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Icarus 229 (2014) 278–294
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Icarus
journal homepage: www.elsevier .com/ locate/ icarus
Stability of rubble-pile satellites
0019-1035/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.icarus.2013.09.023
⇑ Address: Department of Mechanical Engineering, IIT Kanpur, Kanpur 208016,India.
E-mail address: [email protected]
Ishan Sharma ⇑Department of Mechanical Engineering, IIT Kanpur, Kanpur 208016, IndiaMechanics & Applied Mathematics Group, IIT Kanpur, Kanpur 208016, India
a r t i c l e i n f o
Article history:Received 19 April 2013Revised 21 September 2013Accepted 24 September 2013Available online 26 October 2013
Keywords:Satellites, dynamicsSatellites, compositionSatellites, shapesGeophysicsInteriors
a b s t r a c t
We consider the stability of rubble-pile satellites that are held together by their own gravity. A satellite issaid to be stable whenever it is both orbitally and structurally stable to both orbital and structural per-turbations. We restrict attention to satellites whose dimensions are small compared to their respectiveorbital radii and their associated planets’ sizes. In this case, we show that a satellite is stable wheneverit is orbitally stable to orbital perturbations and structurally stable to structural perturbations. Orbitalstability is investigated by a spectral analysis, while structural stability is probed by appropriatelyextending the work of Sharma [Sharma, I., 2012. Stability of rotating non-smooth complex fluids. J. FluidMech. 708, 71–99; Sharma, I., 2013. Structural stability of rubble-pile asteroids. Icarus 223, 367–382]. Thestability test is then applied to planetary satellites of the Solar System that are suspected to be granularaggregates, including many of the recently discovered smaller moons of the giant planets.
� 2013 Elsevier Inc. All rights reserved.
1. Introduction
Many planetary satellites including those of Mars and some ofthe newly discovered moons of the giant planets are suspected tobe granular aggregates held together by self gravity alone. Theequilibrium shapes of these objects have been previously analyzedutilizing volume-averaging by Sharma (2009), henceforth Paper I,assuming them to be of ellipsoidal shape and on tidally-lockedcircular orbits about a massive, possibly oblate, central planet. Inthe context of shapes and failure of solid satellites, we also mentionrelated work of Aggarwal and Oberbeck (1974) who estimated theRoche limit of an elastic and spherical satellite while assuming brit-tle failure, Dobrovolskis (1990) who combined the Navier criterionfor failure of sandy materials with an elastic stress analysis ofellipsoids, Davidsson (1999, 2001) who assumed that the satellitefailed when the maximum value of the average normal stress acrosssome critical plane within the satellite surpassed the constituentmaterial’s tensile strength, and Holsapple and Michel (2006,2008) who employed limit analysis and a pressure-dependentMohr–Coulomb yield condition to investigate the equilibrium ofgranular ellipsoids in the presence of self-gravity and tidal interac-tion. The equilibrium shapes of fluid satellites has, of course, beenextensively studied, see, e.g., Chandrasekhar (1969, Chapter 8).
Stability analyses of planetary satellites has typically focussedon the orbital stability of these objects, disregarding the responseof the satellite as a distributed mass that may possibly yield and
deform significantly. In astrophysical applications, structural sta-bility of orbiting fluid ellipsoids has, however, been investigated.These fall under the stability of Roche and Darwin ellipsoids; seeChandrasekhar (1969, Chapter 8). Some recent advances are dueto Lai et al. (1993) who considered the stability of compressibleinviscid fluid Roche and Roche-Riemann ellipsoids. These authorstested stability by minimizing an energy functional that was al-lowed to depend on parameters such as the ellipsoid’s shape, den-sity, mass, angular momentum, and internal vorticity.
Here, we investigate the stability of granular satellites. A satel-lite will be deemed stable only if its orbit and its structure are bothstable to both orbital and structural perturbations. Here, by struc-ture we mean the collection of material points that constitute thesatellite, and structural stability refers to this collective stayingclose to its equilibrium configuration; we will discuss stabilitymore precisely in Section 5. In the case of satellites whose prima-ries are much more massive, we will show in Section 5.2 thatorbital and structural stability may be considered separately.While orbital stability may then be determined by standard meth-ods, structural stability will be gauged by suitably extending Shar-ma’s (2013) stability analysis of asteroids to the case of satellitesby including the effect of tidal forces. We limit ourselves to homo-geneous velocity perturbations of the equilibrium state. For defi-niteness, at equilibrium the satellite will be assumed to be ofellipsoidal shape and on a circular orbit about a massive primary;see Fig. 1. While the present framework may be utilized for thestability analysis of any satellite system wherein the primary ismuch larger than the secondary, we will present final calculationssuitable for spheroidal primaries and will be most suitable forplanetary satellites. Other shapes of the primary may be probed
e1’ e
1
e2’
e3’
e2
e3
RP
S
Primary
SatelliteOrbital plane eR
Fig. 1. Equilibrium configuration of an ellipsoidal satellite of an oblate planet. Theunit vector eR locates the satellite with respect to the planet’s center.
I. Sharma / Icarus 229 (2014) 278–294 279
by appropriately modifying the tidal shape tensor, and this wouldextend the applicability of the current analysis to asteroidalsatellites.
We first derive the equations for a homogeneously deformingellipsoidal satellite.
2. Satellite dynamics
2.1. Structural deformation
Paper I derives equations for a homogeneously deforming ellip-soidal satellite of an oblate planet. For such a body, a materialpoint’s velocity is linearly dependent on its location with respectto the ellipsoid’s center. In stability investigations, it is expedientto phrase these equations in a frame O rotating at xðtÞ andattached to the satellite’s centroid S. With xðtÞ we associate ananti-symmetric angular-velocity tensor XðtÞ satisfying
x� x ¼ X � x; ð1Þ
where x is a material point’s location relative to S; x is X’s associ-ated axial vector. We employ such rotation rate vectors and theircorresponding tensors interchangeably.
In the rotating frame O, for a homogeneously deforming ellip-soid, a material point’s velocity relative to the ellipsoid’s center is
_x ¼ L � x; ð2Þ
where the dot ð�Þ indicates time derivative with respect to an obser-ver in O and L is the velocity gradient observed in O that dependsonly on time. The tensor L estimates the local rate of change of rel-ative displacement, while its symmetric ðDÞ and anti-symmetricðWÞ parts capture local deformation and rotation rates, respec-tively. The governing equations for homogeneous dynamics asobserved in O were found by Sharma (2013) to be
ð _Lþ L2Þ � I ¼ �rV þMT � _XþX2 þ 2X � L� �
� I ð3aÞ
and _I ¼ L � I þ I � LT ; ð3bÞ
where r is the volume-averaged stress tensor,
I ¼Z
Vqx � xdV ð4aÞ
and M ¼Z
Vqx � bdV ; ð4bÞ
are, respectively, the ellipsoid’s inertia tensor and external momenttensor due to applied body forces b, and q and V are the ellipsoid’sdensity and volume, respectively. In (3a), the last three bracketedterms on the right-hand side are, respectively, angular, centripetaland Coriolis’ accelerations, and act as external moment tensors inthe rotating frame O. Equation (3a) follows L’s evolution in O bybalancing internal stresses, external moments and inertial effects,
while (3b) describes the changing inertia tensor. We note that I isdifferent from the Euler moment of inertia tensor J commonly em-ployed in rigid body mechanics; cf. (25).
The tensor M includes the effect of the satellite’s own gravityand that of the planet. The moment due to the satellite’s self-grav-ity is found by Sharma et al. (2009) to be
MG ¼ �2pqGI � A; ð5Þ
where A is the gravitational shape tensor that captures the effect ofthe satellite’s ellipsoidal shape on its internal gravity. The tensor Ais diagonalized in the satellite’s principal axes coordinate system,and its components in that frame are available in Sharma et al.(2009). The gravitational force exerted on a unit mass within the sa-tellite at X ¼ xþ ReR by an ellipsoidal planet is
bQ ¼ �2pq0GB � X; ð6Þ
where q0 is the planet’s density and B is the tidal shape tensor thatcaptures the effect of the planet’s ellipsoidal shape and dependson X and the planet’s semi-major axes a0i. The tensor B is diagonal-ized in the planet’s principal axes frame and depends on the ellipsoi-dal coordinate of the material point at X; see Eq. (19) of Paper I. Toestimate the tidal moment, we first expand B as
B ¼ Bð0Þ þBð1Þ � xRþ Bð2Þ :
x � x
R2 þ � � � ;
where Bð0Þ;Bð1Þ and Bð2Þ are, respectively, second-, third- and fourth-
order tensors, and, in indical notation, Bð1Þ � x� �
ij ¼ Bð1Þijk xk and
Bð2Þ : x � x� �
ij ¼ Bð2Þijklxlxk. Note that because B is a symmetric tensor,
so too is Bð0Þ, while Bð1Þ and Bð2Þ are symmetric in their first two
arguments, i.e., Bð1Þijk ¼ Bð1Þjik and Bð2Þijkl ¼ Bð2Þjikl. We then substitute theabove expansion in (4b) and compute the resulting integrals, to ob-
tain the tidal moment due to the planet correct up to Oðjxj3=R3Þ as
MQ ¼ �2pq0GI � Bð0Þ þ eR � Bð1Þ� �T
h i; ð7Þ
where eR �Bð1Þ� �
jk ¼ eRiBð1Þijk ; see Paper I for more details. The total
moment is then
M ¼ MG þMQ ¼ �2pqGI � A� 2pq0GI � Bð0Þ þ eR �Bð1Þ� �T
h i: ð8Þ
Equation (3) along with (8) follow a homogeneously deformingellipsoidal satellite’s structural motion relative to frame O.
We now describe the satellite’s orbital motion about the planet.
2.2. Orbital motion
To track the satellite’s orbit, we equate the total force F actingon the satellite to its mass center’s acceleration:
mf€Rþ _XþX2� �� Rþ 2X � _Rg ¼ F; ð9Þ
where m is the satellite’s mass and the left-hand side is the absoluteacceleration of the satellite’s mass center written in the rotatingframe O. The total force F exerted by the oblate planet on the satel-lite is obtained by computing
RqbQ dV with bQ given by (6).
Employing B’s expansion from Section 2.1 of Paper I finds
F ¼ �2pq0GRm Bð0Þ � eR þ1
mR2 Bð1Þ þ eR � Bð2Þ� �
: I� �
;
correct up to Oðjxj3=R3Þ and ðBð1Þ þ eR � Bð2ÞÞ : I� �
j ¼ Bð1Þjkl þ eRiBð2Þijkl
� Ikl.
Because I=mR2 scales as �a2=R2, where �a is the satellite’s averagediameter, we may, for satellite systems where the separation fromthe planet is much greater than the satellite’s size, approximate F as
F � �2pq0GmBð0Þ � R: ð10Þ
CompatibleIncompatible
Yield surfaceσ
2
σ3
σ1
σEσ
0
Δσ
0
D1
D2
D3
σA
σB
DB
Fig. 2. The Drucker–Prager yield surface in principal – stress/strain space appearsas a circle when viewed on a plane perpendicular to the pressure axis r1 ¼ r2 ¼ r3.The point ‘E’ corresponds to the stress state at equilibrium. The strain rate DE
compatible to the stress state rE at ‘E’ is normal to the yield surface there. If theincompatible strain rate D0 is superimposed, the stress state immediately shifts to alocation r0 on the yield surface where the imposed strain rate would be compatible.This leads to jump Dr in the stress state. The stress state rA is not at yield, but maymove to a state rB on the yield surface with corresponding compatible strain rate DB
under the action of external forces; cf. Section 6.6.
280 I. Sharma / Icarus 229 (2014) 278–294
In doing so, we ignore the influence of the satellite’s shape anddeformation on its orbit. Replacing F in (9) we obtain
€Rþ ð _XþX2Þ � Rþ 2X � _R ¼ �2pq0GBð0Þ � R; ð11Þ
which follows the satellite’s motion about an ellipsoidal planet inO.To close the system of Eqs. (3), (8) and (11) we require knowl-
edge of the satellite’s rheology. This we now briefly describe.
