paper dr. ariadna farres - jsforum.or.jp

The Fourth International Symposium on Solar Sailing 2017, Kyoto, Japan. Copyright c©2017 by A. Farrés, S. Soldini and J. Tsuda. Published by theSecretariat of the ISSS 2017, with permission and released to the Secretariat of the ISSS 2017 to publish in all forms.

JAXA’s Trojan Asteroids Mission: Trajectory Design of theSolar Power Sail and its Lander

By Ariadna Farrés Basiana1), Stefania Soldini2) and Yuichi Tsuda2)

1) Institut de Mathemátiques, Universitat de Barcelona, Barcelona, Spain2) Institute of Space and Astronautical Science/JAXA, Sagamihara, Japan

(Received 1st Dec, 2016)

In this paper we use dynamical system tools to design trajectories in the vicinity of Trojan asteroids for both: JAXA’s solarpower sail and its lander. The current JAXA baseline considers a solar power sail hovering a Trojan asteroid at 40 km from itssurface. Which will then descend to 1 km of the surface to release the lander. First we will exploit solar radiation pressure toplace a solar power sail in orbit around the asteroid and illustrate how the effects of changing the sail orientation can enhancethe hovering opportunities. Second we will focus on the lander release, performing a sensitivity analysis on its deploymentvelocity together with possible bouncing trajectory. To model the dynamics of the solar power sail and the lander we use theaugmented Hill three body problem for the far gravity field dynamics and a perturbed two-body problem approximating theasteroid’s as triaxial ellipsoid for the close gravity field dynamics.

Key Words: Solar sail, Far vs near field dynamics, Augmented Hill problem, Triaxial ellipsoid, Lander.

1. Introduction

Jovian Trojans are asteroid located in the L4/L5 equilibriumpoints of the Sun-Jupiter system. Little is known about theirorigin and their evolution as the observations provide limitedclues about the source of these asteroids. Currently, these aster-oids are completely unexplored and the future JAXA’s sampleand return mission to the Trojan asteroids will revolutionise ourunderstanding of these bodies. Trojan asteroids are believed toeither, being captured during the migration mechanism of Gi-ant Planets, or formed during the planetary formation.19 Theirorbits are also destabilised by collision ejection. Thus, Trojanscould also be the remnants of a much more substantial initialpopulation of trapped bodies, or these objects could continuallybe replenished from an unidentified external source.3 A missionto the Trojan asteroids represents the new frontier in mission ex-ploration, and their study would also explain what happened tothe primordial Trojans of Uranus and Neptune.11 With respectto NEOs exploration, a space mission to Trojan asteroids facesthe challenge of reaching the unexplored far side of our solarsystem where large amount of fuel is required and an efficientpower supply is difficult.14 JAXA’s Trojans assessment studyconcluded that the solar power sail is the best way to performsuch a challenging mission.

This paper covers two different aspects of a solar sail missionto the Trojan asteroids, analysing possible trajectories to hoverthe asteroid and the landing of a small probe on the surface ofthe asteroid. Hence, we address both: far and near operationsat the asteroid, dealing with different dynamical models thatinclude the relevant perturbations in each scenario.

The current JAXA’s baseline proposes hovering the asteroidwith an Earth-pointing solar sail.14 However, exploring the ef-fect of Solar Radiation Pressure (SRP) can enhance interestingorbit solution for a solar sail. For this purpose we study the dy-namics of the sail in the far gravity field, where the asteroid isapproximated as a point mass and the effects of SRP and Sun’sgravity are also taken into account, considering the augmented

Hill 3 body problem as a model. In previous works2, 7, 8 termi-nator orbits and other quasi-periodic orbits that appear in thesystem are proposed for hovering and monitoring an asteroid.In Section 3. we will review some of these orbits, moreover, bychanging the sail orientation (keeping it fixed with respect to theSun-sail line) we can artificially displace these orbits changingthe visibility conditions, enlarging the monitoring sites.

