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The gravitational potential energy of two objects is
In this case the differential equation for the radial motion can be written
This can be integrated straightforwardly by changing variables to
.
The result can be written
This is the equation of a conic section. (More information and pictures of these curves can be found in Dave's Math Tables and Xah Lee's visual dictionary.) The shape of the curve is determined by the eccentricity :
The orbit of a bound object () has and so it is an ellipse. The size of the ellipse is specified by its semi-major axis () and its shape is determined by its eccentricity . The centre of force lies at a focus of the ellipse, which is a distance from the centre of the curve. This is Kepler's first law: the orbit of a planet is an ellipse with the Sun at one focus.
An unbound object () has and so follows a hyperbolic orbit. The limiting case is that of an object with only just enough energy to escape (); its orbit is a parabola, with .
The minimum distance of an orbiting object from the centre of force is known as
the pericentre (perihelion, perigee) distance. For a bound object, this
is given by
The total energy per unit mass of the object is
The period of the orbit can be found from its area
(where
is the semiminor axis of the ellipse) by using
Kepler's second law. This gives
David McNamara and Gianfranco Vidali (Syracuse University) have written a JAVA applet to illustrate orbits in a gravitational field.
Textbook references
Home: PC 1672 home page |
Up: 3 Gravity |
Weekly plan |
Help: Guide to using this document |
Next: 3.8 The solar sytem |
Previous: 3.6 Central-force motion |
Mike Birse
27th March 2001