orbital evolution and environmental analyses around asteroid 2008 ev5

AAS 14-360

ORBITAL EVOLUTION AND ENVIRONMENTAL ANALYSISAROUND ASTEROID 2008 EV5

Pedro J. Llanos∗, James K. Miller†, Gerald R. Hintz‡

The orbital evolution and environmental analyses around the target asteroid, 2008EV5, for the ESA’s Marco Polo R mission are assessed during the proximity oper-ations to minimize the efforts towards the improvement of the Guidance, Naviga-tion and Control system. With this analysis, we will extend our knowledge in thedecision-making when the spacecraft is flying under unknown environmental con-ditions, such as the solar radiation pressure and the gravitational harmonics dueto the irregular shape of the body. We compare analyses performed on differentasteroid shape models, such as a parameterized model, an ellipsoid model and ahigh fidelity mascons shape model.

INTRODUCTION

In recent studies,1 we analyzed the key drivers in the landing performances for the optimizationand improvement of the Guidance, Navigation and Control (GNC) system designed for the ESA’sMarco Polo R mission. The objective of this effort was to increase our understanding of missionproximity operations to small bodies and to extend our knowledge for decision-making when thespacecraft is flying under unknown environmental conditions. This paper addresses a detailed analy-sis about the environment and orbital mechanics around the target asteroid, 2008 EV5, for the MarcoPolo R mission so that we can have a better understanding of how to gauge the efforts towards theimprovement of the GNC system.

The orbit and environmental analysis in this paper was performed by assuming different models:1) one that is parameterized, 2) another that is ellipsoidal based on a constant density object and 3) areal shape model based on a high fidelity model of point masses. Besides the shape of the asteroid,the inclusion of high order perturbations, such as the solar radiation pressure and the sphericalharmonics coefficients, were also considered which will help the future Marco Polo R spacecraftduring proximity operations around the target asteroid, 2008 EV5.

PARAMETRIC MODEL FOR 2008 EV5

Harris et al.2 used polar slopes (constrained to a sphere), as displayed in Figure 1, to describethe shape of asteroid 1999 KW4. Asteroid 2008 EV5 and 1999 KW4 share similar shapes although1999 KW4 has an extremely irregular equatorial band while the single asteroid 2008 EV5 has anoticeably smooth surface.3 The black line in Figure 1(b) enclosing asteroid 2008 EV5 corresponds

∗Marie Curie Postdoctoral Researcher Fellow, Madrid, Spain, 28760, Member AAS/AIAA.†Navigation Consultant, Associate Fellow AIAA, Los Angeles, CA, 91326‡Adjunct Professor, Astronautical Engineering Department, University of Southern California, Los Angeles, CA, 90089,Associate Fellow AIAA

1

to the shape obtained with our parameterized model as depicted in Figure 1c by the red contour line.Figure 1c represents the xz-projection of the shape model of asteroid 2008 EV5 shown in Figure 2.

(a) (b) (c)

Figure 1. a: Harris, Fahnestock, Pravec Model (2009) to represent asteroid1999KW4. b: Parametric Model enclosing asteroid real shape of asteroid 2008 EV5.c: Our parametric model representing asteroid 2008 EV5.

Figure 2. Parameterized model of asteroid 2008 EV5.

Instead of using the Itokawa model,1 we employ a high fidelity parameterized model of pointmasses of asteroid 2008 EV5. Our model provides a radial symmetry very similar in shape to theactual shape of 2008 EV5 obtained by radar measurements with a well defined equatorial ridge withabout 30 m of relief above the equator. Our model does not include any craters but we will talkabout this equatorial feature in a later section. For now, we will assume that our parameterizedmodel of 14,000 point masses∗ has several concentric layers as shown in Figure 3.∗We know that this model is preliminary since it may be unpractical for navigation and orbit determination purposes

especially when the spacecraft is very close to one of these point masses.

2

−300 −200 −100 0 100 200 300 −500

0

500

−300

−200

−100

0

100

200

300

Y(m)X(m)

Point−Mass Model of 2008 EV5 (14,000 points)

Z(m

)

Figure 3. Parameterized model composed by 14,000 point masses.

MATHEMATICAL MODEL

We analyze two approaches for controlled motion near the target asteroid 2008 EV5 during closeproximity operations: inertial hovering and asteroid-fixed hovering. Given arbitrary coordinates, ri,and the corresponding point masses, µi, at ri, we can describe a desired target asteroid by rescalingthe target shape and gravitational mass parameter accordingly. This rescaling determines the accel-eration for inertial hovering (gravitational acceleration) and asteroid-fixed hovering (including theinertial forces):

ag = −n∑i=1

µir3i

ri + 2ω × ˙ri + ω × (ω × ri) (1)

where ri = riRmax

Rscaled is the reconfigured position vector defined by the maximum radius value ofthe asteroid, Rmax, and the desired scaled value, Rscaled = 200 m, of our target asteroid. Similarly,we can reconfigure the gravitational mass parameter, µi = µi

