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Dumb-bell-shaped equilibrium figures for fiducial contact-binary asteroids and EKBOs

In this work, we investigate the equilibrium figures of a dumb-bell-shaped sequence with which we are still not well acquainted. Studies have shown that these elongated and nonconvex figures may realistically replace the classic"Roche binary approximation"for modeling putative peanut-shaped or contact binary asteroids. The best-fit dumb-bell shapes, combined with the known rotational period of the objects, provide estimates of the bulk density of these objects. This new class of mathematical figures has been successfully tested on the observed light curves of three noteworthy small bodies: main-belt asteroid 216 Kleopatra, Trojan asteroid 624 Hektor and Edgeworth-Kuiper-belt object 2001 QG298. Using the direct observations of Kleopatra and Hektor obtained with high spatial resolution techniques and fitting the size of the dumb-bell-shaped solutions, we derived new physical characteristics in terms of equivalent radius and bulk density. In particular, the growing inadequacy of the radar shape model for interpreting any type of observations of Kleopatra (light curves, AO images, stellar occultations) in a satisfactory manner suggests that Kleopatra is more likely to be a dumb-bell-shaped object than a"dog-bone".


Introduction
In the past decade, rotational light curves generated by unresolved close-binary systems or by single contact binary asteroids have been systematically modeled with two tightened Roche ellipsoids (Jewitt and Sheppard, 2002, Sheppard and Jewitt, 2004, Takahashi and Ip, 2004, Lacerda and Jewitt, 2007, Descamps et al., 2007a, Lacerda, 2008, Lacerda, 2011, Lacerda et al., 2014. These equilibrium figures of rotating liquid masses were introduced by Edouard Albert Roche  in 1847 to investigate the equilibrium shapes of a tidally locked binary system composed of a triaxial ellipsoidal body orbiting a rigid sphere along a circular Keplerian path. Later, Darwin (1906) extended the problem to a system composed of two mutually interacting fluid ellipsoids. Such a system could exist in equilibrium only for two limit values of the mass ratio q: q = 1 for congruent ellipsoids, and q = 0, if one of them is more massive than the other. More recently, nearly thirty years ago, Leone et al. (1984) revisited the problem for various mass ratios; for the sake of simplicity, they analytically calculated the ellipsoidal figure of each component while keeping the companion spherical. Strictly speaking, this is not the Roche problem but instead the "Roche binary approximation" as it was dubbed by the authors. Indeed, such solutions are pure approximations in the sense that tidal effects of a member's triaxiality on the companion's equilibrium shape are neglected. Moreover, another caveat arises from the assumption of a circular Keplerian orbit. Such an assumption is no longer tenable because the general dynamics of the orbit are not governed by a gravitational potential generated by a spherical mass.
The full and exact problem of equilibrium configurations of synchronous binaries has been numerically investigated only in the past few years (Gnat andSari, 2010, Sharma, 2009). These investigations address the problem differently but reach the same conclusions: the "Roche binary approximation" seriously fails to reproduce the exact equilibrium solutions for small mutual separations. The equilibrium figures of tightly bound binaries are no longer triaxial ellipsoids, and departures from these pure ellipsoidal forms may amount to nearly 20%. The reason is that at closer separation, the bodies are both distorted by their mutual gravitational perturbations, and this distortion is most pronounced along their line of centers. However, the authors independently found that at mutual separations on the order of twice the sum of their mean radius, departures from ellipsoids given by the "Roche binary approximation" are negligible, and the latter can be successfully used for widely separated synchronous systems such as the most puzzling, 90 Antiope (Descamps et al., 2007b, Kryszczyńska et al., 2009, Descamps et al., 2009b, Descamps, 2010. The purpose of this paper is to seek another set of figures able to be reliably substituted for Roche ellipsoids, which are somewhat inappropriate -and unrealistic -when dealing with the so-called contact binary asteroids. In this context, the term, and the model, "contact binary" was first coined by Cook (1971) to account for the unusually large amplitude of the asteroid 624 Hektor. Furthermore, as noted by Weidenschilling (1980), light-curve amplitudes above 0.9 magnitude cannot be produced by rotation of a single ellipsoidal figure, such as an ellipsoid of Jacobi. This is why models of asteroids with two ellipsoids in contact, or nearly in contact, were introduced to satisfactorily match such light curves. In fact, the term "contact binary" is now somewhat misleading. From a simple shape model made of two ellipsoids in contact, the "contact binary" concept gradually changed in terms of phenomenological interpretation. Currently, behind this designation is an implicit scenario of formation lending credence to the idea that a contact binary asteroid originates from low-velocity collisions that allow coalescence of a pair of asteroids into a single body without complete fragmentation. However, caution is needed here before making conclusions too quickly that such a scenario is the major, if not the unique, cause of subsequent formation of such puzzling bodies. Therefore, in the rest of the paper, the term "contact binary" only refers to a shape model of an asteroid with two ellipsoids in contact without any underlying assumption about its origin. Furthermore, another feasible mechanism may exist that presents an opposing viewpoint, i.e., that such highly elongated bodies, with two prominent equal lobes at their ends, could instead be the outcome of an evolutionary process starting from a single body toward a well-detached binary system after undergoing a gradual shrinkage in the middle section. This is the fission hypothesis. In the transient stage, the figures are somewhat peanut-shaped and endowed with a high angular momentum. This family of equilibrium figures exists and can be described through the dumb-bell equilibrium sequence discovered and computed more than three decades ago (Eriguchi et al., 1982). This sequence branches off the Jacobi ellipsoid sequence from a bifurcation point that was noted early by Chandrasekhar (1966) as a point of secular instability. The equilibrium figures of this sequence are peanut-shaped or dumb-bell-shaped bodies composed of a bridge of material separating two equal lobes. This sequence ends by joining the congruent Roche ellipsoid family. Eriguchi and colleagues numerically computed a small sample of figures of the dumb-bell sequence, but this family of concave equilibrium shapes has been sidelined, mainly due to the impossibility of producing a tractable analytical description of such figures that is able to compete with the user-friendly Roche ellipsoids.
In the present work, the dumb-bell sequence is extensively and accurately computed. Each figure is fully characterized by its geometry and its physical properties. They are then compared with the classical Roche approximation as a toolkit used to fit light curves of putative contact binary asteroids.
Fitting a light curve with the help of a dumb-bell-shaped model is equivalent to determining the value of a single parameter: the dimensionless angular velocity that uniquely identifies the dumb-bell figure.
This parameter is a function of the rotational period and of the macroscopic density of the asteroid. In particular, we demonstrate that the dumb-bell model allows avoidance of some bias inherent to the Roche approximation in such a way that it is then possible to fully describe the asteroid not only in terms of overall shape and internal density but also in terms of light-scattering type by its surface without any preliminary, and consequently arbitrary, assumptions. Again, we stress that, in the rest of this article, the terms "dumb-bell-shaped" and "peanut-shaped" do not refer at all to any scenario of formation, whether fission or collision, but only to the best-matching figures derived for some peculiar asteroids. We remind the reader that the main interest in equilibrium shape models is to infer the bulk density of the object under consideration through the determination of the normalized angular velocity Ω.

