Solar Sail Orbit Operations at Asteroids

JOURNAL OF SPACECRAFT AND ROCKETS

Vol 38 No 2 MarchndashApril 2001

Solar Sail Orbit Operations at Asteroids

Esther Morrowcurren

University of California San Diego La Jolla California 92093-0214D J Scheeresdagger

University of Michigan Ann Arbor Michigan 48109-2140and

Dan LubinDagger

University of California San Diego La Jolla California 92093-0221

The inherent capabilities of solar sails and that they need no onboard supplies of fuel for propulsion make themwell suited for use in long-term multiple-objective missions They are especially well suited for the exploration ofasteroids where one spacecraft could rendezvous with a number of asteroids in succession The orbital mechanicsof solar sail operations about an asteroid however have not yet been studied in detail Building on previous studieswe nd both hovering points and orbiting trajectories about various sized asteroids using equations of motion for asolar sail spacecraft The orbiting trajectories are stable and offer good coverage of the asteroid surface althoughrestrictions on sail acceleration are needed for smaller asteroids

Nomenclaturea = solar radiation pressure acceleration vectorap = sail acceleration mms2

Qa p = normalized sail acceleration ap=sup1r iexcl2H

ap0 = sail acceleration at 1 AU or characteristicacceleration mms2

aMp0

= maximum sail acceleration for feasible orbitoperations mms2

amp0

= minimum sail acceleration for feasible orbitoperations mms2

asc = spacecraft orbit semimajor axis kmQasc = normalized spacecraft orbit semimajor axis asc=r0

c = speed of light 3 pound 108 mse = spacecraft orbit eccentricitye = spacecraft periapsis vectorH = spacecraft angular momentum vectori = spacecraft orbit inclination degl = incoming sunlight unit vectorN = angle proportional to asteroid true anomaly about

sun raquont= cos degn = asteroid mean motion about sunn = unit normal vector to spacecraftP = orbit parameter of asteroidR = radial distance of asteroid from sun kmQR = radial distance of asteroid from sun AUR0 = 1 AU measured in km 15 pound 108 kmr = radial distance of spacecraft from asteroid kmr = radius vector from asteroid to spacecraftrH = Hill radius kmrn = nondimensionalized value of r r=rH

r0 = mean asteroid radius kmt = time sW=R2

0 = solar radiation intensity 1368 Wm2 at 1 AUx y z = position coordinates in the rotating frame kmx y z = coordinate vectors in the rotating framexn yn zn = normalized position coordinatesreg = spacecraft pitch angle

Received 7 July 2000 revision received 17 November 2000 acceptedfor publication 22 November 2000 Copyright cdeg 2001 by the AmericanInstitute of Aeronautics and Astronautics Inc All rights reserved

currenResearch Associate California Space Institute Scripps Institute ofOceanograpy 0214 estherarcaneucsdedu Member AIAA

daggerAssistant Professor Department of Aerospace Engineering scheeresumichedu Member AIAA

DaggerAssociate Research Physicist California Space Institute Scripps Insti-tute of Oceanography 0221 dlubinucsdedu

macr = nondimensional sail loading parameter or saillightness number

deg = perturbation anglemicro = spacecraft declination angle measured

from asteroidsup1 = asteroid gravitational parameter km3s2

sup1sun = sun gravitational parameter raquo134 pound 1011 km3s2

ordm ordm0 = spacecraft orbit true anomaly degfrac34 = sail mass to area ratio or sail loading kgm2

frac34 curren = critical sail loading parameter 153 gm2

Aacute = spacecraft yaw angleAtilde = spacecraft right ascension angle measured

from asteroidAuml = magnitude of angular velocity vector

Auml Dp

sup1sun=a3ast siexcl1

X = angular velocity vectorQAuml QAuml0 = spacecraft orbit longitude of the ascending node

relative to the sunndashasteroid line deg = spacecraft orbit argument of periapsis deg

= rst and second time derivatives respectively

Introduction

T HE concept of the solar sail has been known for more than acentury but it has only been with the advent of microtechnolo-

gies and thin lms that solar sailing appears to be a practical modeof space ight1 Because of the long history and proven usefulness ofconventional spacecraft solar sails will never be considered as a re-placement technology for conventional propulsion however thereare some mission applications for which sails are particularly wellsuited A long-term multiple-objective mission such as one to morethan one asteroid is just such an application These missions havenot yet been studied in detail and in particular solar sail behaviorabout asteroids has not been studied This paper is a rst step in thisdirection

Relying on solar propulsion instead of convential propulsion al-lows us exibility in studying asteroids With a solar sail space-craft we are not limited by carrying onboard supplies of fuel forpropulsion Depending on sail performance in the space environ-ment several asteroids could be visited in succession and orbitedfor extended periods of time Thus a spacecraft of this type couldenhance our understanding of asteroids by shortening the period oftime between missions

We derive a model for solar sail dynamics about an asteroid andconsider a number of possible options for operations in the asteroidenvironment We nd limits on feasible sail operations as a functionof sail parameters asteroid parameters and asteroid orbit Bothorbital and hovering options are considered For the purposes of

279

280 MORROW SCHEERES AND LUBIN

this study we focus only on the behavior of the solar sail spacecraftafter rendezvous has been achieved

For our model we assume a spherical point mass asteroid a per-fectly re ecting solar sail and a circular asteroid orbit about the sunThese assumptions can be relaxed in future studies We rst developthe basic equations of motion for the system We then consider theconstraints and feasibility of hovering sail trajectories complement-ing the studies made by McInnes et al2 and McInnes3 We nd acontinuum of hovering points that would allow station keeping tomonitor the asteroid at a particular phase angle for extended periodsof time Next we consider the constraints and feasibility of orbitaloperations about an asteroid from which we develop a number ofbasic criteria for necessary sail acceleration as a function of asteroidparameters The orbital trajectories are found to be stable and to of-fer good coverage of the asteroid surface These trajectories wouldbe very useful for mapping purposes

ModelTo model the dynamics of a sail close to an asteroid we can apply

the Hill approximation with appropriate generalizations to accountfor the solar sailrsquos force vector The Hill approximation applies verywell to the motion of spacecraft about asteroids due to the smallmass of asteroids relative to the sun and due to the proximity that thecraft maintains to the asteroid4 The Hill approximation can also beeasily modi ed to account for the effect of solar radiation pressureleading to the equations of motion5

Rx iexcl 2Auml Py D iexclsup1=r 3x C 3Auml2x C ax (1)

