# What is Binet's Equation

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

Binet's equation allows one to find the acceleration of a body moving in a central force field provided that its trajectory in polar coordinates is known.

Equations

$$\ddot q = \frac{{{h^2}}}{{{q^2}}}\left( {\frac{{{d^2}}}{{d{\phi ^2}}}\left( {\frac{1}{q}} \right) + \frac{1}{q}} \right)$$

$$\ddot q$$ is the magnitude of the acceleration (negative if the force producing it is attractive) of a body relative to the center of the force. $$q = q\left( \phi \right)$$ is the distance from the center of the force to the body given as a function of the body's angular configuration. $$h$$ is twice the sectoral areal velocity of the body with respect to the center of the force (a constant in all central force fields).

Extended explanation

Binet's equation allows one to determine the acceleration of a body in a central force field needed to produce a given orbit in polar coordinates. It's primary utility is proving Newton's laws of gravity from Kepler's laws of planetary orbit. Kepler's second law is equivalent to the statement that the Sun's gravitational force is central. Kepler's first law states that the planetary orbit is elliptical with the Sun as one of its foci. By plugging in the focal polar equation of an ellipse,
$$q = \frac{1}{2}\frac{{a\left( {1 - {\varepsilon ^2}} \right)}}{{1 + \varepsilon \cos \phi }}$$,​
one may prove that the gravitational force exerted by the Sun to one of its planets must be inversely proportional to their seperation.

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