The solar power sail will be equipped with a lander that willbe released at 1 km from the surface of the asteroid. In thiscase, the dynamics of the lander is mainly influenced by theirregular shape of the asteroid.1 Hence, for the near field dy-namics we consider the perturbed two-body problem where weinclude spherical harmonics for the irregular shape of the as-teroid and the SRP effect is neglected. In Section 4. we discushow to select the initial velocity at which the lander is releasedto guarantee a bounded motion of the lander around the aster-oid.16 This will prevent bouncing trajectories of the lander aftertouchdown, as in the Rosetta mission, with the potential risk ofescaping from the asteroid surface.

2. Asteroid Parameters

Little is known on physical properties of Trojans asteroidsdue to the challenges one faces when observing and taking mea-surements of these objects from Earth. In this paper, we con-sider as an example the Trojan asteroid 2001 DY103 to set theorbital period and mean Sun distance (although most asteroidsin the Trojan region present similar values). However, to de-termine the spin rate, shape and density we have used somestatistics on the Trojan population.14 These physical parame-ters are summarised in Table 1. Moreover, we approximate theshape of the asteroid by a triaxial ellipsoid where a, b and c

(satisfying a > b > c) are the semi-major axis of the ellipsoidalong the x, y and z axis respectively using an Asteroid-CentredAsteroid-Fixed (ACAF) reference frame. Table 2 contains thecorresponding Stockes coefficients for this triaxial ellipsoid.


Table 1: Estimated physical parameters for asteroid 2001 DY103.

Orbital period (T ) 4343.87 daysMean Sun-Asteroid Distance (R) 5.21 AURotational Period (τ ) 10 hoursEffective Shape a = 11, b = 10, c = 9 kmEffective Radius (rsb) 9.97 kmDensity (ρ) 2000 kg/m3

µsb = G 43πr3ρ 5.53× 10−4 km3/s2

Table 2: Stockes coefficients for a triaxial ellipsoid with a = 11, b =10, c = 9 (values taken from Table 1).

Cnm Formula15 ValueC20

15r2

(c2 − a2+b2

2) -0.059396640516508

C221

20r2(a2 − b2) +0.010570588566497

C40157

(C220 + 2C2

22) +0.008038790549854

C4257C20C22 -0.000448469606523

C44528C2

22 +0.000069835839151

3. Solar Sail Vicinity Operations

In this first section, we focus on the dynamics of a solar sailthat is orbiting around the asteroid, where the main perturbingforces are the SRP and the Sun and the asteroid gravity field. Aswe are far from the asteroid, we can neglect its shape and ap-proximate it as a point mass. Our goal is to describe the possiblenatural orbits in the system that can be interesting for monitor-ing the asteroid, with a fixed sail orientations.

3.1. Far Field Dynamics: Hill 3-Body Problem

The Augmented Hill 3-Body problem (AH3BP) models themotion of an infinitesimal particle (sail-craft) that interacts witha small mass (asteroid) and is perturbed by a distant large body(the Sun), and takes into account the effect of solar radiationpressure (SRP). As the Sun-sail distance is large compared tothe asteroid-sail distance, the effect of the Sun is included as auniform gravity field and a uniform SRP.7, 9, 13

A rotating reference frame centred on the small body wasconsidered such that: the x direction points from the Sun tothe asteroid; the z direction is aligned with the Sun–asteroidangular velocity; and the y direction completes a positive co-ordinate system. We normalise units in such a way that theunit of distance, L, is (µsb/µsun)1/3R, and the unit of time,T , is 1/ω, where µsun is the gravitational parameter of the Sun(≈ 1.327 × 1011 km3/s2), µsb is the gravitational parameter ofthe smaller body, R is the mean distance between the Sun andthe asteroid, and ω =

√µsun/R3 is the angular velocity. Using

these normalised units, the equations of motion for the AH3BPfor a solar sail10, 18 are:

X−2Y =∂Ω

∂X+aX , Y +2X =

∂Ω

∂Y+aY , Z =

∂Ω

∂Z+aZ .

