Solar Gravity Perturbations to Facilitate Long-Term Orbits: Application to Cassini

AAS 07-275

SOLAR GRAVITY PERTURBATIONS TO FACILITATE LONG-TERM ORBITS: APPLICATION TO CASSINI

Diane Craig Davis,* Chris Patterson,* and Kathleen Howell†

The Sun’s gravitational acceleration is not often a focal point in the design of spacecraft trajectories about an outer planet, but the impact of solar gravity is potentially significant, especially on large orbits. Solar gravity is exploited in the design of trajectory options that support possible end-of-life scenarios for the Cassini spacecraft. Combining solar perturbations with Titan encounters and small maneuvers, the spacecraft can reach various long-term orbits, for example, quasi-circular Saturnian orbits beyond the radius of Phoebe. After a number of Saturn-centered revolutions, other trajectories depart the Saturnian system. A possible return to Saturn within 500 years is considered.

INTRODUCTION

Although the Sun’s gravitational acceleration is not often a focal point in the design of spacecraft trajectories about an outer planet, the impact of solar gravity is potentially significant, especially on large orbits. Once a spacecraft is in orbit about a planet, these perturbations can be exploited for trajectory design to yield options that may otherwise be unavailable. As is well known, incorporating the solar force into the model can yield changes in the orbital characteristics in less than one revolution, without a maneuver. For tours of the outer planets, however, the required apoapsis is large and the time of flight for one revolution may be years. Nevertheless, this strategy expands the number of trajectory options when fuel is very limited.

The use of solar gravity in the design of spacecraft trajectories is familiar, particularly in the Sun- Earth and Earth-Moon systems. Solar gravity clearly plays a major and critical role in multi-body regimes and, as a consequence, in the design of missions incorporating libration point orbits such as SOHO [1], WMAP [2], and Genesis [3]. Inclusion of the gravitational force from the Sun has also enabled the development of low energy concepts in transfers between the Earth and the Moon [4, 5]. Spacecraft in large Earth orbits, including Geotail [6], [7], and Wind [8] also exploit solar perturbations very successfully where requirements rely on orientation within a Sun-Earth rotating frame. Investigations of the perturbing effects of solar gravity on a spacecraft in a large orbit about the Earth include the study by Yamakawa et al. [9], who describe the effects of a net perturbing acceleration as it varies by quadrant in a rotating frame. While solar gravity is clearly an important force in the dynamic models for all missions, the analyses and applications mentioned previously focus specifically on incorporating the Sun’s perturbing force as part of a design methodology to adjust the orbital characteristics of a spacecraft. Results from studies of Sun- planet systems also apply in planet–moon systems. For example, Villac et al. [10] discuss Jupiter’s perturbing effect on a Europa orbiter as a method for facilitating a plane change, using natural forces to reduce ΔV costs.

In the astronomy community, investigations of the influence of solar perturbations on the orbits of moons and natural satellites can lend insight into the behavior of spacecraft in orbit about one of the outer planets. For example, Hamilton and Burns [11] investigate zones of stability for orbital debris about asteroids. The authors describe the components of acceleration due to solar gravitational perturbations as viewed in the rotating frame. In regions where these perturbations are significant, the existence of stable, unstable, and quasi-periodic orbits is discussed. The time histories of the orbital elements for sample orbits

*Ph.D. Student, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall, 701 W. Stadium Ave., West Lafayette, Indiana 47907-2045. †Hsu Lo Professor of Aeronautical and Astronautical Engineering, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall, 701 W. Stadium Ave., West Lafayette, Indiana 47907-2045; Fellow AAS; Associate Fellow AIAA. 1 are also analyzed. Other studies of the behavior of natural satellites include Hamilton and Krivov [12], a further investigation of the stability of natural satellites in various orbits about asteroids, and Sheppard [13], who discusses the dynamics of irregular moons, particularly those of the outer planets.

The current work is focused on solar gravity as a significant perturbing force acting on a spacecraft that is orbiting Saturn. Specifically, some general concepts are investigated to support possible end-of-life options for the Cassini spacecraft. After an impressive tour of Saturn, and as the Cassini mission winds down, safe disposal of the spacecraft is necessary. Because Cassini is already in orbit about Saturn, and because the design of the extended phase is close to completion, the initial conditions for the final end-of-life stage of the mission are constrained. In addition, the available ΔV is limited. Thus, one strategy for the design of the final phase combines solar perturbations with Titan encounters and small maneuvers to enable the spacecraft to reach various long-term orbits within these constraints.

