3. Solutions of the basic equations
We now present the complete set of solutions obtained using fourth order Runge-Kutta method. Fig. 1a shows the variation of the Mach number with radial distance (in logarithmic coordinates in both directions). The solid curves intersecting at the sonic point are the well known Bondi solutions. This has . The arrowed branch which becomes supersonic on the horizon represents the black hole accretion. The long dashed curve (marked NSing) is drawn with (with same ) and represents an accretion on a naked singularity. The accretion flows on neutron star surfaces must be subsonic. The exact subsonic branch depends on the inner boundary condition, i.e., the way matter lands on the staller surface (say, the derivative of the velocity on the surface.). The solid arrowed curves leading to the neutron star surface have more entropy than the curve passing through the critical point (). The difference in entropy must be generated at the standing shocks located at , and respectively. In the present example, , and (for the solid curves drawn in the order inside to outside). The corresponding shock locations are , and respectively. Here, (relativistic flow) and (no excess radiation pressure) is chosen. These correspond to low accretion rate solutions. The short dashed arrowed curves leading to a neutron star have even less entropy than . In this case, they are drawn for and (from bottom to top) respectively. In Fig. 1b, the proton temperatures (in degrees Kelvin) for the corresponding solutions are plotted on a logarithmic scale as a function of the logarithmic radial distance. The black hole and naked singularity solutions have lower temperatures since they are supersonic, while the adiabatic subsonic neutron star solutions have higher temperature, and the distributions are almost independent of the branch that is chosen. As the entropy of the post-shock flow is increased, the shock location comes closer to the neutron star surface and the post-shock temperature is also increased.
In Figs. 2a and b, we include the effect of the radiative force to study the flow properties of the solutions on neutron stars. In Fig. 2a, we plot three solid curves with arrows leading to the neutron star surface. We choose as the post-shock entropy accretion rate on the surface. The vertical arrows are drawn at shock locations and at each shock, the value of C (such as 0, 0.1, and 0.5) is marked. The value of C is made different from zero only in the post-shock region, since the subsonic flow would be maximally affected by the radiation pressure effects. The corresponding shock locations are 91.54 (same as in Fig. 1a), 66.1 and 14.8 respectively. In Fig. 2b, temperature distributions are plotted. Note that as C is increased, the temperature of post-shock region becomes smaller. This is because when the radiation pressure is present, one does not require thermal pressure very much. In the pre-shock region, the ram pressure must increase sufficiently to balance the net pressure. As a result, the shock is located at a place closer to the neutron star surface.
In Fig. 3a and b, we consider the case when the post-shock region is isothermal. In this case, the shock will be in pressure equilibrium only if the neutron star surface is much cooler than the cases mentioned above. The surface temperature of the neutron star (which is also the post-shock temperature of the flow) determines the shock location. Two horizontal lines have sound speeds (lower) and respectively. When the pressure balance condition is satisfied at the shocks, these sound speeds determine the post-shock Mach number variation which go to zero very rapidly close to the surface. As the surface temperature of the star is lowered, the location of the shock, i.e, the width of the terminal boundary layer is also increased since the pre-shock flow temperature is monotonic.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998