Pumping the inclination on the Jupiter equatorfor the Jupiter Magnetospheric Orbiter

Stefano Campagnola∗and Yasuhiro Kawakatsu †

Abstract

This paper study the 3D resonant hopping, which is a sequence of gravity assists at one moonconnecting resonant orbits. The orbits are not necessarily coplanar, and the spacecraft can changeits orbital elements within a wide range using gravity assists. The technique is especially useful formissions targeting high inclinations. However the solution space grows geometrically with the numberof gravity assists, and its full exploration during preliminary design is challenging. In this paper westudy the problem in general and derive some useful analytical formula and the 3D-Tisserand graph,which is an extension of the famed Tisserand graph. We use the formula in a fast branch&bound algo-rithm, with which we compute thousands of solution for the Jupiter Magnetosheric Orbiter, JAXA’splanned mission to explore the Jovian magnetosphere and the plasma torus from high latitudes.

1 Nomenclaturei Inclination

m Number of spacecraft revolutions between two consecutive gravity assists

n Number of moon revolutions between two consecutive gravity assists

r Position vector

ra Apocenter

rp Pericenter

v Velocity vector

v∞ Velocity of the spacecraft relative to the moon

H Gravity-assist altitude

J Pedex, refers to Jupiter

M Pedex, refer to a moon

R Equatorial radius

δ Gravity-assist bending angle

µ Gravitational parameters

ρ Resonant ratio n : m

∗JSPS postdoctoral fellow, JAXA/ISAS, Sagamihara, Japan 252-2510, stefano.campagnola@missionanalysis.org†Associate Professor, JAXA/ISAS.

2 IntroductionSince Mariner10’s first swing-by at Venus in 1974, gravity assists have been used successfully in manymissions. The missions Galileo, Cassini, and the planned Jupiter Europa Orbiter (NASA) and JupiterGanymede Orbiter (ESA) implement several gravity assists connecting resonant orbits to reduce thespacecraft energy. The mission Ulysses, Cassini, and the planned SOLO (ESA) and Solar C (JAXA), usegravity assists to increase the inclination.

JAXA’s planned Jupiter Magnetospheric Orbiter (JMO) is an example of a mission exploring a moonsystems at high latitudes. JMO requires tens of gravity assists to both reduce the apocenter and increasethe inclination. Each gravity assist connects two resonant orbit, so that the entire sequence is an exampleof “resonant hopping”.

The solutions space of such problems is very large and methods are needed to explore it quickly duringpreliminary design. In this paper we study the 3D-resonant hopping strategy in general, and present anautomated design method to compute thousands of trajectories in just a few minutes.

In the first part of this work we show that the Tisserand constant is the main problem parameter,and we introduce the 3D-Tisserand graph, which gives insight to the problem. We also derive some usefulanalytical formula to compute fixed-altitude gravity assists connecting resonant orbits. The graph andthe formula are the first main result of the paper.

In the second part of this work we implement the formula in a branch&bound algorithm, which weapply to the JMO mission. We compute almost ten thousand solutions for some typical initial conditions.The algorithm and the solution space are the second main result of this work. We finally plot the detailsof one solution reaching 48◦ inclination on the Jovian equator in less than 1.5 years.

3 ModelsWe model the spacecraft trajectory using the linked-conic model (or zero sphere-of-influence, patched-conic model). The orbit of the moon is assumed to be a circle. When a gravity assist occurs, the v∞vector is turned by the angle

δ = 2 arcsin

(µM

µM + (RM +H) ‖v∞‖2

)(1)

while ‖v∞‖ is conserved not only before and after one gravity assist, but also during the resonanthopping sequence at the same moon1. In fact we recall[1, 2] that

‖v∞‖ = ‖vM‖√

3− T (2)

where the Tisserand parameter[3] is an approximation of the Jacobi constant (energy) in the restricted,three-body problem2

J ≈ T =2aMra + rp

+ 2 cos i

√2rpra

aM (rp + ra)(3)

The set of all the orbits with a given ‖v∞‖ is the v∞−sphere [4, 5]. Each point on the spheredefines a v∞ vector, thus a unique state vector (r = rM , v = v∞ + vM ) and the corresponding Kepleriancoordinates. Among all possible charts mapping the sphere into Cartesian coordinates, we choose thepump and crank angles [6, 7, 4]

α ∈ [0, π] k ∈ [−π, π]

shown in Figure 1, so that the semi-major axis of an orbit (hence the period and resonant ratio ρ = n : m)depends on one variable only, the pump angle. In particular, it can be shown [4] that

α = arccos

(1− ρ−2/3 − (‖v∞‖ / ‖vM‖)2

2 ‖v∞‖ / ‖vM‖

)(4)

and equivalently1No deterministic maneuvers are introduced in this work.2This formula for the Tisserand constant was first introduced in [2], and is equivalent to the more common definition

involving semi-major axis and eccentricity instead of pericenter and apocenter.

