Tidal disruption of a star in the Schwarzschild spacetime: Relativistic effects in the return rate of debris

Motivated by an improved multiwavelength observational coverage of the transient sky, we investigate the importance of relativistic effects in disruptions of stars by nonspinning black holes (BHs). This paper focuses on calculating the ballistic rate of return of debris to the BH as this rate is commonly assumed to be proportional to the light curve of the event. We simulate the disruption of a low mass main sequence star by BHs of varying masses ($10^5,10^6,10^7{M}_{\odot }$) and of a white dwarf by a $10^5{M}_{\odot }$ BH. Based on the orbital energy as well as angular momentum of the debris, we infer the orbital distribution and estimate the return rate of the debris following the disruption. We find two signatures of relativistic disruptions: a gradual rise as well as a delayed peak in the return rate curves relative to their Newtonian analogs. Assuming that the return rates are proportional to the light curves, we find that relativistic effects are in principle measurable given the cadence and sensitivity of the current transient sky surveys. Accordingly, using a simple model of a relativistic encounter with a Newtonian parametric fit of the peak leads to an overestimate in the BH mass by a factor of $\sim \text{few}\times 0.1$ and $\sim \text{few}$ in the case of the main sequence star and white dwarf tidal disruptions, respectively.

Figure 1

Normalized distribution of the specific orbital energy $\epsilon$ (left), specific orbital angular momentum $\ell$ (middle), and the Mach number $\mathcal{M}$ (right panel) of the debris at $1.5t_p$ after the periastron passage for relativistic model MS3. The density contours (black lines) are of ${\log }_{10}(\rho /{\rho }_c)$, ranging from $-5$ to 0 in steps of 0.5, where ${\rho }_c$ is the initial central density of the star. Arrow indicates the direction from the BH to the star.

Figure 2

Distribution of specific orbital energy and angular momentum of the debris at the end of the encounter for Newtonian and relativistic models MS1, MS2, and MS3. Color indicates the stellar mass fraction enclosed in every contour level.

Figure 3

Distribution of specific orbital energy and angular momentum of the debris for Newtonian and relativistic model WD6 (left) and model WD5 (right) at different points in the encounter: initially (top row), periastron (middle row), end of encounter (bottom row).

Figure 4

Debris streams in the BH frame for relativistic model WD5. Black (red) color marks a portion of the debris stream bound (unbound) to the BH. Note that the streams are overlapping with 45% of the debris bound to the black hole. Star marker follows the initial trajectory of the star.

Figure 5

Periastron location of the debris for Newtonian and relativistic simulations MS1, MS2, and MS3. Periastron location of the initial orbit is indicated by the cross hairs.

Figure 6

Illustration of azimuthal locations of periastrons for orbits of bound debris elements after disruption (not to scale). In Newtonian simulations debris arrives to periastrons with the offset of $\lesssim 2\pi$ in azimuth (dotted line) relative to the periastron of the initial orbit (solid). In relativistic simulations debris arrives to periastrons with the offset of $>2\pi$ in azimuth due to the apsidal advance of the orbit (dashed). The initial orbit is counterclockwise.

Figure 7

Rates of return of the debris to periastrons as a function of time for MS1, MS2, MS3 (left) and WD models (right). Rates from Newtonian (red crosses) and relativistic simulations (green circles) are compared with parametric fits from [12] (solid black line).