2.2. Stellar Disruption

An alternative possibility is that the gas is supplied by disrup- tion of stars in the dense core around the central black hole. Such processes as stellar collisions might occur in a dense stellar system, even if there were not a central black hole [1 - 4]. But one might ask: How does the presence of a massive central black hole affect the surrounding stars? Several authors [16 - 29] have recently investigated some aspects of this phenomenon, motivated by its possible relevance to globular clusters or galactic nuclei. If the stellar distribution is characterized by a "core radius" r_c and a density of n_c, the virial theorem then tells us that the characteristic velocity dispersion is _c (Gm_*n_cr_c²)^1/2, m_* being the typical stellar mass. The presence of a central point mass M_h, such that n_cr_c³m_* >> M_h >> m_*, produces an extra "r^-1" potential well, which can affect the stellar velocity field out to a distance r_h = (GM_h / v_c²).

The effect on the density distribution is less straightforward. However, if the central mass has been present for a time compar- able with the stellar relaxation time (or "reference time") t_R in the core, we expect some kind of stationary state to be established, involving a slow, inward drift of the stars. (In fact the whole core will evolve on a time scale ~ 10t_R so the situation will never be an exactly stationary one.)

A central black hole provides an effective "sink" for stars approaching too close to it - such stars will be swallowed or disrupted. There is therefore no possibility of establishing an "isothermal" distribution where the density n(r) rises exponentially within r_h. This point was first emphasised by Peebles [16] in the context of globular clusters. He conjectured that the distribution of stars in bound orbits followed a power law N(E) E^-p in binding energy E. There is then a power law cusp in the stellar density:

and the velocity dispersion within the cusp scales as r^-1/2.

Insofar as the situation is controlled by stellar-dynamical relaxation processes [18, 19], and provided there has been time for a stationary state to be established, the value of q is ~ 7/4. This is the value such that the rate at which energy is "conducted" outward through the cusp ( n(r) r² t_R^-1) is independent of r. If the stars have a range of masses, the cusp will be populated preferentially by those of higher mass.

Apart from r_s, the following two length-scales, which depend on the physical properties of the stars, are relevant to the problem:

(i) The tidal radius (or "Roche radius") within which a star would be disrupted [21 - 27]. This obviously depends on the type of star (and to some extent on the shape of its orbit around the hole) but for solar-type stars it is

(3)

For other types of stars r_T scales as (r_* / R)(m_* / M)^-1/3. There will be an intermediate range of radii around r_T at which tidal effects would partially disrupt or merely distort the star rather than destroying it completely. This possibility of "partial rip-off" is particularly important for giant stars with dense cores but very extensive atmospheres. These processes may still be able to reduce the orbital energy by an amount sufficient to remove stars from the cusp, and capture them into very tightly-bound orbits passing close to the tidal radius. Such orbits would rapidly circularise, the liberated energy being sufficient to disrupt the star.

(ii) The "collision radius" r_coll at which the velocity dispersion ~ (GM_h/r)^1/2 is comparable with the escape velocity from typical stars [17]. This is significant because the stellar encounters responsible for the relaxation of the velocity distribution, energy diffusion, etc., can be treated as elastic Coulomb-type encounters only outside r_coll: when r r_coll, two stars cannot deflect each other's velocities through a large angle without coming so close that they actually collide. For solar type stars,

(4)

which for other types of stars r_coll scales as (r_* / R) (m_* / M)^-1. A star cannot work its way down into a tightly bound orbit with r << r_coll without colliding with another. Such collisions would lead to coalescence at r r_coll, but to disruption at r << r_coll, because the kinetic energy of a typical impact would greatly exceed that required to unbind the individual stars.

A "cusp" in the stellar distribution, established by stellar-dynamical processes, could exist only at radii between r_h and r_coll. Whereas these radii are very different in globular clusters, for a typical galactic nucleus with v_c 200 km s^-1 they differ (for solar-type stars) by only an order of magnitude. Some authors have neglected the cusp entirely; and, though its presence enhances the tidal capture and stellar collision rate, it does not alter things by an enormous factor [21, 23, 29].

Even if the "cusp" is unimportant, tidal disruption of stars in the core by a massive black hole can provide a supply of gaseous debris. The energy required to disrupt the star comes from the orbital kinetic energy. This means that the gaseous debris will be bound to the hole unless the star enters the hole's "sphere of influence" r r_h with a velocity v_c, exceeding (Gm_* / R_*)^1/2. It would nevertheless be energetically possible for a small fraction of the debris to fall into the hole and release enough energy to expel the remainder completely. For this reason, the rate at which a black hole grows may be below the rate at which tidal debris is produced. Hills [21, 27] has considered the properties of the so-called "debris cloud" and remarks that the expected velocity dispersion within it is com- parable with the velocities inferred from the spectra of type II Seyferts. The material has an initial binding energy corresponding to that of an orbit with major axis ~ r_coll, but its angular momentum is very small: 2 times that of a circular orbit at radius r_T(<< r_coll). The gas from tidally disrupted stars thus moves initially on almost radial orbits. (Debris from a pair of stars destroyed by a collision at r r_coll would have larger angular momentum; but unless the orbits of all stars were similarly oriented, the net specific angular momentum of the whole cloud would again be low.)

The energy released directly by a collision between two solar-type stars in a galactic nucleus or in an r^-7/4 stellar cusp would never be much more than Gm_*² / r_*(~ 10^-5 m_* c²). Really energetic collisions would occur only if compact stars were involved, or if a distribution of stars had managed to form (via, for instance, fragmentation of a massive disk) at radii r_coll.

When M_h grows to exceed 10⁸ M (or 10⁹ M for a Kerr hole), solar.type stars can be swallowed whole, though giants would still have their envelopes ripped off. Recent work by Frank [29] suggests that the mass supply needed for the most powerful quasars cannot come solely from stellar disruption or collisions in the cusp.

Note that the concept of the "accretion radius" GM_h/c_s², where c_s is the sound speed, is never very useful in the context of gas dynamics in galactic nuclei. Not only is the gas likely to be inhomogeneous, but the escape velocity from the region where it is produced, and where the gravitational field of the point mass M_h is dynamically dominant, generally far exceeds c_s. This conclusion is unlikely to be altered even if the radiative heating due to the central source itself (Compton heating by X-rays, photoionization by X-days and UV, and free free and induced Compton heating by the radio and infrared continuum) is self-consistently taken into account.