Radio-X-ray Synergy to discover and Study Jetted Tidal Disruption Events
aa r X i v : . [ a s t r o - ph . H E ] J a n Radio-X-ray synergy to discover and study jetted tidal disruptionevents
I. Donnarumma , E. M. Rossi INAF-IAPS, Via Fosso del Cavaliere 100, 00133, Rome, Italy Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA , Leiden, TheNetherlands [email protected]
Received ; accepted 2 –
ABSTRACT
Observational consequences of tidal disruption of stars (TDEs) by supermas-sive black holes (SMBHs) can enable us to discover quiescent SMBHs, constraintheir mass function, study formation and evolution of transient accretion disksand jet formation. A couple of jetted TDEs have been recently claimed in hardX-rays, challenging jet models, previously applied to γ -ray bursts and activegalactic nuclei. It is therefore of paramount importance to increase the currentsample. In this paper, we find that the best strategy is not to use up-coming X-ray instruments alone, which will yield between several (e-Rosita) and a couple ofhundreds (Einstein Probe) events per year below redshift one. We rather claimthat a more efficient TDE hunter will be the Square Kilometer Array (SKA)operating in survey mode at 1.4 GHz. It may detect up to several hundreds ofevents per year below z ∼ . z ≈ . not t − / , however, is not seen in radio, whose flux is quitefeatureless. Identification therefore requires localization and prompt repointingby higher energy instruments. If radio candidates would be repointed within aday by future X-ray observatories (e.g. Athena and LOFT-like missions), it willbe possible to detect up to ≈
400 X-ray counterparts, almost up to redshift 2.The shortcome is that only for redshift below ≈ .
1. Introduction
Since the late 70s it has been suggested that stars torn apart by the gravitational fieldof a supermassive black hole (SMBH) may be observed as flares from Earth (Hills 1975;Frank & Rees 1976; Rees 1988; Phinney 1989). These are called tidal disruption events(TDEs). These flares would be caused by sudden accretion of the star debris, which wouldfeed the SMBH at an ever decreasing rate, ˙ M ∝ t − / . This theoretical expectation is for acomplete disruption of a star in parabolic orbit, after at least several days from the peak(e.g. Lodato et al. 2009; Hayasaki et al. 2013; Guillochon & Ramirez-Ruiz 2013), and itis expected to be independent on the ratio of pericenter to tidal radius (Sari et al. 2010;Stone et al. 2013).The detection and study of these flares can deliver important astrophysical information.On the one hand, they allow us to detect otherwise quiescent SMBHs and estimate theirmasses. This would inform theory of galaxy-SMBH cosmological co-evolution. On theother, they constitute a unique opportunity to study the – highly theoretically uncertain– formation of an accretion disc and its continuous transition through different accretionstates. As the accretion rate decreases, we can in principle observe a disc which transitsfrom an initial super-Eddington phase, lasting several months, passing through a slim andlater a thin disc regime, and ending its life, years later, in a radiative inefficient state.The super-Eddington phase –which occurs only for SMBH masses < ∼ M ⊙ — is highlyuncertain, but it may be associated with a copious radiative driven wind (Rossi & Begelman2009), which thermally emits ∼ − erg s − , mainly at optical frequencies(Strubbe & Quataert 2009; Lodato & Rossi 2011). The disc luminosity ( ∼ − ergs − ), instead, peaks in far-UV/soft X-rays (Lodato & Rossi 2011). Of paramount theoreticalimportance would also be the possibility to investigate the formation and evolution of anassociated jet, powered by this sudden accretion. There is no specific theory for the jet 4 –emission from TDEs. Astronomers mainly assume a phenomenological description (e.g.Van Velzen et al. 2011; Canizzo et al. 2011) or borrow theory developed for blazars and/or γ -ray bursts (e.g. Metzger et al. 2012; Tchekhovskoy et al. 2013). In general, non-thermalemission in X-rays and radio is the jet signature.Handful of candidate TDEs ( ∼
10) have been detected so far, particularly inROSAT all sky survey (Komossa 2002; Donley et al. 2002), in GALEX Deep ImagingSurvey (Gezari et al. 2009; Gezari et al. 2012; Campana et al. 2011) and in SDSS(Van Velzen et al. 2011a). These “soft” events are believed to be associated with thedisc and wind thermal emission. The presence of a bright optical flare in the initialsuper-Eddington months makes optical surveys a useful tool for discovery. Significantadvances in optical transient surveys are expected to be achieved by the PanoramicSurvey Telescope and Response System (Pan-STARRS) and the Large Synoptic SurveyTelescope (LSST). Two candidates have been claimed in Pan-STARRS data (Gezari et al.2012; Chornock et al. 2014), three in PTF data (Arcavi et al. 2014) and one in ASAS-SN(Holoien et al. 2014), but the total number expected seems to be much higher. For examplein the 3 π Survey, claims in literature range from 200 to ∼ ∼ −
20 should be found (Strubbe & Quataert 2009;Van Velzen et al. 2011a).
Thousands of candidates could be, instead, detected by LSST,with its 6-band (0 . − . . γ -rays revealed a new class of TDEs, where we 5 –are likely observing the non-thermal emission from a relativistic jet. The jet emission isresponsible for the hard X-ray spectrum (with power-law slope β ∼ .
7) and the increasingradio activity (Levan et al. 2011), detected a few days after the trigger.Given the lack of statistics and of a solid theoretical framework for the non-thermalemission, we will take the best studied of these two events, Swift J1644+57 (Sw J1644in short), as a prototype for the study presented in this paper, where we investigate thedetection capability of both SKA and future X-ray observatories.Sw J1644 was hosted by a star forming galaxy at z = 0 .
354 and in positional coincidencewith its center (Zauderer et al. 2011). Its X-ray peak luminosity ∼ × erg s − wasreached after a couple of days from the trigger, and it persisted at the level of > ergs − for about 1 year. During its decay, the X-ray emission was approximately described bya t − / temporal law, the same as that expected for the fallback of stellar debris (see Figure1). After ∼
500 day from the trigger, the X-ray flux declined by two orders of magnitudeand it has been associated with a shut off of the relativistic jet (Zauderer et al. 2013).The modelling of the X-ray luminosity suggests that Sw J1644 is associated with a lightsupermassive black hole < ∼ M ⊙ (e.g. Burrows et al. 2011; Canizzo et al. 2011).Variability at optical wavelengths within the host was not detected, while transientemission was seen in infrared, becoming stronger at longer wavelengths, especially atmillimeter and radio wavelengths. Radio (1.4. and 4.8 GHz) observations from WesterborkSynthesis Radio Telescope (WSRT) showed a bright source. EVLA observations of the radiotransient coincident with the host galaxy were reported, providing an estimate of the bulkLorentz factor Γ ∼ γ -ray bursts by Metzger et al. (2012), andindicate a more complicated jet structure, like perhaps in the magnetically arrested modelproposed by Tchekhovskoy et al. (2013). Snapshot rates of jetted TDEs in radio band havebeen computed for the first time by Van Velzen et al. (2011). Differently from their work,we adopt here a different modelling for the radio lightcurve and a more detailed one forthe black hole mass function, which includes the redshift dependence. We also account fora stellar mass function. We broaden up our investigation to include X-ray detection andfollow-ups.Finally, a 200-s quasi-periodic oscillation (QPO) was detected by both Suzaku andXMM, ∼
10 and 19 days after the Swift/BAT trigger, respectively (Reis et al. 2012). QPOsare regularly detected in stellar mass BHs, but there is no firm physical interpretation ofthese phenomena. However, most models strongly link the origin of high-frequency QPOswith orbits or resonances in the inner accretion disk close to the BH. This may causevariable energy injection into the jet, which consequently results in variability in the X-rayemission. This interpretation led to estimate a BH mass between 5 × M ⊙ and 5 × M ⊙ (Reis et al. 2012).In this paper, we predict the detection rate of jetted TDEs considering current andfuture radio surveys (NVSS + FIRST, VLT Stripe 82, ASKAP, VLASS and SKA) andX-ray instruments (Swift, eRosita, Einstein Probe, Athena, LOFT). In addition, we discussthe ability of these instruments to constrain important physical parameters.The paper is organized as follows. In §
2, we take Swift J1644 as a prototype and wedescribe our phenomenological model for X-ray and Radio emissions. In §
3, we discuss theblack hole distribution functions used in this paper. In §
4, we present our Monte Carlocalculations. Our rates for current and future surveys are presented in §
5. A summary andimplications of our results can be found in §
6. Finally, we draw our conclusions in §
7. 7 –Throughout this paper we use the following cosmological parameters: Ω M = 0 . λ = 0 .