General problem
Let a fluid mass be rotating in space as a rigid body. Let us assume that it is isolated from other bodies and that its particles are subject to mutual attractions according to Newton's law of gravitation. We designate as the z-axis the axis of rotation of the mass and establish the origin of a rectangular system of axes Oxyz at the mass center. The condition fulfilled by an equilibrium figure of a homogeneous, incompressible rotating body is that, at any surface point, there is a perfect balance between self-gravity and the inertial acceleration due to the rotation. In terms of potential, the total potential, which is the sum of gravitational and rotational potentials, must be constant all over the surface. The equilibrium shape is thus an equipotential surface. Since Maclaurin discovered the axially symmetric figure of this equilibrium, many researchers have looked for non-axially symmetric equilibrium figures. It was Jacobi who obtained non-axially symmetric equilibrium figures for the first time: Jacobi ellipsoids. Since this discovery, much effort has been made to find a non-axially symmetric equilibrium with a nonellipsoidal surface. Chandrasekhar (1966) found a neutral point on the Jacobi sequence. It is considered as a point where a secular instability sets in that is stimulated by the presence of viscosity in the fluid, resulting in a slow -or secular -departure from the unperturbed figure at a rate proportional to viscosity. Thus, in a slow evolution, the trajectory of an evolutionary figure would leave the Jacobi family at the point of bifurcation and subsequently follow a new family of non-ellipsoidal figures.
Chandrasekhar did not obtain any new sequence. He was mainly concerned with equilibrium stability.
These figures are algebraic surfaces of the fourth order. Their existence was first proven by Eriguchi et al. (1982). We know that all figures of equilibrium, the existence of which have been demonstrated, possess at least one plane of symmetry. This plane is perpendicular to the axis of rotation. This is the xy plane. Assuming another plane-symmetry about the x-z plane, Eriguchi and colleagues (1982) computed, for the first time, the equilibrium figures of a new sequence that branches off the Jacobi sequence at the bifurcation point found by Chandrasekhar. They demonstrated that the instability would therefore grow slowly with a furrow in the middle and gradually deepen to form non-axisymmetric dumb-bell-shaped figures.
To determine the equilibrium figures, we follow the approach earlier adopted by Eriguchi et al. (1982).
The problem can be posed in the following way: The radius vector of the free surface of an equilibrium figure can be expressed as where φ is the longitude, and μ = cosθ, θ is the colatitude.
A non-axisymmetric equilibrium is assumed, so that the following conditions must be satisfied that suppose the figure to be plane-symmetric about the x-y and x-z planes: If the fluid mass is isolated in space, the boundary surface belongs to the set of surfaces of equal potential. For a given angular velocity, ω, the equilibrium condition is that the total potential, i.e., the sum of the gravitational and rotational potentials, should have the same value everywhere on the surface. It is expressed as Here, Φ (r, μ, φ) and C are the gravitational potential and a constant, respectively. The problem should lead to the determination of a new equilibrium shape in such a way that Eq.
Following the formalism of Eriguchi et al. (1982), the gravitational potential Φ(r, μ, φ) of a homogeneous and incompressible fluid mass of density ρ can be expanded as with the functions f defined for any value of n>0 (with the exception of n=2) by: r R(μ',φ') r, n = 2 Eriguchi et al. (1982) address the problem using the Newton-Raphson iterative scheme: let S 0 be the surface of a figure of equilibrium of a homogeneous liquid mass corresponding to the angular velocity ω 0 ; S 0 is the initial guess, and it is the Jacobi ellipsoid at the bifurcation. After specifying a new value of R(1,φ)/R(0,0) and linearizing Eq.
[3] to first order in the small quantities δR, δω and δC, they solve R(μ,φ), ω and C using successive approximations. The solution is another surface S 1 found in the neighborhood of S 0 for a new value of angular velocity ω 1 = ω 0 + δω differing, though only by a small amount from ω 0 . The solution S 1 is then taken as the initial guess for the next step corresponding to a new value of the ratio R(1,φ)/R(0,0). Successive solutions along the sequence may thus be computed.
The main drawback is that the method fails to provide the solution for any value of ω given a priori because the choice of the initial guess should be very close to the solution. This problem can be circumvented by using a new class of surfaces, dubbed cassinoids, as relevant initial guesses.
Cassinoids are concave figures built from a family of plane curves first introduced by Cassini more than three centuries ago as an alternative for Keplerian elliptic planetary orbits (see Appendix A).
Interest in these figures slowly declined, but now they have unexpectedly found a second life in astronomy. These easy-to-use tridimensional bilobed figures can be mathematically handled through a unique shape parameter. Although cassinoids are axially symmetric about their main axis, with no flattening, they can be successfully applied as initial guesses, and the method converges toward an equilibrium solution after ~80 iterations with all the required precision (10 -4 ) according the following scheme: Let us define the function Γ by For a given angular velocity ω, we wish to solve Γ = 0 for the free surface R(μ,φ) and the constant C.
From an initial guess R 0 , which yields an initial value C 0 from Eq.
[3], we can derive R 1 = R 0 + δR 0 with the Newton-Raphson iterative method. R 1 is the correction of the current guess by a small amount, δR 0 , which is readily given by The same procedure is then iteratively applied and repeated until δC = C-C 0 and δR(μ,φ) become negligibly small to within some numerical threshold fixed to 10 -4 . Each figure is modeled as a polyhedron made up of facets, whose vertices are defined over a square grid in longitude (φ) and latitude (θ), the sides of which are two degrees wide. The system of equations

