Distant Retrograde Orbit Question

[monkey55]

In the circular restricted three body problem (CR3BP) of Sun-Earth system, how can I get the initial conditions for a distant retrograde orbit around Earth? The state vector is X=[x0 0 0 0 vy0 0]. What values do I choose for x0 and vy0, and how do i choose them?

Thanking you in advance

[K^2]

I'm assuming, your coordinate system is relative to Earth's center. In this coordinate system, orbits around the Earth are subject to standard -μ/r potential of Earth's gravity, plus a periodic perturbation due to Sun's gravity and motion around the sun. That perturbation has a non-zero mean, which is why orbital speed is slightly modified.

Because for a circular orbit it will average out exactly the same way, I'm going to ignore the precession of perturbation itself. In that case, a circular orbit would experience potential:

U(r, θ) = -μ/r - MG/(R + r sinθ) - (Ω²(R + r sinθ)²/2)

Here, MG is gravitational parameter for Sun, R is distance of Earth from Earth-Sun's barycenter, and Ω is Earth's angular velocity around the Sun. r and θ describe position of the satellite with respect to Earth's center. Note that I'm assuming that satellite is orbiting in the Earth-Sun plane here. You'll have to add an additional parameter for out-of-plane orbit, but hopefully you can manage that.

At this point, we can start throwing in heavy machinery. We are only interested in average potential energy to make use of Vis Viva. We know that 1/(a + x) = 1/a - x/a² + x²/a³ - ... So by taking only the r-dependence of potential above and taking only the highest even term (odd terms average to zero), we get effective potential.

U*(r, θ) = -μ/r - MG/R³ r²sin²θ - Ω²r² sin²θ / 2

We can further simplify, because MG/R³ = Ω², as Earth itself is in orbit around the Sun, and using the fact that average of sin²θ is 1/2.

U*(r) = -μ/r - (3/4) Ω²r²

And that gives us initial v_y = sqrt(-U*(x)/2) = sqrt(μ/(2x) + (3/8)Ω²x²)

Unless I've screwed something up, that should give you a nice circular motion around Earth, so long as the orbit is co-planar. If the orbit is out of the plane, this can still work under some cases, but will result in some orbital precession as well. Do check all of the above steps, though. I have an odd feeling about that 3/8 factor, but I can't find any errors.

[monkey55]

Thank you K^2

still cant get it. what would i change in the final equation if ti was in a rotational frame? 

[K^2]

2 hours ago, monkey55 said:

Thank you K^2

still cant get it. what would i change in the final equation if ti was in a rotational frame? 

This was in rotating frame, where Sun revolves around Earth. That's where I got the Ω² term in the potential from.

[monkey55]

aah i see! hmm thanks for the explanation. i am still struggling somehow! i will try to sort it out. Thank you!