Optical Signatures of Tidally Disrupted Stars

Optical Flares from the Tidal Disruption of Stars by Massive Black Holes

Linda E. Strubbe, Eliot Quataert
Astronomy Department and Theoretical Astrophysics Center, 601 Campbell Hall, University of California, Berkeley CA, 94720, USA
E-mail:linda@astro.berkeley.edu
Accepted for publication in MNRAS
Abstract

A star that wanders too close to a massive black hole (BH) is shredded by the BH’s tidal gravity. Stellar gas falls back to the BH at a rate initially exceeding the Eddington rate, releasing a flare of energy. In anticipation of upcoming transient surveys, we predict the light curves and spectra of tidal flares as a function of time, highlighting the unique signatures of tidal flares at optical and near-infrared wavelengths. A reasonable fraction of the gas initially bound to the BH is likely blown away when the fallback rate is super-Eddington at early times. This outflow produces an optical luminosity comparable to that of a supernova; such events have durations of days and may have been missed in supernova searches that exclude the nuclear regions of galaxies. When the fallback rate subsides below Eddington, the gas accretes onto the BH via a thin disk whose emission peaks in the UV to soft X-rays. Some of this emission is reprocessed by the unbound stellar debris, producing a spectrum of very broad emission lines (with no corresponding narrow forbidden lines). These lines are the strongest for BHs with and thus optical surveys are particularly sensitive to the lowest mass BHs in galactic nuclei. Calibrating our models to ROSAT and GALEX observations, we predict detection rates for Pan-STARRS, PTF, and LSST and highlight some of the observational challenges associated with studying tidal disruption events in the optical. Upcoming surveys such as Pan-STARRS should detect at least several events per year, and may detect many more if current models of outflows during super-Eddington accretion are reasonably accurate. These surveys will significantly improve our knowledge of stellar dynamics in galactic nuclei, the physics of super-Eddington accretion, the demography of intermediate mass BHs, and the role of tidal disruption in the growth of massive BHs.

keywords:
galaxies: nuclei — black hole physics — optical: galaxies
pagerange: Optical Flares from the Tidal Disruption of Stars by Massive Black HolesReferencespubyear: 2009

1 Introduction

Stellar orbits in the center of a galaxy are not static, and sometimes stars walk into trouble. If an unlucky star passes within of the galaxy’s central black hole (BH), the BH’s tidal gravity exceeds the star’s self-gravity, and the star is disrupted. For BHs with , the disruption of a solar-type star occurs outside the horizon and is likely accompanied by a week- to year-long electromagnetic flare (e.g., Rees 1988).

Gravitational interactions between stars ensure that all supermassive BHs tidally disrupt nearby stars (e.g., Magorrian & Tremaine 1999). The scattering process might be accelerated by resonant relaxation very close to the BH (Rauch & Tremaine, 1996), or interactions with “massive perturbers” like a massive accretion disk (Zhao et al., 2002) or giant molecular clouds (Perets et al., 2007). In addition, the galactic potential may be triaxial so stars need not be scattered at all: they may simply follow their chaotic orbits down to (Merritt & Poon, 2004). Given these uncertainties, predictions for the timescale between tidal disruptions in a given galaxy range from to years. The rate remains uncertain, but tidal disruption must occur.

Indeed, a handful of candidate events have been detected to date. The accreting stellar debris is expected to emit blackbody radiation from very close to the BH, so X-ray and UV observations probe the bulk of the emission. Several candidate tidal disruption events were discovered in the ROSAT All-Sky Survey (see Komossa, 2002) and the XMM-Newton Slew Survey (Esquej et al., 2007); the GALEX Deep Imaging Survey has so far yielded three candidates (Gezari et al., 2006, 2008, 2009). For ROSAT, these detections are consistent with a rate per galaxy (Donley et al., 2002), but the data are sparse. However, we are entering a new era of transient surveys: in the optical, surveys like Pan-STARRS (PS1, then all four telescopes) (e.g., Magnier, 2007), the Palomar Transient Factory (Rau et al., 2009), and later the Large Synoptic Survey Telescope will have fast cadence, wide fields of view, and unprecedented sensitivity. Wide-field transient surveys with rapid cadence are also planned at other wavelengths, including the radio (e.g., LOFAR and the ATA), near-infrared (e.g., SASIR), and hard X-rays (e.g., EXIST). How many tidal flares these surveys find depends on their luminosity and spectra as a function of time.

