BLACK HOLE SPIN IN Sw J1644+57 and Sw J2058+05

A hard X-ray transient event, Swift J16449.3+573451 (Sw J1644+57 hereafter), was discovered by the Swift satellite (Gehrels et al. 2004) on 2011 March 28, initially reported as GRB 110328A (Cummings et al. 2011). It was soon realized that it is not a regular gamma-ray burst (GRB). Long-term follow-up observations with Swift (Burrows et al. 2011) revealed that it has extended emission in the X-ray band without significant decay. A much longer variability timescale δt ∼ 100 s than normal for GRBs (Burrows et al. 2011) as well as its location near the center of a z = 0.354 host galaxy (Levan et al. 2011) link the source to a supermassive black hole (BH) with M_• ∼ 10⁷M_☉ (Burrows et al. 2011; Bloom et al. 2011). The super-Eddington X-ray luminosity (Burrows et al. 2011), bright radio afterglow (Zauderer et al. 2011), and a historical stringent X-ray flux upper limit suggest that the event marks the onset of a relativistic jet from a supermassive BH. The sharp onset and gradual fadeaway of X-ray flux suggest that the transient may be triggered by tidal disruption of a star by the BH (Bloom et al. 2011; Burrows et al. 2011).

A second candidate in the same category, Swift J2058.4+0516 (Sw J2058+05 hereafter), was reported by Cenko et al. (2011), who argued that its observational properties are rather similar to those of Sw J1644+57.

These events are rare (two detected in seven years of Swift operation). Most previous studies of tidal disruption events (TDEs) from supermassive BHs did not expect an associated relativistic jet (e.g., Kobayashi et al. 2004; Strubbe & Quataert 2009; Lodato & Rossi 2011, but see Lu et al. 2008; Giannios & Metzger 2011). The neutrino-annihilation mechanism as invoked in GRBs is inadequate given a much lower accretion rate as compared with GRBs (e.g., Shao et al. 2011). A plausible mechanism to launch a jet from such an event is to tap spin energy of the BH through a magnetic field, which connects the BH event horizon and a remote astrophysical load (Blandford & Znajek 1977, hereafter BZ). This mechanism has been widely invoked to interpret active galactic nucleus (AGN) jets (e.g., Vlahakis & Konigl 2004). Within this scenario, a dominant magnetic jet composition is envisaged. Indeed, modeling the emission of Sw J1644+57 suggests that the jet is highly "particle starved," i.e., most energy is not carried by particles. A natural inference is that the outflow is Poynting-flux-dominated (Burrows et al. 2011).

If one identifies the BZ mechanism as the jet launching mechanism, the BH spin parameter (a dimensionless angular momentum of the BH, which ranges from 0 for no spin to 1 for the maximum spin) can be constrained from the data. This is the subject in this Letter.

The bipolar BZ jet power from a BH with mass M_• and angular momentum J is (Lee et al. 2000; Li 2000; Wang et al. 2002; McKinney 2005)

$$\begin{equation} L_{\rm BZ}=1.7 \times 10^{44} a_{\bullet }^2 M_{\bullet,6}^2 B_{\bullet,6}^2 F(a_{\bullet }) \ {\rm erg \ s^{-1}}, \end{equation} \tag{ 1 }$$

where a_• = Jc/(GM²_•) is the BH spin parameter, M_{•, 6} = M_•/10⁶ M_☉, B_{•, 6} = B_•/10⁶ G, and

$$\begin{equation} F(a_{\bullet })=[(1+q^2)/q^2][(q+1/q) \arctan q-1]; \end{equation} \tag{ 2 }$$

here $q= a_{\bullet } /(1+\sqrt{1-a^2_{\bullet }})$ and 2/3 ⩽ F(a_•) ⩽ π − 2 for 0 ⩽ a_• ⩽ 1. It apparently depends on M_•, B_•, and a_•. However, since an isolated BH does not carry a magnetic field, B_• is closely related to the accretion rate $\dot{M}$ and the radius of the BH, which depends on M_•. Combining these dependences, one finds that L_BZ is essentially independent of M_•, but is rather a function of $\dot{M}$ and a_•. This may be proven with the following rough scalings. For a Newtonian disk, angular momentum equation states $\dot{M}r^2 ({\it GM}_{\bullet }/r^3)^{1/2} \simeq - 4 \pi r^2 \tau _{r\phi } h$, where τ_rϕ is the viscous shear and h∝r is the half disk thickness, with h/r ≪ 1 for a thin disk and h/r ∼ 1 for a thick disk. Adopting the α-prescription for viscosity, the viscous shear can be expressed as τ_rϕ = −αP, where P is the total pressure in the disk. One therefore derives $P \simeq \dot{M} r^2 (GM_{\bullet }/r^3)^{1/2}/(4\pi r^2 \alpha h) \propto (M_{\bullet }/r^5)^{1/2} \propto M_{\bullet }^{-2}$, where we have applied the scaling r∝r_s∝M_• (r_s = 2GM_•/c² is the Schwarzschild radius). To make an efficient BZ jet, the accretion inflow should carry a large magnetic flux (e.g., Tchekhovskoy et al. 2011). It is reasonable to assume that magnetic fields in the disk are in close equilibrium with the total pressure, so that B²_•∝P∝M_•⁻². Inserting this dependence to Equation (1), one finds that L_BZ is essentially independent of M_•.