3. Rheology
Sharma et al. (2006, 2009) and Sharma (2009, 2010) modeled arubble-pile’s yield response by that of a rigid perfectly-plasticcohesionless material obeying a Drucker–Prager yield criterion.Post-yield constitutive behavior was modeled by Sharma et al.(2009) via a non-associative flow rule that preserved volume duringplastic flow. In contrast, Paper I employed an associative flow rule,as non-associative rigid-plastic materials were demonstrated tobe trivially secularly1 unstable. We recall that associative materialsare distinguished by the fact that their yield surface functions alsoas the plastic potential that generates the flow rule. In contrast,non-associative flow rules have plastic potentials distinct from theyield surface; see, e.g., Chen and Han (1988). We will follow thesame rheology as Paper I, which for completeness we now quicklysummarize below.
A granular aggregate is assumed to remain rigid until its stressstate violates the Drucker–Prager yield condition
jsj2 6 k2p2; ð12Þ
where
p ¼ �13
tr r ð13Þ
is the pressure,
s ¼ rþ p1; ð14Þ
is the deviatoric stress tensor with magnitude
jsj2 ¼ sijsij;
utilizing the summation convention, and
k ¼ 2ffiffiffi6p
sin /F
3� sin /F; ð15Þ
in terms of the granular aggregate’s internal friction angle /F . Theyield surface is shown in Fig. 2. Employing the principal shear stres-ses si; i – j – k, we have
jsj ¼ 23
s21 þ s2
2 þ s23
� �; ð16Þ
so that jsj is an estimate of the ‘total’ shear stress. Consequently, theyield criterion (12) limits the shear stress in terms of the pressureand the internal friction angle. This internal friction models the abil-ity of an aggregate to support shear stresses, and depends on boththe usual interfacial friction due to particle sliding, as well as, a geo-metric friction due to interlocking and rearrangement of finite-sizedconstituents. Typical terrestrial soils display /F between 30� and 40�
under standard laboratory conditions; cf. Section 7.2.Once the granular aggregate yields, we employ a flow rule relat-
ing stress and strain increments post-yield. Paper I introduces theassociative flow rule
D ¼ ðsþ �p1Þ _c; ð17Þ
where we recall that the stretching-rate tensor D is the symmetricpart of the velocity gradient L and estimates the local rate of change
1 Systems found stable by the energy criterion are secularly stable, cf. Section 5.2.
of deformation, � ¼ k2=9, and _c is a proportionality constant. The
stress r and strain rate D are said to be compatible whenever r isat yield and D is normal to the yield surface at the point the surfacepasses through r; see Fig. 2. For compatible r and D, (17) holds.
It may happen that perturbations impose a strain rate that isincompatible with the rigid-plastic material’s stress state. Thiscould mean that the material’s stress state r is at yield, but the per-turbing strain rate is not normal to the yield surface at r. Alterna-tively, the rigid-plastic material is not at yield, but is subjected to avelocity perturbation causing plastic flow. Both will cause thematerial’s current stress state to suddenly change to one that iscompatible with the imposed strain rate where (17) holds. This isexplained in Fig. 2. A simple illustrative example is the one-dimensional frictional slider. Suppose such a slider is subjectedto a rightward force till it is just about to slide. The frictional forceacts towards the left. Velocity perturbations towards the right arecompatible to the current frictional state. However, if the slider isgiven an incompatible velocity perturbation towards the left, thefrictional force switches instantly towards the right! Similarly,suppose initially the slider is not at yield, so that its frictionalstrength is not fully mobilized. If now it is made to slide by animposed velocity perturbation, then the frictional force willincrease immediately in magnitude to reach its frictional strength.
The constant _c is found by combining (17) with the yield condi-tion (12). The resultant is then substituted back into (17) to findthe deviatoric stress s post-yield. The stress following yield maythen be written as
r ¼ �pð1þ �Þ1þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 3�2
qDjDj : ð18Þ
We observe that the stress tensor depends on the strain ratethrough the ratio D=jDj, and is reminiscent of dry friction with kand �p playing the role of a friction coefficient and a normal force,respectively. Furthermore, in the above relation and in the flow rule(17), we may employ the stretching-rate tensor as observed in therotating frame O, as pure rotation leaves the symmetric part ofthe velocity gradient unaffected.
1o40o 10o
20o30o
90o
5o3o
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5
Axes ratio, β = a3 /a1
Darwin ellipsoid
Scal
ed d
ista
nce
, q
= R
/a1’
Fig. 3. The equilibrium landscape in b� q0 space as viewed on the section a ¼ 0:5.The number next to a critical curve is the associated friction angle /F . The primary’soblateness b0 ¼ 0:8 and the density ratio g ¼ q=q0 ¼ 2. The axis ratio b’s upper limitis set by the inequality b 6 a. The Darwin ellipsoid at /F ¼ 0� is indicated by a ‘+’symbol. This plot follows Sharma (2009).
I. Sharma / Icarus 229 (2014) 278–294 281
The pressure p remains to be found. Because the materialdilates after yielding at the rate
trD ¼ 3�jDjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 3�2
p ; ð19Þ
obtained by taking the trace of (18), we need to relate the pressureand the dilatation. Paper I postulates the simple linear relationship
_p ¼ �j trD ð20Þ
in terms of the granular aggregate’s plastic bulk modulus j, i.e., bulkmodulus post-yield. Because (19) is positive, the material expandsafter yielding leading to a drop in the pressure. For sands confinedat hundreds of kilo-Pascals, the bulk modulus is of the order of 1MPa; cf. Section 7.2.
We assume the ellipsoid to be an isotropic homogeneous bodythat deforms homogeneously. Thus, the strain rate D, and, conse-quently, the stress field, is uniform across the body. We will inthe following analysis, therefore, employ the constitutive relations(18) and (20) post-yield with r and �p replacing r and p, respec-tively. Finally, we note that for a homogeneously deforming body,its average density q and its volume V change according to
_VV¼ �
_qq¼ trD: ð21Þ
4. Equilibrium shapes
At equilibrium, the ellipsoidal satellite orbits the oblateplanet along a circular orbit lying in the planet’s equatorial planein a tidally-locked configuration with its longest axis pointingtowards the planet’s center; see Fig. 1. Burns and Safronov(1973), Sharma et al. (2005) and Breiter et al. (2012) predict thatenergy dissipation damps a rotating body into a state of purerotation about its shortest axis. We, therefore, further assume thatthe equilibrated satellite rotates at the constant rate xE about itsshortest axis e3 taken perpendicular to the orbital plane; seeFig. 1. It is convenient to take the rotating frame O at equilibriumto coincide with the satellite’s principal axes frame, though it ispossible that these two frame no longer remain aligned followinga perturbation; cf. Section 5.1. Following this choice of O, atequilibrium, the tensors L and its derivative vanish, as do the deriv-atives of XE, and (3a) provides the average stress at equilibrium
rEV ¼ �2pqGA � I � 2pq0G Bð0Þ þ eR �Bð1Þ� �T
h i� I þX2
E � I; ð22Þ
where we have replaced for M from (8), and we recall that XE is theangular velocity tensor related to xE via (1). In a tidally-locked con-figuration, the rotation rate xE coincides with the satellite’s orbitalangular velocity that for motion on a circular orbit is obtained from(11) after setting R and X to be constant. Thus, at equilibrium,
x2E ¼ 2pq0GeR � Bð0Þ � eR: ð23Þ
From (22) and (23), for a given planetary oblateness b0 ¼ a03=a01 anddensity ratio g ¼ q=q0, we may estimate the volume-averagedstress in the satellite as a function of the satellite’s shape and itsseparation from the planet. This estimate is then combined withthe yield criterion (12) to obtain regions in the three-dimensionalseparation q0 ¼ R=a01
� �– shape ða ¼ a2=a1; b ¼ a3=a1Þ space param-
eterized by the friction angle /F wherein a rubble-pile satellite mayexist in equilibrium; see Paper I for more details. The three-dimen-sional region is explored for a choice of g and b0 via planar sectionsobtained by restricting the satellite’s axes ratio a appropriately. Anexample is displayed in Fig. 3, where equilibrium zones are indi-cated in the q0 � b space after fixing a ¼ 0:5. A satellite witha ¼ 0:5 and friction angle /F will remain at equilibrium as long as
its scaled distance from the planet q0 and its axes ratio b are suchthat in Fig. 3 it lies above the curve corresponding to its /F . Thus,an ellipsoidal rubble-pile satellite with /F ¼ 20� will survive onlyif it lies within the shaded region indicated in Fig. 3. The case for/F ¼ 0� corresponds to an equilibrated fluid satellite and representsthe intersection of the first Darwin sequence with the plane a ¼ 0:5;see e.g., Chandrasekhar (1969, p. 221). Analogous equilibrium plotsmay be generated for various choices of a, b0 and g, and are exploredin detail in Paper I.
We now probe the stability of equilibrated rubble-pilesatellites.
5. Stability
According to Lyapunov’s definition of stability, a system islocally stable if small perturbations of the system lead to smalldepartures of its coordinates from its equilibrium values; see,e.g., LaSalle and Lefschetz (1961, p. 28). Sharma (2012, 2013) dis-cuss the two approaches employed to probe Lyapunov stability:(a) the spectral method that investigates the eigenvalues of the sys-tem’s linearized equations of motion, and (b) the energy methodthat tests whether the system’s equilibrium state lies at an appro-priately defined potential energy’s minima; see also Nguyen(2000). Depending on the method employed, a system is said tobe, respectively, spectrally/dynamically or energetically/secularlystable. The system is said to be spectrally/dynamically or energet-ically/secularly unstable in case the corresponding test fails.
Spectral analysis requires that the system’s governing equationsbe amenable to linearization. It cannot, therefore, be employed toinvestigate the stability of non-smooth systems such as rubble-pilesatellites with structures modeled via constitutive laws similar tothe one in Section 3. Thus, to test the stability of these satellites,we will follow Sharma (2012, 2013) and employ an incrementalversion of the energy criterion.
For finite-dimensional systems secular stability guaranteesLyapunov stability. Furthermore, secularly stable systems aredynamically stable, though the converse is not necessarily true asgyroscopic forces may stabilize energetically unstable systems; cf.Section 5.2. These facts may not hold for infinite dimensional con-tinuous systems, and more details are available in Sharma (2012)and references therein, in particular Nguyen (2000). Here, werestrict ourselves to homogeneous deformations of the satellite’sstructure that constitute a finite-dimensional system.
We will require also to consider the stability of the satellite’sorbit. Without appropriate modification, Lyapunov’s definitionimposes unreasonable requirements when investigating the
282 I. Sharma / Icarus 229 (2014) 278–294
stability of orbital motion. Indeed, small differences in initial posi-tion and velocity of the satellite’s center of mass leads to large(periodic) variations in its angular locations, so that the equili-brated and perturbed satellite are seen to physically separate.The orbital motion will then be found Lyapunov unstable. How-ever, in spite of this, it is possible that the perturbed orbit remainsclose to equilibrium one, and this is perfectly satisfactory for mostapplications. Thus, a more useful definition of orbital stability willrequire that we ignore phase separation, and instead simply de-mand that the distance between the equilibrium and perturbed or-bits remains small. The distance between two orbits at a givenangular location of one orbit may be defined as the smallest sepa-ration between the two orbits keeping the angular location fixedfor one orbit. In the current scenario where, as seen in Section 2.2,we ignore the effect of structural deformation on the orbital mo-tion, it is possible to test orbital stability via both energy and spec-tral methods. Here we employ the latter.
We finally note that a spectral stability analysis’ conclusions areindependent of the underlying coordinate system’s rotation, as theframe’s rotation cannot influence the imaginary/real nature of thelinearized system’s eigenvalues. In contrast, the choice of an appro-priate rotating coordinate system is crucial for stability analysis viathe energy method. We now proceed to discuss this.