Figures of the dumb-bell equilibrium sequence
[8] is applied to N = 16471 vertex coordinates R(μ,φ). The two plane symmetries (Eq. [2]) restrict the problem to N/4 vertices distributed on the surface of a quarter-sphere. The geometrical and physical quantities of the solutions are summarized in Table 1. As a triaxial ellipsoid, we characterize the figure by five parameters: the semi-maximum dimensions along the three axes a, b, c -with a > b > c -and the waist polar semi-axis c' (the bilobed shape appears when ′ ≤ ). Values of these geometrical quantities are given for an equivalent radius R e = 1 (radius of a sphere of equal volume). The flattening and the elongation are defined, respectively, by the ratio b/c between the intermediate to short (polar) dimensions and the ratio a/b of the maximum equatorial dimensions. The dimensionless form of the angular velocity, the inertial momentum with respect to the axis of rotation and the angular momentum, Ω, λ and H, respectively, were used. These are defined as follows: The normalized angular velocity Ω (or normalized frequency) is defined by the ratio between the angular velocity and the critical spin rate for a spherical body , which is the maximum spin rate that can be sustained by an undeformable body. At this spin rate, centrifugal forces would equal gravity at the equator of a spherical body.
where G the gravitational constant, and ρ the bulk density.
The non-sphericity parameter λ is defined as the ratio between the moment of inertia of the body with respect to its spin axisconsidered as the maximum moment of inertia axisand the moment of where M is the mass of the body, and h is a coefficient depending on the shape (h=2/5 for a perfect sphere).
The specific angular momentum, H, is obtained by dividing the angular momentum by the following term, with the notation first introduced by Darwin (1887): With the adopted notations, the specific angular momentum can be conveniently written as The point of bifurcation found by Chandrasekhar corresponds, using our notation, to Ω = 0.3995 and H = 0.4841. This is the starting point of the new sequence. The sequence is plotted (crosses) in Fig. 1.
Likewise, the congruent (mass ratio of the components equal to 1) synchronous binaries sequence computed by Gnat and Sari (2010) is shown (diamonds). For the sake of comparison and consistency, the filled circles denote the computations of dumb-bell figures first computed by Eriguchi et al. (1982). 3. Application to some asteroids

Bilobed asteroids
To test the dumb-bell equilibrium figures, we focus, in this section, on three noteworthy bodies, each taken from the main asteroid groups, whose observational characteristics may qualify them for having strongly bifurcated shapes. Confirmation that elongated/bilobed contact binary asteroids, with a central concavity or waist and a modestly asymmetric shape, really exist noticeably came with the first estimations of asteroids' three-dimensional shapes from the radar images of near-Earth objects (Ostro et al., 2000, Benner et al., 2006. They must fulfill at least two inescapable salient conditions, typical of a putative dumb-bell-shaped body: a high degree of elongation, e.g., measured from radar or adaptive optics imagery, with the ratio of its maximum equatorial dimensions of ~2.4-2.8, and a high photometric range light curve (~ 1.2 mag which means that the brightness can change by as much as a factor of 3 during a full rotation) with U-shaped maxima and V-shaped minima of similar depth.
However, the light curve minima become broader with increasing phase angle, which implies that Vshaped minima are highly suggestive of a contact binary asteroid only when viewed near zero phase angle (Lacerda and Jewitt, 2007). As far as the elongation is concerned, we can, based on the data in Table 1, state that the elongation of the dumb-bell-shaped figure varies from 2.8 to 3.6, which is substantially higher (~10-15%) than the most elongated observed asteroids. On the other hand, elongations lower than 3.0 are achieved for scaled angular velocities lower than ~0.30. As such, dumb-bell figures appear to be systematically slightly more elongated than the most elongated asteroids observed thus far.
For this point, we can rightly question the usefulness of fluid equilibrium figures to address real asteroid shapes as has recently been performed (Harris et al., 2009). Indeed, we must pay attention to this crucial point because, as noted earlier, density determination is a highly model-dependent result. It is evident that the theoretically possible figures are determined by a set of conditions that can concern either the physical state of the mass of a celestial body or its kinematical state. As for the physical state, the principal condition is that a celestial body is a fluid body. The fact that most asteroids are now considered as rubble piles, meaning that they are composed of solid boulders, as they appear to be, will not diminish the importance of results based on the fluidity hypothesis. Furthermore, some recent works tend to demonstrate that actual asteroid shapes are consistent with the evolution of aggregates tending toward minimum free energy states (Tanga et al., 2009a, Comito et al., 2011. These weak aggregates are held together by self-gravity. They seemingly have zero tensile strength, but this does not mean they are "fluid." However, they have the ability to re-shape if pushed to rapid rotation rates. Within this granular scheme, rubble-pile asteroids may exhibit many permissible spin and shape combinations (Holsapple, 2004). Yet Tanga and colleagues (2009a), from their numerical simulations, conclude that "when reaching a potential valley, the objects are very close in potential energy to both the Maclaurin and the Jacobi sequence. They thus can be stable and close to fluid equilibrium, despite being fairly different in shape." From subsequent simulations of the re-accumulation following a catastrophic disruption, Tanga et al. (2009b) demonstrate that natural shapes close to fluid equilibrium shapes are preferentially produced. They stress the importance of other non-disruptive shaping factors, such as minor impacts, tidal forces and seismic shaking, during the lifetime of rubble pile asteroids. In other words, it is not surprising not to find perfect equilibrium fluid figures because most of the solar system small bodies are not fluid but rather a collection of aggregates with some level of internal friction. Bodies' gravity tends to dominate over their internal strength, such that they gradually take the forms dictated by gravity and appropriate for their angular velocity. These forms, on the whole, are therefore well accounted for by equilibrium figures, but, locally, they depart from them with lumps and ridges that carve them out to produce their final bifurcated shapes.
Most of the time, we do not have observations other than those provided by photometry. A typical case of this is that first considered in the relation with the Edgeworth-Kuiper belt object 2001 QG 298 . The two following test-asteroids, 216 Kleopatra -the famous "dog-bone-shaped" asteroidand 624 Hektor, take advantage of high-resolution imagery (radar and/or adaptive optics), which undoubtedly have revealed their bilobed form. Moreover, they contain at least one satellite in orbit, which allows a direct measurement of their bulk density provided that their sizes are sufficiently well known, and consequently, their angular velocity Ω. The asteroids 624 Hektor and 216 Kleopatra may thus be directly compared face-to-face to the dumb-bell model. Several light curve measurements are available for consideration for inclusion in this study: Sheppard and Jewitt (2004) and Lacerda (2011) for 2001 QG 298 , Descamps et al. (2011) for Kleopatra and Dunlap and Gehrels (1969) for Hektor. The light curves are of suitable quality (RMS < 0.07 mag) for use in the analysis. All light curves are double peaked with a symmetrical shape, i.e., with both peaks and both minima having essentially the same brightnessa classic signature of a both elongated and bilobed body. Table 2 summarizes the results obtained for each object taken in consideration in the present paper in terms of angular velocity Ω and resulting bulk density ρ.

