Lunar Transfer Trajectories

Introduction

When discussing paths by which an object, which we will term a spacecraft, can travel from one of the principal bodies of the Terra-Luna system to the other, we can recognize two basic categories permitted by the laws of orbital mechanics, which we will describe as ballistic & non-ballistic trajectories. A ballistic trajectory trajectory is one in which the spacecraft receives a single instantaneous momentum input, or impulse, which imparts (as kinetic energy) the total mechanical energy required for its subsequent motion, which is shaped only by gravitational forces & momentum ; a non-ballistic trajectory is one which does not fit this description. Within the category of ballistic trajectories, there are four classes : low-energy, first-revolution elliptical, parabolic, & hyperbolic (the parabolic class being trivial). The non-ballistic category comprises two classes, constant-thrust & multiple-impulse.

Ballistic Trajectories

No real trajectory corresponds with the definition of a ballistic trajectory which we have used above. Even a projectile fired from a gun acquires its momentum in a non-zero period of time, & most actual lunar transfers down to the present time have incorporated some form of midcourse correction maneuver. Nevertheless, if the time duration of the initial impulse is brief compared to the transit time, & the velocity increment of the of the initial impulse is large compared to the midcourse maneuver, the ideal ballistic trajectory proves to be a good approximation to the actual behaviour of the spacecraft. The first condition is normally fulfilled by the behaviour of a high-thrust chemical rocket engine, which may burn for a few hundred seconds at the beginning of a transit lasting a hundred hours. The fulfillment of the second condition is largely dependent upon the accuracy of the initial burn, but has proven practicable.

Low-energy Trajectories

Low-energy ballistic trajectories rely on momentum transfer between the spacecraft & celestial bodies to move the spacecraft from an orbit centered on one body to an orbit centered on another body. The classic low-energy trajectory in the Terra-Luna system is based on the "Jacobi integral", a particular solution of the equations of motion in a rotating system. It has been shown The Deep Space One spacecraft provides a good example of a "classical" low-thrust mission. The more recent Smart-1 lunar probe, using a Hall-effect thruster, combined elements of constant-thrust, multiple-impulse, & low-energy trajectories to transfer from geosynchronous transfer orbit (having been "piggy-backed" on a communications satellite) into lunar orbit over the course of more than a year.

Multiple-impulse Trajectories

A multiple-impulse trajectory has several discrete momentum inputs, each of which is brief compared to the total duration of the trajectory. While such a trajectory may be arbitrary in form, for our purposes, the most relevant category begins with a high-eccentricity elliptical orbit & increases the eccentricity by engine burns at successive periapses until the desired apoapsis altitude (substantially the same as that in the first-revolution elliptical ballistic case) is achieved. Owing to the dynamics of a rocket-propelled spacecraft (with its constantly-changing mass) in a gravity field, an engine burn at peripasis adds more velocity than it would in free space. The higher the apoapsis, the greater the advantage. The Chandrayaan-1 probe is currently employing this type of maneuver to transfer into lunar orbit.