# Lunar Transfer Trajectories

## Contents

## 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^{[1]} that, for transit from Terra to Luna, the shape of this trajectory is a high geocentric ellipse, the apogee of which gradually increases under the influence of lunar gravity, until it passes through the Lagrangian point L1 after a period of several months & enters a selenocentric trajectory. Other sets of low-energy trajectories have been developed, based for example on weak boundary capture using solar as well as lunar perturbations, & a family of special trajectories involving the Lagrangian points has become known as the Interplanetary Transport Network, but in general they are similar to the Jacobi-integral trajectory.

### First-revolution Elliptical Trajectories

A first-revolution elliptical ballistic trajectory is, in essence, a single-impulse elliptical orbit about one of the principal bodies, which intersects the position of the other principal body at some point between the first and second periapses. It is obvious that the other body will be reached if the apoapsis altitude of this orbit is the same as the altitude of the body's orbit, & lies in the same direction at the same time. If the apoapsis altitude is greater & the angular orientation is different, an "ascending" & "descending" interception will occur, depending on whether the interception occurs before or after apoapsis. What is not quite so obvious is that interception can also occur in an orbit with an apoapsis somewhat lower than the altitude of the body to be reached, as that body's gravity will draw the spacecraft in (assuming proper angular orientation — otherwise, angular momentum considerations may cause the spacecraft to be ejected in an unexpected direction). The lowest-velocity Terra-to-Luna first-revolution elliptical trajectory requires an initial velocity of approximately 10.9 km/s & occupies a transit time of approximately 5 days. (It is possible to construct an elliptical trajectory which will intercept the body to be reached after a number of revolutions larger than one, but this is seldom called for.)

### Parabolic Trajectories

If the velocity of the spacecraft with respect to the body it is departing is exactly equal to the velocity of liberation of that body, its path with respect to that body will mathematically be a parabola, hence the term parabolic velocity. This ideal case, however, is mathematically strictly true only for a two-body system. Due to inexactness in velocity & the presence of perturbations, the unique parabolic trajectory shades imperceptibly into the family of elliptical trajectories on the one hand, & of hyperbolic trajectories on the other. For this reason it may be considered a trivial case.

### Hyperbolic Trajectories

When the velocity attained by the spacecraft is greater than the velocity of liberation of the body to be departed, the orbit of departure takes the form of a hyperbola. The principal difference from the first-revolution ellipse, other than the greater velocity involved, is that interception must be of the "ascending" type, as the apoapsis & descending leg do not exist.

## Non-Ballistic Trajectories

### Constant-thrust Trajectories

The constant-thrust trajectory is one for which the addition of momentum to the spacecraft is protracted. While the thrust may be continuous during the trajectory, or held at a constant level, it need not be. This mode of operation is generally associated with the use of low-thrust, high-exhaust-velocity propulsion systems such as electric thrusters. These systems are attractive because of the large maneuvers possible with very low propellant mass fractions, but the low acceleration results in major changes to the rules of orbital dynamics ; as an example, the total velocity needed to escape the gravitation of a body increases goes from 1.4 times the body's orbital velocity to 2 times.^{[2]} 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.