Circumlunar Free Return Trajectory

Robert A. Braeunig

The following is an animated graphical representation of a computer simulated circumlunar trajectory with free return to Earth. 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 red 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.

Below the animation is a table giving the elapsed time since translunar injection, the geocentric distance to Earth, the lunar centric distance to the Moon, the spacecraft velocity relative to Earth, and the velocity relative to the Moon. Each four-hour increment is included as well as a few other important points in the 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 orbit perturbations are included.
• TLI delta-v is applied instantaneously.

 Time(hhh:mm:ss) Distance to Earth(km) Distance to Moon(km) Velocity, Earth relative(m/s) Velocity, Moon relative(m/s) Comments 000:00:00 6,563 388,677 10,943 11,629 Translunar Injection, Note 1 004:00:00 63,723 322,544 3,292 2,871 008:00:00 102,344 289,876 2,474 2,050 012:00:00 133,629 264,655 2,074 1,680 016:00:00 160,632 243,133 1,817 1,466 020:00:00 184,660 223,822 1,630 1,328 024:00:00 206,427 205,947 1,485 1,235 028:00:00 226,380 189,041 1,367 1,170 032:00:00 244,825 172,804 1,267 1,126 036:00:00 261,980 157,024 1,181 1,095 040:00:00 278,012 141,553 1,106 1,075 044:00:00 293,052 126,276 1,039 1,063 048:00:00 307,207 111,104 979 1,057 052:00:00 320,567 95,963 925 1,057 056:00:00 333,214 80,788 876 1,062 060:00:00 345,227 65,506 832 1,073 060:17:41 346,089 64,374 829 1,074 Equigravisphere, Note 2 064:00:00 356,694 50,026 794 1,093 068:00:00 367,738 34,191 764 1,134 072:00:00 378,602 17,635 760 1,248 075:32:51 387,587 3,184 998 2,021 Pericynthion, Note 3 076:00:00 387,045 4,086 933 1,845 080:00:00 376,208 21,483 752 1,207 084:00:00 365,388 37,818 766 1,122 088:00:00 354,309 53,551 798 1,087 090:32:27 347,042 64,374 823 1,075 Equigravisphere, Note 2 092:00:00 342,772 68,975 838 1,069 096:00:00 330,668 84,222 883 1,060 100:00:00 317,911 99,379 933 1,056 104:00:00 304,423 114,513 989 1,057 108:00:00 290,123 129,691 1,050 1,064 112:00:00 274,916 144,989 1,118 1,078 116:00:00 258,691 160,499 1,196 1,100 120:00:00 241,313 176,340 1,284 1,132 124:00:00 222,605 192,672 1,387 1,180 128:00:00 202,334 209,717 1,510 1,249 132:00:00 180,173 227,802 1,662 1,349 136:00:00 155,635 247,436 1,859 1,498 140:00:00 127,923 269,484 2,136 1,734 144:00:00 95,528 295,676 2,583 2,154 148:00:00 54,631 330,780 3,594 3,183 151:10:03 6,500 387,407 10,998 11,595 Entry Interface, Note 4

NOTES:

1) Translunar Injection, or TLI, is a propulsive maneuver used to set a spacecraft on a trajectory that will arrive at the Moon. Prior to TLI the spacecraft is in a low circular parking orbit around Earth. In this example, we have assumed a parking orbit altitude of 185 kilometers and a TLI delta-v of 3,150 m/s.

2) The Equigravisphere, or "sphere of influence", is the boundary where the spacecraft trajectory is considered to transition from earth-centered to moon-centered, which NASA defines as being 40,000 statute miles (64,374 kilometers) from the center of the Moon. This arbitrary definition is not to be confused with the commonly held definition of the equigravisphere being all points in space where Earth and lunar gravity are equal, the so-called "neutral point."

3) Pericynthion is the point in the spacecraft's trajectory that is nearest the Moon. For a free return trajectory, the altitude at pericynthion is typically about 100 to 1,500 nautical miles (185 to 2,800 km) - see diagram. The pericynthion altitude in this example is 1,446 kilometers.

4) Entry Interface is the point at which the first effects of Earth's atmosphere are encountered, defined as an altitude of 400,000 feet (121,920 meters). An entry flight path angle of -5.3 to -7.7 degrees is required for a survivable reentry. In this example, the entry angle is -6.46 degrees.

Here is an illustration of the same circumlunar free return trajectory as seen from a Moon-centered perspective: Lunar-Centric View.

Also see the following simulation: Lunar Hybrid Profile with LOI and TEI.