(1)with

Ω(X,Y, Z) =1

r+

1

2(3X2 − Z2),

r = (X,Y, Z) denotes the position of the solar sail in the rotat-

ing frame, r =√X2 + Y 2 + Z2 is the distance between the so-

lar sail and the centre of the asteroid, and asail = (aX , aY , aZ)

is the acceleration given by the solar sail.In this paper, the solar sail is perfectly reflecting, i.e. we only

consider the effect of the reflected photons on the surface ofthe sail. To have a more realistic model one should also haveto include the absorption properties of the sail material.4 Theacceleration due to a perfectly reflecting solar sail is:

asail =2PA

m〈rs,n〉2n, (2)

where, P = P0(R0/R)2 is the SRP magnitude at a distanceR from the Sun (P0 = 4.563 µN/m2, the SRP magnitude atR0 = 1AU), A is the area of the solar sail, rs is the direction ofSRP (rs = (1, 0, 0) in the rotating reference frame) and n is thenormal direction to the surface of the sail.

The sail’s orientation is parameterised through two an-gles α and δ, which represent the horizontal and verti-cal displacement of n with respect to rs. Hence n =

(cosα cos δ, sinα cos δ, sin δ). Notice that asail cannot pointtowards the Sun, hence α, δ ∈ [−π/2, π/2].

The sail’s efficiency is measured in terms of the sail lightnessnumber β = 2PA/m. As we use in Eq.(1) normalised units fordistance and time these parameter β has to be re-scaled,10 hav-ing β = β(µsbω

4)−1/3 as the normalised sail lightness number.Following the definition of Broschart et al.2 and Giancotti et al.8

we have β = K1(A/m)µ−1/3sb where K1 = 7.6874 when A is

given in m2 and m in kg. Through this paper we consider thatthe solar sail has an total area of 50× 50 m2 and a total mass of1300 kg that orbits around 2001 DY103. Taking the asteroid’sparameters from Table 1 we have that β = 179.44.

3.2. The Jacobi constant

An important property of the AH3BP is that the system isHamiltonian and admits an integral of motion, also known asthe Jacobi constant.

Jc = X2+ Y 2+Z2−2Ω(X,Y, Z)+aXX+aY Y +aZZ. (3)

For a given initial condition, Jc is preserved through time. Thisconstant is related to the energy of the system and it provides aninsight of different types of motion, as well as the regions wherethe motion is possible. Notice that Jc = ||v||2−2Ω+〈asail, r〉,hence the regions of possible motion are delimited by the zerovelocity surfaces Z(X,Y, Z) = (X,Y, Z) ∈ R3|Jc + 2Ω −〈asail, r〉 = 0. In Fig. 1 we show these curves for a solar sailthat is perpendicular to the Sun-sail line (i.e. α = δ = 0) andβ = 179.44. Notice how, for low energy levels, Jc ≤ −60, themotion around the asteroid and inside the asteroid are separated,but as this one increases, Jc ≥ −50, a gap is opened at L2

connecting the inner and outer regions.

3.3. Equilibrium Points

We recall that when we neglect the effect of the solar sail(β = 0) the system has two equilibrium points L1 and L2

symmetrically located around the origin, whose coordinates are(±3−1/3, 0, 0). When we include the SRP (β 6= 0) and the sail


Zero Velocity Curve β = 179.44

SL1

0.00-10.00-20.00-30.00-40.00-50.00-60.00-70.00-80.00

-1000 -500 0 500 1000 1500 2000 2500

X (km)

-1500

-1000

-500

0

500

1000

1500

Y (

km

)

Fig. 1: Zero Velocity curves for β = 179.44 and α = δ = 0.

is kept perpendicular to the Sun–line direction (α = δ = 0),the position of L1 and L2 move along the x-axis towardsthe Sun as we increase the value of β.16 For β = 179.44

SL1 = (−59.8134, 0, 0) UD and SL2 = (0.07461, 0, 0) UD,with SL2 closer to the asteroid than SL1.

