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Imagine a shuttle moving in an orbit near the earth surface,

its total energy is "-GMm/2r", so its total energy is directly proportional to -1/r

In order to spiral into another orbit of smaller radius which mean it would have a larger angular speed, the shuttle should increase or decrease its orginal speeed so that it can get into a lower orbital ?

1) increase

As orbital speed = (GM/r)^1/2, speed should increase in order to get a smaller r.

2) decrease

As total energy directly proportional to -1/r,

and total energy = PE + KE

As r decrease, total energy is more negative and KE should decrease

So the speed should decrease.

Are there anything wrong in these two contradicting concept ?

thank you!