DISRUPTION OF A RED GIANT STAR BY A SUPERMASSIVE BLACK HOLE AND THE CASE OF PS1-10jh

Stars on highly eccentric orbits can venture far from the central SMBH and experience a gravitational influence of other stars in the nuclear cluster at apocenter, where gravitational influence of the SMBH is at the minimum. Their orbital energy and angular momentum changes as a consequence of these interactions and therefore it is necessary for such spatially extended orbits to consider the contribution of the cluster stars to the gravitational potential of the central SMBHs.

In crowded stellar environments, in the centers of nuclear star clusters, the initial energy and angular momentum of an RG star can also be altered by physical collisions with other stars (Frank & Rees 1976). Collisions between RGs and stellar mass black holes have indeed been suggested as a plausible mechanism that can deplete the low-mass giants and asymptotic giant branch stars in the mass range 1–3 M_☉ within the 0.4 pc radius in the Galactic Center (Dale et al. 2009; Davies et al. 2011). These studies indicate that the probability that an RG star will encounter a stellar mass black hole (or with a lesser probability a main sequence star) at least once during its lifetime is substantial. Specifically, as the stellar black hole passes close to the RG, the core is deflected from its original orbit retaining some fraction of the envelope. A scattering (and even more so a physical collision) with a massive perturber can, however, lower the initial eccentricity of the RG orbit and place it on a bound orbit around the SMBH, even if it initially originated beyond the sphere of gravitational influence of the SMBH (Perets et al. 2009). We thus conjecture that the RG remnant underwent such an interaction and consider its end as a starting point of our investigation, given that the RG remnant has been deflected onto a new orbit with the pericenter in the range given by Equation (11) and an arbitrary apocenter somewhere within r_h (Equation (6)).

We borrow from the investigations of EMRIs to elucidate the necessary conditions for an RG remnant to spiral in close to the SMBH where it can be disrupted or become a source of GWs. EMRI studies indicate that the evolution of compact objects with the range of pericenters shown in Equation (11) is dominated by a combination of relativistic precession effects, GW energy loss, but also vectorial resonant relaxation and two-body relaxation (Amaro-Seoane et al. 2013b). Recent work on stellar dynamics has shown that the formation rate of EMRIs sensitively depends on their eccentricity. For example, the rates of the EMRIs with semi-major axes a_sb ∼ 10¹⁵ cm (1 − e²)^−1/3 can be strongly suppressed by the presence of what is usually referred to as the Schwarzschild barrier (Merritt et al. 2011; Brem et al. 2014). This barrier is a consequence of relativistic precession, which becomes pronounced as the compact object finds itself sufficiently deep in the potential well of the SMBH. The precession effectively blocks the gravitational influence of cluster stars that would otherwise build up over many approximately coherent orbits (Rauch & Tremaine 1996) and drive compact objects into the GW dominated regime. In the context of the RG remnant the relevant question is: can the core continue to spiral in toward the SMBH given the presence of the Schwarzschild barrier? Along similar lines, vectorial resonant relaxation has been found to change the orientation of the stellar orbit relative to the SMBH spin, causing the compact object that was initially outside of the ISCO at one instance of time to plunge into the black hole in the next, thus limiting any tidal disruption or GW signal.

It has been shown that stellar objects with a > a_sb, on highly eccentric orbits can evolve into EMRIs or tidal disruptions without danger of premature plunge into the SMBH or blockage by the Schwarzschild barrier (Merritt et al. 2011; Brem et al. 2014; Amaro-Seoane et al. 2013b). We use this criterion to further constrain the orbit of the RG remnant. For these objects, the two-body relaxation and emission of gravitational radiation remain the main drivers of orbital evolution. Which mechanism dominates is determined by the density and distribution of stars in the nuclear star cluster, the mass of the SMBH, and the initial orbit of the RG core.

The two-body relaxation is a diffusive (random walk) process resulting from small-angle gravitational scattering following the encounter of two stars (Chandrasekhar & Bok 1942), which can change both the orbital energy and angular momentum of the scattered star. The relaxation in angular momentum occurs on a timescale that is (1 − e²) times shorter than that for the energy, implying that stars on very eccentric orbits can significantly change their eccentricity while maintaining approximately the same semi-major axis (Amaro-Seoane et al. 2012). In the case of an RG remnant which is undergoing two-body scattering close to the apocenter of its orbit, this would cause the pericentric distance to change. Since we are interested in following the cores that approximately maintain their pericentric radius over multiple passages by the SMBH, we need to determine which orbits are likely to be affected by two body relaxation as opposed to the emission of GWs.

We use Equation (6) from Amaro-Seoane et al. (2012) and estimate the critical pericentric distance for return orbits at which the timescale for two-body relaxation in angular momentum is equal to the timescale for emission of gravitational radiation to shrink and circularize the orbit,

$$\begin{equation} R_p^{\rm crit} \sim 10^{12}\,{\rm cm} \left(\frac{a}{10^{18}\,{\rm cm}}\right)^{-1/2}\,M_6^{5/4} \end{equation} \tag{ 12 }$$

for a cluster with the assumed stellar mass density profile ρ(r)∝r^−7/4 (Bahcall & Wolf 1976). The orbits with pericenters below this critical value will evolve predominantly due to the emission of GWs and those above it will evolve due to the two-body interactions. Note that a large portion of the parameter space outlined in Equation (11) (but not all of it) resides within the GW-dominated regime as long as a < 10¹⁸ cm. Hence, there is an allowed region of the orbital parameter space where the RG remnant will continue to spiral into the SMBH due to the emission of GWs, such that 10¹⁵ < a < 10¹⁸ cm and pericenter is in the range given by Equation (11). In the rest of the paper we focus on RG remnants which find themselves on such orbits.

