Tidal Disruption Event (TDE) Demographics

0.64 MB·

by  C. S. Kochanek

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed27January2016 (MNLATEXstylefilev2.2)
Tidal Disruption Event (TDE) Demographics
C. S. Kochanek1,2
1 Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH 43210
2 Centerfor Cosmology and AstroParticle Physics, The Ohio State University,191 W. Woodruff Avenue, Columbus OH 43210
27January2016
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0
ABSTRACT
2
We surveythe propertiesofstarsdestroyedinTDEsasa functionofBHmass,stellar

n mass and evolutionary state, star formation history and redshift. For MBH<∼107M⊙,
a the typical TDE is due to a M∗∼0.3M⊙ M-dwarf, although the mass function is
J relativelyflatfor M∗<∼M⊙.The contributionfromoldermainsequencestarsandsub-
5 giantsissmallbutnotnegligible.FromMBH≃107.5-108.5M⊙,thebalancerapidlyshifts
2
to higher mass stars and a larger contribution from evolved stars, and is ultimately

dominatedbyevolvedstarsathigherBHmasses.Thestarformationhistoryhaslittle
]
E effect until the rates are dominated by evolved stars. TDE rates should decline very
H rapidlytowardshigherredshifts.ThevolumetricrateofTDEsisveryhighbecausethe
BH mass function diverges for low masses. However, any emission mechanism which
.
h is largely Eddington-limited for low BH masses suppresses this divergence in any
p observed sample and leads to TDE samples dominated by MBH≃106.0-107.5M⊙ BHs
- with roughly Eddington peak accretion rates. The typical fall back time is relatively
o
long,with16%havingt <10−1 years(37days),and84%havinglongertime scales.
r fb
t Many residualrate discrepanciescanbe explained if surveysare biasedagainstTDEs
s
a withtheselongertfb,whichseemsveryplausibleiftfbhasanyrelationtothetransient
[ risetime.ForalmostanyBHmassfunction,systematicsearchesforfainter,fastertime

scale TDEs in smaller galaxies,and longer time scale TDEs in more massive galaxies
1
are likely to be rewarded.
v
7
Key words: stars: black holes – quasars: supermassive blackholes
8
7
6
0
1. 1 INTRODUCTION 1978, Magorrian & Tremaine 1999, Wang& Merritt
0 2004, Merritt & Wang 2005, Brockamp et al. 2011,
6 Tidal Disruption Events (TDEs) occur when a star passes Vasiliev & Merritt2013).Thesestudiesgenerallyconsidered
1 sufficiently close to a supermassive black hole for the only main sequence stars with a common mass and struc-
: tidal fields to destroy (or severely maim) the star (Hills ture, although Magorrian & Tremaine (1999) discusses the
v
1975, Lacy et al. 1982, Carter & Luminet 1983, Rees 1988, effect of a mass function on the rates, while Syer& Ulmer
i
X Evans& Kochanek 1989). In the last ∼ 10 years, signifi- (1999) and MacLeod et al. (2012) considered the relative
r cant numbers of TDEs have begun to be discovered (see, rates for main sequence and evolved stars at a fixed mass
a e.g., the reviews by Gezari 2012 and Komossa 2015). The but not general populations. Strubbe& Quataert (2009)
first candidates were mostly found as either X-ray or UV used an evolving black hole mass function and a model for
flares in archival data(see the summary in Komossa 2015). theobservationalpropertiesofTDEstoestimateratesfora
More recently, large scale transient surveys like ASAS- range of observational surveys. Mageshwaran & Mangalam
SN (Shappeeet al. 2014), PTF (Law et al. 2009) and Pan- (2015) and Stone & Metzger (2016) considered a pop-
STARRS(Kaiser et al.2002)havefoundincreasingnumbers ulation of main sequence stars mainly following the
of TDEs in real time, allowing more detailed photometric Kroupa (2001) initial mass function (IMF) truncated at
andspectroscopicfollowupstudies(e.g.,Holoien et al.2014, M⊙ to mimic an old stellar population but otherwise
Holoien et al. 2016, van Velzen et al. 2011, Gezari 2012, ignored star formation histories and stellar evolution.
Arcaviet al. 2014). Mageshwaran & Mangalam (2015) focused on absolute
Demographic studies of TDEs have largely focused on rate estimates in various scenarios, while Stone & Metzger
the dynamical problem of understanding the rate at which (2016) also examine the distribution of event properties.
stars can be placed into low angular momentum orbits Broadly speaking, most theoretical studies predict TDE
that will pass sufficiently close to the black hole to be dis- rates of order 10−4 yr−1 gal−1 for MBH <∼ 107.5M⊙,
rupted (e.g., Lightman & Shapiro 1977, Cohn & Kulsrud while many observational rate estimates are closer to
(cid:13)c 0000RAS
2 C. S. Kochanek
10−5 yr−1 gal−1 (e.g.Donley et al.2002,Gezari et al.2008, smaller than the Schwarzschild radius, we assume the star
van Velzen & Farrar 2014). Holoien et al. (2016), however, fallsintotheblackholeandisabsorbedwithoutaluminous
found a somewhat higher rate in the All-Sky Automated transient. Arguably, we might instead use the radius of the
Surveyfor Supernovae(ASAS-SN). laststableorbit.Dependingonthestructureofthestarand
There are extensive numerical studies of the hy- theexactpericenter,thestarmaybefullydestroyedoronly
drodynamics of TDEs (e.g., Evans & Kochanek 1989, stripped of all or part of its envelope (e.g. MacLeod et al.
Lodato et al. 2009, MacLeod et al. 2012, Dai et al. 2012). In addition to the mass of the black hole, the de-
2013, Guillochon & Ramirez-Ruiz 2013, Hayasaki et al. tailed limits (η etc.) depend weakly on the structure of the
2013, Guillochon et al. 2014, Shiokawa et al. 2015, star (e.g. MacLeod et al. 2012, Guillochon & Ramirez-Ruiz
Bonnerot et al. 2016, Sadowski et al. 2015) and semi- 2013) and the properties of the black hole (e.g. Kesden
analyticmodels of theirobservational properties(e.g., Rees 2012).
1988, Cannizzo et al. 1990, Kochanek 1994, Loeb & Ulmer WeassumeaKroupa(2001)initialmassfunction(IMF)
1997, Strubbe& Quataert 2009, Syer& Ulmer 1999, extending from 0.08M⊙ to 100M⊙. This makes the IMF a
Kasen & Ramirez-Ruiz 2010, Lodato & Rossi 2011, broken power law, (dn/dM∗)IMF ∝ (M∗/0.5M⊙)−α with
Strubbe& Quataert 2011, Stone et al. 2013, Miller α = 1.3 for M∗ < 0.5M⊙ and α = 2.3 for M∗ > 0.5M⊙.
