Kepler's First Law of Planetary Motion:

The orbits of the planets are elliptical in shape with the sun at one focus.

          Start by assuming a system of two bodies, the larger of which has mass M, and the small of which has mass m, which are separated by a distance r.  Next assume an attractive force between the two bodies, F(r) which depends solely on the distance separating the bodies.  The Principle of Least Action states that

must be an extrema.  From the conservation of energy,

  ,

which can be re-written to show

  .

A look at a diagram of the situation gives

  .

  .

To find where this is an extrema, plug this into the Euler-Lagrange Equation, yielding

  ,

which means

where l is a constant and turns out to be the total angular momentum of the system.  This can be re-written as

  .

Squaring both sides gives

    .

Solving for q ’ yields

  .

At this point it is necessary to determine the form of U.  Since the force governing the actions of the system is gravity, the force is of the form

where k is a constant.  From the relation between force and potential energy,

  .

Plugging this into the equation for q ’ gives

  .

This differential equation can be solved to give the result

  ,

which is of the form

  .

It is easy to recognize that C is just a rotation angle of the function, which means the shape of the orbit is the same regardless of it’s value; it only affects to orientation of the curve.  For simplicity, choose C=0, which gives

  , where  ,  .

            This expression for r is the generic form of a conic section with one focus at the origin in polar coordinates.  The exact shape of the curve depends on the eccentricity, e.  In reality an orbit can be in the shape of any conic section.  For the curve to be a closed orbit, however, the shape must be either a circle or an ellipse.  If the shape is a parabola or hyperbola, the object will make one pass by the central body, and then never return since it would be an open orbit.  In the case of planets, the orbits are closed.  As for whether the shape is circular or elliptical, it is almost always elliptical.  This is because in order to be circular, the speed, momentum, energy, etc. must be exactly right in order to have the force of gravity PERFECTLY balance the centrifugal force.  If this condition is not exactly met, the resulting orbit will be an ellipse.  The fact that planet’s orbits are ellipses with the sun at one focus is Kepler’s First Law of Planetary Motion.

Background image courtesy of NASA's Planetary Photojournal.