1. Relativistic cross sections of tidal disruption events
Pavel Ivanov
Lebedev Physical Institute

2. A simple formulation of the problem
In a very crude approach, one may say that the star is tidally disrupted
when its periastron, rp, is smaller than the tidal radius:
When the tidal radius is smaller than the gravitational radius
the star is swallowed by the black hole without disruption. This happens
when M > 108 M☼ . The strength of tidal interaction is characterized
by either a parameter η=(r p/rT)3/2 or by penetration factor β=r T/rp .

3. Orbit’s parameterization in GR
The orbital parameters of a highly elongated orbit can be characterized by three numbers since the orbital energy (per unit of mass, in the natural units) is approximately 1. One can specify e.g. the inclination with respect to the rotational axis, θ∞, at the periastron, and components of specific angular momentum in θ and φ directions: j θ and j φ .
Alternatively, one can use
projection of the angular momentum
onto the rotational axis:
Lz = jθsin θ∞ and the Carter integral
Q= j φ2+ j θ2 cos θ∞.
Since Q > 0 for nearly parabolic orbits, we use q=√Q below.

4. A computationally efficient model of a tidally disrupted star.
It has been developed in papers Ivanov & Novikov 2001, Ivanov, Novikov & Chernyakova 2003, Ivanov & Chernyakova 2006.
1) The model is a generalization of the affine model of Carter and Luminet. The star is assumed to consist of elliptical shells. Unlike the affine model the shell are not
self-similar, their parameters are functions of time and a Lagrangian coordinate (say, mass enclosed in a shell)
2) Dynamical equations are derived for so-called virial
relations written for every shell.
3) It is a one-dimensional model. Therefore, it is numerically much faster than the 3D schemes.

5. Testing the model: in general our simple model demonstrates rather good agreement with more numerically expensive three
dimensional hydrodynamical schemes. For example, below
it is shown the amount of mass lost in our model (black circles)
and in the recent model of Guillochon and Ramirez-Ruiz 2012
against the penetration parameter β, for stars with polytropic
indices n=1.5 and 3.

6. a=0.75. The angle θ∞ =π/2.

7. a=0.75. The angle θ∞ =π/2.

9. The case of angle θ∞ not equal to π/2.
When cross sections are plotted on the plane (Lz, q)
this dependence is practically absent!

11. Other degeneracies
Our results suggest that the levels of equal mass loss on the plane (Lz, q) are very close to segments of circles. Therefore, for
a given value of mass loss, M, a, the cross
section may be characterized by only one
number!, the shift of the circle with respect to the coordinate origin. The length of the circle may be determined by its intersection with the know cross section of direct capture. In my opinion,
It’s important to check this statement with help of
more advanced numerical schemes.

12. Conclusions/Discussion
1) The anisotropy of the tidal disruption cross sections can, in principal, manifest itself in an induced anisotropy of distribution of stars around SMBH. This could test the spin of SMBH.
2) It would be very interesting to check whether these degeneracies of the cross sections are present in more realistic models of tidally disrupted star/more realistic
stellar models.
3) The reported results deal with ‘the first passage problem’. But, assuming that the orbit of the star is approximately unchanged after a partial mass loss, it could be easier to destroy the remnant during next periastron passages. Thus, the whole business should be generalized on the multipassage case.
5) In general, the problem should be considered together with orbital evolution, due to tides and due to star-star scatterings. The results could be different in the regimes of ‘empty and full loss cones’. This could have some implications on the regimes of disc formation (whether the disc is formed after a partial mass stripping or full disruption). Also, there is a possibility of survival of the remnants, especially in the case of full loss cone. A possible discovery of such remnants could test the whole paradigm.
6) Details are in Ivanov & Chernyakova, A&A, 448, 843, 2006.