BACKGROUND

Definition of Rotating Frame and Solar Quadrants

The impact of the Sun as a perturbing force can be significant on large orbits, even at Saturn. The direction of the perturbing acceleration that originates with solar gravity depends on the orientation of the spacecraft’s orbit relative to the Sun and Saturn. Thus, to investigate this force and exploit it for trajectory design, observations in terms of a coordinate frame that rotates with Saturn about the Sun are insightful. A Saturn-centered rotating frame is defined to facilitate the analysis. Let the xˆ − yˆ plane represent Saturn’s orbital plane. The xˆ -axis is fixed along the Sun-Saturn line and the yˆ -axis is perpendicular to the xˆ -axis, defined as positive in the general direction of Saturn’s orbital velocity. Four quadrants, centered at Saturn, are defined in the rotating frame, and appear in Fig. 1. The quadrants are defined in a counterclockwise fashion, with quadrant I on the far side of Saturn and leading Saturn in its orbit. When the spacecraft orbit is viewed in this rotating frame, its orientation is defined by the quadrant that contains the orbit apoapsis. In the current work, the angle of orientationφ within each quadrant is defined as the angle from the Sun- Saturn line. The positive sense of the angle in each quadrant is defined in Fig. 1.

Fig. 1. Quadrants and orientation angles as defined in the rotating frame.

As discussed by Yamakawa et al. [9], the direction of the perturbing acceleration acting on the spacecraft is dependent upon the orientation of the orbit within a quadrant. For insight into the perturbing accelerations, assume a rotating frame consistent with the circular restricted three-body problem. Then, in the rotating frame, along with the direct acceleration due to the central body (Saturn), three perturbing accelerations must be considered: the solar gravitational term and the centripetal acceleration, together known as the tidal acceleration, and the Coriolis acceleration. The total acceleration acting on a spacecraft orbiting Saturn as viewed in the Sun-Saturn rotating frame is written

rd 2r − GM ⎛ − r r ⎞ sat−sc = sat r + GM ⎜ sun−sc − sat−sun ⎟ − 2ω ×r v − ω × ()ω × r (1) 2 3 sat−sc sun ⎜ 3 3 ⎟ sat−sc dt rsat−sc ⎝ rsun−sc rsat−sun ⎠

2 where the position vectors r are defined in Fig. 2, r v is the velocity of the spacecraft relative to Saturn as observed in the rotating frame, and ω is the angular velocity of the Sun-Saturn system. In terms of the relative velocity, the expression for the Coriolis acceleration is 2 ω×r v = 2ω (r yxˆ−rxyˆ) (2) For a prograde orbit, the Coriolis acceleration term in equation (1) is always directed outward and perpendicular to the spacecraft velocity vector at any point along the path. The magnitude of the Coriolis acceleration is always largest near periapsis. The centripetal acceleration is evaluated as

2 ω × (ω × r ) = ω (− xsat−sc xˆ − ysat−sc yˆ) (3) where, assuming Keplerian motion of the primaries, the magnitude of the angular velocity is

2 GM sun + GM sat ω = 3 (4) rsun−sat

If the direct acceleration due to Saturn and the term corresponding to Coriolis acceleration are removed from the expression for the total acceleration, the tidal acceleration remains. It is written as

tidal d 2 r GM GM sat−sc = sun − x xˆ − y yˆ + sun x xˆ 2 3 ()sun−sc sun−sc 3 ()sun−sat dt rsat−sc rsun−sat (5) GM GM + sun x xˆ + y yˆ + sat x xˆ + y yˆ 3 ()sat−sc sat−sc 3 ()sat−sc sat−sc rsun−sat rsun−sat

Fig. 2. Position vectors.

The direction of the tidal acceleration appears in Fig. 3. In Fig. 3a, the mass of the smaller primary has been adjusted and the yˆ -component of the acceleration has been exaggerated to clarify the detail of the acceleration pattern. In the actual Sun-Saturn system, in the region of interest surrounding Saturn, the tidal acceleration is heavily dominated by the xˆ -component and is directed outwards from the yˆ -axis, as indicated in Fig. 3b. The symbol Rs denotes the radius of Saturn. In the rotating frame, Saturn is stationary and, at the origin, the tidal acceleration is equal to zero; the centripetal acceleration term (negative in equation (1)) is equal in magnitude and opposite in direction to the solar gravitational acceleration. Along the xˆ -axis but closer to the Sun (see the red X in Fig. 3), the solar gravitational acceleration is greater than the centripetal acceleration, and the tidal acceleration is directed towards the Sun. Along the xˆ -axis but on the far side of the Sun (see the blue X in Fig. 3), the centripetal acceleration term is of greater magnitude than the solar term, and the tidal acceleration is directed away from the Sun. Along the yˆ -axis, the xˆ -components of the two forces nearly cancel, and the tidal acceleration is directed away from Saturn (see the orange X’s in Fig. 3.)

X Quadrant II X Quadrant I Quadrant II Quadrant I ˆ xˆ -axis To Sun x -axis To Sun X X X Primary X Saturn

Quadrant III Quadrant IV Quadrant III X Quadrant IV X

Fig. 3a. Solar tidal acceleration as viewed Fig. 3b. Solar tidal acceleration as viewed in the rotating frame; artificially large primary in the rotating frame; actual Sun-Saturn and yˆ - component exaggerated. system.

In quadrants I and III, the effects are greatest near apoapsis and the direction of the perturbing tidal acceleration generally opposes the direction of motion in a prograde orbit. In quadrants II and IV, on the other hand, at apoapsis the net perturbing acceleration is in the same direction as the motion along a prograde orbit. As a result, the tidal effects are similar in diagonal quadrants. For example, solar gravity perturbations tend to circularize prograde orbits lying in quadrants II and IV and to elongate prograde orbits with an apoapsis in either quadrant I or quadrant III.