2015, Piran et al. 2015, Strubbe& Murray 2015, ThecompleteKroupa(2001)IMFbreakstoastillshallower
Metzger & Stone 2015). However, it is fair to say that α=0.3power-lawat0.08M⊙ andextendsdownto0.01M⊙,
thesestudieshaveyettoconvergeonapredictivemodelfor butweignorethisextensiontobrowndwarfs.Observational
TDE properties. The fundamental difficulty is that TDEs selectioneffectswilldisfavorfindingsuchlowmassTDEsin
are a three dimensional radiation hydrodynamics problem. any case. The mass function at any given time, dn/dM∗,
Simulationsarestillchallengedbythelargerangeofspatial is not the IMF, due to the combined effects of stellar evo-
scales and do not yet include the effects of radiation, while lution and the star formation history. Where stellar mass
semi-analytic models are not well-suited for transients with functions have previously been used in TDE rates studies
necessarily complex spatial structures. (Magorrian & Tremaine 1999, Mageshwaran & Mangalam
Our goal in this paper is to examine the demographics 2015, Stone& Metzger 2016), they assume an old stellar
ofTDEs.Giventhelackofanyreliablepredictivemodelfor population by simply truncating the IMF at a maximum
observables,wefocusonasimple,genericmodelforselection massofM⊙.Whilethisisareasonablemodelforthepresent
effectsthatcanbeobservationallycalibrated.Inourmodels, day main sequence population of an early typegalaxy, it is
we include not only an initial mass function for stars, but not a good characterization of the central regions of the
also star formation histories and complete models of stellar Milky Way. Pfuhl et al. (2011), for example, find that the
evolution. We use a mass function for the black holes as Galactic center population is mostly old (80% formed 5-
well as estimates of its evolution with redshift. In §2 we 10 Gyr ago), but the remainder is in a very young popula-
describe our model for the stellar populations, disruption tion(20%formedinthelast∼0.1Gyr).IfthetypicalTDE
rates andtheblack hole mass function.In §3we surveythe occurs in a galaxy with MBH ∼ 106 to 107M⊙, the mixed
expected demographics of TDEs as afunction of black hole stellarpopulationweseeintheMilkyWaymaybemorerep-
mass, stellar mass, stellar evolutionary state, and redshift resentative than a purely old population. It is also natural
both globally and for a specific observational case. We will to include broader models of star formation histories since
discusstheimplicationsanddirectionsforfurtherinquiryin weincludestellarevolutionandwillexploretheevolutionof
§4. TDE rates with redshift,
We considered two basic star formation histories, a
1 Gyr burst and continuous star formation, with the star
formation rateconstant duringthestarforming period. We
2 MODEL DESCRIPTION
examine the resulting TDE rates and properties at ages of
1, 3 and 10 Gyr. The two histories are the same at an age
Inthissectionweoutlinethemodelwewilluseforthisstudy.
of1Gyr,sotherearereally5distinctcases.Forexample,if
We start with the criteria for disrupting (or stripping suffi-
cient mass to cause a flare) a star of mass M∗ = M∗⊙M⊙ thelife time of a star is t∗(M∗), then
andradiusR∗ =R∗⊙R⊙.Thenweintroduceamassfunction dn dn
forthestars,starformationhistoriesandamodelforstellar ∝ min(t,t∗(M∗)) (2)
dM∗ dM∗ IMF
evolution. Next we estimate the rates of disruptions for a (cid:16) (cid:17)
bulge with velocity dispersion σ = 200σ200 km/s contain- is themass function at time t for a constant star formation
ing a black hole of mass MBH =107MBH7M⊙, and discuss rateandignoringmassloss.ItfollowstheIMFuntilthemass
blackholemassfunctions.Finally,weexamineseveralphysi- where t=t∗(M∗) and is then cut off because only star for-
calpropertiesofdisruptionsandintroduceasimpleselection mation in thelast t∗(M∗) contributestothemass function.
effects model. Theburstmodelsimplyrequiresmoreaccountingtoinclude
Weassumethataneventoccurswhenastarapproaches theeffectsofthecutoffinstarformation.Wedefinethemass
closer to theblack hole than functionsothatitisnormalized, dM∗dn/dM∗ ≡1,andit
RT =R∗ η2MMB∗H 1/3 (1) issquuaseWrfeuemltaousssdeehfiMnt∗2ehietheomldeearnsPtaeldlauraRmastseslhlaMr∗iisaoncdhrtohneemseaonf
(cid:16) (cid:17)
where η ≃1. If the pericentric radius R is larger than the Marigo et al.(2008)becausetheyincludethethermallypul-
p
Schwarzschild radius, R = 2GM /c2, but smaller than sating AGB (TP-AGB) phase of stellar evolution. We con-
S BH
R , then we assume there is some form of TDE. If it is sideredonlytheSolarmetallicitymodels.TheMarigo et al.
T
(cid:13)c 0000RAS,MNRAS000,000–000
TDE Demographics 3
(2008) tracks start at M∗ = 0.15M⊙, so we extended α = −0.164. For comparison, Wang& Merritt (2004) ulti-
them down to M∗ = 0.08M⊙ by logarithmically extrap- mately adopt r0 = 3.7×10−4/year and α = −0.25 based
olating physical quantities (luminosity, temperature) with on numerical fits to their results for individual galaxies.
mass. The exact details are not critical – the primary goal Stone& Metzger(2016),basedonsimilarmodelsofalarger
is simply to better estimate the absolute numbers of low number of galaxies and averaging over a stellar mass func-
mass stars. We also tracked the population of stellar rem- tion,findr0=7.4×10−5/yearandα=−0.404.Usingacom-
nants.Weusedtheinitial-to-finalwhitedwarfmassrelation pletelydifferentapproach,Brockamp et al.(2011)findr0 =
MWD =0.109MZAMS+0.394M⊙ fromKalirai et al.(2008) 8.3×10−5/yearandα=+0.446orr0=1.0×10−4/yearand
forM∗ <8M⊙,neutronstarmassesof1.4M⊙ from8M⊙ to α= +0.353 depending on their choice of MBH-σ relations.
