How to calculate the size and orientation of a tidal bulge

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To calculate the size of a tidal bulge created by the moon, one must consider the gravitational forces between the Earth and the moon, factoring in the opposing gravitational pull of the Earth’s mass. The calculation involves determining the net gravitational force and how it affects the distribution of water, including the loss of compression in the bulge area. Additionally, due to the Earth's rotation, the tidal bulge will be slightly ahead of the moon, which can be quantified by analyzing the gravitational influence of the bulge on the moon's acceleration. For broader applicability, the formulas used can be adapted to other celestial bodies by adjusting the parameters for their specific masses and distances. Understanding these dynamics is crucial for accurate tidal predictions.
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How would one go about calculating the size of a tidal bulge created by the moon? I know how to calculate the gravity between two points and to calculate the attractive force of matter to the center of the Earth in the bulge you would subtract its opposite direction attraction to the matter, but how would you do this with a large amount of matter and be able to calculate the loss in compression in that area? Also, because the Earth spins faster than the moon the bulge would be ahead of the the moon causing it to speed up due to a slight increase in gravity from the direction of the bulge. How would you be able to calculate how far ahead of the moon the bulge would be? I am specifically asking about the Earth and moon but would appreciate it if the formulas could be expanded so that they could be viable for other orbiting bodies.
 
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