Surface Gravity of Rotating Dumbbell Shapes
SSURFACE GRAVITY OF ROTATINGDUMBBELL SHAPES
Wai Ting Lam ∗ , Marian Gidea ∗ , and Fredy R Zypman † Yeshiva University, 2495 Amsterdam Avenue,New York City, NY 10033
Abstract
We investigate the problem of determining the shape of a rotatingcelestial object – e.g., a comet or an asteroid – under its own gravita-tional field. More specifically, we consider an object symmetric withrespect to one axis – such as a dumbbell – that rotates around a secondaxis perpendicular to the symmetry axis. We assume that the objectcan be modeled as an incompressible fluid of constant mass density,which is regarded as a first approximation of an aggregate of particles.In the literature, the gravitational field of a body is often describedas a multipolar expansion involving spherical coordinates (Kaula, 1966).In this work we describe the shape in terms of cylindrical coordinates,which are most naturally adapted to the symmetry of the body, andwe express the gravitational potential generated by the rotating bodyas a simple formula in terms of elliptic integrals. An equilibrium shapeoccurs when the gravitational potential energy and the rotationalkinetic energy at the surface of the body balance each other out. Suchan equilibrium shape can be derived as a solution of an optimizationproblem, which can be found via the variational method. We givean example where we apply this method to a two-parameter familyof dumbbell shapes, and find approximate numerical solutions to thecorresponding optimization problem. ∗ Research partially supported by NSF grant DMS-0635607 and DMS-1814543. † Corresponding author: [email protected]. Partially supported by NSF grant CHE-1508085 a r X i v : . [ a s t r o - ph . E P ] F e b eywords: Potential theory, planetary gravitation, solar system,asteroids, mathematical astronomy, axisymmetric celestial objects,dumbbell. Modeling the gravitational fields produced by celestial bodies in the SolarSystem has been of interest since the time of Isaac Newton, Alexis ClaudeClairaut, and George Gabriel Stokes. Their work was focused on determiningthe shape of the Earth. The more general problem is to determine all possibleequilibrium shapes of rotating, homogeneous fluid bodies. Such equilibriumshapes are given by the condition that the total energy, i.e., the sum of thegravitational potential energy and the rotational kinetic energy, should havethe same value at any point of unit mass on the surface of the body. Therotational speed is a parameter of the problem, with different rotational speedsyielding different equilibrium shapes. Maclaurin (Maclaurin, 1742) discovereda family of equilibrium shapes – referred to as Maclaurin spheroids –, whichare ellipsoids that are symmetric with respect to the axis of rotation. As therotational speed of the body is increased, the family of spheroid equilibriumshapes branches off at some value of the rotational speed, giving rise to afamily of tri-axial ellipsoids, referred to as the Jacobi ellipsoids. Further,the family of Jacobi ellipsoids branches off into two families of equilibriumshapes at two distinct values of the rotational speed. One of the branchesconsists of pear-shaped equilibrium figures, referred to as Poincar´e figures.The second branch consists of dumbbell-shaped equilibria. The correspondingbranching point for this latter family was found in (Chandrasekhar, 1967).The dumbbell sequence was first computed in (Eriguchi, Hachisu, & Sugimoto,1982). One should note here that these dumbbell shapes are not given byclosed form equations, but they are computed numerically, for example viaiterative methods.Additionally, the gravitational fields of oblate planets have been of interestto understand the motion of satellites, in particular artificial satellites orbitingaround the Earth (Vinti, 1966; Lara, 2020). More recently, general shapes havebeen considered with the aim of providing accurate dynamics of interactinggravitational bodies that can be monitored experimentally to high precision(Dirkx, Mooij, & Root, 2019; Celletti, Gales, & Lhotka, 2020). In suchproblems, multipolar expansions of the gravitational potential play a key role2n the analysis.The