Effects of Solar Gravity by Quadrant

The influence of solar gravity on the characteristics of an orbit about Saturn must be quantified. This is necessary to incorporate solar gravity as part of any design algorithm. It is particularly challenging for this application since any future trajectory to be considered for the Cassini spacecraft is constrained by the actual orbit of the spacecraft as well as the ΔV available for maneuvers. To exploit solar perturbations on the orbit in support of the mission objectives, a large orbit in the desired quadrant must be achieved. This can be accomplished via a series of Titan flybys. Thus, the initial state for the analysis in the current work is a post-flyby state in the vicinity of Titan. This post-flyby state possesses a particular V∞ with respect to Titan, a quantity that is used to characterize the orbit about Saturn. The quantities that are significant for design purposes are the orbital elements. Characteristics of the post-flyby orbit, including V∞ and the orbital elements with respect to Saturn, are then influenced by solar gravity during the subsequent journey around the planet. A comparison between the characteristics of the spacecraft orbit from one revolution to the next demonstrates the solar influence. Consider a prograde orbit large enough to be impacted significantly by the Sun but sufficiently small such that the solar gravitational perturbations do not cause the orbit to become retrograde or to escape. If the apoapsis originally lies in quadrant I or III, solar gravity will lower the periapse radius, increase eccentricity, and increase V∞ with respect to Titan at a subsequent encounter. Alternatively, if apoapsis lies in quadrant II or IV, solar gravity will raise the periapse radius, decrease eccentricity, and decrease V∞ with respect to Titan. These results are summarized in Table 1.

For a given orbit at a specified orientation angle, the solar gravitational perturbations have maximum effect when the orbit lies in the ecliptic plane. Note that Saturn’s equatorial plane is inclined at about 26.7º with respect to the ecliptic. Also, within each quadrant, solar perturbations are at a maximum

Table 1. Effects of solar gravitational perturbations, relative to the previous orbit. Quadrants I and III Quadrants II and IV Semi-major axis decreases increases Periapse radius decreases increases Eccentricity increases decreases

V∞ wrt Titan increases decreases

when apoapsis lies at approximately 45º from the Sun-Saturn line (see Fig. 1), although the precise value varies as the period of the orbit changes. Especially for large orbits, the orientation of the Sun-Saturn line with respect to the spacecraft orbit line of apsides changes due to Saturn’s motion about the Sun while the spacecraft is in transit about the planet, affecting the value of the optimal orientation. For example, consider an orbit in the ecliptic plane with a periapse radius of 5.56 x 105 km, or 9.23 Rs (Saturn radii), and with a period of 173 days. Assuming that the orbit lies in quadrant I, solar perturbations are maximized when apoapsis is oriented at 45º. When the orbit is modified and the period is increased to 756 days, however, the maximum solar perturbations occur when apoapsis lies at 42.3º. Similar effects are reproduced in the other quadrants. The changes in semi-major axis, periapse radius, and eccentricity from one revolution to the next as functions of the angle φI in quadrant I for a 957-day orbit appear in Fig. 4.

Of course, once φI > 90º, the orbit has shifted into another quadrant. Thus, the quantities increase and decrease consistent with the quadrant. This information is utilized in Yam et al. [14] to facilitate a Saturn impact from a quadrant III orbit.

Fig. 4. Changes in orbital elements from one revolution to the next

as a function of quadrant angle φI : 957-day orbit.

The solar effects on an orbit obviously increase as the apoapse radius increases. Thus, for long- period orbits, the effects of solar gravity can be significant. As is well known for spacecraft or natural objects in orbit about a planet, as the size of the orbit increases and the orbit becomes irregular, a prograde orbit may at first remain stable. Escape criteria are available from both analytical and numerical investigations for prograde circular orbits [11,15,16]. Let rH be the Hill sphere radius with a value for the 7 Sun-Saturn system that is equal to the following: rH =×6.518 10 km = 1081.5 Rs . Incorporating only tidal accelerations, it is suggested from various sources that the critical radius for prograde circular orbits, that is, the lower limit for escape, is ~ .49 rH. This value is equal to 530 Rs, i.e., larger than the orbit of

5 Phoebe, defined in terms of its semi-major axis, i.e., arPhoebe∼ .20 H (= 214Rs) . As the size is further increased, however, the orbit enters a chaotic region, and finally, sufficiently large orbits are unstable and escape. An orbit that is not circular may initially be sized and oriented such that the semi-major axis is below the critical value, but evolves due to solar perturbations. Aided by the outward Coriolis acceleration term, if an eccentric prograde orbit evolves to a sufficient energy level, it also escapes. Depending on the characteristics of the unstable orbit, however, escape can occur immediately, after 100 years, after 10,000 years or longer.