General photometry of a dumb-bell-shaped figure
Theoretical light curve computations of polyhedral models were performed to compare their respective morphology and main features with the observed light curves as a function of the sole normalized rotational velocity Ω. From a 3D shape model, once the line of sight and the direction of Sun are known, it suffices to select the facets that are both visible by the observer and illuminated by the Sun.
The total reflected light is then computed by adding the contribution of each of these active facets. In the case where dumb-bells exhibit strong concavities, self-shadowing may occur at non-zero phase angles. This effect is taken into consideration because it represents a non-negligible amount of the overall brightness variation. Each facet reflects the solar light according to its orientation with respect to the Earth and Sun as well as according to the adopted scattering law. We adopted an empirical scattering model for the surfaces first proposed by Kaasalainen et al. (2001) combining, through a weight factor k, a lunar-type reflection -described by the Lommel-Seeliger law appropriate for low albedo rocky surfaces -and an icy-type law -Lambertian or diffuse reflection suitable for high-albedo surfaces with multiple scattering.
In the above equation, μ 0 and μ are the cosines of the angles between the surface normal and the incidence and emission directions, respectively. A pure Lambert scattering is obtained with k = 1.0; this is the case of bright bodies with high limb darkening. Conversely, it is expected that a low-albedo body, with no multiple scattering and negligible limb darkening, has a small k value. The sudden nonlinear brightening toward small phase angles close to oppositionknown as the opposition surge or opposition effectis not taken into account. As the opposition effect occurs at very small phase angles for loosely packed regolith (phase angle α < 1°) and most data are recorded at higher phase angles, light curves are weakly affected by the opposition effect. Because a dark surface would be dominated by single scattering, the k values are expected to be in the order of 0.1 (Kaasalainen et al., 2001).

Fig. 4: To be inserted
The light curve range varies between 1.0 mag and 1.4 mag. In addition to the large range of variability, light curves exhibit some general common features such that maxima are more smoothly rounded (Ushaped) than are sharp-edges minima (V-shaped). The neck between the two lobes plays a role in the overall shape of the light curve observed, forcing it to be more U-shaped than V-shaped. Several main trends can be drawn from these observations:  Amplitudes become larger with increasing Ω values;  At the same angular velocity and for the same weighting parameter, the depth of the two minima is always deeper with the dumb-bell-shaped figure. The gap can only be filled by the Roche model with a higher value of the angular velocity;  For a given angular velocity, both models generate light curves of the same amplitude, provided that the weighting parameter used with the Roche model is significantly higher.

Amplitude-aspect method for shape and pole determination
In light curve fitting, we treat, as three independent parameters, the scaled angular velocity Ω, the scattering parameter k and the aspect angle ψ, which is the angle of then spin vector with respect to the line of sight (ψ = 90° when the object is observed exactly equator-on and the aspect angle varies from 0° to 180°). As the amplitude of the light curve depends on these three parameters, we work out a specific method suitable for high-amplitude light curves with characteristic shapes to discriminate and quantify the effect of each of them. The method is analogous to the classical amplitude-aspect method (Zappalà, 1981, Pospieszalska-Surdej andSurdej, 1985), which is based on the assumed triaxial ellipsoidal model for the shape of the asteroid with a geometrical scattering law (in fact, it is supposed that the observed brightness of an asteroid is proportional to the instantaneous cross-section seen by a distant observer) and on the relationships between the aspect angle, the light curve amplitude and the asteroid magnitude at the light curve maximum, all obtained in at least three oppositions. These methods assume a triaxial ellipsoidal model for the shape of the asteroid. In the present method, the main difference is in the absence of any preliminary assumptions about the scattering properties of the asteroid surface and a shape model reduced to only one parameter, Ω, instead of three (the three semiaxes a > b > c of the ellipsoid).
To this end, we look for a dumb-bell solution with the appropriate surface scattering and space orientation that minimizes the difference between the observed and synthetic light curve characterized by the following goodness-of-fit criterion χ 2 : Where σ describes the assumed common uncertainty in each point (typically ~0.02-0.05 mag), L measure,i is the value of the i th data point and L model,i the value of the modeled brightness for a given dumb-bell solution.
If we define the number of degrees of freedom ν = N-m, for N data points and m parameters (m = 3), we can introduce the reduced chi-square: which provides a way to obtain the root mean square error or RMS error in our measurements. To find the best solution, in the sense of minimum χ 2 , we use a grid-searching method. We construct a library of light curves for comparison with observation through the entire parameter space studied. All 5°. It appears that the best aspect angle, in the sense of the smallest χ 2 , while the weighting parameter k is held fixed, is roughly independent of Ω and k. The best aspect angle ψ best is thus taken as the mean value of ψ for which confidence levels in the space parameter Ω-k are then computed and plotted from the smallest value χ 2 of the grid position.
To avoid the photometric effects due to the phase angle, light curves must come from observations made near opposition, for phase angles close to zero. To find the 1σ (corresponding to the 68.4% probability of finding the true values of the three parameters), 2σ (95.4%) and 3σ (99.7%) standard deviation regions encompassed by the joint variation of the three free parameters (ψ, Ω, k), we draw the contour plots corresponding to the values of χ 2 increased, respectively of 3.53 (solid line), 8.02 and 14.16 (dashed lines) from the χ 2 minimum. As will be demonstrated, the best solution is not necessarily given by the smallest χ 2 . In fact, confidence levels isolate one or more solution regions in the space (Ω, k) called "islets." In the case of several such regions, the best final solution can be univocally set out from at least two light curves by taking the overlapping islet. Reliable errors on the parameters are estimated from the full range of the outer limit of the largest overlapping 1σ islet.
Only two observations made at two different oppositions are needed to both solve for the shape solution and the scattering parameter, which are supposed to be identical in each case, whereas the aspect angle must preferably be changing to better constrain the final solution. Furthermore, the derived aspect angles allow inferring two opposite pole solutions. Indeed, we have the well-known relationship between the aspect angle ψ, the geocentric ecliptic longitude and latitude of the asteroid λ a β a , and the ecliptic longitude and latitude of the spin axis λ p and β p (Zappalà, 1981):   cos sin sin cos cos cos The knowledge of the cosine of the aspect angle for a single observation means that all the possible rotational axes lie on a cone forming an angle ψ with respect to the direction Earth-asteroid. At least two light curves obtained on different dates are needed to provide information about the changing aspect of the asteroid as it moves around its orbit. From these observations obtained, we may derive two pole solutions that define an identical orientation of the rotation axis. One, (λ p0 , β p0 ), is the true pole. The other, (λ p0 +π, -β p0 ), is a "spurious" pole that is the 180° opposite pole solution. For a low inclination orbit (β a~0°) , which is the most common case among asteroids, these two pole solutions equally satisfy Eq.
[20], and the ambiguity in the pole solution cannot be eliminated from this method alone. This ambiguity is reflected in the value of the phase angle and, unless otherwise stated, the phase angle is always displayed between 0 and 90°. Several pairs of solutions, obtained in different oppositions, produce a set of pole coordinates, whose mean is adopted as the direction of the rotational axis. The standard deviation from the mean can provide a check of the obtained accuracy.

