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When a satellite in a circular orbit around a planet is given a sudden speed boost, it tends to settle into an elliptical orbit, with the planet at one focus. Why does it settle into an elliptical orbit rather than a circular one with the corresponding, appropriate radius? Also, why does the planet have to be at one focus of the ellipse?

I have worked through the calculations, and I understand that when the speed boost is given, the satellite is moving too fast to remain in the current circular orbit, and hence deviates from the circle. What I want to understand is why doesn't its deviation lead it to relocate to another circular orbit further away from the planet? Also, moving to a higher orbit requires positive work, which is done by whatever provided the speed boost, so the satellite should be able to join a higher circular orbit.

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  • $\begingroup$ Before I answer, have you looked at the effective potential and, in particular, the point on the effective potential curve for an orbit with constant radius? $\endgroup$ – Alfred Centauri Mar 4 '18 at 14:23
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The nature of the orbit is dictated by the position and velocity. When velocity is increased for instance the object at that position loses the required velocity for the circular orbit. Morover it will at some later stage come back to that point. Therefore it cannot have a circular orbit after the change in velocity.

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  • $\begingroup$ "moreover it will at some later stage come back to that point" this is really important! In an orbit, if you change velocity at one point you CANNOT change location too. Changing velocity at point X changes the location of precisely every other point except X in the orbit. $\endgroup$ – bendl Mar 4 '18 at 15:08
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For simplicity, I assume the plane of the orbit is unchanged, the change in velocity is essentially instantaneous, and the change in velocity is less than required to reach escape velocity, i.e., the new orbit is closed.

Focus on the point in space where the velocity change occurs. Through that point are an infinity of closed orbits (in the chosen plane) and each is characterized by a velocity vector there.

There are only two circular orbits through that point; the velocity vectors are equal in magnitude and opposite in direction (one orbit is 'clockwise' while the other is 'counterclockwise'). The remaining orbits through that point are elliptical.

Thus, if the object is initially in a circular orbit, there is only one other circular orbit available through that point to choose from. Unless the change in velocity simply reverses the velocity vector, the change in velocity leaves the object in an elliptical orbit.

Later, at another point, another change in velocity can be made that will leave the object in another circular orbit.

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When the satellite is in a circular orbit the speed of the satellite and the its rate of falling are proportional and are not fluctuating. But, at a lower orbit if the satellite is accelerated by the on-board boosters then the energy that the satellite has due to the velocity is more than the energy of that orbit. So, the satellite to compensate for this moves far away from the planet to compensate. When in an elliptical orbit far away from the planet the speed is less and when it comes close the speed increases.

Just imagine it like the pendulum was still ( circular orbit) when you you give it a push the pendulum oscillates (elliptical orbit -speed).

This imbalance is used to place the satellite in the proper phasing angle and at the correct altitude. For more information you can see the Holman Transfer. This is one of the methods to dock at the ISS.

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