Caption: A diagram illustrating the tidal force caused by an (external) gravitational field source (off the image to the right) on a free falling spherical astro-body. The astro-body could be free falling directly to the gravitational field source, but in actual cases is much more liking to be orbiting the gravitational field source.
In brief, the tidal force is a stretching force due to the varying gravity of an external gravitational field source.
Features:
The residual external gravitational force is called the tidal force.
The lower panel of the image illustrates the tidal force and its stretching effect.
If the tidal force gets too strong relative to the internal forces, the astro-body can be disrupted. This happens to moons that get too close to their host planet.
Also the tidal force can prevent a planetary ring from coalecsing into a moon under its self-gravity.
The general formula for the tidal force per unit mass in a celestial frame is effective tidal field which for sufficiently symmetric celestial frames is the difference between the external gravitational field and the external gravitational field at the center of mass. An celestial frame that is nearly spherically symmetric (like the astro-body) in the image is usually sufficiently symmetric.
This relatively rapid fall off of the tidal force plus cancelation between whole surrounding array of external gravitational field sources, means that in many cases the tidal force will be negligible. For example, for relatively planetary systems (e.g., the Solar System) the tidal force is negligible.
In many cases, the tidal force is NOT negligible at least on some scale. For example the tidal forces of the Moon and, secondarily, the Sun cause the tides, the Earth tide, and atmospheric tide on Earth. Of course, these tides are very small compared to the Earth, but NOT compared to humans.
For more on the tides, see tide_ideal.html.
For example, artificial satellite have absolutely negligible tidal force.
We assume both astro-bodies are spherically symmetric.
For simplicity and also because it is usually approximately true in such cases, we assume the rotation axis is perpendicular to the line between the centers of the astro-bodies.
Let ρ be the radius of the secondary.
The centrifugal force per unit mass is (ω**2)*ρ, where ω is the angular frequency of the axial rotation (i.e., its frequency times 2π). The centrifugal force per unit mass points radially outward from the center of the secondary.
However, because of the tidal locking ω is also the angular frequency of the circular orbit (i.e., its frequency times 2π).
This means that (GM/R**2) = (ω**2)*R so that the gravitational force of the primary is canceled by the centrifugal force of the secondary orbital motion: i.e., the orbital centrifugal force.
We can now write rotational centrifugal force per unit mass as
However, the rotational centrifugal force is probably best thought of as causing the tidal bulges of the secondary and NOT as contributing effectively to the tidal force stretching. However, the above analysis shows that tidal force and the rotational centrifugal force are comparable for the simple case of tidal locking we have analyzed.
Credit/Permission: ©
William M. Connolley
(AKA User:William M. Connolley),
2009
(uploaded to Wikimedia Commons
by User::Justass,
2009) /
Creative Commons
CC BY-SA 3.0.
Image link: Wikipedia:
File:Tidal-forces.svg.
Local file: local link: tidal_force.html.
File: Mechanics file:
tidal_force.html.