In this paper, we predict the light curves and spectra of tidal disruption events as a function of time. Since the early work on tidal disruption (e.g., Rees, 1988), it has been well-appreciated that the bulk of the emission occurs in the UV and soft X-rays, with a possible extension to harder X-rays.111Such a hard X-ray component may be detectable with upcoming all-sky X-ray surveys like the proposed EXIST mission (Grindlay, 2004). However, we choose not to include predictions for hard X-rays in our calculations: one could draw analogy to the hard X-ray power-law tail observed in AGN spectra, but since the origin of this feature is uncertain, it is difficult to make firm theoretical predictions for tidal disruption events. Taking into account only this emission, optical wavelengths are not the most promising for detecting tidal flares, because the blackbody temperature of the inner accretion disk is . We show, however, that there are two additional sources of optical emission that likely dominate the optical flux in many, though not all, cases: (1) emission produced by a super-Eddington outflow at early times and (2) emission produced by the irradiation and photoionization of unbound stellar debris (see also the earlier work of Bogdanović et al., 2004). In the latter case, much of the optical emission is in the form of very broad emission lines, while in the former it is primarily continuum (although some lines may also be present). Throughout this paper, we typically discuss these two sources of emission separately, largely because the physics of the photoionized stellar debris is more secure than that of the super-Eddington outflows.

The remainder of this paper is organized as follows. In §§2.1, 2.2 and 3, we describe our models for the polar super-Eddington outflow, accretion disk and the equatorial unbound material, respectively; then in §4 we calculate the luminosity and spectral signatures of tidal disruption events. We predict detection rates in §5, and summarize and discuss our results in §6. §§5.2 and 6 include a discussion of our models in the context of ROSAT and GALEX observations of tidal flare candidates.

2 The Initially Bound Material

We consider a star approaching the BH on a parabolic orbit with pericenter distance . Once the star reaches the vicinity of the tidal radius (), the tidal gravity stretches it radially and compresses it vertically and azimuthally. The acceleration is and acts for a dynamical time near pericenter, resulting in velocity perturbations , where is the star’s radius and is the star’s escape velocity. The change in velocity is smaller than the star’s orbital velocity at pericenter, , by a factor of .

Because is at least as large as the sound speed inside the star, the stellar gas may shock vertically and azimuthally (e.g., Brassart & Luminet, 2008; Guillochon et al., 2008). Once the shredded star passes through pericenter, the compression subsides and the star re-expands, cooling adiabatically; thermal pressure becomes negligible and the particles travel away from the BH ballistically. We assume that the particle trajectories become ballistic when the star passes through pericenter. At that time, the particles have perturbed azimuthal, vertical, and radial velocities .

The particles have a range in specific energy (e.g., Lacy et al., 1982; Li et al., 2002), due to their relative locations in the BH’s potential well and differences in their azimuthal speeds. Initially, approximately half of the stellar mass is bound and half is unbound (Lacy et al., 1982; Evans & Kochanek, 1989). After a time

the most bound material returns to pericenter. (Here we have defined , , and .) Less bound gas follows, at a rate

(2)

(Rees, 1988; Phinney, 1989). There will be some deviations from this canonical scaling at early times, depending on the precise structure of the star (Lodato et al., 2009; Ramirez-Ruiz & Rosswog, 2009), but we use equation (2) for simplicity. As matter returns to pericenter, it shocks on itself, converting most of its bulk orbital energy to thermal energy (see Kochanek, 1994). The viscous time is typically shorter than the fallback time, so at least some of the matter begins to accrete.

For , the mass fallback rate predicted by equations (2) & (2) can be much greater than the Eddington rate for a period of weeks to years; here , is the Eddington luminosity, and 0.1 is the assumed efficiency of converting accretion power to luminosity. The fallback rate only falls below the Eddington rate at a time

(3)

where . While the fallback rate is super-Eddington, the stellar gas returning to pericenter is so dense that it cannot radiate and cool. In particular, the time for photons to diffuse out of the gas is longer than both the inflow time in the disk and the dynamical time characteristic of an outflow. The gas is likely to form an advective accretion disk accompanied by powerful outflows (e.g., Ohsuga et al., 2005), although the relative importance of accretion and outflows in this phase is somewhat uncertain (see §6). Later, when (), the outflows subside, and the accretion disk can radiatively cool and becomes thin.

