I see a similar question: Change of orbit with a radial impulse. The answer to that question seems fine to me. In the answer, mention is made of this similar question, but it is not fully treated.

So, suppose a satellite is in a perfectly circular orbit. If you impart a brief forward impulse (in the direction of the orbit), what now happens to the orbit? (And what about a backward impulse?)

  • $\begingroup$ Have you seen this one? physics.stackexchange.com/q/480904 $\endgroup$ – Gert May 24 '19 at 16:29
  • $\begingroup$ @Gert, I'm familiar with the qualitative aspects of adding velocity to a circular orbit. However, I have never seen an equation for calculating the shape of the resulting elliptical orbit. Does such an equation exist? $\endgroup$ – David White May 24 '19 at 16:34
  • $\begingroup$ @DavidWhite: I believe it would be very complicated. $\endgroup$ – Gert May 25 '19 at 17:19

See my answer here. Consider a satellite whose speed changes as $\vec {v}=k\vec {v}$ at some point of the trajectory. Figure 1 shows the trajectory before the speed has changed (blue) and after (orange). We see an elliptical trajectory that is getting farther from the central body at $k>1$ ant it is getting closer to the central body with decreasing $k$. When $k\ge \sqrt {2}$, the satellite leaves the central body.

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  • $\begingroup$ Very nice. The diagrams you gave answer exactly what I was wondering, and additional cases as well. I guess it would be irresponsible to just believe you. After some search, I found what is called the vis-viva equation. Is that what you used, or did you derive these in some other way? $\endgroup$ – Mark Goldfain May 26 '19 at 2:44
  • $\begingroup$ @MarkGoldfain This is the result of numerical simulation. $\endgroup$ – Alex Trounev May 26 '19 at 2:58
  • $\begingroup$ Okay, so you simulate a central gravitational pull or acceleration and different initial velocities. Thanks for the answer. $\endgroup$ – Mark Goldfain May 26 '19 at 3:15
  • $\begingroup$ @MarkGoldfain You're welcome. I solved the problem as it is. At the beginning, the satellite moves in a circular orbit, then the satellite speed suddenly changes according to the law $\vec {v_1}=k\vec {v_0}$. As a result, the satellite moves to an elliptical orbit. $\endgroup$ – Alex Trounev May 26 '19 at 10:43

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