5.1. Coordinate system
As discussed in Sharma (2012, 2013), to obtain physicallymeaningful stability results via the energy criterion, it is necessaryto exercise care when selecting the underlying rotating frame O.For the case of a body with a fixed mass center, this coordinate sys-tem was found to be the well-known Tisserand’s mean axis of thebody. It is not necessary that this same choice be suitable for satel-lites whose mass center orbits a planet, as in this case we have toconsider both the stability of the satellite’s structure and its orbitallowing for both structural and orbital perturbations. However,as we argue in Section 5.2, for the Solar System’s planets it is pos-sible to decouple the stability problem in a manner that reducesthe stability of planetary satellites to a consideration of their struc-tural stability to structural perturbations alone. This in turn is anal-ogous to the stability of granular asteroids, and may, therefore, beexplored in the Tisserand’s mean axis of the body, and this definesthe coordinate system O.
Tisserand’s mean axis of the body rotate in a manner so that thesatellite’s structural angular momentum relative to it vanishes, i.e.,in O,
Hrel ¼Z
Vx� vdV ¼ 0;
where v is a material point’s velocity observed inO. Recall from Sec-tion 4 that at equilibrium the frame O is aligned with the ellipsoidalsatellite’s principal axes and rotates at the equilibrium rate xE.
We will investigate a satellite’s structural stability by providingmaterial points an initial velocity perturbation, and then followingthe subsequent growth/decay of the system’s kinetic energy rela-tive to O. As Sharma (2012, 2013) show, it is necessary to ensurethat the perturbations do not introduce additional angularmomentum, otherwise, to keep Hrel zero, O’s rotation rate x willchange abruptly from xE just before perturbation at t ¼ 0� to x0
just after at t ¼ 0þ; see Sharma (2013). Systems with such a jumpin x are found to be trivially unstable.2 Thus, we restrict ourselvesto angular momentum preserving perturbations, so that x0 ¼ xE.
2 Though mathematical correct, this may not reflect a physically relevant instabil-ity. A stability criterion that treats perturbations that introduce angular momentummeaningfully requires further work. In the past such perturbations were ignored.
This also ensures that the velocity perturbation observed in O att ¼ 0� is the same as the one observed in O at t ¼ 0þ.
Post-perturbation, the frame O’s rotation rate x may need tochange to keep Hrel zero. Indeed, the necessary rate of change forisolated bodies were obtained by Sharma (2012). In the case of sat-ellites, the planet typically exerts a torque T ¼
RV qx� bQ dV about
the satellite’s mass center. To find O’s rate of change, we note fromSharma (2012, Eq. (4.2)) that the satellite’s total angular momen-tum H about its center of mass and Hrel are related as
H ¼ Hrel þ J �x; ð24Þ
where
J ¼ trI � I ð25Þ
is the Euler’s moment of inertia. From Euler’s law of conservation ofangular momentum, see, e.g., Greenwood (1988, p. 390), the rate ofchange of H with respect to an observer in the rotating frame O is
_H ¼ T�x� H:
Differentiating (24), setting _Hrel to zero, as Hrel is constant in O, andfinally replacing for _H in the previous equation, we obtain O’s angu-lar acceleration:
_x ¼ J�1 � T�x� J �x� _J �x�
The tensor _J may be obtained by differentiating (25). For the specialcase when the frame O’s rotation rate post-perturbation is about aprincipal axes of inertia, the above formula simplifies to
_x ¼ J�1 � ðT� _J �xÞ: ð26Þ
To summarize, we will investigate the structural stability of rubble-pile satellites in a coordinate system O that rotates in a manner thatkeeps the body’s relative angular momentum zero. We now adaptthe energy criterion to satellites whose dimensions are much smal-ler than the size of their orbit and the parent planet.
5.2. Energy criterion
According to the energy criterion developed in Sharma (2012,2013), a system is secularly stable if in an appropriately rotatingframe O the system’s total relative kinetic energy Ek following avelocity perturbation decreases over a small time dt, i.e., if
dEk ¼Z dt
0
_Ek dt < 0:
The kinetic energy has contributions from both structural and orbi-tal motions. Indeed, noting that the total velocity of a material pointrelative to O is u ¼ _Rþ v, we have
Ek ¼12
ZVqu � udV ¼ 1
2
ZVqv � vdV þ 1
2mR � R :¼ Es
k þ Eok; ð27Þ
where the mixed-terms involving both _R and v drop out in homoge-neous deformation, and Es
k and Eok are, respectively, structural and
orbital relative kinetic energies. A satellite will be termed stableonly if its structure and its orbit are found stable to both structuraland orbital perturbations. This general analysis is rather compli-cated and will be pursued in a future work. Here our interest liesprimarily in satellites of Solar System’s planets that are character-ized by extremely small satellite to planetary and orbital size ratios.In this case, we show next that the stability problem may be splitinto an analysis of orbital stability following only orbital perturba-tions and structural stability in the presence of structural perturba-tions alone.
Consider an equilibrated tidally-locked ellipsoidal satelliteorbiting at the rate xE at a distance RE, and assume that thesatellite’s average size �a is much smaller than the planet’s �a0. Let
3 In Section 6.5 we will see that compatible perturbations automatically preserveangular-momentum.
I. Sharma / Icarus 229 (2014) 278–294 283
the satellite be perturbed structurally at t ¼ 0 by homogeneousvelocity perturbations and subsequently deform also homoge-neously. Let its orbit be also perturbed. We study stability in thecoordinate system defined in Section 5.1. Finally, structural pertur-bations are assumed to not add angular momentum, and are takento be compatible to the equilibrium stress. Thus, both the frameO’srotation rate and the satellite’s stress state do not jump acrosst ¼ 0.
We obtain estimates for the change in kinetic energies Esk and Eo
k
over the short time dt after the satellite is perturbed. In O these
energies scale as Esk �
RV j _xj
2dV � IL2=2 – from (2) and (4a) – and
Eok � m _R2=2, where I and L are, respectively, the magnitudes of
the inertia and velocity gradient tensors and _R is the mass center’s
speed. Then dEk �R dt
0 IL _Lþm _R€Rdt. Consider first the termR dt0 m _R€Rdt. We estimate €R from (11). While doing so we recognize
that the term �ðX2 þ 2pqGBð0ÞÞ � R vanishes at equilibrium and, so,may be assumed to be small for small time dt over which weintegrate. Furthermore, the Coriolis’ term does not provide any
power. Thus,R dt
0 m _R€Rdt �R dt
0 m _R _xRdt. Now, (26), along with (25)and (3b) provides the estimate _x � xL, so that, finally,R dt
0 m _R€Rdt �R dt
0 mxLR _Rdt.
A similar analysis helps gaugeR dt
0 IL _Ldt. We again recognize that
over a short time dt, �r � I�1V þMT �X2 � L�
will be small; see
(22) and (8). The term involving 2X � L is derived from Coriolis’acceleration and so, being gyroscopic, does not contribute anypower. Similarly, it may be shown that _X’s contribution to thepower is zero whenever perturbations don’t add angular momen-tum; see Sharma (2012, Section 4.3). Combining these consider-
ations with (3a) leads to the estimateR dt
0 IL _Ldt �R dt
0 IL3 dt.Thus, dEk �
R dt0 IL3 þmxLR _Rdt. This may be approximated by
expanding the integrand in a Taylor series in time about t ¼ 0þ
and integrating each term:
dEk � ðIL3 þmxLR _RÞ���0þ
dt þ ddtðIL3 þmxLR _RÞ
����0þ
dt2
2þ � � � :
The orbit’s radius just after perturbation will remain at its equilib-rium value RE. At equilibrium the satellite is taken to be in a tidally-
locked configuration orbiting at xE �ffiffiffiffiffiffiffiffiffiGm0p
=RE, where m0 is the pla-net’s mass. We assume that the frame O’s rotation before perturba-tion to be xE and have taken its angular velocity at t ¼ 0þ to be ofthe same order as xE. Then the ratio of the changes in structural andorbital kinetic energy at OðdtÞ scales as IL2
0=mxERE_R0 �
�a2L20=
ffiffiffiffiffiffiffiffiffiGm0p
_R0 � �a2L20=
ffiffiffiffiffiffiffiffiffiffiffiffiffiq0G�a03
p_R0 1, as a a0. For the class of
perturbations for which the OðdtÞ perturbations vanish identically,stability is decided at Oðdt2Þ. Expanding the derivative in the secondterm in dEk’s approximation, we again find that changes instructural and orbital kinetic energies scale in the ratioIL2
0=mx2ER2
E � �a2L20=q0G�a02, which again is a small quantity. Thus, at
both first- and second-order we observe that changes in the orbitalkinetic energy dominate the corresponding changes in the struc-tural kinetic energy.
Now, the discussion in Section 2.2, and (11) in particular, indi-cates that for small satellite to orbital size ratios, the satellite’sorbital motion is to leading order decoupled from its structuraldeformation. Thus, a satellite’s orbital stability may be investigatedin exclusion of its overall stability. Combining this with the previ-ous paragraph’s scaling analysis, we conclude that an orbitallyunstable satellite with �a �a0 will be unstable. However, if such asatellite is orbitally stable, then for its overall local stability we willadditionally require that it be structurally stable when orbital pertur-bations are absent. This latter requirement may be checked by a
slight extension of the energy criterion as it was employed in thecase of asteroids in Sharma (2013).
The satellite’s structural stability will be tested via the energycriterion in the frame O, introduced in Section 5.1, to structuralperturbations alone. The perturbations will be restricted to angu-lar-momentum preserving homogeneous motions. Such velocityperturbations may be represented as
v0 ¼ Do � x; ð28Þ
where Do is an arbitrary symmetric tensor that is diagonalized inthe ellipsoidal satellite’s principal axes frame.
According to the energy criterion the satellite is structurally sta-ble if
dEsk ¼
Z dt
0
_Esk dt < 0;
where Esk was defined in (27) as the kinetic energy observed in O to
be stored in the satellite’s structural deformations. Sharma (2013)shows that for homogeneous motions
dEsk ¼
Z dt
0�rV þMT � ð _XþX2Þ � In o
: LT dt; ð29Þ
with _X being the anti-symmetric tensor associated with _x and Mnow containing contributions from both self-gravity and tidal inter-action. Note the absence of the Coriolis force that being gyroscopiccontributes no power. Within the confines of a local stability anal-ysis, where dt is small, dEs
k is estimated by expanding it aboutt ¼ 0þ. Thus, the instant after v0 is imposed, we write
dEsk ¼ dð1ÞEs
kdt þ dð2ÞEskdt2
2þ Oðdt3Þ;
where, with subscript ‘0’ indicating evaluation at t ¼ 0þ,
dð1ÞEsk ¼ �r0V þMT
0 �X2E � I
n o: Do ð30aÞ
and dð2ÞEsk ¼
ddt
�rV þMT � ð _XþX2Þ � In o
: LTh i����
0; ð30bÞ
and in (30a) we have recognized that L0 ¼ Do and X0 ¼ XE. Forsmall dt; dEs
k’s sign is regulated by the first non-zero term in dEsk’s
expansion, and secular stability is assured if that term is negativefor all permissible Do.