In §2.1, we describe our model for the super-Eddington outflows, and in §2.2, we describe our model for the accretion disk. We discuss uncertainties in these models in §6.

2.1 Super-Eddington Outflows

When the fallback rate to pericenter is super-Eddington, radiation produced by the shock and by viscous stresses in the rotating disk is trapped by electron scattering. By energy conservation, this material is initially all bound to the BH, but it is only weakly bound because the radiation cannot escape and because the material originated on highly eccentric orbits. Some fraction of the returning gas is thus likely unbound (see, e.g., the simulations of Ayal et al., 2000), with energy being conserved as other gas accretes inward (Blandford & Begelman, 1999). If the outflow’s covering fraction is high, most of the radiated power will be emitted from the outflow’s photosphere, which can be far outside (Loeb & Ulmer, 1997). We now estimate the properties of this outflowing gas (see Rossi & Begelman 2009 for related estimates in the context of short-duration gamma-ray bursts).

In our simplified scenario, stellar debris falls back at close to the escape velocity and shocks at the launch radius , converting bulk kinetic energy to radiation:

(4)

where is the temperature at , is the density of gas at , and . Outflowing gas is launched from at a rate

(5)

and with terminal velocity

(6)

We approximate the outflow’s geometry as spherical, with a density profile

(7)

inside the outflow where ; the density falls quickly to zero at . We define the trapping radius via (where is the opacity due to electron scattering): inside , the gas is too optically thick for photons to escape and so the outflowing gas expands adiabatically. Because the outflow remains supported by radiation pressure, . The photosphere of the outflow is where . Because is likely not much smaller than , ; we thus neglect any deviations from adiabaticity between and .

At the earliest times for small and , the fallback rate can be so large and the density so high that the edge of the outflow limits the location of the photosphere to be just inside . In that case, the density of the photosphere is still given by ; lacking a detailed model, we assume that the photospheric gas near the edge is on the same adiabat as the rest of the gas, so that

Note that the photospheric temperature during the edge-dominated phase is essentially independent of all parameters of the disruption (e.g., , , etc.), and is only a weak function of time. The total luminosity during this phase grows as while the luminosity on the Rayleigh-Jeans tail increases even more rapidly, . After a time

(9)

the density falls sufficiently that the photosphere lies well inside ; the photosphere’s radius is then

(10)

and its temperature is

(11)

The adiabatically expanding outflow preserves the photon distribution function generated in the shock and accretion disk close to the BH. Estimates indicate that this gas is likely to be close to thermal equilibrium and thus we assume that the escaping photons have a blackbody spectrum

(12)

When the photosphere lies inside the edge of the outflow (i.e., so ), equations (10) and (11) imply that the total luminosity of the outflow is

(13)

The total luminosity of the outflow is thus of order the Eddington luminosity: see Figure 4, discussed in §4.1. Note that the total luminosity decreases for lower outflow rates, , because the photosphere’s surface area is smaller. The luminosity on the Rayleigh-Jeans tail (generally appropriate for optical and near-infrared wavelengths) declines even faster for lower , scaling as

(14)

These relations only apply if because otherwise the outflow is optically thin; we impose this lower limit to in our numerical solutions described later.

2.2 The Accretion Disk

We now consider the bound stellar debris that accretes onto the BH. After shocking at pericenter, this gas circularizes and viscously drifts inward, forming an accretion disk. The disk extends from down to the last stable orbit, . We expect the viscous time in the disk to be substantially shorter than the fallback time for at least a few years (see Ulmer 1999 for the case of , assuming a thick disk), and check this expectation at the end of §2.2; we thus assume that accretion during this period proceeds at . During the super-Eddington phase, the time for photons to diffuse out of the disk is longer than the viscous time, and so the disk is thick and advective. In contrast, at later times when , the disk is thin and can cool by radiative diffusion. We derive an analytic “slim disk” model (similar to the numerical work of Abramowicz et al., 1988) to describe the structure of the disk in both regimes.

