# Restricted circular 3 body problem RC3BP

kristian jerpetjøn
Hello everyone i am new to these forums
As a part of my master thesis i am trying to construct transfer orbits based on pathces of the restricted 3 body problem.

However i keep meeting several walls

One is how to calculate the optimal orbit from the earth orbit to the sun earth L1 point
The next is propagating this untill it falls back towards the earth then patch it to the earth moon restricted problem and fall trough earth moon L2 into a low lunar orbit

All help is welcome
I have the book from belbruno covering much of the math involved but i am having problems constructing a program from it.

Staff Emeritus
kristian jerpetjøn said:
Hello everyone i am new to these forums
As a part of my master thesis i am trying to construct transfer orbits based on pathces of the restricted 3 body problem.

However i keep meeting several walls

One is how to calculate the optimal orbit from the earth orbit to the sun earth L1 point
The next is propagating this untill it falls back towards the earth then patch it to the earth moon restricted problem and fall trough earth moon L2 into a low lunar orbit

All help is welcome
I have the book from belbruno covering much of the math involved but i am having problems constructing a program from it.

http://www.cds.caltech.edu/~shane/papers/gomez-et-al-2004.pdf [Broken]
http://etd.caltech.edu/etd/available/etd-05182004-154045/unrestricted/rossthesis_5_11.pdf [Broken]

The way I found this was to start at the Wikipedia link

http://en.wikipedia.org/wiki/Interplanetary_Superhighway

and take a look at the very first external link on this page under "papers" which led me here

http://www.cds.caltech.edu/~shane/papers/ [Broken]

At this point I looked for some theory-oriented titles.

At a very basic level you would want to start with the Hamiltonian of the third body

$$H(x,y,p_x,p_y)= {{{\it p_x}}^{2}/2m+{\it p_x}\,y\omega+{{\it p_y}}^{2}/2m-{\it p_y}\,x\omega+V \left( x,y \right)$$

These give the equations of motion of the third body directly by Hamilton's equations

http://mathworld.wolfram.com/HamiltonsEquations.html

here $\omega$ is the angular frequency of rotation of the two massive bodies M1 and M2 around their common center of mass, and the potential function V(x,y) is - GmM1/r1 - gmM2/r2, where r1 and r2 are the distance from the third body m to the massive bodies M1 and M2 respectively.

The above equation is for the restricted circular planar 3-body problem.

Also useful is the Hamiltonian's little brother, the energy function 'h'
$$h(x,y,\dot{x},\dot{y})=1/2\,m{{\it \dot{x}}}^{2}+1/2\,m{{\it \dot{y}}}^{2}-1/2\,m{y}^{2}{\omega}^{2 }-1/2\,m{x}^{2}{\omega}^{2}+V \left( x,y \right)$$

The quantity h is a conserved quantity, it is proportional to the "Jacobi intergal function". It is the Hamiltonian re-expressed in different variables. h is conserved because the Hamiltonian in the rotating coordinate system is not a function of time.

http://www.geocities.com/syzygy303/

has some plots of the inequalities that setting h <= some number gives on the (x,y) plane. These plots are inequalities because you can't plot $\dot{x}$ or $\dot{y}$ on the same graph, so you get the minimum value of h when $\dot{x}=\dot{y}=0$

These plots illustrate why low energy orbits must pass through the L1 and L2 plots. You'll probably see plots like these a lot (with prettier graphics) if you read through some of the literature.

This is all very basic, I'm afraid, but perhaps the background info will prove useful.

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