Black-hole spin dependence in the light curves of tidal disruption events

A star orbiting a supermassive black hole can be tidally disrupted if the black hole’s gravitational tidal field exceeds the star’s self gravity at pericenter. Some of this stellar tidal debris can become gravitationally bound to the black hole, leading to a bright electromagnetic flare with bolometric luminosity proportional to the rate at which material falls back to pericenter. In the Newtonian limit, this flare will have a light curve that scales as $t^{-5/3}$ if the tidal debris has a flat distribution in binding energy. We investigate the time dependence of the black-hole mass accretion rate when tidal disruption occurs close enough the black hole that relativistic effects are significant. We find that for orbits with pericenters comparable to the radius of the marginally bound circular orbit, relativistic effects can double the peak accretion rate and halve the time it takes to reach this peak accretion rate. The accretion rate depends on both the magnitude of the black-hole spin and its orientation with respect to the stellar orbit; for orbits with a given pericenter radius in Boyer-Lindquist coordinates, a maximal black-hole spin anti-aligned with the orbital angular momentum leads to the largest peak accretion rate.

Figure 1

The distribution $dm/dE$ of the specific binding energy $E$ for the tidal debris of a star on an initially parabolic orbit. The solid black curve corresponds to a adiabatic index $\gamma =5/3$ appropriate for a fully convective star, while the dashed blue and dotted red curves correspond to $\gamma =1.4$ and $4/3$, respectively.

Figure 2

The rate $dm/dt$ at which the debris of a tidally disrupted star falls back to pericenter and is subsequently accreted. The time $t$ is given in units of $t_0\equiv 2\pi (r_p^3/GM{)}^{1/2}$, the period of an orbit with semi-major axis $r_p$. As in Fig. 1, the solid black, dashed blue, and dotted red curves correspond to adiabatic indices $\gamma =5/3$, 1.4, and $4/3$, respectively. The dot-dashed green line shows the canonical $t^{-5/3}$ dependence predicted by Ref. 7, 46.

Figure 3

The SBH is located at the origin with its spin $\mathbf{S}$ pointing along the z axis. The inclination $\iota$ is the angle between $\mathbf{S}$ and the star’s orbital angular momentum ${\mathbf{L}}_N$, and ${\theta }_p$ is the angle between $\mathbf{S}$ and the position ${\mathbf{r}}_p$ of the star at pericenter.

Figure 4

The widths ${\sigma }_E$ and ${\sigma }_{L_z}$ of the projected distributions $dm/dE$ and $dm/d{L}_z$ as functions of the pericenter radius $r_p/M$. These are normalized by the widths $E_{\mathrm{tid}}$ and $L_{z,\mathrm{tid}}$ of the corresponding Newtonian distributions. The solid red curves are for nonspinning black holes, while the dashed orange, dot-dashed green, and dotted blue curves are for dimensionless spins $a/M=0.5$, 0.9, and 0.99, respectively. For each value of the spin, the curve extending to smaller values of $r_p$ corresponds to prograde ($\iota =0°$) orbits while the other curve corresponds to retrograde ($\iota =180°$) orbits.

Figure 5

The ratio ${\sigma }_E/E_{\mathrm{tid}}$ for orbits with $r_p=4M$, $a/M=0.99$, and $\iota \le 90°$ as a function of inclination $\iota$ and latitude ${\theta }_p^{\prime }=90°-{\theta }_p$. The yellow region above the diagonal is unphysical as it corresponds to a portion of parameter space forbidden by the inequality (25a).

Figure 6

The ratio ${\sigma }_E/E_{\mathrm{tid}}$ for orbits with $r_p=6M$, $a/M=0.99$, and $\iota >90°$ as a function of supplementary inclination ${\iota }^{\prime }=180°-\iota$ and latitude ${\theta }_p^{\prime }$. The portion of the plot above the diagonal is unphysical.

Figure 7

The ratio ${\sigma }_{L_z}/L_{z,\mathrm{tid}}$ for orbits with $r_p=4M$, $a/M=0.99$, and $\iota \le 85°$ as a function of inclination $\iota$ and latitude ${\theta }_p^{\prime }$. The region of the plot above the diagonal is unphysical.

Figure 8

The ratio ${\sigma }_{L_z}/L_{z,\mathrm{tid}}$ for orbits with $r_p=6M$, $a/M=0.99$, and $\iota \ge 95°$ as a function of supplemental inclination ${\iota }^{\prime }=180°-\iota$ and latitude ${\theta }_p^{\prime }$. The region of the plot above the diagonal is unphysical.

