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The following has been bugging me recently and I can’t figure it out:

The orbital equation of an object of negligible mass is:

$$v = \sqrt[2]{MG \over r}$$

So in other words, v decreases as r increases.

Yet I know that in order to increase it's orbital radius (let’s assume a circular starting orbit to keep things simple) a spaceship must accelerate at any point for a given amount of time, then again at once it reaches it’s apoapsis (in order to re-circularise the orbit).

So the spacecraft has gone through two burns in order to accelerate and achieve an orbit of larger radius, yet its final orbital velocity is lower!?!

What happened to the missing velocity?

Chopo87
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  • What's so hard about an acceleration producing a reduction (in the magnitude of) the velocity? It happens all the time when you apply the brakes on your car. – The_Sympathizer Aug 08 '13 at 23:41
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    Velocity is not conserved, so it can't go "missing". – dls Aug 08 '13 at 23:44
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    @mike4ty4: In the case of breaking the acceleration is negative along the velocity vector resulting in deceleration. In an burn to increase orbital radius it is in the same direction as that of travel. – Chopo87 Aug 08 '13 at 23:49
  • @dls: true, but we are still accelerating along the velocity vector so it must go into potential energy or something... – Chopo87 Aug 08 '13 at 23:54

2 Answers2

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Check out this site http://www.rcptv.com/spacetec-ita/changing_orbit.htm

It helps to conceptually explain whats going on. The short version is that the energy from accelerating is converted to gravitational potential energy. So the net energy is greater even though the kinetic energy is smaller.

If you fire your rockets towards the way you are going to slow down your orbit will drop but your orbital velocity will increase because don some of your gravitational potential energy is being converted to kinetic energy.

russloewe
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This is more physics than maths, but the answer is that the acceleration is to act in opposition to gravity - while "acceleration" in plain english usually refers to an increase in speed/velocity, in physics it merely refers to a change in velocity, and can affect both the magnitude and direction of the velocity vector. In the case of the first burn, it changes the velocity from that of the circular orbit to the more elliptical one, and then in the second burn it adjusts to the new orbital velocity.

ConMan
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  • Ok, I think I'm starting to understand, so correct me if I'm wrong: On the first burn, the velocity at the point that will become the new elliptical orbit's periapsis does in fact increase, however The resultant apoapsis will have a lower orbital velocity. And then, even when we increase v at the apoapsis to circularise the orbit, the final v required to maintain the new circular orbit is less than that of the original. – Chopo87 Aug 09 '13 at 00:02