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I have read that the predicted re-entry speed on the Artemis I mission for the Orion spacecraft into the Earth's atmosphere is about 24,500 mph or 10.9 km/s. Why is it almost the same as Earth's escape velocity (11.2 km/s)?

Is the destination influencing both speeds? For example, is this the speed to just make it to the Lagrange Point of Earth and Moon? What if a spacecraft leaves Earth in the direction of another planet or return from it, are escape and re-entry speeds the same as when aiming for the Moon?

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A good question, and you are thinking along the right lines. Think about gravity as a hill, you use the same amount of energy climbing up a hill as you get back going down it. Sending a spacecraft to the moon and back is very similar to going up and down a hill.

When sending a spacecraft to the moon is like going uphill; the spacecraft is going just fast enough to get to the point where the moon's gravity becomes stronger than the Earth's gravity, any more velocity is a waste of energy as you have to slow down more to land on the moon. The reverse is the same, you just need to get to the point where the Earth's gravity is stronger than the moon's, then it's like going downhill. Gravity acts the same no matter which way you go, so the result is the spacecraft will be going about as fast on re-entry from the moon as the velocity it took to get to the moon. There's no reason to want to slow down as a good thermal protection system is all you need to re-enter.

With other planets the principle hold the same: it takes the same amount of delta-V to go to a higher orbit as a lower one. However exit velocity and re-entry velocity will likely be different because there are other factors at work:

  • Gravity assists: spacecraft visiting the outer planets often use flybys to increase or decrease velocity and change direction
  • The moon has a relatively constant orbit around the Earth and follows the same orbit at the Earth, on the other hand planets change position relative to each other as they are in different orbits. For example a Mars mission would be launched when the Earth is moving towards Mars, a return mission would most likely be launched when the Earth is moving away from Mars, so would be catching up. This would make the exit and re-entry velocities of the spacecraft different
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  • $\begingroup$ Nice answer, but perhaps a volcano would be a better analogy: once you reach the rim, you have a small descent to the crater floor. On your return trip you need to climb out of the moon and then come back to earth downhill. $\endgroup$ Feb 2, 2021 at 9:31

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