# Change in velocity of a circular orbit?

For a body in a circular orbit will any (i.e. however small) decrease in velocity mean it falls to earth or will it go into an elliptical orbit? I originally thought that it was the first (i.e. falls to earth) but this the answer by User58220 in this question What may be effect of air friction to the velocity of satellite? seems to take the latter point of view. Here are my reasoning,

1. The force is central and therefore never does work on the tangential component of velocity.
2. This means that it will never gain the tangential velocity it has lost and therefore never be able to return to a stable orbit and will continually gain a radial velocity towards the earth until it crash.

So which is it? Could you also give the relevant equations, thanks

If the velocity decreases by a little bit, it goes into an elliptical orbit with the apogee the same as the original orbit. The orbit will be stable if no other changes happen. Only when the velocity decreases to the point where the elliptical orbit intersects the Earth's atmosphere will the object crash into the Earth.

• Typed faster than me! -_- Nov 8 '14 at 16:40
• @Cheeku No, I was just finished before you. :)
– LDC3
Nov 8 '14 at 16:41
• @LDC3 Would this orbit not need a negative eccentricity, given by $$\sqrt{1+\frac{2EL^2}{m\alpha^2}}$$ since in the case for a circular orbit it is already 0 and E is decreasing more?
– user43487
Nov 8 '14 at 17:01
• @Joseph The eccentricity is defined as a ratio of 2 positive numbers; how can it be negative? The only reason you think it is negative is that you have the distances reversed.
– LDC3
Nov 8 '14 at 18:35
• @LDC3 Sorry I meant imaginary, simply from the formula I have given above
– user43487
Nov 8 '14 at 18:58