As discussed in detail in Sharma (2012, 2013), the leading-orderterm dð1ÞEs
k vanishes only when systems whose equilibrium stressstate is on the yield surface – these are called critically equilibratedsystems – are perturbed by compatible angular-momentum pre-serving velocity perturbations3. That perturbations preserve angularmomentum was already presumed, allowing us to set X0 equal to XE
above; see also Section 5.1. Recall from the discussion following (17)and Fig. 2 that a velocity perturbation is compatible to the equilib-rium stress rE if it is normal to the yield surface at the point thatit passes through rE; thus, it is implicit that the system is criticallyequilibrated. In that case the perturbation does not cause the stressstate to jump and r0 ¼ rE. Consequently, dð1ÞEs
k vanishes as (22)holds at equilibrium, and dEs
k’s sign, and hence stability, will bedecided by dð2ÞEs
k.On the other hand, imposed incompatible perturbations will
cause the stress to change suddenly to r0 – rE, and we invoke(22) in (30a) to obtain
dð1ÞEsk ¼ � r0 � rEð Þ : DoV :
The maximum dissipation postulate now ensures stability at firstorder; see also Hill (1957). This postulate hypothesizes that given
284 I. Sharma / Icarus 229 (2014) 278–294
a stress state, say, r1 that may or may not be on the yield surface,and any another stress state, say, r2 that is yielding at a compatiblestrain rate D2,
ðr2 � r1Þ : D2 P 0;
see Fig. 2 after identifying subscript ‘1’ and ‘2’ with, respectively, ‘E’and ‘0’, or, ‘A’ and ‘B’. Applying the above inequality to dð1ÞEs
k’s expres-sion with rE ¼ r1;r0 ¼ r2, and Do ¼ D2, we see that dð1ÞEs
k 6 0, i.e.,the satellite is stable to incompatible perturbations. The maximumdissipation postulate holds for materials in which the plastic flowis normal to the yield surface, i.e., associativity is assumed. Physi-cally, the postulate reflects the impossibility of obtaining usefulwork from plastic materials by cyclically applying and removingexternal forces, i.e., plastic materials dissipate, or at best conserve,4
energy during closed cycles; see, e.g., Lubliner (1990, p. 117).To summarize, we claim a satellite to be stable only when its
structure and orbit are simultaneously stable to both structuraland orbital perturbations. We recognize that whenever a satellite’ssize is small in comparison to its orbit, we may consider its orbitalstability separately. Additionally, when the satellite is also muchsmaller than its planet, then the satellite will be unstable if it is orb-itally unstable. However, an orbitally stable satellite is stable onlywhen it is also structurally stable to structural perturbations alone.To investigate structural stability, we restrict ourselves to initialhomogeneous velocity perturbations (28) that do not add angularmomentum. Of this subset, the asteroid is secularly structurallystable at first-order to perturbations that are incompatible to theequilibrium stress state. It then remains to investigate structuralstability to compatible angular-momentum preserving perturba-tions as dictated by dð2ÞEs
k’s sign. We now note that in (30b) thederivatives of the tidal moment tensor MQ contained in M will van-ish when only structural perturbations are present, as MQ dependson the planet’s shape and the satellite’s location. Thus, the struc-tural stability computation for a satellite whose size is much smal-ler than its parent planet’s is essentially the same as that ofasteroids, with the only difference being that the satellite’s equilib-rium stress is modified by tidal moments.
We now test the stability of rubble-pile satellites.
6. Example: Rubble-pile satellites
We model satellites as self-gravitating tidally-locked rubble-pile ellipsoids orbiting an oblate planet on a circular path androtating about their shortest axis e3 that is normal to the orbitalplane; see Fig. 1. The rubble-pile is assumed to follow the rheologyof Section 3. A satellite is considered stable only if it is structurallyand orbitally stable. We restrict attention to satellites whose sizesare much smaller than their orbital radius and their associated pla-net’s size, as is the case in the Solar System, so that orbital stabilitymay be considered independently of structural motions. Then, asdiscussed in Section 5.2, the satellite is unstable whenever it isorbitally unstable. However, the satellite’s stability requires bothorbital stability, as well as, structural stability to structural pertur-bations. Orbital stability is determined in Section 6.3 by a spectralanalysis, while structural stability is explored via the energy crite-rion in Section 6.4.
First, we introduce a non-dimensionalization that will help sim-plify various equations.
6.1. Non-dimensionalization
We will frequently non-dimensionalize various quantities andequations by rescaling time by 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffi2pqG
p, stress by ð3=20pÞ
4 In case the stresses do not cause yielding.
ð2pqGmÞð4p=3VÞ1=3 with q being the satellite’s density at equilib-rium, and inertia by ma2
1=5. For example, the non-dimensionalaverage stress at equilibrium is found from (22) as
rE ¼ �ðabÞ�2=3X2
E þ Aþ 1g
Bð0Þ þ eR �Bð1Þ� � �
� Q ; ð31Þ
where g ¼ q=q0;Q is a non-dimensional tensor obtained from I that,in the satellite’s principal axes coordinate system, takes the form
½Q ¼1 0 00 a2 00 0 b2
264
375; ð32Þ
and XE and rE now represent non-dimensional quantities. Considernext the evaluation of various tensors in the stability calculation inthe frame O.
6.2. Components
We investigate the satellite’s equilibrium shape and its struc-tural stability in the rotating coordinate system O defined in Sec-tion 5.1. At equilibrium O is aligned with the ellipsoid’s principalaxes coordinate system. We observe that equations governing bothequilibrium through (31), and it stability via (30), contain quanti-ties that are evaluated at equilibrium or just after perturbation.At both these instances, though O’s rotation may change, its align-ment remains unchanged. Thus, we have
½A ¼A1 0 00 A2 00 0 A3
264
375; ½XE ¼
0 �xE 0xE 0 00 0 0
264
375 and rE½ ¼
�r1 0 00 �r2 00 0 �r3
264
375; ð33Þ
with �ri to be found in the following section, and for angularmomentum-preserving homogeneous perturbations,
½Do ¼D1 0 00 D2 00 0 D3
264
375: ð34Þ
Note that ½Q is already available from (32).
6.3. Orbital stability
The evolution of the satellite’s orbit is governed by (11).Employing the definition of orbital stability discussed in Section 5’sintroduction, we see that following a perturbation we need onlyconcentrate on the satellite’s separation from the planet to confirmthe stability of its orbit. To this end, we first employ cylindricalcoordinates e0r � e0h � e03
� �to separate (11) into its radial ðrÞ, tangen-
tial ðhÞ and out-of-plane ðzÞ components:
€r � r _h2 ¼ �2pq0GRe0r � Bð0Þ � eR;
r€hþ 2_r _h ¼ �2pq0GRe0h � Bð0Þ � eR;
and €z ¼ �2pq0GRe03 � Bð0Þ � eR:
The tensor Bð0Þ above is evaluated at the satellite’s perturbed loca-tion ReR; see Fig. 4. Noting the planet’s symmetry, e01 can alwaysbe chosen along e0r allowing us to write ReR ¼ RE þ rð Þe01 þ ze03, wherer ¼ r � RE is the deviation of the projection onto the planet’s equa-torial plane of the perturbed orbit from the circular equilibrium one.As, the perturbations are small, i.e., r; z RE, we approximate Bð0Þ
by its equilibrium value:
Bð0ÞðRÞ ¼ Bð0Þðr; zÞ ¼ Bð0ÞðRE;0Þ þ@Bð0Þ
@r
�����ðRE ;0Þ
r þ @Bð0Þ
@z
�����ðRE ;0Þ
zþ � � � ;
because of the planet’s symmetry, Bð0Þ doesn’t depend on h. Evalu-ating the above in the planet’s principal axes coordinate system,we obtain
e3’
er’eθ’ z
r
R
eR
e1
e2
e3
S
P
e2’
RE
Equilibrium orbit
Perturbed orbit
Fig. 4. The perturbed configuration of an ellipsoidal satellite of an oblate planet.Both the orbit and the structure are perturbed. The unit vector eR locates thesatellite with respect to the planet’s center.
I. Sharma / Icarus 229 (2014) 278–294 285
Bð0Þ � B0ð0Þ1 � 2b0qq03E q0o
� e01 � e01 þ B0ð0Þ1 � 2b0q
q03E q0o
� e02 � e02
þ B0ð0Þ3 � 2b0qq0Eq03o
� e03 � e03:
where the B0ð0Þi refer to the principal components of Bð0ÞðRE; 0Þ in theplanet’s principal axes frame P, and we have non-dimensionalizedas per Section 6.1 with q ¼ r=a01 and q0o ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq02E � 1þ b02
q.
Employing Bð0Þ’s approximation in the orbit’s governing equa-tions above, and rescaling according to Section 6.1, we obtain
€�r � �r _h2 ¼ � q0
gB0ð0Þ1 � 2b0q
q03E q0o
� cos w
�r€hþ 2 _�r _h ¼ 0
and €�z ¼ � q0
gB0ð0Þ3 � 2b0q
q0Eq03o
� sin w;
where �r ¼ r=a01; �z ¼ z=a01, and w ¼ tan�1 z=r is the perturbed satel-lite’s inclination with respect to the equatorial plane, so thateR ¼ e01 cos wþ e03 sin w. The middle equation’s right hand side van-ishes, reflecting the fact that an oblate planet’s external gravita-tional field is radial, i.e., points towards the planet’s center ofmass. This equation may now be integrated to yield the dimension-less specific orbital angular momentum about e03 of the satellite inits perturbed orbit :
�h ¼ �r2 _h; ð35Þ
recall that the satellite’s orbital motion is not influenced by itsstructural dynamics, and it orbits as a point mass. We will assumethat no angular momentum is added to the orbital motion by orbitalperturbations, so that �h equals the angular momentum of theunperturbed circular orbit. Eliminating _h in terms of �h in the gov-erning equations finally provides
€�r ��h2
�r3 ¼ �q0
gB0ð0Þ1 � 2b0q
q03E q0o
� cos w and
€�z ¼ � q0
gB0ð0Þ3 � 2b0q
q0Eq03o
� sin w:
To investigate the satellite’s orbital stability, we linearize the aboveequation about the satellite’s scaled equilibrium radius q0E. We thus
write �r ¼ q0E þ q, with q q0E, note that q0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq0E þ q� �2 þ �z2
q;�z q0E,
w 1, and expand the previous equations in terms of q=q0E and �z=q0E,retaining terms till O q2=q02E ;�z
2=q02E� �
:
€qq0E�
�h2
q04E1� 3
qq0E
� ¼ �1
gB0ð0Þ1 � 1
gB0ð0Þ1 � 2b0
q02E q0o
� qq0E
and€�zq0E¼ �1
gB0ð0Þ3
�zq0E:
We obtain B0ð0Þ3 from Eq. (45b) of Paper I, and see that B0ð0Þ3 is positive,so that the second equation above reduces to that of a simple har-monic oscillator. Thus, �z remains small for all time following a smallout-of-plane perturbation.
Turning to the equation for q, we see that at leading order werecover the orbital angular momentum of the circular equilibriumorbit:
�h2 ¼ 1g
q4EB0ð0Þ1 ;
cf. (23) and (35). At first order in q=qE we obtain the equation gov-erning small departures from equilibrium:
€qþ 2g
2B0ð0Þ1 � b0
q02E q0o
� q ¼ 0;
where we have replaced for �h2 from the preceding equation.Employing Eq. (45a) of Paper I for B0ð0Þ1 , we confirm that q’s coeffi-cient is always positive for 0:56 < b0 < 1 and 1 < q0E. Thus, forb0 P 0:56 the above again represents a simple harmonic oscillator,and �r remains confined to q0E’s neighborhood. For b0 6 0:56, orbitsvery close to the planet start becoming unstable to in-plane pertur-bations. However, Solar System planets have b0 P 0:9, and we neednot concern ourselves with small b0. Thus, for physically relevant b0,both in-plane and out-of-plane motions following an infinitesimalperturbation remain small, and the equilibrium circular orbit inthe oblate planet’s equatorial plane is always locally spectrallystable.
Before continuing on to structural stability a comment is in or-der. The equations governing the satellite’s orbit are decoupledfrom its structural deformation and, hence, conserve energy. Fur-thermore, the linearized equations for q and z do not contain anygyroscopic terms. This implies that were we to investigate orbitalstability via the energy criterion we would arrive at the same con-clusions as above. An energy analysis would begin by first formingthe energy stored in radial and out-of-plane motions and thenevaluating their incremental change following velocity perturba-tions in those directions.
6.4. Structural stability
We now consider the structural stability of satellites followingstructural perturbations alone. This is determined by the changedEs
k in the kinetic energy stored in structural deformations. Wesaw in Section 5.2 that the satellite is structurally stable at first-order to all structural angular-momentum preserving homoge-neous velocity perturbations, except those that are also compatibleto the equilibrium stress state. The stability to these latter pertur-bations is decided at the second order by dð2ÞEs
k’s sign; cf. (30b). Thiswe now probe for the case of rubble-pile satellites.