More precisely, we adopt the following prescription to treat the problem. The total pressure in the disk can be expressed as P = P_rad + P_gas + P_B, where P_rad = aT⁴/3, P_gas = nkT, and P_B = B²_disk/8π are radiation, gas, and magnetic pressure, respectively. Here T is the temperature of the disk, n is the gas particle number density, a is the radiation density constant, and k is the Boltzmann constant. We denote P_B = βP and take β ∼ 0.5 in this work. This corresponds to a maximized magnetic flux. A smaller β would demand an even larger a_• to reach the same BZ power. So taking β ∼ 0.5 gives an estimate of a conservative lower limit of a_•. For the two sources (Sw J1644+57 and Sw J2058+05) we are interested in, the accretion rate is estimated close to or even larger than the Eddington accretion rate (Burrows et al. 2011; Cenko et al. 2011). In this regime, a thin disk model may be adequate to describe the disk, and we apply it to estimate disk pressure for simplicity. The disk pressure peaks in the inner region where radiation pressure may dominate. As a result, for β ∼ 0.5, one has

$$\begin{equation} \frac{B_{\rm disk}^2}{8\pi } \sim \frac{1}{3} a T^4, \end{equation} \tag{ 3 }$$

where the temperature of the disk (a_•- and r-dependent) can be written as

$$\begin{equation} T(a_\bullet,r) = \left(\frac{3G M \dot{M}}{8\pi r^3 \sigma } f\right)^{1/4}, \end{equation} \tag{ 4 }$$

where σ is the Stephan–Boltzmann constant and the general relativistic correction factor (Page & Thorne 1974)

$$\begin{eqnarray} f(a_\bullet,r) & = & \frac{\chi ^2}{(\chi ^3-3\chi +2a_{\bullet })} \left[ \chi -\chi _{\rm in} -\frac{3}{2}a_{\bullet } \ln \left(\frac{\chi }{\chi _{\rm in}}\right) \right. \nonumber \\ & &- \frac{3(\chi _1-a_{\bullet })^2}{\chi _1(\chi _1-\chi _2)(\chi _1-\chi _3)} \ln \left(\frac{\chi -\chi _1}{\chi _{\rm in}-\chi _1}\right) \nonumber \\ & & - \frac{3(\chi _2-a_{\bullet })^2}{\chi _2(\chi _2-\chi _1)(\chi _2-\chi _3)} \ln \left(\frac{\chi -\chi _2}{\chi _{\rm in}-\chi _2}\right) \nonumber \\ & & \left. -\,\frac{3(\chi _3-a_{\bullet })^2}{\chi _3(\chi _3- \chi _1)(\chi _3-\chi _2)} \ln \left(\frac{\chi -\chi _3}{\chi _{\rm in}-\chi _3}\right) \right], \qquad \end{eqnarray} \tag{ 5 }$$

where χ = (r/r_g)^1/2, χ_in = (r_in/r_g)^1/2, and r_g = GM_•/c². For a Kerr BH, the disk inner edge r_in is expressed as (Bardeen et al. 1972)

$$\begin{eqnarray} &&r_{\rm in}/r_g = 3+Z_2 -\left[(3-Z_1)(3+Z_1+2Z_2)\right]^{1/2}, \end{eqnarray} \tag{ 6 }$$

for 0 ⩽ a_• ⩽ 1, where Z₁ ≡ 1 + (1 − a²_•)^1/3[(1 + a_•)^1/3 + (1 − a_•)^1/3] and Z₂ ≡ (3a²_• + Z²₁)^1/2. In Equation (5), χ₁, χ₂, and χ₃ are the three roots of χ³ − 3χ + 2a_• = 0, i.e., $\chi _1= 2 \cos (\frac{1}{3} \cos ^{-1}a_{\bullet } - \pi /3)$, $\chi _2= 2 \cos (\frac{1}{3} \cos ^{-1}a_{\bullet } + \pi /3)$, and $\chi _3=- 2 \cos (\frac{1}{3} \cos ^{-1}a_{\bullet })$. It is easy to check that f(r = r_in) = 0 and f(r ≫ r_in) ≃ 1. For a Newtonian disk, f can be simply written as $f = 1-\sqrt{r_{\rm in}/r}$.