21.4M⊙, and black hole masses of 7M⊙ for higher masses. Therearethenfurthersystematicuncertaintiescomingfrom
Thesechoicesarebroadlyconsistentwiththeobservedprop- thetreatmentoftheangularstructureofthestellarcore(e.g.
erties of supernova progenitors (Smartt et al. 2009), typi- Magorrian & Tremaine 1999, Vasiliev & Merritt 2013) and
calblackholemasses(e.g.O¨zel et al.2010,Kreidberg et al. the role of binary black holes (e.g. Merritt & Wang 2005,
2012, Kochanek 2015) and estimates of the fraction of core Chen et al.2008,Li et al.2015).Asacompromiseoverthese
collapses leading to black holes (∼ 25%, Kochanek 2015). various results, we adopt a rate model of
Thedetaileddistributionofneutronstarandblackholeout-
c2o0m15e)s,wbuitthusntiemllaprormtaanstshiseraen. open question (see Kochanek ddMr∗ ≃r0MB1h/MH47∗2M⊙i∗13⊙//182RhM∗1/⊙4∗⊙iddMn∗. (5)
We track the evolutionary state of the stars in five
bins using the tags supplied by the Marigo et al. (2008) with r0 = 10−4/year and α = −1/4, where the ∗⊙ sub-
isochrones. We track main sequence stars (MS, up to the script indicates that the quantity is in Solar units. We
turn off tag TO), sub-giant stars (from TO to the base of also neglect the process of a star “evolving” into its loss
the red giant branch, RGBb), red giants (RGB stars, from cone due to its increasing radius as it ascends the gi-
RGBb to helium ignition, BHeb), horizontal branch stars ant branch (Syer& Ulmer 1999) since it is sub-dominant
(HB stars, from helium ignition BHeb to core helium ex- (Magorrian & Tremaine 1999). The effects of changing r0
haustion,EHeb),asymptoticgiantbranchstars(AGB,from on our results are trivial, and we will explore the conse-
coreheliumexhaustiontocarbonignitionCb),andstarsaf- quencesofchangingthedependenceon theblackholemass
ter carbon ignition. In practice, this latter phase makes a (α) below.
negligible contribution and can simply be ignored. We will Equations 3 and 5 depend on the mean stellar mass
always use the description of the phases as MS, sub-giant, hM∗iandthemeansquaremasshM∗2i.ThehM∗itermrepre-
RGB,HBandAGB.Thecategoriesarenotfullycorrectfor sentsthechangeintheratewiththenumberofstarsatfixed
themost massive stars (and thefull sequence of tags is not total mass. The rate increases if the mass is divided over
present),buttheyareappropriatefortheintermediatemass largernumbersofstars.ThehM2itermisduetothedepen-
∗
stars that dominate the disruption rates of evolved stars. denceoftheorbitdiffusionratesonstellarmass–thehigher
Usingapower lawfit tothemain sequenceturnoff age, we themean square stellar mass at fixed total mass, the faster
also track the elapsed main sequence lifetime fraction fMS the diffusion times, because the gravitational potential is
of each star. becomingmore“granular”(Magorrian & Tremaine1999).If
We start from the TDE rate estimate of weexaminejustthehM∗2i3/8/hM∗ifactorinEquation3,the
Wang & Merritt (2004) modified to include the effects inclusion of amass function (excludingremnants) increases
of a mass function (see Magorrian & Tremaine 1999), the rates by roughly a factor of 1.7 and depends little on
the age or star formation history cases we consider. This is
dr ≃ 7.28hM∗2i3/8R∗1/4η1/6σ7/2 dn . (3) consistentwithStone & Metzger(2016),althoughtheyalso
dM∗ G5/4MB11H/12M∗1/12hM∗i dM∗ note that the factor increases significantly for very young
(∼ 100 Myr) stellar populations. If we include stellar rem-
The disruption rate for an individual star scales as r ∝
R∗1/4M∗−1/12 ∝ RT1/4 as noted by MacLeod et al. (2012). rneamntnsainntshMm∗2aike(tahenyegshlioguibldlencootnbtreibinuctliuodnetdoindihsMru∗pitisoinncreatthees
In this expression, we have neglected an additional term
due to their high densities), the rate increase is modestly
of the form (lnΛ/lnB)3/4 ∼ 1 where Λ = 0.4MBH/M0, higher, at a factor of roughly 2.0. Stone& Metzger (2016),
B = r /4R and r = GM /σ2 for simplicity. Since the
h T h BH using theoretical models of theblack hole mass function by
η1/6 dependenceonthedimensionlessfactorsettingthedis-
Belczynski et al. (2010), found modestly larger effect, and
ruption boundary from Equation 1 is very weak, we simply
thedifferenceslikelylieinourusingalower,observationally
set η ≡ 1. This rate estimate also neglects any role of stel-
driven, choice for the typical black hole mass. Since the ef-
lar collisions in suppressing disruptions of giant stars (see
fectsofremnantsinourmodelsaresosmall(10-20%effects)
MacLeod et al. 2012). Wang et al. (2012) combine Equa-
we neglect them for simplicity.
tion 3 with theM -σ relation
BH To examine the overall rates of TDEs we need the
MBH ≃1.5×108σ240.605M⊙ (4) black hole mass function n(MBH) as a function of red-
shift. We consider the models of Hopkinset al. (2007)
toyieldaratescalingasr∝σ−3/4orr∝M−1/6 withblack and Shankaret al. (2009), focusing on the more recent
BH
hole mass. Shankaret al. (2009) models. Both models are based on
Using Solar values for all the stellar variables in Equa- using quasar luminosity functions and estimates of merger
tions 3 and 4, the absolute scale of the rate is of the form rates to model the growth of black holes, constrained by
dr/dM∗ =r0MBαH7dn/dM∗ with r0 =4.7×10−4/year and the requirement of matching local estimates of the black
(cid:13)c 0000RAS,MNRAS000,000–000
4 C. S. Kochanek
dius R , usually expressed as the ratio β = R /R . Since
T T p
we are simply adopting η=1 in Equation 1, β is restricted
to therange 1<β<β where
m
-2 βm = RRT =5.1R∗1M∗−11/3MB−H2/73 (6)
S
is the point where the star passes through the horizon (ar-
guably, we could also use the last stable orbit). The dis-
tribution of encounters in β is non-trivial and cannot sim-
ply be derived from the r ∝ R1/4 scaling of Equation 3.
-4 T
If Equation 3 really was directly related to the distribu-
tion of pericentric radii for disrupting stars, it would im-
ply an unphysical differential distribution of dP/dR ∝
p
R−3/4 that is dominated by strongly radial orbits. The
p
same holds for the dP/dR ∝ 1/R distribution adopted
p p
-6 byStrubbe& Quataert (2009).
TDEratesaredominatedbytwolimiting regimes(see,
e.g.,Wang & Merritt2004).Formoredistantorbits,theor-
bitalangularmomentumchangesrelativetothescaleneeded
to pass close to the black hole faster than the orbital time
scale.Inthis“pinhole”limit,theangularmomentumforan
-8
encounterisrandom.Forcloser orbits,theangularmomen-
tum changes slowly compared to the orbital time scale. In
this “diffusion” limit, an orbit slowly approaches the crit-
Figure1. TheevolutionoftheShankar etal.(2009)blackhole ical angular momentum needed to reach R . In the “pin-
T
mass function, dn/dlogMBH, defined by the number of black hole” limit, dP/dRp is constant once we include the ef-
holes per (base 10) logarithmic mass interval from the present fectsofgravitational focusing, andthusdP/dβ ∝β−2 (e.g.,
(top)toz=6(bottom)instepsof∆z≃0.25.TheHopkinsetal.
