# Mathematical Derivation

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Tides are caused by two things: the gravitational attraction of bodies such as the Moon and Sun, and the effects of their rotation about the Earth, which is a consequence of that attraction. There are just two equations which govern these effects. The first is Newton's law of gravitation , where is the gravitational force between bodies with masses and , is the distance between them, and is the universal gravitational constant.

The second is the formula for the centrifugal force on a rotating object, , where is its mass, is the distance from the centre of rotation, and is the angular velocity.

Consider the Earth (mass , radius ) and Moon (mass , distance ): Both orbit monthly about their centre of mass with angular velocity . The centre of mass is at a distance from the centre of the Earth, where , so .

The radial component of the gravitational force of the Moon on a mass at angle to the Earth-Moon line is .

The radial component of the centrifugal force is . Added to these there are the gravitational force due to the Earth, and the centrifugal force (where is the angular velocity of the diurnal rotation) due to the Earth's daily rotation about its own axis, but these are independent of . To combine these forces, we need to calculate the orbital angular velocity. The centrifugal forces on the Earth and Moon themselves must exactly balance the gravitational force between them, so , giving . Putting all this together, the total radial force is  , so the terms cancel and the only remaining angle-dependent term is symmetrical between the far and near sides (and less than times the Earth's own gravitational force.) The shape of the resulting equipotential surface is given by setting the integral with respect to height constant: . Then (using the binomial approximation and neglecting various small terms,) , so the equipotential surfaces are prolate ellipsoids. Filling in actual numbers we find that the maximum value of is about 0.35m for the Moon and 0.16m for the Sun. The sum of these is about 0.5m, which is the expected spring tidal range in deep oceans away from land.

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