Ry C 2Auml Px D iexclsup1=r 3y C ay (2)

Rz D iexclsup1=r 3z iexcl Auml2z C az (3)

where the origin of the rotating reference frame is centered at theasteroid with the positive x axis in the antisolar direction the z axisnormal to the asteroid orbit ecliptic and the y axis according tothe right-hand rule This frame rotates about the z axis with angu-lar velocity Auml D

psup1sun=a3

ast where sup1sun is the sunrsquos gravitationalparameter and aast is the asteroidrsquos heliocentric orbit radius in kilo-meters The gravitational parameter of the asteroid is denoted as sup1

The spacecraft is propelled by solar radiation ux incident on thesail which is herein assumed to be perfectly re ecting and planarThe sail receives anaction force from incident sunlight and a reactionforce from the re ected light so that the net force is directed normalto the surface of the sail and pointing away from the sun The acceler-ation vector of this force is denoted as a D [ax ay az] D apl cent n2n

The unit vector l de nes the direction of the incident light and nis the unit normal to the sail In the rotating reference frame l will al-ways equal [1 0 0] and n will be [cos Aacute cos reg sin Aacute cos reg sin reg]where reg is the sail pitch angle with respect to the sun line and Aacuteis the orientation angle about the z axis as shown in Fig 1 The

Fig 1 Model schematic for a solar sail spacecraft about an asteroid

direction of the solar radiation pressure (SRP) force can never bepointed toward the sun and so the sail attitude is constrained suchthat l cent n cedil 0

The parameter ap is called the sail acceleration and is representedas

ap Diexcl2W

macrcR2

0

centiexclR2

0

macrR2

cent1=frac34 (4)

for a perfectly re ecting solar sail where W=R20 is the solar radiation

intensity c is the speed of light R0 is the distance from the sun to theEarth R is the distance from the sun to the sail in kilometers and frac34is the mass to area ratio of the sail or sail loading For a nonperfectsail there will be an additional component of acceleration transverseto the sail normal At 1 AU

ap D ap0 Diexcl2W

macrcR2

0

cent1=frac34 (5)

and in general

ap D ap0

iexclR2

0

macrR2

cent(6)

Thus SRP varies as 1=R2 about the sun (but see discussion byMcInnes1)

Multiplying and dividing by the sunrsquos gravitational parametersup1sun we have

ap D 2W=csup1suniexclsup1sun

macrR2

cent1=frac34 (7)

or

ap D macriexclsup1sun

macrR2

cent(8)

where macr D frac34 curren=frac34 is the nondimensional sail loading parametercalled the sail lightness number de ned to be the ratio of theSRP acceleration to the solar gravitational acceleration The criticalsail loading parameter frac34 curren D 2W=csup1sun is a constant whose value isapproximately 153 gm2

The value of the sail acceleration varies with both the ef ciencyand size of the sail1 For example McInnes1 shows that a typicalvalue for a 98 pound 98 m square sail that carries a payload of 25 kg is1 mms2 At this acceleration a solar sail spacecraft has trip timesto asteroids comparable to conventional spacecraft This spacecraftdesign yields a sail loading parameter of 912 gm2 or a sail lightnessnumber of 01678 This parameter can be reached with a sail lmthickness of 2 sup1m

Solar Sail Hovering Constraints and DynamicsGiven the basic equations of motion we search for conditions

under which the spacecraft can be placed into an arti cial equilib-rium point found by setting all of the time derivatives to zero and nding the combination of SRP acceleration and natural forces thatwill yield no net acceleration acting on the sail The result is a spe-cialization of the analysis carried out by McInnes2 for the restrictedthree-body problem now applied to the Hill problem Followingthe Hill equations we nd the equations of motion of a solar sailspacecraft to be as in Eqs (1ndash3)

Because the sail normal is required to point in the antisolar direc-tion we are limited to iexclfrac14=2 middot reg middot frac14=2 The problem is symmet-ric about the angle reg D 0 however and so we need only consider0 middot reg middot frac14=2 When reg D frac14=2 the sail is edge on to the sun In thissituation we effectively are in irons that is we have no SRP forceat all Although there may be instances when this effect would bedesirable we do not consider this case explicitly Thus we limit ourdiscussion to 0 middot reg lt frac14=2

Likewise to satisfy the constraint l cent n cedil 0 we only consider Aacutelimited to iexclfrac14=2 middot Aacute middot frac14=2 The problem is also symmetric aboutAacute D 0 therefore we consider Aacute such that 0 middot Aacute lt frac14=2 for the samereasons stated for reg

MORROW SCHEERES AND LUBIN 281

The preceding de nitions lead us in the most general case tode ne the equilibrium equations for hovering points to be

0 D iexclsup1=r 3x C 3Auml2x C ax (9)

0 D iexclsup1=r 3y C ay (10)

0 D iexclsup1=r 3z iexcl Auml2z C az (11)

with the components of a as

ax D ap cos2 Aacute cos2 reg cos Aacute cos reg (12)

ay D ap cos2 Aacute cos2 reg sin Aacute cos reg (13)

az D ap cos2 Aacute cos2 reg sin reg (14)

and the vector r as

r D r[cos micro cos Atilde cos micro sin Atilde sin micro ] (15)

where micro is the declination angle measured from the xy plane towardthe z axis and Atilde is the right ascension angle measured in the xyplane from the Cx axis as shown in Fig 1

Hovering Points with No SRP ForceWe rst consider the hovering points for the system with no SRP

force In this case ax D ay D az D 0 It is obvious that y and z mustalso be 0 to satisfy Eqs (10) and (11) respectively Thus fromEq (9) sup1=r 3 D 3Auml2 at equilibrium Solving for r we nd

r D sup1=3Auml213 (16)

This value is called the Hill radius and denoted rH herein We use thisradius to normalize the hovering point radii throughout our analysis

Hovering Points Along the x AxisWe consider next the simplest situation for the model when the

sail is along the x axis Here Aacute Atilde and reg are all 0 because there areno components in the y or z directions In this case the equationsof motion become

0 D iexclsup1=r 3x C 3Auml2x C ap (17)

which becomes

x=jx j3 D ap=sup1pound1macriexcl

1 iexcl r 3n

centcurren(18)

where rn is the nondimensionalized value for r normalized by theHill radius rH that is rn D r=rH

In this case there are two equilibrium points one for x positiveand one for x negative When x is positive that is on the asteroidnight side rn must be less than 1 which means that the hoveringradius is less than the Hill radius When x is negative (on the asteroidday side) rn must be greater than 1 which means that the hoveringradius is greater than the Hill radius

