APPENDIX M: CROSSING THE BOUNDARY
It is very easy for matter to cross the boundary from the inside. In that case, the boundary provides an impulse to increase the radial velocity of particles on the way out. The impulse can be calculated using the techniques of section VI(A). The solution gives the radial velocity for matter crossing the boundary,
\[\begin{array}{l}\beta (V) = {v_r}(V){c^{ – 1}},\\\beta (V) = \pm Q{( – V)^{ – 1/2}}Q(V){[Q{(V)^{ – 1}} – \Delta ]^{1/2}},\end{array}\tag{M.1}\]
where \(Q(V)\) is given by Eq. (6.10) through Eq. (6.13). \(\Delta \) is well approximated as a constant across the boundary even during the (slow) gravitational transition in spring and fall. Eq. (7.9) leads one to a relation between the velocity on each side of the boundary,
\[\begin{array}{l}{\beta _ > }^2 = a + d{\beta _ < }^2,\\a = {\varepsilon _{SL}}{(2 + {\varepsilon _{SL}})^{ - 1}}[1 - {\varepsilon _{SL}}{(1.5\,{\delta _{SL}})^{ - 1}}],\\d = {(2{\varepsilon _{SL}})^2}{(3\,{\delta _{SL}})^{ - 2}},\end{array}\tag{M.2}\]
where \(c{\beta _ > }\) is the velocity on the \(V > – 1\) side of the boundary, and \(c{\beta _ < }\) is the velocity on the \(V < - 1\) side. The universal constants, \({\varepsilon _{SL}}\) and \({\delta _{SL}}\), are tabulated in Appendix Z. The betas are positive for outbound matter and negative for inbound matter. Since the impulse is a function only of universal constants, Eq. (7.10) can be applied to a wide range of problems. Furthermore, \({\varepsilon _{SL}}\) and \({\delta _{SL}}\)are determined by other considerations, e.g., the mass gap between the heaviest CCNS and the lightest CCBO. In some problems, \({\beta _ < }\) can be neglected and \({\beta _ > } = \pm 0.3590\)—a very robust impulse.
A brief review is useful here. Eq. (7.9) results from the Hamilton-Jacobi equations—valid for a single particle orbiting in a static central field. The boundary thickness is roughly one meter—small compared to the radius of the boundary (approximated by a Dirac delta function). The orbit for a particle at rest at a large distance from the center of the source (\({R_0}\)) has three outcomes. If the (conserved) angular momentum (A) is larger than a small value (\({A_0}\)), the particle will simply bounce off of the boundary and return to \({R_0}\) (but not where it started). If \({A_0} > A > {A_1}\) (where \({A_1}\) is very tiny), the particle will cross the boundary. Once inside, the particle will travel near the boundary and recross the boundary (returning to \({R_0}\)). If \({A_1} > A\), the particle will cross the boundary and collide with the source—be absorbed by the source. The probability of a particle being absorbed is very small (if heading toward the source from outside the boundary). Conversely, the probability of a particle escaping from within the boundary is very good. The radial component of the particle velocity is subject to Eq. (7.9).
The application of Eq. (M.2) depends on the problem. For core collapse, all the matter is moving with a radial velocity across the boundary, so Eq. (M.2) applies to the whole mass. For an accretion of diffuse gas, there would be a pressure discontinuity across the boundary.
It is important to note that a boundary is ephemeral if the material source of the gravitational potential is a function of physical time. A boundary can move, disappear, or appear in a new place. If \({V_{00}}(\tau ,x,y,z) = – 1\), then that point lies on a boundary.