An example of a trajectory affected by solar gravity appears in Fig. 5. Immediately after the final Titan encounter, the quadrant II orbit, in blue, is initially characterized by a periapse radius of 5.56 x 105 km (9.23 Rs) and a Keplerian period of 1478 days. After one revolution of Saturn, solar gravity has raised the periapsis to a radius of 2.3550×107 km (391 Rs). A maneuver of 120 m/s is added at the new periapsis, which occurs about 8.25 years after the initial state and is marked in Fig. 5 as a black dot. The spacecraft then continues in its orbit about Saturn, as seen in red in Fig. 5, and is propagated for 90 years. The resulting large orbit continues to be affected by solar gravity, and obviously evolves over time, but remains bounded in its orbit. In the rotating frame, the orbit shifts back and forth along the Sun-Saturn line as the trajectory passes through the quadrants, behavior that is very characteristic of a prograde orbit in this regime [11, 12]. Qualitatively, the orbital evolution, as observed in the rotating frame in Fig. 5, is explained by the acceleration terms in equation (1). The orbit is eccentric immediately after the maneuver and quickly shifts into quadrant I. The path is then elongated by the tidal acceleration term. As the orbit becomes more eccentric, the Coriolis term increases in importance. The Coriolis acceleration becomes very large near periapsis and this term acts to circularize the orbit as the Coriolis effect overwhelms the tidal acceleration. The elongation slows and the orbits contracts. As the orbit becomes more circular, tidal effects again eventually dominate and the cycle repeats. Thus, the perturbing accelerations continually modify the semi-major axis and eccentricity. The time histories of various osculating orbital elements associated with the post-maneuver (red) trajectory appear in Fig. 6. Certain frequencies are obvious in semi-major axis and eccentricity. A short period, corresponding to the spacecraft’s orbital period (ranging from 5.6 to 7.4 years, with an average of 6.6 years), clearly appears. A long period, corresponding to Saturn’s orbital period (29.4 years), is also clearly visible in semi-major axis and eccentricity. It requires approximately 2 periods of Saturn (~ 60 years) for the apoapsis to rotate through all four quadrants. Inclination displays a period of approximately twice the orbital period of the spacecraft. The regular appearance of the orbit as viewed in the rotating frame is enhanced when the two dominant frequencies are commensurate.

Fig. 5. Trajectory in the inertial (left) and rotating (right) frames.

Fig. 6. Osculating orbital elements as a function of time for the post-maneuver trajectory. Maneuver occurs at t = 8.25 years.

All numerical simulations are based on the relative equations of motion as formulated in an inertial frame. The trajectory propagation is accomplished using MATLAB and its ode45 integrator. Tolerances (absolute and relative) are fixed at 1×10-12. Initial conditions and GM values for the Sun and Saturn are extracted from the JPL de408 ephemeris. Saturn’s orbit about the Sun is integrated together with the spacecraft state in the Saturn orbital frame and equinox of epoch inertial reference frame.

LARGE QUASI-CIRCULAR ORBITS ABOUT SATURN

For various scientific reasons, one potential final orbit to keep the Cassini spacecraft within the Saturnian system is a quasi-circular orbit beyond the irregular, retrograde orbit of Phoebe. Phoebe’s semi- major axis is 1.3×107 km (214 Rs), which corresponds to a period of 1.5 years, and its orbit is inclined at ~173º with respect to Saturn’s ecliptic plane. A transfer from some likely spacecraft trajectory at end-of- mission into a long-term trajectory outside of the orbit of Phoebe cannot be accomplished solely with Titan encounters and ΔV, since the required maneuvers are prohibitively large. Instead, solar perturbations are employed as an aid to reduce the required ΔV. In quadrants II and IV, solar gravity decreases the eccentricity of a prograde orbit and raises its periapsis. Thus, solar gravity perturbations may be exploited to design large orbits in these quadrants. Given the characteristics from the previous section, different maneuver times and magnitudes will deliver a range of Saturn orbits.

Effects of Initial Sun-Saturn Orientation

The impact of the perturbing acceleration due to the Sun depends on the initial orientation of the orbit in the Sun-Saturn frame. Titan encounters can deliver the vehicle into any quadrant. Assume that a sample prograde orbit is defined in quadrant II. One such orbit is initially inclined at 26.4º with respect to Saturn’s ecliptic plane, thus, approximately in Saturn’s equatorial plane. The Keplerian period, or the period as calculated from the osculating semi-major axis, is 1478 days. With an initial state near Titan’s orbit, the spacecraft possesses an initial periapse radius of 5.56 x 105 km, or 9.23 Rs. The spacecraft subsequently passes through apoapsis after about 3.2 years. Although there is no ΔV applied at apoapsis, the characteristics of the orbit noticeably change in its vicinity. Solar gravitational effects raise the next periapsis significantly. At this second periapsis, the tangential ΔV that is required to circularize the orbit in a Keplerian model is applied to the trajectory. Solar gravity continues to act on the large circularized orbit and yields a trajectory that is circular in nature but not perfectly repeating. One example of such a long- term trajectory appears in Fig. 7. It is propagated here for 500 years. The orientation angle φII