Hovering Points in the xz PlaneNow consider when the sail is in the xz plane Here Aacute and Atilde are

both 0 whereas reg varies from 0 to frac14=2 as discussed In this case theequations of motion are as in Eqs (9ndash11) where

ax D ap cos2 reg cos reg (19)

ay D 0 (20)

az D ap cos2 reg sin reg (21)

We see immediately that y must be 0 from Eq (10)Letting x D r cos micro and z D r sin micro we can solve for angle reg and

ap in terms of r and micro When we rewrite Eqs (9) and (11) theybecome

sup1=r 3 iexcl 3Auml2r cos micro D ap cos2 reg cos reg (22)

sup1=r 3 C Auml2r sin micro D ap cos2 reg sin reg (23)

which gives

tan reg Dmicro

sup1=r 3 C Auml2

sup1=r 3 iexcl 3Auml2

paratan micro (24)

Multiplying the numerator and denominator by 3r 3=sup1 we have

tan reg D

Aacute1 C 1

3r 3n

1 iexcl r 3n

tan micro (25)

which gives the sail orientation as a function of sail positionWhen we use the identity sec2 reg D 1 C tan2 reg Eq (9) becomes

sup1=r 2iexcl1 iexcl r 3

n

centcos micro D ap1 C tan2 regiexcl 3

2 (26)

and substituting in Eq (25) and solving for ap we have

apDpoundsup1

macrr 2

iexcl1iexclr 3

n

cent2cos2 micro

currenpoundiexcl1iexclr 3

n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (27)

which gives the necessary sail acceleration as a function of sailposition We can normalize ap by dividing by sup1=r 2

H We then nd

Qa p D ap

macriexclsup1

macrr 2

H

cent(28)

Qa p Dpound1macr

r 2n

iexcl1iexclr 3

n

cent2cos2 micro


n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (29)

Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n

cent2currenpoundiexcl1 iexcl r 3

n

cent2C 8

3 rnz2n

iexcl1 iexcl 1

3 r 3n

centcurren 32 (30)

where xn D rn cos micro and zn D rn sin micro When rn iquest 1 (ie the solar sail is close to the asteroid compared

to its ideal Hill radius) then

tan reg raquo tanmicroiexcl1 C 4

3r 3n C cent cent cent

cent(31)

ap raquo sup1=r 2 cos2 micro pound1 C 4 sin2 micro iexcl 1r 3

n C cent cent centcurren

(32)

In this case the perturbation due to the solar tidal terms has a smalleffect on the motion of the spacecraft and is characterized by thevalue of r 3

n

Out-of-Plane Hovering PointsFor out-of-plane hovering points we must take into consideration

the most general form of the equations of motion at equilibrium withthe equations for SRP and the equation for r as given earlier namely

0 D iexclsup1=r 3r cos micro cos Atilde C 3Auml2r cos micro cos Atilde C ax (33)

0 D iexclsup1=r 3r cos micro sin Atilde C ay (34)

0 D iexclsup1=r 3r sin micro iexcl Auml2r sin micro C az (35)

with




Dividing Eq (34) by Eq (33) we nd that

tan Aacute Dpound1macriexcl

1 iexcl r 3n

centcurrentan Atilde (39)

From this equation we see that angle Aacute is related to angle Atilde alonewhich indicates that as the sail moves out of the xz plane it mustalso turn so that the sail normal is always pointing approximatelyradially outward from the asteroid


Next we want to solve for angle reg and ap as in the last sectionand so we again use the identity 1 C tan2 Aacute D sec2 Aacute to nd that

sec Aacute D

iexcl1 iexcl r 3

n

cent2C tan2 Atilde

iexcl1 iexcl r 3

n

cent2

12

(40)

Dividing Eq (35) by Eq (33) we arrive at the relation

tan reg Diexcl1 C 1

3 r 3n

centpoundiexcl1 iexcl r 3

n

cent2C tan2 Atilde

curreniexcl 12 secAtilde tan micro (41)

Solving for ap from Eq (33) we have

sup1=r 3 iexcl 3Auml2r cos micro cos Atilde D ap1 C tan2 Aacuteiexcl 32 1 C tan2 regiexcl 3

2

(42)

which becomes

ap Dpoundsup1

macrr 2

iexcl1 iexcl r 3

n

cent2cos2 micro cos2 Atilde

currenpoundiexcl1 iexcl r 3

n

cent2

C 83 r 3

n sin2 microiexcl1 iexcl 1

3r 3n

centC r 3

n

iexcl2 iexcl r 3

n

centsin2 Atilde cos2 micro

curren 32 (43)

and after normalizing by dividing by sup1=r 2H as before we have

Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2 C 83rnz2

n

iexcl1 iexcl 1

3r 3

n

cent

C rn y2n

iexcl2 iexcl r 3

n

centcurren 32 (44)

again giving the sail acceleration and orientation as a function ofsail position In this case however we de ne the normalized po-sition coordinates as xn D rn cos micro cos Atilde yn D rn cos micro sin Atilde andzn D rn sinmicro

Once again when rn iquest 1 we can expand each function about rn

and nd

tan Aacute raquo tan Atildeiexcl1 C r 3

n C cent cent centcent

(45)

tan reg raquo 13 tan micro

pound3 C r 3

n 1 C 3 cos2 Atilde C cent cent centcurren

(46)

ap raquo sup1=r 2 sec2 Atilde sec2 microcopy1 C r 3

n [iexcl1 C 3 sin2 Atilde

C 1 C 3 cos2 Atilde sin2 micro ] C cent cent centordf

(47)

Notice that when Atilde D 0 the approximations for tan reg and ap revertback to those found in the xz plane

Practical Consideration of the Hovering PointsThe existence of these equilibrium points is of interest for scien-

ti c missions One application is to use these points to monitor anasteroid at a constant phase angle for an extended length of time Toimplement such a design however it is necessary to estimate thenormalized sail acceleration Qa p before sail construction and launchThus we rst look in detail at what parameters are needed to esti-mate the value of Qa p at a given asteroid In particular we can showthat this value depends only on the sailrsquos acceleration ap0 and theasteroidrsquos gravitational parameter and is independent of the aster-oidrsquos distance from the sun We see this by expanding the de nitionfor Qa p

Qa p D ap

macr9sup1Auml4

13 (48)

which when we combine with Eq (6) gives

Qa p Dap0 R2

0

9sup1Auml4 R613

(49)

