APPENDIX H: SOURCE MASS SPECTRUM DETAILS
It is necessary to convert Eq. (6.7) into dimensionless form for a numerical solution. MKS units of density (\({\rho _c}\)) and pressure (\({P_c}\)) are required. They can be arbitrary, but it is useful if they correspond to a specific point in the EOS that will be used (Table 3 and Table 5 of [9]). I have chosen the following MKS values,
\[\begin{array}{l}{\rho _c} = 1.0748x{10^{18}},\\{P_c} = 1.6930x{10^{34}}.\end{array}\tag{H.1}\]
The following dimensionless set will be used:
\[
\begin{array}{l}
r = {R_c}x{\rm{ ,}}\\
\rho = {\rho _c}{\kern 1pt} \theta (x){\rm{ ,}}\\
{R_c}^{ – 2} = 8\pi G{\rho _c}{\kern 1pt} {c^{ – 2}},\\
P = {P_c}\psi (x){\rm{ ,}}\\
{\chi _c} = {P_c}/{\rho _c}{\kern 1pt} {c^2}{\rm{ ,}}\\
{M_c} \equiv 4\pi {\kern 1pt} {\rho _c}{R_c}^3 = {\kappa _c}{\rm{ (solar~masses).}}
\end{array}\tag{H.2}
\]
The numerical values follow:
\[\begin{array}{l}{R_c} = 7.0606{\rm{ km,}}\\{\kappa _c} = 2.3908,\\{\chi _c} = 0.17526{\rm{ }}{\rm{.}}\end{array}\tag{H.3}\]
The dimensionless form for Eq. (6.7) follows:
\[\begin{array}{l}Q(V) \equiv 1 + H(V),\\U = – x{\kern 1pt} (\partial V/\partial x),\\(\partial V/\partial x) = – {x^{ – 2}}\int_0^x {dx’\,{{x’}^2}} \theta (x’){(\partial {Q^{ – 1}}/\partial V)_{x’}}{\rm{ ,}}\\\partial {\rm{ln}}\theta {\rm{/}}\partial x = – {[2{\chi _c}(\partial \psi /\partial \theta )]^{ – 1}}(\partial V/\partial x)[Q{( – U)^{ – 1}}(\partial Q/\partial V)]{\rm{ ,}}\\{M_{gravity}}(x) = {\kappa _c}({x^2}\partial V/\partial x){\rm~{ solar~masses}}{\rm{.}}\end{array}\tag{H.4}\]
The EOS information is confined to \(\partial \psi /\partial \theta \) (a function of \(\theta \)). There are three EOS regions of interest that allow \(\partial \psi /\partial \theta \) to be formed into a connected function of theta. The lowest density region consists of EDM—EOS defined by Eq. (8.5), Eq. (8.15), and Eq. (8.16) of [14]. The middle density region consists of neutron star matter—EOS defined by table 3 and table 5 (cgs units) of [9]. The (hypothetical) highest density region consists of a quark-gluon phase outlined in [15]. There are changes of phase between each region, so \(\partial \psi /\partial \theta \) will have discontinuities between the regions. The quark-gluon phase would only play a role in the source for an MBO of incredible mass (\( > 3{\rm{x1}}{{\rm{0}}^{10}}\) solar masses). Such objects are thought to exist (possibly at the center of a very massive MDGO within a heavy canister).
The boundary conditions are \(U(0) = 0\), \(V(0) = {V_0}\), and \(\theta (0) = {\theta _0}\). A trial solution is generated using \({V_0}\) and \({\theta _0}\). A correct solution is found for that value of \({\theta _0}\) for which \(0 \le 1 – (U({x_{edge}})/V({x_{edge}})) < \varepsilon \), where I have chosen \(\varepsilon = {10^{ - 6}}\). The numerical solutions use the constants of Eq. (6.13). The gravitational mass of an object is evaluated at the edge, \(M = {M_{gravity}}({x_{edge}})\).
Numerical solutions yield the following set of figures for three types of (cold) summer sources. In these figures, mass is in units of solar masses, size is in km, and rho0 is the density at the center (\({\rm{kg/}}{{\rm{m}}^3}\)) . The mass and size are single valued functions of the (negative) gravitational potential at the center (\({V_0}\)). The bottom end of each figure represents the least negative potential, and the top end is the most negative potential (deepest in the potential well). The lower end of Fig. H.2 is a limit. Both ends of Fig. H.3 are limits. For Fig. H.4 the lightest mass is to the right, and mass increases moving to the left. For MBOs the gravitational mass is equal to the inertial mass.