We assume that the strength of the magnetic field threading the BH is comparable to the largest field strength in the disk, i.e., B_• ∼ B^max_disk. For given $\dot{M}$ and a_•, we solve numerically the temperature profile of the thin disk and identify the radius r_peak, where T reaches the maximum. It is found that r_peak is very close to r_in, the innermost radius of the accretion disk. We then calculate B^max_disk from Equation (3), which is assigned to B_•. Applying Equations (1) and (2), one then obtains L_BZ once $\dot{M}$ and a_• are specified.

Observationally, a time-dependent isotropic X-ray luminosity L_{X, iso} was measured for the two sources. This luminosity can be connected to the BZ power through

$$\begin{equation} \eta L_{\rm BZ}=f_b L_{\rm X,iso}, \end{equation} \tag{ 7 }$$

where η is the efficiency of converting BZ power to X-ray radiation,

$$\begin{equation} f_b = \frac{\Delta \Omega }{4\pi } = {\rm max} \left(\frac{1}{2\Gamma ^2}, \frac{\theta _j^2}{2}\right)< 1 \end{equation} \tag{ 8 }$$

is the beaming factor of the jet (Burrows et al. 2011), ΔΩ is the solid angle of the bipolar jet, and Γ and θ_j are the Lorentz factor and opening angle of the jet.

In this work, we adopt η ∼ 0.5, motivated by the potentially high radiation efficiency of a magnetically dominated jet (e.g., Drenkhahn & Spruit 2002; Zhang & Yan 2011). Again this gives a conservative lower limit of the inferred a_•, since a less efficient jet would demand an even higher spin rate in order to interpret the same observed luminosity.

The parameter f_b has a large uncertainty. Based on the current data, one cannot precisely measure Γ and θ_j. Here, we apply a statistical method to estimate the range of f_b within the TDE framework (see also Burrows et al. 2011). First, the facts that two such events (Sw J1644+57 and Sw J2058+05) were detected by Swift in ∼7 yr and that the field view of the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) is ∼4π/7 sr suggest that the all-sky rate of such events is R_obs ∼ 2 yr⁻¹, with a 90% confidence interval of (0.44–5.48) yr⁻¹ (Kraft et al. 1991). Next, the TDE rate is estimated as ∼10⁻⁵ to 10⁻⁴ yr⁻¹ galaxy⁻¹ based on observational (Donley et al. 2002; Gezari et al. 2009) and theoretical (Wang & Merritt 2004) studies. The galaxy number density is n_gal ∼ 10⁻³ to 10⁻² Mpc⁻³ (Tundo et al. 2007). Sw J2058+05 was marginally detected at z = 1.1853 (Cenko et al. 2011). We then obtain the total TDE event rate within the volume (z ⩽ 1.1853) R_tot ∼ 10⁴–10⁵ yr⁻¹. Considering that only ∼10% of the population can launch a jet (the "radio-loud" AGN fraction; Kellerman et al. 1989; Cirasuolo et al. 2003), the beaming factor can be estimated as

$$\begin{equation} f_b \sim \frac{R_{\rm obs}}{10\% R_{\rm tot}} \in (4.4\times 10^{-5}, 5.5 \times 10^{-3}). \end{equation} \tag{ 9 }$$