75 and H = 70 km / s / Mpc.
2. Modelling the Lightcurve
A tidal disruption event of a star by a SMBH causes a transient accretion disc toform, whose accretion rate is set by the rate at which the stellar debris falls back to theblack hole under its gravitational pull. How matter circularizes to form a disc and whetherthis process is accompanied by outflows and their characteristics are subject to intenseinvestigations, as mentioned above. From phenomenology and theory, we know that in thepresence of an accretion disc and some ordered magnetic field, matter and energy outflowsin form of (relativistic) jets are produced. In the absence of fully consistent simulations ofjet production by a tidal disruption event, we use below a simplified description for the jetenergy content as a function of time. This is partially supported by analytical and numericalcalculations (see references above) and partly by the observed features of the X-ray emissionof Sw J1644. In particular, its temporal decay ( ∼ t − / ) suggests that at least in thisoptically thin regime, the X-ray luminosity scales as the accretion rate. As a consequence,it supports a scenario in which the star was completely tidally disrupted, since partialdisruption would lead to a shallower decay of the fallback rate (Guillochon & Ramirez-Ruiz2013). Moreover, a partial disruption is difficult to reconcile with a long lasting superEddington accretion phase, which may be needed to power the jet for its total duration of ∼
500 days. Finally, the modelling of the X-ray luminosity suggests that Sw J1644 is theconsequence of a disruption of a roughly one solar mass star by a light supermassive blackhole < ∼ M ⊙ (e.g. Burrows et al. 2011; Canizzo et al. 2011). 8 – We work in the framework of two identical jets, with θ j < / Γ. The total energyinjected in the two jets is L j = ǫ j ˙ M fb c , where ǫ j is the jet production efficiency, which weassume constant in time, and the gas fall back to form a disc occurs at a rate ˙ M fb . For acomplete disruption of a star in parabolic orbit the fallback rate can be approximated by˙ M fb (¯ τ ) = ˙ M p (cid:18) t min + ¯ τt min (cid:19) − / , (1)(Rees 1988; Phinney 1989). The lag time “¯ τ ” is the time from the beginning of the debrisaccretion, that roughly happens after a time t min ≈ M / m / ∗ , day , from the star disruption, in the galaxy rest frame. More precisely, t min is the minimumtime it takes the most bound debris to come back to pericenter after the star hasbeen torn apart. Here and in the following, M is the BH mass in units of 10 M ⊙ and m ∗ , the mass of the disrupted star in units of 1 M ⊙ . The peak of the accretionrate is quite intuitively the mass of the star divided by the characteristic timescale,˙ M p ≈ (1 / m ∗ /t min ≈ . × M − / m / ∗ , g s − . In our description, the jet is launched atthe onset of accretion (¯ τ = 0), as there are no strong theoretical reasons why it should bedelayed. The temporal evolution of the jet energy is thus L j (¯ τ ) = L j , p (cid:18) t min + ¯ τt min (cid:19) − / , (2)where L j , p = ǫ j ˙ M p c ≈ . × erg s − (cid:16) ǫ j . (cid:17) M − / m / ∗ , . (3) In the formula used in this paper, we assume the standard linear relation between massand radius of the star. See eq.6 in (Lodato & Rossi 2011). 9 –Note that the larger the black hole mass, the lower the peak luminosity, because thecharacteristic timescale increases. Viceversa, the jet luminosity decreases with m ∗ . The unabsorbed ≈ τ ≈ after the actual disc and jet formation. The observed time interval τ is related tothe rest frame analogous quantity by τ = ¯ τ (1 + z ) and in this case ¯ τ ≈ L x , iso (∆ t ) ≈ . × erg s − (cid:18) τ + ∆t τ (cid:19) − / , (4)(Fig. 1, solid line). Specifically, L x , iso is an isotropic equivalent luminosity, computedfrom the X-ray flux. Note that here τ = 3 is a fixed time delay, unlike ¯ τ in eqs.1 and 2.Superimposed to this baseline trend, there is a complex structure of flares and dips wherethe flux oscillates within two orders of magnitude in the first ten days of observations. It isclear that eq.4 does not capture this large variability, possibly associated with jet precessionand nutation (Saxton et al. 2012; Stone et al. 2012). But in absence of a compelling theoryfor these sudden X-ray variations, we prefer to reproduce the upper part of the envelopethat contains the initial variability, since the BAT instrument was triggered by one of thepeaks in the lightcurve. We will discuss later how this choice affects our X-ray TDE rateestimates.The Swift/XRT (0.3-10 keV) spectrum of Sw J1644+57 is well described by an absorbed 10 –power-law with a photon index β ≈ . − . N H ≈ × cm − (Burrows et al. 2011).The observed BAT spectrum at early times and its count rate later on (up to the beginningof June) are consistent with an extrapolation at higher energies of the XRT spectrum(Burrows et al. 2011). This suggests that we are observing the same component in both softand hard X-ray bands. The average spectrum is consistently hard (1 . < ∼ β < ∼ .
7) duringthe whole emission, although a spectral softening is observed during the short dips in theinitial variable phase (Saxton et al. 2012). The radiation efficiency in 1-10 keV band (i.e.the fraction of the total luminosity emitted in that band) is ǫ x ≈ .
20 (Burrows et al. 2011).With this last information, we can calculate the associated jet kinetic luminosity from theobserved light curve, once we assume a jet opening angle θ j and a Doppler factor δ , L j = L x , iso (1 − cos θ j ) / ( ǫ x δ ) . With the highest probability, our line of sight is at an angle ∼ Γ − (i.e. the inverse ofthe Lorentz factor Γ) that grazes the relativistic beam, and δ ≈ Γ. The fact that there areno sharp breaks in the lightcurve may indicate that the whole emitting area was visible,i.e. Γ − > θ j . Therefore, we further assume a jet opening angle of a similar size of therelativistic beaming, say θ j ≈ Γ − /
2, and we get a jet power at the trigger time (∆ t = 0)of L j (¯ τ ) ≈ . × erg s − (Γ / − . Since L j (¯ τ ) = ǫ j ˙ M fb (¯ τ ) c , it turns out that to havean efficiency ǫ j greater than 1% requires m ∗ ≤ M ⊙ , for Γ ≤
5. In particular, m ∗ = 1 M ⊙ gives efficiency between roughly 1% and 37% for 2 ≤ Γ ≤
5, that are in agreement withnumerical simulations of jets from highly super-Eddington accretion discs S¸adowski et al.(2014). Lower mass stars would give a higher efficiency range. We therefore assume in thefollowing that Sw J1644 is the result of the disruption of a solar mass star. However, it isclear that this is just a tentative, though reasonable, choice, since the stellar mass cannotin fact be univocally determined, unless we can actually measure θ j .Assuming Sw J1644 as a prototype, we can adopt a general description of the X-ray 11 –lightcurve in the 1-10 keV band, when we catch the flare after a time τ from the beginningof the event, L x , iso (∆ t ) = L x , t (cid:18) τ + ∆t τ (cid:19) − / . (5)The (isotropic equivalent) luminosity L x , t at the time of the trigger (∆t = 0) is L x , t = L j (¯ τ ) ǫ x δ / (1 − cos θ j ) ≃ L j (¯ τ ) ǫ x /θ j ) , which can be written more explicitly as L x , t ≈ . × erg s − M − / m / ∗ , (cid:18) ǫ x ( z )0 . (cid:19) × (cid:18) t min + ¯ τt min (cid:19) − / , (6)where ¯ τ = τ / (1 + z ) and the radiation efficiency ǫ x ( z ) varies because of the spectral shiftingwith redshift, ǫ x ( z ) = 0 .