µ µscaled, where µ is the average of allthe gravitational point masses and µscaled = 7.0 m3/s2 is the desired gravitational parameter ofour target asteroid. The spin rotation period of the asteroid is 3.725 h. In our simulations, we alsoincluded the force on the spacecraft by the solar radiation pressure (SRP). We assumed that the areaof the spacecraft is 10 m2, the mass of the spacecraft is 1000 kg and the solar radiation pressurecoefficient is 2. The heliocentric distance of the asteroid is assumed to be constant for short periodsof times (a few days) with a value of 0.958 AU. As shown in Figure 4, the solar radiation pressure ismore dominant than the gravitational acceleration of the asteroid when increasing the distance withrespect to the centre of mass of the asteroid as shown beyond a radius distance of 8390 m from thecenter of mass. Similarly, we included other external perturbations, such as the effects due to theSun-tide and the Venus-tide. For a 1-km orbit around asteroid 2008 EV5, the Sun-tide accelerationis of the order of 10−10 m/s2 whereas the accelerations due to Venus and Jupiter tides are almostnegligible and of the order of 10−14 m/s2 and 10−15 m/s2, respectively.

In Figure 5, we illustrate the external gravitational potential on a 225-m radius sphere enclosingthe asteroid 2008 EV5. The density of the sphere was assumed to be 3 g/cm3 and the mass pa-

3

102

103

104

105

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

Distance to body center of mass (m)

Acc

eler

atio

n (m

/s2 )

Acceleration in function of distance to asteroid center of mass

Asteroid−Sphere

Sun−Tide

Asteroid’s Gravity

SRP

Venus−Tide

Jupiter−Tide

Figure 4. Accelerations as a function of the distance from the center of mass of 2008EV5. Distances run from 200 m (surface of asteroid) to 50 km.

rameter 6.67 m3/s2. The gravitational potential map values are between 0.025 m2/s2 and 0.033m2/s2. As a test, we checked the gravitational map obtained with our parameterized model with thegravitational map obtained by the DLR (German Space Agency)4 team for a mass parameter of 4.9m3/s2, density of 2.11 g/cm3 and a radius of 230 m. Our preliminary parameterized model yieldsvery similar results and deviates from their model an average of about 8 %. Note that Reference 3presents values of 2-4 g/cm3 for the density. Also, uncertainty in the mass of the asteroid3 willhave an impact on the gravitational field of the asteroid.

Table 1 shows the position change (second column) and velocity change (third column) due to thedifferent acceleration magnitudes (first column). For example, during the radio science phase, thespacecraft will be required to stay in orbit (1-2 km) around 2008 EV5 for about 2 weeks. Therefore,an acceleration due to the sun-tide of the order of 10−10 m/s2 translates into a position error ofover 10 m and a velocity error of about 0.12 mm/s. During the far global characterization phase(≈ 20 km), the sun-tide acceleration can be of the order of 2 × 10−9 m/s2 yielding errors inposition and velocity of about 15 m/day and 0.17 mm/s/day, respectively. The sun-tide acceleration(3.85 × 10−9 m/s2) becomes comparable to the asteroid gravity at about 42.66 km orbital radiusobtained numerically.

Table 1. Accelerations acting on asteroid 2008 EV5

Acceleration (m/s2) ∆r (m/day) ∆V (mm/s/day)

10−7 746 8.6410−8 74.6 0.8610−10 0.74 ≈ 0.0110−12 ≈ 0.01 0.0001

4

Figure 5. Gravitational Potential on a 225 m radius sphere around target asteroid 2008 EV5

CONTROLLED MOTION STRATEGIES AND DELTA-V ANALYSIS RESULTS

The station-keeping by inertial hovering5 strategy could be a good method for placing a spacecraftin a low orbit around the asteroid before the sampling mission phase given that the ∆V required forstation-keeping is relatively low. For example, an acceleration of 8 × 10−6 m/s2 (see Figure ??)would result in about 7 m/s to keep the spacecraft at 1-km orbit during 10 days. Note that, dependingon the size of the asteroid, different control strategies may work better for a specific mission. In ourcase, we are dealing with a small asteroid.

But there is another important factor to consider for the asteroid-fixed hovering strategy sincethe spacecraft will go through solar occultation cycles. This means that the spacecraft would needbatteries or some kind of power generation when it it stays on the surface of the asteroid for longperiods of time during operations. When comparing our shape model with one from NASA,6 thepercentage difference in the average for the station-keeping performance is about 1 % at a radiusof 260 m from the center of the asteroid and less than 0.05 % at a radius of 312 m. Our modelrepresents a preliminary high fidelity model of point masses. When comparing our shape modelwith one from NASA,6 the difference in the maximum value of ∆V can be of 1.1% at a radius of1 km and 17 % at 312 m. These deviations show how important the shape of the asteroid (NASAshape model has craters) is as we get close to the surface of the asteroid. The uncertainties in thedensity and mass of the asteroid may not be reduced until the actual spacecraft visit to the asteroid.The corresponding station-keeping maps are displayed in Figures 6 and 7 for specified orbit radiusvalues.

