Kepler Orbit - The Osculating Kepler Orbit

The Osculating Kepler Orbit

For any state vector the Kepler orbit corresponding to this state can be computed with the algorithm defined above. First the parameters are determined from and then the orthogonal unit vectors in the orbital plane using the relations (56) and (57).

If now the equation of motion is

where

is a function other than

the resulting parameters

defined by will all vary with time as opposed to the case of a Kepler orbit for which only the parameter will vary

The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" (59) at time t is said to be "osculating" at this time.

This concept is for example useful in case

where

is a small "perturbing force" due to for example a faint gravitational pull from other celestial bodies. The parameters of the osculating Kepler orbit will then only slowly change and the osculating Kepler orbit is a good approximation to the real orbit for a considerable time period before and after the time of osculation.

This concept can also be useful for a rocket during powered flight as it then tells which Kepler orbit the rocket would continue in in case the thrust is switched-off.

For a "close to circular" orbit the concept "eccentricity vector" defined as is useful. From (53), (54) and (56) follows that

i.e. is a smooth differentiable function of the state vector also if this state corresponds to a circular orbit.