20 ( E (1 + z )) − β +2 − ( E (1 + z )) − β +2 ( E (1 + z sw )) − β +2 − ( E (1 + z sw )) − β +2 , (7)where we assume β = 1 . E = 1 keV, E = 10 keV and z sw = 0 .
35. We note that thiscorrection is in the source rest-frame and applies to unabsorbed fluxes.In eq.6, we set ǫ j Γ /θ ≈ .
9. Indeed, any combination of these quantities that gives afactor ≈
24, allows us to reproduce the Sw J1644 X-ray luminosity at the trigger time. Thedegeneration should then be lifted, when we need to choose a Lorentz factor to compute theTDE rates. From the X-ray luminosity, the flux is easily computed, F ν = L x , iso πD , where D is the luminosity distance. In this section, we first reproduce the lightcurve at 1.4 GHz of Sw J1644 and then wegeneralize it to events at different redshifts and with different stellar and black hole masses. 12 –The radio emission is synchrotron emission and the low energy spectrum can bedescribed with the following broken power-law F ν = F ν ( ν a ) "(cid:18) νν a (cid:19) − s + (cid:18) νν a (cid:19) − s / − /s × (cid:20) νν m s (cid:21) − /s , (8)(Granot & Sari 2002) where ν a < ν m are respectively the absorption and peak frequency, s , s are smoothing factors and the electron power-law index has been assumed to be 2 . F ν ( ν a ) ≡ F ν,sw ( ν a , sw ), and characteristicfrequencies ν a ≡ ν a , sw and ν m ≡ ν m , sw , in several snapshots that cover the evolution of thelightcurve up to ∼
220 days after the trigger. Later, Zauderer et al. (2013) extended theperiod of the radio monitoring up to ∼
600 days. The first observation is at ∼ τ ≃ ∼
600 days (Berger et al. 2012; Zauderer et al. 2013), while theX-ray emission has been observed up to ∼
500 days. This mismatch, however, is not aproblem, since we are interested in modelling the lightcurves only up to one year after theexplosion, when is already too dim to be detected by an X-ray survey in most cases.Using the available data and eq.8, we can therefore model the temporal evolution ofthe flux at any radio frequency. In Figure 2, we show the lightcurve of Sw J1644 at 1.4GHz, and its comparison with data. A smooth temporal behavior has been obtained bylinearly interpolating the flux between data points.We now need to generalize our prototypical lightcurve to a generic TDE. The mainuncertainty is how the flux scales with black hole and stellar masses. A first possibility isto describe the jet evolution with a Blandford Mckee (thereafter “BM model”) solution,usually adopted for γ -ray burst afterglows (e.g. Metzger et al. 2012; Berger et al. 2012). 13 –Frequencies below 5 GHz are in the self-absorbed part of the synchrotron spectrum, for thewhole observed duration of the event (see fig.3 in Berger et al. 2012). The observed specificluminosity in this regime ( ν < ν a < ν m ) is given by the Raleigh Jeans part of the BlackBody spectrum B ( ν/δ ) δ ∝ k b T ( ν/δ ) δ (see eq. 8), with a kinetic temperature given by3 k b T = m e c γ min , where the minimum Lorentz factor for the shocked accelerated electronsis γ min ∝ Γ. Therefore the specific radio luminosity is L ν ∝ B ( ν/δ ) δ ( rθ j ) ∝ ( r Γ) , (9)where ( rθ j ) is the emitting area, and we are assuming Γ − > ∼ θ j . In the blast wave modellingof J1644, the external medium swept up by the jet is better described by a power-lawdensity decay that goes as r − , rather than a constant density environment (Zauderer et al.2011). This implies Γ ∝ E / and r ∝ E / , where E j ≈ L j , p t min ∝ m ∗ is the total jet energy.Therefore eq. 9 becomes, L ν ∝ ( L j , p t min ) / ∝ m / ∗ , where there is no dependence on theblack hole mass, but only on the stellar mass.The simple blast wave solution, however, does not describe the whole evolution ofthe radio spectrum (Berger et al. 2012). Therefore, we also consider a simpler approach.In line with our treatment of the X-ray flux, we may assume that the radio luminosity isproportional to the jet peak luminosity L ν ∝ L j , p ∝ M − / m / ∗ , rather than to its totalenergy, (see the X-ray analogous, eq.6, which bears the same mass dependencies). As anextra motivation, this prescription may be justified in the context of the “magneticallyarrested” jet model (e.g. Narayan et al. 2003). We will call this prescription “the MassDependent Luminosity” model (thereafter MDL model).The scaling of the peak flux for sources at different redshift, with different black holeand stellar masses (but at the same observed time τ from the beginning of the event) wouldbe F ν ( ν a , τ ) = F ν,sw ( ν a , sw , τ sw ) × m / ∗ , (cid:18) z z sw (cid:19) (cid:18) D sw D (cid:19) , (10) 14 –for the BM solution and F ν ( ν a , τ ) = F ν,sw ( ν a , sw , τ sw ) × M − / m / ∗ , (cid:18) z z sw (cid:19) (cid:18) D sw D (cid:19) , (11)for our second approach. The equivalent delay at which we need to calculate the flux of SwJ1644 is τ ws ≡ τ z sw (1+ z ) .The characteristic frequencies need to be redshifted according to ν a ( τ ) = ν a , sw ( τ sw ) (cid:18) z z sw (cid:19) − , and ν m ( τ ) = ν m , sw ( τ sw ) (cid:18) z z sw (cid:19) − . In all cases, the flux F ν,sw ( ν a , sw , τ sw ) and the characteristic frequencies at any time τ sw areobtained by linearly interpolating the available data. For τ sw <
3. Black hole mass functions
The mass distribution of black holes as a function of redshift is an essential ingredientto calculate TDE rates. Since black holes grow mainly by efficient accretion (Soltan 1982),one can calculate these functions using the mass continuity equation, given a radiationefficiency and distribution of Eddington ratios. In this paper, we use the results fromShankar et al. (2013). In particular, we consider the two accretion models which yield the Formally, one would need to consider the transformation due to different Doppler factorsbetween jets. However, we here assume that all jets have approximately the same Lorentzfactor Γ and viewing angle of nearly θ o ≈ / Γ. The latter is because the viewing angleprobability ( ∝ θ o , between 0 < θ o < Γ − ) is the highest at Γ − . 15 –largest and the lowest black hole comoving number density φ ( M, z ), and are still consistentwith the quasar bolometric luminosity functions and the local black hole mass function(models labeled G and G ( z ) in Shankar et al. 2013). In this way, we can estimate theuncertainty due to the black hole mass distribution of our expected TDE rates. In Figure3 upper panel, we show the mass distribution functions and their uncertainty strips as afunction of redshift, for M = 10 M ⊙ and M = 10 M ⊙ black holes. In Figure 3, instead, weshow the “intrinsic” TDE rate as a function of redshift, R ( z ) = Z M max M min φ ( M, z ) V ( z ) N tde dM, (12)where we denote with V ( z ) the comoving cosmological volume. N tde = 10 − yr − is ourfiducial TDE rate per galaxy: this value is in the range of theoretical expectations (Merritt2013) and observational claims (Donley et al. 2002; Gezari et al. 2009; Van Velzen et al.2011).The minimum black hole mass (here and thereafter in our calculations) is M min = 10 M ⊙ , as just a few SMBHs have been observed with a lower mass.