Furthermore, we compared our numerical values with an analytical approach7 for different radiias we illustrate in Table 2. The spacecraft will perform its correspondent duties at each of these radiiduring the different science mission phases. It is thought3 that asteroid 2008 EV5 may trap small

objects around the equilibrium point req =(µω2

)1/2≈ 312 m. If there is any companion3 around

2008 EV5, it would have a size of about 30 m. Therefore, in the presence of a secondary body, verysmall tidal accelerations may take place on the target asteroid 2008 EV5 as we will see in a later

5

Figure 6. Left: Station-keeping by inertial hovering at a distance of 312 m, which isthe equilibrium point req . Right: Station-keeping by inertial hovering at a distance of1000 m.

Figure 7. Left: Station-keeping by inertial hovering at a distance of 2000 m. Right:Station-keeping by inertial hovering at a distance of 5000 m.

section. During the global characterization phase (at 5 km), the spacecraft will obtain a completemapping of the gravity field of the asteroid during a couple of weeks. During the radio sciencephase (at 1-2 km), the spacecraft may spend about two weeks to obtain high-spatial resolutionobservations to identify possible landing sites. During the local characterization phase, at a radiusof 460 m, the spacecraft will gather information on the gravity field of the asteroid that will helpsubsequent proximity operations in the following weeks of science data-collection. In this phase, thespacecraft will obtain information about the site from where the spacecraft will collect the samplesof the asteroid. This mission phase can last about a month. In Table 2, we show four columns: thefirst column shows the science mission phase; the second column is the distance from the center ofmass of the asteroid; the third column shows the ∆V station-keeping (determined analytically); andthe fourth column displays the ∆V for station-keeping (determined numerically). The values in thethird column were obtained for a mass parameter of 7 m3/s2 whereas the values in column 4 were

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Table 2. Numerical and analytical station-keeping at different radii.

Science Phase Radius (m) ∆VSK,analytical(m/s/day) ∆VSK,numerical(m/s/day)

Trapped Objects 312 5.9172 6.2348 , 5.9409Local Characterization 460 2.7221 2.8609 , 2.7261

Radio Science 1000 0.5760 0.6053 , 0.5767Radio Science 2000 0.1440 0.1520 , 0.1449

Global Characterization 5000 0.0230 0.0275 , 0.0265

obtained for a mass parameter of 6.67 m3/s2. The analytic formula used here is:

∆V ≈ 43ρR(Rr

)2m/s/day (2)

where ρ is the density of the asteroid, R is the radius of the asteroid and r is the distance of the

Figure 8. Left: Station-keeping in asteroid-fixed hovering at a distance of 500 m.Right: Station-keeping by inertial hovering at a distance of 1000 m.

spacecraft from the center of mass of the asteroid. Our results show that the error between thestation-keeping obtained analytically7 using Equation (2) and the station-keeping obtained numer-ically using Equation (1) is about 5% when assuming a mass parameter of 7 m3/s2. When usinga mass parameter of 6.67 m3/s2, based on the data from Reference 3, we determine that the dif-ference in the station-keeping performance is of a few mm/s/day for most cases as indicated by thevalues from Table 2.

Finally, we show some results on the asteroid-body fixed as a second controlled strategy forhovering. As we can see in Figure 8, this second approach is less attractive given the large ∆Vvalues required for station-keeping when the spacecraft is at a fixed position relative to the rotatingasteroid. Although, this approach is relatively expensive for orbital radii of 1 km or larger, thisstrategy may be acceptable at an orbital radius of 500 m (local characterization phase) and it willallow the spacecraft to measure the gravity field of the asteroid from a fixed position at a cost ofabout 9.53 m/s down to 8 m/s along the equator and mid-latitudes, respectively and at a lower costas we move towards the poles of the asteroid with a minimum of 0.32 m/s at the poles.

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TIDAL ACCELERATIONS RESULTS

Recent observations3 have revealed the possibility of asteroid 2008 EV5 (Alpha) having a sec-ondary companion (Beta) of about 30 m in diameter. In this section, we analyze how the tidal effectsof this binary system can levitate particles from the surface of Alpha and re-deposit them into an-other place on its surface in the presence of a secondary body. This process is called ”tidal saltation”as explained in Reference 2 in which some regolith from the surface of Alpha may be transported toother parts of the equator of the asteroid in the presence of this secondary body. Given the diameterof the secondary asteroid and assuming it is a spherical body, the mass of Beta,m, is about 3.4×108

kg and orbits the primary (Alpha) with an orbital period Torb. Alpha and Beta are assumed to beseparated by 90 m. The mass,M , and rotational period,Prot, of Alpha are 1011 kg and 3.725 h,respectively.