Luminet & Barbuy 1990). In the diffusion limit, all stars
(2007) (H07) mass function used by Strubbe&Quataert (2009)
disrupt very close to R , so dP/dR ∝ δ(R −R ) and
is shown by red dotted lines at z =0 and 6, and the local mass T p p T
dP/dβ ∝ δ(β −1). Following Stone& Metzger (2016) we
functionusedbyStone&Metzger(2016)(SM16)isshownbythe
model this as
reddashedline.
dP f β−2 1−β−1 1<β≤β
≃ pin m m (7)
hole mass function. The Shankaret al. (2009), mass func- dβ (cid:26) (1−fpin(cid:0)) (cid:1) β=1
tion is defined per (base 10) logarithmic mass interval, where a reasonable match to their estimates of the fraction
n(MBH) = dn/dlogMBH, over the mass range 5.0 < of “pinhole” mergers is
logMBH/M⊙ < 9.6 and the redshift range 0 < z < 6 as −1
shown in Figure 1. The number of lower mass black holes f ≃ 1+M1/2 . (8)
is divergent (∼ M−3/2 for low M ), although the total pin BH7
BH BH (cid:16) (cid:17)
mass in black holes is convergent. This means that the ob- Lowmass blackholeshavecorestructuresfavoring encoun-
servedrateofTDEsfromlowmassblackholesiscontrolled terscloserthanRT inadditiontohavinglargeβm,allowing
by selection effects. Strubbe& Quataert (2009) used the such encounterswhile remaining outside theblack hole.
Hopkinset al. (2007) models and these are very similar, as The importance of β = RT/Rp is presently under dis-
illustrated by their structure at z = 0 and 6 in Figure 1. cussion. The simplest physicalpicture of a TDE is that the
Mageshwaran & Mangalam (2015) used an evolving quasar star reaches pericenter intact and then is disrupted with a
luminosityfunctionrescaledbyadutycycleestimate,while spread in orbital binding energy set by the tides across the
Stone& Metzger(2016)startfromalocalgalaxyluminosity star,
functionandthenpopulatethegalaxieswithblackholesus- δǫ≃ GMBHR∗βn (9)
ingtheMcConnell & Ma(2013) bulge/blackholemasscor- R2
T
relation combined with a bulge mass-dependent black hole
occupation fraction.1 This mass function, which has ashal- with RS ≤ Rp ≤ RT (1 ≤ β ≤ βm = RT/RS)
lower divergence ∼ M−1.07 for low black hole masses, is and n = 2 (following Stone et al. 2013). However,
BH Guillochon & Ramirez-Ruiz(2013)(alsoseeHayasaki et al.
alsoshown inFigure1.AsemphasizedbyStone& Metzger
2013) found that this was not the case in their numerical
(2016), the degree to which the divergence of the number
simulations and that the spread in energy was essentially
density for small M is real controls the absolute volu-
BH
independent of β, implying n = 0. This was further con-
metric rate of TDEs.
firmed by the semi-analytical study of Stoneet al. (2013).
Some properties of TDEs may depend on the orbital
Inessence,thestarceasestobeboundasitcrosses R and
pericentric radius R relative to the nominal disruption ra- T
p
the debris already proceeding on independent orbits before
it approaches pericenter.
1 In Equation 31 of Stone&Metzger (2016), the differential Theconsequenceofn=0ratherthann=2canbeseen
should be dMBH rather than dlnMBH (Stone, private commu- inhowtheenergyspreadthendeterminesthecharacteristic
nication). fall back time,
(cid:13)c 0000RAS,MNRAS000,000–000
TDE Demographics 5
tfb=0.36β−3n/2MB1/H27m−∗11R∗3/12 years. (10) (νL ) =ǫL 15 hν 3 (12)
ν peak peakπ4 kT
For a given fall back time, there is a characteristic peak (cid:18) peak(cid:19)
accretion rateof M˙peak =M∗c2/3tfb.Compared to theEd- where ǫ is some dimensionless efficiency factor. This re-
dington rate, this accretion rate is processing model is different from Lodato & Rossi (2011),
wheretheopticalemissionisthetailofdirectemissionfrom
M˙Mp˙eEak ≃4.4β3n/2MB−H3/72M∗21R∗−13/2, (11) wanildalcycraestioansdyisstke.mTaoticthfeunexcttieonnt othfaptaTrapmeaketedroses(MnoBtHv,arβy,
etc.), and L is Eddington limited, then there is a very
assuming a radiative efficiency of η = 10% (L = ηM˙c2). peak
simple model for the relative survey volumes to be asso-
This is an overestimate of the accretion rate if only (part
ciated with different events. If an Eddington limited event
of) the envelope is stripped, as can occur for evolved stars
(seeMacLeod et al.2012).Thereisabigdifferencebetween from a MBH = 107M⊙ black hole would be detected in a
local survey out in volume V0, then any other event would
the n = 2 scaling, where close encounters produce short
bedetected in volume
(t ∝β−3), high peak accretion rate events (M˙ ∝β3),
fb peak
and n = 0, where the time scale and peak accretion rates V min M˙peak,M˙E(MBH) 3/2
are independentof β. = . (13)
Forsurveys,thediscoveryofaTDElargelydependson V0 (cid:0)M˙E(107M⊙) (cid:1)!
tfbandM˙peak–therisetimemustbeshortenoughtotrigger This provides a simple approach to reasonably estimating
a detection, and the peak luminosity must be high enough the differential contributions of events to a survey, while
toallowdetectionofeventsinalargeenoughvolume.Over- avoiding the very much harder problem of determining
all event durations for TDEs are long enough to be unim- Lpeak/Tpeak and hence the volume V0 in which the fiducial
portant factors for discovery. If n = 2, the importance of event would be detectable. More importantly, V0 can sim-
“pinhole”encountersisgreatlyenhanced.First,eventswith plybeestimatedempiricallyfromthepropertiesofobserved
longtfbatβ =1becomeshortenoughtotriggeratransient transients,aswewilldocrudelyfortheASAS-SNsurveyin
survey. Second, and more importantly, if peak luminosities §3.3.Equation13isappropriateforlowredshiftsurveyslike
determine detectability and scale as Lpeak ∝ M˙peak ∝ β3, ASAS-SNwherecosmology andevolutioncanbeneglected.
