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Let's say you have an elliptical orbit, if you burn prograde at the periapsis, you extend the apoapsis by some factor(according to vis-viva equation). But if you burn prograde on somewhere during the elliptical orbit, the orbit will be "tilted", so the semi-major axis would be angled. But by how much? I want to actually calculate the new location of the apsis with equations.

Edit: How would the true anomaly of the orbit change numerically (in a cartesian coordinate system) I know that the periapsis will rotate toward the spacecraft but let's say:

  • My current velocity is $3000 \ m/s$ tangent to the orbital trajectory
  • I fire my engines so my velocity increases by $25\%% $ still tangent to the trajectory and I do that in 1 second(assume instantaneous burn).
  • The mass of the spacecraft is 10 metric tons(forget realistic, just for the sake of simple calculations)
  • current semi-major axis: 3.5 AU
  • eccentricity: 0.82 pointing from apoapsis to periapsis
  • current true anomaly: 139.5 $^\circ $ counter-clockwise
  • the position vector $vec{v}$ is 2.9 AU with a frame of reference as the sun(focus).

With this information, can I use any equations to compute the new location of the apoapsis as well as the new orbital state vectors? Thanks

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To calculate the new orbit, assuming an instantaneous burn you would:

  • calculate the state vectors for the burn point
  • add the velocity burn (assuming a prograde burn it'll be in the direction of the velocity state vector)
  • calculate the orbital parameters from the state vectors

If the burn isn't instantaneous, you'll have to do some calculus for the duration of the burn.

Generally speaking, however, a prograde burn burn rotates the apoapsis away from the burn (and the periapsis towards it) so on your diagram the apoapsis would move above the horizontal axis and the periapsis below it, but if the object completed a half orbit and was above the horizontal axis, then it would be reversed

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