The Edgeworth-Kuiper-belt object 2001 QG 298
In 2003, the EKBO 2001 QG 298 was found to have an extremely large light variation of 1.14 ± 0.04 mag and a relatively long period of 13.7744 ± 0.0004 h (Sheppard and Jewitt, 2004). It was the first EKBO suspected of being a putative contact binary asteroid. Since then, the light curve obtained was successively fitted into the framework of the Roche binary approximation (Takahashi an Ip, 2004, Lacerda andJewitt, 2007) and from the exact solutions of equilibrium configurations of tidally locked homogeneous binaries (Gnat and Sari, 2010). The scaled angular velocity Ω was derived, and, from the knowledge of the rotation period, the bulk density was determined to a value ranging from 0.59 to 0.72 g/cm 3 (Table 2). Seven years later, another light curve was measured showing a peakto-peak photometric range of 0.7 ± 0.1 mag, which is significantly lower than in 2003 (Lacerda, 2011).
This observed decrease in amplitude, caused by a change in viewing geometry, was interpreted by a large obliquity near 90 ± 30° with an aspect angle falling from ψ = 90° in 2003 (equator-on) to nearly ψ = 78° in 2010.  Figure 6 shows the confidence levels derived from each light curve. The contours were calculated by holding the aspect angle fixed at its optimum value -ψ = 83.5 ± 1.0° in August 2003 and ψ = 68.5 ± 1.0° in August 2010 while varying Ω and k. Leftmost plots show how the best dumb-bell model (solid curve) compares to the light curve data (points). Several islets appear to be permissible solutions. Although the 1σ contour is highly spread for the confidence levels related to the 2010 light curve, due to a poor RMS of 0.07 mag, there appears to be only one common islet that requires a unique solution given by the center of the islet: Ω = 0.318 ± 0.001 and k = 0.14 ± 0.03 (see Table 2). It stems from the fact the bulk density is 0.56 ± 0.02 g/cm 3 . This is not a measurement of the density but just a model-dependent density, which is the result of a relevant model used to interpret the photometric data, and, as such, the associated uncertainty is internal uncertainty, which includes only measurement uncertainty and reliability of the applied shape model. The external uncertainty should include all input uncertainties, including for sidereal period, aspect angle, weighting parameter and, above all, shape model.
Our solution can be compared to the previous attempts to derive the physical properties. Takahashi and Ip (2004) could reproduce the observation only for a nearly equatorial view with an aspect angle of approximately 90°. Their best-fitted k value of the scattering parameter was k ~ 0.6-0.8. Smaller k values could not fit the observed light curve and amplitude too well. Likewise, Gnat and Sari (2010) cannot derive a solution with an angular velocity Ω greater than 0.282: the maximum value of their shape solutions. Consequently, they are forced to adopt a high value of the weighting parameter k to match the observed amplitude. Lacerda and Jewitt (2007), in their simulations, found that different surface propertieslunar or icy-type surface lawdo not significantly change their density estimates. This is roughly true, albeit in Fig. 6, we can see that some preferred solutions, materialized in the form of islets, are well-identified for a set of discrete value of the weighting parameter. Lacerda and Jewitt adopted an equatorial-on geometry and found two solutions for, respectively, a lunar-type surface and an icy-type surface with no possibility of giving priority to one solution over another. In Fig. 7, we display the observed light curve (crosses) in 2003 along with the best-fit models coming from the dumb-bell solution (solid line) and the Roche solution (dashed line) found by Takahasi and Ip (2004) with a secondary to primary mass ratio q = 0.7. The RMS error is of 0.055 mag with the dumb-bell solution, whereas it is of 0.063 mag with the Roche solution. Thus, we may conclude that the dumb-bell model fits the data better than the Roche approximation. Furthermore, it leads to more realistic physical properties. The weighting parameter, for example, is in good agreement with what it is expected for a dark surface, k ~ 0.1 (Kaasalainen et al., 2001). In addition, there are no preliminary assumptions about the aspect angle, which is univocally determined by our amplitude-aspect method.