then the survey volume scales as V ∝ L3/2 ∝ β9/2. Since Other effects may also suppress the observed contributions
peak
dN/dβ∝β−2,thecontributionofeventsatagivenβ scales fromlowermassblackholes.Forexample,Stone & Metzger
as dr/dβ ∝ VdN/dβ ∝ β1/2 and events with β ≃ β (2016) explore a model where only relatively close encoun-
m
(modulo the Eddington limit) will dominate the observed ters(Rp <6RS)circularizethedebrisrapidlyandproducea
rates.2 If, on the other hand, n = 0, then the time scales strong flare, which means that only high β pinhole encoun-
and peak luminosities are independent of β and the rela- ters contribute to the TDE rate as MBH decreases below
tivedetectability of eventswith differingβ must dependon ∼107M⊙.
higher order effects than the basic time and accretion rate
scales. To avoid considering too many cases, we will follow
Guillochon & Ramirez-Ruiz (2013), Stone et al. (2013) and 3 TDE DEMOGRAPHICS
Stone& Metzger (2016) and assume n=0 for our primary
Inthissectionweexplorevariousaspectsofthedemograph-
discussion in §3. We illustrate some consequences of n = 2
ics of TDEs. In §3.1 we examine the effects of the star for-
in theAppendix.
mationhistoryonthemassesandevolutionarystatesofthe
If we focus on the UV/optical TDEs, the observed op-
disrupted stars as a function of black hole mass. In §3.2 we
tical/UV spectra are broadly consistent with black bodies
examinetheeffectschangingthescalingofthecapturerates
(Holoien et al.2014,Holoien et al.2016),althoughtheturn
in Equation 5 with black hole mass. In §3.3 we survey the
over at short wavelengths is not always observed and it
propertiesoflocalTDEsinstellarmass,evolutionarystate,
is clear that some TDEs with thermal optical/UV prop-
peak accretion rate (M˙ /M˙ ), fall-back time (t ) and
erties have significant non-thermal emission components peak E fb
pericentric distance (β = R /R ). In §3.4 we examine the
(Holoien et al. 2016). The general assumption for these T p
evolution of TDE rates with redshift.
events is that the observed emission is reprocessed emis-
sion from an underlyingaccretion disk (e.g., Loeb & Ulmer
1997, Strubbe& Quataert 2009, Guillochon et al. 2014, 3.1 Stellar Mass and Evolutionary State
Strubbe& Murray 2015, Stone& Metzger 2016) rather
than direct emission from an accretion disk (e.g., Figures 2 and 3 show the integral, r(> M∗), and differen-
Strubbe& Quataert 2009, Lodato & Rossi 2011), although tial,dr/dlogM∗,TDEratesasafunctionofstellarmassfor
other hypotheses have been advanced (e.g., Svirski et al. black hole mass bins centered at MBH =106,107,108, and
2015). To the extent this is true and the observed temper- 109M⊙ and thetwo star formation histories (a 1 Gyr burst
atureis hot compared totypical surveybands(e.g. V-band or constant) at ages of 1, 3 and 10 Gyr. At 1 Gyr, the two
for ASAS-SN),theobserved peak luminosity is star formation histories are identical. The rate estimates at
MBH ∼106M⊙ of roughly 10−4 year−1 are consistent with
earlier results, as they must be given their underlying de-
pendenceon the Wang& Merritt (2004) models.
2 NotethatStoneetal.(2013)usetheintegraldistributionP(> Forthetwolower blackhole mass ranges, therates are
β)∝β−1 intheirdiscussionratherthanthedifferentialdistribu- dominated by lower mass, main sequence stars, as was al-
tiondP/dβ∝β−2. readywell known.Theratesincrease slightly towardslower
(cid:13)c 0000RAS,MNRAS000,000–000
6 C. S. Kochanek
-3 -3
-4 -4
-5 -5
-6 -6
-7 -7
--38 -3
-4 -4
-5 -5
-6 -6
-7 -7
-8
0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10
Figure 2. Integral TDE rates, r(> M∗), per black hole as a Figure3. DifferentialTDErates,dr/dlogM∗,perblackholeas
function of stellar mass M∗ for black hole mass ranges of 108.5- afunctionofstellarmassM∗ forblackholemassrangesof108.5-
109.5 (top left), 107.5-108.5 (top right), 106.5-107.5 (lower left), 109.5 (top left), 107.5-108.5 (top right), 106.5-107.5 (lower left),
105.5-106.5 (lower left) and either 1 Gyr burst (black, solid) or 105.5-106.5 (lower left) and either 1 Gyr burst (black, solid) or
constant (red, dashed) star formation models. The present ages constant (red, dashed) star formation models. The present ages
(frommosttofewesthighermassstars)are1,3and10Gyr.The (frommosttofewesthighermassstars)are1,3and10Gyr.The
twostarformationmodelsareidenticalat1Gyr. twostarformationmodelsareidentical at1Gyr.
M∗ >∼ M⊙. The total rates for the different star formation
black hole masses because of the weak black hole mass de- histories differby only ∼25%.
pendence of Equation 5. Because the Kroupa (2001) mass For the highest mass black holes, the mass function of
functionisalmostlogarithmicallyflatatlowmassesanditis the disrupted stars and the overall rates depend strongly
somewhateasiertodisrupthighermass,lowermeandensity onthestarformation historybecauseitisincreasingly only
main sequence stars, the logarithmic rates are almost con- evolved stars that can undergo a TDE. The absolute rates
stantforM∗ <0.5M⊙ andthendropslowlyuptoM∗ ≃M⊙ now vary by a factor of ∼ 3 between the star formation
dueto the break in the slope of the IMF. The typical TDE historiesandthetypicaldisruptedstarhasthemassofastar
is of an M∗ ≃ 0.3M⊙ main sequence M-dwarf. The star at the MS turn off. For star formation histories where the
formation history completely controls the rates for higher durationofthestarformationperiodisstillrelevant,thereis
mass stars, but affects the total rates little since only 14% apeakwithapower-lawdeclinetowardshighermasses.For
of stars on the IMF have M∗ > M⊙. The rapid decline in the older burst populations, the mass function increasingly
stellarlifetimeswithstellarmassleadstoverysharpbreaks looks like a delta function because the range of stellar ages
in the mass dependent rates for the burst star formation corresponds to a negligible spread in MS turn off masses.
models at late times. Forthehighermassblackholes(MBH >∼108M⊙)itislikely
In the next higher mass range, centered on M = that only the burst models are relevant at lower redshifts
BH
108M⊙,webegintoseetheeffectsofthedroppingstrength becausethehostswillbeearly-typegalaxieswitholdstellar
of the tidal gravity at the event horizon as the black hole populations.