For a fixed β, the position of the equilibrium points can beartificially displaced by changing the sail orientation. Changesin α shifts the equilibrium point to one side or the other of theSun–asteroid line, while variations in δ displace the equilibriumpoint above or below the ecliptic plane.7, 9, 13

If we look at the stability of these equilibria we can see thatall the equilibrium points on the SL2 family are linearly un-stable and their linear dynamics is centre×centre×saddle. Thismeans that under generic non-resonant conditions, two familiesof periodic orbits emanate from each fixed point (Lyapunov’sCentre Theorem). Each family is related to one of the two pairsof complex eigenvalues (±iω1,±iω2) and the period of theorbits tends to 2π/ωi when we approach the equilibrium point.Furthermore, the coupling between the two frequencies givesrise to a Cantorian family of invariant tori, or Lissajous-typeorbits.

3.4. Periodic Orbits

Previous works2, 7, 8 show that around the displaced equilibria(SL2) we can find periodic and quasi-periodic orbits that spendmore than half of the time on the day side of the asteroid, whichwould be useful for mission operations. Here we want to brieflydescribe some of those orbits for β = 179.44. We will also seehow changes of the sail orientation affect this orbits.

When we consider the sail perpendicular to the Sun-sail line,we are essentially changing the gravitational pull from the Sunan the sail-craft. The non-linear dynamics close to the displacedequilibrium point is very similar to the no-sail case. The motionon the Z = 0 plane is decoupled from the rest, and one of thefamilies that emanates from the equilibrium point is completelycontained in the plane. We call them planar Lyapunov orbits.The other family oscillates above and bellow the Z = 0 plane,and we call them vertical Lyapunov orbits. As it happens in theclassical Hill problem, when we move along the planar family,

at some point we have a 1:1 resonance between the orbit andits normal frequencies and the Halo family of periodic orbitsappears. In Fig. 2, we show the xyz projection of the orbitsin these three families. Notice how some of the orbits in boththe planar and vertical Lyapunov families spend most of theirorbital period on the day side of the asteroid. These kind oforbits can be used to observe the asteroid. On the other hand,although the Halo get very close to the asteroid they remain onthe dark side of the asteroid. If we look at the linear stability ofthese orbits we can see that most of them are linearly unstableexcept the final orbits on the Halo family that are linearly stable.The stable ones are known in the literature as Terminator orbitsand are plotted in blue in Fig. 2. Broschart et al.2 saw that alongthe Terminator orbits there are several n:m resonances that giverise to periodic orbits that spend most of their time on the dayside of the asteroid.

-800

-400

0

400

800

-1000

-500

0

500

1000

-0.5

0

0.5

Planar Lyap

X (km)Y (km)

Z (

km

)

-800-400

0 400

800

-40-20 0 20 40

-800

-400

0

400

800

Vertical Lyap

X (km)Y (km)

Z (

km

)

0 400

800

-400 0

400

-400

0

400

Halo Orbits

X (km)Y (km)

Z (

km

)

Fig. 2: Projection on the xyz space of the planar Lyap. (left), vertical Lyap.(middle) and Halo (right) family of orbits for β = 179.44 and α = δ = 0.

As it happens with the equilibrium points, we can artificiallydisplace the periodic orbits by changing the sail orientation.Changes in α induce an extra force on the xy direction and theorbits are displaced to one side or the orbits of the Sun-asteroidline. Changes in δ induce an extra force on the z direction anddisplace the orbits above and below the asteroid’s orbital plane.Both changes will break, in some cases, the symmetry of theorbits.

