APPENDIX F

APPENDIX F: CANISTER DETAILS

Physics for the MDGO plasma is complicated by the extremely high temperature and extremely low density. One would find only a few thousand hydrogen and helium nuclei per cubic meter and an equal number of electrons—the mean free path for collisions between particles would be on the order of light year fractions. The high temperature (roughly 1 Kev energy) generates soft Xray photons (thermal bremsstrahlung) which interact with matter via Compton scattering. The photons will lose more energy in collisions with electrons compared to hydrogen or helium. Thus, the opacity of the gas is dominated by free electrons, but the tiny density of electrons causes the gas opacity to be nearly zero—a telescopic photograph of the gas will show an Xray expanse.
The energy source for the MDGO has four possible components, proton-proton fusion (in red or blue summer), gravitational Kelvin contraction (in all seasons), The NG version of a super-massive black-hole heat source within the MDGO, or shock waves from a collision between MDGOs in a container.
In a MDGO (free of shock waves), Kelvin contraction rules. That situation is studied below. A NG version of a super-massive black hole has a negligible effect on the MDGO, but it is much brighter and hotter than the Kelvin radiation of the MDGO. The Chandra Wolter-1 telescope had ten times better resolution compared to the best of the three Wolter-1s of the Newton observatory spacecraft. The Kelvin radiation can be obscured by a poorly resolved (active) black hole source in a Newton Xray camera shot of a distant canister. By comparison, a Chandra camera shot can show the Kelvin radiation more clearly. The radius of the MDGO (radius of the Kelvin radiation) is important below, so the limited number of Chandra camera shots is the only source of information about the size of MDGOs.
Kelvin contraction is a diffuse energy source. Thus, the three modes of heat transfer (conduction, convection, radiation) become marginalized (confined to eliminating local micro-imbalances). There are negligible gradients of pressure, temperature, and density within the MDGO. A good approximate model for MDGO density is \(\rho (r,\tau ) = {\rho _0}(\tau )\) for \(r < {R_E}(\tau )\) (else = 0), where \(\tau \) is physical time and derivatives of \({\rho _0}(\tau )\) and \({R_E}(\tau )\) are very small.
A solution for Eq. (5.5) uses the following functions: \[\begin{array}{l} b(q,r,\tau ) = \int_0^r dr’\, r’\rho(r’,\tau)\sinh(r'{L_{gq}}^{-1}),\\ c(q,r,\tau ) = \int_r^\infty dr’\, r’\rho(r’,\tau)\exp(-r'{L_{gq}}^{-1}),\\ B(r,\tau ) = \sum\limits_{q=1}^{{N_q}} Z_q L_{gq}\,b(q,r,\tau)\exp(-rL_{gq}^{-1}),\\ C(r,\tau ) = \sum\limits_{q=1}^{{N_q}} Z_q L_{gq}\,c(q,r,\tau)\sinh(rL_{gq}^{-1}),\\ {\mathcal{A}_N}(r,\tau ) = -4\pi G r^{-1}[B(r,\tau ) + C(r,\tau )],\\ {V_{00}}(r,\tau ) = 2c^{-2}{\mathcal{A}_N}(r,\tau ). \end{array}\tag{F.1}\] There is a constraint for Eq. (F.1), \[ \begin{aligned} M &= 4\pi \int_0^\infty r^2 \, \mathrm{d}r \, \rho(r,\tau) = \frac{4\pi}{3} \, \rho_0(\tau) \, R_E(\tau)^3, \\ \partial_\tau \ln \rho_0(\tau) &= -3 \, \partial_\tau \ln R_E(\tau). \end{aligned} \tag{F.2} \] Eq. (F.1) can be solved in closed form, \[\begin{array}{l} B(r,\tau ) = {\rho _0}\sum\limits_{q = 1}^{{N_q}} Z_q L_{gq}^3\, f_q(r)\,\exp(-rL_{gq}^{-1}),\\ f_q(r) = (rL_{gq}^{-1})\cosh(rL_{gq}^{-1}) – \sinh(rL_{gq}^{-1}){\rm~for~} r < R_E(\tau),\\ f_q(r) = (R_E(\tau)L_{gq}^{-1})\cosh(R_E(\tau)L_{gq}^{-1}) - \sinh(R_E(\tau)L_{gq}^{-1}){\rm~for~} r \ge R_E(\tau),\\ C(r,\tau ) = {\rho _0}\sum\limits_{q = 1}^{{N_q}} Z_q L_{gq}^3\, g_q(r)\,\sinh(rL_{gq}^{-1}),\\ g_q(r) = (1 + rL_{gq}^{-1})\exp(-rL_{gq}^{-1}) - (1 + R_E L_{gq}^{-1})\exp(-R_E L_{gq}^{-1}){\rm~for~} r < R_E(\tau),\\ g_q(r) = 0{\rm~for~} r \ge R_E(\tau),\\ {\mathcal{A}_N}(r,\tau ) = -4\pi G r^{-1}[B(r,\tau ) + C(r,\tau )]. \end{array}\tag{F.3}\] The radius of an equilibrium MDGO (\({R_E}\)) is an observable since it can be inferred from soft Xray photography. The gravity field of this type of MDGO becomes a function of M and \({R_E}\). The velocity dispersion for galaxies orbiting within the field determined by Eq. (F.1) and Eq. (F.3) should provide a method for estimating M. As shown in section V(B), the virial theorem is unreliable for gravitating objects of galactic size or larger, so a new approach is required (beyond the scope of this document).