mass increases. The lower mass M dwarfs can no longer be Figure 4 shows the rates as a function of the evolu-
disrupted,leadingtoanorderofmagnitudedropintheTDE tionary state of the star and the black hole mass for the
rate. The mass function of the disrupted stars is strongly 3 Gyr and 10 Gyr old stellar population models. For sim-
truncated near M∗ ≃ 0.3M⊙, so the typical mass of a dis- plicity we simply show the results for all stars, MS stars,
rupted star increases and now depends more on the star sub-giants,starsthathaveevolvedpastthebaseoftheRGB,
formationhistory.Whiletheexistenceofthesharpcutoffat andhorizontalbranchandlaterstars.MacLeod et al.(2012)
low masses is generic, its exact location will be sensitive to illustrate relative rate distributions over these later evolu-
the radius used to define the boundary between disruption tionary phases for a range of stellar masses. Figure 4 also
andabsorption.Kesden(2012)exploressomeoftheseissues compares the estimates to the two recent rate estimates by
for rotating black holes and findsthat they haveonly mod- van Velzen & Farrar (2014) and Holoien et al. (2016). Nei-
est effects. The primary effect of the star formation history ther study differentiates by black hole mass, but typical
isstilltomodifythemassfunctionofthedisruptedstarsat black hole mass estimates for observed optical/UV TDEs
(cid:13)c 0000RAS,MNRAS000,000–000
TDE Demographics 7
-3 0.5
-4
-5 0.4
-6
-7 0.3
-3
-4 0.2
-5
-6 0.1
-7
0
0 0.2 0.4 0.6 0.8 1
Figure4. Ratesperblackholefordifferentstellarevolutionary Figure5. Integraldistributionn(>fMS)ofmainsequencestars
phasesasafunctionofblackholemassforthe1Gyrburstmodel inthe fractionofelapsed mainsequence lifetimefMS atdisrup-
at3Gyr(topleft)or10Gyr(topright)andthecontinuousstar tion for black hole mass ranges of 106.5-107.5 (black lower/left)
formation model at 3 Gyr (lower left) or 10 Gyr (lower right). and 107.5-108.5M⊙ (red, upper/right). The solid curves are for
Thereddashedlineshowsthetotalrate,andthesolidlinesshow the1Gyrburstmodelat1(lower),3(middle)and10Gyr(top),
(from top to bottom) the rates for MS stars, sub-giants, red gi- andthedashedcurvesarefortheconstantstarformationmodel
antandlaterevolutionarystatesandhorizontalbranchandlater at3(lower)and10Gyr(top).Thedistributionsarenormalizedto
evolutionarystates.Thetwopointswitherrorbarsaretheratees- n(>fMS =0)=1but the upper halves of the distributions are
timatesbyvanVelzen&Farrar(2014)(vV14)andHoloienetal. notshowntomakethedistributionnearfMS =1morevisible.
(2016)(H16).Theirlocationinblackholemassisintherangeof
theBHmassestimates forobservedTDEs.
massitdoesnottakeahugeincreaseintheblackholemass
to absorb a sub-giant as compared to a MS turn off star.
are of order MBH = 106 to 107.5M⊙ (see below). The two Second, the subgiants are associated with the highest mass
rate estimates are in mild conflict, but they illustrate the MS stars in the stellar population, because the lower mass
continuingtensionbetweenobservedandtheoreticaldisrup- stars have not had time to evolve. As a result, the sub-
tion rates. giantcontributiontotheratesdropsrapidlyatmassesonly
For lower black hole masses where all stars disrupt slightlyabovetheblackholemasswheretheyareasimpor-
(MBH <∼ 107M⊙), the rate steadily rises for lower masses tantasMSstars.Finally,astheblackholemassapproaches
because of the MB−H1/4 scaling of the adopted rate (Equa- MBH >∼ 108.5M⊙, only the giant stars contribute to the
tion 5). Disruptions are overwhelmingly dominated by MS rates.Sincesuchevolvedstars arerareinall starformation
stars,withevolvedstarsrepresentingonly∼3%oftherate. histories, the expected rates are 2 to 3 orders of magnitude
The rate for evolved stars is dominated by sub-giants, then lower than for MBH <∼107.5M⊙ where all MS stars will be
redgiants,andthenalllaterphases.Theaveragemassofthe disrupted.
disrupted evolved stars will be much higher than the main Kochanek (2015) noted that M∗ >∼ M⊙ stars develop
sequence stars, so selection effects can significantly modify significantly depressed (enhanced) carbon (nitrogen) aver-
theobserved ratios, as we will discuss in §3.3. ageabundancesevenbyfractionf ≃0.1oftheirMSlife-
MS
Starting around MBH >∼ 107.5M⊙, an increasing frac- timeduetoCNOreactions.Alongwiththeirslowlyincreas-
tion of lower mass MS stars are absorbed rather than dis- ing helium mass fractions, this means that stellar evolution
rupted, leading to a very rapid drop in the TDE rate at canleadtoabundanceanomalies inTDEdebrisandpoten-
higher black hole masses. At roughly MBH = 108M⊙, the tially their spectra. This is a different concern from studies
rates for evolved and MS stars are comparable, with the ofnuclearreactionstriggeredbydeeplyplungingTDEorbits
subgiants still dominating the rates for evolved stars. The (e.g., Luminet & Pichon 1989). Figure 5 shows the integral
blackholemassscaleoftherapiddropintherateswillshift distributionn(>f )forthetwostarformationmodelsand
MS
in direct proportion to changes in the criterion for absorp- ages of 1, 3 and 10 Gyr for black hole mass ranges of 106.5-
tion over disruption. At slightly higher black hole masses, 107.5 and107.5-108.5M⊙.Lowerblackholemassrangeshave
thesub-giant contribution also drops rapidly dueto acom- distributions very slowly shifting to having fewer evolved
bination of two factors. First, subgiants are not tremen- stars, while higher mass black hole ranges quickly shift to
dously larger than MS turn off stars, so for fixed stellar beingdominatedbystarsclosetotheMSturnoff.Forblack
(cid:13)c 0000RAS,MNRAS000,000–000
8 C. S. Kochanek
TDEs with black holemass estimates drawn from theliter-
ature3. The results for the Hopkinset al. (2007) black hole
mass function are very similar.
5
The scaling of the rates with M in Equation 5 was
BH
something of a compromise. Many possible changes have
little consequence. For example, the original rate scaling
4 in Equation 3 depends on the bulge velocity dispersion
as σ7/2, which when combined with the M -σ relation
BH
M ∝ σ4.65 in Equation 4 has a black hole mass de-
BH
pendence of σ7/2 ∝ M3/4. Examples of other recent es-
3 BH
timates of the exponent of the M -σ relation are 5.13
BH
(Graham et al. 2011), 4.32 (Schulze& Gebhardt 2011) and
5.64 (McConnell & Ma 2013). As also shown in Figure 6,
2 varying the exponent of the MBH-σ relation by ±0.5 has
very little effect on the predictions since it changes the de-
pendenceof the rate on black hole mass by a factor of only
aboutr∝M±0.1.Similarly,Stone& Metzger(2016),based
1 on McConnelBlH& Ma (2013),used r∝M−0.404,which leads
BH
to changes only slightly larger than the example of using a
steeper M -σ relation.