The three-dimensional relativistic hydrodynamic simulations of single encounter between a white dwarf and an intermediate mass black hole by Cheng & Evans (2013) indicate that during partial disruption most of the tidal energy is deposited into the surface layers of the star by shock heating. Radiation can escape the surface layers of the star easier than its denser central parts, which implies that shocks during repeated pericentric passages should cause a characteristic brightening of the RG remnant. As a result, the luminosity, temperature, and timescale for radiation to diffuse out of the surface layers of the star would be similar to those estimated for the fallback of nearly unbound material from the tidal tails in Section 3. In cases when the estimated radiation diffusion timescale (t_diff ≈ 7 × 10⁸ s) is longer than the orbital period of a returning orbit, the rate of tidal heating will be higher than the rate of radiative cooling leading to the net heating of a star. The orbital period of a Keplerian orbit, $2\pi \sqrt{a^3/GM}$, is

$$\begin{eqnarray} P_a &=& 5\times 10^8\,{\rm s}\; \beta ^{-3/2} \left(\frac{1-e}{0.01}\right)^{-3/2} \nonumber\\ &&\times\left(\frac{R_*}{10^{12}\,{\rm cm}}\right)^{3/2} \left(\frac{M_*}{1\,{M_{\odot }}}\right)^{-1/2}, \end{eqnarray} \tag{ 13 }$$

and hence P_a < t_diff for e ≲ 0.99 orbits around 10⁶ M_☉ black holes. The implication of this effect is the increase in luminosity of the stellar remnant as well as the expansion and degeneracy lifting in the outer layers. Because the luminosity of the shocked outer layers of perpetually heated star can become comparable or larger than the stellar Eddington luminosity after only a few orbits (see Equation (7)), the combined effect of radiative pressure and expansion will drive the increased mass-loss of the remainder of the stellar envelope. These two regimes have been previously described in some detail by Alexander & Morris (2003), who studied a class of tidally heated stars called squeezars. Thus, a majority of tidally stripped RG remnants may lose all of their bound hydrogen envelope after several orbital passages by the SMBH. A more precise timescale on which this happens depends on a complex interplay of radiative processes, hydrodynamics, and orbital dynamics and can only be captured though simulations.

We carry out a simple estimate to determine the relative importance of the disruption of the envelope versus the GW emission for the evolution of the remnant's orbit in the case where the stellar envelope is removed gradually. We assume that the envelope is stripped over N_s orbits, and write the necessary energy budget as

$$\begin{equation} \Delta E_s \sim \frac{1}{N_s}\,\frac{\eta _{\rm env} GM_{\rm core}\,M_*}{R_{\rm *}}, \end{equation} \tag{ 14 }$$

where η_env = M_env/M_*. We evaluate the energy loss per orbit due to the emission of GWs as

$$\begin{equation} \Delta E_{\rm gw} = -\frac{64\pi }{5} \frac{G^{7/2}\,M^2\,M_{\rm core}^2\, (M + M_{\rm core})^{1/2}}{c^5\,a^{7/2}} f_1(e), \end{equation} \tag{ 15 }$$

where f₁(e) = (1 + 73e²/24 + 37e⁴/96)/(1 − e²)^7/2 (Peters 1964). In the limit when e → 1 this expression becomes

$$\begin{equation} \lim _{e\rightarrow 1}\Delta E_{\rm gw} = -\frac{85 \pi }{12 \sqrt{2}} \frac{G^{7/2}\,M^2\,M_{\rm core}^2\, (M + M_{\rm core})^{1/2}}{c^5\,R_p^{7/2}} \;. \end{equation} \tag{ 16 }$$

For most orbital parameters considered here the energy loss due to the emission of gravitational radiation is well approximated by the solution based on semi-Keplerian orbits as described in Peters (1964). However, for cores that make very close pericentric passages by the SMBH (R_p ≲ 10 r_g), this approximation ceases to be accurate. Martell (2004) compared approximations equivalent to the approach by Peters (1964) and more accurate prescriptions and found that the difference in the GW energy loss amounts to a factor between 0.8 and 1.3. This level of approximation is appropriate for our study and we therefore use the Peters approximation.