To calculate the disk’s properties, we solve the equations of conservation of mass, momentum, and energy:

(15)
(16)
(17)

where we have approximated the radial entropy gradient as . Here is the accretion rate, is the cylindrical distance from the BH, is the disk scale height, is the density, is the radial velocity, and is the midplane temperature. The no-torque boundary condition at the inner edge of the disk implies . We neglect gas pressure, since radiation pressure is dominant throughout the disk for at least a few years; we further assume that the viscous stress is proportional to the radiation pressure, so that (Shakura & Sunyaev, 1973) with sound speed and , where . Simulations indicate that this assumption is reasonable and that such disks are thermally stable (Hirose et al., 2009). The vertically integrated heating and cooling rates are given by and , where the half-height optical depth is and is the electron scattering opacity. These relations form a quadratic equation for the dimensionless quantity ,

(18)

Solving equation (18) yields the effective temperature of the disk,

(19)

Combining this relation with equation (2), we calculate the luminosity and spectrum of the disk as a function of time, modeling it as a multicolor blackbody.

The solution to (18) also yields the disk scale height ratio,

(20)

The scale height while , and decreases as at fixed at later times. The viscous time at a radius in the disk is

(21)

which is times the local dynamical time during the super-Eddington phase, and later increases as at fixed . For , the viscous timescale evaluated at the disk’s outer edge is shorter than the time since disruption for ; our assumption of steady-state accretion during this period is thus consistent.

3 The Equatorial Unbound Material

While half of the initial star becomes bound to the BH during the disruption, the other half gains energy and escapes from the BH on hyperbolic trajectories. From the viewpoint of the BH, this unbound material subtends a solid angle , with a dispersion in azimuth and a dispersion in orbital inclination. This material absorbs and re-radiates a fraction of the luminosity from the accretion disk.222The polar outflow could also irradiate the unbound material, but it will have less of an effect because its spectrum is softer and its luminosity declines more rapidly. We now estimate the dimensions of the unbound wedge.

Figure 1: Spatial diagram of the equatorial stellar debris in the plane defined by the original orbit of the star. The unbound material is shown 0.1 yr, 0.3 yr, and 1 yr after the tidal disruption of a solar-type star by a BH at . The dashed and dotted curves indicate the incoming parabolic trajectory of the star and its continuation if the star were not disrupted. The debris also has an inclination dispersion perpendicular to this plane of .

In the orbital plane at a fixed time , the unbound stellar debris lies along an arc, as the spread in specific energy produces a spread in radius and azimuthal angle (see Fig. 1). The most energetic particles escape on a hyperbolic orbit with eccentricity . These particles race away from the BH at a substantial fraction of the speed of light,

(22)

(ignoring relativistic effects) and lie furthest from the BH at a distance

(23)

They also have the smallest angle away from stellar pericenter, , where obeys so that (see also Khokhlov & Melia, 1996). Particles with lower energies and thus smaller eccentricities are closer to the BH and make a larger angle relative to pericenter, up to . This produces an azimuthal dispersion .

Figure 2: Spectral energy distributions for our three fiducial tidal disruption flares at several different times after disruption. These spectra include only the emission from the super-Eddington outflows (for and ; see eqns [5] & [6]), which dominate the emission at early times (see Fig. 3). For and (right panel), the disk dominates the emission for days (Fig. 3); this is why we do not plot the outflow emission at later times.
Figure 3: Light curves at - (solid) and -band (dashed) for our three fiducial models. In each panel, times to the left of the vertical dotted line (; eq. [3]) have super-Eddington fallback rates and an outflow dominates the optical emission; at later times, the fallback rate is sub-Eddington and the emission is produced by the accretion disk and the photoionized unbound material (see §4.2). In the leftmost panel, the emission rises at early times because the edge of the outflow limits the size of the photosphere. The optical emission then declines until the end of the outflow phase as the photosphere recedes and the photospheric temperature rises (Fig. 2).

Particles having the maximum vertical velocity have vertical specific angular momentum , and total specific angular momentum

(24)

to lowest order in . The orbital inclination is given by , so . The resulting inclination dispersion is (our result is consistent with Evans & Kochanek 1989 but we disagree with Khokhlov & Melia 1996).

The finite inclination dispersion produces a vertical wall of debris whose inside face scatters, absorbs, and re-radiates a fraction of the disk’s emission. This face subtends a solid angle

The number density of particles in the unbound wedge is , where is the radial dispersion of the material at fixed . This dispersion is due to differences in the particles’ radial velocities and azimuthal positions when the star passes through pericenter. Particles at travel on orbits whose pericenter is shifted from the star’s pericenter by an angle , which produces a spread in radial position . The number density is then

(26)

and the radial column density seen by the black hole is

(27)

As the unbound material expands, it cools very quickly; after at most a few weeks, the gas would all be neutral if not for the disk’s ionizing radiation. This radiation ionizes the surface layer of the unbound material. The ionized gas in turn emits via bremsstrahlung, radiative recombination, and lines. The physical conditions and processes here are similar to those in the broad line region of an active galactic nucleus (AGN).

