Let us presume that we have a scenario with ballistic objects. As an example, one object $m_1$ is launched at time $t_0$ from Earth Location $a$ towards Earth Location $b$, both of which are in the same hemisphere, positioned so that a minimum time trajectory must go over one polar region, and are between 60° and 120° apart. Then at some time $t_1$ between the first launch and its impact at $t_2$, a ballistic object $m_2$ is launched from $b$ toward the first object's launch site $a$, to impact at $t_3$.

Both ballistic objects are on flight paths with approximately the same eccentricity, chosen to minimise flight time as far as practical.

While the objects will both pass roughly over the pole, the trajectories are not superimposed since the Earth is rotating, and at closest approach the objects are a considerable distance apart.

However, while launched from $a$ in such a manner that it should not have to use it to reach $b$, $m_1$ has an on-board capability to generate delta-V, and if it has sufficient delta-V, it has the capability of targeting and manoeuvring to generate a collision with $m_2$.

Question: How can I calculate the minimum amount of delta-V required to alter the trajectory of $m_1$ starting at $t_1$ so that it will impact $m_2$ at some time between $m_2$'s launch at $t_1$ and $m_1$'s impact at $m_2$'s launch site $b$ at $t_2$ or $m_2$'s impact at $m_1$'s launch site $a$ at $t_3$?

$m_2$ should be considered to have no way of having its trajectory altered once launched.

My thinking so far:

Obviously, if $t_1$=$t_2$, then $m_1$'s delta-V required will be minimal. I am more concerned with solutions in the range $t_1 = 0$ to $t_1 = t_0 +(t_2-t_0)/2$, i.e. $m_2$'s launch at $t_1$ occurring with $m_1$ half-way along its trajectory.

Since this is a missile counter-battery question, this is not a zero-velocity rendezvous, but should be an impact that takes place at as high a closing velocity as practical, effectively meaning that no deceleration or course-matching by $m_1$ to meet $m_2$ is necessary.

I have access to AGI's free STK 11.2 software package, but I have none of the paid add-ons that would make this question unnecessary, since I am not only short of funds, but I am also not American, and hence prohibited from receiving the most relevant of AGI's packages even as a trial due to US export restrictions, and I have also exhausted the use of trial licenses for the paid packages that I am able to receive.

On the other hand, I can write my own code, either stand-alone or interop with STK, but I lack knowledge of the math involved.

Additionally, how can I calculate the impact velocity and the maximum extent of the debris field in the event of a successful intercept?

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    $\begingroup$ Actually, this question could probably be asked in six or eight carefully written sentences. Site 1 launches ballistic missile to hit Site 2, Site 2 detects it early in flight and launches ballistic missile to hit Site 1. Because of the Earth's rotation and different launch times, they will not intercept each other ballistically, but Site 1's missile has thrust capability! Once SIte 1's missile sees the other missile in flight, it will divert to intercept it. How can I calculate the necessary $\Delta v$ for this maneuver? or something similar. $\endgroup$
    – uhoh
    Sep 12 '17 at 12:30
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    $\begingroup$ Minimizing eccentricity doesn't necessarily minimize velocity. See hopsblog-hop.blogspot.com/2014/06/travel-on-airless-worlds.html $\endgroup$
    – HopDavid
    Sep 12 '17 at 12:49
  • $\begingroup$ @uhoh, this scenario is specifically New York City to central China, and is part of a future chapter of my The Juggernaut series in Worldbuilding SE's Universe Factory on Medium medium.com/universe-factory/the-juggernaut-26073bba4e37 where alien automaton Jormungandr is sitting in NYC, firing self-guided railgun rounds at China's ICBMs, and the Chinese, warned by the US after they lose their own land-based ICBMs, decide to use them rather than lose them. Still, I expect that the math would be the same regardless of the exact source of M1 and M2, and only the numbers would change. $\endgroup$
    – Monty Wild
    Sep 12 '17 at 15:25
  • $\begingroup$ @uhoh, it doesn't have to be that exact scenario, I can plug in my own numbers, I just don't know the math to use. If you and others would prefer a more generic question, then I'm happy to oblige, and I'll take specific locations out of the question entirely. I'd be quite happy with a generic answer with no specific numbers, since I can put my numbers in for myself. $\endgroup$
    – Monty Wild
    Sep 12 '17 at 15:42
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    $\begingroup$ @uhoh, I've put in a few relevant constraints, and added a sub-question about impact velocity and debris. $\endgroup$
    – Monty Wild
    Sep 12 '17 at 16:51

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