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We can find out how much work or energy needs to be added to an orbiting mass so that it lands at a new orbit with a different orbital radius, provided we know the new and old orbital radii and the mass of the planet it's orbiting: $$E_1 + W = E_2 $$ where $E_1 = U_1 + K_1 $ , $E_2 = U_2 + K_2$, and $W$ is the work or energy required.

If we suppose that the energy required is added as kinetic energy, that the orbital path of the mass is circular, and that the velocity is given only in the direction pointing outside of the circle (in a right angle to the current velocity, which is tangent to the circular path of the mass), how do we find out how much velocity needs to be added?

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  • $\begingroup$ energy conservation $\endgroup$ Commented Jan 27, 2022 at 13:30
  • $\begingroup$ Applying more velocity at right angles to the current velocity is an extremely inefficient way to move to a higher orbit, and regardless of how you want to move your spacecraft, you cannot go from a lower circular orbit to a higher circular orbit without at least two impulses, because your starting orbit and your destination orbit do not intersect. $\endgroup$
    – notovny
    Commented Jan 27, 2022 at 14:21
  • $\begingroup$ I should have clarified that I'm studying this at the high school physics level right now, so I don't know what an impulse is. I'm just not sure how I would go about finding the velocity in this way, regardless of how efficient it is @notovny $\endgroup$
    – Orian
    Commented Jan 27, 2022 at 14:26
  • $\begingroup$ @BullyMaguire fixed, thank you $\endgroup$
    – Orian
    Commented Jan 27, 2022 at 14:26
  • $\begingroup$ As notovny said, using a right angle is a bad idea. The minimum energy solution uses thrusts in line with the craft's current velocity, as described in R. W. Bird's answer. See en.wikipedia.org/wiki/Hohmann_transfer_orbit But see here for what happens with a right angle burn: physics.stackexchange.com/q/677465/123208 $\endgroup$
    – PM 2Ring
    Commented Jan 27, 2022 at 16:58

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For a mass in orbit around a much larger body, the total energy is (1/2)m$v^2$ - GMm/r. (The potential energy reference is at very large (r).) Conservation of energy gives the change in (v). In practice, to go up, you increase the horizontal speed to go into an elliptical orbit which reaches the desired radius. Then you increase the speed (which has now dropped) again to make the new orbit circular.

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