4. Monte Carlo calculations
Assuming the X-ray and radio modelling described in §
2, we perform Monte Carlosimulations (MCs) to derive the number of jetted TDEs to be detected per year, for givenflux limit and sky coverage.Beside the BH mass, the main ingredients of our MCs are the trigger lag time, τ , and The recent discovery of TDEs in dwarf galaxies ( < ∼ M ⊙ ) (Donato et al. 2014;Maksym et al. 2014a,b) seems particularly promising in overcoming this limit and use TDEsto find lower mass BHs 16 –the mass of the disrupted stars, m ∗ . The former is randomly extracted from a uniformdistribution between 0 and 1 yr . The latter follows a Kroupa Initial Mass Function, (IMFKroupa 2001), f ( m ) ∝ m − . , . ≤ m ∗ ≤ . ,m − . , . ≤ m ∗ ≤ . ,m − . , m ∗ > . . (13)In fact, for each black hole mass, the minimum stellar mass is set by the requirement thatthe tidal radius should be greater than the last stable orbit (we assume a non-spinning BH).This requirement implies that m ∗ min = max[0 . , . M ]. Note that for M = 10 M ⊙ , theminimum mass is m ∗ min = 4 . M ⊙ . Therefore, events associated with high BH masses aresuppressed in numbers by the steepness of the IMF, as only 0 .
4% of all stars have m ∗ > m ∗ is larger.In our simulation, we start by considering the intrinsic rate R ( z ) (eq. 12) properlymodified by accounting for the relativistic beaming, which results in a reduction by a factorof 2 π Γ − / (4 π ) = 1 / (2Γ ): this is the fraction of solid angle subtended by the emission,when considering a two sided jet. Our fiducial value for the jet Lorentz factor is Γ = 2, asinferred by radio observations (Γ ≈
2, Zauderer et al. 2013; Berger et al. 2012). If the jetdecelerates, this value has to be intended as an average one, over the observation period.However, we note that this is a geometrical scaling factor and our results may be easilyre-scaled by assuming different values of the jet bulk Lorentz factor. In addition, R ( z ) isscaled for the fraction of the sky surveyed by the assumed instrument. In the calculationof R ( z ) we have adopted both G and G ( z ) models in order to account for the systematic we do not use longer time lags because any extrapolation beyond the currently avail-able radio data would make our estimates more model-dependent, since there is no hydro-dynamical model that can reproduce the whole radio behavior of J1644. 17 –uncertainties in the mass function modellings. The number of trials in MCs is properlyfixed by requiring a high statistics level in each mass and redshift bin (typically ≥ ).
5. Results
In this section, we first validate separately our emission models for X-ray and radiolight curves, by comparing our predicted rates with current instruments and survey results.In fact, we find that current data do not put strong constraints on our modelling, as we willexplain in the following. Future data have instead a greater potential. In the SKA era, wepropose that a strategy where radio will be triggering X-ray facilities can allow us not onlyto detect but also to identify and investigate jetted TDEs in a multi-wavelength fashion. Inthe following, if not otherwise mentioned, our results are derived adopting Γ = 2.
So far, only two jetted TDE candidates have been detected by BAT, implying anobserved rate of ∼ . − .Since BAT is not operating in survey mode, it is not straightforward to compareobservations with our predictions, i.e. it is difficult to chose sky coverage and detectionlimit, because they are not univocally determined. The two TDE candidates were detectedin two different modes: Sw J1644 was triggered onboard, while Sw J2058 was discovered bystacking 4-day integration images (Krimm et al. 2011; Cenko et al. 2012). In both modes, itis hard to define a survey flux limit, the key ingredient of our MCs. Indeed, Swift has over
500 onboard trigger criteria in different modes which makes the use of a flux limit survey a 18 –rather simplified approach. The same applies to possible discoveries of fainter TDEs withlonger integration times, by applying the image mosaics technique (Krimm et al. 2013).These have to be promptly followed-up by XRT for their identification: monitor the softX-ray emission and then measure the characteristic temporal slope of TDEs. In this way,a further efficiency accounting for any reason preventing XRT to monitor the event hasto be included in our rate calculations (e.g., the stochastic nature of the Swift pointingplan, the target visibility and the mission schedule; Krimm, private communication). Suchan efficiency is hard to quantify and any assumption would be arbitrary and would biasour discussion on the comparison between the predicted and observed rates. In additionto that, our soft X-ray modelling assumes a total disruption of the star (see § caveats . In fact,any reliable prediction based on on-board triggers would require complex simulations asdone by (Lien et al. 2014) for GRB rates. We therefore set to achieve a less ambitiousaim at predicting indicative rates, which should be considered most likely as upper limits.Specifically, we adopt the BAT daily sky coverage reported in (Krimm et al. 2013) and fixa unique “survey” flux limit to be consistent with the detection of the Sw J1644. We detailthe procedure in the following.Sw J1644 was detected with an on-board BAT image trigger (Cummings et al. 2011).In this trigger mode, we assume a flux limit of 2 . × − erg cm − s − in the 15 −
150 keVband, which is consistent with the faint tail of the observed GRB rate (Lien et al. 2014)and the detection of Sw J1644 (Burrows et al. 2011). We adopt a daily sky coverage of 85%(Krimm et al. 2013) and apply an efficiency of ∼ ∼ . τ with our flux threshold. We obtaina TDE rate of ≈ −
20 events yr − (see Table 1). The rate distribution with redshiftextends up to z max ≈ . and peaks at z ≈ .
2. The peak value ranges between 1 − − (see Table 1). The peak of the corresponding BH mass distribution is at 10 M ⊙ andcontains ∼
23% of all events.Given our predicted mass and z distributions for the observed TDEs, an event like SwJ1644 has a chance probability which is a factor of ∼
10 lower than that of an event at thepeak rate. Therefore, it is not an unlikely event, but a lower redshift object would havehad a higher probability. At this point, it is unclear to us if this result is more due to oursimplified treatment of the BAT trigger, to our assumption of a constant jet luminosity fora given BH and stellar mass. Both are very likely to have a role. But since we can not trustat this level our trigger modelling and the paucity of detected events does not constrain apossible luminosity function, we do not attempt here to modify our X-ray model to fit thisobserved distribution. When more events will be identified, our procedure can be refined toaccount for a TDE variety. A comparison with future, easier to model, surveys (see Sec.5.2.1) will also help constructing a more robust emission model.Interesting, these rates are actually up to two order of magnitude higher than that(0.3 yr − ) derived from BAT observations, but a key role is played by the low value of Γconsidered. We will elaborate on this point in section 6.1. Here and in the following, we define z max as the redshift at which the expected rate is0 . − . 20 – We compare our predictions with constraints on the jetted TDE rate derived fromcurrent radio surveys (Bower 2011). In the following radio estimates, we will require a 5- σ flux limit to claim detection.We first consider the combined catalogs of VLA First and NVSS at 1.4 GHz. Thecombined sky coverage is 0.19 sr with a flux limit of 6 mJy at 1.4 GHz. The analysis ofthese catalogs didn’t yield any TDE candidate.To derive our predictions, we adopt the radio modelling described in section 2.3, andfor each event (i.e. for each set of τ , black hole and stellar mass, and redshift), we calculatethe average flux over a period of one day from the trigger. This is compared with a 6 mJyflux threshold. Rescaling our all sky results for the catalogue sky coverage, we obtain anobserved rate that even in the most favorable case ( ∼ . − using BM model, eq.10) isconsistent with Bower (2011) and Frail et al. (2012) results. To strengthen this conclusion,we note that our assumption of a 6 mJy threshold per day combined with a sky coverage of0.19 sr may be considered already rather optimistic, since both values are referred to a 1 yrsingle epoch.In the near future, the VLA Stripe 82 survey may constrain jetted TDE models thanksto the improved sensitivity (50 µ Jy rms) at 1.4 GHz over a FoV of 90 deg (Hodge et al.2013). By assuming a 5- σ threshold of 0.25 mJy, our modelling predicts a number of a fewobjects to be detected per year. Significant advances in TDE detections are expected tocome from on-going wide radio surveys at both low (see e.g. MWA and LOFAR) and highradio frequencies (e.g. ASKAP and VLASS). Since our radio modelling was constrainedby observations at higher ( > . § deg reaching a sensitivity of 0 . ∼
14 TDE yr − , consistent with expectations from Murphy et al. (2013). Forcomparison, Frail et al. (2012) obtain a value of ∼
82 yr − by considering longer integrationtime ( ∼ ten days). In the case of the VLA Sky Survey (Hallinan et al. 2013, VLASS), weconsider a sky coverage of 10 deg with ∼ . Currently, the only two jetted TDE candidates were discovered in X-rays, where thecharacteristic t − / decay slope has been observed. Therefore, we first discuss the discoverypotential of future X-ray surveys. We then predict the expected rate of TDEs for the SKA1.4 GHz wide survey. Finally, we derive the properties and rate of TDEs that can bedetected in radio with SKA and subsequently identified in X-rays. The rate estimates provided in this section are based on a unique observing strategyaimed at detecting and providing a first identification of the transient as a TDE. Weassume that a given fraction of the sky is covered in 1 day at a flux threshold defined bythe requirement to follow the typical TDE decay over 4 lightcurve bins, each with S / N ≥ σ flux limit of each survey, then tracing back the t − / decay in order to obtain the flux over the 4 time bins and then compute the average 22 –flux over that period. This average flux defines the identification flux threshold. We willgive values for both Γ = 2 and Γ = 20 and we will justify this choice and elaborate on thecomparison in section 6.The all sky survey mission eRosita (Merloni et al. 2012) is expected to detect jettedTDEs, in its “hard” X-ray band (2 −
10 keV). We apply our methodology to the eRositasurvey, properly re-scaling the sky coverage achieved in a 6-month scan to 1 day. Wederive the identification flux threshold for our observing strategy from the 2 −
10 keV 5- σ sensitivity of ∼ − erg cm − s − , corresponding to ∼
250 s exposure (Merloni et al. 2012)as foreseen for each point in the sky. We calculate the corresponding un-absorbed fluxand then we extrapolate it in the energy range 1 −