The main-belt asteroid 216 Kleopatra
The asteroid 216 Kleopatra has the largest amplitude of the large main-belt asteroids. Kleopatra was observed, for the first time in 1980 (Tholen, 1980), with an amplitude between 1.3 and 1.4 magnitude over the course of its 5.3853 ± 0.0003 h rotational period (Pilcher and Tholen, 1982). Twenty years later, early radar observations helped to develop the first polyhedral shape model of this M-type asteroid, which appeared as a double-lobed, narrow-waisted object (Ostro et al., 2000). In addition, recent adaptive optics imagery, conducted in connection with its unprecedented close opposition of September 2008, disclosed two small moons orbiting Kleopatra, which allowed the measurement of a bulk density of 3.6 ± 0.2 g/cm 3 (Descamps et al., 2011). As Kleopatra is classified as an M-type asteroid, this bulk density is suggestive of a highly unconsolidated rubble-pile internal structure if we adopt a metallic grain density of ~7 g/cm 3 (see discussion in Descamps et al., 2011). Its normalized angular velocity Ω was then determined at a value of 0.318 ± 0.045, which, once combined with a nonsphericity parameter of 3.65 inferred from its radar shape model, yields a specific angular momentum of 0.47. This immediately places Kleopatra near the dumb-bell equilibrium sequence (Descamps and Marchis, 2008a). Kleopatra is thus a worthy object to be tackled from the perspective of the dumb-bellshaped model approach.  Table 2) for an aspect angle of 79.5 ± 1° in agreement with the value (81.8°) given by the pole solution λ = 76 ± 3° and β = 16 ± 1° in J2000 ecliptic coordinates (Descamps et al., 2011). To check the validity of this solution, we applied it to the observation performed on 1 August 2008. At this time, the high phase angle (α = 19.93°) was responsible for the observed high amplitude of ~1.3 mag because of the shadows cast successively by each lobe. This self-shadowing effect has been taken into account for deriving the synthetic light curves generated by each model. Figure 9 clearly shows that the radar shape model fails to satisfactorily reproduce the observed light curve, possibly due to the high asymmetry between its lobes, asymmetry which is not present in the observed minima. In contrast, the dumb-bell-shaped solution perfectly fits not only the observed amplitude but also the whole light curve. The intensity of the self-shadowing effect is maximum when the brightness is minimum and is rendered in Fig. 10. We have also computed the light curve from the other solution given by k = 0.41 and Ω = 0.288. The RMS error is 0.0294 mag, larger than the RMS of 0.0255 mag obtained with the previous solutions.  of Kleopatra. With a solid line, R e = 67.5 ± 3 km, the IRAS radius confirmed by the 2008 campaign (Descamps et al., 2011). With a dashed line, R e = 62.5 ± 3 km, the best-fitted value of the equivalent radius derived through a specific method described in Descamps et al. (2009a). This shortening of the equivalent radius comes from the longer form of the dumb-bell-shaped model (see Fig. 11). In passing, it emphasizes the high degree of dependence of the measured size on the adopted shape model.  Dunham et al. (2013). Figure 13 displays the resulting occultation chords for the event derived by D.
Herald with Occult 4.0 software (http://www.lunar-occultations.com/iota/occult4.htm). The aspect is basically the same as that of Fig. 12. In this case, the phase plays no part in the result. Among the observed chords, Kleopatra signals its presence with its fine-tapered silhouette, which seen from its longer side has a dumb-bell shape. Both shape models were fitted and applied to the silhouette. The best equivalent radius for the dumb-bell model was again adjusted to 62.5 ± 3 km. The dumb-bell outline is approximately 250 km long and 70 km wide, whereas the radar model outline is approximately 230 km long and slightly more than 80 km wide. The narrow waist of the silhouette is clearly visible and is better rendered by the dumb-bell model. Consequently, the AO image as well as the stellar occultation indicates that Kleopatra may be more of a dumb-bell than a dog-bone, which is not surprising as the overall uncertainty on the radar shape, as reported in Ostro et al. (2000), is large (20-25%). Ultimately, we see that Kleopatra appears to have a more lengthened and narrow form than generally believed through its radar model. The second point is that its equivalent radius should be revised to a smaller value of 62.5 ± 5 km. This pushes the bulk density, measured from the mass of Kleopatra derived from Kepler's third law (Descamps et al., 2011), to a value of 4.4 ± 0.4 g/cm 3 , which is in excellent agreement with the value inferred from the angular velocity of the associated dumb-bell model (Table 2). In addition to the radar model, several other shape models have been proposed. Tanga et al. (2001) derived from observations made with the HST/FGS interferometer a shape with two ellipsoidal components in contact. Adopting a lunar-type law, the overall size of their model is 273 ± 7 km x 75 ± 2 km x 51 ± 13 km. The large error for the minor axis c is due to the lack of sensitivity on this axis due to the nearly pole-on geometry of the asteroid during the observation. The inferred elongation (3.64 ± 0.20) is notably greater than the radar result but comparable to the elongation of the dumb-bell model, 3.2 (see Table 1). Their flattening is equal to 1.47 ± 0.36, which is also on the same order of the dumb-bell  (Table 2). They notably reported that "we could not find a proper k value smaller than 0.1, and furthermore, with such small k values, no Roche binary models can explain all the observations." This makes it impossible of reconciling the inferred k value with the photometric properties of the dark surface of Kleopatra. This apparent incompatibility is eliminated through our dumb-bell-shaped solution. Eventually, both solutions have the same amplitude of 1.25 mag at 90° of aspect and 0° phase. This correspondence has already been raised in Fig. 4. 3.6. The Trojan asteroid 624 Hektor Cook (1971) appears to have been the first to speculate that 624 Hektor was a close binary, but that was based on the high amplitude light curve. Cook's work followed Dunlap and Gehrels' discovery of the light curve's high amplitude and their correct identification of an elongated shape as the cause of the light curve inferred from the Hektor's rotational pole, which lies near the ecliptic (Dunlap and Gehrels, 1969). Depending on the orbital configuration, the light curve range varies between 0.1 and 1.2 mag with V-shaped minima and U-shaped maxima. At this time, most of the known highly elongated asteroids were among the smallest known asteroidsonly a few kilometers in lengthand believed to be plausible huge splinter-shaped fragments of a larger body. In contrast, Hektor, among the large asteroids, ranked third in the highest light curve amplitude. The fragment explanation was no longer ad hoc. Hartmann and Cruikshank (1978) made the first observations using simultaneous photometry and infrared (IR) bolometry to demonstrate that the maxima of the visual light curve correlates with maxima of IR thermal light curve, proving that the amplitude is caused by an elongated shape rather than being caused by albedo variations (which would produce anti-correlation). More than three decades later, in 2006, the final confirmation of the highly elongated/bilobed shape came from AO high-resolution imaging made at the Keck-II telescope (Marchis et al., 2006).

Fig. 13: To be inserted
We selected three light curves (Fig. 14) from the extensive dataset presented in Dunlap and Gehrels (1969) and used by Lacerda and Jewitt (2007) to fit a binary Roche solution. The differences in light curve range from one observation to another reflect the change of the observational geometry mainly through the aspect angle ψ. We obtain only one unique solutionor one common confidence isletable to simultaneously best fit the observations at three observing campaigns (see Table 2). The weighting parameter, k = 0.12 ± 0.06, indicates a dark lunar-type surface in agreement with a low albedo of ~0.06 (Fernández et al., 2003). Although the angular velocity determined by each of the solutions, Roche and dumb-bell model, is nearly the same, the light curve amplitude can be rendered with the Roche approximation only with a significantly higher value of the weighting parameter, k = 0.5. In fact, Lacerda and Jewitt made extensive use of a uniform scattering lawconsidered as equivalent to a lunar-type law with no limb darkening (k = 0) at small phase anglesbut we were unable to replicate their results unless an increased k value is used. They report a maximum amplitude of 1.2 mag at 90° of aspect and 0° phase, but we found it equal to 0.89 mag with k = 0 and 1.2 mag with k = 0.5. This moderate scattering agrees with its D-type and its red spectral slope characteristics of a rocky surface (Emery et al., 2011). Hektor is a member of the redder spectral group among Trojan asteroids. Patroclus, another Trojan asteroid, known as a nearly equal-sized binary with a density of 1.08 ± 0.33 gcm -3 (Marchis et al., 2006b), is a member of the less-red spectral group resulting in a very different intrinsic composition. Using the three positions of Hektor in its orbit about the Sun and the three derived aspect angles (see Fig. 14  In the same way as for Kleopatra previously, the reliability of dumb-bell-shaped solutions derived for Hektor was tested on an AO image taken with the 10-m Keck telescope on 16 July 2006 at 13:50 UTC and published in Marchis et al. (2014). Figure 15 shows two overall contours superimposed on the AO image. Each contour corresponds to a value of the equivalent radius, 92 ± 5 km (dotted line)the best fitted value -and 112 km (solid line). Marchis et al. (2014) derived a radius of 128 ± 3 km, which appears to be clearly overestimated. With such a size and from the new orbital parameters of the moonlet, they infer an extremely low bulk density of 1.0 ± 0.3 gcm -3 . There is a typo in the reported value of the semi-major axis in Marchis et al. (2014), and the right value is a = 957.5 ± 55.3 km (F. Vachier, private communication). With an orbital period P = 2.965079 ± 0.000288 days, the measured mass is therefore M = 7.914215 ± 1.408682 10 18 kg, which gives a bulk density of 1.32 ± 0.53 gcm -3 for an object radius of 112.74 ± 17.86 km. With our new size of 92 km, which is 18% smaller, Hektor is approximately 416 x 131 x 120 km in size and the measured bulk density increases to the value 2.43 ± 0.35 gcm -3 , which is in very close agreement with the value derived from the knowledge of the scaled angular velocity (Table 2), 2.56 ± 0.05 gcm -3 .