Fig 2 Spacecraft position contours for different values of Aumlap allparameters are nondimensionalized

For an asteroid in a circular orbit however we have Auml Dpsup1sun=R3 Therefore the relation becomes

Qa p Dap0 R2

0iexcl9sup1sup12

sun

cent 13

(50)

or in terms of the sail lightness number from Eq (8)

Qa p D macrsup1sun=9sup113 (51)

This form of Qa p relates the normalized value of ap to the asteroidmass and the sailrsquos acceleration Once a target asteroid is identi edits gravitational parameter sup1 can be constrained by ground-basedobservations of its size and spectral type (which will put constraintson its density) This estimate may allow suf cient accuracy for pre-liminary mission planning incorporating suf cient reserves to coverthe likely uncertainty range of the asteroidrsquos mass

In choosing the sail acceleration parameter it will be necessaryto evaluate the possible hovering positions in the asteroid frame Inthe following we only consider hovering points in the xz-plane thesuitable generalizations can be found for hovering positions out ofthis plane as well Figure 2 presents a contour plot of normalizedsail acceleration as a function of position in the xnzn -plane Theseresults are similar to the contours of sail lightness numbers thatMcInnes2 found in the restricted three-body problem but may havea signi cantly different form for a nonperfectly re ecting sail3 Forcomparison purposes the asteroids that we use for this paper haveQa p values of approximately 170 for Vesta 533 for Eros and 1331for Ida with corresponding macr values of 01686 00169 and 00843respectively

A sail with a given sail acceleration (which is transformed into Qa p

at a given asteroid) can ideally move along lines of constant contourby modifying the orientation of the sail Thus the line of constantQa p de nes the hovering positions that a sail can have at an asteroidMcInnes2 determined that all such hovering points were unstable inthe Lyapunov sense but were controllable using feedback controlto sail attitude alone (see also McInnes6 ) Instability and controlla-bility can be established similarly here because McInnesrsquos2 resultssatis ed equations that took into account a general form for thepotential

In planning multiple-objective missions to asteroids it would bebest to determine the sail acceleration necessary for the desiredradial distance from the largest asteroid rst Once the maximumsail acceleration is chosen nothing can be done to increase it shortof ejecting mass At the same time the necessary sail accelerationat the smallest target asteroid must be considered to design a usefulhovering position at that body In general we see from Fig 2 thatfor large values of Qa p (which can correspond to small asteroids) itis always possible to nd a hovering position that corresponds tothat larger value In general the sail will be forced to hover outof the asteroid orbital plane (ie at nonzero values of zn ) to also


Table 1 Asteroid parameters maximum and minimum sail accelerationsand normalization factors for asteroids Vesta Eros and Ida

Asteroid r0 km rH km sup1 km3 s2 QR AU aMp0

mms2 amp0

mms2 sup1=r2H mms2

Vesta 24427 116680 142568 236 8317 00494 00105Eros 887 2308 00005 146 079 00012 00009Ida 1565 9064 00038 286 783 00035 00005

ensure that its hovering radius is suf ciently large as compared tothe asteroidrsquos radius

Finally we discuss some particulars of the contour plot in Fig 2First we focus on the asteroid night side (0 middot micro lt frac14=2) FromEq (30) we see that it takes an in nitely large Qa p to hover in theregion where rn D 1 unless zn D 0 in which case the necessary Qa p

will drop to 0 (this situations occurs at the equilibrium point alongthe Cx axis which lies in the asteroidrsquos shadow) We can understandhow Qa p approaches 1 as rn goes to 1 by inspection of Eq (25) asrn approaches 1 tan reg tends to 1 which implies that reg goes to frac14=2as we approach this radius As this case occurs we lose the net solarradiation pressure force on the sail although we still must generateforce to make the sail hover Thus a larger sail acceleration is neededto counter this effect which leads to in nite values at rn D 1 We canalso see from Eq (30) that it is impossible to hover outside of theHill radius rn D 1 as the tidal and SRP forces then cause the sail toescape Conversely on the asteroid day side (frac14=2 lt micro middot frac14 ) rn mustbe greater than 1 to achieve equilibrium because tan micro is negativein this region The situation is reversed here because the sail cannothover close to the asteroid but must maintain some distance fromit Again as rn approaches 1 the necessary acceleration becomeslarge for reasons similar as described

Orbital Constraints and DynamicsWe now explore orbital options for the sail about the asteroid We

look for orbits that are stable in the sense that they neither impactthe asteroid nor escape Here we use results from previous investi-gations to identify a family of stable orbits for a solar sail7iexcl9 Thereare constraints on the orbit geometry and orbit size (as a function ofthe sail acceleration) for these trajectories to be feasible These con-straints are presented as a function of basic asteroid parameters andheliocentric orbit When these constraints are met and the orbits arefeasible they have a number of features that make them attractiveincluding better asteroid coverage and nearly trivial station-keepingcosts In this section we will assume reg D Aacute D 0 so that the sail isface-on to the sun For this case then a D [ap 0 0]

Sail Constraints for Bound OrbitsA suf cient condition for the orbit semimajor axis of a spacecraft

subject to large solar radiation pressure perturbations to be orbitallybound to an asteroid can be derived to be1011

asc lt QR=4p

sup1=ap0 (52)

where asc is the orbit semimajor axis and QR is the asteroidndashsundistance in AU This equation places a restriction on the size of anorbit as a function of the asteroid mass distance from the sun andsail acceleration Spacecraft with semimajor axes greater than thislimit are subject to being stripped from the asteroid and injected intoa heliocentric orbit

De ning the normalized semimajor axis Qasc D asc=r0 where r0 isthe mean asteroid radius we can derive the maximum sail acceler-ation for feasible orbital operations for the case when Qasc D 1

aMp0

D QR2=16iexclsup1

macrr 2

0

cent(53)

This equation is clearly anupper boundwhereas in generalwe wishto limit the sailrsquos acceleration as a function of the orbit semimajoraxis Qasc

ap0 ltiexcl1macr

Qa2sc

centaM

p0(54)

where Qasc gt 1 in general Some characteristic values of the param-eter aM

p0are shown for different asteroids in Table 1 These values

indicate that smaller asteroids will require low values of sail accel-eration for orbital options to be feasible about them

For a sail orbit that is bound according to the preceding ex-pression it has been found9 that the semimajor axis of the orbit isconstant on average The other mean orbit elements are not con-stant however and in general will have large secular variationsover time We con rm these ndings in our numerical integrationsas well