● Saturn To Sun To Sun * Post-Flyby state (at ΔV) (at apoapsis) x Apoapsis ● ΔV ▬ Orbit of Phoebe ▬ Pre -ΔV traj ▬ Post-ΔV traj → Saturn-Sun direction at time of apoapsis → Saturn-Sun direction at time of ΔV

Fig. 7. ΔV = 172 m/s, trajectory propagated for 500 years. Initial time: 18-Aug-2009 8:52:14.

corresponding to apoapsis is 32.8º, relative to the Sun-Saturn line in quadrant II. The ΔV applied in this case is 172 m/s. The maneuver occurs about 8.25 years after the final (inbound) Titan flyby, at a new periapse radius of 2.3550×107 km (391 Rs). The period of the resulting near-circular orbit is approximately 3.7 years and a ~ .36 rH. Over 500 years, the radius remains fairly tightly bounded.

The Titan post-flyby conditions will yield different point solutions if the orientation angle at apoapsis, φII , is rotated. By changing this orientation, the magnitude of the solar perturbations also increases or decreases. The resulting periapse radius and required ΔV can therefore be adjusted simply by rotating an orbit within a quadrant. This reorientation is achieved by shifting the Julian Date of the post- flyby state, and therefore the relative positions of the Sun and Saturn. If the start date associated with the post-flyby state (and hence the orientation of apoapsis relative to the Sun and Saturn) is shifted by just 23 days, the different orientation within quadrant II produces a larger radius at the second periapsis; once the ΔV is applied, a larger periodic orbit results. The trajectory is propagated for 500 years and appears in Fig.

8. The quadrant II apoapse orientation angle φII in this case is 32.4º from the Sun-Saturn line, and the second (new) periapse radius is 2.4924×107 km (414 Rs). Note that the orientation angle has shifted by less than half a degree. The velocity of the spacecraft at periapsis is closer to the velocity of a circular orbit at this radius, so the required ΔV, 150 m/s, is lower than in the previous example. This ΔV still delivers the spacecraft to a circular orbit in a Keplerian model such that ar∼ .38 H . The ΔV is implemented 8.6 years after the final Titan flyby, a slightly longer wait than in the previous example. The period of the resulting orbit is 4.0 years. However, the radius of this trajectory is more variable. This is due to the greater solar effects acting on the larger orbit. Note that some structure is visible in this trajectory — two small regions exist near the xˆ -axis through which the trajectory has not passed after 500-years of propagation.

Effects of ΔV Adjustment

The orbits in the above discussion are achieved by application of a tangential ΔV at the new, higher periapsis. The magnitude of the ΔV’s in Figs. 7 and 8 equal the value required to circularize the orbit in a Saturn-only force model. Different orbits are produced by changing the epoch of the initial

● Saturn To Sun * Post-Flyby state (at ΔV) To Sun (at apoapsis) x Apoapsis ● ΔV ▬ Orbit of Phoebe ▬ Pre -ΔV traj ▬ Post-ΔV traj → Saturn-Sun direction at time of apoapsis → Saturn-Sun direction at time of ΔV

Fig. 8. ΔV = 150 m/s, trajectory propagated for 500 years. Initial time: 26-Jul-2009 8:52:14. conditions. If, on the other hand, the starting epoch is held constant and the magnitude of the ΔV is adjusted, the resulting trajectories will also differ.

Consider the trajectory appearing in Fig. 7. The ΔV added in this case is 172 m/s. A lower ΔV, still applied tangentially at periapsis, yields an initially eccentric trajectory that remains beyond Phoebe’s orbit for the full 500 years. When the ΔV is decreased to 150 m/s, for example, the resulting trajectory possesses similar characteristics and appears in Fig. 9. The main difference between the trajectories in Figs. 7 and 9 is the increased variability of the Saturn-spacecraft distance. However, this variability is still

Fig. 9. ΔV = 150 m/s, trajectory propagated for 500 years. Initial time: 18-Aug-2009 8:52:14.

9 less than that in the orbit in Fig. 8, also created using a 150 m/s ΔV, but with a different start date and higher initial rp. Again starting with the orbit in Fig. 7, and decreasing the ΔV further to 120 m/s, however, significant changes occur in the orbit. An approximate resonance is achieved between the motion of the line of apsides in the spacecraft orbit about Saturn and Saturn’s orbital period. This relationship results in a quasi-periodic spacecraft orbit that appears ‘locked’ in a 13:2 near-commensurability with Saturn’s orbit about the Sun. A 500-year propagation appears in Fig. 10. This particular trajectory does descend below the orbit of Phoebe, but the closest approach to Phoebe in the 500-year time frame is 7.65×105 km (12.7 Rs). When the orbit is plotted in the Sun-Saturn rotating frame in Fig. 11, an elongation along the Sun- Saturn line is evident, consistent with other analyses [11,12]. Along astronomical timelines, this elongation

● Saturn To Sun (at ΔV) To Sun * Post-Flyby state (at apoapsis) x Apoapsis ● ΔV ▬ Orbit of Phoebe ▬ Pre -ΔV traj ▬ Post-ΔV traj → Saturn-Sun direction at time of apoapsis → Saturn-Sun direction at time of ΔV

Fig. 10. ΔV = 120 m/s, trajectory propagated for 500 years. Initial time: 18-Aug-2009 8:52:14.

