# Restricted circular 3 body problem RC3BP

• kristian jerpetjøn
In summary, the conversation discusses someone's difficulties in constructing transfer orbits based on patches of the restricted 3 body problem for their master thesis. They mention specific challenges such as calculating the optimal orbit from Earth to the Sun-Earth L1 point and propagating it to fall back towards Earth and then patching it to the Earth-moon restricted problem. They ask for help and mention a book they have on the topic. Some helpful resources and equations are suggested, such as the Hamiltonian and energy function, as well as links to papers and plots that illustrate the concept of low energy orbits passing through L1 and L2 points.
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 until 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.

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 until 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
http://etd.caltech.edu/etd/available/etd-05182004-154045/unrestricted/rossthesis_5_11.pdf

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/

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.

Last edited by a moderator:

## What is the Restricted Circular 3 Body Problem (RC3BP)?

The Restricted Circular 3 Body Problem (RC3BP) is a simplified version of the classical 3-body problem in celestial mechanics. It involves three masses (usually a larger primary mass and two smaller secondary masses) moving in a circular orbit around their center of mass, with the assumption that the secondary masses have negligible influence on each other's motion.

## Why is the RC3BP important in astrodynamics?

The RC3BP is important in astrodynamics because it allows for the study of the dynamics of a system with three or more bodies, which is a common scenario in celestial mechanics. It also serves as a useful approximation for more complex systems, such as planetary orbits in our solar system.

## What are the key equations and assumptions of the RC3BP?

The key equations of the RC3BP include Newton's laws of motion, Kepler's laws of planetary motion, and the equations for circular motion. The main assumption is that the secondary masses have negligible influence on each other's motion, which allows for the simplification of the equations and the study of the system as a two-body problem.

## What are some real-world applications of the RC3BP?

The RC3BP has many real-world applications, including the study of binary star systems, satellite orbits around Earth, and the motion of planetary moons. It is also used in the design of spacecraft trajectories and in the study of gravitational interactions between celestial bodies.

## What are some challenges in studying the RC3BP?

One of the main challenges in studying the RC3BP is the complexity of the equations and the difficulty in finding exact analytical solutions. This often requires the use of numerical methods and computer simulations. Additionally, the assumption of circular orbits may not accurately represent real-world scenarios, leading to some discrepancies between theoretical predictions and observed data.

• Astronomy and Astrophysics
Replies
3
Views
1K
• Astronomy and Astrophysics
Replies
3
Views
359
• Astronomy and Astrophysics
Replies
1
Views
879
• Astronomy and Astrophysics
Replies
2
Views
966
• Astronomy and Astrophysics
Replies
5
Views
2K
• Classical Physics
Replies
4
Views
708
• Astronomy and Astrophysics
Replies
10
Views
2K
• Astronomy and Astrophysics
Replies
3
Views
915
• Introductory Physics Homework Help
Replies
12
Views
1K
• Special and General Relativity
Replies
10
Views
1K