BH
0 Thedominanceoflowmassblackholesislargelydriven
by the relatively steep slope of the black hole mass func-
tion, (n ∝ M−3/2 for Shankaret al. 2009), rather than
BH
the mass dependence of the rate. Even a large change in
Figure 6. Effects of the black hole mass function, MBH-σ re- the mass dependence of the rates has difficulty suppressing
lation and the dependence of TDE rates on MBH on the dis- the contribution from low mass black holes. For example,
tribution of TDEs in black hole mass. The histogram shows the
Brockamp et al.(2011)deriverateexpressionsfromaseries
distribution of black hole mass estimates for a sample 12 opti-
of N-body experiments that scale as r ∝ M0.446 or M0.353
cal/UVTDEs(seefootnote).Thesolidblackcurveisourfiducial BH BH
model based on Equation 5 and the local Shankaretal. (2009) dependingontheirchoiceofanMBH-σrelation.Asshownin
black hole mass function. The dotted lines bracketing it show Figure6,thisleadstoalocalpeaknearMBH ∼107M⊙ and
theeffectofvaryingthelogarithmicslopeoftheMBH-σrelation a reduced but still significant contribution from low mass
MBH ∝ σα over the range α = 4.65±0.50 assuming the TDE black holes.
ratescalesasσ7/2 asinEquation3.Adoptingthemuchstronger Without an even stronger black hole mass dependence
Brockampetal.(2011)ratescalingsofr∝MB0.H35 toMB0.H45 with than found by Brockamp et al. (2011), the divergence of
blackholemassdoessignificantlyalterthepredictions,asshown
the volumetric rates for low black hole masses is an in-
bythe black dashed lines.These areallforthe 1Gyr burststar
evitable consequence of the divergent number of low mass
formationmodelatanageof10Gyr.
galaxies/halos. For the remainder of the paper we sim-
ply use our fiducial model, combining Equation 5 with the
holemassesMBH <∼107M⊙,thehugenumbersoflonglived, Shankaret al. (2009) black hole mass function. We show
low mass MS stars which have had no time to evolve dom- most of the subsequent distributions as a function of black
inate the TDE rates. As a result only some 10-20% of MS hole mass, making it relatively easy to evaluate the con-
TDEs will have significantly anomalous carbon and nitro- sequences of changing either the black hole mass function
gen abundances,and only ∼5% will havefMS >∼0.5 where or the black hole mass dependence of the TDE rates. We
the helium enhancement begins to become significant. To alsoexaminetheconsequencesofoursimpleselectioneffects
this can be added the ∼ 3% contribution from sub-giants, model.
which are also likely to be fully disrupted in a TDE. Later
evolutionarystates(asidefrom therareAGBstars) areless
likely to produce anomalous abundance signatures because 3.3 Local Demographics
thematerialinsidethehydrogenburningshellissequestered
Next we explore the distribution of TDE properties as a
inaveryhighdensitycorethatislargelydecoupledfromthe
function of black hole mass for the local Shankaret al.
envelope. As the black hole mass increases, the fraction of
(2009) mass function and (as limiting cases) the burst and
TDEsassociated withmoreevolvedstarssteadilyincreases.
continuous star formation models at an age of 10 Gyr. The
results for other star formation histories can be approxi-
3.2 The Local Mass Function and M -σ Relation mately inferred from the earlier figures including all 5 star
BH
Figure6showstheexpecteddistributionoflocalTDEsasa
functionofM forthe10Gyroldburstmodelasafunction
BH 3 We included ASASSN-14ae (Holoienetal. 2014), ASASSN-
of black hole mass for the Shankaret al. (2009) mass func-
14li(Holoienetal.2016),TDE1/TDE2(vanVelzenetal.2011),
tion.Toemphasizehowstronglyeventsfromlowmassblack
PS1 10jh (Gezari 2012), PS1 11af (Chornocketal. 2014),
holesarefavored,wehaveusedalinearscaleforthenumbers PTF09g/PTF09axc/PTF09djl (Arcavietal. 2014), and the
of events per logarithmic mass interval. The distributions GALEXeventsfromGezarietal.(2006),Gezarietal.(2008)and
arenormalizedforcomparisontoasampleof12optical/UV Gezarietal.(2009).
(cid:13)c 0000RAS,MNRAS000,000–000
TDE Demographics 9
-5 -5
-6 -6
-7 -7
-8 -8
-6 -6
-7 -7
-8 -8
-9 -9
Figure 7. Volumetric (top) and observed (bottom) TDE rates Figure 8. Volumetric (top) and observed (bottom) TDE rates
as a function of stellar mass M∗ for the burst (left) and con- asafunctionofstellarevolutionarystatefortheburst(left)and
tinuous (right) star formation models at an age of 10 Gyr and continuous(right)starformationmodelsatanageof10Gyrand
the local Shankar etal. (2009) black hole mass function. In or- thelocalShankaretal.(2009)blackholemassfunction.Inorder
derofincreasingstellarmassanddiminishingTDEratesforlow ofincreasingstellarageanddiminishingTDErateforlow black
blackhole masses,the solidcurves areforM∗ =0.08-0.25,0.25- hole masses, the solid curves are for stars with main sequence
0.50,0.50-0.75and0.75-1.0M⊙,andthedottedredcurvesarefor lifetimefractionsoffMS =0-0.25,0.25-0.50,0.50-0.75and0.75-
M∗ =1.0-1.5,1.5-2.0and>2.0M⊙.Thedashed curves givethe 1.00andthedottedredcurvesareforpost-MSstars.Thedashed
total rate. In the lower panels the rate scales with the fiducial curvesgivethetotalrate.Inthelowerpanelstheratescaleswith
volumeV0 (inMpc3,seetext). thefiducialvolumeV0 (inMpc3,seetext).
formation histories. In each case, we show the “true” volu- massstarsareabsorbedratherthancaptured.Forthehigh-
metricrateperlogarithmicblackholemassintervalandthe est mass black holes, the volumetric rates are driven by
“observed” rate per unit fiducial volume V0 (in Mpc3) as- the M∗ >∼ M⊙ stars, but only the continuous star forma-
sumingthatn=0sothathighβ disruptionshavethesame tion models have any events from stars significantly more
fall backtimes andpeak luminosities asthoseat β ≃1(see massive than M∗ ≃ M⊙. For our simple selection effects
thediscussion in §2 and theAppendix). model(Equation13),theobservedcontributionsfromlower
As a reminder, the fiducial volume V0 is nominally the mass black holes are strongly suppressed because the Ed-
volume in which an Eddington-limited TDE for a 107M⊙ dingtonlimitontheluminosityrestrictsthesurveyvolume,
black hole would be detected. We are simply going to give V ∝L3/2 ∝M3/2. Thecontribution at high masses is sup-
Edd BH
arough empirical calibration based ontheASAS-SNTDEs pressed by the steeply falling mass function (Figure 1) and
(Holoien et al. 2014, Holoien et al. 2016), which are associ- because the longer fall back time scales and roughly fixed
ated with black holes roughly in this mass range and have stellarmassesincreasingly limittheaccretion ratetobebe-
peakluminositiesthatarereasonablyclosetotheEddington lowEddington.Asaresult,theobservedeventsarestrongly
limit.Forexample,ASASSN-14ae(Holoien et al.2014)had peaked near 107M⊙ rather than having the distribution of
an observed peak of M ≃ −19.5 mag (the true peak was Figure 6.
