2. Basic equations
We simplify our hydrodynamic calculations around a black hole and a neutron star by choosing the Paczyski-Wiita (1980) pseudo-potential,
which mimics the geometry around a compact star quite well. Around the naked singularity we choose an ordinary Newtonian potential
allowing matter to flow arbitrarily close to . The main conclusions drawn in this paper are not affected by these simplifying assumptions. However, intuitively, we do assume that the inner boundary condition on accretion flows into a naked singularity is supersonic, which is valid if the flow does not violate causality (velocity of sound less than the velocity of light).
In an adiabatic Bondi flow, the conserved specific energy is given by (C90b)
using , where G is the gravitational constant, is the mass of the compact object, and c is the velocity of light, so that the units of mass, length and time are , and respectively. Here u is the radial velocity, x is the spherical radial coordinate, a is the adiabatic sound speed, and is the adiabatic index, P is the pressure, is the mass density, , and is either 2 or 0 depending on the nature of the compact object. Equation (2) is obtained by integrating the radial momentum equation. Note that we have weakened the gravitational potential by which essentially represents the outward radiative force . Here, C is chosen to be independent of x, for convenience. Roughly speaking, on a neutron star accretion, C scales with accretion rate: . In a black hole or naked singularity accretion, radiation could be partly trapped and drawn in radially, and therefore there is no simple relation between C and the accretion rate. On the right hand side we have which is the sound speed at a large distance. If the flow originates from a Keplerian flow (C90ab, C96ab), it may have to be pre-heated (either by radiation or by magnetic dissipation) to make , in order that it passes through a sonic point at a large distance.
The mass flux is obtained as,
which could be re-written in terms of 'entropy-accretion' flux
Here, is a constant and a measure of entropy and therefore K can change only if the flow passes through a shock. Thus, is a measure of entropy and mass flux. From Eqs. 2 and 3,
At the sonic surface, where numerator and denominator vanish, one must have,
We have ignored the other positive root for , as it is nearly zero and therefore is inside the horizon or the star surface. Flow can be sonic at for a naked singularity ().
Since two conditions (5a) and (5b) are to be satisfied, while only one extra unknown (namely, ) is introduced, clearly, both the specific energy and entropy-accretion flux cannot be independent. Indeed, for a given energy , the critical entropy-accretion flux is,
It is clear that if the flow is hot at a large distance (), i.e., , then the sonic surface is located at a finite distance provided the polytropic index is suitable (roughly, . See, C90b). Once the flow passes through the sonic surface it will continue to remain supersonic if the central object is a black hole or a naked singularity, but the flow has to pass through a standing shock (at , say) and become subsonic if the central object is a neutron star. On the other hand, a neutron star accretion can be subsonic throughout if
If on the surface of the neutron star,
then the flow must have a standing shock and the entropy generated at the shock must be such that the post-shock flow has
This condition along with the pressure balance condition
determines the location of the standing shock.
The Eqs. 7a, 7b and 8 can be translated in many ways in terms of the injection speed, temperature, the gradient of velocities at the outer or inner boundary, or the location of the sonic point by employing any of the definitions given by Eq. 2, Eq. 3b, Eq. 4 or Eq. 6.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998