, where 
is the gravitational force between bodies with masses 
and 
, 
is the
distance between them, and 
is the universal
gravitational constant.
(This page is full of equations, so it's essential to read it with images turned on. None of them is bigger than about 3k, so it shouldn't take long to load.)
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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