V
likely slightly higher). ASAS-SN can detect such transients Similarly,aslongastheratesaredominatedbythelow
out to a comoving distance of roughly 200 Mpc and moni- massdwarfs,mostTDEsarefromstarsthatareveryyoung
torsroughlyone-thirdof theskyafterclippingtheGalactic compared to their overall MS lifetimes. The contribution
plane and fields that are just rising or setting. Thus, a rea- fromstarspastthemid-pointoftheirMSlifetimesisnearly
sonable, empirical estimate for theASAS-SNsurveyis that ∼30timeslower,Fortheselowermassblackholes,thecon-
V0 ≃107 Mpc3. This is meant to provide a rough guide for tributionsfrompost-mainsequencestarsandstarsnearthe
interpreting rates rather than as a formal estimate. endoftheirMSlifetimesarecomparable.Asthedwarfsare
First, in Figures 7 and 8, we show the distributions increasinglyabsorbedratherthandisruptedforhighermass
in stellar mass and evolutionary state. In the volumetric blackholes,thedistributionofeventsinstellaragebecomes
rates, we again see that the low mass dwarfs completely muchmoreuniform.Andthen,finally,theTDEsassociated
dominate the rates for MBH <∼ 107M⊙, and then the dom- with the highest mass black holes are increasingly due to
inant mass rapidly shifts to higher stellar masses as lower evolved stars. In the observed distributions, the balance is
(cid:13)c 0000RAS,MNRAS000,000–000
10 C. S. Kochanek
-5 -5
-6 -6
-7 -7
-8 -8
-6 -6
-7 -7
-8 -8
-9 -9
Figure 9. Volumetric (top) and observed (bottom) TDE rates Figure10. Volumetric(top)andobserved(bottom)TDErates
as a function of the peak accretion rate in Eddington units as a function of the fall back time tfb for the burst (left) and
M˙peak/M˙E for the burst (left) and continuous (right) star for- continuous (right) star formation models at an age of 10 Gyr
mation models at an age of 10 Gyr and the local Shankar etal. and the local Shankar etal. (2009) black hole mass function. In
(2009)blackholemassfunction. Inorder ofincreasingaccretion orderofincreasingaccretionrateand(generally)blackholemass,
rate and lower black hole mass, the solid curves are accretion the solid curves are for tfb < 10−1.5 years and 10−1.5-10−1.0,
ratesofM˙peak/M˙E <10−1.5,10−1.5-10−1.0,10−1.0-10−0.5,and whilethe dotted redcurves arefortfb =10−1.0-10−0.5,10−0.5-
10−0.5-100.0 while the dotted red curves are for accretion rates 100.0,100.0-100.5,100.5-101.0 and>101.0 years.Eventswiththe
of 100.0-100.5, 100.5-101.0 and >101.0. The dashed curves show time scales of the solid curves are very likely to trigger present
thetotalrate.Inthelowerpanelstheratescaleswiththefiducial daytransientsearches,whilethetimescalescorrespondingtothe
volumeV0 (inMpc3,seetext). dotted red curves areincreasingly likelyto be ignored. Note the
contribution of long time scale events for low mass black holes
due to the disruption of evolved stars. The dashed curves show
thetotalrate.Inthelowerpanelstheratescaleswiththefiducial
modestly shifted towards older stars as the fall back time volumeV0 (inMpc3,seetext).
scales become longer and the numerous low mass dwarfs
can nolongersupport Eddington-limitedaccretion. Inboth
the stellar mass and evolutionary state distributions, the implies a TDE rate in ASAS-SN of order 15/year. In
differencesbetweentheburstandcontinuousstarformation practice, ASAS-SN is finding roughly one TDE per year
modelsarerelativelysubtle,withobviousdifferencesonlyfor (Holoien et al. 2014, Holoien et al. 2016). Onepossibility is
high mass black holes where the TDE rates are dominated that ourTDE model is overestimating therate byan order
by evolved stars. ofmagnitude,butthedistributioninfallbacktimesseenin
Figure9and10showthedistributionsofeventsinpeak Figure 10 suggests an alternate explanation.
accretionrate,M˙ /M˙ ,andfall-backtime,t ,assuming Thefallbacktimeislargelysetbytheblackholemass,
peak E fb
n=0inEquations10and11.Inavolumelimitedsampleof and spans a range from 10 days or less at MBH = 105M⊙
TDEs, the spread in the black hole masses is so large com- uptodecadesat109M⊙.Ifweconsiderthefourlowesttime
paredtothespreadinstellarmassesthatthereisatightcor- scale bins in Figure 10, t <10−1.5, 10−1.5-10−1.0, 10−1.0-
fb
relation of M˙ /M˙ with M even when a stellar mass 10−0.5 and 10−0.5-100.0 years (< 12 days, 12-37 days, 37-
peak E BH
functionisincluded.Lowermassblackholeshavehigherac- 116 days and 116-365 days) they contain roughly 2%, 14%,
cretionrates,andFigure9mayunderestimatethetrendbe- 61%,22%and1%oftheobservedevents.Totheextentthat
causewesimplyusedM∗ forthedisruptionsofevolvedstars thefall backtimeisareasonableproxyforeventrisetimes,
(rather than a partial stripping model, e.g., MacLeod et al. it is likely that most transient surveys will increasingly re-
2012). If, however, the luminosities are Eddington limited, ject sources with time scales longer than t >10−1.0 years
fb
then the visibility of the high M˙ /M˙ events associated that are located at the centers of galaxies because of AGN
peak E
withlowermassblackholesisgreatlyreduced,andthesam- variabilityandotherpotentialfalsepositives.Ifwerequired
pleofobservedTDEswill befairly tightlyclustered around t < 10−1.0 years as a selection limit, we would have only
fb
M˙ /M˙ ∼1. 16%ofthepotentiallyobservableTDEs,leadingtoarateof
peak E
With the simple selection effects model, the expected only2.5/year inASAS-SNthatisfarmorecompatiblewith
observed TDE rate is roughly 1.5 × 10−6V0/year. Given observed discovery rate. This is not a panacea since such
our rough estimate of V0 ≃ 107 Mpc3 for ASAS-SN, this a cut on the time scales truncates the expected black hole
(cid:13)c 0000RAS,MNRAS000,000–000