The equations of motion2 when levitation motion on the surface of the target asteroid 2008 EV5is considered are given as:

r = r(ω + θ2)− (1 + δ)ω2R(Rr

)2+ Tradial(r) (3)

θ = Ttangential(r)− 2r

r(ω + θ) (4)

where Tradial and Ttangential are the radial and tangential tidal accelerations which are expressedas:

Tradial(r) = −εω2R

(cosα+

ra − cosα[

1 +(ra

)2− 2(ra

)cosα

]3/2

)(5)

Ttangential(r) = εω2(Rr

)sinα

(1− 1[

1 +(ra

)2− 2(ra

)cosα

]3/2

)(6)

where r and θ are radial and tangential coordinates, respectively; ω and Ω are the spin angular ve-locity and the orbit period, respectively; α is the angle, (ω−Ω)t+θ, from overhead of the spacecraftat time t; and ε and δ are two parameters that can be obtained from the following two equations:

ε =m

M +m

a

R

(ProtPorb

)2(7)

g ≈ (1 + δ)ω2R(Rr

)2(8)

Figure 9 illustrates a plot of the radial and tangential accelerations assuming the target asteroid2008 EV5 has a companion of about 30 m. For this case, we assume that the ratio between the semi-major axis of the orbit of the companion and the radius of 2008 EV5 is a/R = 1.55. We observethat the radial tidal accelerations are more dominant on the companion side of the target asteroidbut both the tangential and radial tidal accelerations are of the same order (tangential accelerationslightly larger) on the opposite side of its companion.

Assuming the surface of the asteroid is composed of loose rubble, sand or loose rock, this materialcan in some scenarios slide on the sloppy terrain. This sliding material is confined to the surface

8

50 100 150 200 250 300 350

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−8

Frequency Angle, (ω−Ω) t (deg)

Tid

al A

ccel

erat

ion

(mm

/s2)

Equatorial Tidal Accelerations, asecondary

≈310 m

Radius Orbit/Radius Asteroid = a/R = 1.55T

r

Tθ

Figure 9. Radial (red) and tangential (blue) accelerations at the equator of asteroid2008 EV5. The ratio between the radius orbit and the asteroid radius is 1.55, that is,the secondary is assumed to be at 310 m from the center of mass of the primary.

such that the dynamics that govern this sliding motion can be written as:

r = R(ω + θ2)− (1 + δ)ω2R(Rr

)2+ Tradial(r = R) (9)

θ = Ttangential(r = R)− 2r

R(ω + θ) + µkr

θ

|θ|(10)

where the last term on the right-hand side of Equations (9) and (10) represents the resisting acceler-ation in the tangential direction where µk is the sliding coefficient, which we assumed to be 0.7 inour simulations. From Equation (8), we obtain that the values for ε and δ are 0.005358 and 0.0769,respectively.

In Figure 10, we show an example of the motion of particles along the equator of the main targetasteroid. Some of these particles reach equatorial altitudes of about 90 cm, then drop to about 70cm and finally they are ejected into orbit. As for the second case, particles reach an altitude of about80 cm and after getting as close as the surface of the asteroid, most particles escape into orbit butsome particles impact the surface of the asteroid. These tidally unstable configurations may impactthe surface of the asteroid and therefore should be taken into account during proximity operations.

Figure 11 illustrates the equatorial motion of particles on the main target asteroid. In this caseand assuming a constant angle, α, we observe that there are both escaping and sliding particles. Theinitial velocity of these particles was about 5.2 mm/s. Those particles that do not have enough energyto escape are confined to the surface. When decreasing the initial velocity of these test particles to5.05 mm/s, we see that the dynamics of the test particles is governed by a sliding motion. Thismotion is well reflected in the blue box in Figure 11 where some particles approach the surface ofthe asteroid and follow an oscillatory motion until reaching a steady sliding motion. During thisoscillatory motion, some of these particles bounce up to 3 cm slowly reducing their velocity untilreaching a sliding motion state. This oscillatory motion can last from a few tens of seconds to abouta minute. For these sliding motion simulations, θ 6= 0.

Figure 12 displays a plot where we compare the dynamics of test particles for different initial

9

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4

200

200.2

200.4

200.6

200.8

201

R θ (m)

r (m

)

Equatorial Motion of Particles, α=π

ASTEROIDSURFACE

Escaping Particles

(a)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4199.9

200

200.1

200.2

200.3

200.4

200.5

200.6

200.7

200.8

200.9

R θ (m)

r (m

)

Equatorial Motion of Particles, α=5π/4

ASTEROIDSURFACE

Escaping Particles

Impacting Particles

(b)

Figure 10. a: Escaping particles from the surface of Alpha for α = π. b: Escapingparticles from the surface of Alpha for α = 5π/4. Most particles escape but some ofthese particles impact the surface of the asteroid.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4

200

200.1

200.2

200.3

200.4

200.5

200.6

200.7

200.8

200.9

201

R θ (m)

r (m

)

Equatorial Motion of Particles, α=π/4

ASTEROIDSURFACESliding

Particles

Escaping Particles

(a)

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

200

200.1

200.2

200.3

200.4

200.5

200.6

200.7

R θ (m)

r (m

)


−0.24 −0.22 −0.2

200

200.01

200.02

200.03

ASTEROIDSURFACESliding

Particles

(b)

Figure 11. a: Example where there are escaping particles from the surface of Alphaand some other sliding particles for α = π/4. b: Only sliding particles from thesurface of Alpha for α = π/4.

conditions. The particles (red) are assumed to have no initial tangential velocity, that is, their initialtangential velocity is zero and the particles (blue) are assumed to have a non-zero initial tangentialvelocity of about 1.1× 10−3 mm/s. Since the particles motions have been exaggerated 20 times tobe more visible, the added tangential velocity results in a visible hop. At the beginning, the particlesmove slightly toward the right slowly rising up and moving back towards the left, then they fall backto the surface to the left. After reaching the surface, particles take different motions. Some of theparticles with initial tangential velocity impact the surface, others have a sliding motion and stillothers escape into orbit. These particles have slightly longer excursions (of the order of one meter)than the particles with no initial tangential velocity.