● Saturn * Post-Flyby state ● ΔV ▬ Pre -ΔV traj ▬ Post-ΔV traj

Fig. 11. ΔV = 120 m/s, trajectory propagated for 500 years, rotating view. Initial time: 18-Aug-2009 8:52:14.

10 can signal an eventual instability. However, in the 500-year time frame considered in this work, this irregular orbit remains bounded at Saturn. In fact, after a propagation of over 2200 years, the orbit characteristics are essentially the same.

A similar type of trajectory is produced by application of a slightly larger maneuver, that is, a ΔV of 137 m/s. The post-Titan orbit, orientation angle, and maneuver date remain fixed. This trajectory, plotted in Fig. 12, remains significantly above the orbit of Phoebe for the 500 years of propagation. The orbit also displays the quasi-periodic, near-commensurate behavior between the period of rotation of the line of apsides of the spacecraft orbit and Saturn’s orbital period.

To Sun To Sun ● Saturn (at ΔV) (at apoapsis) * Post-Flyby state x Apoapsis ● ΔV ▬ Orbit of Phoebe ▬ Pre -ΔV traj ▬ Post-ΔV traj → Saturn-Sun direction at time of apoapsis → Saturn-Sun direction at time of ΔV

Fig. 12. ΔV = 137 m/s, trajectory propagated for 500 years. Initial time: 18-Aug-2009 8:52:14.

Sample Orbit that Escapes

For the particular baseline orbit from Fig. 12, if the applied ΔV is sufficiently decreased (thus, sacrificing any commensurate motion), the spacecraft will escape from its Saturn-centered orbit within the 500-year time frame. One such example, propagated for 255 years, appears in Fig. 13. A ΔV of 95 m/s is applied tangentially at periapsis. After approximately 250 years, the spacecraft departs the vicinity of Saturn in the direction of the inner solar system. When the orbit in Fig. 13 is viewed in a Sun-Saturn rotating frame (Fig. 14), it is apparent that the elongation along the Sun-Saturn line is increasing as the orbit evolves. The ends of the oval that bounds the trajectory arcs become more sharply defined over time tending to “flatten”. Eventually, the elongation is sufficient to cause escape. These low energy escapes always occur close to the xˆ -axis in the rotating view and pass by the L1 or L2 collinear libration point. The spacecraft in this example departs the vicinity of Saturn through the gateway near the Sun-Saturn L1 7 libration point. Note that, at the time of escape, semi-major axis aa.r=×4.267 10 km = 708 Rs (∼ 65H ) . Observation from numerical studies here and in other investigations [11-13,16] suggest that a reasonable expectation of imminent escape exists for a prograde orbit in this regime when the semi-major axis evolves to a value such that . ar≥ .65 H

11 To Sun ● Saturn (at ΔV) * Post-Flyby state To Sun (at apoapsis) x Apoapsis ● ΔV ▬ Orbit of Phoebe ▬ Pre -ΔV traj ▬ Post-ΔV traj → Saturn-Sun direction at time of apoapsis → Saturn-Sun direction at time of ΔV

Fig. 13. ΔV = 95 m/s, trajectory propagated for 250 years. Initial time: 18-Aug-2009 8:52:14.

● Saturn * Post-Flyby state ● ΔV ▬ Pre -ΔV traj ▬ Post-ΔV traj

Fig. 14. ΔV = 95 m/s, trajectory propagated for 255 years, rotating view. Initial time: 18-Aug-2009 8:52:14.

LONG-TERM HELIOCENTRIC ORBIT

Because the hardware on the Cassini spacecraft possesses a limited lifespan, a ΔV that is eight or more years after the last Titan flyby may not be feasible; it may not be possible to rely on the spacecraft operational capability for such an extended period of time. If no ΔV is applied, the spacecraft is not likely to remain in the Saturnian system for the full 500 years; solar gravity may be sufficient to cause escape. Several solutions are examined, some of which remain in a heliocentric orbit for 500 years without

12 reencountering Saturn or approaching other planets. Of course, with sufficient information, it is also possible to efficiently design a planned escape.

One example appears in Fig. 15. The initial conditions are the same as those used to create the trajectories in Fig. 7 and Figs. 9-12. The orientation angle of the initial apoapsis is 32.8º in quadrant II. In this case, however, no ΔV is applied to the trajectory. After approximately 20 years in a large orbit about Saturn, the spacecraft departs the neighborhood of the planet. In the rotating view, the trajectory exits the Saturnian system towards the outer solar system through the gateway near the Sun-Saturn L2 libration point. This is not unexpected since the osculating semi-major axis has quickly evolved to a value such that a ~ .69 rH. This trajectory is then propagated for 500 years including the gravitational effects of the Sun, Saturn, Titan, Hyperion, Iapetus, Phoebe, Jupiter, Uranus, and Neptune, in a true ephemeris model using the JPL de408 and sat242l data. The integrator used for the heliocentric propagations is part of the Purdue University Generator-C software package [17] and utilizes a Runge-Kutta-Verner 8/9 procedure with an absolute tolerance value equal to 1×10-11. The heliocentric trajectory appears in Fig. 16 in a projection onto

● Saturn * Post-Flyby state ▬ Orbit of Phoebe ▬ Spacecraft traj

Fig. 15. Initial conditions propagated for 20.5 years with no ΔV. Initial time: 18-Aug-2009 8:52:14.