The gravitational potential of the mass within an equilibrium MDGO, \[\mathcal{E}_G(\tau ) = 4\pi {\rho _0}(\tau )\int_0^{{R_e}(\tau )} {{r^2}dr} \,\mathcal{A}_N(r,\tau ),\tag{F.4}\] can be calculated in closed form using the equations above. Furthermore, the power balance for Kelvin contraction is as follows, \[\;d\mathcal{E}_G/d\tau = – \;{\varsigma _X}{L_{sun}},\tag{F.5}\] where the MDGO Xray luminosity is proportional to solar luminosity (\({L_{sun}}\)). The dimensionless factor \(\;{\varsigma _X}\) is an observable. The potential energy is as follows, \[\begin{array}{l}{\mathcal{E}_G} = – 9G{M^2}{R_E}^{ – 1}\sum\limits_{q = 1}^3 {{Z_q}} {x_{Eq}}^{ – 5}I({x_{Eq}}),\\ I({x_{Eq}}) = a({x_{Eq}}) – b({x_{Eq}})\exp ( – 2{x_{Eq}}),\\ a({x_{Eq}}) = 0.5 + ({x_{Eq}}^3/3) – ({x_{Eq}}^2/2),\\ b({x_{Eq}}) = 0.5 + {x_{Eq}} + (5{x_{Eq}}^2/8),\end{array}\tag{F.6}\] where \({x_{Eq}} = {R_E}{L_{gq}}^{ – 1}\). Plausible units are needed to estimate the rate of contraction for a heavy MDGO: \[\begin{array}{l}{M_{GB}} = {10^{14}}{\rm{ solar masses,}}\\ {R_{GB}} = {10^6}{\rm{ light years,}}\\ M = {\varsigma _M}{M_{GB}},\\ {R_E}(\tau ) = {\varsigma _R}(\tau ){R_{GB}},\\ {\tau _{GB}} = G{M_{GB}}^2{R_{GB}}^{ – 1}{L_{sun}}^{ – 1} = 2.310\;x{10^{22}}{\rm{ years,}}\\ {\mathcal{E}_G} \equiv – {\varsigma _M}^2{\tau _{GB}}{L_{sun}}{F_{GB}}({R_E}).\end{array}\tag{F.7}\] The equations above lead one to a Xray luminosity relation for this type of MDGO, \[{\varsigma _X} = {\varsigma _M}^2({\tau _{GB}}/\Delta \tau )[{F_{GB}}({R_E} – \Delta {R_E}) – {F_{GB}}({R_E})],\tag{F.8}\] where \(\Delta \tau \) is a duration of interest (\(\Delta {R_E}\) is the shrinkage). For example: use Eq. (5.10), \(\Delta \tau = 6.4x{10^{10}}\) years (summer duration), \({\varsigma _M} = 1\), \({\varsigma _R} = 2\), and \(\Delta {\varsigma _R} = 2{\rm{x}}{10^{ – 6}}\), then \({\varsigma _X} = 8.66x{10^{10}}\) (\({\varsigma _X} = \Delta {\varsigma _R}\,4.33x{10^{16}}\)). Contraction is an extremely slow mechanism that produces a vast amount of energy for gravitating objects of canister size and huge mass—MDGOs being the most massive objects in the universe. Contraction will be revisited in section VIII about NCNG stability.
Using the same parameters as the preceding example, the red summer solution of Eq. (F.1) gives some important characteristics for a single heavy MDGO canister. The size of the canister is \({R_{nul}} = 11.78\) MLY. The height of the (summer) repulsive barrier corresponds to 1480 km/s—only objects with radial velocities greater than 1480 km/s can enter the summer canister. A negligible amount of matter will enter during summer. The gravitational redshift at the center of the MDGO relative to the outside of the canister is \({z_{gravity}} = 0.00202\). The escape velocity for matter located at \({R_E}\) is 15013 km/s (0.050 c). It is obvious that negligible matter can enter or escape this canister in summer. These values show that this gravitational well is very deep. The doppler red shift of galaxies in the canister will be of order \({z_{gravity}}\), so one must account for the radial distance of galaxies when studying velocity dispersion. The study of dispersion is complicated by the fact that our earthly observer is in a canister of unknown position, size, or depth. Note that velocities are directly proportional to MDGO mass (other factors being equal). In some cases where there is more than one MDGO in a canister, I believe that the solution above can be used to obtain a canister field that is the superposition of the separate MDGO fields.