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I understand that the Earth moves through the solar system at an extremely fast rate, but a question occurred to me as I reflected on this piece of knowledge: Does the speed of the Earth traveling at through the cosmos have any effect on the gravitational pull of the planet? Or, in other words, Wouldn't the speed of the Earth cause one side of the Earth's gravitational pull to be stronger and the other side to be equal due to inertia?
(By the way, I don't know that much about astrophysics.)

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From the point of view of someone standing on Earth the gravitational field of the Earth is due to a stationary object, so the velocity of Earth relative to something else does not matter.

Earth's gravitational field is not exactly spherically symmetric because Earth is not exactly spherically symmetric either.

Also note that as the Earth rotates about it's own axis there is an additional outward component of centrifugal force outward. As this depends on the position you are at on Earth (because it depends on distance from the axis of rotation) that creates an additional change in the net force you feel.

So there are reasons why the gravitational field of the Earth is not the same everywhere on it's surface and not perfectly spherically symmetrical.

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