Replacing in (30b) for _L from (3a), and for M and _M from,respectively, (8) and its derivative, _q and _V from (21), post-yieldstress’ derivative _�r after differentiating (18) and invoking (20),we obtain after simplification:
dð2ÞEk ¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 3�2
qjDoj � ð1þ �ÞtrDo
� �trDoV
� rEtrDo � 2rE � Doð Þ : DoV
� 2pqG _A0 � AtrDo
n oþ 4XE � _X0
h i� I : Do � _X2
0 : I:
In arriving at the form above, we have recognized that for angular-momentum preserving structural perturbations Do’s and the satel-lite’s principal axes coincide. The derivatives of tensors A and X
are provided by, respectively, Sharma (2012) and (26), while theaverage equilibrium stress rE is obtained from (22). Note that thetorque T in (26) vanishes at equilibrium. As mentioned in Section 5.2,
5 The polytropic index n ¼ 0 corresponds to incompressible fluids.
286 I. Sharma / Icarus 229 (2014) 278–294
the derivatives of the tidal moment tensor MQ vanish when onlystructural perturbations are admitted.
Following Section 6.1, we may non-dimensionalize dð2ÞEk to find
dð2ÞEk ¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þ3�2
qjDoj� ð1þ�ÞtrDo
� �trDo� rEtrDo�2rE �Doð Þ : Do
� �abð Þ2=3
� _A0�AtrDo
n oþ4XE � _X0
h i�Q : Do� _X2
0 : Q ; ð36Þ
where j; Do; rE and XE are now dimensionless quantities, and thederivatives are with respect to non-dimensional time. We need onlycheck dð2ÞEk’s sign for compatible structural angular-momentumpreserving velocity perturbations. Comparing (36) with the corre-sponding expression for asteroids in Sharma (2013), we note thatthe only difference is that rE now includes tidal effects. We nextevaluate the asteroid’s equilibrium stress state and thereby identifycompatible velocity perturbations.
6.5. Equilibrium and compatible structural perturbations
The equilibrium stress rE’s non-dimensional components in therotating frame O that coincides at t ¼ 0þ with the ellipsoid’s prin-cipal axes system are found from (22) to be
�r1 ¼ x2E � A1 �
1g
Bð0Þ11 þ Bð1Þ111
� � �ðabÞ�2=3 ð37aÞ
�r2 ¼ a2 x2E � A2 �
1g
Bð0Þ11
� ðabÞ�2=3 ð37bÞ
and �r3 ¼ �b2 A3 þ1g
Bð0Þ33
� abð Þ�2=3
; ð37cÞ
where we have adopted Section 6.1’s scaling; note that shear stres-ses are absent. The equilibrium pressure �pE may now be obtainedfrom (13).
Velocity perturbations are compatible with the stress state onlyif their associated strain rates are normal to the yield surface. Thus,the strain rates must satisfy the flow rule (17). Eliminating the pro-portionality constant _c, we find that a homogenous perturbation(28) is compatible when
D2
D1¼
�r2 þ �pEð1þ �Þ�r1 þ �pEð1þ �Þ
andD3
D1¼
�r3 þ �pEð1þ �Þ�r1 þ �pEð1þ �Þ
;
note that the shear strain rates are zero. Thus, a compatible struc-tural perturbation automatically preserves the equilibrium valueof the satellite’s angular momentum about its mass center.
6.6. Local stability
We saw in Section 6.3 that satellites with dimensions smallerthan their orbital radius and their parent planet’s size were orbital-ly stable, as long as the planet’s b0 P 0:56. This includes the case ofthe Solar System’s planetary satellites. In these cases, the satellite’sstability/instability rests on it being stable/unstable to structuralperturbations alone. Thus, henceforth, we will not distinguishbetween a satellite’s stability and its structural stability.
Before considering rubble satellites, it is instructive to makecontact with inviscid incompressible fluid satellites that areobtained as a special case of granular aggregates when the internalfriction angle goes to zero. In particular, we will consider thestability of fluid Roche ellipsoids. Chandrasekhar (1969, p. 205,Table XX), reports that the Roche ellipsoids became dynamicallyunstable at qC ¼ 2:4554, a separation larger than the Roche limitqR ¼ 2:4552. Earlier, Jeans (1961, p. 229), following an energyapproach, posited that the Roche ellipsoids became secularlyunstable at the Roche limit. The reasons underlying this inconsis-tency were explored by Lebovitz (1963). He showed that the Rocheellipsoids went dynamically unstable at qC without going througha stationary mode, i.e., the linearized equations had a zero
eigenvalue at qC to which were associated no polynomially grow-ing solutions. In Chandrasekhar’s (1969) analysis, the mode thatgoes unstable oscillates prior to becoming unstable. Such a transferto instability cannot be identified by an energy analysis that re-quires that at the point of instability the perturbed solution be alsostationary. Lebovitz (1963) further showed that while perturbedsolutions are stationary at the Roche limit, this does not indicateinstability. In fact, homogeneous perturbations only displace theequilibrium solution further along the Roche series. Recently, Laiet al. (1993) also followed an energy approach to investigate thestability of a binary sequence wherein the satellite’s mass is a frac-tion p of the planet’s. Lai et al. (1993) claimed that, in general, sec-ular instability set in before the Roche limit is reached. This is notthe case when the primary is infinitely more massive than the sa-tellite, and secular instability is seen to occur exactly at the Rochelimit; see entry corresponding to p ¼ 0 and n ¼ 05 in Lai et al.(1993, Table 10).
Specializing our stability criterion to the case of inviscid incom-pressible fluids, we find that Roche ellipsoids become secularlyunstable at q� ¼ 2:6007, well before the Roche limit qR. The dis-crepancy with aforementioned results is easily reconciled oncewe recognize that all previous authors had defined stability/insta-bility on the basis of deviations in the ellipsoid’s shape. This wasmeasured in terms of variations of the inertia tensor by Chandrase-khar (1969) and by the growth/decay of the ‘‘energy’’
Pda2
i , whereai are the ellipsoid’s semi-major axes, by Jeans (1961) and Lai et al.(1993). Both these definitions ignore energy contained in internalmotions that do not affect the shape. In contrast, we define struc-tural stability on the basis of the increase/decrease post-perturba-tion of the structural kinetic energy dEs
k relative to an appropriateframe O; cf. (29). There is no reason to believe that different inter-pretation of what constitutes stability/instability will identify thesame instability points. Indeed, numerical integration of the under-lying dynamical equations (3) confirm that, while
Pda2
i grow onlyfor q < qC (Chandrasekhar, 1969, p. 203), dEs
k begins to grow forq < q�. We note finally that different choices of energies was thereason why, in spite of following the energy criterion, Riemann(1860) and Sharma (2012) arrived at different points of instabilityfor the Maclaurin spheroids; see Sharma (2012) for greater detail.
In contrast to fluids, see Chandrasekhar (1969, Chapter 8),where the equilibrium fluid satellites lie on a three-dimensionalcurve in shape (a� b) – distance (q0) space, a rubble-pile satellitecan exist in equilibrium within a three-dimensional volumeparameterized by the satellite’s internal friction angle /F . Thus,we have to test stability for granular satellites that lie on and with-in its associated equilibrium zone’s boundaries. A satellite lying onits associated equilibrium region’s boundary will be identified as acritically equilibrated satellite.
In the discussion following (30) we showed that satellites thatare not critically equilibrated are locally stable at first order tocompatible structural perturbations. It remains, therefore, to con-sider the structural stability to compatible perturbations of onlycritically equilibrated granular satellites. We will explore this onappropriate two-dimensional sections of the satellite’s equilibriumlandscape.
Figures 5 and 6 plot the stability zones for various values of thescaled plastic bulk modulus j for ellipsoidal satellites with, respec-tively, a ¼ 0:75; a ¼ 1 (oblate), a ¼ b (prolate), and a ¼ ð1þ bÞ=2(average triaxial). The planet is taken to be a spheroid withb0 ¼ 0:8 and half the density of the satellite, i.e., g ¼ 2. For eachj, there is associated a shaded region within which all criticallyequilibrated satellites are secularly structurally stable followingstructural perturbations, and, hence, locally stable. Equivalently,
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’=
R/a
1’
10 1001
10000
1o
40o
3o
5o10o20o
30o
90o
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’ =
R/a
1’
1o
3o20o
90o
1o
20o
30o
3o
30o
10o
10o
10 1001
1000
0
1
10100
0
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’ =
R/a
1’
1o
40o
3o5o10o
20o
30o90o
10 10011000
0
0
Fig. 6. Continued from Fig. 5. Stability regions of oblate, prolate and average triaxialellipsoidal granular satellites displayed on associated a�sections of the three-dimensional a� b� q0 space. Fluid satellites with these shapes are not possible. Seealso Fig. 5’s caption.
I. Sharma / Icarus 229 (2014) 278–294 287
satellites that lie on their corresponding equilibrium zone’sboundary, outside the shaded stability region associated with theirplastic bulk modulus, will be secularly structurally unstable, and solocally unstable. They will, nevertheless, continue to be orbitallystable as b0 ¼ 0:8 > 0:56; cf. Section 6.3.
As an example, consider a rubble ellipsoidal satellite S witha ¼ 0:75 and b ¼ 0:4, and moving on an orbit with q ¼ 3 aboutan oblate primary with b0 ¼ 0:8 and g ¼ 2. The satellite may belocated in Fig. 5 as shown. Let us assume that S’ internal frictionangle /F ¼ 30�. Its location in Fig. 5 indicates that it is not criticallyequilibrated, so that S will be locally stable at first order to homo-geneous velocity perturbations. However, were S0 closer to theplanet, say at q � 1:4, then it would lie on the boundary of the/F ¼ 30� equilibrium region, as shown by the filled circle inFig. 5, and thus be critically equilibrated. However, this locationlies outside the stability regions corresponding to any choice ofplastic bulk modulus j. Our analysis will then find the satellite Sto be locally unstable, and would raise doubts about S beinggranular. Alternatively, it could be that S’ internal friction is higher,so that even at q � 1:4 it is not critically equilibrated, and is, there-fore, locally stable. It is interesting to note that for current choicesof b0 and g, except at low b, the equilibrium curve corresponding toa friction angle /F of 30� or more typically lies outside any of theshaded local stability regions. This indicates that for the planet inquestion, any rubble-pile satellite of it with a /F P 30� cannot becritically equilibrated.
For each case considered above, we observe that as the plasticbulk modulus is lowered, the stability zones expand. Stabilityregions associated with lower j contain the one corresponding tohigher j as a subset. This was also observed in the case of asteroidsby Sharma (2013), and we adapt the explanation offered thereinfor satellites. If the pressure within a critically equilibrated satellitewas to suddenly reduce, the satellite will attempt to move withinan equilibrium region corresponding to a lower friction angle, asthe maximum frictional resistance that the satellite can mobilizedecreases; see (12). We expect that the satellite’s departure fromits equilibrium will be slow and short for small pressure drops.On the other hand, were the pressure to be significantly lowered,the satellite will move faster and closer to a state with low shearstress, i.e., to a state that lies within a equilibrium zone corre-sponding to a lower /F . Now, (19) requires that the materialexpand post-yield and, subsequently, (20) will lead to a loweringin the pressure from its equilibrium value. The pressure drop’sextent is proportional to the plastic bulk modulus j. Therefore,
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.75Axes ratio, β = a3 /a1
Scal
ed d
ista
nce
, q
= R
/a1’
1010
0
100Darwin ellipsoid
1000
1o
40o
3o
10o
20o
30o
90o
S
Fig. 5. The stability of a granular satellite with a ¼ 0:75 is explored on theassociated planar a�section of the three-dimensional a� b� q0 space. The planet isoblate with b0 ¼ 0:8 and g ¼ q=q0 ¼ 2. Stability regions are shaded and identified bynumbers corresponding to non-dimensional plastic bulk moduli j. Boundaries ofequilibrium zones for various internal friction angles /F are shown by dashed lines;these follow Fig. 5 of Sharma (2009). The Darwin ellipsoid obtained at /F ¼ 0� isindicated by a ‘+’ symbol. The abscissa’s upper limit is set by the requirement b 6 a.The satellite ‘S’ is an example discussed in the text.
for large j, satellites critically equilibrated at large /F will deviatesignificantly from their equilibrium state, and, so, will be foundunstable. Equivalently, ellipsoids with smaller plastic bulk moduliwill have larger stability regions.