10 keV (used in our X-ray modelling).We predict a maximum of ∼
15 TDE per year to be detected up to z ≈ .
5, although z max ≈ .
4. The peak rate is between 0.15 and 0.5 yr − at z ≈ . z ≈
2, therate is < × − yr − . If a larger value of Γ = 20 is considered, the rate decrease by twoorders of magnitude (see Table 1) with a maximum total rate of ∼ .
15 yr − and peakrate of only 4 × − yr − . We therefore predict both higher (Γ = 2) and lower (Γ = 20)rates than those previously published by Khabibullin et al. (2014, 1 object to be detectedper 6-month long scan), but we definitively reach a much lower redshift ( z max = 0 . z = 4 . ≈
150 events per scan byconsidering the number of jetted TDEs to be a 1/5th of their “soft” TDE sample ( ≈ σ ∼ − × − erg cm − s − (a few mCrab) in 2 −
50 keV energyband. We estimate tens of objects per year up to z max ≈ .
6. The peak rate is ∼ − at 23 – z = 0 .
2. These numbers imply a total rate of 0.7 yr − for Γ = 20 with a peak rate of only6 × − yr − .Finally, we consider Einstein Probe, which is expected to monitor 1 / . − σ sensitivity of ∼ − erg cm − s − in eachpoint (1 ks exposed) of the sky (EP, W. Yuan private communication). In this case, MCswere adapted in order to extend our X-ray modelling to this energy range. This requiresto first estimate the un-absorbed flux limit by accounting for both the Galactic and theintrinsic absorption (Burrows et al. 2011) and then calculate a proper radiation efficiencyby extrapolating from the value inferred in 1-10 keV. We estimate a number between ∼ −
240 yr − to be detected below z max ≈ ≈
15 yr − at z = 0 .
3. Inthe case of Γ = 20, a few objects are expected to be detected per year, with a peak rate of ∼ . − . A summary of the actual numbers can be found in Table 1.Inspecting the trigger time distributions (see top panel in Figure 5), we find that up to z max ∼ Presently, the most ambitious and revolutionary project in radio astronomy is theSquare Kilometer Array (SKA Carilli & Rawlings 2004) planned to operate in 2020. SKA,in survey mode (SKA1-Survey, Dewdney et al. (2013)), is able achieve a half sky coverage 24 –(20,000 deg ) with a 2-day cadence at a 5 − σ flux limit of 90 µ Jy (Donnarumma et al.2014; Feretti et al. 2014). These unprecedented sky coverage and sensitivity make SKA anoptimal radio transient hunter.Differently from X-ray searches, in radio, we cannot have a first identification basedon the lightcurve, since the 1.4 GHz radio emission of a TDE is not particularly differentfrom those of other radio transients (e.g. GRB, blazars). Therefore, we consider a differentstrategy. In our MC simulations, we directly assume the SKA 5 − σ flux limit in orderto claim the detection of a transient event. The identification strategy will fully rely onthe multi-frequency follow-up of the trigger event as it will be discussed at the end of thissection.We calculate the predicted average flux over 2 days from the trigger and then wecompare it with the SKA flux limit. The results are shown in Figure 4. The upper panelsare derived using the BM model (eq.10) for the radio lightcurve modelling, while the lowerpanels use the MDL model (eq.11). There, we show the distribution of the TDE rate as afunction of z (right panels) and their BH mass distribution (left panels) for the two BH massfunctions described in § M ⊙ . Both radio models produce redshift distributions peakingaround z ≈ .
4, regardless of the BH mass function. The peak rates are roughly between6 and 40 yr − (see also Table 1). Events with BH mass lower than 10 M ⊙ dominate thedistributions at all redshifts in the MDL model, while this only happens at z < . − M ⊙ areequally probable in the BM model (because the flux is BH mass independent, eq.10), BHswith mass < × completely dominate the observed sample in the MDL model. As aconsequence, BM model distribution extends to higher redshifts ( z max ≈ . z max ≈ . M ⊙ interact with higher mass stars (higher m ∗ , min ) andproduce intrinsically brighter flares. This will allow us to study TDEs close to the peak ofcosmic star formation.In Table 1, we also report the total rates obtained by integrating these distributionsin z and M BH . We obtain yearly rates of the order of a few to several hundreds. Theseresults are not consistent with those that can be derived by using eq.4 in (Van Velzen et al.2013): inserting our SKA survey parameters, we obtain thousands of events per year. Thisdiscrepancy is due to our inclusion of the stellar mass dependence, that modulates the TDEluminosity for a given BH mass: the lower m ∗ , the dimmer the event. In the assumption ofa Kroupa IMF, the bulk of the events are caused by the disruption of stars with m ∗ < z < . z . Thisearly period coincides with the rise of the radio light curve. Although we expect this gap indetection, the exact epoch at which it occurs depends on the detailed behavior of the lightcurve during this undetected rise. Our extrapolation at earlier times is quite steep and weconsider 10 days as an upper limit for the initial gap in detection.So far we focused on detection of TDEs with SKA, that, depending on the observing 26 –strategy, will only be a fraction of a noticeable sample of slow radio transients. Asmentioned earlier, we cannot use radio properties or variability alone to distinguish a TDEcandidate from neither a slowly variable AGN or a GRB. A possibility for identification that we explore below is through quick follow-ups at higher energies, particularly in X-rays.A first pre-screening of the radio candidates could be done by cross-correlating the radiotransient positions with deep AGN catalogues, expected to be provided in the near futureby optical surveys (e.g. LSST) or the SKA precursors (e.g. ASKAP). However, we expecta larger degree of contamination of the TDE sample to come from transient sources suchas GRBs. Since, unlike GRBs, most of TDEs should have a nuclear origin, it is mandatoryto quickly identify the host galaxy. An accurate localization of the radio transient in thecore of galactic nuclei, helping to assess the nuclear origin, will therefore play a major rolein the screening of the radio transient sample. This means that first the host galaxy has tobe found by a rapid optical follow-up and after the brighter transients could be localized bySKA with a precision of ∼
100 milliarcsecond (mas) essential to separate nuclear transientsfrom other phenomena (e.g., GRB).For details see Donnarumma et al. (2014). X-ray follow-up will have a major role in the identification of the TDE candidatedetected by SKA because of the possibility to detect the characteristic t − / decay. Apossible X-ray follow-up strategy aimed at identifying and then characterizing the eventconsists in a fast repointing of the transient detected by SKA. We consider a 1-day delay in this can be achieved thanks to the resolution of about 2 arcsec of SKA1-SUR and 0.6arcsec or better of SKA1-MID (Dewdney et al. 2013) 27 –the X-ray repointing and require a set of X-ray observations spread over a few days in orderto follow the characteristic temporal decay of the TDE. We foresee an observing strategywhich is similar to the one adopted in the case of future X-ray surveys (see section 5.2.1):four observations spread over , with S/N ratio ≥
