# Why do we prefer using the rotating reference frame rather than the Inertial frame in the three-body problem?

As the title says, I wanted to ask why do we prefer using the rotating reference frame rather than the Inertial frame in the three-body problem to place the primaries?

And how is it better for describing the motion of the third body (i.e. spacecraft)?

• Perhaps just easier?
– uhoh
Mar 16 '20 at 11:18

The rotating reference frame is usually preferable over an inertial reference frame when analyzing three-body motion. Two main reasons for this:

1. The motion is determined by the two primary bodies, so it makes sense to use these to define the reference system
2. The typical three-body dynamics for which you would use such a reference system, such as Libration point orbits arise from the equations of motion in the rotating reference frame. The Libration points are only defined in their classical way in the circular restricted three-body problem.

This of course doesn't mean that Inertial can't be used. You could recreate the same orbit in an inertial reference frame, the main problem is that the dynamics will not be recognizable as for example a libration point orbit. A Sun-Earth L2 Halo orbit will look like a plain heliocentric orbit if you look at it in an inertial sun-centered reference frame.

As a final note, libration points and their respective orbits are defined in the Circular Restricted Three-Body Problem. The real-life situation is as you might expect radically different from the CRTBP. You have eccentric orbits, leading to both a change in the distance between the primaries as a non-uniform rotation rate. On top of this you have all kinds of perturbations from other solar-sytem bodies, solar radiation pressure, non-uniform gravity,...

For recreation of CRTBP dynamics in the real-life situation, something interesting to look at is a Rotopulsating Reference frame. The reference frame is centered at the barycentre of the two primary bodies and rotates along with the line connecting them both. On top of this, the distance is scaled so that the distance between the two primaries is always equal to 1. This translation of the CRTBP reference frame to a real solar system model is unfortunately highly non-newtonian because of its non-uniform rate of rotation (Roto-) and changing definition of distance (Pulsating). This makes the derivation of the equations of motion quite the challenge but if you are interested, check out this thesis for a detailed explanation and derivation.

• Very informative! Thank you very much it makes sense now.
– John
Mar 16 '20 at 20:26
• Rotopulsation? Perhaps that's the answer to What is a “synodic frame”? Can one be defined for an elliptical orbit? and perhaps the orbits in the GIF in Does the shape of Cruithne's orbit change within one period? are rescaled frame-by-frame to null the "pulsation".
– uhoh
Mar 18 '20 at 18:05
• Synodic only refers to the rotation going along with the sun-earth axis. Concerning the cruithne orbit I believe the strange behaviour you see in the GIF is indeed due to the reference frame (apparently) keeping the distance between Sun and Earth constant. Mar 18 '20 at 18:18