Lambert’s theorem

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 r₁ be the distance to the first observation and r₂ the distance to the second observation, and C the length of the chord between the two observations. We don’t need to know r₁ and r₂ individually but only their sum r₁ + r₂. 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: r₁ + r₂ 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

$\sqrt{\frac{\mu}{a^3}} (t_2 - t_1) = \alpha - \beta - (\sin \alpha - \sin\beta)$

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

$\begin{align*} \sin\frac{\alpha}{2} = \frac{1}{2}\left( \frac{r_1 + r_2 + C}{a} \right)^{1/2} \\ \sin\frac{\beta}{2} = \frac{1}{2}\left( \frac{r_1 + r_2 - C}{a} \right)^{1/2} \\ \end{align*}$

Hyperbolic orbits

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

$\sqrt{\frac{\mu}{a^3}} (t_2 - t_1) = \gamma - \delta - (\sin \gamma - \sin\delta)$

where

$\begin{align*} \sinh\frac{\gamma}{2} = \frac{1}{2}\left( \frac{r_1 + r_2 + C}{a} \right)^{1/2} \\ \sinh\frac{\delta}{2} = \frac{1}{2}\left( \frac{r_1 + r_2 - C}{a} \right)^{1/2} \\ \end{align*}$

Parabolic orbits

The parabolic case was worked out by Newton:

$t_2 - t_1 = \frac{1}{6\sqrt{mu}}\left( (r_1 + r_2 + C)^{3/2} - (r_1 + r_2 - C)^{3/2}\right)$

Elliptical orbits

Hyperbolic orbits

Parabolic orbits

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