Combining Equations (14) and (16) in the limit of e → 1 we obtain

$$\begin{eqnarray} \frac{\Delta E_s}{\Delta E_{\rm gw}} &\sim& 3 \times 10^4 \,\eta _{\rm env} \,\beta ^{-7/2} \left(\frac{N_s}{10^4} \right)^{-1} \left(\frac{R_*}{10^{12}\,{\rm cm}} \right)^{5/2} \nonumber \\ &&\times\left(\frac{M_*}{1\,{M_{\odot }}} \right)^{-1/6} \left(\frac{M_{\rm core}}{0.3\,{M_{\odot }}} \right)^{-1} M_6^{-4/3}. \end{eqnarray} \tag{ 17 }$$

Therefore, hydrodynamic effects related to the stripping of the envelope are the dominant driver of the orbital evolution in a majority of scenarios. It is worth noting that if most of the tidal energy (Equation (14)) is absorbed by the core, then repeated tidal interactions can in principle supply a significant fraction of energy necessary to lift the degeneracy of the core. This follows from the energy budget necessary to generate thermal pressure comparable to the degenerate pressure of the core, which can be estimated from the virial theorem of a white dwarf in equilibrium. We consider the tidal heating of the core in more detail in the next section.

It is worth noting that there is an additional mechanism that may drive an increased mass loss from the surface of the star. We do not consider the Roche lobe overflow in this work, but it has been shown by MacLeod et al. (2013) that RG stars on eccentric orbits continue to lose mass from their envelope at each pericentric passage as they overflow the Roche lobe for a short time. For an RG on a relatively stable orbit, the mass loss eventually ceases due to the stabilizing gravitational influence of the compact core.

Recent simulations also indicate that tidal disruption is an inherently asymmetric process and that unequal fractions of the debris are launched toward the SMBH and unbound from the system (Cheng & Evans 2013; Manukian et al. 2013; Kyutoku et al. 2013). This asymmetry is more pronounced (in geometric sense) for the larger mass ratios (q ≳ 10⁻⁴) and arises from the fact that more material escapes from the surface of the star facing the SMBH, relative to the surface that faces away (see Figures 6 and 7 in Cheng & Evans 2013). Since unequal portions of the debris carry unequal amounts of linear momentum, the conservation of linear momentum then requires that the remnant core is deflected from its initial orbit. We neglect this effect which stems from the hydrodynamic interaction of the star with the tides of the SMBH and assume that the core continues to orbit along the geodesic previously occupied by the center of mass of the RG star.

In this section, we consider the effect of tidal interaction on the helium core. The SMBH tides drive oscillations which if dissipated within the core lead to a gradual circularization of its orbit. It can be shown that the amount of energy available from circularization is much larger than the gravitational energy of the star (Ivanov & Papaloizou 2007). Namely, the energy released from circularization of an orbit with initial pericentric radius R_p and eccentricity e → 1 is ΔE_circ ≈ GMM_core/4R_p. Comparing that to the gravitational energy of the core, $E_{\rm core} \approx GM_{\rm core}^2/R_{\rm core}$, yields

$$\begin{eqnarray} \frac{\Delta E_{\rm circ}}{E_{\rm core}} &&\nonumber \approx \frac{1}{4}\beta _{\rm core}\left(\frac{M}{M_{\rm core}}\right)^{2/3} \\ && \approx 6\times 10^3 \beta _{\rm core}\left(\frac{M_{\rm core}}{0.3\,{M_{\odot }}} \right)^{-2/3}\,M_6^{2/3} \;. \end{eqnarray} \tag{ 18 }$$

Even if a small fraction of this energy is deposited into the remnant it is in principle sufficient to lift the degeneracy of the core and possibly dissolve it before the core reaches the tidal radius or plunges into the SMBH (Ivanov & Papaloizou 2007). Recall, however, that at the same time the emission of gravitational radiation is concurrently acting to decrease the orbital eccentricity of the core. If tidal heating is the faster of the two processes, the remnant will be destroyed before the orbit is circularized. If, however, the GW emission is more efficient, the remnant will find itself on a quasi-circular orbit, gradually inspiralling into the SMBH and possibly forming an EMRI (see, for example, Zalamea et al. 2010, who studied mass transfer between an inspiralling white dwarf and a low-mass SMBH in this regime).

To estimate the timescale on which the tides are depositing energy onto the star, we rely on the linear theory of tidal interactions developed for stars and planets on eccentric orbits (Press & Teukolsky 1977; Leconte et al. 2010). The linear theory applies to weak encounters, rather than disruptive ones when description of the tidal interaction as a low level perturbation ceases to be accurate. Consequently, the linear theory can be used to estimate the energy deposited onto a compact core, for which β_core ≪ 1, but is not applicable to the stellar envelope that is in the β ≳ 1 regime. In this calculation we therefore consider the tidal heating of the compact core only and do not account for the effects of the envelope (which are discussed in Section 4.2 as an order of magnitude estimate). This is a conservative assumption which leads to a slower orbital evolution of the core since in absence of the envelope $\dot{E}_{\rm tid}$ is only a lower limit on the rate of energy deposition into the remnant. Similar to Amaro-Seoane et al. (2012), we adopt the expression from Leconte et al. (2010) for the rate at which the orbital energy is lost to the excitation of oscillatory modes within the star

$$\begin{equation} \dot{E}_{\rm tid} = 2K_p\left[ N_a(e) - \frac{N^2(e)}{\Omega (e)} \right], \end{equation} \tag{ 19 }$$

where the expression in the brackets only depends on eccentricity and can be simplified to 7e²/2 for e < 1. This is an adequate approximation for the orbital evolution of the helium core, which starts on a highly eccentric orbit and is gradually circularized by tidal heating and GWs. Using their expression for K_p

$$\begin{equation} K_p \approx \frac{9}{4}Q^{-1}\, \left(\frac{GM_{\rm core}^2}{R_{\rm core}}\right) \left(\frac{M}{M_{\rm core}}\right)^2 \left(\frac{R_{\rm core}}{a}\right)^6 \left(\frac{GM}{a^3}\right)^{1/2}. \end{equation} \tag{ 20 }$$

