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When planning space missions the aim of the mission designer is to create a safe and cost effective mission plan in a short time frame. A significant part of any mission design is the trajectory the spacecraft will take on its journey. The trajectory dictates how much energy will be required, how long the mission will take and what conditions the spacecraft will encounter. The trajectory ultimately dictates how soon a return, in terms of science or money, will be delivered. The return of the mission makes the trajectory a very important part of any mission plan and therefore makes trajectory design a valuable research area. Creating trajectories that are cost effective means creating a trajectory that will take a short amount of time and also be energy efficient. Such trajectories cannot simply be chosen, trajectories have to be carefully designed within the framework of celestial mechanics. The area of celestial mechanics is complex and many different theories exist such as the Three Body Problem, The Restricted Three Body Problem, the n-Body Problem and more complex chaotic dynamics. From the study of celestial mechanics an interesting equation of energy appears, this is the Jacobi Integral equation also called the integral of relative energy.
The purpose of this report is to examine the uses of the Jacobi equation in the planning of space missions. The report provides a background of previous work on trajectories specifically efficient low energy trajectories and looks at real life missions in which new mission planning techniques have been used. A study on the Jacobi equation is carried out to identify methods that could be used in trajectory planning. The report attempts to identify how the Jacobi equation relates to trajectory planning and how it could be used in the future to develop new methods. Methods discovered by the study are described before being demonstrated via a computer program. The computer program assembly is described through its testing and development. The results of the program are looked at and the findings related back to the method descriptions given earlier. It is hoped that by drawing results from the methods more detail can be gained about the implications and outcomes of planning trajectories via these methods. This paper will give an indication of how useful the Jacobi equation is in planning trajectories. As part of the trajectory program creation the overall method of investigation will be discussed at the end of the document. This will include a look at the errors in the program method and the accuracy of the results. As part of an analysis the results found via the program will be compared with existing results to verify them.
To focus the study the Earth moon system will be the basis of all calculations. Specific focus will also be given to the energy requirements of any trajectories investigated. Using the results from the computer program and various orbits of spacecraft, values of the Jacobi constant C will be compared. Comparing these values will give an indication of the energy requirements on the trajectories between the Earth and the Moon. This will further indicate the usefulness of these new trajectories.

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