10

−10 −8 −6 −4 −2 0

200

201

202

203

204

205

206

R θ (m)

r (m

)


Escaping Particles

Escaping Particles

ASTEROID SURFACE

(a)

−5.5 −5 −4.5 −4 −3.5 −3199.8

200

200.2

200.4

200.6

200.8

201

201.2

R θ (m)

r (m

)


ASTEROID SURFACE

Impacting Particles

Escaping Particles

Sliding Particles

(b)

Figure 12. a: Comparison of the dynamics of particles when no tangential velocityis imparted to particles (red) and when tangential velocity is imparted (blue) for α =π/4. b: Enlarged section as indicated by the green box. Particles either impact thesurface of the asteroid, escape or slide on the surface of the asteroid.

As for the particles with no initial tangential velocity, they start rising up in the radial direction andslowly moving towards the left until they branch out into particles sliding on the surface and otherparticles escaping into orbit. There are no particles (with no initial tangential velocity) impactingthe surface for this scenario. The timescale of particles levitating above (about 70 cm) the surfaceof the asteroid is of the order of ten minutes before they either slide on the surface, escape or impactthe rotating asteroid.

TRIAXIAL ELLIPSOID MODEL: IVORY’S APPROACH

We used the analytical gravitational potential of a homogeneous triaxial ellipsoid.8 This potentialcan be simplified if we assume that two of the semi-axes are the same, α = β > γ and greater thanthe third one. For now, we will assume that the density of the ellipsoid is constant. The shape ofthis ellipsoid can be described by:

x2

α2+y2

β2+z2

γ2− 1 = 0 (11)

where x, y and z are the coordinates at which we want to compute the potential. Then, the Ivory′sequation follows as:

Ψ =x2 + y2

α2 + u+

z2

γ2 + u− 1 = 0 (12)

The potential8 is given as:

V (x, y, z) =3µ2R

H0 −3µ

4R3(x2H1 + y2H2 + z2H3) (13)

11

where:

H0 =1λ

(14)

H1 = H2 =1λ3

(arctan(λ)− λ

1 + λ2

)(15)

H3 =2λ3

(λ− arctan(λ)) (16)

λ =

√α2 − γ2

R(17)

R =√γ2 + u (18)

Using Mathematica c© software (version 8), we obtain the following partial derivatives of the po-tential with respect each coordinate:

Vx =3yµ

4√

2D5

(4(E + F )Jy2

− 4√

2D(G+H + I)ξCJ3/2

)(19)

Vy = Vx (20)

Vz = 3zµ(−√

2JD

+ξ

J3/2

)(21)

where:

u =12

(A∓

√A2 + 4B

)(22)

A = x2 + y2 + z2 − α2 − γ2 (23)

B = z2α2 + (x2 + y2 − α2)γ2 (24)

C =√

4y4 + (z2 + J2) + 4y2(z2 − J) (25)

D =√C + 2y2 + z2 − J (26)

E = 8y6 + (α− γ)2(α+ γ)2(C − z2 − J) + 4y4(C + 2z2 − 3J) (27)

F = 2y2(z4 + J(−2C + 3J + z2(C − J))) (28)

G = 8y6 + z6 − z2J2 + z4(C + J) (29)

H = (α− γ)2(α+ γ)2(C − J) + 4y4(C + 3z2 − 3J) (30)

I = 2y2(3z4 + J(−2C + 3J) + 2z2(C − J)) (31)

J = α2 − γ2 (32)

ξ = arctan(√2JD

)(33)

As a check, the reader should note that the accelerations due to the ellipsoid potential have theproper units of m/s2 assuming the mass parameter, µ, has units of m3/s2 and the coordinatesx, y, z and the semi-axes of the ellipsoid are also given in meters. The parameter u, defined byEquation (22), is the maximum real root. The ridge or prominent surface feature that appears on theequator of asteroid 2008 EV5 is closer in shape to the ridge9 of asteroid 1994 CC than to the ridgeof asteroid 1999 KW4.

12

(a)

−200−100

0100

200 −200

0

200−200

−100

0

100

200

Y(m)

Real Shape 2008 EV5

X(m)

Z(m

)

(b)

Figure 13. a: Comparison of parameterized model of asteroid 2008 EV5 with ellipsoidmodel, single asteroid 2008 EV5 and triple asteroid 1994 CC. b: Real shape of asteroid2008 EV5.

HIGH FIDELITY SHAPE MODEL FOR 2008 EV5

The second approach is to use the gravitational potential as a function of the spherical harmonicsas expressed in Equation (34):

V (r, θ, φ) =GM

r

[1 +

∞∑n=2

n∑m=0

(Rasterr

)nPnm(sin θ)(Cnmcos(mφ) + Snmsin(mφ))

](34)

where r, θ and φ are the radius, latitude and longitude, respectively. Raster is the largest equato-rial radius of the asteroid, Pnm are the Legendre polynomials, and Cnm and Snm are the Stokescoefficients of the potential.