Sail Orbit Dynamics About AsteroidsAssuming we have a solar sail with sail acceleration less than

the bound given in Eq (54) it becomes feasible to discuss orbitaloperations of the sail about the asteroid The general orbital dynam-ics of such a bound highly perturbed orbit are discussed in greaterdetail in Refs 7ndash9 In Ref 9 a particular class of stable orbits inthe highly perturbed SRP problem was identi ed that is particularlywell suited to solar sails These orbits lie in the plane perpendicu-lar to the asteroidndashsun line nominally have their periapsis aligned90 deg above or below the orbit plane and have their eccentricitychosen according to the equation

e D cos deg (55)

tan deg D 32ap0 R2

0

pasc=Psup1sup1sun (56)

where R0 is one astronomical unit in kilometers and P is the or-bit parameter of the asteroid equal to the heliocentric radius for acircular orbit We can rewrite this result as

tan deg Dp

Qascap0

macram

p0(57)

amp0

D 23

qsup1sun

macrR3

0

qQRsup1=r0 (58)

amp0

raquo 133 pound 10iexcl7

qQRsup1=r0 (59)

where QR is the asteroidndashsun distance in astronomical units Somecharacteristic values of am

p0for some select asteroids are shown in

Table 1 From these values we note that tan deg Agrave 1 in general indi-cating that the stable orbits will have near zero eccentricity

Shown in Figs 3ndash7 is a sample of numerically integrated sailtrajectories about an asteroid with the sail placed in a stable orbitas described Note that the integration occurs in the frame rotatingwith the asteroidndashsun line This condition implies that these or-bits are sun-synchronous in that they remain xed in the sun planeof sky as shown in Fig 4 We also note that the trajectory oscil-lates about a circular orbit in general but is stable as shown inFig 5

These trajectories were calculated using the ode45 algorithm fromMATLABreg (The MathWorks Inc) on the state equations taken fromthe equations of motion A relative tolerance of 10iexcl6 and an absolutetolerance of 10iexcl8 was used each time Each trajectory was computedover a different time interval for the sake of clarity in the image

We can extend the analytical solution of the sail trajectory aboutthe asteroid beyond this simple circular orbit relying on an analyti-cal solution to the averaged solar radiation pressure problem formu-lated by Richter and Keller8 In their paper they nd the averagedangular momentum and periapsis vector of the orbit as a functionof asteroid true anomaly with respect to the sun Taking this generalsolution we specialize it to our particular case by assuming that the


perturbation angle deg frac14=2 We then nd the angular momentumvector and periapsis vector to be8

H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN

C5 cosN iexcl C6 sinN

3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)

Fig 3 Integrated orbit for a sail about Vesta using orbital elementsasc = 13r0 e = = ordm0 = 0 and i = AumlX 0 = frac142 and parameters r0 =24427 km 0 lt t lt 116 days and ap0 = 1 mms2 spacecraft position unitsare in kilometers trajectory is shown in the coordinate system rotatingwith the asteroid about the sun

Fig 4 Integrated orbit for a sail about Vesta in the xz plane usingorbital elements and parameters of Fig 3 trajectory is shown in thecoordinate system rotating with the asteroid about the sun

Fig 5 Integrated orbit for a sail about Vesta in the yz plane usingorbital elements and parameters of Fig 3 trajectory is shown in thecoordinate system rotating with the asteroid about the sun

Fig 6 Integrated orbit for a sail about Eros with asc = 4r0 e i AumlX 0and ordm0 the same as in Fig 3 r0 = 887 km 0 lt t lt 12 days and ap0 =01 mms2 trajectory is shown in the coordinate system rotating withthe asteroid about the sun

Fig 7 Integrated orbit for a sail about Ida with asc = 6r0 e i AumlX 0and ordm0 the same as in Fig 3 r0 = 1565 km 0 lt t lt 24 days and ap0 =05 mms2 trajectory is shown in the coordinate system rotating withthe asteroid about the sun

where N is an angle proportional to the asteroid true anomaly aboutthe sun The constants Ci can be related to the initial osculatingorbital elements of the sail orbit as

C1 Dp

1 iexcl e2 sin i sin QAuml (62)

C2 D iexcle sin i sin (63)

C3 D iexclp

1 iexcl e2 sin i cos QAuml (64)

C4 D iexcle[cos QAuml cos iexcl cos i sin QAuml sin ] (65)

C5 Dp

1 iexcl e2 cos i (66)

C6 D iexcle[sin QAuml cos C cos i cos QAuml sin ] (67)

where e is the orbit eccentricity i is the orbit inclination measuredfrom the asteroid ecliptic plane QAuml is the orbit longitude of the as-cending node measured from the sunndashasteroid line and is the ar-gument of periapsis From this solution it is possible to evaluate theevolution of the mean orbit elements as a function of N raquo nt= cos deg where n is the asteroidrsquos mean motion about the sun t is time and degis the perturbation angle again We note that for 0 lt frac14=2 iexcl deg iquest 1which is our case the angle N will increase very rapidly with timeindicating that the mean solutions will oscillate rapidly This re-sult is not necessarily desirable as can be seen if we take as initialconditions i D 0 and e D 0 giving C1 D C2 D C3 D C4 D C6 D 0 andC5 D 1 The solution then becomes

e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)

meaning that the orbit will remain in the ecliptic plane and theeccentricity will repeatedly pass through a value of unity eventuallycausing the sail to impact on the asteroid surface

If instead we take initial conditions i D frac14=2 e D 0 and QAuml D QAuml0we nd the solution

i D frac14=2 (71)

D sectfrac14=2 (72)

e D jcos QAuml0jjsinN j (73)

tan QAuml D tan QAuml0=cosN (74)

Now the orbit plane remains normal to the asteroid ecliptic the lon-gitude of the ascending node varies between QAuml0 middot QAuml middot frac14 iexcl QAuml0 andthe eccentricity varies between 0 middot e middot jcos QAuml0j These conditionsare a potentially useful generalization of the stable circular orbit dis-cussed earlier that allows the orbit plane to move out of the sun planeof sky while bounding the maximum value of eccentricity In Fig 8we show another plot of a numerically integrated orbit that showsthis characteristic solution Note that even though the oscillationsin mean elements occur rapidly it is not valid to average over thembecause the amplitude of the oscillations are large In Fig 9 weshow how eccentricity varies with QAuml This particular type of orbitallows the asteroid to be imaged from a range of phase angles