The ionized gas can reach photoionization equilibrium provided conditions change more slowly than the hydrogen recombination rate . The recombination coefficient for hydrogen is , and is the electron number density. In the ionized region, , as we show below. The material can remain in equilibrium for at least a few years, until :

(28)

The column depth of the ionization front is , where is the ionization parameter,

(29)

The electron density in the ionized layer is and the fractional depth of the ionization front is

so the ionized layer is typically thin and highly ionized.

4 Predicted Emission

We use the results of §§2 and 3 to calculate the emission due to the tidal disruption of a solar-type star as a function of time and wavelength. We consider a solar-type star because the stellar mass-radius relation and typical stellar mass functions imply that these stars should dominate the event rate. The two key parameters we vary are the star’s pericenter distance and the BH mass . We consider the mass range .

4.1 Super-Eddington Outflows

Early on, when (), outflowing gas likely dominates the emission. We calculate its properties using results from §2.1.

In Figure 2 we plot the spectral energy distribution at various times during the outflow phase, for three fiducial models: and ; and ; and and . We take nominal values of and ; we discuss the uncertainties in these parameters in §§5.3 and 6. The photosphere lies well inside the edge of the outflow at all times shown in Figure 2. The emission from the outflow has a blackbody spectrum, initially peaking at optical/UV wavelengths. As time passes and the density of the outflow subsides, the photosphere recedes and the emission becomes hotter but less luminous.

In Figure 3, we plot - (4770Å) and - (7625Å) band light curves for the three fiducial models. In the leftmost panel at , the edge of the outflow limits the size of the photosphere, so the photosphere initially expands, following the edge of the outflow. After a time (eq. [9]), however, the photosphere begins to recede inside the edge of the outflow and the luminosity declines. In the middle and rightmost panels, the photosphere lies well inside for virtually the entire outflow phase. The optical emission decreases as the photosphere’s emitting area decreases and the temperature rises only slowly. As Figures 2 & 3 demonstrate, the peak optical luminosity of the outflow is substantial, , comparable to the optical luminosity of a supernova. The color of the emission is quite blue (). To illustrate how the peak luminosity depends on the parameters of the disruption event, Figure 4 shows the peak bolometric and -band luminosities of the outflow as a function of , for and . For sources at cosmological distances (which are detectable; §5), the negative k-correction associated with the Rayleigh-Jeans tail implies that the rest-frame -band luminosity in Figure 4 underestimates the peak optical luminosity visible at Earth.

Figure 4: Peak bolometric (heavy lines) and -band (light lines) luminosities of the early-time super-Eddington outflows as functions of , for (dashed) and (solid), assuming that and . Figure 11 shows the duration of this phase.

4.2 Disk and Photoionized Unbound Debris

When , a fraction of the falling-back gas is blown away while the remainder likely accretes via an advective disk (§2.2). As the fallback rate declines below Eddington, the photons are able to diffuse out of the region close to the BH and the disk cools efficiently, but also becomes less luminous. The vertical dotted line in Figure 3 delineates the super-Eddington fallback (and outflow) phase from the sub-Eddington fallback phase.

The accretion disk irradiates the surface of the equatorial unbound stellar material (§3). In this section we calculate the combined emission produced by the accretion disk and the irradiated stellar debris. In order to isolate the more theoretically secure emission by the disk and photoionized material, we do not consider the emission from super-Eddington outflows in this section. We show results for the disk and photoionized material at both and ; depending on the geometry of the outflow, and the viewing angle of the observer to the source, it is possible that all three emission components could be visible at early times. Because the mass driven away by outflows during the super-Eddington phase can also be photoionized by the central source at times , our emission line predictions are likely a lower limit to the total emission line fluxes (§6).

Figure 5: Spectral energy distributions for tidal flares around a non-rotating BH, 30, 100, 300, and 1000 days after disruption. Emission from the accretion disk dominates at short wavelengths. The photoionized unbound stellar debris absorbs and re-radiates some of the disk’s emission, producing optical-infrared emission. These spectra do not include the emission from super-Eddington outflows at early times; see Figs. 2 & 3 for this emission.
Figure 6: UV to near-infrared spectra for tidal flares around a non-rotating BH, 30, 100, 300, and 1000 days after disruption. The spectra are the sum of contributions from the accretion disk and the photoionized unbound material, but do not include the emission from super-Eddington outflows, which likely dominate at early times (Fig. 3). The linewidths and line strengths are both larger for smaller and smaller (eq. [22]).

