Change of ellipse while accelerating the rocket

[NODARman]

TL;DR Summary

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When the rocket accelerates in space does its trajectory which is an ellipse change in size and not the focal points because the Earth is still in one of two and also the current height doesn't increase, right?

 

[andrewkirk]

Yes the Earth's centre of mass remains the ellipse's focus that is closest to the rocket's trajectory. The ellipse changes in both shape and size as the speed increases. When the rocket accelerates past escape velocity the trajectory changes from an ellipse to a hyperbola, so it ceases to be an orbit.
I don't know what you mean by "the current height".

 

[cjl]

When a rocket accelerates (in any direction, not just in the direction of travel), its new path will share the following characteristics:

1) The new ellipse (or hyperbola) will intersect with the old one at the location where the acceleration was applied, so yes, the height at that location in the orbit will remain unchanged

2) The Earth (or body being orbited) will still be a focus, yes.

There's no requirement for the second focus to remain fixed though, and in general the second one will move (aside from a pure inclination change while at either periapse or apoapse)

If you're curious to play around with orbits, the game Kerbal Space Program is a really good way to get a feel for how orbits behave when you thrust in various directions, and how to maneuver in space. It does strictly two body mechanics (using patched conics with spheres of influence to handle travel between different bodies), but it's great for at least getting a feel for the basics, and it does so in a much more intuitive way than just looking at the math in my opinion.

 

[sophiecentaur]

When a rocket accelerates (in any direction, not just in the direction of travel), its new path will share the following characteristics:

I would say that the trajectory is not elliptical whilst the rocket is actually accelerating because there is not just one central attracting force. This could be more and more relevant when propulsion systems start to use low thrust - long acceleration times.