Tidal force

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Comet Shoemaker-Levy 9 after breaking up under the influence of Jupiter's tidal forces.

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.

1 Explanation
2 Effects of tidal forces
3 Mathematical treatment
4 See also
5 External links

[edit] Explanation

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. This causes strains on both bodies and may distort them or even, in extreme cases, break one or the other apart. These strains would not occur if the gravitational field is uniform, since a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.

Saturn's rings are inside the orbits of its moons. Tidal forces prevented the material in the rings from coalescing gravitationally to form moons.

The figure shows Comet Shoemaker-Levy 9 after it had broken up under the influence of Jupiter's tidal forces. The comet was falling into Jupiter, and the parts of the comet closest to Jupiter fell with a greater acceleration, due to the greater gravitational force. From the point of view of an observer riding on the comet, it would appear that the parts in front split off in the forward direction, while the parts in back split off in the backward direction. In reality, however, all parts of the comet were accelerating toward Jupiter, but at different rates.

[edit] Effects of tidal forces

In the case of an elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid, with two bulges, pointing towards and away from the other body. This is essentially what happens to the Earth's oceans. Although the Earth is not falling along a line directly toward the moon, the Earth is continuously accelerating due to the moon's gravitational forces, causing it to wobble around their common center of mass. All parts of the Earth accelerate in response to the moon's gravitational forces, but to an observer on the Earth, it appears that the Earth's center remains at rest, while water in the oceans is redistributed to form bulges on the sides near the moon and far from the moon.

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io.

[edit] Mathematical treatment

For a given (externally generated) 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 used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant.

Graphic of tidal forces; the gravity field is generated by a body to the right. The top picture shows the gravitational forces; the bottom shows their residual once the field at the centre of the sphere is subtracted; this is the tidal force. See calculated tidal forces for a more exact version

Tidal acceleration does not require rotation or orbiting bodies; e.g. the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

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. The derivation of the equation of tidal force proceeds as follows:

The tidal force is the difference between the gravitational forces at 2 different distances:

$F_{tidal} =\frac{GMm}{R^2}-\frac{GMm}{\left( {R+\Delta r} \right)^2}$

Find a common denominator

$F_{tidal} =\frac{GMm\left( {R+\Delta r} \right)^2-GMmR^2}{R^2\left( {R+\Delta r} \right)^2}$

FOIL the $(R + Δ r) 2$ 's

$F_{tidal} =\frac{GMm\left( {R^2+2\Delta rR+\Delta r^2} \right)-GMmR^2}{R^2\left( {R^2+2\Delta rR+\Delta r^2} \right)}$

Factor out GMm

$F_{tidal} =GMm\frac{R^2+2\Delta rR+\Delta r^2-R^2}{R^4+2\Delta rR^3+R^2\Delta r^2}$

$R 2$ and $- R 2$ cancel each other

$F_{tidal} =GMm\frac{2\Delta rR+\Delta r^2}{R^4+2\Delta rR^3+R^2\Delta r^2}$

Eliminate that which is not significant. Take advantage that R >> r

$F_{tidal} =GMm\frac{2\Delta rR}{R^4}$

Cancel the R in the numerator with one of the R's in the denominator

$F_{tidal} =GMm\frac{2\Delta r}{R^3}$

Rearrange:

$F_{tidal} =\frac{2GMmr}{R^3}$

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 r ≪ R 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 $F t / 2$ 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, in combination with centripetal forces, create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and the Sun.

Tidal forces are also responsible for tidal locking.