Fig 8 Integrated orbit for a sail about Vesta with asc e i ordm0 andap0 the same as in Fig 3 AumlX 0 = 45 deg and 0 lt t lt 116 days trajectoryis shown in the scoordinate system rotating with the asteroid about thesun

Fig 9 Eccentricity vs longitude of the ascending node relative to thesunndashasteroid line for the orbit about Vesta depicted in Fig 8

This family of solutions provides a viable family of stable sun-synchronous sail orbits about asteroids that require no active controlto maintain and that allows the sail to view the asteroid over a rangeof phase angles for long periods of time Because these orbits arestable we do not need to apply any closed-loop control to stabilizethe sail which implies that the orbital operations will be safer andrequire less operational effort

For future studies it will be of interest to include the effects ofan asteroidrsquos nonspherical gravity eld in the sail orbital dynamicsThe effects of these perturbations are dif cult to predict but couldlead to situations where the orbital dynamics become signi cantlymodi ed perhaps leading to the destabilization of the orbital mo-tion in some cases In general however it is possible to mitigatethe effects of the asteroid gravity eld on orbital motion by mov-ing to a larger semimajor axis although this will further constrainthe sail acceleration [Eq (54)] In the simplest case the effect ofthe oblateness of the asteroid will cause the orbit plane to precessabout the asteroid pole This precession will ght with the preces-sion of the orbit plane caused by the solar radiation pressure withthe net effect of this interaction being an open question for futurestudy

ConclusionsWe have shown that both hovering and orbital options are avail-

able with a solar sail spacecraft operating in close proximity to aspherical asteroid in a circular orbit There are a continuum of hov-ering positions available near the asteroid These hovering pointsdepend on sail attitude asteroid mass and the acceleration of thesail For a given distance from the asteroid and sail acceleration thesail attitude can be determined to maintain that position for extendedperiods of time

It has also been shown that several orbital options are availablethat offer good coverage of the asteroid and that neither impact thesurface nor escape These orbits are stable and sun synchronous Asail could maintain orbit for extended periods of time in such anorbit Some asteroids (such as Eros) require small sail accelerationsfor orbital operations to be feasible In such cases as these whereit may be necessary to reduce the sail acceleration we can makeadjustments with trims in the sail area With the possible excep-tion of spin-stabilized sails the deployment mechanism could bereversed so that the sail could be partially refurled In this way wecould decrease the sail area and thereby decrease ap making orbitaloperations about smaller asteroids possible

Solar sails offer the advantage that long-duration missions can beplanned visiting several solar system objects without the spacecraftmass and complexity involved with conventional propulsion and fuelsupply For example after one asteroid has been investigated thespacecraft is capable of escaping from orbit in principle under sailpropulsion alone and traveling to another asteroid or returning toEarth This capability offers the potential for low-cost and exiblesolar system exploration

AcknowledgmentsResearch at the University of California San Diego was sup-

ported by the National Science Foundation NSF-OPP 98 06941Research at the University of Michigan was supported by theTelecommunications and Missions Operations Directorate Tech-nology Program by a grant from the Jet Propulsion LaboratoryCalifornia Institute of Technology which is under contract withNASA

References1McInnes C R Solar Sailing Technology Dynamics and Mission

Applications Praxis Chichester England UK 1999 pp 1ndash552McInnes C R McDonald A J C Simmons J F L and MacDonald

E W ldquoSolar Sail Parking in Restricted Three-Body Systemsrdquo Journal ofGuidance Control and Dynamics Vol 17 No 2 1994 pp 399ndash406

3McInnes C R ldquoArti cial Lagrange Points for a Partially Re ecting FlatSolar Sailrdquo Journal of Guidance Control and Dynamics Vol 22 No 11999 pp 185ndash187

4Scheeres D J ldquoThe Restricted Hill Four-Body Problem with Applica-tions to the EarthndashMoonndashSun Systemrdquo Celestial Mechanics and DynamicalAstronomy Vol 70 No 2 1998 pp 75ndash98


5Scheeres D J and Marzari F ldquoSpacecraft Dynamics Far from aCometrdquo Journal of Astronautical Sciences (submitted for publication)

6McInnes A I ldquoStrategies for Solar Sail Mission Design in the CircularRestricted Three-Body Problemrdquo MS Thesis School of Aeronautics andAstronautics Purdue Univ West Lafayette IN Aug 2000

7Mignard F and Henon M ldquoAbout an Unsuspected IntegrableProblemrdquo Celestial Mechanics Vol 33 No 2 1984 pp 239ndash

2508Richter K and Keller H U ldquoOn the Stability of Dust Particle Or-

bits Around Cometary Nucleirdquo Icarus Vol 114 No 2 1995 pp 355ndash

371

9Scheeres D J ldquoSatelliteDynamics About SmallBodies AveragedSolarRadiation Pressure Effectsrdquo Journal of the Astronautical Sciences Vol 47Nos 1 and 2 1999 pp 25ndash46

10Hamilton D P and Burns J A ldquoOrbital Stability Zones About Aster-oids II The Destabilizing Effects of Eccentricities and of Solar RadiationrdquoIcarus Vol 96 No 1 1992 pp 43ndash64

11Scheeres D J ldquoAnalysis of Orbital Motion Around 433 Erosrdquo Journalof the Astronautical Sciences Vol 43 No 4 1995 pp 427ndash452

A C TribbleAssociate Editor


this study we focus only on the behavior of the solar sail spacecraftafter rendezvous has been achieved

For our model we assume a spherical point mass asteroid a per-fectly re ecting solar sail and a circular asteroid orbit about the sunThese assumptions can be relaxed in future studies We rst developthe basic equations of motion for the system We then consider theconstraints and feasibility of hovering sail trajectories complement-ing the studies made by McInnes et al2 and McInnes3 We nd acontinuum of hovering points that would allow station keeping tomonitor the asteroid at a particular phase angle for extended periodsof time Next we consider the constraints and feasibility of orbitaloperations about an asteroid from which we develop a number ofbasic criteria for necessary sail acceleration as a function of asteroidparameters The orbital trajectories are found to be stable and to of-fer good coverage of the asteroid surface These trajectories wouldbe very useful for mapping purposes

ModelTo model the dynamics of a sail close to an asteroid we can apply

the Hill approximation with appropriate generalizations to accountfor the solar sailrsquos force vector The Hill approximation applies verywell to the motion of spacecraft about asteroids due to the smallmass of asteroids relative to the sun and due to the proximity that thecraft maintains to the asteroid4 The Hill approximation can also beeasily modi ed to account for the effect of solar radiation pressureleading to the equations of motion5

