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The spacecraft is on a LEO orbit with known keplerian elements. The equation of motion for spacecraft is also known.

I have to intersect (just intersect!) the given coordinates $X$, $Y$, $Z$ on near, LEO orbit. Departure and Arrival times are given. $\Delta V$ is applied in an instant. X,Y,Z is on different plane.

How to calculate the required $\Delta V$?

I made research about the 'Hohmann transfer orbit' and 'vis-viva equation'. However, I can't understand how to intersect a point on another orbit exactly on time $T$?

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  • $\begingroup$ I'd recommend that you first take a look around at the other questions and answers here, you can click the tags that you've chosen (orbital-maneuver, orbital-mechanics, delta-v) and see the level of discussion and math involved in working on even one small part of this. This is like going to stackoverflow and saying "I have a core-i7 processor, 12 Volts AC, and a six-pack of Mountain Dew. I need an operating system, ideas?" :) $\endgroup$ – uhoh Apr 16 '17 at 14:21
  • $\begingroup$ İ thougt my question is clear enough :) İ'm looking on other questions now $\endgroup$ – Tarlan Mammadzada Apr 16 '17 at 14:26
  • $\begingroup$ Oh it is very clear, but it is just a huge question! Try to narrow down to some small part of the problem, and ask something more specific. $\endgroup$ – uhoh Apr 16 '17 at 14:29
  • $\begingroup$ Edited the question. The question should have analytic solution, so I'm looking for ideas $\endgroup$ – Tarlan Mammadzada Apr 16 '17 at 14:30
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    $\begingroup$ Pointing to other answers, or giving references, where I could read exactly about this would be Perfect! $\endgroup$ – Tarlan Mammadzada Apr 16 '17 at 14:35
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Finding the orbit that connects two points with a given transit time is known as Lambert's Problem, and has a small set of solutions. Once you have that orbit, you just need to subtract the velocities of the old and new orbit at the starting point of the new orbit.

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  • $\begingroup$ Thanks! I'll research now. Is this solution is better than building the Hohmann transfer orbit, which would comply with my transfer time? $\endgroup$ – Tarlan Mammadzada Apr 16 '17 at 18:06
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    $\begingroup$ It is the only solution. The constraints you stated are the Lambert problem constraints, and they result in unique solutions to meet those constraints. This has nothing to do with Hohmann transfers. $\endgroup$ – Mark Adler Apr 16 '17 at 18:10
  • $\begingroup$ Another question: How the problem would change, if the TRANSFER TIME is unknown, but the DELTA-V is minimal? $\endgroup$ – Tarlan Mammadzada Apr 16 '17 at 18:11
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    $\begingroup$ Then you would generate the Lambert solutions for a range of transfer times, and look for a minimum in the $\Delta V$ magnitude. That would be a one-dimensional version of a porkchop plot, where in two dimensions the Lambert solutions between two moving planets is plotted as a function of departure and arrival times. $\endgroup$ – Mark Adler Apr 16 '17 at 18:15
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    $\begingroup$ The point is that your last problem is not well posed. It can be solved with no change in orbit at all. $\endgroup$ – Mark Adler Apr 16 '17 at 22:29

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