Kepler Orbit - Determination of The Kepler Orbit That Corresponds To A Given Initial State

Determination of The Kepler Orbit That Corresponds To A Given Initial State

This is the "initial value problem" for the differential equation (1) which is a first order equation for the 6-dimensional "state vector" when written as

(48)

(49)

For any values for the initial "state vector" the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm:

Define the orthogonal unit vectors through

(50)

(51)

with and

From (13), (18) and (19) follows that by setting

(52)

and by defining and such that

(53)

(54)

where

(55)

one gets a Kepler orbit that for true anomaly has the same r, and values as those defined by (50) and (51).

If this Kepler orbit then also has the same vectors for this true anomaly as the ones defined by (50) and (51) the state vector of the Kepler orbit takes the desired values for true anomaly .

The standard inertially fixed coordinate system in the orbital plane (with directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation

(56)

(57)

Note that the relations (53) and (54) has a singularity when and

i.e.

(58)

which is the case that it is a circular orbit that is fitting the initial state

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