6.7. Stability to finite structural perturbations
The local analysis of the previous section was applicable to crit-ically equilibrated satellites, declaring them stable whenever theleading estimate of the relative structural kinetic energy dEs
k’schange is negative following infinitesimal structural perturbations.However, given the improbability of finding critically equilibratedsatellites, it is of interest to investigate satellites whose stress stateat equilibrium lies within the yield surface. Such satellites were de-clared structurally stable at first-order by the maximum dissipa-tion postulate. However, if structural perturbations are finite,then it may happen that dEs
k increases following an initial declinecausing structural instability. We make a first attempt to analyzethis. We assume that the orbital perturbations remain infinitesi-mal. Furthermore, because, as in the previous section, the satellite’s
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’=
R/a
1’
50
100
1
1000
0 1o
40o
3o
5o20o
30o90o
1
0
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’ =
R/a
1’
10000
50
100
0
1o
3o
20o
90o
1o
20o
30o
3o
30o
40o
1
1
2
3
4
5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
Scal
ed d
ista
nce,
q’ =
R/a
1’
50
100
1
1000
0
0
1o
40o
3o
5o
20o
30o
90o
S
0
Fig. 7. Regions where dð2ÞEsk is negative for oblate, prolate and average triaxial
ellipsoidal granular satellites displayed on associated a�sections of the three-dimensional a� b� q0 space. The internal friction angle /F ¼ 30� . Each shadedregion is identified by its corresponding choice of non-dimensional plastic bulkmodulus j as indicated. Satellite ‘S’ in the bottom figure is an example discussed inthe text.
288 I. Sharma / Icarus 229 (2014) 278–294
dimensions are much smaller than its orbital radius and theplanet’s size, and the planet’s b0 > 0:56, the satellite’s overall sta-bility/instability to finite structural perturbations is equivalent toits structural stability/instability to these disturbances.
Consider a satellite whose equilibrium stress is within yield, i.e.,its location in shape–distance space is within its associated equi-librium region’s boundary. For such satellites dð1ÞEs
k was shown tobe negative. However, if perturbations are large enough to ensurethat the subsequent motion persists for times longer thandt� � 2 dð1ÞEs
k=dð2ÞEs
k
�� ��, then dð2ÞEsk’s contribution dð2ÞEs
kdt2=2 wouldovercome dð1ÞEs
k’s. Consequently, if dð2ÞEsk > 0, it can lead to dEs
k
becoming positive, and the satellite, though locally stable, maybe unstable at longer times. Here, for the sake of definiteness, wewill assume that satellites lying within their associated equilib-rium zones will be unstable if dð2ÞEs
k > 0. This assumption will bebetter the greater dð2ÞEs
k is in comparison to dð1ÞEsk, as will typically
hold for satellites that are close to being critically equilibrated.Following (36)’s derivation, we find that for satellites whose
average stress state rE at equilibrium does not lie on the yieldsurface,
dð2ÞEsk ¼ j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 3�2
qjDoj � ð1þ �ÞtrDo
� �trDo
�
� r0trDo � 2rE � Doð Þ : DoðabÞ2=3
� _A0 � AtrDo þ 4XE � _X0
� � Q : Do � _X2
0 : Q
þ Dr : D2o þ Q�1 � Dr
� ðabÞ2=3
; ð38Þ
where the stress’ change from its equilibrium estimate rE to itspost-perturbation value r0,
Dr ¼ r0 � rE ¼ sE þk�pE � jsEjjsEj
sE;
is obtained by scaling the equilibrium deviatoric stress sE so that s0
lies on the yield surface. This also ensures that there is no change inthe pressure. The tensor Do is compatible with r0. This choice guar-antees the smallest possible value for dð1ÞEs
k. Note that sE is availablefrom (37) and (14) whenever the satellite’s shape and rotation ratexE are known. In the above, the states ‘E’ and ‘0’ correspond, respec-tively, to locations A and B in Fig. 2.
Figure 7 displays regions in shape–spin space where dð2ÞEsk given
by (38) is negative for, oblate, prolate and average-triaxial ellipsoi-dal rubble satellites with an internal friction angle /F of 30�. Asbefore, the planet is oblate with b0 ¼ 0:8 and g ¼ 2. The regionsare parameterized by the non-dimensional plastic bulk modulusj. Because we have taken the satellites to have /F ¼ 30�, theseregions do not extend beyond the /F ¼ 30� equilibrium zone. Weobserve that, typically, as j changes from 0 to about 10, the regionwhere dð2ÞEs
k < 0 shrinks. For some j this zone vanishes. As jincreases beyond 10, these regions reappear and grow in size, sat-urating at about j ¼ 1000. At large j, the first term in dð2ÞEs
k’sexpression that controls its sign is regulated by j
As an example, consider an average-triaxial satellite S at a dis-tance q ¼ 1:8 from the planet with b ¼ 0:4; /F ¼ 30� and g ¼ 2. Itwill be located as shown in Fig. 7(c). Assume further that the satel-lite has a plastic bulk modulus of 1 MPa and density of2000 kg m�3. Taking the satellite’s largest semi-major axis to be10 km, its non-dimensional plastic bulk modulus j is about 300.Because S is within the /F ¼ 30� equilibrium region, the previoussection’s results guarantee its local secular stability at leading-order to infinitesimal perturbations. However, its location inFig. 7(c) confirms that dð2ÞEs
k > 0, so that there is a possibility thatS may deform appreciably following a finite structural perturba-tion. On the other hand, were S less slender, and have a b P 0:5,then it would lie within the shaded region corresponding to j of
100, and dð2ÞEsk would be negative. In that case, the satellite S would
be stable to finite, and not just infinitesimal, structuraldisturbances.
Similar analyses may be done to investigate the effect of evenhigher-order corrections to dEs
k in the presence of finite structuralperturbations.
7. Application: Planetary satellites
We now consider the stability of the satellites of Mars, andsome of the recently discovered satellites of the giant planets thatare suspected to be granular aggregates. These objects are orbitallystable, as for Solar System’s planets, b0 P 0:9 > 0:56; cf. Section 6.3.Consequently, given that these objects’ dimensions are smallerthan both their orbital radius and their parent planet’s size, werecall from Sections 5.2 and 6.4 that the stability/instability ofthese satellites is equivalent to them being structurally stable/unstable to structural perturbations alone. Furthermore, rather
I. Sharma / Icarus 229 (2014) 278–294 289
than employing the scaled distance q to report our results, we willutilize the more convenient equivalent distance
�q0 ¼ q0gb0
� 1=3
;
as is appropriate for systems with b0 � 1 and/or large enough q. Thenecessary formulation may be obtained by expanding Bð0Þ and Bð1Þ
everywhere in powers of 1=q; see Section 4.1 of Paper I.We now gather together relevant data.
7.1. Data
Data for satellites suspected of being rubble piles was collectedin Sharma (2009) from various references, and is repeated inTables 1–5 for convenience, after including these satellites’ equiv-alent diameters and the average pressures within each calculatedemploying (13) and (22). Estimates of the equivalent diametersof Mars’ moons are taken from Thomas (1989). The size of Jupiter’smoon Amalthea was estimated by Anderson (2005) and for theremaining Jovians by Thomas et al. (1998). In the case of Saturn,we utilize data of Thomas (2010) for all except Anthe, for which,we turn to Cooper et al. (2008). Thomas (2010) also updates theshapes of several of Saturn’s satellites. For Uranus’ moons, weutilize Karkoschka (2001)’s size estimates for all except Cupid,Mab and Perdita, for which Showalter and Lissauer (2006) provide.Finally, Karkoschka (2003) furnishes approximate sizes of all ofNeptune’s moons listed in Table 5. We will locate these satelliteson two-dimensional sections of the a� b� �q0 space correspondingto a ¼ 0:6;0:7;0:8;0:83;0:9 and 1, and a ¼ b.
7.2. Material parameters
In Section 3 we modeled granular aggregates like soils via twomaterials parameters, viz., the internal friction angle /F and theplastic bulk modulus j. To apply our stability results to planetarysatellites we will require estimates of both /F and j for suchobjects. Sharma (2013) discusses in detail tests by which theseparameters are measured for terrestrial soils. These are the hydro-static compression test that provides j and the triaxial test that esti-mates /F; see also Bolton (2003, p. 189). Both /F and j depend onthe pressure at which the aggregate is confined, and the void ratiobetween the aggregate’s void space and the grains’ volume.
It is generally observed that when sheared, dense soils display amaximum friction angle before settling into a lower limiting valuethat defines the soil’s critical state. The maximum corresponds tothe soil’s peak strength, while the final limiting value is the soil’sultimate strength. On the other hand, in loose soils the peak andultimate strengths typically coincide. Now, the critical state of asoil is found insensitive to its initial sate. Moreover, the criticalstate changes little from one soil to another. Given the regularityin a soil’s critical state, and the contrasting variability in its peakstress, we follow Sharma (2013) and identify /F with the soil’sultimate friction angle.
Most laboratory tests are done at confining pressures of one ormore MPa and sand is found to have a /F between 30� and 40�
and a plastic bulk modulus j of 1 MPa. Barring Jupiter’s Adrasteaand the Saturnians Daphnis, Methone, Anthe, Pallene and
Table 1Satellites of Mars. For Mars, q0 ¼ 3934 kg m�3 and b0 � 1. The satellite’s axes ratios a and bdimensional rotation rate xE , and orbital period P (in days) are given. For each q0;xE is fo
Satellite a b q0 g ¼ q=q0
Phobos 0.83 0.70 2.76 0.48Deimos 0.83 0.72 6.91 0.38
Polydeuces, all other satellites are confined by average internalpressures �pE of about 1 kPa or more. All of Uranus’ and Neptune’smoons have �pE’s of the order of tens of kPa. Only Neptune’s moonProteus is large and distant enough to have an internal averagepressure beyond 1 MPa. Thus, we require information of thebehavior of granular aggregates at confining pressures rangingfrom hundreds of Pa to about a MPa.
The triaxial tests of Ponce and Bell (1971) and Fukushima andTatsuoka (1984), and the micro-gravity tests of Sture et al.(1998) are the few tests that report internal friction angles of soilsbelow a confining pressure of 100 kPa. Similarly, hydrostatic com-pression tests of Lancelot et al. (2006) and Lade et al. (2009) arepossibly the only current sources that report the plastic bulk mod-uli j of lightly-confined soils. These tests are discussed in detail inSharma (2013). Based on the discussion therein, we will assumethat a rubble satellite with average internal pressure �pE above10 kPa will have an internal friction angle /F lying between 30�
and 40�, and a plastic bulk modulus j of about 1 MPa. For granularsatellites confined by lower �pE, following Sharma (2013), we willexplore /F over the broader range of 20–50�. The lower limit ismotivated by the fact that dissimilar soils may respond very differ-ently, e.g., natural sandy gravel has a /F of 25�, and, furthermore, itis possible for previously worked soils to have residual frictionalstrengths as low as 10�; see Bolton (2003, Section 8.7.3, p. 55).Similarly, given the paucity of data for the plastic bulk moduli ofgranular aggregates lightly confined at tens of Pa, we will admitthe possibility of j being of the order of a kPa. Finally, we note thatgiven that most satellites considered here have diameters greaterthan 10 km, it is unlikely that internal cohesion as discussed byScheeres et al. (2010) will play a significant role.
We now investigate the stability of these satellites.
7.3. Local stability
We will investigate the local stability to infinitesimal homoge-nous velocity perturbations of the satellites of Mars, Jupiter, Sat-urn, Uranus and Neptune by locating them on appropriate planarsections of the three-dimensional a� b� �q0 space. This is done inFigs. 8–11 employing data of Tables 1–5.