To show this phenomena we have taken one orbit from eachfamily that is close to the asteroid, and used a continuationmethod to compute their displaced equivalents. We have var-ied α ∈ [−15o, 15o], δ = 0 and α = 0, δ ∈ [−15o, 15o] keepingthe same orbital period for all the displaced orbits. Fig. 3 showsthe results for the three different families. Notice how the dis-placed Terminator orbits spend part of their orbital period onthe day side.

Being able to displace the periodic orbits with a solar sail isinteresting for two main factors. The first one is that this will in-creases the visibility properties of the orbits and their potentialuse for observational missions. Moreover, we can use an ac-tive control law to drift along these families of displaced orbitsfollowing the ideas presented in Farrés and Jorba.5, 6


-800

-600

-400

-200

0

200

-1000

-500

0

500

1000

-0.5

0.5

Planar Lyap wrt α

X (km)Y (km)

Z (

km

)

-800-600

-400-200

0 200

-1000

-500

0

500

1000

-150

-50

50

150

Planar Lyap wrt δ

X (km)Y (km)

Z (

km

)

-800-600

-400-200

0 200

-150

-50

50

150

-600

-200

200

600

Vertical Lyap wrt α

X (km)Y (km)

Z (

km

)

-800-600

-400-200

0 200

-60

-20

20

60

-800

-400

0

400

800

Vertical Lyap wrt δ

X (km)Y (km)

Z (

km

)

-40-20

0 20

40

-150-50

50 150

-150

-50

50

150

Halo Orbits wrt α

X (km)Y (km)

Z (

km

)

-40-20

0 20

40

-150-50

50 150

-150

-50

50

150

Halo Orbits wrt δ

X (km)Y (km)

Z (

km

)

Fig. 3: Periodic orbits for α ∈ [−15o, 15o], δ = 0 (left) and α = 0, δ ∈[−15o, 15o] (right) of the planar Lyap. (top) vertical Lyap. (middle) andHalo (bottom) families.

4. Landing Operations

The focus of this section is to investigate the dynamics of thelander after its deployment from the solar power sail. In par-ticular, we want to explore how to confine the lander dynamicsin the case of bouncing motion, to prevent the falling off of thelander from the asteroid surface.15 In this case, the lander isin the proximity of the asteroid (closed field dynamics) wherethe SRP can be neglected. The lander dynamics is influencedmainly by the irregular shape of the asteroid here approximatedby a triaxial ellipsoid. JAXA’s solar power sail will be firstplaced hovering at 40 km from the asteroid surface (Home Po-sition (HP) 1), pointing at the Earth to enhance communication.The solar power sail will then descend to 1 km from the sur-face (HP 2) to deploy the lander as shown in Fig. 4. We haveexplored the possibility of a free-falling trajectory for the sailfrom 40 km to 1 km, considering the sail to be Sun-pointing.Notice that when the Earth-Asteroid-Sun angle is small the HPz-axis is opposite to the Hill x-axis and the sail is now Sun-pointing. In this case, the sail would reach 1 km distance fromthe surface with a residual velocity of 8.42 m/s. However, acontrolled descend of the sail is likely to be adopted for safetyreasons.

4.1. Near Filed Dynamics: Perturbed 2-Body Problem

The equations of motion in the close vicinity of the asteroidare given in the two-body problem with the gravity harmonicsexpressed in the rotating (ACAF) frame, where the spin axis isaligned with the z-axis:16

x− 2ωsby = −µsbr3

x+ ω2sbx+

∂B

∂x,

y + 2ωsbx = −µsbr3

y + ω2sby +

∂B

∂y,

z = −µsbr3

z +∂B

∂z,

(4)

1 k

m

40 k

m

HP1

HP2

Fig. 4: Landing phase mission architecture: HP1 at 40 km (sail and lander)and HP2 at 1 km (lander deployment).

where ωsb is the angular velocity of the spin axis (2π/τ in Ta-ble 1) andB is the potential associated to the asteroid harmonicsdefined as:17