Table 3. Spherical Harmonics

Degree (n), Order (m) Cnm Snm

2,0 -9.89732E-3 0.02,1 -1.11327E-4 7.62812E-52,2 1.73162E-3 4.92091E-53,0 -4.09758E-3 0.03,1 4.39585E-4 7.18890E-43,2 -2.80316E-3 -2.53398E-33,3 1.82728E-4 -1.08893E-34,0 5.82646E-3 04,2 -1.22417E-5 -4,4 5.35451E-7 -

Equation (35) is an extension of the potential,8 obtained also with Mathematica c© software,for the special case of an axisymmetric ellipsoid that includes the first few degrees of the sphericalharmonics. In Reference 8, the potential was expanded to fourth order where only the spherical

13

harmonics, C20, C22, C40, C42 and C44 were considered.10 In our study, we also included the otherStokes coefficients: S21, S22, S31, S32 and S33. The other coefficients, S42, S44 are not availablein this study but we will compute higher order degree coefficients in a later study. We used the thenormalized coefficients4 and other spherical coefficients (C42 and C44) computed analytically8 asshown in Table 3. Note that the coefficients S42 and S4,4 are not used.

V (r, θ, φ) =µ

r

(1 +

(Rr

)2[12C20(3sin2φ− 1) + 3cosφsinφ(C21cosθ + S21sinθ)+

+3cos2φ(C22cos2θ + S22sin2θ)]

+(Rr

)3[12C30(−3sinφ+ 5sin3φ)+

+32cosφ(−1 + 5sin2φ)(C31cosθ + S31sinθ) + 15cos2φ(C32cos2θ + S32sin2θ)+

+15cos3φ(C33cos3θ + S33sin3θ)]

+(Rr

)4[18C40(3− 30sin2φ− 35sin4φ)+

+52C41cosφcosθ(−sinφ+ 7sin3φ) +

152C42cos2θcos2φ(−1 + 7sin2φ)

+105C44cos4θcos4φ])

(35)

Figure 13 shows a high fidelity model of asteroid 2008 EV5 obtained by the radar6 science teamfrom NASA, which we used for our simulations that we will show in the next section.

ORBIT ANALYSIS RESULTS

In this section, we analyze the orbital evolution around the asteroid, 2008 EV5. The objective ofthis analysis is to provide a good understanding on how a test particle moves around this irregularobject to obtain information that will help in navigating the future European Marco Polo-R space-craft. We will show results that compare the dynamics of orbits around this target asteroid usingthe different models explained before. Several orbit scenarios were examined for different orbitalradii associated with some of the science mission phases of the Marco Polo-R mission. These mis-sion phases include the local characterization phase and the radio science phase that will be crucialfor the landing site selection and determining the gravity map more accurately to characterize theasteroid and its environment.

In Figure 14 and Figure 15, we illustrate some orbits using our preliminary high fidelity parame-terized model of point masses.

Figure 14 displays several orbits that can be used for proximity operations to asteroid 2008 EV5.These orbits with orbital radii of 400 m (blue), 500 m (magenta) and 600 m (black) are good optionsduring the local characterization phase. These orbits are equatorial orbits from where the spacecraftwill be able to cover most of the asteroid surface. Higher latitude regions will not be as visible tothe spacecraft.

Figure 15 depicts a very interesting stable orbit at an orbital radius of 1 km integrated over oneyear from where the spacecraft will be able to perform observations to analyze possible landing sitesduring the radio science phase. This orbit geometry allows the spacecraft to perform science whenorbiting both the dark side and lit side of the asteroid, that is, there is no restriction to only daylightobservations. The orbits obtained in Figure 14, Figure 15 and Figure 16 were subject to the solarradiation pressure but not to the spherical harmonics.

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Figure 14. Orbits around asteroid 2008 EV5 at different radii for the local char-acterization phase in an inertial frame. The red star indicates the beginning of theintegrated trajectory and the red circle is the end of the integration. Blue orbit at 400m. Magenta orbit at 500 m. Black orbit at 600 m.

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Figure 15. a: XZ projection of orbit around asteroid 2008 EV5 (real shape) at 1000m for the radio science phase in inertial frame. The red star indicates the beginningof the 1-year integrated trajectory and the red circle is the end of the integration. b:XY projection in the rotating frame.

In Figure 16, we show the comparison of an integrated trajectory of a stable elliptical orbit with asemi-major axis of about 1 km for our parameterized shape model and the real model with the solarradiation pressure acting on the spacecraft. The difference in the orbital position between whenusing our parameterized model and using the real model is about tens of meters.

Now, our parameterized model used to obtain the orbits in Figure 14 had no craters. If we assumethe same model of point masses but with some craters so that they mimic the shape of the realasteroid 2008 EV5, the dynamics of the spacecraft is slightly different as we can observe in Figure17.

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Figure 16. a: XZ projection of the orbit around asteroid 2008 EV5 (parameterizedshape in red, real shape in black) at 1000 m for the radio science phase in the rotatingframe. b: XY projection in the rotating frame.