We calculate the photoionization properties of the unbound material using version 07.02.02 of the publicly available code Cloudy, last described by Ferland et al. (1998). We simplify the geometry: the unbound spray traces out a widening spiral shape with most of the area coming from close to , so we approximate it as a cloud of area located a distance from the ionizing source. Our model cloud has constant density (eq. [26]), column depth (eq. [27]), and is irradiated by the accretion disk having the luminosity and spectrum described in §2.2. The total emission calculated here is the sum of the emission from this photoionized layer and the emission from the central accretion disk. We focus on non-rotating BHs (), although we quote results for rapidly rotating holes () as well.

In Figure 5 we plot the spectral energy distribution 30 days, 100 days, 300 days, and 1000 days after disruption, for our three fiducial models: and ; and ; and and . The early-time short-wavelength peaks at with luminosity are emission from the disk. After a time , the mass fallback rate declines below the Eddington rate, and the disk begins to cool and fade. For and , the optical light is dominated by lines and continuum from the photoionized material. For larger (and larger ), the equatorial debris subtends a smaller solid angle (see eq. [3]) and the disk’s luminosity is larger, so the disk dominates the optical emission.

Figure 7: Line strength evolution for the three fiducial models. Results are shown for Ly (solid), CIV 1548+1551 (dotted), H (dashed), H (dot-dashed), and Pa (triple-dot-dashed). The quantity is the ratio at line center of the line intensity to continuum intensity, taking into account emission from both the disk and the photoionized material, but not the early-time super-Eddington outflows.
Survey Cadence Rate: D+UB Rate: Outflows
Pan-STARRS Survey 23 AB mag (-band) 0.75 6 months 200
Pan-STARRS Medium Deep Survey (MDS) 25 AB mag (-band) 4 days 20
Palomar Transient Factory (PTF) 21 AB mag (-band) 0.2 5 days 300
Large Synoptic Survey Telescope (LSST) 24.5 AB mag (-band) 0.5 3 days 6000
Synoptic All-Sky Infrared Survey (SASIR) 23.5 AB mag (-band) 0.03? 10 days? 100
ROSAT All-Sky Survey 1 N/A
GALEX Deep Imaging Survey 25 AB mag () N/A

Rates for the emission from the disk and unbound equatorial debris; the range corresponds to (low), (high). This emission is relatively faint in the optical/infrared and may be difficult to detect relative to the host bulge (see §6).
Rates for the emission from super-Eddington outflows, restricted to , for and (see Fig. 13).
The survey strategy for SASIR has not yet been finalized.
These parameters are for comparison with the all-sky rate calculation by Donley et al. (2002).
ROSAT and GALEX do not have regular cadences.
ROSAT and GALEX have insufficient cadence and/or sky coverage to detect flares from super-Eddington outflows.

Table 1: Assumed parameters for transient surveys and predicted rates. Our results can be scaled to other survey parameters and model assumptions using eq. (31).

Figure 6 zooms in on the UV/optical/near-infrared spectra for our three fiducial models. The emission lines characteristic of the broad line region of an AGN are typically the strongest lines here as well: e.g., Ly, CIV 1548+1551, H, and H. In most cases, these lines are optically thick for more than a year. The lines are extremely broad, since the marginally bound gas has a speed close to zero while the most energetic gas leaves the BH at , , and for , ; , ; and , , respectively (see eq. [22]). In addition, the mean velocity along our line of sight will usually be substantial, so the lines should have a large redshift or blueshift on top of the galaxy’s redshift. For clarity, we plot the spectra with a mean redshift of zero.

Figure 7 focuses on the evolution of five strong lines, plotting the ratio at line center for each.333By comparing the results of different versions of Cloudy, we find that the results for line strengths can be uncertain by up to factor few. The quantity is the line intensity at line center accounting for the significant broadening. As the surface area of the equatorial wedge grows in time, line luminosities grow until and irradiation by the disk subsides. The quantity is the continuum intensity, which includes the contributions of both the disk (blackbody) and the photoionized unbound material (bremsstrahlung and radiative recombination)—again, the emission from the super-Eddington outflows is not included in . The lines remain prominent for a few years, and are strongest and broadest for small and small . The UV lines are the strongest lines when the unbound material dominates the continuum (left panel), while the near-infrared lines are the strongest when the disk dominates the continuum (middle and right panels).

