TAB 9

FREE RETURN MARS TRAJECTORY

ABSTRACT

The Mars capture trajectory is configured to seek a special type of incoming flight path called a "free return trajectory." That is, the Apollo missions to the moon were designed to approach the moon along a flight path that, if something went wrong, would bring the spacecraft right back toward Earth - no thrusts or maneuvers were needed for the spacecraft to return. It was a safety precaution, and it is also a very efficient trajectory.

The conjunction to Mars part of the trajectory is optimized by adjusting the velocity component of the initial conditions so that the trajectory ends just outside of Mars' orbit. Then, unlike how the conjunction to Earth flight path was configured, the initial conditions of the problem must be adjusted so that the spacecraft actually rendezvous with Mars. That is, the starting position of Mars is adjusted by a few days so that the spacecraft and Mars both arrive at the same time when their paths intersect near apoapse of the transfer ellipse.

The first few iterations target Mars' Sphere of Influence (SOI). Subsequent iterations continue integrating the flight path until it either intersects Mars or goes beyond Mars.

The program output shows both stages of the process - the coarse, heliocentric targeting of the SOI, then the fine geocentric targeting of Mars itself. The integration is stopped as soon as the range to Mars starts increasing. This is the closest point of approach to Mars, and a tangential thrust is applied to put the spacecraft in a circular orbit. The spacecraft must still reach the 100 Km parking orbit, so additional thrust is added for a simple Hohmann Transfer to the lower orbit.

NOTE: In this "pure" Hohmann scenario, the optimal trajectory begins at the 100 Km parking orbit itself even though the optimization technique calculates an additional small Hohmann for prior iterations that do not terminate at the parking orbit; the program still needs a total thrust for each pass.

The user defined options allow more complex conditions. One scenario is for the spacecraft to go beyond Mars, to a higher altitude orbit, and to begin the final trajectory from there, slightly behind Mars. In this scenario, the spacecraft has a small gravity assisted flyby. Another option allows the periapse of this hyperbola to drop as low as 5 Km above Mars. In either case there will be two thrusts calculated for the maneuver: The large thrust to enter orbit around Mars, and a second small thrust to go to the final 100 Km parking orbit.

Mars Approach Geometry

The optimization of the user-defined scenarios is more complex than just stopping the integration when the range starts opening. These options are intended to explore the possibility of letting the spacecraft go closer to Mars to take best advantage of Mars' gravity in slowing down the orbit, and adjusting the geometry of the flight path so it's easier to enter a circular orbit around Mars. So, once the range begins opening, the program calculates the total thrust to reach the 100 Km parking orbit at each step of the integrator. The lowest final thrust among these values is saved as the global minimum is the optimal thrust for that particular pass. Additional passes are done until the closest point of approach to Mars on the trajectory is 100 Km (or less, as specified by the user). The lowest total thrust among all these passes is the global minimum for the Mars half of the flight path.

Targeting the Free Return Approach to Mars

The first objective is to adjust the phase angle of Mars at conjunction so the spacecraft path intersects Mars’ SOI. This is a highly nonlinear problem and it cannot be solved by direct method used for the trajectory to Earth. Instead, once the state vector of the spacecraft is known at SOI, it is assumed that a small thrust can be done in real time to target any point on the SOI in the immediate neighborhood. Modifying the tolerances in the code can make the solution exact, at the cost of perhaps doubling the computational time. This was not done in the distribution copy of the program so the program can run on a PC in less than 60 seconds to get the approximate solution.

A Free Return Loop in the Earth-Moon System

 

The maximum possible error is 4 hours in the total time of flight of 260 days (± .001% ~ this decreases by at least one order of magnitude with each iteration, better if a sophisticated one dimensional search routine is used). The total thrust is not affected because of how the algorithm is structured.

Analysis

A phase path separating locally bounded motion and locally unbounded motion is called a separatrix (see the figure below). A sepatrix always passes through a point of unstable equilibrium (e.g. the L2 Lagrange Point). Motion near a sepatrix is very sensitive to initial conditions because points on either side of the separatrix have very different trajectories.

The L2 Lagrange Point as a Separatrix

The plot given is a zero velocity curve, for the Three Body Problem with equal masses. The potential function is such that for such an equipotential surface. That is, t he (gravity) field does no work on a body moving along an equipotential surface (e.g. a body on the L2 free return loop orbit between the two bodies will remain their indefinitely, with no other forces acting).

It is interesting to note that because the masses are equal and the paths or symmetric, it is possible to create a complete 3D map by rotating the above graph along the line shown. That is, the above map is identical for any section through the 3D map which includes both masses.

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© 2004 WH Clark