B(r, φ, λ) = −µsbr

∞∑n=1

(rsbr

)n [ n∑m=1

Pnm(sinφ)

[Cnm cos(mλ) + Snm sin(mλ)]]

,

(5)where n and m are respectively the order and the degree of theharmonics taken into account; r, φ (latitude) and λ (longitude)are the coordinates of the satellite in spherical coordinates withrespect to the ACAF frame; Pnm are the Legendre polynomial;while Cnm and Snm are the Stockes coefficients. Note that Bis expressed in spherical coordinates (r, φ, λ), while the accel-erations in Eq.(4) are expressed in Cartesian ACAF coordinates(x, y, z). Thus, a transformation from spherical to Cartesian co-ordinates is required.17

The energy of the spacecraft is given by:

E =1

2(x2 + y2 + z2)− 1

2ω2sb(x

2 + y2)− µsbr−B. (6)

Fig. 5 shows the shape of the potential energy (when the veloc-ity in Eq.(6) is equal to zero) as a function of r and λ for φ = 0

(i.e. at the equator).

Longitude (deg)

Ra

diu

s (

km

)

0 100 200 30024

25

26

27

28

29

−3.215

−3.21

−3.205

−3.2

−3.195

−3.19

−3.185

−3.18

−3.175x 10

−5

Fig. 5: Potential Energy as function of r and λ for φ = 0.

Notice that Eq.(4) is an autonomous system that admits fourequilibrium points (1:1 resonances between the asteroids spinrate and the equatorial orbit around the asteroid).12 For a tri-axial ellipsoid, the four equilibrium points lie on the equator(φ = 0) and are located at λ equal to 0, 90, 180 and


270.15 The equilibrium point can be found by imposing thevelocities and accelerations equal to zero in Eq.(4). The co-ordinates of the equilibrium points are shown in Fig. 6 whereP1,3 = (±26.61, 0, 0) km with E = −3.184 × 10−5 andP2,4 = (±26.466, 0, 0) km with E = −3.172× 10−5.

As for the Hill problem, the Zero Velocity Curves (ZVC) ofthe perturbed two-body problem (also known as Roche Lobe16)give qualitative information on the lander motion. We recallthat, if the energy of the lander is less than the energy of P1,3,then the ZVC separate the inner from the outer motion. In thiscase the motion around the asteroid will be confined if the lan-der is inside this region. On the other hand, when the energyof the lander increases, a gap close to P1,3 is opened, connect-ing the inner and outer regions (green curves in Fig. 6). In thiscase the trajectory of the lander can escape. Note that the en-ergy correspondent to an escape trajectory (i.e. green curves inFig. 6) is associated to a velocity of the lander that is usuallylower then the asteroid escape velocity.15 In Fig. 6 we showthe ZVCs for different energy values: in black we have the en-ergy value for P1,3; the red and blue lines correspond to energyvalues lower than P1,3 (i.e. −3.438 × 10−5 and −4.9 × 10−5

respectively); and the green curve represent the case where theenergy is higher than the one at P1,3.

Finally, when the ZVCs are closed, we denote as altitude ofthe ZVC the point on the ZVC that intersects the negative x-axisand is on the right-hand side of P3. This definition is used whenexploring the possible release velocity of the lander as a func-tion of the maximum altitude within the lander can bounce. Forexample, P3 is the minimum altitude to close the ZVC and itsZVC altitude corresponds to -26.61 km (black point on Fig. 6),while the red point corresponds to a ZVC altitude of -20 km andthe blue point correspond to a ZVC altitude of -12.5 km.

Fig. 6: Zero Velocity Curves for a triaxial ellipsoid.

4.2. Lander Trajectory

The altitude of deployment is fixed at 1 km, and given thefact that the maximum radios for the asteroid is of 11 km, wewill consider that the initial condition for the deployment is at(−12, 0, 0) km. Then, the deployment and touchdown velocities

of the lander can be determined as a function of the altitude ofthe ZVC (or the energy) as defined in the previous section.

