Bringing your Analogue past into our Digital future

Lunar Trajectories, Interim Report
17/11/2004

1 Introduction
This project is looking into methods of trajectory plotting, specifically with the trajectories between the earth and the moon. There are numerous methods of plotting interplanetary trajectories for spacecraft. Most of these methods simplify the actual celestial situation in order to give an approximation of the actual trajectory. This project is looking specifically at the jacobi integral. Studying how this energy equation can be used to find highly efficient trajectories for low power spacecraft.

2 Background
This project is concerned with using the jacobi integral to provide us with information about trajectories in the solar system. It looks specifically at those trajectories that can be used to send a spacecraft from the earth to the moon, with the maximum efficiency. The efficiency being a rating based on the amount of energy used and the speed of transfer. A low expenditure of energy with a high rate of transfer being the ideal transfer.
A simplified method of finding a low energy trajectory is the patched conic method. First it is assumed that only two of the three bodies are interacting at any time. For instance the (earth, spacecraft) and (moon, spacecraft) systems. This then allows the problem to be looked at as a two body problem. We can make this kind of assumption as it can be seen that each massive body in a system has its own sphere of gravitational influence in which its gravity is the overriding influence on a spacecraft, the gravity of the other massive body can be neglected. To make this method apply to the entire system the patched conic method is used. The Patched conic method is named with patched referring to joining the two systems at the point where there spheres of influence interact. Conic to the two orbits being conic sections, or Keplerian orbits. The sphere of influence around the earth is larger than that of the moon because the earth is much more massive than the moon. The earths sphere of influence is is 6.5∙104km. The distance between the earth and the moon is 3.8∙105 showing how much more massive the earth is when compared to the moon. The patched conic method is an inaccurate method which does not provide the maximum efficiency for a spacecraft. It is appropriate to use the energies and transfer speeds of an orbit plotted using the patched conic method as a comparison with a trajectory plotted using the alternative methods below.
This project is focusing on the Jacobi integral method. The Jacobi integral is an equation that gives us the relationship of the energies a spacecraft has while in an orbit.
The Jacobi integral is written as follows: [2]
The Lagrangian points are points in the vicinity of two massive bodies with circular orbits around there center of mass, where the gravitational and centrifugal forces exactly balance.
There are five Lagrangian points. If a small body is placed at a Lagrangian point it will remain there with respect to the the motion of the massive bodies. The first three Lagrangian points are located on a line that passes through the centers of the two bodies in a rotating frame of reference.
L4 and L5 are located at the points of equilateral triangles created between the bodies. It can be shown that L1, L2 and L3 are unstable points at which a body would not remain if placed. L4 and L5 can be shown to be stable. If a body is perturbed away from one of these points it will pick up speed and due to the Coriolis effect will enter into a halo orbit of the point.[3]

2 Work Carried Out
In order to understand the problem information about the earth moon system needs to be looked at. The mass of the earth is 5.9742∙1024kg and the mass of the moon is 7.39∙1022kg.[4] Using the simplification that the moon is on a circular orbit around the earth then the distance between the earth and the moon is 384400km. Calculating the center of mass gives us a distance of 4671.6km from the earth as a point mass. The radius of the earth is 6378.14km therefore in reality the center of mass of the system is within the earths body. This demonstrate how large the earth is compared to the moon. In our model however it can be assumed that both the earth and the moon rotate around the center of mass of the system.
In this project the jacobi equation is to be used for orbit plotting. The first us of this equation can be seen when the energy of the spacecraft in question reaches zero, this equates to the spacecraft having zero velocity allowing the velocity terms to be removed. If the equation is then rearranged to make C the subject it can be seen that for certain values of C a graph of the equation will provide surfaces on which the velocity of the spacecraft will be zero. This will give a graph showing for certain values of C areas that are impassible to the spacecraft. Plotting these will allow us to find a specific value of C when the spacecraft can just pass between the earth and the moon. Also plotting the graphs for various values of C will show us where the exact locations the Lagrangian points reside, this will be useful later in the project.
In order to plot the surfaces of zero velocity the equation must be solved for certain given values of C. There is no obvious numerical solution to the equation so alternative methods have to be found. A first method to try is, for given values of x,y values of C can be found. Matching these values will give us the graphs. This can easily be performed in a spreadsheet program. On attempting this method it was discovered that in order to find values of C that were equal a very large number of values of x,y were required. To obtain accurate plots the number of x,y values became too large for the spreadsheet program. Plots were found using this method but were too inaccurate to be of use. The second method was to give a value of C and x then using an iterative Fortran program an initially guessed value of y could be iterated to give an accurate value of C. Using the program many values of x,y could be found, easily providing accurate plots. Using the original method as a guide to values graphs of surfaces of zero velocity were plotted for various values of C between 2.9 and 3.5 this gave use a set of useful graphs from which the Lagrangian points can be derived.
3 Plans For The Project
On obtaining plots of the surfaces of zero velocity and thus finding the jacobi constants the next stage will be to accurately use these graphs to find the exact locations of the Lagrangian points. These points can then be used when looking for trajectories. Specifically, for the earth moon system, at a certain value of C the surfaces of zero velocity on a plane of x,y will just allow a passage from the area around earth to that around the moon passing L1. Trajectories can be plotted, using invariant manifold theory, passing L1 to find a low energy trajectory.[5] However this trajectory will be very slow. Much slower than the lowest energy trajectory given by the patched conic method. To find the manifolds around L1 a computer program must be written. [6]

4 Final Aims
In the shorter term the project hopes to demonstrate the usefulness of the jacobi equation and its subsequent methods in this relatively new field of advanced highly efficient trajectory plotting.
The overall aim of this project is to look into ways of improving the efficiency with which new space missions are created, specifically in the area of trajectory plotting. Creating a computer program capable of providing suitable efficient trajectories for a range of missions. The program will help in giving one overall solution to the problem of having to plot new trajectories for each mission.
5 References
[1] Principles of Spacecraft design lecture notes section 2,3
[2] Orbital Motion A E Roy 1988 Third Edition
[3] http://en.wikipedia.org/wiki/Lagrangian_point
[4] http://www.freemars.org/jeff/planets/Luna/Luna.htm
[5]Interplanetary and lunar transfers using libration points, Francesco Topputo, Massimiliano Vasile and Franco Bernelli-Zazzera, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa, 34 - 20156- Milan
[6]http://www.jpl.nasa.gov/releases/2002/release_2002_147.html