MULTIPLE EARTH AND EARTH LOOPS
The simple two Earth-Mars synodic period cycler. In the circular coplanar model it has a period P=l.348 years, a radius of aphelion R~ = l .15 A U and the V, at Earth is 5.6 M s . For the "Up" transfer, the Earth-Mars transfer is Type I or I1 and the Mars-Earth leg is Type VI. The trajectory departs the Earth with the V, inward of the Earth's velocity vector taking it through a perihelion of about 0.93 AU, crossing the Earth's orbit ahead of the Earth and outward to Mars' orbit. As seen from Figure 1 the transfer to Mars is about 225 degrees and takes a little over nine months. The trajectory continues onward making three complete orbits about the Sun without coming near either the Earth or Mars again until passing through its original starting point on the Earth's orbit for the third time, somewhat behind the Earth and finally encountering the Earth 2/7 of a revolution about the Sun (102.9 deg.) from the starting point. The cycler has made 3 2/7 complete orbits about the Sun while Earth has made 4 2/7. The Earth flyby must now rotate the incoming V, vector, which is outward, to the symmetrically inward orientation to begin the next cycle. Unfortunately, the rotation angle required is approximately 135 degrees and with a V, of 5.65 km/s the Earth can only rotate the V, vector about 82 degrees. Now in the actual Solar System, the orbit of Mars is elliptical with a semi-major axis of 1.524 AU, a perihelion of 1.381 AU and an aphelion of 1.666 AU. Thus the simple Case 1 cycler does not quite reach Mars' average distance from the Sun. It is thus clear that a real world version of the Case 1 cycler would require AV to make up for the inability of the Earth to rotate the V, vector, as well as for the fact that over the course of seven cycles, of two synodic periods each, the Case 1 cycler will not make it to Mars' orbit more than one half of the time. The real value of Case 1 is as a basis for variations that can address these deficiencies.
Modifylng Case 1 by introducing another Earth flyby, approximately six months and 180 degrees after the first, changes the situation somewhat. This six month, 180 degree transfer, or "backflip" trajectory, was first introduced for lunar trajectories by U p h ~ f f . ~ The "Up" trajectory for this version leaves the Earth with a Type I or I1 short transfer to Mars and a Type V transfer back to Earth. This transfer to the first Earth encounter makes 2 11/14 revolutions about the Sun in 3 11/14 years. The Earth flyby then puts the vehicle onto a heliocentric orbit with a period of one year which re-encounters the Earth approximately six months and 180 degrees later, completing the 3 217 revolutions in 4 2/7 years. This second Earth flyby then sends the vehicle on to the next Mars encounter, continuing the cycle. Figure 2 shows this cycler trajectory. Note that the first Earth encounter is in the lower portion of the plot. The backflip trajectory is not shown since its difference from the Earth's orbit is primarily in the z-direction. The second Earth flyby and departure point for the second cycle is indicated slightly left of straight up on the Earth's orbit. In the circular co-planar model the Earth-Mars-Earth trajectory has a period P=l.325 years, a radius of aphelion R~ z l . 4A5 U and the V, at Earth is 4.15 MSF.or Case 2, the transfer does not reach Mars' orbit in the circular co-planar model, but in the real world does reach Mars when Mars is near its perihelion.
The lower V, for Case 2 enables the Earth to rotate the V, vector as much as about 102 degrees, thus easily enabling the first Earth flyby to rotate the incoming V, to the required near polar orientation required for the backflip trajectory outgoing V, as well as the second earth flyby to rotate the near polar incoming V, to the outgoing V, required for the transfer to the next Mars, Thus, although Case 2 has many desirable characteristics, it cannot be used for an entire seven cycles. If fact it will reach Mars for at most two of the seven cycles without propulsive AV to augment the gravity assists.
Modifying Case 2 to introduce a third Earth flyby in addition to the "backflip" adds additional flexibility. This is accomplished by adding a one year Earth-Earth loop either before or after the backflip. The order of the one year loop and the "backflip" can be chosen to best advantage in the real world. The TJp" trajectory for this version leaves the Earth with a Type I short transfer to Mars and a Type I11 or IV transfer back to Earth. This transfer to the first Earth encounter makes 1 11/14 revolutions about the Sun in 2 11/14 years. The Earth flyby the puts the vehicle onto a heliocentric orbit with a period of one year which re-encounters the Earth approximately six months and 180 degrees later and then re-encounters the Earth one year later, or vice versa. The final Earth flyby then sends the vehicle on to the next Mars encounter. Figure 3 shows this cycler trajectory. Again as in Case 2, the backflip trajectory is not seen. The one year Earth-Earth loop is also not shown. In the circular co-planar model the Earth-Mars-Earth trajectory has a period P=l.484 years, a radius of aphelion R~=l .65A U and the V, at Earth is 5.4 km/s.
In this case the transfer reaches an aphelion approximately equal to Mars' aphelion and will thus always cross Mars orbit in the real world. Analysis of Case 3 with the actual ephemerides of Earth and Mars is considered in more detail below.
Modifying Case 1 to introduce one or two one year Earth-Earth loops or even a two year
Earth-Earth loop without a backflip is also possible, it leads however, to much higher
V,'s less desirable characteristics that any of Cases 1,2 or 3, or the Aldrin Cycler for that matter.
A detailed analysis of Case 3 was performed using the actual ephemerides of the Earth and Mars. The trajectories were modeled as Sun-centered point-to-point conics connecting the Earth and Mars flybys. The flybys were modeled as instantaneous V m rotations. This ― V m -matching‖ model gives excellent insight into both the heliocentric and planetocentric trajectories and sufficient accuracy for developing long term trajectory scenarios that can be closely reproduced with fully numerically integrated trajectory models. The Table shows data for a full cycle of seven two-synodic period cyclers (30 years). This should approximately repeat since the Earth and Mars are very nearly at the same inertial positions every 15 years.The choice of one year loop or backflip and whether the backflip is ―north‖ or ―south‖ needs to be made in each case to make best use of the arrival and departure V,‘s to minimize the required bending by the Earth and potential required AV. The Mars flybys (given to the nearest 1000 km) are all at reasonably high altitudes. Whereas in the circular co-planar analysis the Mars flybys are arbitrarily high, in the real world the Mars gravity assist must control the inclination of the heliocentric orbit as well as adjust the energy slightly to properly phase for the next encounter. The Mars V,‘s vary between about 3 km/s and 8 km/s which compares to the value of 5.3 km/s in the circular coplanar case. The Earth V,‘s vary between about 4 km/s and 7.5 km/s which compares to 5.4 km/s.