10 in each. A high
S/N is requiredin order to characterize both the temporal and spectral behavior of the source.For each event in the MCs, we calculate the average X-ray flux over the 4 days afterthe repointing and compare it with the identification flux threshold derived as explainedin section 5.2.1, with the only difference of a
S/N ≥
10 requested in each observation.Practically, τ in eq.5 has to be the radio trigger time-lag, plus an extra delay of one day forrepointing, and 0 ≤ ∆ t ≤
3. In this way, we derive the properties of samples of TDEs whichare first detected in radio and promptly followed-up in X-rays.In Fig. 6, we show the fraction of SKA candidates that can be identified as afunction of the X-ray (1-10 keV) unabsorbed flux limit. A rapid X-ray follow-up will beable to detect a complete radio-selected sample provided that the instrument sensitivityis close to F lim . − erg cm − s − in the energy. In fact, a moderate sensitivity ∼ − − − erg cm − s − is already enough to detect equal or a larger number of eventsthan with X-ray wide sky instruments alone. It is therefore clear that a radio trigger is amore efficient way to build up a large X-ray sample of TDEs. Rates reported in that Figureassume a fast (1 day) X-ray repointing and F lim reached with an integration of ∼ F lim or in the case of longer repointing time. This is a natural consequence of the decreasingtrend of the X-ray light curve.When considering an actual follow-up strategy, the values reported in Figure 6 shouldbe scaled by the fraction of sky accessible to the X-ray instrument considered. In general,the X-ray follow-up will provide us with a sub-sample of radio triggered TDEs, defined by 28 –the target accessibility, the repointing chance of the X-ray satellite and the sensitivity ofthe instruments. Since TDEs also emit in hard X-rays, a trade-off between sensitivity, skycoverage and a broad energy range is foreseen. In particular, the broader is the energy rangethe better the characterization of the non-thermal process and of the jet energy budget.Future X-ray experiments like Athena (Nandra et al. 2013) and a LOFT-like mission(Feroci et al. 2012) could offer a unique chance to follow-up and characterize SKA triggeredTDEs. Moreover, if Swift were still operating in the 2020s, XRT will have a great potentialin following-up the radio candidates.Athena sensitivity lies in the saturation branch of Figure 6, which implies that theobserved rate of X-ray jetted TDEs will be crucially linked to its follow-up efficiency.This is mainly influenced by the Athena sky accessibility which is of the order of ∼ z max ≈ ≥ − erg cm − s − in the 2-10 keV band, for a 100 ks exposure. The LOFT pointingvisibility will assure a sky accessibility for these targets of ∼ F lim ∼ − erg cm − s − in the 2-10 keV band,which was then translated in the corresponding un-absorbed value in the 1-10 keV band(the energy range adopted in our modelling). Again, we assume a 1-day repointing delay.Figure 7, shows the expected rate of jetted TDEs for a LOFT-like mission as a functionof redshift (right panel) and their mass distribution (left panel). The rate distributionsare calculated under the BM model (top panels) and MDL model (bottom panels) for theradio modelling. In both cases, we found that the redshift distribution extends above z = 1( z max ≈ . − .
7) (see Table 1), with most of the TDEs expected around z ⋍ . ≈ ∼
25% ofall events have BHs with masses ≈ M ⊙ , and events with BH masses < M ⊙ dominatethe redshift distribution at all epochs. In the BM model, instead, the TDE rate peaks at ≈ M ⊙ (see left panel in Figure 7), with lighter BHs dominating at z ≤ z peak (yellowlines in Figure 7, right panel). The behavior at higher z fully reflects the one observed inthe BM radio rates (see fig. 4). In total, a LOFT-like mission should be able to detect asub-sample of radio TDEs between ≈
130 and ≃
350 yr − . Instead, very few objects peryear are predicted if Γ = 20. For these events, the mission broad energy band (2 −
50 keV)should enable us to put tighter constraints on the energy budget of the X-ray component,than possible with Athena instrument.The price to pay for detecting more X-ray TDEs with a follow-up strategy is illustratedin Figure 5 upper panel, where we compare the trigger distribution for the EP (black lines)and the LOFT radio triggered (blue lines) samples. Most of LOFT events are observedafter 10 days from the beginning of the emission . In particular, high redshift z ≈
6. Discussion
The Swift/BAT discovery of Sw J1644 opened a window on a new class of X-ray andradio transients, which are optimal targets for future radio and X-ray surveys/instruments.The study of these objects allows us to investigate the formation of transient jets in See discussion in Sec. 5.2.2 30 – R R z peak R R z max yr − yr − yr − yr − Radio selected sampleSKA BM
226 468 0 . . MDL
327 770 0 . BM
128 (2.5) 305 (6.5) 0 . MDL
135 (1.3) 352 (3.5) 0.4 8 (0.08) 22 (0.2) 1.2Athena BM
113 (1) 234 (2.3) 0.3 3 (0.03) 8.5 (0.09) 2Athena
MDL
163 (1.6) 385 (4) 0.4 7 (0.07) 20 (0.2) 1.4X-ray surveysBAT . − . . and are for the Gz and G MFs, respectively), 3rd column redshift atthe peak rate, 4th and 5th columns maximum peak rate and 6th column maximum redshift,defined as the z where the rate is 0.5. BAT : calculation for an on-board image trigger.X-ray and Radio expected rates are derived for Γ = 2. X-ray rates are also reported forΓ = 20 in parenthesis. 31 –extra-galactic sources. Moreover, there is the potential to discover quiescent SMBHs indistant galaxies and constrain the SMBH mass function. In this section, we qualitativelydiscuss our results and what we may learn from them. Any quantitative parameterinvestigation (for instance with a Fisher Matrix technique) is beyond the scope of thispresent paper, and will be presented in a follow-up work. So far, only two jetted TDEs have been detected, while the thermal candidates, relatedto the presence of an accretion disk, have been more numerous. The question then ariseswhether this is due to observational biases, highly collimated jets or to an intrinsic lowefficiency of transient accretion disks to produce (luminous) jets.To try and address this question, we could compare our predictions to the Swift/BATobserved rate ( ≈ . − ): our lower limit ( ≈ − ) is a factor of 30 higher. It is tempting— and indeed it has been done in the literature — to reconcile this discrepancy by invokinga jet production efficiency of a few percent, since our calculations assume that each TDE isaccompanied by a jet. However, there are several reasons why this inference should not be drawn. First, asdiscussed in Section 5.1.1, it is absolutely non-trivial to describe the characteristics (e.g. fluxlimit and sky coverage) of an effective Swift/BAT survey. We believe that our assumptionsfor the trigger, together with a 100% identification efficiency gives rates that are indicative In our simplified description here, there are only two kinds of possible events: Sw J1644with its own jet luminosity (i.e. a given jet energy efficiency ǫ j ) and events with no jet (i.e. ǫ j very small). In reality, there must be an intrinsic distribution of ǫ j , with a tail of lowenergy events that cannot be detected or failed to be launched at relativistic speeds. 32 –of an upper limit. Second, BAT rate predictions, unlike those of other X-ray instrumentsconsider here, strongly depend on the modelling of the early stage variability of the X-raylightcurve (see § ∼ −