Modifying further to relate a to the orbital pericenter and orbital period, we obtain

$$\begin{equation} \dot{E}_{\rm tid} \approx \frac{63\pi }{2} Q^{-1}\,e^2\,\beta _{\rm core}^6 \left(\frac{GM_{\rm core}^2}{R_{\rm core}}\right)\,P_a^{-1}, \end{equation} \tag{ 21 }$$

where Q is the quality factor, a parameterization of the coupling of the tidal field to the stellar core. The magnitude of Q is quite uncertain and commonly assumed in the literature to be in the range ∼10⁵–10⁶. However, a recent work by Prodan & Murray (2012) indicates that the quality factor of helium white dwarfs may be even higher (they find Q ≳ 10⁷) but these types of measurements are complicated by the fact that the value of Q can only be determined as a lower limit and is coupled to other (uncertain) parameters of the system.

We calculate the characteristic time for circularization of the RG core orbit assuming that tidal dissipation within the remnant is efficient and obtain

$$\begin{equation} \tau _{\rm tid} = \frac{\Delta E_{\rm circ}}{\dot{E}_{\rm tid}} = \frac{1}{126\pi } \frac{Q}{\beta _{\rm core}^6\, e^2} \left(\frac{M}{M_{\rm core}} \right)^{2/3}P_a. \end{equation} \tag{ 22 }$$

We then calculate the amount of time it takes to circularize the orbit due to the emission of GWs (Peters 1964)

$$\begin{equation} \tau _{\rm gw} = \frac{e}{\dot{e}} = \frac{15}{608\pi }\,f_2(e)\,\beta _{\rm core}^{-5/2} \left(\frac{R_{\rm core}}{r_g}\right)^{5/2} \left(\frac{M}{M_{\rm core}}\right)^{11/6}P_a, \end{equation} \tag{ 23 }$$

where f₂(e) = (1 + 121e²/304)⁻¹. Comparing the two timescales yields

$$\begin{equation} \frac{\tau _{\rm tid}}{\tau _{\rm gw}}\approx 2\times 10^3 \,\frac{\beta _{\rm core}^{-7/2}}{e^2 f_2(e)}\, Q_6 \left(\frac{R_{\rm core}}{10^{9}\,{\rm cm}} \right)^{-5/2} \left(\frac{M_{\rm core}}{0.3\,{M_{\odot }}} \right)^{7/6} M_6^{4/3}, \end{equation} \tag{ 24 }$$

where 10⁶Q₆ = Q. Hence, the gravitational radiation will always circularize the core's orbit on a timescale much shorter than that for the tidal circularization. This implies that only a fraction of the energy available from orbital circularization will be deposited into the remnant. How large that fraction is in terms of the remnant's binding energy depends on the properties of the core, its orbit, and the mass of the black hole. Equations (18) and (24), for example, indicate that close to disruption (β_core ∼ 1), the core can in principle absorb ∼6 × 10³/2 × 10³ ∼ few times its binding energy by the time its orbit is circularized due to the emission of GWs. However, the compact core considered here reaches the maximum penetration factor, $\beta _{\rm core}^{\rm max}\approx 0.5 <1$ at the event horizon of a 10⁶ M_☉ SMBH, where the amount of energy deposited into it is at maximum ∼0.08E_core.

This is illustrated in Figure 1 where the color bar shows the maximum tidal energy deposited into the inspiralling stellar core as a function of the initial orbital eccentricity (e_i) and strength of the tidal encounter (β and β_core). This energy will be stored within the core because it is deposited on a timescale shorter than that on which it can be radiated (i.e., the Kelvin–Helmholtz timescale). Since τ_KH = E/L (where E is the thermal energy of the white dwarf-like core) the more luminous cores tend to cool faster. Indeed, theoretical calculations of white dwarf cooling sequences show that a 0.3 M_☉ white dwarf with luminosity between 0.3 > L > 10⁻⁴ L_☉ cools on a timescale 10⁸ < τ_KH < 10¹⁰ yr, respectively (Driebe et al. 1998; Nelemans et al. 2001). For the cores with a representative cooling timescale of τ_KH ∼ 10⁹ yr ≫ τ_gw, the evolution may proceed similar to squeezars, tidally powered stars that find themselves on non-disruptive orbits (Alexander & Morris 2003).

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We now consider the interaction of the RG remnant on a return orbit with the accretion disk that forms from the tidally stripped hydrogen stellar envelope. By the time the core returns, the inner parts of the debris disk have had time to complete multiple orbits and start circularizing (Ulmer 1999). As a consequence, the debris is more spatially extended than the core's orbit at separation ∼R_T from the SMBH leading to the intersection of their orbits. Based on considerations in Section 2 we estimate that a core encounters the debris disk with radius ∼2R_T and mass in the range M_disk ≈ 0.2–0.7 M_☉, where the lower bound corresponds to parabolic encounters at the threshold of disruption and the upper bound corresponds to β > 1 encounters where the RG star is initially placed on moderately eccentric, bound orbit.