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Figure 17. a: Integrated orbits over a time span of 60 days around asteroid 2008EV5 assuming a cratered (red) and non-cratered (black) asteroid. b: Comparison ofthe difference in orbit position due to harmonics for both cratered and non-crateredasteroid with the ellipsoid model (blue) during a time span of 30 days.

The cratered-asteroid was obtained by removing some point masses. The orbits illustrated inFigure 17 result in escape from the asteroid vicinity when including the Cnm spherical harmonics asshown by the red (cratered asteroid) and black (non-caretered asteroid) orbits. These perturbationsyield deviations in the orbit position of the spacecraft of the order of tens of meters after one month.These deviations are much less (of the order of five meters after one month orbiting the asteroid)when considering the ellipsoid model as shown in Figure 14 by the blue orbit.

The next step was to include both Cnm and Snm spherical coefficients into our simulations toobserve the orbit deviations as displayed in Figure 18. Adding the Snm andCnm harmonics seems tohave a dominant effect in the deviation of the orbit position (more than if we only consider the Cnm

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Figure 18. a: Deviation in the orbit position when orbiting asteroid 2008 EV5 con-sidering both Cnm and Snm coefficients. b: Amplified section of the deviation in orbitposition.

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Figure 19. a: Integrated elliptical orbits around asteroid 2008 EV5 in inertial frameunder the influence of the spherical harmonics. The red star indicates the beginningof the 512-days integrated trajectory and the red circle represents the end of the tra-jectory. b: YZ-projection of elliptical orbits around asteroid.

coefficients only) during early stay times around the asteroid. The inclusion of the Snm coefficientsdisrupts the long-term stability of the orbit as indicated by the deviation in the orbit position of thespacecraft of over 100 m difference after over a 2-month stay around the asteroid. Also, during thetime span of a month, the deviations are of the order of about 30 meters and less than 20 meters fora time span of about 2 weeks. This weakly instability for short periods of time can be corrected withsome station-keeping maneuvers. These perturbations must be taken into account during proximityoperations to the asteroid where the spacecraft will require station-keeping maneuvers if it were toorbit asteroid 2008 EV5 more than a few weeks.

In Figure 18, we see that Snm must be considered when performing science operations in the

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proximity of the asteroid. For example, the difference in the orbit (in black) position when consid-ering the Cnm coefficients only and considering both the Cnm and Snm coefficients (magenta orbit)is about 20 meters over the time span of three weeks, which translates into a deviation rate in theorbit position of almost one meter per day in average or about 0.1 mm/s per week. The positionuncertainty grows with time and over the time span of 5 months or about 150 days, the deviation inthe position of the spacecraft averages about 5.5 m per day which translates to about 1 mm/s everytwo weeks.

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Figure 20. a: Quasi-periodic orbit of 20 revolutions around asteroid. b: Quasi-periodic orbit of 40 revolutions around asteroid. c: Quasi-periodic orbit of 36 revolu-tions around asteroid. d: Quasi-periodic orbit of 44 revolutions around asteroid.

Figure 19 displays an orbit around asteroid 2008 EV5 given its most realistic shape6 of pointmasses. The spacecraft orbits around the asteroid in elliptical orbits with a radius of periapsis of500 m and a radius of apoapsis of 2000 m. The trajectory was integrated over 512 days and includedthe solar radiation pressure and both Cnm and Snm coefficients. These orbits are good candidatesfrom where the spacecraft can gather critical information about the gravity field map of the asteroid(during periapsis passages) and information in regards to the landing site (during apoapsis passages)

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before the spacecraft lands or performs a touch-and-go maneuver1 on the surface of the asteroid tograb a sample.

The final example is an integrated trajectory using the parameterized and real shape models asillustrated in Figures 20(a), 20(b), 20(c) and 20(d). The orbit in blue represents the trajectoryobtained when only the solar radiation pressure is present. The red trajectory is assumed to includethe solar radiation pressure and the spherical harmonics. In general, all these orbits show goodstability for many revolutions around the target asteroid 2008 EV5. Figure 20(a), displays a 400-mquasi-periodic orbit covering equatorial and most parts of the polar regions. The spacecraft leavesthe asteroid vicinity after 21 revolutions. This same orbit (see Figure 20(b)) shows signs of higherstability for more than 40 revolutions when it orbits the asteroid at lower latitudes, that is aroundthe equator. Figure 20(c) illustrates a 900-m quasi-periodic orbit where the satellite escapes theasteroid gravity field after 36 revolutions. In Figure 20(d), the spacecraft slowly gains gravity assistwhen orbiting the irregular body because of the large harmonics until it escapes the asteroid after44 revolutions.

The spacecraft is influenced mainly by the asteroid’s gravity around 3 km as we can observe inFigure 4. The asteroid’s gravity is still one order of magnitude larger than the effect of the solarradiation. The closer the spacecraft is to the asteroid, the larger the uncertainty in both positionand velocity. Therefore, during proximity operations at the asteroid, the spacecraft may requiretight control strategies before its final approach to grab a sample. Having a more accurate gravitymap of the asteroid will minimize these dynamical uncertainties, which will allow sufficient time toperform station-keeping maneuvers when required during the different mission phases.

