Is there a way to get the angle of approach of a spacecraft with respect to a planet velocity vector?

Question

I apologize if my question is badly formulated I know it's really specific but here what I would like to know :

I managed to compute the turning angle of the velocity for a body along a hyperbola trajectory thanks to the following equation :

$\delta = 2 \arcsin(1/e)$

This equation give me the turning angle at the end of the flyby. What I would like to know is the original angle, at start, of that spacecraft with respect to the planet velocity vector.

I thought using the HORIZONS Web-Interface and put the settings as following :

• Ephemeris Type : VECTORS
• Target Body : The spacecraft (e.g. Voyager 1)
• Coordinate origin : The planet (e.g. Jupiter)
• Time Span : Start of fly-by (e.g. 1979-Mar-4 / 1979-Mar-5 )

But then I'm completely lost in the generated tables ! Am I at the right place to find what I'm looking for ?

Thanks for your help

Answer 1

Since I had nearly the same problem, here is what worked for me:

1. Get two vector ephemerides from Horizons, one for Voyager and one for Jupiter. Choose the Sun as the Coordinate origin.

2. Subtract the VX, VY and VZ values of Jupiter from the corresponding values of the spacecraft to get a velocity relative to Jupiter.

3. Calculate the angle between that vector and the Jupiter velocity vector, good old $\cos \theta = \frac{\vec{u} \cdot \vec{v}}{||\vec{u}||||\vec{v}||}$

That should give you the angle between the Jupiter-centric spacecraft velocity, and the Jupiter orbital velocity vector. There might be other ways to get this directly, but this method only involves some minor coordinate math.