The core can only experience significant orbital energy losses if it encounters the amount of disk mass comparable to its own. Since the debris disk is spread to a very low density, the remnant will not be able to sweep a mass equivalent to its own even after many passages and thus, the effect of the disk on the orbit of the remnant can be neglected. This outcome is guaranteed given a relatively short estimated timescale for accretion of the debris, t_acc ∼ few × P_m (see Section 3), indicating that the debris disk will be accreted and dispersed before it can impact the RG core in any way.

Along similar lines, given a relatively short lifetime of the transient disk, it is also unlikely that the core-disk collisions can give rise to more than a few quasi-periodic luminosity outbursts.

We now consider a recently discovered tidal disruption candidate PS1-10jh described by Gezari et al. (2012). PS1-10jh was discovered as an optical transient in the Pan-STARRS1 Medium Deep Survey and independently as a near-ultraviolet source within the GALEX Time Domain Survey. Based on the modeling of the multi-band photometry and optical spectra of this object, Gezari et al. (2012) proposed that the disrupted object is a tidally stripped helium core of an RG star that initially had mass M_* ≳ 1  M_☉. They adopt properties of the core consistent with those measured for an RG stellar remnant stripped in a stellar binary system (Maxted et al. 2011), where M_core ∼ 0.23 M_☉ and R_core ∼ 0.33 R_☉, and use the local scaling relations and tidal disruption light curve to constrain the mass of the SMBH to M ≈ 2.8 × 10⁶ M_☉.

Note that the radius of the remnant adopted by Gezari et al. (2012) is an order of magnitude larger than that of the "bare" helium core we consider in previous sections (based on models of Hamada & Salpeter 1961; Althaus & Benvenuto 1997). This is because a low-mass RG remnant created through stripping and mass transfer in a stellar binary is expected to retain some fraction of its hydrogen envelope in which hydrogen burning is maintained in a shell. The luminosity contribution due to the nuclear burning drives the expansion of the envelope and degeneracy lifting in the exterior shells of such white dwarf and results in the radius of the remnant typically a few times larger than that of a pure helium core (Driebe et al. 1998, 1999).

The choice of the RG remnant structure has important implications for the outcome of the tidal interaction. Specifically, a bare helium core considered previously is too compact in size to be disrupted by the SMBH (Figure 1), while the core with the radius ≈0.33 R_☉ considered by Gezari et al. (2012) can be disrupted after multiple passages (this will be shown in the next section). What is a more realistic choice for the structure of the remnant is a question of some subtlety. Even if the RG core initially had a radius of ∼10⁹ cm, given a sudden loss of 30%–50% of the RG mass during its first pericentric passage (see Section 2), the core will have to adjust to a new equilibrium by expanding beyond its original radius. We estimate the change in the radius of the core from the mass–radius relation for the adiabatic evolution of a nested polytrope, as outlined by MacLeod et al. (2013). Depending on the adopted value of the polytropic index, one finds that the new radius of the remnant can be up to 5.5 times larger.⁸ Therefore, we expect that the radius of the remnant core in the case of PS1-10jh to be larger than that of the core in an intact RG and for the purpose of the analysis in this section adopt the mass and radius suggested by Gezari et al. (2012).

Several observational properties of PS1-10jh offer important clues about the remnant.

• 1.  
  The temporal decay of the light curve closely follows the canonical prediction L_fb∝t^−5/3 indicating that the incoming core must have been on a highly eccentric orbit prior to disruption such that at least one portion of it ended up being gravitationally unbound from the system. It has otherwise been shown that lower eccentricity encounters in which 100% of the debris is gravitationally bound to the black hole have the fallback accretion rate significantly different from ∝t^−5/3 (Hayasaki et al. 2013; Dai et al. 2013, see however the end of this section for further discussion of this point).
• 2.  
  An interesting property of PS1-10jh are its broad high-ionization He ii lines (with wavelengths λ = 4686 Å and λ = 3203 Å) which are contrasted by the absence of the hydrogen emissions lines. Gezari et al. (2012) find that the lack of hydrogen Balmer lines requires a very low hydrogen mass fraction of <0.2, thus supporting the picture that the disrupted star is a helium core. The lack of any emission lines from the hydrogen stellar envelope, which is expected to be several times more massive than the core in the progenitor RG star, is, however, puzzling and is the key to understanding the nature of this event.
• 3.  
  The characteristic temperature of the emitted radiation from PS1-10jh has remained relatively uniform before and after the disruption (T ≳ 3 × 10⁴ K). The lack of evolution of the spectrum indicates that the observed radiation may have been reprocessed by the obscuring layer or a shell, enshrouding the inner debris disk. Because the luminosity for this system can only be determined as a lower limit, ≳ 0.6 L_Edd (Gezari et al. 2012), it follows that both the radiation forces and relativistic frame-dragging are viable candidates for a mechanism that can distribute the debris across the sky and create obscuration.

The expectation that the fallback accretion rate differs significantly from ∝t^−5/3 for disruption of stars on initially bound orbits with moderate eccentricities (e ≲ 0.99) is based on theoretical studies by Dai et al. (2013) and Hayasaki et al. (2013), who carried out the particle and hydrodynamic simulations of this scenario, respectively. There is, however, an inherent uncertainty in interpretation of simulated accretion rates in terms of the observed tidal disruption light curves, since these simulations do not account for the effects of radiative feedback nor do they model accretion light curves in a given wavelength band. We acknowledge this uncertainty in the step where we suggest that an observed optical light curve of PS1-10jh with behavior close to L_fb∝t^−5/3 indicates a similar power-law behavior in the underlying accretion rate.

One possible explanation for the absence of hydrogen emission lines is that most of the tidally stripped envelope was accreted by the black hole before the helium core disruption (see the next subsection for discussion about other possible explanations). This scenario could play out if the RG core completed multiple orbits before being disrupted by the SMBH during which time the envelope was nearly completely stripped and accreted. We consider the orbital properties of the incoming RG star which could have given rise to such a sequence of events and can simultaneously satisfy the described observational properties. A condition for partial disruption which must be satisfied in order to have the RG envelope stripped by the tidal forces and the compact core intact is β ⩾ 1 and β_core < 1. We assume that the stellar core had a relatively common RG progenitor star with mass and radius M_* = 1.4 M_☉ and R_* = 10¹² cm. Note that these choices are uncertain even though they are physically motivated, as the evolutionary sequence of the helium remnant cannot be constrained more precisely from the available data. Based on this adopted structure and Equation (3), we find that the penetration factor of the compact core is β_core ≈ β/24 and thus there is a range in β-parameter space for which partial disruption can be achieved.

In order for most of the hydrogen envelope to be accreted, it must be placed on bound eccentric orbits around the SMBH. It is possible to estimate from analytic arguments the critical orbital eccentricity below which the entire envelope is bound. From the condition that the orbital energy of the RG envelope (and the star) is equal to the energy spread in the tidal field of the black hole at the point on the orbit when the envelope is removed we find

$$\begin{equation} \frac{GM(1-e)}{2R_p} = \frac{GM\,R_{\rm *}}{R_T^2}. \end{equation} \tag{ 25 }$$

From this condition, it follows that the initial eccentricity of the RG stellar orbit must be smaller than $e_{\rm crit}^{\rm env} \approx 1 - (2/\beta) (M_*/M)^{1/3}$ in order for the entire envelope to be bound to the SMBH (Hayasaki et al. 2013). We derive a similar condition for the first pericentric passage of the RG core,

$$\begin{equation} \frac{GM(1-e)}{2R_p} = \frac{GM\,R_{\rm core}}{R_p^2}. \end{equation} \tag{ 26 }$$

Note that in this case the spread in energy reaches maximum at the pericenter of the orbit, rather than at the tidal radius of the envelope and that $R_p = R_T/\beta = R_T^{\rm core}/\beta _{\rm core}$. This implies a critical eccentricity value of $e_{\rm crit}^{\rm core} \approx 1 - (2 \beta _{\rm core}) (M_{\rm core}/M)^{1/3}$ and is valid when β_core < 1.

The ∝t^−5/3 temporal behavior of the disruption light curve indicates that at least one portion of the tidally disrupted core is not bound to the SMBH. It follows that the eccentricity of the first orbit of the RG star must satisfy the condition $e_{\rm crit}^{\rm core} < e_i < e_{\rm crit}^{\rm env}$. This criterion is satisfied only when $e_{\rm crit}^{\rm core} < e_{\rm crit}^{\rm env}$ leading to

$$\begin{equation} \beta \beta _{\rm core} > \left(\frac{M_*}{M_{\rm core}}\right)^{1/3} \end{equation} \tag{ 27 }$$

and, consequently, 6 < β < 24, or written in terms of the core penetration factor, 0.25 < β_core < 1. Hence, there is a portion of the parameter space conducive to disruption of the RG envelope followed by the disruption of the core upon completion of some number of orbits by the core. In such a case, the hydrogen envelope ends up bound to the SMBH in its entirety, while the debris created by the subsequent disruption of the helium core is only partially bound. It is important to point out that the exact bounds within which this region lies are approximate, as they are derived from simple analytic considerations. However, a realization that such a region must exist for highly eccentric orbits, naturally follows from the difference in the relative spread in energies of the core and envelope.

The inspiralling core will be subject to tidal heating and emission of gravitational radiation, as discussed in Section 4.3. Taking the ratio of Equations (18) and (24), we calculate the maximum tidal energy that can in principle be deposited into the remnant before gravitational radiation circularizes its orbit. For the assumed mass and radius of the helium core disrupted in PS1-10jh and the inferred mass of the SMBH we find

$$\begin{equation} \frac{E_{\rm dep}}{E_{\rm core}}\sim 5\times 10^3 \, e^2\, f_2(e)\; Q_6^{-1}\beta _{\rm core}^{9/2} \; M_{\rm core}^{-11/6} \; R_{\rm core}^{5/2}\; M^{-2/3}, \end{equation} \tag{ 28 }$$

where M_core, R_core, and M have been normalized to the appropriate values for this system. The amount of the deposited tidal energy is a strong function of β_core. Given that we are considering the β_core < 1 portion of the parameter space, where the core gradually inspirals into the SMBH, the ratio of E_dep/E_core will generally be lower than the estimate in Equation (28).

