Kepler Orbits

The Kepler orbits for a two-body system under a central gravitational force can be represented by a polar curve that relates the distance between the two bodies with the angle from the axis. The shape of the orbit depends on its eccentricity, which is determined by the energy of the system.
Vary the eccentricity to see how the energy and the shape of the orbit changes. The center of attraction is shown by a point at the origin.


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1. Equation of Motion for the Two-Body Problem
To get the equations of motion, write the Lagrangian for a two-body system:
where μ is the reduced mass of the system, is the distance between the two bodies, is the angular velocity and is the potential energy of the system.
The equations of motion are obtained by differentiating the Lagrangian with respect to the two generalized coordinates and . The equation gives:
where the constant is the angular momentum.
The equation (radial equation) is given by:
which further gives:
( equation).
2. Solution to the Radial Equation
The solution to the radial equation gives as a function of time . In order to get the equation of the orbit in terms of the polar equation (i.e. as a function of ), recast it in a slightly different form using the variable substitution . Without going into the details of the transformation, the result is:
where is a constant, equal to ; is the force constant, equal to for the gravitational potential ; and is the eccentricity of the curve.
3. Eccentricity and Energy
Eccentricity and energy are related by
Also, for bound orbits, the minimum and maximum radial distance are
and .
Snapshot 1: circular orbit having eccentricity and minimum energy; bound system
Snapshot 2: elliptical orbit having negative energy; bound system (Earth orbiting the Sun)
Snapshot 3: parabolic orbit having and energy = 0; the system is just free
Snapshot 4: hyperbolic orbit having and positive energy; system is unbound
[1] J. R. Taylor, Classical Mechanics, Sausalito, CA: University Science Books, 2005.
[2] D. Morin, Introduction to Classical Mechanics: With Problems and Solutions, New York: Cambridge University Press, 2008.
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