Uranus Saturn

Jupiter

Fig. 16. Initial conditions propagated for 500 years with no ΔV, heliocentric trajectory. Earth equator and equinox of J2000 frame.

13 the ecliptic plane. Once the spacecraft leaves the neighborhood of Saturn, it remains between the orbits of Saturn and Uranus without collisions: the closest subsequent approach to Saturn is about 1.7×108 km (1.1 AU). Note that the number of revolutions within the Saturnian system and the resulting specific heliocentric orbit depend on the solar perturbation through the size of the orbit about the planet as well as the quadrant and orientation of the orbit prior to the escape.

By altering the initial conditions, the length of time that the spacecraft remains in the Saturnian system can be changed, as is clear from the example in Figs. 17 and 18. Recall that modifying the date of the post-flyby state shifts the orientation angle of the orbit. This rotation of the orbit modifies the solar perturbations and the subsequent motion. Given the original trajectory in Fig. 15, shifting the orientation angle corresponding to the first post-Titan apoapsis to 31.3º (a decrease of 1.5 ), retains the spacecraft in an orbit about Saturn for just over 65 years. Not surprisingly, this trajectory is sensitive to perturbations and, thus, is model-dependent. The trajectory plotted in Fig. 17 is the result of a propagation over 65.75

● Saturn * Post-Flyby state ▬ Orbit of Phoebe ▬ Spacecraft traj

Fig. 17. Initial conditions propagated for 65.75 years; no ΔV. Initial time: 4-May-2009, 22:34:11.

Uranus

Saturn

Jupiter

Fig. 18. Initial conditions propagated for 500 years with no ΔV; heliocentric trajectory. Earth equator and equinox of J2000 frame.

14 years in a Sun-Saturn ephemeris gravity model. The spacecraft escapes towards the inner solar system through the gateway near the Sun-Saturn L1 libration point. Note the relatively fast elongation of the orbit along the xˆ -axis in the rotating view. A 500-year propagation initiated from the same set of initial conditions is completed using an ephemeris gravity model including the effects of the Sun, Saturn, Titan, Hyperion, Iapetus, Phoebe, Jupiter, Uranus, and Neptune. The heliocentric trajectory is plotted in Fig. 18. The spacecraft remains between the orbits of Jupiter and Saturn throughout the 500-year propagation. Once escaped, the closest subsequent approach to Saturn is about 2.2 x 108 km (1.5 AU).

From Figs. 15 and 17, it is apparent that slight changes in the initial conditions can modify the length of time that the vehicle remains in the Saturnian system before the low energy escape. The subsequent heliocentric orbits differ but are bounded between the orbits of the outer planets, that is, Jupiter and Saturn or Saturn and Uranus.

IMMEDIATE ESCAPE TO HELIOCENTRIC ORBIT

By adjusting the orientation and size of the spacecraft orbit about Saturn, the trajectory can be directed toward one of the libration point gateways for an immediate escape to a heliocentric orbit with less time spent in the vicinity of Saturn. These escapes are initially characterized by a periapsis at or below the orbit of Titan and a distant apoapsis oriented near the middle of either quadrant II or quadrant IV. An orientation in quadrant II allows an immediate escape, but with the lowest possible energy, through the L1 gateway, while orientation in quadrant IV escapes via L2. Orbits large enough to escape immediately will 6 generally possess Keplerian apoapse radii larger than 51 x 10 km (846.2 Rs = .782 rH). The mechanism for this escape is demonstrated in Fig. 19. In the figure, the orbits are propagated first in a two-body conic model (red) and then in a three-body circular restricted Sun-Saturn model (blue). The results are viewed in the rotating frame along with the quadrant numbers and the region of exclusion shaded in yellow. The region of exclusion is defined as the region of space that cannot be reached by an orbit with a given value of Jacobi Constant in the circular restricted three-body problem. In the figure, the region of exclusion possesses an opening in the area around L2 that defines the gateway allowing escape. A similar opening exists around L1. For smaller orbits with higher values of Jacobi Constant, this opening is closed and escape is prevented. In Fig. 19, the orbits have zero inclination with respect to Saturn’s orbital plane. The 6 smallest orbit initially possesses an apoapsis radius of only 9.2 x 10 km (152.65 Rs = .14 rH). Thus, the solar perturbing effect on apoapsis distance is not immediately noticeable. The initial apoapse radius

Fig. 19. The large orbit in the middle of quadrant IV escapes due to the solar perturbation.

