Resonant Flybys

ˆ N = rbody× vbody krbody× vbodyk (6.7) ˆ C = ˆT × ˆN (6.8)

Flybys that result in a change only to the pump angle (∆θ) will alter the inertial velocity of the spacecraft with respect to the central body. This means that the semimajor axis, and therefore the orbital period of the post-flyby trajectory, will also be adjusted. Gravity-assists that increase the pump angle are referred to as “pump downs” as the inertial velocity of the spacecraft is decreased, and the semimajor axis reduced. Conversely, “pump up” maneuvers increase the semimajor axis as a result of the larger v∞ decreasing the

pump angle.

A gravity-assist that changes only the crank angle will alter the inclination and eccentricity of the spacecraft’s orbit, but preserve the size of the semimajor axis. If the inclination of the spacecraft’s pre-flyby orbit plane is small (i.e. near the central body’s equator), then the inclination change will be large, while the eccentricity change will not be very significant. The opposite is true if the pre-flyby inclination is large. A cranking flyby at high inclinations will not result in a large inclination change, however, the eccentricity change will be more pronounced. The procedure for performing a DSM trajectory search is summarized in Algorithm 4.

6.6.2

Resonant Flybys

Two bodies are in orbital resonance if there exists an integer relationship between the periods of their orbits about the central body. In other words, p : q resonance exists between a spacecraft and a potential flyby target if the spacecraft completes p revolutions about the central body in the time it takes the flyby target to complete q revolutions. Incorporating resonant transfers into multiple gravity-assist trajectories is beneficial for several reasons.

Algorithm 4 Departure to a DSM 1: inputs are φ, θ, T OF , v_∞+, tDSM 2: procedure DSM 3: compute v+ ∞ 4: compute TICRF RTN 5: compute v_s/c+

6: if departing from flyby then 7: compute required turn angle

8: if required turn angle > maximum possible turn angle then

9: return

10: end if

11: end if

12: propagate spacecraft’s post-flyby state (r+_s/c, v+_s/c) for ∆tDSM1 =⇒ r

− DSM v

− DSM

13: solve Lambert transfer from rDSM to rarrival with transfer fime ∆tDSM2

14: if Lambert solution exists then

15: compute the magnitude of the DSM using Eq. (6.2) 16: save DSM transfer

17: end if 18: end procedure

Resonant orbits set up repeat encounters with a gravitating body, which is useful for queuing future flyby maneuvers, but also for scientific reasons. When investigating a body, such as a planetary satellite, repeat visits to that body are beneficial as they offer additional opportunities to make observations and collect scientific data. For example, repeat encounters with a gravitational body allow a spacecraft to refine its measurements of the gravity field and multiple ground tracks are a necessity for generating surface maps.

Including resonant transfers in a Lambert-based pathfinding algorithm is challenging as Lambert’s prob- lem is singular for transfers of 0 and 180 degrees. Furthermore, in the context of a real ephemeris model, exact orbital resonances do not exist. With this in mind, this study implements near-resonant transfers by inserting a small impulsive maneuver near the midpoint of the transfer in order to target a precise flyby of the target body. The target arrival date is computed using the following equation:

tarr= tdep+ qTbody, (6.9)

where tdepis the flyby departure epoch, and Tbody is the orbital period of the target flyby body, calculated

at the departure epoch. Incorporating near-resonances into the flyby search in this manner is convenient as these trajectories are a subset of the DSM transfers described in the last section. In particular, a near- resonance transfer is one where the flyby pump angle α remains unchanged and the change in crank angle φ can vary on the interval [0, 2π].

repeat-encounter trajectory of Saturn’s moon Enceladus. The tree search problem setup is summarized in Table 6.1.

Table 6.1: Lambert tree search settings for repeat Enceladus encounter sequence.

Parameter Value Start date (JD) 2463598.5 min. departure C3 1.96 km2/s2 max. departure C3 4.0 km2/s2 ∆C3 0.1 km2/s2 ∆φ 1 degree max. revs. 1

min. flyby altitude 20 km max. flyby altitude 100 km

In this example, the spacecraft begins at Enceladus with a relatively low velocity relative to the satellite, and then executes ten consecutive π transfers, however, since linearly interpolated SPICE ephemeris data is used, these transfers are not exact π transfers. The minimum ∆v ten-encounter trajectory identified by the tree search is shown in Fig. 6.8.

Figure 6.8: Repeat Enceladus backflip encounters using the Lambert tree search and linearly interpolated SPICE data. Dots indicate flyby encounters and maneuver locations. Plot is shown in the J2000 ecliptic frame.

The repeat “backflip” encounters with the moon accumulate a significant amount of ∆v during the Lambert search. In order to increase the ephemeris fidelity, and reduce the propellant cost, the tree search results are input to a local-optimizer using the MGAnDSMs transcription. The optimizer is free to adjust the epoch, magnitude and direction of the targeting maneuvers as well as the epochs of the encounters slightly to reduce the cost function (total ∆v). Local optimization using MGAnDSMs reduced the required ∆v for the complete ten encounter sequence from 293 m/s to 195 m/s. An example Enceladus-to-Enceladus transfer from the optimal trajectory is shown in Fig. 6.9.

Figure 6.9: Example Enceladus to Enceladus π transfer locally optimized using the MGAnDSMs transcrip- tion. Ephemeris data is spline-fit SPICE (SplineEphem). The original Lambert transfer used 36.71 m/s of ∆v. Plot shows the trajectory in the J2000 Saturn equatorial frame.