Summary
Contact-binary asteroids with a bimodal appearance were suspected long ago based on their high photometric range light curves. The only tool available for modeling such unusual light curves relied on the figures drawn from the "Roche binary approximation" within a fully synchronized binary system, which is able to account both for the bilobed form and the distinguishable light curves.
However, there is a serious caveat tied to their very conditions of use, i.e., when they are nearly in contact. In this case, the Roche binary approximation fails to properly represent the exact solution of the full Roche problem. However, the inferred solutions, albeit with these reservations, allowed the first coarse estimations of their bulk density. In the present work, this issue has also been tackled from another conceptual approach, taking advantage of the benefits provided by a new class of equilibrium figures: the dumb-bell-shaped figures. For the first time, we have reckoned and independently validated the dumb-bell sequence found by Eriguchi et al. (1982) in a more detailed and precise manner. In particular, we better assessed biases involved in the Roche approximation. Roche solutions lead to a systematic overestimate of the surface-scattering behavior (identified by the weighting parameter k) coupled with a smaller elongated shape. They are more often inconsistent with the global reflectance properties. Inferred results on surface properties should therefore be interpreted with caution and even not interpreted at all. Furthermore, quite obviously, the appearance of Roche solutions is clearly unrealistic, and, as such, they cannot be used to match resolved images or outline observations such as those provided by stellar occultations. Ultimately, solutions provided by the Roche approximation are useful only to the extent that they can be used as tractable models for fitting light curves and retrieving a first estimate of the bulk density and nothing else.
A more appropriate way to pursue the modeling of misshapen objects is through the bilobed models We can also question the usefulness of using equilibrium figures in comparison with the powerful results achieved in the last ten years with the light curve inversion method (Kaasalainen and Torppa, 2001). Tridimensional shape models of asteroids can be reconstructed from the analysis of a set of light curves taken with various viewing and illumination geometries. However, the derived shape models are limited to global convex shapes. The difficulty is to tackle strongly nonconvex shapes, which can be dealt with only by using complementary multimodal data such as adaptive optics imagery, stellar occultations, interferometric HST/FGS observations, spacecraft flybys and thermal radiometry Viikinkoski, 2012, Carry et al., 2012). These promising techniques are presently limited to a few targets, which should ideally have a large enough apparent diameter to be resolved and imaged beyond their star-like appearance. Only a few targets fulfill these conditions hitherto: 21 Lutetia (Carry et al., 2012), 22 Kalliope (Descamps et al., 2008b), 121 Hermione and216 Kleopatra (Kaasalainen andViikinkoski, 2012, Descamps et al., 2009). Kuiper belt objects are typical objects for which most of the available observations are nothing more than light curves. Accordingly, the equilibrium model remains the only solution, in the first instance, that is able to provide a first insight into an unknown distant object.
The figure of contact binary objects actually differs, slightly, from the figures given by the solutions of the problem of an isolated fluid mass in uniform rotation. Such a result opens up a new issue to be addressed: the dynamic reshaping under the action of various external effects, through a continuous adaptation of the shape to a new figure of equilibrium resulting from the displacement of the axis of rotation with respect to the axes of inertia.

Appendix A: Cassinoids
The Cassinian ovals (or ovals of Cassini) were first proposed in the late seventeenth century by Giovanni Domenico Cassini (1625-1712) as a model for describing the movement of the Earth relative to the Sun. He rejected the Keplerian ellipses and believed instead that the Sun travelled around the Earth on one of these ovals, with the Earth at one of its foci.

Fig. A1: To be inserted
A Cassinian oval is a plane curve, locus of all points P such that the product of the distances of P from two fixed points has some fixed value, that is 1 ̅̅̅̅̅ 2 ̅̅̅̅̅ = 2 , where 1 and 2 are two points located in (a, 0) and (-a, 0), respectively, and b is a constant. Note the analogy with the definition of an ellipse (where product is replaced by sum). As for the ellipse, the two points 1 and 2 are called the foci of the oval. The shape of the curve depends on the ratio = ⁄ . Furthermore, we will assume that < < √2. The case =≥ √2 yields an oval, and the case = gives another well-known curve, the Bernoullian lemniscate described by Jacob Bernoulli. The final case > reduces to two disjoint ovals. The oval may be transposed to the tridimensional case after rotating this curve around its main axis joining the foci. The condition for a point P=(x, y, z) to lie on the oval becomes √( − ) 2 + 2 + 2 . √( + ) 2 + 2 + 2 = 2 As for the ellipse, the origin O is a symmetry center, and Ox and Oy are twofold symmetry axes. On squaring the two sides, we end up with the following quartic polynomial equation for the tridimensional surface of Cassini: ( 2 + 2 + 2 + 2 ) 2 = 4 + 4 2 2 This surface is dubbed a "cassinoid" to echo "asteroid." It can be expressed in a spherical coordinate system by substituting = sin cos = sin sin = cos where θ and φ are, respectively, the colatitude and the longitude of the radius vector R (θ,φ). The parametric surface R (θ,φ) of a cassinoid eventually reads ( , ) 2 = 2 [2 2 2 − 1 + √(2 2 2 − 1) 2 + 4 − 1] For convenience and without loss of generality, we may admit as unit length the half separation between foci so that a = 1. As long as 1 < < √2, the cassinoid describes a purely non-convex surface. As the parameter e of the cassinoid gets smaller the region in the middle, near x = 0, gets thinner. Figure A1 shows the cassinoid solution for an arbitrary value of the parameter e = 1.04. The cassinoid has a longitudinal extension along its main axis of 2√ 2 + 1 with a waist in the middle of 2√ 2 − 1 in diameter; the two lobes have a maximum transversal size equal to 2 , which gives the general expression of the elongation of a cassinoid: The maximum elongation is reached for e = 1 and amounts to 2√2 = 2.83.