Fig. 7(a) shows the deployment velocity for altitude of theZVC that increases from -12 km (deployment point) to -26.61km (P3). When the altitude of the ZVCs is at -12 km, the de-ployment velocity is 0 m/s, which represents potentially the bestcase scenario. However, a higher velocity can be required tocompensate uncertainties in the solar power sail position or forlanding time constrains. Thus, Fig. 7(a) tells the maximum po-tential bouncing altitude of the lander as a function of the de-ployment velocity when the velocity is given perpendicular tothe asteroid surface (nominal case). Fig. 7(b) shows the cor-respondent touchdown velocity when the deployment velocityis set. The worst case scenario of elastic bouncing (no dissi-pation) is here considered where the touchdown velocity corre-sponds to the take-off velocity after the first touchdown. Alongboth curves in Fig. 7 we marked the case highlighted in Fig. 6,where the blue star correspond to a ZVC altitude of -12.5 km,the red star to -20 km, while the black star is the minimum ZVCaltitude that guarantees bounded motion corresponding to P3.

∆ V (@ 12 km) [m/s]0 1 2 3 4 5 6 7

Altitude Z

VC

[km

]

12

14

16

18

20

22

24

26

28

(a) Deployment velocity.

∆ V (@ Touchdown) [m/s]6 6.5 7 7.5 8 8.5 9

Altitude Z

VC

[km

]

12

14

16

18

20

22

24

26

28

(b) Touchdown velocity.

Fig. 7: Deployment velocity and touchdown velocity as a function of ZVCaltitude.

Note that the velocities in Fig. 7 are express in the Asteroid-Centred Inertial (ACI) frame. For the case of small Earth-Asteroid-Sun angle, we can directly correlate the ACI framewith the Hill coordinates. Thus, the ∆V given is the one re-quired by the solar power sail. We recall that the rotation be-tween the Asteroid-Centred Asteroid-Fixed (ACAF) frame andthe Asteroid-Centred Inertial (ACI) frame is given by a rota-tion along the z-axis, Rz(φ), with φ = ω · t + φ0 so thatrACAF = Rz(φ) · rACI .17

The lander deployment velocity is here explored for differentinitial conditions including when the ZVC can or cannot boundthe motion of the spacecraft around the asteroid as shown inFig. 8. In Table 3 we summarise some of the cases that we havestudied where in the first column we show the initial velocities(t0) at which the lander is realised and in the second columnwe show the final velocities at touchdown (tf ). In Fig. 8 weshow the trajectory that the lander follows for the different ini-tial velocities. In Fig. 8, two different trajectory are shown: thenominal trajectory when the velocity of the lander is along thex-axis (light blue line) and one general case where the velocitycomponents of the lander have x and y components in a 45

direction (black line) and an elastic bouncing is also consid-ered. This case was selected to demonstrate the importance ofhaving bounded motion. Each row of figures in Fig. 8 corre-sponds to a different initial deployment velocity (first column


(a) Deployment velocity of the lander 2.14 m/s. Left zoom relevant region.

(b) Deployment velocity of the lander 5.82 m/s. Left zoom relevant region.

(c) Deployment velocity of the lander 6.237 m/s. Left zoom relevant re-gion.

(d) Deployment velocity of the lander 6.24 m/s. Left zoom relevant region.

(e) Deployment velocity of the lander 8.88 m/s. Left zoom relevant region.

(f) Deployment velocity of the lander 9.94 m/s. Left zoom relevant region.