Finally, in order to infer a_• from L_{X, iso}, f_b (given β = 0.5 and η = 0.5), one needs to know the accretion rate $\dot{M}$. This is an even more loosely constrained parameter. However, if one assumes that the luminosity history of the light curve delineates the accretion history of the BH (noticing $L_{\rm BZ} \propto \dot{M}$) well, one can normalize the accretion rate using the total accreted mass based on the observed flux and fluence of the event. For Sw J1644+57, since the source has entered a decaying phase, during which the residual fluence no longer contributes significantly to the total fluence, one can take the current total X-ray fluence as a good proxy of the total mass of the tidally disrupted star. Taking the peak flux as an example, the peak accretion rate can be estimated as

$$\begin{equation} \dot{M}_{\rm peak} = \frac{F_{\rm X}^{\rm peak}(1+z)}{S_{\rm X}} M_* =\frac{L_{\rm X,iso}^{\rm peak}}{E_{\rm X,iso}} M_*, \end{equation} \tag{ 10 }$$

where F^peak_X is the peak X-ray flux, S_X is the total X-ray fluence, L^peak_{X, iso} is the peak isotropic X-ray luminosity, E_{X, iso} is the isotropic X-ray energy, and M_* is the total mass of the star. The factor (1 + z) was applied to convert the observed time to the time in the source rest frame. The accretion rate at other epochs can be estimated similarly. The range of M_* may be between 0.1 M_☉ and 10 M_☉. One can then derive a mass-dependent constraint on a_•.

Now we apply the above method to the two sources.

According to Burrows et al. (2011), during the first 50 days after the first BAT trigger the total X-ray energy corrected for live-time fraction for Sw J1644+57 is E_{X, iso}(J1644) ∼ 2 × 10⁵³ erg in the 1.35–13.5 keV rest-frame energy band (corresponding the energy band of the Swift X-Ray Telescope, Burrows et al. 2005). The peak luminosity in the same energy band is L^peak_{X, iso}(J1644) ∼ 2.9 × 10⁴⁸ erg s⁻¹. The accretion rate (Equation (10)) can be estimated as

$$\begin{equation} \dot{M}_{\rm peak} ({\rm J1644}) \simeq 1.45 \times 10^{-5} M_* {\rm s}^{-1}. \end{equation} \tag{ 11 }$$

Figure 1(a) presents the constraint on a_• for Sw J1644+57 as a function of M_*, with β = 0.5 and η = 0.5. The two boundary lines correspond to two ends of the range of f_b, with the lower and upper boundaries corresponding to f_b = 4.4 × 10⁻⁵ and f_b = 5.5 × 10⁻³, respectively. The middle dashed line corresponds to the most probable value f_b ∼ 10⁻³. One can see that the supermassive BH must have a moderate to high spin rate. Given the standard stellar initial mass function, the number of low-mass stars is much more abundant than the number of high-mass stars. If one takes M_* = 1 M_☉, the required range of a_• is (0.23, 0.85), with the most probable value a_• = 0.63. For M_* = 0.1 M_☉ (more probable), the range of a_• is (0.51, 0.98) with the most probable value a_• = 0.90.

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Sw J2058+05 was discovered by Swift/BAT through a four-day (2011 May 17–20) integration. A subsequent target-of-opportunity (ToO) observation eight days after the end of the four-day integration (2011 May 28) revealed an X-ray source that behaves very similarly to Sw J1644+57 (Cenko et al. 2011), suggesting that it is very likely another Sw J1644+57-like event. The 0.3–10 keV peak flux is F^peak_{X, iso} ≃ 7.9 × 10⁻¹¹ erg cm⁻² s⁻¹, corresponding to a peak luminosity of L^peak_{X, iso} ≃ 3 × 10⁴⁷ erg s⁻¹. The total X-ray fluence from the beginning of the ToO observation to 2011 July 20 is S_X ≃ 1.0 × 10⁻⁴ erg cm⁻² (Cenko et al. 2011). Since we did not catch the X-ray emission during the first 11 days (May 17–27) when it is supposed to be much brighter, the registered X-ray fluence only corresponds to a small fraction of the total mass of the star. One can still apply Equation (10) to estimate the accretion rate, except that one should replace M_* by ζM_*, where ζ < 1 is the fraction of stellar mass that is accreted after May 28. Noticing z = 1.1853 (Cenko et al. 2011), this gives a peak accretion rate

$$\begin{equation} \dot{M}_{\rm peak} ({\rm J12058}) \simeq 1.72 \times 10^{-7} \zeta _{-1}M_* {\rm s}^{-1}, \end{equation} \tag{ 12 }$$

where ζ = 0.1ζ₋₁ has been adopted.

The constraint on BH spin for Sw J2058+05 is shown in Figure 1(b). We find that the demand for BH spin is even more stringent for this source. For M_* = 1 M_☉, the required range of a_• is (0.49, 0.98) with the most probable value a_• = 0.89. For M_* = 0.1 M_☉ (more probable), the range of a_• is (0.81, 0.998) with the most probable value a_• = 0.99.