In Figure 2 the color bar shows the maximum tidal energy deposited into the inspiralling stellar core as a function of the initial orbital eccentricity (e_i) and strength of the tidal encounter (β and β_core). In this parameter space β_core > 0.2 encounters can in principle deposit the amount of tidal energy larger than E_core prior to orbital circularization and thus lift the degeneracy of the core and lead to its disruption before it reaches the tidal radius (note that this is different from the outcome shown in Figure 1). This can be accomplished as long as the tidal energy is stored in the core over many orbits and not efficiently radiated. The initial orbital period of the core is shown in seconds as a function of e_i and β_core with black contour lines. This indicates that the majority of this parameter space comprises of initial orbital periods <τ_KH. The portion of the parameter space where e ≈ 1 is characterized by orbital periods longer than the Hubble time is indicated by the letter "H."

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In Figure 2 we also mark the region within which the criterion for partial disruption described by Equation (27) is satisfied (blue dash–dotted line). The region is confined to a narrow strip of high eccentricities with values e > 0.991, characteristic of stellar objects that fall toward the black hole on a nearly parabolic orbit. The incoming RG star must occupy the region below the upper dash–dotted line in order for its hydrogen envelope to be entirely bound to the SMBH after it is stripped. The observed temporal behavior of the light curve in PS1-10jh on the other hand indicates that at the point of disruption the helium remnant must still have occupied a very eccentric orbit, somewhere above the lower dash–dotted line. This region is characterized by orbits with the period in the range ∼10⁷–10⁸ s, GW circularization times in the range ∼10⁷–10⁹ yr, and intense tidal interaction with maximum deposited tidal energy in the range ∼10–5 × 10³E_core. The core occupying this portion of the parameter space would therefore have had time to complete many of orbits before the deposited tidal energy is radiated on a Kelvin–Helmholtz timescale. It is thus likely that such a core would have its degeneracy lifted by perpetual episodes of tidal heating, leading to premature disruption. An early disruption would also ensure that the core remains on a highly eccentric orbit, before the GW emission had time to circularize it significantly.

The observed light curve of PS1-10jh also indicates that the white dwarf-like core would have to undergo a single disruption event at the end of the tidal heating period, rather than produce a series of quasi-periodic accretion events as the core overflows its Roche lobe at every pericentric passage. The latter scenario has been modeled by Zalamea et al. (2010), who find that the resulting accretion curve consists of multiple exponentiating flares very distinct from the $\dot{M}\propto t^{-5/3}$ behavior. We propose that the core undergoes a single disruption scenario if the imparted tidal energy can be redistributed over the volume of the remnant, maintaining the evolution through a series of quasi-equilibria and avoiding rapid expansion of the outer layers and stripping.

The timescale on which the tidal energy can be transported from the surface of the remnant inward is given by its thermal diffusion timescale, $\tau _{\rm cond} = R_{\rm core}^2/\chi _{\rm th}\sim 10^6$ yr, where χ_th is the diffusivity constant (Padmanabhan 2001). The hierarchy of timescales P_a ≪ τ_cond ≪ τ_KH implies that tidal interactions establish a temperature gradient between the surface and the center of the remnant that last over many orbits but that the tidal energy is diffused throughout it before the remnant has a chance to cool radiatively. It is thus plausible in this regime to tidally heat the remnant gradually, although some amount of tidal stripping from the surface of the remnant is likely at the rate lower than that found by Zalamea et al. (2010).

In the mean time, the stripped stellar envelope forms an accretion disk with maximum initial mass M_{disk, i} = M_* − M_core ≈ 1.2 M_☉, which circularizes and accretes onto the SMBH. The observations of PS1-10jh indicate that at the moment of disruption of the helium core the mass fraction of hydrogen must have been more than five times lower relative to helium and other atomic species thus, placing the upper limit on the final mass of the hydrogen disk to M_{disk, f} < 0.2M_core ≈ 0.05  M_☉. Integrating over the accretion rate in Equation (5), we find that this portion of the disk would be accreted after t_acc ∼ P_m. For the parameters considered here, the orbital period of the debris deepest in the SMBH potential well is P_m ∼ 2 × 10⁸ s and is shorter for more bound configurations of the envelope, corresponding to RG orbits with lower initial eccentricity. This implies that with a moderate or low level radiative feedback the gas would be swallowed by the black hole on a timescale ∼10 yr. Alternatively, strong radiative feedback would promptly disperse and ionize the debris. This timescale places a lower limit constraint on the time between the disruption of the hydrogen envelope and disruption of the helium core. Note that PS1-10jh was not detected in 2009 in the deep coadd of the GALEX time domain survey with a ∼3σ upper limit of >25.6 mag (Gezari et al. 2012). In the context of our model this indicates that by that point most of the disk must have been accreted or dispersed.