In Fig. 8, both Phobos and Deimos lie within the equilibriumzone corresponding to /F ¼ 10�, so that they are not criticallyequilibrated. Thus, as discussed in Section 6.6’s third paragraph,both these moons will be locally stable to small perturbations.Consider now the effect of the decay in Phobos’ orbit assuming itto be a granular aggregate with friction angle /F ¼ 40�. Shor(1975) reports that Phobos is moving closer to Mars at the rateof about 4 m per century. Thus, in about 2000 years its �q0 willreduce to 1.54, corresponding to nearly two martian radii. At thisstage, a granular Phobos will reach the boundary of the /F ¼ 40�
equilibrium region. This location lies outside the stability regionsassociated with any choice of the plastic modulus, and suggestingthat local instabilities may cause Phobos, were its /F ¼ 40�, toreshape significantly. Similar arguments may be made for otherchoices for Phobos’ internal friction angle.
Of Jupiter’s four satellites, both Thebe and Amalthea lie withinthe /F ¼ 20� equilibrium region; see, respectively, Fig. 9(b) andFig. 10(c). Thus, they will probably be locally stable. Adrastea in
, along with its scaled and equivalent distances from Mars, q0 and �q0 respectively, non-und from (23) assuming the satellite to be tidally locked.
�q0 ¼ q0g1=3 xE P ðdÞ �d (km) �pE (kPa)
2.16 0.257 0.319 22.2 14.24.98 0.073 1.262 12.2 3.2
Table 2Satellites of Jupiter. For Jupiter, q0 ¼ 1300 kg m�3 and b0 ¼ 0:935. Data for the four inner moons are displayed. See also Table 1’s caption.
Satellite a b q0 g ¼ q=q0 �q0 ¼ q0 g=b0ð Þ1=3 xE P ðdÞ �d (km) �pE (kPa)
Metis 0.7 0.6 1.79 0.65 1.59 0.412 0.295 43.4 5.2Adrastea 0.8 0.7 1.80 0.65 1.59 0.407 0.298 16.5 1.2Amalthea 0.6 0.5 2.54 0.65 2.24 0.244 0.498 167 91.4Thebe 0.8 0.7 3.10 0.65 2.75 0.180 0.675 98.5 60.5
Table 3Satellites of Saturn. For Saturn, q0 ¼ 690 kg m�3 and b0 ¼ 0:902. Data for the fourteen inner satellites are displayed. See also Table 1’s caption.
Satellite a b q0 g ¼ q=q0 �q0 ¼ q0 g=b0ð Þ1=3 xE P ðdÞ �d (km) �pE (kPa)
Pan 0.9 0.6 2.22 0.6 1.93 0.306 0.575 28.2 0.96Daphnis 1 0.7 2.26 0.5 1.85 0.325 0.594 7.6 0.06Atlas 0.9 0.5 2.28 0.67 2.07 0.276 0.602 30.2 1.2Prometheus 0.6 0.4 2.31 0.68 2.11 0.268 0.613 86.2 5.5Pandora 0.8 0.6 2.35 0.73 2.19 0.253 0.629 81.4 11.9Epimetheus 0.9 0.8 2.51 0.93 2.54 0.195 0.694 116.2 67.9Janus 0.9 0.8 2.51 0.92 2.52 0.204 0.695 179 135.5Methone 1 1 3.22 1.45 3.77 0.111 1.010 3.2 0.14Anthe 1 1 3.28 1.45 3.84 0.109 1.037 1.8 0.04Pallene 1 0.7 3.52 1.45 4.12 0.097 1.154 5 0.27Telesto 0.7 0.6 4.89 1.45 5.73 0.060 1.888 24.8 4.7Calypso 0.8 0.5 4.89 1.45 5.73 0.060 1.888 21.4 3.3Polydeuces 0.8 0.7 6.26 2.17 8.39 0.034 2.737 2.6 0.14Helene 0.9 0.6 6.26 1.45 7.33 0.041 2.737 35.2 11.2
Table 4Satellites of Uranus. For Uranus, q0 ¼ 1270 kg m�3 and b0 ¼ 0:977. Data for the innerthirteen satellites are displayed. We assume all satellites to have a ¼ b and q ¼ q0 , sothat g ¼ 1 and �q0 ¼ q0=b01=3. See also Table 1’s caption.
Satellite a ¼ b q0 �q0 ¼ q0=b01=3 xE P ðdÞ �d(km)
�pE
Cordelia 0:7� 0:2 1.95 1.97 0.298 0.335 40 18.2Ophelia 0:7� 0:3 2.10 2.12 0.265 0.376 43 21.9Bianca 0:7� 0:2 2.31 2.33 0.229 0.435 51 32.1Cressida 0:8� 0:3 2.42 2.44 0.215 0.464 80 98.8Desdemona 0:6� 0:2 2.45 2.47 0.211 0.474 64 40.2Juliet 0:5� 0:1 2.52 2.54 0.202 0.493 94 64.5Portia 0:8� 0:1 2.59 2.61 0.194 0.513 135 285.4Rosalind 1 > 0:8 2.74 2.76 0.179 0.558 72 113.1Cupid 1 2.91 2.93 0.163 0.618 18 7.1Belinda 0:5� 0:1 2.94 2.96 0.160 0.624 90 63Perdita 1 2.99 3.01 0.156 0.638 26 14.9Puck 1 > 0:93 3.36 3.39 0.131 0.762 162 581.2Mab 1 3.82 3.85 0.108 0.923 24 12.8
290 I. Sharma / Icarus 229 (2014) 278–294
Fig. 9(b) and Metis in Fig. 10(b), however, lie well outside any ofthe shaded stability zones, and very nearly on the /F ¼ 40�
equilibrium curve. This suggests that if Adrastea and Metis aregranular aggregates, then their internal friction angle has to begreater than 40�, else, being locally unstable to infinitesimalperturbations, they could not have retained their current structure.Given the discussion in Section 7.2 and the fact that Table 2 reportsthat both Metis and Adrastea are confined at average pressures ofless than 10 kPa, this is not an unlikely possibility. Alternatively,perhaps, both these moons are not rubble piles. We note that these
Table 5Satellites of Neptune. For Neptune, q0 ¼ 1600 kg m�3 and b0 ¼ 0:98. Data for the six inner
Satellite a b q0 g ¼ q=q0 �q0 ¼
Naiad 0.6 0.5 1.95 0.73 1.9Thalassa 0.9 0.5 2.02 0.73 1.8Despina 0.8 0.7 2.12 0.73 1.9Galatea 0.9 0.7 2.50 0.46 1.9Larissa 0.9 0.8 2.97 0.73 2.6Proteus 0.9 0.9 4.75 0.73 4.3
conclusions about Adrastea and Metis are quite robust to errors inestimates of the axes ratios of these objects’ best-fit ellipsoids.Indeed, except at small b, the /F ¼ 40� curve lies outside stabilityregions shown in Figs. 8–11.
Most of Saturn’s moons fall within the equilibrium zone corre-sponding to /F ¼ 20�, and this makes them locally stable. Anexception, perhaps, is Daphnis in Fig. 9(a) that is very nearly onthe /F ¼ 20�, and right on the edge of the stability region associ-ated with j ¼ 1. Daphnis is a 7.6 km diameter object with a lowdensity of :340 kg m�3 and confined by an average pressure of amere 60 Pa. As discussed in Section 7.2, it is, therefore, conceivablefor it to display low frictional resistance, and, so, be critically equil-ibrated. But, in that case, it would require a scaled plastic bulkmodulus of about one or lower in order to be locally stable, as a lo-cally unstable granular Daphnis would not have survived. ForDaphnis, a j of one is equivalent to a dimensional plastic bulkmodulus of approximately 114 Pa. This appears unreasonablylow, and we are forced to conclude that either Daphnis’ frictionalresistance is greater than 20�, or it is perched precariously closeto an unstable boundary. Similar arguments may be made aboutthe lightly-confined Saturnians Pan in Fig. 10(a) and Prometheusin Fig. 10(c). Interestingly, Pan is about the size of Eros, but has amuch lower internal pressure due to the presence of Saturn’s tidalforces.
Neptune’s satellites are reasonably big, with the smallest, the66 km sized Naiad, confined most lightly by an average pressureof about 21.8 kPa. At such pressures, as discussed in Section 7.2,we may take the aggregate’s internal friction angle /F to bebetween 30� and 40� and its plastic bulk modulus to be at least 1
most moons are displayed. See also Table 1’s caption.
q0ðg=b0Þ1=3 xE P ðdÞ �d (km) �pE
5 0.348 0.294 66 21.83 0.329 0.311 82 50.72 0.307 0.335 150 230.94 0.303 0.429 176 134.79 0.185 0.555 194 542.31 0.091 1.122 420 2884.9
40o
20o
30o
Phobos
Deimos
90o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.831
2
3
4
Axes ratio, β = a3 /a1
5
10o 5o
3o
1o101001
1000
0
Dis
tanc
e in
q’ =
q‘(η
/β’)1/
3
Fig. 8. The stability of the two martians is explored on the planar section a ¼ 0:83of the three-dimensional a� b� �q0 space. Stability regions are shaded andcorrespond to different values of the non-dimensional plastic bulk modulus j.Boundaries of equilibrium zones for various internal friction angles /F are shown bydotted lines; these follow Fig. 9 of Paper I. When it exists, the Darwin ellipsoidobtained at /F ¼ 0� is indicated by a ‘+’ symbol. The abscissa’s upper limit is set bythe requirement b 6 a.
0.1 0.2 0.3 0.4 0.5 0.6 0.73 Axes ratio, β = a /a1
1
2
3
4
5
6
7
8
0.8
Daphnis
Helene
Methone
Anthe
Equi
vale
nt d
ista
nce
, q’ =
q‘(η
/β’)1/
3
0.7 1
101
0
50
40o
3o5o
10o
20o
30o
90o
1o
1000
100
Pallene
Calypso
0.1 0.2 0.3 0.4 0.5 0.6 0.7Axes ratio, β = a3 /a1
1
2
3
4
5
6
7
8
9
0.8
Despina
Thebe
Adrastea
Equi
vale
nt d
ista
nce,
q’ =
q‘(η
/β’)1/
3
40o
3o
5o
10o20o
30o
90o
1o
10100
1
1000
0
500
Polydeuces
Pandora
Fig. 9. The stability of the inner moons of Neptune, Saturn and Jupiter is exploredon appropriate a�sections of the three-dimensional a� b� �q0 space. The bound-aries of equilibrium zones shown by dashed curves follow Fig. 12 of Paper I. See alsoFig. 8’s caption.
0.90.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Axes ratio, β = a3 /a1
1
2
3
4
5
Thalassa
Larissa
Pan
Atlas Epimetheus
1o
40o
3o
10o20o
30o
90o
101001
1000
0
500
Dis
tan
ce in
q’ =
q‘(η
/β’)1/
3
Proteus
Janus
Galatea
0.70.1 0.2 0.3 0.4 0.51
2
3
4
Axes ratio, β = a3 /a1
5
0.6
MetisEqui
vale
nt d
istan
ce, q
’ = q
‘(η/β
’)1/3 6 Telesto
1o
40o
3o
5o
10o20o
30o
90o
10
100
1
1000
0
0
500
0.60.1 0.2 0.3 0.4 0.51
2
3
4
Axes ratio, β = a3 /a1
5
Naiad
q’ =
q‘(η
/β’)1/
3
1o
40o
5o
10o20o
30o
90o
10
100
11000
0
0Amalthea
Prometheus
Fig. 10. Continued from Fig. 9. The stability of the inner moons of Neptune, Saturnand Jupiter is explored on appropriate a�sections of the three-dimensionala� b� �q0 space. The boundaries of equilibrium zones shown by dashed curvesfollow Fig. 13 of Paper I. See also Fig. 8’s caption.