Figure 1: v∞−sphere, pump and crank angles, and maximum inclination.

ρ =(

1− 2 ‖v∞‖ / ‖vM‖ cosα− (‖v∞‖ / ‖vM‖)2)−3/2

(5)

3.1 Energy as parameterThe energy of the three-body problem acts as the main parameter for the resonant hopping: T uniquelydefines ‖v∞‖ (from Eq. 2) and the maximum gravity-assist deviation angle (from Eq. 1). It alsodetermines the maximum inclination achievable by the resonant hopping sequence [7]. In fact Figure 1shows that the maximum inclination is

imax = arcsin‖v∞‖‖vM‖

(6)

for which α = π/2 + imax and k = π/2.

3.2 Gravity-assist mapWe now explain how the gravity assist maps the incoming direction (αin, kin) into the set of possibleoutgoing directions (αout, kout), for a given gravity assist altitude or bending angle. We first consider thecase

αin ∈ (δ, π − δ) (7)

Then the outgoing pump angle varies in the interval

αout ∈[αin − δ, αin + δ

]and the corresponding crank angle is found imposing δ as the angular distance between the incoming andoutgoing directions

cos δ =⟨vin∞, v

out∞⟩/ ‖v∞‖2 = · · · =

= cosαin cosαout + sinαin sinαout cos(kout − kin)

which yields to

kout = kin ± arccos

(cos δ − cosαin cosαout

sinαin sinαout

)(8)

Some additional consideration is necessary when Eq.7 does not hold. If αin − δ < 0 , the minimumpump angle is δ − αin(lower pump angles can only be achieved by reducing δ); similarly if αin + δ > π,the maximum pump angle is 2π − (αin + δ). In particular, if αin = 0 (or αin = π), then αout = δ (orαout = π − δ) and kout ∈ [−π, π]. Also, if sinαout = 0, then kout is not unique (in practice we can pickkout = 0).

Figure 2: Level set of inclination, resonances, argument of the pericenter, 15RJ pericenter and 40RJapocenter in a crank angle vs. pump angle graph , with v∞ = 4 km/s at Callisto. The picture also showsthe maximum changes in pump and crank angles for a 100 km altitude gravity assist, starting from a4 : 1 resonance and approximately 10◦ inclination.

Summarizing, a gravity assist maps(αin, kin; δ

)into (αout, kout) with the equations

sinαin = 0 −→ αout = π −∣∣π − αin − δ∣∣ ;

kout ∈ [−π, π]sinαin 6= 0 −→ αout ∈ [

∣∣αin − δ∣∣ , π − ∣∣π − αin − δ∣∣];kout =

{kin ± arccos

(cos δ−cosαin cosαout

sinαin sinαout

)if sinαout 6= 0

any if sinαout = 0

(9)

Figure 2 shows a (α, k) chart of the v∞-sphere for a spacecraft with v∞ = 4 km/s at Callisto. Weplot the level-set rp = 15RJ , and the ρ, i and argument of the pericenter level-sets. We also plot the setof possible αout, kout that can be reached with a 100 km altitude gravity assist at Callisto, starting fromαin = 68◦ (corresponding to ρ = 4 : 1) and kin = 30◦. The arrow represents a gravity assist that bothincreases the inclination and reduces the period, leading to a 2:1 resonant orbit with i = 15◦.

3.3 3D-Tisserand graphThe Tisserand graph is a graphic tool [8, 9] to design planar, gravity-assisted trajectories in the linked-conic model. Each point of the two-dimensional graph represents an orbit. Gravity assists at one moonare simple shifts along v∞ level sets, which are plotted for different moons. In this paper we extend theTisserand graph to non-planar orbits.

For a given T (or a given ‖v∞‖ or imax), the level set T (a, e, i) = T can be written explicitly byrewriting Eq. 3 as

i(ra, rp;T

)= arccos

(T − 2aM/(ra + rp)

2√

2rpra/aM (ra + rp)

)(10)

Figure 3 shows the 3D-Tisserand graph with the imax = 40◦ level set for Ganymede and Callisto.For both moons the figure also shows a specific point (orbit) on the level set, and a corresponding closedcurve of the orbits reachable with a 100 km altitude gravity assist. In this example, the graph clearlyshows that high inclination orbits are reached at low-energies and high pericenters. The 3D-Tisserandgraph can be seen as the equivalent of the 3D T − P graph[2] in the linked-conic model.

The section i = 0◦ of the 3D Tisserand graph corresponds to the traditional (planar) Tisserand graph,except that the v∞ level sets are replaced by the imax level sets. Figure 4 shows the section i = 0◦ withdifferent imax level sets for Ganymede and Callisto. The graph also shows constant-period level sets[10],where the period is expressed through the resonant ratio ρ. The graph reveals which planar initialconditions can lead to high-inclination orbits through resonant hopping.

