## 2. Basic equationsWe 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 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 At the and, 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 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 |