I. Sharma / Icarus 229 (2014) 278–294 291
MPa. Despina in Fig. 9(b), and Larissa, Galatea and Proteus inFig. 10(a) are all comfortably within the equilibrium zonecorresponding to /F ¼ 30�. All of these should, therefore, be locallystable at first order. Thalassa in Fig. 10(a) is an 82 km sized objectthat lies outside the stability region and close to the /F ¼ 30� equi-librium zone’s boundary. Considering that the shapes of Neptune’smoons are not well estimated, it is possible that errors in shapesmay locate Thalassa on or outside the /F ¼ 30� curve. But, becauseequilibrium boundaries for /F P 30� typically fall outside theshaded stability regions, we will require that Thalassa not be crit-ically equilibrated, else it would have been locally unstable andwould not have survived. This will force us to conclude that Tha-lassa has an internal friction angle greater than the /F of the criticalequilibrium surface that passes through its location in a� b� �q0
space. Thus, estimates of errors made in its shape prediction canallow us to provide lower bounds to Thalassa’s internal frictionangle. Finally, Naiad in Fig. 10(c) lies almost on the boundary ofthe /F ¼ 40� equilibrium region and well outside all the shadedstability regions. This suggests that either that Naiad is not a gran-ular aggregate or its internal friction angle is at least 40�. The
CupidRosalind
Puck
Belinda
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Axes ratio, β = a3 /a1
1
2
3
4
5
Mab
Perdita
Cordelia
JulietPortia
Bianca Ophelia
Equi
vale
nt d
ista
nce,
q’ =
q‘(η
/β’)1/
3
1o
10o
90o30o
10o
3o
40o20o
3o
1o
20o
Desdemona
10100
1
1000
0
1
10
100
0
Cressida
0
Fig. 11. The stability of the thirteen small inner moons of Uranus shown on the a-section a ¼ b of the three-dimensional a� b� �q0 space. The boundaries of equilib-rium zones shown by dashed curves follow Fig. 14 of Paper I. The error bars depictuncertainty in the axes ratio listed in Table 4. See also Fig. 8’s caption.
40o
20o
30o
Phobos
Deimos
90o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.81
2
3
4
Axes ratio, β = a3 /a1
5
10o
5o 3o 1o
50100
1000
Dis
tanc
e in
q’ =
q‘(η
/β’)1/
3
Fig. 12. The two martians are located in the regions where dð2ÞEsk is negative on the
planar section a ¼ 0:83 of the three-dimensional a� b� �q0 space. The internalfriction angle /F ¼ 40� , as indicated by the heavy black line. Each shaded region isidentified by its corresponding choice of non-dimensional plastic bulk modulus j.The boundaries of equilibrium zones shown by dotted lines follow Fig. 9 of Paper I.When it exists, the Darwin ellipsoid associated with /F ¼ 0� is indicated by a ‘+’symbol. The abscissa’s upper limit is set by the requirement b 6 a.
0.1 0.2 0.3 0.4 0.5 0.6 0.7Axes ratio, β = a3 /a1
1
2
3
4
5
6
7
8
0.8
Daphnis
Methone
0.7 1
050
1000
100
AnthePallene
3o5o
20o
30o
90o
1o
1
0
Helene
10oD
ista
nce
in q
’ = q
‘(η/β
’)1/3
0.1 0.2 0.3 0.4 0.5 0.6 0.7Axes ratio, β = a3 /a1
1
2
3
4
5
6
7
8
9
0.8
Adrastea
50100
1000
Pandora
Polydeuces
Thebe
Despina
10o
Calypso
50o
3o
5o
20o
30o
90o
1o
Dis
tanc
e in
q’ =
q‘(η
/β’)1/
3
Fig. 13. The inner moons of Neptune, Saturn and Jupiter are located in regionswhere dð2ÞEs
k is negative on appropriate a�sections of the three-dimensionala� b� �q0 space. The internal friction angle /F is set to 20� and 50� in the top andbottom figures, respectively, and indicated by a heavy black line. The boundaries ofequilibrium zones shown by dotted lines follow Fig. 12 of Paper I. See also Fig. 12’scaption.
292 I. Sharma / Icarus 229 (2014) 278–294
experiments discussed in Section 7.2 report that soil confined atpressures seen in Naiad display friction angles not much greaterthan 40�. This suggests that Naiad is located rather close to anunstable boundary.
Finally, in the case of Uranus, from Fig. 11 we observe that itsmoons are unlikely to be critically equilibrated. Thus, they are alllocally stable.
7.4. Stability to finite structural perturbations
We consider now the stability of planetary satellites listed inTables 1–4 to finite structural perturbations following Section 6.7’sanalysis. Figures 12–14 locate these satellites on plots of regionswhere dð2ÞEs
k < 0 on appropriate planar sections of the three-dimensional a� b� �q0 space while fixing the internal friction angle/F at some suitable value. Recall that in Section 6.7, stability tofinite structural perturbations was linked to the second-ordercorrection dð2ÞEs
k to the structural kinetic energy change beingnegative. As before, we will limit the friction angle /F of rubblesatellites to between 20� and 50�. Finally, recall from Section 6.7that the further a satellite is from its critical equilibrium curve,greater will be the structural perturbation required to make ityield. Because larger perturbations are less probable than smallerones, we will assume that a satellite that lies well within theequilibrium region corresponding to a plausible choice of its /F isnot susceptible to finite structural perturbations, notwithstandingdð2ÞEs
k’s sign. Therefore, in the following we will focus typically onthose satellites that are close to boundaries of their associatedequilibrium zones.
From Fig. 8 we observe that both of Mars’ moons lie comfort-ably within the equilibrium region associated with /F ¼ 20�, andso will be considered stable to finite structural perturbations. Nev-ertheless, Fig. 12 reports that were these moons to have internalfriction angles of 40�, they will have dð2ÞEs
k < 0 as long as theirscaled plastic bulk moduli j are about 50 or higher. For Phobosand Deimos this corresponds to dimensional plastic bulk modulibeing greater than 1 MPa and 0.25 MPa, respectively. Theserequirements appear reasonable given their average internal pres-sures reported in Table 1 and the discussion in Section 7.2. Wehave commented previously about the possibility of Phobosbecoming locally unstable when in about 2000 years its orbit’s size�q0 reduces to below 1.54 and it crosses the /F ¼ 40� critical curve.
0.90.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Axes ratio, β = a3 /a1
1
2
3
4
5
GalateaPanAtlas
50100
1
10000 Proteus5o
1o
40o
3o
10o
20o
30o
90o Thalassa
Epimetheus
Larissa
Janus
Dis
tan
ce in
q’ =
q‘(η
/β’)1/
3
0.70.1 0.2 0.3 0.4 0.51
2
3
4
Axes ratio, β = a3 /a1
5
0.6
Metis
6
50
1000
100
Telesto
10o
1o
50o
3o5o
20o
30o
90o
Dist
ance
in q
’ = q
‘(η/β
’)1/3
0.60.1 0.2 0.3 0.4 0.51
2
3
4
Axes ratio, β = a3 /a1
5
50100
10001o
40o
5o
20o
30o
90o
10o
PrometheusAmalthea
Naiad
0
Dis
tanc
e in
q’ =
q‘(η
/β’)1/
3
Fig. 14. Continued from Fig. 13. The inner moons of Neptune, Saturn and Jupiter arelocated in regions where dð2ÞEs
k is negative on appropriate a�sections of the three-dimensional a� b� �q0 space. The internal friction angle /F is set to 20�;50� and 40�
in the top, middle and bottom figures, respectively, and indicated by a heavy blackline. The boundaries of equilibrium zones shown by dotted lines follow Fig. 13 ofPaper I. See also Fig. 12’s caption.
I. Sharma / Icarus 229 (2014) 278–294 293
However, Fig. 12 indicates that dð2ÞEsk becomes positive much
before �q0 decreases to 1.54. It is, therefore, likely that in the pres-ence of finite structural perturbations Phobos will become unstablevia a structural instability even earlier.
Jupiter’s moons Thebe and Amalthea in, respectively, Fig. 9(b)and Fig. 10(c), lie well within the /F ¼ 20� curve, so that their caseis similar to those of Mars’ satellites. Consider now Adrastea andMetis that, as we saw in Section 7.3, needed a /F > 40� to be locallystable as rubble piles. Table 2 reports that these two satellites areconfined by pressures of a few kPa. Hence, we suppose that theseobjects have /F ¼ 50�. Figure 13(b) and Fig. 14(b) locate, respec-tively, Adrastea and Metis on plots of regions where dð2ÞEs
k is nega-tive. We observe that both lie just within their correspondingequilibrium zone, but in a region where dð2ÞEs
k is positive for allchoices of j. This indicates that while they may be locally stable,they will be unstable to finite structural perturbations no matterwhat be their plastic bulk modulus.
Most of Saturn’s satellites fall well within plausible equilibriumzones. Possible exceptions are Daphnis in Fig. 9(a), Pan and Atlas inFig. 10(a), and Prometheus in Fig. 10(c) that lie close to the
/F ¼ 20� critical equilibrium curve. We will explore the stabilityto finite structural perturbations of Daphnis, Pan and Atlas, and asimilar analysis will hold for Prometheus. Figure 13(a) andFig. 14(a) locate, respectively, Daphnis, and Pan and Atlas, on plotsof regions where dð2ÞEs
k is negative. We observe that Daphnis, Panand Atlas will have have dð2ÞEs
k > 0 for all choices of their plasticbulk moduli. They may, therefore, succumb to structural perturba-tions of finite magnitude, if they had a low /F of 20�.
All of Neptune’s moons are confined by an average pressuregreater than 20 kPa. Following the discussion in Section 7.2, weassume that their /F P 30�. With the exception of Thalassa andNaiad, the other Neptunians all lie comfortably within the/F ¼ 30� equilibrium zone. As we saw in the previous sections,Naiad needs an internal friction angle of at least 40� to be locallystable; see Fig. 10(c). From Fig. 14(c) we observe that Naiad witha /F ¼ 40� has dð2ÞEs
k > 0 for all possible choices of its plastic bulkmodulus, thereby making it susceptible to finite perturbations.This suggests that for Naiad to have survived as a rubble pile, itshould have a /F much greater than 40�, or, alternatively, Naiadmay not be a granular aggregate. Thalassa’s case is similar.
Finally, except for Cordelia that lies close to the /F ¼ 20�
equilibrium curve, Uranus’ other moons are well in the interiorof the equilibrium region associated with /F ¼ 20�. However, fromTable 4, we see that Cordelia is confined by an average pressure ofnearly 20 kPa, and the soil tests of Section 7.2 indicate its frictionangle to be greater than 30�. In that case, Cordelia is comfortablywithin its corresponding equilibrium zone, and we are reasonablyassured of its stability to finite homogeneous structuralperturbations.
8. Conclusion
In this paper, we considered the local stability of rubble-pilesatellites via an energy criterion. A satellite was defined to be(totally) stable/unstable provided both its orbit and its structurewere stable/unstable to both orbital and structural perturbations.We narrowed our focus to satellites whose dimensions were muchsmaller than both their orbital radii and their parent planets’ sizes,as is true for planetary satellites in the Solar System. In this limit,we saw that orbital motion remained unaffected by structuraldeformation, so that orbital stability could be considered indepen-dently. We then showed that an orbitally unstable satellite wouldbe totally unstable, but an orbitally stable satellite would also needto be structurally stable to structural perturbations alone in orderto be deemed totally stable. Orbital stability was tested by a spec-tral analysis, while structural stability to structural perturbationswas probed by an extension of the structural stability analysis ofrubble-pile asteroids of Sharma (2013). Finally, we also consideredthe stability to finite structural perturbations of satellites that areclose to the boundaries of their respective equilibrium zones.
We applied our results to moons of Mars and the inner satellitesof the giant planets that are often hypothesized to be granularaggregates. We were able to relate the impact of the satellites’material parameters, such as internal friction and plastic bulkmodulus, on their local stability. We also found that it is possiblefor satellites that are found to be locally stable to be, in fact, unsta-ble to finite structural perturbations. This allowed us to predictthat moons such as Adrastea, Metis and Naiad either have highinternal friction angles or are not cohesionless granular aggregates.
In the future, we aim to extend the stability test to granularsatellites whose dimensions are comparable to the central planet’sand/or are close to the latter. In this case, orbital motion will beinfluenced by structural deformation, and orbital stability cannotthen be investigated independently. Such an analysis would thenculminate naturally with an investigation into the stability ofrubble-pile binaries.
294 I. Sharma / Icarus 229 (2014) 278–294
Acknowledgments
I thank members of the Mechanics & Applied MathematicsGroup at IIT Kanpur for helpful discussions. This work was doneunder PLANEX grant.
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