Another possible explanation for the weakness or absence of the hydrogen emission lines is that the debris hydrogen envelope may be present at the disruption site in the vicinity of the SMBH, but is completely ionized or produces a negligible amount of hydrogen emission. Tidal disruptions have indeed been proposed to create a highly ionized nebular emission from the ISM in the inner kiloparsec of their host galaxies (Eracleous et al. 1995). This scenario is considered by Guillochon et al. (2014), where three-dimensional hydrodynamic simulations show the disruption of a low mass main sequence star (∼0.2 M_☉) with debris forming a finite size BLR, truncated at the outer radius. The compact BLR consists of gas from the disrupted star including hydrogen, but effectively behaves as an H ii region and consequently emits weak hydrogen emission lines below the current detection threshold. He II emission-line on the other hand remains present in the spectrum longer due to the higher ionization potential of this element. In this scenario, the unbound tidal stream has a negligible surface area and makes negligible contribution to either the continuum or line emission. This explanation is consistent with the finding that the luminosity of the hydrogen Balmer lines from the tidal disruption of a solar type star by a 10⁶  M_☉ SMBH is ≲ 10³⁹ erg s⁻¹ (Bogdanović et al. 2004), and thus, below the detection threshold of the available spectroscopic data for PS1-10jh. This work, however, does not address the intensity of the helium emission-lines and hence does not constrain the He/H line ratio. Recently, Gaskell & Rojas Lobos (2014) carried out the one-dimensional photoionization modeling of the emission properties of gas in truncated BLRs⁹ and concluded that strong helium emission does not require depletion of hydrogen. Specifically, they find that the He ii λ4686/Hα line ratio peaks at the value as high as ∼4.0 when the gas density is ∼10¹¹ cm⁻³, given a solar abundance ratio of the two elements. This line ratio is a sensitive function of the gas density, however, and the work by Gaskell & Rojas Lobos (2014) also indicates that unless most of the solar metallicity debris resides in the density range 10¹⁰ < n < 10¹² cm⁻³, the value of the line ratio He ii λ4686/Hα ≲ 1. Stellar debris produced in tidal disruptions is indeed characterized by a wide range of densities and is spatially distributed over eccentric orbits, allowing multiple density phases of gas to reside at the same distance from the ionization source (Bogdanović et al. 2004; Strubbe & Quataert 2009, 2011). The non-axisymmetric distribution of the gas results in a more complex emission stratification than in the BLR regions of AGN that does not a priory preclude the emission of the broad hydrogen lines. Indeed, more recent and detailed calculations indicate that it is difficult to suppress them to the level consistent with the observed properties of PS1-10jh (L. E. Strubbe & N. Murray 2014, in preparation).

Given the difficulty to "hide" the hydrogen emission lines in the presence of the illuminated hydrogen rich stellar debris, in this work we consider a family of models that lead to a near complete accretion rather than dispersal, of the hydrogen envelope of the partially disrupted RG star. In the context of this hypothesis, at a later point in time when the helium core is disrupted, helium dominates the optical emission line spectrum, as predicted by the photoionization calculations of the spectrum from a tidally disrupted white dwarf by Sesana et al. (2008). This is also consistent with the work Clausen & Eracleous (2011), who find that the UV and optical spectrum of a tidally disrupted carbon–oxygen white dwarf is dominated by these species and characterized by the overall lack of hydrogen emission lines. It is also worth noting that have the multiple and more varied emission line features been seen in the spectrum of PS1-10jh, it would be possible to put stronger constrains on the structure of the disrupted RG star and the mass of the black hole. This is illustrated by Clausen et al. (2012) who found a good agreement of their modeled emission line spectrum from the disruption of a horizontal branch star by a BH with that observed from the globular cluster NGC 1399 which hosts the ultraluminous X-ray source CXOJ033831.8-352604.

The favored portion of the parameter space, 6 < β < 24 marked in Figure 2, maps into pericentric radii in the range ∼13–52 r_g. At this distance the helium core and its debris are subject to relativistic effects such as pericenter precession (see Guillochon et al. 2014, for discussion of this effect) and precession of the orbital plane, if the orbital axis and spin of the SMBH are misaligned.

Evidence that relativistic effects can lead to the extended distribution of the debris can be found in work by Haas et al. (2012), who simulated the disruption of a white dwarf star by a spinning IMBH with arbitrarily oriented spin axes. They find that as a consequence of frame-dragging, the stellar debris forms an optically and geometrically thick torus rather than an accretion disk. This leads to the obscuration of the inner fallback disk by the outflowing debris. As a consequence, the spectrum from the debris is softer because it is mostly emitted by the material further away from the black hole. Along similar lines, Loeb & Ulmer (1997) have calculated the temperature of radiation reprocessed and re-emitted at the Eddington limit by an optically thick, static shell of gas with mass ∼0.1 M_☉ enshrouding the inner debris accretion disk

$$\begin{equation} T_{\rm shell} \approx \left(\frac{L_E}{4\pi R_{\rm out}^2 \sigma }\right)^{1/4} = 2.5\times 10^4\,{\rm K}\,\left(\frac{M}{10^7 M_{\rm env}} \right)^{1/4}, \end{equation} \tag{ 29 }$$

where R_out is defined as the photospheric radius—the radius at which the optical depth to Thomson scattering τ_T = 1. T_shell calculated in Equation (29) is comparable to the characteristic temperature of radiation maintained by PS1-10jh throughout the disruption. This finding, combined with the fact that the range for the peak bolometric luminosity of PS1-10jh inferred from observations permits values lower than the Eddington luminosity (Gezari et al. 2012), leaves room for relativistic effects as a plausible explanation for the configuration and emission properties of the debris in PS1-10jh.