Rx iexcl 2Auml Py D iexclsup1=r 3x C 3Auml2x C ax (1)

Ry C 2Auml Px D iexclsup1=r 3y C ay (2)

Rz D iexclsup1=r 3z iexcl Auml2z C az (3)

where the origin of the rotating reference frame is centered at theasteroid with the positive x axis in the antisolar direction the z axisnormal to the asteroid orbit ecliptic and the y axis according tothe right-hand rule This frame rotates about the z axis with angu-lar velocity Auml D

psup1sun=a3

ast where sup1sun is the sunrsquos gravitationalparameter and aast is the asteroidrsquos heliocentric orbit radius in kilo-meters The gravitational parameter of the asteroid is denoted as sup1

The spacecraft is propelled by solar radiation ux incident on thesail which is herein assumed to be perfectly re ecting and planarThe sail receives anaction force from incident sunlight and a reactionforce from the re ected light so that the net force is directed normalto the surface of the sail and pointing away from the sun The acceler-ation vector of this force is denoted as a D [ax ay az] D apl cent n2n

The unit vector l de nes the direction of the incident light and nis the unit normal to the sail In the rotating reference frame l will al-ways equal [1 0 0] and n will be [cos Aacute cos reg sin Aacute cos reg sin reg]where reg is the sail pitch angle with respect to the sun line and Aacuteis the orientation angle about the z axis as shown in Fig 1 The

Fig 1 Model schematic for a solar sail spacecraft about an asteroid

direction of the solar radiation pressure (SRP) force can never bepointed toward the sun and so the sail attitude is constrained suchthat l cent n cedil 0

The parameter ap is called the sail acceleration and is representedas

ap Diexcl2W

macrcR2

0

centiexclR2

0

macrR2

cent1=frac34 (4)

for a perfectly re ecting solar sail where W=R20 is the solar radiation

intensity c is the speed of light R0 is the distance from the sun to theEarth R is the distance from the sun to the sail in kilometers and frac34is the mass to area ratio of the sail or sail loading For a nonperfectsail there will be an additional component of acceleration transverseto the sail normal At 1 AU

ap D ap0 Diexcl2W

macrcR2

0

cent1=frac34 (5)

and in general

ap D ap0

iexclR2

0

macrR2

cent(6)

Thus SRP varies as 1=R2 about the sun (but see discussion byMcInnes1)

Multiplying and dividing by the sunrsquos gravitational parametersup1sun we have

ap D 2W=csup1suniexclsup1sun

macrR2

cent1=frac34 (7)

or

ap D macriexclsup1sun

macrR2

cent(8)

where macr D frac34 curren=frac34 is the nondimensional sail loading parametercalled the sail lightness number de ned to be the ratio of theSRP acceleration to the solar gravitational acceleration The criticalsail loading parameter frac34 curren D 2W=csup1sun is a constant whose value isapproximately 153 gm2

The value of the sail acceleration varies with both the ef ciencyand size of the sail1 For example McInnes1 shows that a typicalvalue for a 98 pound 98 m square sail that carries a payload of 25 kg is1 mms2 At this acceleration a solar sail spacecraft has trip timesto asteroids comparable to conventional spacecraft This spacecraftdesign yields a sail loading parameter of 912 gm2 or a sail lightnessnumber of 01678 This parameter can be reached with a sail lmthickness of 2 sup1m

Solar Sail Hovering Constraints and DynamicsGiven the basic equations of motion we search for conditions

under which the spacecraft can be placed into an arti cial equilib-rium point found by setting all of the time derivatives to zero and nding the combination of SRP acceleration and natural forces thatwill yield no net acceleration acting on the sail The result is a spe-cialization of the analysis carried out by McInnes2 for the restrictedthree-body problem now applied to the Hill problem Followingthe Hill equations we nd the equations of motion of a solar sailspacecraft to be as in Eqs (1ndash3)

Because the sail normal is required to point in the antisolar direc-tion we are limited to iexclfrac14=2 middot reg middot frac14=2 The problem is symmet-ric about the angle reg D 0 however and so we need only consider0 middot reg middot frac14=2 When reg D frac14=2 the sail is edge on to the sun In thissituation we effectively are in irons that is we have no SRP forceat all Although there may be instances when this effect would bedesirable we do not consider this case explicitly Thus we limit ourdiscussion to 0 middot reg lt frac14=2

Likewise to satisfy the constraint l cent n cedil 0 we only consider Aacutelimited to iexclfrac14=2 middot Aacute middot frac14=2 The problem is also symmetric aboutAacute D 0 therefore we consider Aacute such that 0 middot Aacute lt frac14=2 for the samereasons stated for reg










and the vector r as










which becomes


1 iexcl r 3n

centcurren(18)






ay D 0 (20)






which gives

tan reg Dmicro

sup1=r 3 C Auml2


paratan micro (24)


tan reg D

Aacute1 C 1

3r 3n

1 iexcl r 3n

tan micro (25)



n


2 (26)


apDpoundsup1

macrr 2

iexcl1iexclr 3

n

cent2cos2 micro


n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (27)


H We then nd

Qa p D ap

macriexclsup1

macrr 2

H

cent(28)

Qa p Dpound1macr

r 2n

iexcl1iexclr 3

n

cent2cos2 micro


n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (29)

Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2C 8

3 rnz2n

iexcl1 iexcl 1

3 r 3n

centcurren 32 (30)





cent(31)



(32)


n






with






1 iexcl r 3n





sec Aacute D

iexcl1 iexcl r 3

n

cent2C tan2 Atilde

iexcl1 iexcl r 3

n

cent2

12

(40)


tan reg Diexcl1 C 1

3 r 3n


n

cent2C tan2 Atilde




2

(42)

which becomes

ap Dpoundsup1

macrr 2

iexcl1 iexcl r 3

n



n

cent2

C 83 r 3


3r 3n

centC r 3

n

iexcl2 iexcl r 3

n


curren 32 (43)


Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2 C 83rnz2

n

iexcl1 iexcl 1

3r 3

n

cent

C rn y2n

iexcl2 iexcl r 3

n

centcurren 32 (44)



and nd



(45)


pound3 C r 3


(46)




(47)




Qa p D ap

macr9sup1Auml4

13 (48)


Qa p Dap0 R2

0

9sup1Auml4 R613

(49)