interest in modeling and computing the gravitational field created byshapes beyond spheroids is stimulated by the fact that many asteroids andcomets have irregular shapes. Dumbbell shapes are among the shapes thathave been observed for comets and asteroids, making them both astronomicallyand mathematically interesting.Examples of astronomical dumbbells include the Jupiter Trojan Asteroid624 Hektor (Descamps, 2015), the comet 8P/Tuttle (Groussin et al., 2019),the comet 103P/Hartley (Harmon, Nolan, Howell, Giorgini, & Taylor, 2011),and the transneptunian object 486958 Arrokoth/Ultima Thule (Amarante& Winter, 2020). Astronomically, these dumbbell shapes can originate fromfusion of ellipsoidal precursors, or from elongation. We do not quest hereinto the mechanisms of formation, rather concentrating on the self-gravity ofbodies, and searching for possible shapes that can occur in practice.A related problem, of astrodynamics interest, is to understand the dy-namics of an infinitesimal mass (e.g, a spacecraft) near a dumbbell shapedasteroid (e.g., the Trojan Asteroid 624 Hektor), under the additional influenceof other planetary bodies (e.g., Sun, Jupiter). See, e.g., (Burgos-Garc´ıa,Celletti, Gales, Gidea, & Lam, 2020).For spheroid shapes the gravitational field can be expressed as a sphericalharmonic expansion. However, this method is not particularly suitable fornon-spheroidal bodies. Other types of expansions have been proposed. Forinstance, ellipsoid harmonic expansions (Romain & Jean-Pierre, 2001) arebetter fitted for bodies that can be approximated by an ellipsoid rather thanby a sphere.In this work we provide a general expression for the gravitational potentialproduced by a body of constant mass density, whose boundary surface isdescribed by revolving the graph of a single-valued function about an axis,which becomes the symmetry axis of the body.Using cylindrical coordinates, which are adapted most naturally to thesymmetry of the problem, we obtain a general formula for the gravitationalpotential as a one-dimensional integral of a closed form expression given interms of elliptical functions.When the rotation of the body about an axis perpendicular to the symme-try axis is considered, the total energy of a unit mass particle at the surfaceof the body is expressed as the sum of the gravitational potential energy andthe rotational kinetic energy.Then we formulate the problem of finding shapes of axisymmetric rotating3odies that are approximate equilibrium shapes of the total energy, in thesense that the surface of the body is as close as possible to a level set ofthe total energy function. We formulate this problem as a two-dimensionaloptimization problem, which can be solved numerically using variationalmethods. We illustrate this approach via an example consisting of a two-parameter family of dumbbell shapes, for which we find numerical solutionsto the corresponding optimization problem.The physical justification that the surface of an equilibrium shape isapproximately a level set of the total energy function relies on viewing thebody as either a solidifying fluid, or a collection of solid particles. In thecase of asteroids, this assumption is partially justified by the fact that manysmall bodies in the solar system are believed to be rubble piles, that is,collections of smaller particles. There are models that analyze in detail thegranular structure of asteroids, and study the tidal stress corresponding todifferent particle shapes; see, e.g., (Goldreich et al., 2009). However, numericalsimulations show that such granular structures preferentially assume shapesthat are close to fluid equilibrium shapes (Tanga et al., 2009). Nevertheless,perfect equilibrium fluid shapes are not attained since the bodies are nottruly fluid but subject to some level of inter-particle friction. The fluidityhypothesis-based approach was recently used in (Descamps, 2015) to find,through an iterative scheme, a family of shapes that fit the observed lightcurves of some small bodies in our Solar System, for example of the TrojanAsteroid 624 Hektor.The paper is divided into three main sections. Section 2 provides aformula for the gravitational