15 6 corresponding to the next largest orbit is 37 x 10 km (613.9 Rs = .56 rH) and the effect of the solar perturbation on the orbit is apparent, yet the perturbation does not immediately cause an escape. The 6 largest orbit possesses an initial apoapsis radius of 54 x 10 km (896 Rs = .83 rH) in the conic model and achieves an escape via the L2 gateway when the solar perturbation is included.

Additional examples of immediate escape appear in Figs. 20 and 21. Three trajectories are plotted in the rotating frame in Fig. 20 and all are escaping via the L1 gateway in quadrant II. Each case is

Periapse Apoapse φII ( x 106 km) ( x 106 km) ▬ 0.55572 54.5169 35o ▬ 0.55755 59.8746 39 o ▬ 0.55893 64.7085 43 o

Fig. 20. Immediate escapes via the L1 gateway.

Fig. 21. Conditions for immediate escape via L1 propagated for 500 years with no ΔV; heliocentric trajectory; Earth equator and equinox of J2000 frame.

16 generated with a series of Titan flybys to achieve the appropriate initial state from an assumed Cassini extended mission end-state. The initial state corresponds to a semi-major axis value such that the apoapsis distance rraH> .8 in each case. The orientation angles are all close to the center of the quadrant; thus, the perturbing influence from solar gravity is near its maximum value. A 500-year propagation is completed in an ephemeris model including the gravity of the Sun, Jupiter, Saturn, Uranus, Neptune, Titan, Hyperion, Iapetus, and Phoebe. The three escaping trajectories from Fig. 20 are then plotted in a heliocentric frame in Fig. 21. All three depart the Saturnian system via L1 and the spacecraft subsequently remains between Jupiter and Saturn without collisions. The closest subsequent approach to Saturn is about 1.7×108 km (1.1 AU) for the orbit in red, about 3.4×108 km (2.3 AU) for the orbit in magenta, and about 1.4×108 km (.94 AU) for the orbit in blue.

The heliocentric orbits in Figs. 16, 18, and 21 do not return to the vicinity of Saturn. However, direct departure via a libration point gateway allows for the possibility of a return through the same gateway. A maneuver far beyond Saturn’s moons may prevent return. Analysis in the circular restricted three-body model lends insight into this strategy. Since a maneuver alters the shape of the region of exclusion, the existence and characteristics of the libration point gateways can be adjusted. A strategy for escaping without a return then involves a coast phase to reach a location past the libration point gateway and then, once the spacecraft has completely left the vicinity of Saturn, adding a maneuver to modify the shape of the region of exclusion such that a return to the vicinity of Saturn is no longer possible within the context of this model. The cost of such a maneuver depends on the maneuver time, that is, the further outside the Saturnian system (i.e., the longer the delay before the maneuver), the less costly the ΔV. For the escaping trajectory from Fig. 19, the cost of a maneuver to close the L2 gateway as a function of time after the final Titan encounter appears in Fig. 22. A significant amount of time is required to coast beyond the gateway itself. With a limit of 100 m/s or less, it is necessary to wait at least 10.2 years to move sufficiently far such that a maneuver will close the gateway and prevent a return.

Fig. 22. Maneuver to guarantee no Saturn return as a function of time-of-flight along the escape arc.

In the event that a particular escape is highly likely to return to the vicinity of Saturn, a compromise is possible that involves a timelier maneuver that does not close the gateway but delays the chance of return. In Fig. 23, a trajectory (green) escapes via the L2 gateway in the rotating frame. It returns in 420 years through the same gateway, enters the vicinity of Saturn, and passes within 2.6 x 106 km (431.4 Rs) of the planet. However, a ΔV of 50 m/s, one year after the final Titan encounter, is sufficient to alter the trajectory significantly such that the return does not occur within 500 years (red).

Fig. 23. Return is prevented with a 50 m/s ΔV one year after the final Titan encounter. The original return path with no maneuver is in green; the non-return path after the maneuver is in red.

CONCLUDING REMARKS

The impact of solar gravity on a vehicle in orbit about an outer planet can be significant and can be exploited for trajectory design. For application to Cassini, the design of a long-term orbit beyond Phoebe requires the use of the solar perturbing force to facilitate delivery to such an orbit. Escape within a 500-year window is possible, but careful selection of initial orientation relative to a Sun-Saturn rotating frame, as well as the initial orbital elements, can yield a trajectory that remains in the Saturnian system for over 2000 years. The trade-off to achieve such a satisfactory trajectory beyond the orbit of Phoebe is a 7-8 year delay after the final Titan encounter prior to the final maneuver. Immediate escape can also be designed such that the vehicle remains in a heliocentric orbit and does not encounter any other planets or return to Saturn within the 500-year window.

ACKNOWLEDGEMENTS

The authors would like to thank Masaki Kakoi for providing long-term integrations in the ephemeris model. Valuable discussions with Nathan J. Strange and Jerry B. Jones (Technical Manager) are appreciated. Portions of this work were supported by Purdue University and the Jet Propulsion Laboratory, California Institute of Technology, under Contract Number 1283234 with the National Aeronautics and Space Administration.

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