We next describe the broadband optical evolution of a tidal disruption event. Figure 3, also discussed in §4.1, plots the optical light curve for each fiducial model, showing the total emission at both - and -bands. For (left of the dotted lines), the emission is dominated by the super-Eddington outflows, while for the emission is dominated by the accretion disk and photoionized equatorial debris. Once , i.e., , the disk’s optical luminosity falls off gently, approximately as : although the bolometric luminosity is declining as , the optical emission lies on the Rayleigh-Jeans tail. Increasing and/or increases the disk’s luminosity by up to two orders of magnitude because of the disk’s larger emitting area and/or because rises. At all times, the disk emission is quite blue ().

For large and/or large (middle and right panels in Fig. 3), the disk outshines the photoionized material at optical wavelengths, and the light curves and color evolution are determined by the disk emission alone. By contrast, for and (left panel), the photoionized material’s optical line emission is initially an order of magnitude brighter than the disk. As the illuminating power of the disk declines but the unbound debris becomes less dense, different lines wax and wane. The significant redshift or blueshift of the unbound material further complicates the photometry by altering which lines contribute in which wavebands (again, our figures assume a mean redshift of zero). These effects can produce a non-monotonic light curve and a complicated color evolution, depending on the exact redshift of the source and the velocity of the equatorial debris.

5 Predicted Rates

We use our calculated spectra and light curves to predict the number of tidal disruption events detectable by observational surveys. We focus on an (almost) all-sky optical survey like the Pan-STARRS PS1 survey, but we also predict results for surveys with more rapid cadence (e.g., PTF and LSST) and discuss the results of our models compared to ROSAT and GALEX observations. Our assumed survey parameters are listed in Table 1, along with some of our results.444These are intended to be illustrative, and may not correspond precisely to the true observational survey parameters, although we have attempted to be as accurate as possible. Our results can readily be scaled to other surveys using equation (31) discussed below.

To predict rates, we use the redshift-dependent BH mass function given by Hopkins et al. (2007). At , the BH density is for and gently falling at higher masses; as rises to 3, the BH number density falls by . We assume that the BH number density for is the same as for , although it is poorly constrained observationally. We do not consider tidal disruption events beyond . (Our results can easily be scaled to other assumed BH mass densities; see eq. [31]).

The rate of tidal disruptions within a single galaxy is . To predict detection rates, we assume that is independent of BH mass. We adopt as found by Donley et al. (2002) using the ROSAT All-Sky Survey, which is also in line with conservative theoretical estimates. We further assume that this rate is distributed equally among logarithmic bins of stellar pericenter distance , so that . In the limit of , the equation for the predicted rate is

(30)

where is the fraction of sky surveyed; when necessary we use the generalization of equation (30) that includes cosmological effects. When the duration of a flare is shorter than the cadence of the survey , we approximate the probability of detection as .

We start by considering emission from only the accretion disk and photoionized equatorial debris. Then in §5.3, we include the emission from super-Eddington outflows, where the physics is somewhat less certain, but the observational prospects are particularly promising.

5.1 Disk and Photoionized Material

For all but the largest , the duration of peak optical emission for the accretion disk and photoionized material is (eq. [3]). This timescale depends on the BH’s mass and the star’s pericenter distance, as shown in Figure 8. For , the flare lasts for and then decays only gently since the disk dominates the optical emission. However, for and , the optical flare is shorter, , and then the emission decays more quickly since irradiation of the unbound material—the dominant source of optical emission—subsides. For , (eq. [2]) because the fallback rate in these systems is never super-Eddington.

Figure 9 shows our calculated rates for optically-detected tidal flares (for a survey like the Pan-STARRS 3 survey), for both non-rotating and rapidly rotating BHs. For , the disk contributes most of the emitted power, so the rates increase with as increases. The rates are dominated by and (non-rotating BHs) and (rapidly rotating BHs). Since most of the flares that dominate the rates have relatively long durations (Fig. 8), imperfect survey cadence only modifies the detection rates by . At and small , the photoionized material re-emits a relatively large fraction of the disk’s power in the optical and boosts the detection rates significantly.

