Hybrid Lunar Profile with LOI and TEI

Robert A. Braeunig

The following is an animated graphical representation of a computer simulated hybrid lunar profile with lunar orbit insertion and transearth injection. Spacecraft position is shown relative to a fixed Earth, with Earth being located at the origin of the axes and represented by the blue disc. Location of the spacecraft is shown in four-hour increments with the path traversed since translunar injection represented by the trailing line. The yellow line is the Moon's orbit and the Moon is shown as a gray disc. The distances shown are in units of kilometers and the illustration is drawn to scale. The time scale is one second equals four hours.

A hybrid profile begins as a free return trajectory and transitions to a non-free return following a midcourse transfer maneuver (see diagram). The red line represents the free return portion of the trajectory (this is the same trajectory as modeled here). After completion of the transfer maneuver, the trajectory line changes to blue. As the spacecraft swings behind the Moon, a lunar orbit insertion (LOI) maneuver is performed. For this demonstration four lunar orbits are completed*, which are represented by the magenta line. To set the spacecraft on a trajectory that will return it to Earth, a transearth injection (TEI) maneuver is performed. The green line represents the transearth trajectory.

The following simplifying assumptions have be made:

• The Moon and spacecraft orbits are coplanar.
• The Moon's orbit is circular with a radius equaling 384,403 km.
• Other than Earth and lunar gravity, no perturbations are included.
• All velocity changes are applied instantaneously.

(* Four orbits is an arbitrary selection; any other number could just as easily been used.)

Event Timeline and Propulsive Maneuvers

 Event Time(hhh:mm:ss) Distance to Earth(km) Distance to Moon(km) Delta-V(m/s) Velocity, Earth relative Velocity, Moon relative before after before after Translunar Injection 000:00:00 6,563altitude = 185 km 388,677 +3,150 7,793 10,943 8,489 11,629 Midcourse Transfer 028:00:00 226,380 189,041 -9.74 1,367 1,357 1,170 1,164 Equigravisphere 060:26:14 345,293 64,374 n/a 816 1,068 Pericynthion,Lunar Orbit Insertion 075:33:27 386,224 1,849altitude = 111 km -880.0 1,511 646 2,508 1,628 Transearth Injection 083:24:22 386,252 1,849altitude = 111 km +923.9 605 1,529 1,628 2,552 Equigravisphere 097:23:49 339,598 64,374 n/a 970 1,172 Entry Interface 152:06:16 6,500altitude = 122 km 387,651 n/a 11,007Entry angle = -6.50o 11,639

Compare this simulation's delta-V figures to Apollo 12:

 Maneuver Delta-V (ft/s) Apollo 12 Simulation TLI 10,515 10,335 MCC 61.8 32.0 LOI 2,889.5 2,887.2 TEI 3,042.0 3,031.3

Also see the following simulation: Circumlunar Free Return Trajectory.