al- tidal disruption occurs deep enough in the SBHs poten-
(cid:12)
though TDEs are very rare for large SBH masses, they tial well that Newtonian gravity is no longer valid. Fur-
are still possible for M < M (cid:39) 109M . Since SBHs thermore, there is no reason to expect the orbital angu-
max (cid:12)
with masses M (cid:39)109M predominantly live in galaxies larmomentaoftidallydisruptedstarstoalignwithSBH
(cid:12)
with cored profiles, Fig. 4 may somewhat underestimate spins. For both these reasons, accurate calculations of
TDE rates at these masses since the stellar density will TDEratesrequireevaluationoftherelativistictidalten-
not fall as steeply with r as the single isothermal profile sorC onarepresentativesampleofgenericallyoriented
ij
of Eq. (27). Kerr geodesics.
9
We have performed a series of Monte Carlo simula- spinning (a/M (cid:38) 0.9) SBHs will be able to produce ob-
tions that provide this required sample. We use this servable TDEs. Theory [31] and observation [32, 33]
sample to calculate TDE rates for spinning SBH as a suggest that most SBHs may have such large spins, but
function of their mass M, both in constant-density cores further observations are needed to investigate this pos-
and in isothermal spheres that approximate real power- sibility. A future survey like the Large Synoptic Survey
law galaxies. We find that for M (cid:38)107M , a significant Telescope[19]thatfindsthousandsofTDEsmayprovide
(cid:12)
fraction of stars will be directly captured by the SBH’s important constraints on the distribution of SBH spins.
event horizoninstead ofbeingtidallydisruptedand sub-
sequently accreted. This will reduce the observed TDE Acknowledgements. We would like to thank Mike
rate assuming that directly captured stellar debris will Blanton, Glennys Farrar, Andrei Gruzinov, David Mer-
not have the chance to radiate appreciably before being ritt,MaryamModjaz,SterlPhinney,andScottTremaine
swallowed by the SBH. Above M (cid:39)108M , only highly for useful conversations.
(cid:12)
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