Fig. 8: Deployment velocity of the lander at 1 km from the surface: lightblue trajectory when the velocity is along the x-axis and black trajectorywhen the velocity has both x and y components.

in table 3). The solutions in Fig. 8(a), Fig. 8(b) and Fig. 8(c)correspond to the blue, the red and the black stars in Fig. 7 re-spectively. The other three figures (Figs. 8(d), 8(e) and 8(f))correspond to initial velocities where the ZVC are open and apriory we cannot guarantee bounded motion. Finally, as onecan see, if the velocity of touchdown is above the escape veloc-ity vf =

√2µsb/rsb = 10.54 m/s, the lander is likely to fall off

from the asteroid. However, the energy of the lander associatedto Fig. 8(d), Fig. 8(e) and Fig. 8(f) can potentially lead to escapemotion even if the velocity of the lander is initially below vf .15

Table 3: Deployment and Touchdown velocities for the cases in Fig. 8

V (t0) [m/s] V (tf ) [m/s]2.14 6.655.82 8.66.237 8.876.24 8.878.88 10.899.94 11.77

Note that the ZVCs are an important dynamical tool whenchoosing the landing site. In case the when the ZVCs touchthe surface, it is possible to constrain the bouncing motion ofthe lander on one side of the asteroid for example the day side.Fig. 9 shows the case in which the lander is drooped from thepositive y-axis at 1 km from the surface with an initial velocityof 2.75 m/s. In this case, the lander motion is constrained in thepositive coordinates of the y-axis. Hence, for a uniformly rotat-ing asteroid, the ZVCs are an interesting tool for the selectionof the landing point.15 This type of study can be generalised tomore complex shape through the definition of the potential forpolyhedron. The ZVCs were studied for Phobos where the theminimum energy for bounded motion of Phobos shows that sur-face is prone to escape dynamics. Scheeres15 studied the ZVCfor Eros and Itokawa to show the velocity distribution on thetheir irregular surface. These considerations should be kept inmind when selecting the target asteroid for landing.

Fig. 9: ZVC where the altitude at x = -11 km (semi-major axis in x of thetriaxial ellipsoid) touches the asteroid.

We want to mention that, the elastic bouncing was computedby introducing a local horizon frame. The ACAF frame withx and y coordinates, while the local horizon frame is given by


x′ and y′ (see Fig. 10). Where V1 is the velocity at touchdownwhile V2 is the velocity at lift off. We call α the angle betweenthe touchdown position (d) and V 1, and γ the angle betweenthe touchdown position (d) and the local horizon x coordinate.So:

α = cos−1(

V 1 • d||V 1 • d||

), V 2 = ||V 1||

[cos(α)

sin(α)

]. (7)

[x

y

]=

[cos(γ) sin(γ)

− sin(γ) cos(γ)

] [x′

y′

]. (8)

Fig. 10: ACAF (left) and local horizon (right) reference frame at the touch-down position.

5. Conclusions

This paper focuses on the JAXA’s solar power sail mission toTrojan asteroids with particular interest on the asteroid vicinityoperations. The dynamics of the solar power sail and of the lan-der are here investigated following the assumption that the solarpower sail is mainly influenced by the SRP effect while the lan-der dynamics are mainly perturbed by the irregular shape of theasteroid. Thus, it is possible to distinguish between the asteroidfar and close field dynamics. Under this assumption, the sail-craft follows the far field dynamics approximated in the Hillproblem while the lander is operating in the close field dynam-ics where its dynamics are approximated with a perturbed twobody problem. For the case of the sail, it is investigated orbit so-lutions as an alternative of JAXA’s hovering baseline where theorbits are obtained as a function of the attitude of the sail. Thismethodology adds mission flexibility depending on the missionconstraints. The lander operation has been studied looking atthe ZVCs as an important tool for landing decision making andfor confining the lander trajectory by preventing the fall-off ofthe lander from the asteroid surface after bouncing.

Acknowledgements

The research of A.F. has been supported by the Spanish grantMTM2015-67724-P (funded by MINECO/FEDER, UE) andthe Catalan grant 2014 SGR 1145. The research of S.S. hasbeen supported by the Japan Society for the Promotion of Sci-ence grant.

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paper dr. ariadna farres - jsforum.or.jp

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