Yesterday we considered the Cartesian expression of the pursuit problem with relativistic time-delay. For an eight light-minute comms delay this looked best-modeled at the edge of the solar-system. Now, let's consider a light-minute example closer to home: intercepts to Venus/Earth transfer-orbits. Or Earth/Mars. Say we're denied clearance for Zubrin's torch.
Most supply-runs will be Hohmann, usually one-way trips. Earth/Mars, rather famously, affords also (theoretic) Cycler Orbits, the best of which may be semipermanently colonised with a "castle" each. Between Venus and Earth(ish) Hohmann is almost a cycler itself. I have identified two more: one between Venus and STL1 repeating two synods with four revolutions so "2L4"; another repeating three synods passing both planets - Earth once Venus twice - "3-0-2-9". [LAPLACE 12/6/23: And now, the circular Hohmann.] But here we'll start with Earth/Mars (or HEO/Deimos, whatevs). If only because it's that bit further from the Solar well.
Stuff happens in space and, when it happens, the ailing transport-vessels will be calling the Lewis and Clark. The merchant is in a free Marsbound Hohmann without propellant. The pursuer starts from a castle at some arbitrary point between Earth and Mars. Again, we assume that the pursuer can get initial-velocity from a coilgun, Laurence Fishburne not minding the G-force because he's awesome. Pursuer keeps the same speed, as it locks onto where it sees the merchant... with time retardation. Formulate the equations of motion.
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