Adam Hibberd
A spacecraft is travelling on a very hyperbolic orbit w.r.t. an object X (possibly a star) which has gravitational mass, μ, meaning the spacecraft is only slightly deflected from its direction of motion. Our task is to quantify the errors in velocity, both longitudinal and transverse, associated with this encounter compared to simply travelling in a straight line without the star being present.
The situation is depicted in figure 1.
We see that the distance at closest approach (or Periastron) is denoted P, and the spacecraft is deflected by an angle, α . Also Vi and Vo are the arrival and exit velocities respectively. Let us define the speed V= ║Vi║=║Vo║
The following two equations will hold for the encounter, with e as the eccentricity of the hyperbolic orbit:
and:
Assuming the deflection angle α is small, then we find:
Since we are on a hyperbolic orbit, we know that
since this is the square of the circular orbital speed at P.
Thus we can write:
Now the difference in longitudinal velocity ΔVx, of the spacecraft compared to travelling in a straight line (with no gravitational field) is as follows:
Thus we can approximate:
Similarly, the transverse velocity error ΔVy, of the spacecraft compared to travelling in a straight line is as given by:
Substituting (4) into (6) and (7), we can derive the following results:
