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If I have a body $\text{A}$, which is orbiting a more massive body $\text{B}$, and I increase $\text{A}$'s velocity, I'd like to be able to calculate $\text{A}$'s new position after some time.

I have all the orbital elements of the previous orbit, and I can calculate orbital period $T$, semi-major axis $a$, eccentricity $e$ of the new orbit from the new initial position and increased velocity vectors $V$.

So, given ...

  • a known position along a planet's orbit $P$,
  • a velocity vector $V$,
  • the semi-major axis $a$,
  • the eccentricity of the orbit $e$, and
  • the period of orbit $T$

... is it possible to calculate the new position of $\text{A}$ given its old position in the orbit and after some time $t$?

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Yes. You have the position and the $\Delta V$. From the orbital elements, calculate the velocity at that position. Add the $\Delta V$. You now have a new position and velocity. Throw out the old orbital elements -- they are no longer useful. Using the position and velocity, derive new orbital elements. The velocity and radius can give you the energy and angular momentum, and from those you can get the new $a$ and $e$. You will need to solve for the orientation of the ellipse (in 2D or 3D), and the true or eccentric anomaly of the position. From that, you can compute the new position and velocity at any time.

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  • $\begingroup$ As I understand the eccentric anomaly is dependent upon the mean anomaly (via Keplers equation), which has a time dependency. This time, t, is the time since the last periapsis? So the position in this new orbit (with velocity V + $\Delta V$) has an associated time. If I want to get the next position after $\Delta t$, wont I need to know the time since last periapsis + this $\Delta t$? $\endgroup$ – iasksillyquestions Feb 25 '14 at 9:44
  • $\begingroup$ Finding the true, eccentric, or mean anomaly of the current position in the orbit is equivalent to finding the time since periapsis. It is a simple formula to calculate that time from the mean or eccentric anomaly. $\endgroup$ – Mark Adler Feb 25 '14 at 14:52
  • $\begingroup$ I think I'm being unclear! I have a position. I increase the velocity. As a result I have a new orbit and therefore new orbital elements. I want to get the position after an additional dt. $\endgroup$ – iasksillyquestions Feb 25 '14 at 15:00
  • $\begingroup$ That was clear, and I have provided the answer. $\endgroup$ – Mark Adler Feb 25 '14 at 15:07
  • $\begingroup$ Sorry I pressed "Add Comment" before I'd completed $\endgroup$ – iasksillyquestions Feb 25 '14 at 15:13

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