Qa p Dap0 R2

0iexcl9sup1sup12

sun

cent 13

(50)











mms2 amp0

mms2 sup1=r2H mms2

Vesta 24427 116680 142568 236 8317 00494 00105Eros 887 2308 00005 146 079 00012 00009Ida 1565 9064 00038 286 783 00035 00005








asc lt QR=4p

sup1=ap0 (52)



aMp0

D QR2=16iexclsup1

macrr 2

0

cent(53)


ap0 ltiexcl1macr

Qa2sc

centaM

p0(54)







e D cos deg (55)

tan deg D 32ap0 R2

0



tan deg Dp

Qascap0

macram

p0(57)

amp0

D 23

qsup1sun

macrR3

0

qQRsup1=r0 (58)

amp0


qQRsup1=r0 (59)









H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN


3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)







C1 Dp



C3 D iexclp



C5 Dp




e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371














and the vector r as










which becomes


1 iexcl r 3n

centcurren(18)






ay D 0 (20)






which gives

tan reg Dmicro

sup1=r 3 C Auml2


paratan micro (24)


tan reg D

Aacute1 C 1

3r 3n

1 iexcl r 3n

tan micro (25)



n


2 (26)


apDpoundsup1

macrr 2

iexcl1iexclr 3

n

cent2cos2 micro


n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (27)


H We then nd

Qa p D ap

macriexclsup1

macrr 2

H

cent(28)

Qa p Dpound1macr

r 2n

iexcl1iexclr 3

n

cent2cos2 micro


n

cent2C 8

3 r 3n sin2 micro

iexcl1iexcl 1

3r 3n

centcurren 32 (29)

Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2C 8

3 rnz2n

iexcl1 iexcl 1

3 r 3n

centcurren 32 (30)





cent(31)



(32)


n






with






1 iexcl r 3n





sec Aacute D

iexcl1 iexcl r 3

n

cent2C tan2 Atilde

iexcl1 iexcl r 3

n

cent2

12

(40)


tan reg Diexcl1 C 1

3 r 3n


n

cent2C tan2 Atilde




2

(42)

which becomes

ap Dpoundsup1

macrr 2

iexcl1 iexcl r 3

n



n

cent2

C 83 r 3


3r 3n

centC r 3

n

iexcl2 iexcl r 3

n


curren 32 (43)


Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2 C 83rnz2

n

iexcl1 iexcl 1

3r 3

n

cent

C rn y2n

iexcl2 iexcl r 3

n

centcurren 32 (44)



and nd



(45)


pound3 C r 3


(46)




(47)




Qa p D ap

macr9sup1Auml4

13 (48)


Qa p Dap0 R2

0

9sup1Auml4 R613

(49)



Qa p Dap0 R2

0iexcl9sup1sup12

sun

cent 13

(50)











mms2 amp0

mms2 sup1=r2H mms2

Vesta 24427 116680 142568 236 8317 00494 00105Eros 887 2308 00005 146 079 00012 00009Ida 1565 9064 00038 286 783 00035 00005








asc lt QR=4p

sup1=ap0 (52)



aMp0

D QR2=16iexclsup1

macrr 2

0

cent(53)


ap0 ltiexcl1macr

Qa2sc

centaM

p0(54)







e D cos deg (55)

tan deg D 32ap0 R2

0



tan deg Dp

Qascap0

macram

p0(57)

amp0

D 23

qsup1sun

macrR3

0

qQRsup1=r0 (58)

amp0


qQRsup1=r0 (59)









H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN


3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)







C1 Dp



C3 D iexclp



C5 Dp




e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371







sec Aacute D

iexcl1 iexcl r 3

n

cent2C tan2 Atilde

iexcl1 iexcl r 3

n

cent2

12

(40)


tan reg Diexcl1 C 1

3 r 3n


n

cent2C tan2 Atilde




2

(42)

which becomes

ap Dpoundsup1

macrr 2

iexcl1 iexcl r 3

n



n

cent2

C 83 r 3


3r 3n

centC r 3

n

iexcl2 iexcl r 3

n


curren 32 (43)


Qa p Dpound1macr

x2n

iexcl1 iexcl r 3

n


n

cent2 C 83rnz2

n

iexcl1 iexcl 1

3r 3

n

cent

C rn y2n

iexcl2 iexcl r 3

n

centcurren 32 (44)



and nd



(45)


pound3 C r 3


(46)




(47)




Qa p D ap

macr9sup1Auml4

13 (48)


Qa p Dap0 R2

0

9sup1Auml4 R613

(49)



Qa p Dap0 R2

0iexcl9sup1sup12

sun

cent 13

(50)











mms2 amp0

mms2 sup1=r2H mms2

Vesta 24427 116680 142568 236 8317 00494 00105Eros 887 2308 00005 146 079 00012 00009Ida 1565 9064 00038 286 783 00035 00005








asc lt QR=4p

sup1=ap0 (52)



aMp0

D QR2=16iexclsup1

macrr 2

0

cent(53)


ap0 ltiexcl1macr

Qa2sc

centaM

p0(54)







e D cos deg (55)

tan deg D 32ap0 R2

0



tan deg Dp

Qascap0

macram

p0(57)

amp0

D 23

qsup1sun

macrR3

0

qQRsup1=r0 (58)

amp0


qQRsup1=r0 (59)









H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN


3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)







C1 Dp



C3 D iexclp



C5 Dp




e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371








mms2 amp0

mms2 sup1=r2H mms2

Vesta 24427 116680 142568 236 8317 00494 00105Eros 887 2308 00005 146 079 00012 00009Ida 1565 9064 00038 286 783 00035 00005








asc lt QR=4p

sup1=ap0 (52)



aMp0

D QR2=16iexclsup1

macrr 2

0

cent(53)


ap0 ltiexcl1macr

Qa2sc

centaM

p0(54)







e D cos deg (55)

tan deg D 32ap0 R2

0



tan deg Dp

Qascap0

macram

p0(57)

amp0

D 23

qsup1sun

macrR3

0

qQRsup1=r0 (58)

amp0


qQRsup1=r0 (59)









H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN


3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)







C1 Dp



C3 D iexclp



C5 Dp




e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371







H Dp

sup1asc

2

4C1

C2 sinN C C3 cosN


3

5 (60)

e D iexcl

2

4C4

C5 sinN C C6 cosN


3

5 (61)







C1 Dp



C3 D iexclp



C5 Dp




e D jsinN j (68)


D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371






D sectfrac14=2 (69)

i D 0 (70)



i D frac14=2 (71)

D sectfrac14=2 (72)

























371











371





Solar Sail Orbit Operations at Asteroids

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Transcript of Solar Sail Orbit Operations at Asteroids