Sw J1644+57 and Sw J2058+05 are the first two proto-type objects in this newly identified astrophysical phenomenon, namely, a relativistic jet associated with a TDE from a supermassive BH. A straightforward question is why only some TDEs launch jets. Based on observational properties, it has been argued that the jets must be Poynting-flux-dominated (Burrows et al. 2011; Shao et al. 2011). Invoking the BZ mechanism as the power of the jet, we show here that both events need to invoke a BH with a moderate to rapid spin in order to interpret the observations: the most probable values are a_•(J1644) = 0.63, 0.90 and a_•(J2058) = 0.89, 0.99 for M_* = 1, 0.1 M_☉, respectively. We therefore suggest that BH spin is the key factor behind the Sw J1944+57-like events, although other factors may also play a role (e.g., Cannizzo et al. 2011).

An elegant feature of the method we employ is that the inferred BH spin parameter a_• essentially does not depend on the BH mass. On the other hand, knowing a_• leads to a better constraint on BH mass based on the variability argument. For example, the observed minimum variability timescale of J1644+57 is δt_{obs, min} ∼ 100 s. If one relates δt_min = δt_{obs, min}/(1 + z) to the timescale defined by the innermost radius of the accretion disk, r_in/c, one can derive the BH mass of Sw J1644+57:

$$\begin{equation} M_{\bullet,6} \simeq 15 \left(\frac{r_{\rm in}}{r_g}\right)^{-1} \left(\frac{\delta t_{\rm obs,min}}{100\,{\rm s}}\right). \end{equation} \tag{ 13 }$$

One has 2.5 < M₆ < 15 for 0 ⩽ a_• ⩽ 1, with the most probable value M₆ = 6.5 for a_• = 0.9. This is consistent with the constrained BH mass from the M–L_bulge relation, which gives an upper limit of 2 × 10⁷ M_☉ (Burrows et al. 2011).

From Equation (1), one can also infer the strength of the magnetic field at the BH horizon:

$$\begin{equation} B_{\bullet,6} \simeq 131 f_b^{1/2} \eta ^{-1/2} F(a_{\bullet })^{-1/2} a_{\bullet }^{-1} M_6^{-1}. \end{equation} \tag{ 14 }$$

Taking BH mass M_• = 6.5 × 10⁶ M_☉ and the most probable value for BH spin a_• = 0.9, one finds that the magnetic field threading BH would be B_• ∼ 1.1 × 10⁶ G, which is much higher than the average field strength of a typical main-sequence star (<10³ G). The accumulation of magnetic flux by accretion flow and instability in the disk may account for such high magnetic field strength (e.g., Tchekhovskoy et al. 2011).

For Sw J2058+05, due to the low X-ray flux at the late epochs a much looser constraint on variability, δt_{obs, min} < 10⁴ s, was obtained (Cenko et al. 2011), so that the precise values of M_• and B_• cannot be derived.

An alternative scenario to interpret the Sw J1644+57-like event may be the onset of an AGN (Burrows et al. 2011). This scenario, which predicts that the two sources will be active at least in the following millennium, may be less favorable in view of the rapid onset of emission in Sw J1644+57 and the gradual decay in both events, but is not ruled out. Our method can be applied to this scenario as well, except that f_b becomes much larger (due to the intrinsic rarity of AGN onset events), and M_* is no longer limited to the range of 0.1–10 M_☉. Our analysis suggests that the BZ power is likely not adequate to interpret the data (because of the large emission power demanded by the small f_b factor) even for maximum spin (a_• ∼ 1), unless the accretion rate is much higher, so that the total amount of fuel M_* ≫ 1 M_☉. This is not impossible since the fuel in the AGN onset scenario is from a gas cloud near the BH, whose mass is not specified.

Krolik & Piran (2011) proposed a model for Sw J1644+57 invoking a white dwarf being tidally disrupted by a smaller BH (M_• ∼ 10⁴ M_☉). Regardless of how this model may interpret the δt_min and the apparent association of the source with the center of host galaxy, the derived a_• range also applies to their model for the M_* range of a white dwarf since our constraint is M_•-independent.

Finally, in our calculations we did not consider the evolution of a_• during the accretion phase. This is justified given the large M_•/M_* ratio.

This work is supported by NSF under grant No. AST-0908362, by NASA under grant No. NNX10AD48G, by the National Natural Science Foundation of China under grants 11003004 and 10873005, and the National Basic Research Program ("973" Program) of China under grant No. 2009CB824800. W.-H.L. acknowledges a Fellowship from the China Scholarship Program for support.