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