Figure 3: ra − rp − i Tisserand graph with the energy level-sets imax = 40◦ for Ganymede and Callisto.The two loops are the points reachable with a 100 km-altitude gravity assist starting from the large dotswithin.

Figure 4: ra − rp Tisserand graph with imax and ρ level-sets. Two consecutive dots along one imax levelset shows the shift due to a 100 km-altitude gravity assist. The star is a typical initial condition for theJMO resonant hopping.

4 Resonant hoppingIn this section we use the formula and graph introduced previously to automatically design resonanthoppings.

During a resonant hopping, each gravity assist occurs at the same location in the moon’s orbit andthe spacecraft trajectory can be described with the set of outgoing v∞ directions αouti , kouti , i = 1, . . . , n.Starting with some fixed initial conditions, the number of possible resonant hopping grows geometricallywith the number of gravity assists [11]. A full exploration of the solution space is only possible using abranch&bound technique [12], as explained for instance in [13] for planar resonant hoppings. The resonanthopping design presented in this paper is fully automated once the user define:

• the initial conditions ‖v∞‖, αin1 ,kin1

• some target conditions on αN , kN (N unknown),

• the maximum transfer time between two consecutive gravity assists ∆tmax (or equivalently nmax =∆tmax/PM )

• the maximum total transfer time

Additional constrains can be introduced. A brief description of the algorithm follows.From the initial conditions, we compute the minimum and maximum αout1 and the corresponding ρ1

using Eq.9 and Eq.5. For all the possible n1,m1 (hence ρ1) satisfying

n1 < nmax ; ρ1,min < (n1/m1) < ρ1,max

we compute (αout1 ,kout1 ) with Eq.4 and Eq.9 (branching). In particular, if we want to raise the inclination,we choose the positive sign in Eq. 9, and if we want to decrease the energy, we pick αin1 as lower bound forαout1 . Each possible outgoing direction αout1 , kout1 is then used as the new incoming direction (αin2 , k

in2 )

Figure 5: Solutions to the resonant hopping problem.

for the next gravity assist. The same procedure is repeated recursively, with each incoming directiongenerating several possible outgoing directions (branches). If the target conditions are met, the currentsequence is stored. If the constraint on the maximum transfer time is violated, the current sequence isdiscarded (bounding) and the search proceeds to the other branches.

4.1 Jupiter Magnetospheric orbiterWe now use the graph and the resonant hopping algorithm to design the JMO resonant hopping. In atypical mission scenario, the orbit insertion maneuver and the pericenter raise maneuver place the space-craft into a 200-day orbit [14], which is the starting conditions for the resonant hopping. In particular, inthis paper we consider an initial orbit on a 12:1 resonance with Callisto and with rp = 16RJ . This initialcondition is marked with a star in Figure 4, and can lead to inclination higher than 50◦. The targetorbit is a 6:7 resonance, with an inclination greater than 30◦. We consider a maximum time of flight of2 years and we include a constraint on the minimum pericenter to mitigate the radiation exposure (formore details see [14]). Summarizing, the input parameter for the automated search are:

• ‖v∞‖ = 6.717 km/s, αin1 = 85◦, kin1 = 0◦

• ρ(αN ) = 6 : 7, i(αN , ρN ) > 30◦

• ∆tmax =85 days (slightly above 5 Callisto revolutions)

• Maximum transfer time of 2 years.

• Minimum pericenter rp > 12RJ

The algorithm computes almost 10,000 feasible solutions in just two minutes on a laptop machine. In theactual JMO design, the initial conditions are looped, longer transfer time are allowed, and the minimumpericenter constraint is relaxed; as a result the number of feasible solutions grows to several hundredthousand, but the computation time remain manageable (<12 hours).

Figure 5 shows the time of flight and the final inclination of the solution space computed by thealgorithm. The example solution in the circle reaches 48◦ in 1.5 years. For this solution, Figure 6shows the sequence of (α, k) on a close-up of the pump-crank angle chart, while Figure 7 shows thecorresponding (ra, rp, i) on the 3D-Tisserand graph. Finally Figure 8 shows a close-up of the solution inan inertial reference frame centered in Jupiter.

5 ConclusionIn this paper we study the 3D-resonant hopping, showing that the Tisserand constant is the main problemparamete. To support the design of resonant hopping transfers, we derive some analytical formula andthe-3D Tisserand graph, which are the first main result of this paper. We use the formula in a fastbranch&bound algorithm, with which we compute thousands of solution for the Jupiter MagnetosphericOrbiter. The algorithm and the JMO solution space are the second main result of the paper. We plotthe solution space, and some detail of a sample transfer.

Figure 6: Sequence of pump and crank angles for a sample solution.

Figure 7: Sequence or apocenter, pericenter and inclination for a sample solution.

Figure 8: Sample solution for JMO in a Jupiter-centered reference frame. The initial condition is the200-day orbit (thick line).

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