APPENDIX G: FILAMENT DETAILS
The fact that \({V_{00}}\) is a function of two variables dooms any possibility for familiar or easy to calculate orbits. There are only two special cases (\(r = 0\) or \(z = 0\)) which yield (complicated) results that don’t require a super-computer. The probability of a special case is negligible.
The canister-sources are assumed to be near the bottoms of their gravitational wells—small orbits with small velocities. \({V_{00}}\) is proportional to the source mass, (\({\xi _S}\) solar) and for this example, \({\xi _S} = {10^{12}}\), a modest mass. The special cases follow.
For \(r = 0\), one has the following:
\[\begin{array}{l}{V_{00}}(0,z) = {V_{00}}(0,0) + \alpha {z^2},\\\alpha = 1.6576x{10^{ – 19}}{\rm{ l}}{{\rm{y}}^{ – 2}},\\\partial _t^2z(t) = – .5\,{c^2}{({\partial _z}{V_{00}})_{z = z(t)}} = – {\omega ^2}z(t),\\{\omega ^2} = {c^2}\alpha ,\\period = 2\pi \,{c^{ – 1}}{\alpha ^{ – 1/2}}{\rm{ years,}}\\{V_{00}}(0,0) = – 4.590x{10^{ – 6}},\end{array}\tag{G.1}\],
where z has units of light-years. The period is 15.43 billion years. A similar calculation for \(z = 0\) yields a period of 1.24 trillion years. These results indicate that canister-sources in the frame chain are weakly bound.
A rogue passing near a canister-source will have a greater velocity than the canister-source. This meeting will provide an impulse to the canister-source—a new velocity in some random direction. Dislodgement of a canister-source can only be the result of a random walk. The orbits of rogues within the chain gravity need to be calculated to establish the mean time between dislodgements for a canister-source. This problem is hopelessly complex. For example, one must include the collisions between orbiting rogues (more likely than a collision with a canister-source).
There is another complication. It is possible for a durable equilateral triangle of canister-sources to form on a chain—one of the triangle sides are two pre-existing members of the chain with one new outlier canister-source. The outlier can be the start of a new chain heading in a random direction (a kink). The probability of such a formation is very small, but there are eons available to wait. There is no end to the complexity of this topic.