SUMMARY

This work was devoted to the orbital and environmental analysis in the proximity of asteroid 2008EV5 for the ESA’s Marco Polo R mission.

To better understand the unknown environment of the non-spherical body, which is crucial forproximity operations to the asteroid, we evaluated different perturbations around the body, such assolar radiation pressure, third body perturbations and tidal accelerations (in the presence of a sec-ondary body). We further analyzed inertial hovering and asteroid-fixed body strategies as possiblecontrolled strategies during proximity operations. Our numerical analysis was compared both withdifferent asteroid shape models and with an analytical approach for several test scenarios yieldingdifferences within only a few percent.

In these simulations, the solar radiation pressure plays an important role in the dynamics of thespacecraft around the asteroid, especially as the orbital distance increases from the center of theasteroid. Similarly, the harmonics also are responsible for increasing the dynamics uncertaintywhen orbiting the asteroid, especially at very close orbital distances from the asteroid. The tidalaccelerations are very small, but if there is a secondary body orbiting the primary, the spacecraftmay be subject to being impinged by escaping particles from the surface of the asteroid.

We also analyzed the stability of possible orbits around 2008 EV5 for different science phases ofthe Marco Polo R mission. Our numerical results were also compared with different asteroid shapemodels. These results show discrepancies in the orbits, especially when orbiting at a small orbitaldistance from the asteroid, accentuated by the presence of spherical harmonics.

Future work will be devoted to obtaining a higher degree gravity field from which we can derivehigher degree spherical harmonics to have a more accurate orbit representation. Irregular bodies

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have a tremendous amount of non-linearity and this may cause harmonics of very small coefficientsto yield very tiny perturbations that may affect the dynamics of the spacecraft around the asteroid.Also, a proper analysis of the orbit determination will be considered to have a robust understandingof the dynamics around the asteroid. In this work, we assumed that the density of the object wasconstant and in the future, we will address a model with a non-constant density.

ACKNOWLEDGMENTS

This work has been partially supported by the Marie Curie fellowship PITN-GA-2011-289240Astronet-II. Part of this work has been performed at the G.M.V. Aerospace and Defense, S.A.

REFERENCES[1] P. J. Llanos, M. D. Domenico, and J. G. Fernandez, “Advanced GNC technologies for proximity oper-

ations in missions to small bodies,” AAS Guidance and Control Conference, Breckenridge, Colorado,January 31-February 5, 2013.

[2] A. W. Harris, E. G. Fahnestock, and P. Pravec, “On the shapes and spins of ”rubble pile” asteroids,”Icarus, No. 199, 2, 2009, pp. 310–318.

[3] M. W. Busch, S. J. Ostro, L. A. M. Benner, M. Brozovic, J. D. Giorgini, J. S. Jao, D. J. Scheeres,C. Magri, M. C. Nolan, E. S. Howell, P. A. Taylor, J. L. Margot, and W. Brisken, “Radar observationsand the shape of near-Earth Asteroid 2008 EV5,” Icarus, No. 212, 2011, pp. 649–660.

[4] X.Shi, H. Hussmann, K. Wickhusen, J. Oberst, F. Damme, and F. Ludicke, “The Dynamical Environ-ment of Asteroid 2008 EV5, The Science of Marco Polo-R, ESTEC, 04/06/2013,” 2013.

[5] M. J. S. Belton, T. H. Morgan, N. Samarasinha, and D. K. Yeomans, Mitigation of Hazardous Cometsand Asteroids. The Pitt Building, Trumpington Street: Cambridge University Press, 1st ed., 2004.

[6] http://echo.jpl.nasa.gov/asteroids/shapes/. Accessed July 12, 2013.[7] D. J. Scheeres, Orbital Motion in Strongly Perturbed Environments: Applications to Asteroid, Comet

and Planetary Satellite Orbiters. Berlin Heidelberg: Springer-Verlag, 1st ed., 2012.[8] A. Rossi, F. Marzari, and P. Farinella, “Orbital Evolution Around Irregular Bodies,” Earth, Planets and

Space, Vol. 51, 1999, pp. 1173–1180.[9] M. Brozovic, L. A. M. Benner, P. A. Taylor, M. C. Nolan, E. S. Howell, C. Magri, D. J. Scheeres, J. D.

Giorgini, J. T. Pollock, P. Pravec, A. Galad, J. Fang, J. L. Margot, M. W. Busch, M. K. Shepard, D. E.Reichart, K. M. Ivarsen, J. B. Haislip, A. P. LaCluyze, J. Jao, M. A. Slade, K. J. Lawrence, and M. D.Hicks, “Radar and optical observations and physical modeling of triple near-Earth Asteroid (136617)1994 CC,” Icarus, No. 216, 2011, pp. 241–256.

[10] G. Balmino, “Gravitational potential harmoncis from the shape of an homogeneous body,” CelestialMechanics and Dynamical Astronomy, Vol. 60, 1994, pp. 331–364.

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orbital evolution and environmental analyses around asteroid 2008 ev5

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