# Find keplarian orbital elements between orbits

Say I have two different orbits: orbit A and orbit B , defined from these Keplerian orbital elements, which I know relative to some refence plane and reference direction (which are the same for these two orbits):

$i$: Inclination
$\Omega$: longitude of ascending node
$\omega$: argument of periapsis
$v$: true anomaly of at a specific time

The elements are illustrated here:

Is there a way for me to figure out these orbital elements of orbit B but with orbit A as reference plane, and the true anomaly of orbit A as the reference direction.

I would prefer a simple calculation, but since I am going to use this in a space flight simulation a recursive algorithm solution should also be okay.

• It sounds like you're asking how to specify a satellite orbiting A. Are you asking because you have the parameters of B and they are given in terms of A, or because you want to specify B in terms of A in order to, for example, render an epoch into real space? Jan 30, 2017 at 17:43
• @omaha Because i want to specify B in terms of A, to later find a transfer orbit from A to B Jan 30, 2017 at 17:54

I'm also not sure I totally understand the question, so correct me if I misunderstood you. Here's what I understood: you have the orbital elements of two orbits A and B, and would like to define orbit A as the new reference frame, and express the orbit B with respect to that reference frame A.

If so, yes, that's possible, and not to difficult. In effect, an orbit is simply a rotation. We can compute rotations in different ways. I prefer Modified Rodriguez Parameters, but these aren't that popular yet, so let's use Euler Parameters (also called Euler Angles).

Let me explain. Start from an Earth Inertial Frame. First, do a rotation by the Z (or third) axis of your right angle of the ascending node. Then do a rotation by the X axis (or first axis) by your inclination. Finally, do a rotation about the Z axis (again) of your true anomaly. So, in terms of Euler parameters, you've done a 3-1-3 rotation (i.e. multiply the R3, R1 and R3 matrices together computed with the correct angles to get your direct cosine matrix (DCM)).

Using DCMs, you just need to multiply both DCMs to change the reference frame, and of course multiply that with the R vector of your spacecraft.

(I highly recommend "Analytical Mechanics of Space Systems (AIAA Education Series) 2nd Edition" by Dr. Schaub and Dr. Junkins for any astrodynamics question by the way. The above computation is thoroughly explained in chapter 3.)

• Are there any good online resources for working through this sort of math? I'm wanting to do something similar, and your answer is incredibly dense to me. Apr 24, 2017 at 5:00
• @KeithB I am not aware of any online resource dedicated to this. However, it might be possible to find a copy of the Schaub and Junkins book online. If you're interested in astrodynamics, this book is very much a reference: it includes everything from rigid body kinematics (ie how things rotate on themselves) to controlling the attitude of spacecraft, to the definition of orbits, rendez-vous scenarios, spacecraft formations, and so on. It also includes a very large number of exercises and examples so you can really thoroughly understand the subject. Apr 24, 2017 at 5:03
• I'm mostly interested in orbital mechanics. The main problem I'm trying to puzzle through is how to figure out the nodes for transits between say, Jupiter and Mars, given that I only know their orbits relative to the ecliptic. Apr 24, 2017 at 5:21
• @KeithB so you would like to solve a problem where you have a departure and arrival point and would like to find a way to connect both points? Apr 24, 2017 at 5:23
• No, I'm not dealing with spacecraft at all. I want to know e.g. how many days after the Northern Vernal Equinox on Jupiter its orbit "crosses" Mars' orbit, such that it's when transits can occur. In much the same way that we know that Mercury transits relative to Earth occur in early May and early November, because those are where the ascending/descending nodes are relative to Earth's orbit. Apr 24, 2017 at 5:29