5) value. However,hard X-ray observations are consistent with larger Lorentz factors (Γ ≤
20 Burrows et al.2011), which will bring down our rates to the observed value (see Tab.1). The consequencewould be that the simultaneous hard X-ray and the radio emissions need to come fromdifferent regions — as already claimed (e.g. Zauderer et al. 2011). The picture may bethat while the radio emission is produced from further out, after the jet has substantiallydecelerated, X-rays probes regions much closer to the central engine (Bloom et al. 2011). Ifthat was true, X-ray detections and follow-ups would be further suppressed with respect tothe expected SKA performance.Unlike the previous comparison with BAT results, our predictions of the radio ratesare consistent with the upper limits derived using with the NVSS + FIRST catalog (Bower2011), for any Γ ≥
2. As a consequence, this comparison cannot provide us with furtherconstraints on either Γ or the jet efficiency. In the next future, surveys such as VLA Stripe82, ASKAP and VLASS will give tighter constraints on jetted TDEs thanks to the improvedsensitivity (50 µ Jy rms, Hodge et al. 2013) of the former and the wide field of view of thelatter two surveys. In this case, our radio modelling predicts a number of a few objects yr − (a few tens yr − ) to be detected by assuming Γ = 2. Comparing predictions with (positive)observations will thus constrain possible combinations of Γ and jet production efficiency. 33 –As already discussed, the optical transient surveys Pan-STARRS and LSST areexpected to make significant advances in the study of TDEs. LSST will be a realbreakthrough in this respect, surveying 2 × square degrees of the southern sky. Thousands of objects are expected to be discovered at z < not expectedto be relativistically beamed nor to be connected with jet emission. These features implythat a comparison between optical, X-ray and radio selected samples can help constrainingboth the TDE efficiency to produce jets and the relativistic Lorentz factor. This latter,when an X-ray sample is available, will help assessing the jet energy efficiency ǫ j . To understand supermassive BH cosmic growth and their connection with the hostgalaxy, it is necessary to have a good understanding of which mass can be found in whichgalaxy and, more broadly, of the SMBH mass function as a function of redshift.The detection and light modelling of a TDE event is a unique way to constrain the massof an otherwise quiescent BH, that is too distant to be detected by stellar dynamics. Anattractive feature is that TDEs may occur in any type of galaxy, allowing for the detection 34 –of a broader range of SMBH hosts. For the lightcurve modelling, a multi-wavelengthapproach can yield tighter constraints on the mass, since other parameters such as the jetenergy, Lorentz factor and the stellar mass need to be simultaneously determined.A perhaps more direct measurement of the BH mass can come from very fast X-rayvariability, as the quasi periodic oscillation (QPO) observed in Sw J1644 (Reis et al. 2012).The prospect for detection of QPOs in such events is quite favorable for both Athena and aLOFT-like mission. If QPOs in TDEs were associated with the Keplerian frequency at theinnermost stable orbit (as discussed in Reis et al. 2012), the highest rest frame frequencyshould be of the order of 200 s for a BH mass of 10 M ⊙ . This QPO frequency is easily withinreach of both LOF T -like and Athena instruments (Feretti et al. 2014; Nandra et al. 2013).Longer oscillations are expected for more massive BHs ( ∝ p / M BH ), whose detectabilitycould be more complicated due to satellite orbit constraints (e.g. Earth occultation, SouthAtlantic Anomaly). However, providing that the QPO is persistent over a long period andthe source is bright enough to remain above threshold for several cycles, a direct measureof such a QPO is also possible.An other method to constrain the mass function may be to compare our ratedistributions with future SKA triggered observations. As shown by Figure 3 upperpanel, there are still uncertainties in the BH mass function, which in turn affect our ratepredictions (see Figure 3 lower panel, Figure 4 and Figure 7).
7. Conclusions
We have investigated the best strategies to increase the sample of the new class ofTDEs, which was recently discovered by BAT. These events emitted non-thermal emissionin X-ray and radio bands, probing a relativistic jet. Given the lack of statistics and 35 –of a solid theoretical framework for their non-thermal emission, we adopted a ratherphenomenological approach to model their lightcurve. We fit the behavior of the beststudied candidate, Sw J1644, in both radio (1.4 GHz) and X-rays (1-10 keV), and weused the classical theory of TDEs to rescale the emission for different black hole and starmasses. In the radio band, we also considered, in alternative, the blast wave model, usuallyadopted for GRBs. We then used a Monte Carlo code to compute their expected rate asa function of redshift and black hole mass. We considered both current and future radioand X-ray surveys/instruments. Since the characteristic temporal decay of a TDE eventcan be observed in X-ray, an identification is claimed only when the X-ray emission canbe sampled in at least 4 lightcurve bins with high signal to noise ratio,
S/N ≥
5. Whenthe TDE is detected in radio, we investigated a follow-up strategy for identification whichrequired X-ray detectors to sample the lightcurve with the almost the same requirements asabove (but with a
S/N ≥
10 ). To concretely explore future possibilities, we investigatedin particular the expected performance of eRosita, Einstein Probe, Athena, a LOFT-likemission and SKA operating in survey mode (SKA1-SUR).Our major findings can be summarized as follows: • results from current instruments (such as BAT and NVSS + FIRST catalogues) donot provide constraints on jet parameters or the jet production efficiency; • However, to reconcile BAT predictions with observations a Γ ≈
20 may be adopted,consistently with hard X-ray observations (Burrows et al. 2011). If this were true,X-ray and radio emissions should come from two different regions, as already suggestedon different bases (Zauderer et al. 2011). The predicted X-ray rates would also besuppressed by (2 / with respect to those in the radio band. • In the near future, VLA Stripe 82 survey, VLASS and ASKAP-VAST may providefrom a few to ten events yr − , putting some constraints on possible combinations of 36 –bulk Lorentz Γ and jet production efficiency; • Hundreds (Γ = 2) of Sw J1644-like objects per yr are expected to be within reach ofSKA1-SUR at 1.4 GHz. They can probe the distant Universe up to z ∼ .
5. Theseresults differ from previous, more optimistic, predictions of thousands yr − (for Γ = 2Van Velzen et al. 2011) • Future X-ray surveys will provide a more modest sample, between several (eRosita)to a maximum of ≈
240 (EP) jetted events per year. With a highly collimated jet,with Γ = 20, these numbers drop to a maximum of a few. • X-ray detections can be substantially enhanced, if a prompt follow-up of SKAcandidate is adopted with an instrument with flux limit < ∼ − erg cm − s − in the1-10 keV band over 4-day timescale. With that flux limit each SKA triggered eventcan have in principle an X-ray counterpart (see Fig.6). A suppression factor shouldbe adopted if the X-ray emitting region would be moving with a larger Lorentz factor. • The sample of SKA preselected X-ray events can extend up to redshift ∼ z < ∼ • Despite the several advantages of a radio trigger, direct X-ray detections are the onlyway to study the early stages ( <
10 day) of the flare (see Figure 5).Once TDE samples in different bands have been built up, the synergy between radio,X-rays and optical can in principle constrain important physical quantities such as thejet luminosity, bulk Lorentz factor, the jet production efficiency and the black hole massfunction. These findings will inform theories of jet and disc formation from sudden accretionevents and, on the other hand, of SMBH cosmological evolution. 37 –We would like to thank Francesco Shankar for providing us with the black hole massfunctions, I. Prandoni and H. Krimm for the interesting discussions about SKA and BATobserving strategies, respectively. Finally, we thank S. Van Velzen for the careful reading ofthis paper and his helpful comments. We thank the anonymous referee for her/his helpfulsuggestions. This work made use of data supplied by the UK Swift Science Data Centre atthe University of Leicester. I. Donnarumma is grateful for support by ASI (under contractI/021/12/0-186/12), INAF (under contract PRIN-INAF-2011 “Strong Gravity”). 38 –
REFERENCES
Arcavi I. et al. 2014, Ap. J., 793, 38Berger, E., Zauderer, A., Pooley, G. G., Soderberg, A. M., Sari, R., Brunthaler, A. &Bietenholz, M. F., 2012 Ap. J.,748, 36Bloom, J. S. et al. 2011, Science, 333, 203Bower, G. C. 2011 ApJ, 732, L12Burrows, D. N. et al. 2011, Nature, 476, 421Campana, S. 2011, Nature, 480, 69Cannizzo J. K., Troja E., & Lodato G., 2011, Ap. J., 742, 32Carilli C. L., Rawlings S., 2004, New A Rev., 48, 979Cenko S. B., et al., 2012, Ap. J., 753, 77Chornock R., et al., 2014, Ap. J., 780, 44Cummings J. R., et al., 2011, GCN, 11823Dewdney P. E., et al., 2013, SKA1 System Baseline Design. Tech. rep., SKA ProgramDevelopment, Document Number: SKA-TEL-SKO-DD-001Donato, D. et al. 2014, Ap. J.781, 59Donley, J. L. ,Brandt, W. N., Eracleous, M. & Boller,Donnarumma, I. et al. 2014, The Transient Universe with the Square Kilometre Array,in proceedings of Advancing Astrophysics with the Square Kilometre Array,PoS(AASKA14)054 39 –Feretti, L. & I. Prandoni, et al. 2014, ”Italian SKA White Book”, Eds. L. Feretti, I.Prandoni et al., INAF Press, ISBN 978-88-98985-00-5Feroci M., et al. 2012, Experimental Astronomy, 34, 415Frail D., et al. 2012, Ap. J.747, 70Frank, J., & Rees, M. J., 1976, MNRAS, 176, 633Gezari S. , HeckmanT. , S. B. Cenko, M. Eracleous, K. Forster, T. S. Gon¸calves, D. C.Martin, P. Morrissey, S. G. Neff, M. Seibert, D. Schiminovich, & T. K. Wyder, 2009,Ap. J.698, 1367Gezari, S. et al. Nature, 485:217Granot J. & Sari R., 2002, Ap. J., 568, 820Guillochon, J. & Ramirez-Ruiz, E. 2013,Ap. J., 767, 25Hayasaki, K. and Stone, N. and Loeb, A. 2013, Mon. Not. Roy. Astro. Soc.434, 909Hallinan, G. et al. 2013, VLASS White PaperHills, J G. 1975, Nature, 254, 295Holoien, T. W.-S. et al. 2014, arXiv1405.1417Hodge, J. A., Becker R. H., White R. L., Richards G. T., 2013, Ap. J., 769, 125Kesden, M., 2012, PhRvD, 85, 4037Khabibullin I., Sazonov S., Sunyaev R., 2014, Mon. Not. Roy. Astro. Soc., 437, 327Kroupa, P. 2001, Mon. Not. Roy. Astro. Soc., 322, 231Komossa, S., 2002, Reviews in Modern Astronomy, 15, 27 40 –Krimm H. A. et al., 2011, Astronomer Telegrams, 3384Krimm H. A. et al., 2013, Ap. J. Suppl., 209, 14Levan, A. J et al. 2011, Science, 333, 199Levan, A. 2012, In
European Physical Journal Web of Conferences , 39, 2005Lien A., Sakamoto T., Gehrels N., Palmer D., Barthelmy S., Graziani C., Cannizzo J.,2014, Ap. J., 783, 24Lodato G., King A. R., & Pringle J. E., 2009, Mon. Not. Roy. Astro. Soc., 392, 332Lodato G. & Rossi, E. M., 2011, Mon. Not. Roy. Astro. Soc., 410, 359Maksym,W. P. and Lin, D. and Irwin, J. A. 2014, Ap. J. Lett., 792, 29Maksym,W. P. et al. 2014, AAS 223, 406Merloni A. et al., 2012, ArXiv e-prints, arXiv1209.3114MMerritt D.,2013, Classical and Quantum Gravity, Volume 30, Issue 24, article id. 244005Metzger, B. D., Giannios, D. &Mimica, P., 2012, Mon. Not. Roy. Astro. Soc., 420, 3528Murphy, T. et al. 2013, PASA, 30, 6Nandra P., et al., 2013, arXiv1306.2307NNarayan R., et al., 2003, PASJ, 55, L69Phinney, E. S. Nature, 1989, 340, 595Rees, M. J., 1988, Nature, 333, 523Reis, et al., 2012, Science, 337, 949 41 –Rossi, E. M. & Begelman, M. C. 2009 Mon. Not. Roy. Astro. Soc.,392, 1451S¸adowski A., Narayan R., McKinney J., Tchekhovskoy A., 2014, Mon. Not. Roy. Astro.Soc., 439, 503Sari, R, Kobayashi, S., & Rossi, E.M. 2010, Ap. J., 708,605Saxton C. J., Soria R., Wu K., Kuin N. P. M., 2012, Mon. Not. Roy. Astro. Soc., 422, 1625Shankar F., Weinberg D. H., Miralda-Escud´e J., 2013, Mon. Not. Roy. Astro. Soc., 428, 421Soltan, A. 1982, Mon. Not. Roy. Astro. Soc., 200, 115Stone, N. and Loeb, A. 2012, Physical Review Letters 108,1302Stone, N. and Sari, R. and Loeb, A. 2013 Mon. Not. Roy. Astro. Soc., 435,1809Strubbe L. E. & Quataert, E. 2009, Mon. Not. Roy. Astro. Soc., 400, 2070Tchekhovskoy, A., Metzger, B. D., Giannios, D. & Kelley, L. Z., 2013, Mon. Not. Roy.Astro. Soc., accepted.van Velzen S., et al., 2011, Ap. J., 741, 73van Velzen S., Frail D., K¨ording E., Falcke H., 2011, Mon. Not. Roy. Astro. Soc., 417, 51van Velzen S., Frail D., K¨ording E., Falcke H., 2013, Astron. Astrophys., 552, A5Zauderer, B. A., Berger, E., Margutti, R., Pooley, G. G., Sari, R. , Soderberg, A. M.,Brunthaler, A. & Bietenholz, M. F., 2013, Ap. J., 767, 152Zauderer, B. A.,et al., 2011, Nature, 476, 425This manuscript was prepared with the AAS L A TEX macros v5.2. 42 –Fig. 1.— The X-ray (0.3-10 keV) lightcurve of J1644 as a function of time from the X-raytrigger: data (absorbed flux, circle marks taken from Publicy available XRT lightcurves)versus our modelling (solid line). After a few days, the temporal decay approaches t − / . 43 –Fig. 2.— The radio (1.4 GHz) light curve of SwJ 1644 as a function of time from theradio trigger (5 days after the X-ray first detection): data (circle marks) from (Berger et al.2012; Zauderer et al. 2013) versus our modelling (solid line). While our modelling wellreproduce higher radio frequencies lightcurves (see fig. 1 in (Berger et al. 2012)), it slightlyunderpredicts the 1.4 GHz one. In this respect our flux modelling is conservative. 44 –Fig. 3.— Upper panel: BH number density as a function of z for 10 (black shaded area) and10 M ⊙ (blue shaded area). Lower panel: intrinsic rate of TDEs as a function of redshift. Arate of 10 − yr − per galaxy is assumed. Most of the events are expected below z ∼
2. 45 –Fig. 4.— Rate of events predicted for SKA in wide survey mode at 1.4 GHz as a functionof redshift (right panels) and their distribution as a function of BH mass (left panels), fortwo different black hole distribution functions (black solid line: G model black dotted line:Gz) model. Rates associated to events with BH masses lower than 10 M ⊙ are also shown(yellow lines). Upper panels:
BM model for the jet evolution; lower panels , MDL model. 46 –Fig. 5.— Cumulative distributions of delays in detecting the TDE from the explosion time,for different redshifts.
Top panel:
EP (black lines) and LAD follow-ups of radio triggeredTDEs (blue lines). The different line styles are for z = 0 . , . , . , . Bottom panel: the same as above but for SKA BM model and z = 0 . , . , . , . , . , < ∼ −11