Integrated over , our estimated rates for the Pan-STARRS 3 survey, assuming non-rotating BHs, are and in -band and -band, respectively. The mass range contributes another (both -band and -band), assuming (probably optimistically) that and are the same at as at . If the BH is rotating faster, can be smaller. This allows an accretion disk to form for even , widens the disk for all , and increases the solid angle of the unbound material. Indeed for and , the unbound material covers a quarter of the sky! (At this point so our approximations begin to break down.) These effects raise the total predicted rates for rapidly rotating BHs to .

Figure 10 plots the detection rates as a function of for , , and , for the disk alone (light lines) and for the disk plus photoionized material (heavy lines). The photoionized material enhances detection rates significantly for most at and , but has little effect for . The rates decrease substantially for because then the outer radius of the disk is at the last stable orbit; our disk model assumes a no-torque boundary condition at , implying that there is essentially no emission from the disk when .

Although the rates quoted above and in Figures 9 and 10 are for a survey covering 3/4 of the sky, assuming constant and constant BH mass density below , our predicted rates can be scaled to other assumed parameters:

(31)

assuming that the sources are at , where .

Figure 8: Duration of maximum luminosity during the late-time accretion disk phase, plotted as a function of for and . Most flares last longer than a month. The super-Eddington outflow at early times produces a shorter flare that should precede this emission (Figs. 3 & 1113). (The fallback rates for are never super-Eddington, so the flare duration in these systems is .)
Figure 9: Predicted detection rates as a function of BH mass for the Pan-STARRS survey (see Table 1). Results are shown for non-rotating BHs (left panel) and rapidly rotating BHs (right panel). The rates shown here can be scaled to other surveys and other model parameters using eq. (31). These rates do not include the emission from super-Eddington outflows at early times; see Figs. 12 & 13 for these results.

The Pan-STARRS MDS, PTF, and LSST will have cadences of only a few days, giving them sensitivity to all events within their survey volumes. However, for detecting this relatively long-lived emission, the advantage of fast cadence is only minor; more significant is the spatial volume probed by the survey (see Table 1). The MDS will deeply () image a relatively small (84-deg) region, and so will detect as many events as the Survey, per year. PTF will image a larger region () but less deeply (), and so will have detection rates similar to the MDS. The emission from early-time super-Eddington outflows may significantly increase these rates, as we discuss in §5.3. In the next decade, LSST will image a large region (20,000 ) deeply (), and so will discover hundreds of tidal flares. MDS and LSST will detect events at cosmological distances (), where the negative k-correction of the disk’s blackbody peak enhances rates by a factor of a few. By co-adding images up to , these short-cadence surveys will also be able to raise their detection rates of events having by a factor of a few. Also in the next decade, SASIR plans to deeply image 140 each night in the near infrared (to at -band; Bloom et al. 2009). Their observing strategy is not yet finalized; assuming a 10-day cadence covering 1400 deg (a good strategy for detecting flares during the super-Eddington outflow phase; see §5.3), this survey should detect a flare every few years from the accretion disk plus photoionized debris.

5.2 Current Observational Constraints

Figure 10: Predicted detection rates as a function of pericenter distance for the Pan-STARRS survey at -band (see Table 1). Results are shown for (solid), (dashed), and (dot-dashed), assuming a non-rotating BH. The thin/light lines are for the disk emission alone, while the thick/heavy lines include the emission from both the disk and the photoionized unbound material. The photoionized material significantly increases the rates for low and small . These rates do not include emission from super-Eddington outflows at early times; those results are shown in Figs. 12 & 13. The small fluctuations in the results for are due to difficulties in performing Cloudy calculations at very high densities.

Figure 11: Duration of peak luminosity during the early super-Eddington outflow phase, as a function of for and (for and ; see eqs. [5] & [6]). The vertical dotted lines mark the boundary between events where the edge of the outflow limits the size of the photosphere (lower BH masses) and where it does not (higher BH masses). The flares from super-Eddington outflows typically last longer than the few-day cadence of surveys like PTF, Pan-STARRS MDS, and LSST, but they are often short enough that they would not be detected in surveys optimized solely for supernovae.

Figure 12: Predicted detection rates for emission from super-Eddington outflows at early times (for and ; see eqs. [5] & [6]). Results are shown as a function of for , , and , all for a Pan-STARRS -like survey at -band (see Table 1).

As a check on our model, we have calculated rates for the ROSAT All-Sky Survey for comparison with Donley et al. (2002)’s result of 42 events per year over the whole sky for