If the earth is stationary, how do engineers calculate spacecraft trajectories and gravity-assist maneuvers?
A viewer says he is stuck on the operation of space technology: if the earth is stationary and the sun orbits it, how do engineers calculate the trajectories of spacecraft, and how do gravity-assist ('slingshot') maneuvers work for probes like Voyager or Mars missions if a fixed earth has zero orbital kinetic energy?
Answer
Your engineers do not actually use the heliocentric system, where the earth goes around the sun at about a hundred thousand kilometers per hour, to set the trajectories of their satellites and probes, because the math is too difficult. They would much rather use an earth-centered system, and there are two they use: the earth-centered inertial frame and the earth-centered earth-fixed frame. NASA, JPL, and the people navigating at sea all use an earth model. Using it does not make the earth model right; what it shows is that the earth-centered and sun-centered models work exactly the same, otherwise they could not switch back and forth between them. All the distances to the planets and all the speeds are the same in both systems; if they were different, the bodies would be crashing into each other. As for the gravity assist or slingshot: in the heliocentric system the earth is moving, say counterclockwise, so a probe sent in that direction gets a boost from the earth's motion. In the geocentric system the earth is fixed, but the universe is moving. You could have the earth fixed and the whole universe rotating the other way, or any combination, but either way there is a difference in speed between the probe and its surroundings. The heliocentric model gets its boost from the speed of the moving earth; the geocentric model gets the same boost from the speed of the universe moving in the opposite direction, because motion is relative. That is the one thing Einstein did for us: he showed that motion is relative unless you have an absolute fixed reference. Before Newton and Galileo everyone treated the earth as that fixed reference, a yardstick that never moved, and you could measure everything against it. That is exactly why NASA likes the earth-centered earth-fixed frame: it treats the earth as fixed, and that is easy to work with.