# Tidal force

The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational field is not constant across a body's diameter. When a body is acted on by the gravity of another body, the gravitational acceleration can vary significantly between the near side and the far side. This tends to distort the shape of the body without altering its volume; supposing it was initially a sphere, the tidal force will tend to distort it into an ellipsoid, with two bulges, pointing towards and away from the other body.

## Tidal force due to variations in gravity

For a given gravitational field, the tidal acceleration at a point with respect to a body is obtained by vectorially subtracting the gravitational acceleration at the center of the body from the actual gravitational acceleration at the point.

Correspondingly the term tidal force is common. The tidal force within the body tends to distort its shape without altering its volume; supposing it was initially a sphere, the tidal force will tend to distort it into an ellipsoid, with two bulges, pointing towards and away from the other body.

File:Tidal-forces.png
Graphic of tidal forces. See calculated tidal forces for a more exact version

There is not necessarily a rotation, the body may e.g. be freefalling in a straight line under the influence of the field.

Suppose that the gravitational field is due to one other body. Linearizing Newton's law of gravitation around the centre of the reference body yields an approximate inverse cube law. Along the axis through the centers of the two bodies, this takes the form.

$F_t = \frac{2GMmr} {R^3}$

where G is the gravitational constant, M is the mass of the body producing the field, m is the mass on which the tidal force acts, R is the distance between the two bodies and rR is the distance from the reference body's center along the axis. This tidal force acts outwards both at the near side and at the far side of the body, leading to a bulge on both sides.

The tidal forces can also be calculated away from the axis connecting the bodies. In the plane perpendicular to the axis, the tidal force is directed inwards, and its magnitude is Ft / 2 in the linear approximation (1).

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces, including the additional term explained in the next section, are also responsible for the oceanic tides, where the reference body is the Earth with the water in its oceans, and the attracting bodies are the Moon and the Sun. Tidal force is responsible for tidal locking.

For two bodies rotating about their barycenter, the variation in centripetal force required for this motion adds to the tidal force. Consider for simplicity circular orbits. Again subtracting the value at the center of one body we get

$F_t = \omega^2mr + \frac{GMmr} {R^3}$

(where ω is the angular frequency), i.e. one half of the other effect.

This applies regardless of whether the barycenter is inside the body, as in the case of considering the tidal effect on Earth due to the Moon.

Laterally rotation has no such effect.