At the start of this series we pointed out that a conic section has five degrees of freedom, and so claims to be able to determine an orbit from less than five numbers must be hiding information somewhere. That is the case with Lambert’s theorem which reportedly determines an orbit from two numbers.

Lambert’s theorem says that the time between two observations of an object in an elliptical orbit can be determined by the sum of the distances to the two observations, the length of the chord connecting the two observations, and the semi-major axis of the ellipse. From this one can determine the equation of the ellipse.

Let *r*_{1} be the distance to the first observation and *r*_{2} the distance to the second observation, and *C* the length of the chord between the two observations. We don’t need to know *r*_{1} and *r*_{2} individually but only their sum *r*_{1} + *r*_{2}. And if we know the angle between the two observations we can use the law of cosines to find *C*. However we get there, we have two lengths: *r*_{1} + *r*_{2} and *C*. We also know *a*, the length of the semi-major axis. This gives us three lengths.

The observations are taken from the body being orbited, so we implicitly know one of the foci of the ellipse. That’s four pieces of information. The fifth is the mass of the object being orbited.

Lambert’s theorem divides into three cases: elliptic, parabolic, and hyperbolic.

## Elliptical orbits

For an object in an elliptic orbit, Lambert’s theorem becomes

where μ is the standard gravitational parameter, equal to *GM* where *G* is the gravitational constant and *M* is the mass of the body orbited. The angles α and β are given by

## Hyperbolic orbits

The hyperbolic case is very similar, replacing sine with hyperbolic sine.

where

## Parabolic orbits

The parabolic case was worked out by Newton:

## Other posts in this series

- Five points determine a conic section
- Gibbs method of determining an orbit
- Determining a conic from mix of points and tangents

Source: Adventures in Celestial Mechanics by Victor Szebehely