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Let's assume we're talking about circular orbits for simplicity. Low Earth Orbit is approximately 400 miles away from Earth's surface, and the required velocity to maintain it is 7.5 km/s (roughly 17,000 mph). Lunar orbit is approximately 250,000 miles away from Earth's surface, but it only requires a measly 1 km/s in velocity in comparison. These are the tangential velocities, of course.

Why then, for example, in order to increase the ISS's orbital distance from the Earth, NASA increases its tangential velocity by boosting it with rockets? Does that added tangential velocity at the time of rocket burn somehow translate into an increase in orbital distance, eventually decreasing back to lower than its previous tangential velocity after a greater orbital distance has been achieved?

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You can't solve this problem using only circular orbits.

When you add velocity to the ISS to move it away from it's initial circular orbit, you're actually moving it into an elliptical orbit. This is typically done as a transfer orbit, to shift from one circular orbit to another, rather than to enter an elliptical orbit permanently.

So you're not talking about the same process when you talk about changing the ISS orbit (using a transfer orbit) in comparison to the initial and final orbits of the ISS.

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