Aknowledgements
The   (Ω) Dumb-bell solution 0.318 ± 0.003 0.56 ± 0.02 0.14 ± 0.03 Fig.1: Dimensionless angular velocity Ω vs. dimensionless angular momentum H for the dumb-bell sequence (crosses). The Jacobi and the congruent synchronous binaries sequences are displayed as well. The physical properties of the synchronous binaries sequence shown with diamonds are taken from Gnat and Sari (2010). The filled circles denote the computations of some dumb-bell figures made by Eriguchi et al. (1982). Physical properties of each of these points refer to the values given in Table 1. The Roche approximation of synchronous twin ellipsoids (dashed line) departs drastically from the exact solution (diamonds) at angular velocities higher than 0.2. No Roche solution can exist beyond Ω ~ 0.325. The transition between the dumb-bell sequence and the binaries sequence is smooth and continuous. Fig.2: Sketch of various equilibrium shapes along the dumb-bell sequence for different values of the dimensionless angular velocity Ω. The left uppermost figure is the bifurcation Jacobi ellipsoid (at Ω = 0.3995, H = 0.484) where the Jacobi sequence becomes dynamically unstable. At this point, the dumb-bell shaped sequence branches off smoothly. From this departure ellipsoid, the middle portion of the dumb-bell shape narrows gradually as the angular velocity decreases. From Ω ≅ 0.33, two equal lobes appear at each end with a furrow in the middle of the figure which then looks like a peanut. The family terminates when the furrow shrinks to nothing (Ω = 0.2815, H = 0.502). The shape resembles to the starting equilibrium configuration of synchronous rotating bodies with two tight equal-mass components (see Fig. 12 of Gnat and Sari, 2010). The continuity between both families is clearly seen on Fig. 1.  Fig.3: Light curves of a dumb-bell-shaped figure for different values of the scaled angular velocity Ω (0.2815, 0.32). The scattering parameter k is fixed to 0, the phase angle is zero and the aspect angle is 90°. The light curve obtained with a pair of twin ellipsoids in the framework of the Roche approximation for Ω = 0.32 (close to the limiting case) is superimposed (dashed line). Light curve amplitudes are greater than 1.0.

Fig.4:
Comparison between light curves of equal amplitude derived for a same scaled angular velocity Ω = 0.32 at zero phase and for an aspect angle of 90°. In solid line, the light curve generated by the dumb-bell-shaped figure with k = 0. The dot-dashed line results from the Roche approximation adopting k = 0.4.

Fig.5:
For visualization purposes, dumb-bell and Roche models images are displayed equatorial-on at opposition with their true rendering according their respective scattering law. Shape models have the same volume and are computed for the same angular velocity Ω = 0.32. The scattering parameter k of the Roche model is adjusted to a value giving the same light curve amplitude (see Fig. 4). A significant limb darkening is visible on the Roche model (k = 0.4). All other things being equal (amplitude, reduced velocity, aspect angle), the dumb-bell model is always slightly more elongated than its Roche counterpart. Fig.6: Set of confidence levels for the Kuiper belt object 2001 QG 298 . The contours were calculated by holding the aspect angle fixed at its optimum value while varying Ω and k. The small contours, drawn with solid lines, give the region of the Ω-k space in which there is a 68.4% probability of finding the true values of the two parameters (1σ region). The large contours, shown as dashes lines, correspond to the regions 2σ (95.4%) and 3σ (99.7%). Observed light curve for the Kuiper belt object 2001 QG 298 (data points) and best fit solution (overlapping 1σ region in solid line) is obtained for Ω = 0.318 ± 0.001 and k = 0.14 ± 0.03. With an assumed uncertainty of 0.05 mag, this solution yields in 2003 χ 2 = 187.7 for ν = 167 degrees of freedom which gives a reduced χ 2 of 1.12; in 2010, χ 2 = 112.1 for ν = 60 degrees of freedom which gives a reduced χ 2 of 1.86. Fig.7: Equatorial view of the best shape models of 2001 QG298 according to the dumb-bell or the Roche approach. The Roche solution is taken from Takahashi and Ip (2004) with a secondary to primary mass ratio q = 0.7. Resulting light curves have been superimposed to the 2003 observation: in solid line, the light curve caused by the dumb-bell model, and in dashed line the light curve caused by the Roche solution. The RMS error is σ = 0.053 mag for the dumb-bell model and 0.063 mag for the Roche model.  Fig. 6 for the general description of the contours. Two possible solutions appear. The solution Ω = 0.298 ± 0.002 and k = 0.14 ± 0.05 is preferred because it gives a moderately low scattering parameter. With an assumed uncertainty of 0.025 mag, this solution yields χ 2 = 446.5 for ν = 446 degrees of freedom which gives a reduced χ 2 of 1.00. Fig. 9: Observed light curve for Kleopatra (data points) taken on 1 August 2008. The best fit solution (solid curve) was obtained with a dumb-bell model for Ω = 0.298 and k = 0.15. The dashed curve comes from the dogbone radar model with the same scattering law applied to the surface. The high amplitude of ~ 1.3 magrarely observedis due to a huge self-shadowing effect yielded by the lobes at a high phase angle of 19.9° (see Fig. 10). The radar shape model obviously fails to satisfactorily reproduce the observed light curve, possibly owing to the high asymmetry between both lobes. Fig. 10: Effect of self-shadowing on both models (dumb-bell and radar) on 1 August 2008. Fig. 11: The "dog-bone" radar model and the dumb-bell-shaped model of Kleopatra displayed at the same scale. and east is to the left. A laplacian filter was applied in order to highlight the edge of Kleopatra. Two extracted contours of the dumb-bell-shaped model, each corresponding to a value of the equivalent radius (R e = 67.5 km in solid line, and R e = 62.5 km in dashed line), were superimposed to the AO image. Fig. 13 : Stellar occultation of TYC 4909-00873-1 by Kleopatra on 2009 December 24 th at 11:59 UTC. Kleopatra's silhouette figures like Chinese shadows for each model with an equivalent radius adjusted to 62.5 km. Dumb-bell outline is about 250 km long and 70 km wide. Radar model outline is about 230 km long and slightly more than 80 km wide. The adaptive optics observations as well as the stellar occultation indicate that Kleopatra may be more of a dumb-bell than a dog-bone.  Fig. 6 for the general description of the contours. Phase angle (α) and aspect angle (ψ) are given for each geometry. The best solution (overlapping 1σ region in solid line) is obtained for Ω = 0.298 ± 0.003 and k = 0.12 ± 0.05. With an assumed uncertainty of 0.05, 0.02 and 0.03 mag respectively in 1957, 1967 and 1968, this solution yields in 1957 χ 2 = 81.3 for ν = 65 degrees of freedom which gives a reduced χ 2 of 1.32; in 1967, χ 2 = 102.7 for ν = 65 degrees of freedom which gives a reduced χ 2 of 1.58; in 1968, χ 2 = 188.8 for ν = 119 degrees of freedom which gives a reduced χ 2 of 1.58.   A1: Geometrical description of a cassinoide of parameter e. The cassinoide is computed and displayed for e = 1.04. The elongation of a cassinoide is given by 2√1 + 2 2 ⁄ . Owing to its axisymmetry, the flattening is always zero. The angle θ 0 defines the cassinoide parameter e such that = √2sin 0 . The limiting case is obtained with e = 1 which gives an elongation of 2√2 = 2.83 and a minimum value of θ 0 = 30°.