Figure 9

The ratio ${\sigma }_Q/Q_{\mathrm{tid}}$ for orbits with $r_p=4M$, $a/M=0.99$, and $\iota \le 90°$ as a function of inclination $\iota$ and latitude ${\theta }_p^{\prime }$. The region of the plot above the diagonal is unphysical.

Figure 10

The ratio ${\sigma }_Q/Q_{\mathrm{tid}}$ for orbits with $r_p=6M$, $a/M=0.99$, and $\iota \ge 90°$ as a function of supplemental inclination ${\iota }^{\prime }=180°-\iota$ and latitude ${\theta }_p^{\prime }$. The region of the plot above the diagonal is unphysical.

Figure 11

The dimensionless fallback accretion rate $(t_{\mathrm{tid}}/m_{*})dm/dt$ as a function of time $t/t_{\mathrm{tid}}$ following the tidal disruption of a stars on equatorial orbits with $r_p=6M$ by SBHs with $M={10}^8{M}_{\odot }$. The smooth, solid black curve is the Newtonian prediction shown in Fig. 2. The solid red curve is the relativistic prediction for a nonspinning SBH. The dashed orange (dotted blue) curves correspond to spin magnitudes $a/M=0.5(0.99)$, respectively. The curves of each type with lower (higher) peak accretion rates correspond to orbital inclinations $\iota =0°(180°)$.

Figure 12

The dimensionless fallback accretion rate $(t_{\mathrm{tid}}/m_{*})dm/dt$ as a function of time $t/t_{\mathrm{tid}}$ following the tidal disruption of a stars on equatorial orbits with $r_p=6M$ by SBHs with $M={10}^6{M}_{\odot }$. The smooth, solid black curve is the Newtonian prediction. The solid red curve is the relativistic prediction for a nonspinning SBH. The dashed orange, dot-dashed green, and dotted blue curves correspond to spin magnitudes $a/M$ of 0.5, 0.9, and 0.99, respectively. The curves of each type with higher (lower) accretion rates at late times correspond to orbital inclinations $\iota =0°(180°)$.

Figure 13

The dimensionless fallback accretion rate $(t_{\mathrm{tid}}/m_{*})dm/dt$ as a function of time $t/t_{\mathrm{tid}}$ following the tidal disruption of a stars on equatorial orbits with $r_p=12M$ by SBHs with $M={10}^7{M}_{\odot }$. The smooth, solid black curve is the Newtonian prediction. The solid red curve is the relativistic prediction for a nonspinning SBH. The dashed orange (dotted blue) curves correspond to spin magnitudes $a/M$ of 0.5 (0.99). The curves of each type with lower (higher) peak accretion rates correspond to orbital inclinations $\iota =0°(180°)$.

Figure 14

The dimensionless fallback accretion rate $(t_{\mathrm{tid}}/m_{*})dm/dt$ as a function of time $t/t_{\mathrm{tid}}$ following the tidal disruption of a stars on inclined orbits with $r_p=6M$ by SBHs with $M={10}^8{M}_{\odot }$ and $a/M=0.99$. The smooth, solid black curve is the Newtonian prediction. The remaining curves correspond to relativistic predictions with polar angle ${\theta }_p=90°$ but different values of the orbital inclination $\iota$. The peak accretion rate $d{m}_{\mathrm{peak}}/dt$ increases with $\iota$; with increasing values of $d{m}_{\mathrm{peak}}/dt$, the dotted blue, dot-dashed green, dashed orange, solid red, dotted purple, and dot-dashed cyan curves correspond to $\iota =0°$, 45°, 75°, 105°, 135°, and 180°.

Figure 15

The dimensionless fallback accretion rate $(t_{\mathrm{tid}}/m_{*})dm/dt$ as a function of time $t/t_{\mathrm{tid}}$ following the tidal disruption of a stars on inclined orbits with $r_p=6M$ by SBHs with $M={10}^8{M}_{\odot }$ and $a/M=0.99$. The smooth, solid black curve is the Newtonian prediction. The remaining curves correspond to relativistic predictions with inclination $\iota =80°$ but different values of the polar angle at pericenter ${\theta }_p$. The peak accretion rate $d{m}_{\mathrm{peak}}/dt$ increases with ${\theta }_p$; with increasing values of $d{m}_{\mathrm{peak}}/dt$, the dotted blue, dashed orange, and solid red, curves correspond to ${\theta }_p=30°$, 60°, and 90°.