potential of a solid of revolution in terms ofelliptical functions. This formula provides an economical method to exactlycompute the gravitational field inside and outside the body. Section 3 givesan integral equation that any rotating such body, assuming an equilibriumshape, must satisfy. This result serves as a practical starting point to explorepossible equilibrium shapes that can be attained from suitably chosen familiesof profiles. Section 4 presents an application to a parameterized family ofdumbbells, which uses an optimization process to identify the best fit shapes.Section 5 presents the conclusions. Light curves are measurements of the brightness of a celestial body as a function oftime. They are used for example to determine the shape, rotation period, as well as otherparameters of an asteroid (Kaasalainen & Torppa, 2001). AXISYMMETRIC BODY IN ITS OWNGRAVITATIONAL FIELD
We consider an axisymmetric body relative to the z -axis, obtained by revolvingthe graph of single-valued function s max = f ( z ) ≥ z -axis. Theorigin of the ( z, s )-coordinate system is set at the center of mass. See Figure 1.In this section, we consider shapes that have only this symmetry. InSection 4, we will restrict to a family of shapes that have an additionalsymmetry, namely that the graph of the function f ( z ) is symmetric aboutthe s -axis.Figure 1: Sagittal section of body of revolution around the z -axis. All pointsinside the object are within − z ≤ z ≤ z . For a fixed z , the points are on acircular disc of radius s max .In cylindrical coordinates ( s, φ, z ), the inside of the body is defined by − z ≤ z ≤ z , 0 ≤ s ≤ s max = f ( z ), 0 ≤ φ ≤ π . The function f ( z ) definesthe shape of the body. The points (cid:126)r (cid:48) inside the body produce a gravitationalpotential at a point (cid:126)r = z ˆ z + f ( z )ˆ s on the surface that is given by U G ( (cid:126)r ) = − G (cid:90) Body ρ d (cid:126)r (cid:48) | (cid:126)r − (cid:126)r (cid:48) | , (1)where ρ is the constant mass density of the object, G is the universal gravi-tational constant, and the caret stands for the unit vectors in the axial and5ransverse directions. We note that the integrand in (1) is undefined when (cid:126)r (cid:48) = (cid:126)r , yet the value of the potential at any point (cid:126)r on the surface of the bodyis well defined owing to the three-dimensional nature of the body.We begin by considering the explicit form of equation (1) in cylindricalcoordinates (Skelton, 1982), (Conway, 2000): U G = − Gρ (cid:90) (cid:90) (cid:90) dz (cid:48) ds (cid:48) dφ (cid:48) s (cid:48) + ∞ (cid:88) m = −∞ (cid:90) + ∞ dk e im ( φ (cid:48) − φ ) J m ( ks ) J m ( ks (cid:48) ) e − k | z − z (cid:48) | . (2)Here we will use the primed variables to refer to the sources and the unprimedfor the observation point. The integrals in equation (2) are in the variables z (cid:48) ∈ [ − z , z ], s (cid:48) ∈ [0 , f ( z (cid:48) )], and φ (cid:48) ∈ [0 , π ], respectively.The triple integral in (2) can be rearranged as a sum of an infinite seriesas follows + ∞ (cid:88) m = −∞ (cid:90) + ∞ dk (cid:90) z − z dz (cid:48) e − k | z − z (cid:48) | (cid:90) f ( z (cid:48) )0 ds (cid:48) s (cid:48) J m ( ks ) J m ( ks (cid:48) ) (cid:90) π dφ (cid:48) e im ( φ (cid:48) − φ ) , (3)where the variable s is evaluated at the surface of the body. The integral in φ (cid:48) in equation (3) is equal to 2 πδ m , where δ is the Kronecker function. Thus,equation (2) becomes U G = − πGρ (cid:90) + ∞ dk (cid:90) z − z dz (cid:48) e − k | z − z (cid:48) | (cid:90) f ( z (cid:48) )0 ds (cid:48) s (cid:48) J ( ks ) J ( ks (cid:48) ) . (4)We now recall the identity (Watson, 1922) s (cid:48) J ( ks (cid:48) ) = 1 k dds (cid:48) [ s (cid:48) J ( ks (cid:48) )] (5)which allows us to evaluate the integral in s (cid:48) and obtain U G = − πGρ (cid:90) + ∞ dk (cid:90) z − z dz (cid:48) e − k | z − z (cid:48) | J ( ks ) k f ( z (cid:48) ) J ( kf ( z (cid:48) )) (6)where we have used that J (0) = 0.For clarity, equation (6) is rearranged as U G = − πGρ (cid:90) z − z dz (cid:48) f ( z (cid:48) ) (cid:90) + ∞ dk J ( kf ( z (cid:48) )) J ( kf ( z )) k e − k | z − z (cid:48) | , (7)6here we have explicitly set s = f ( z ) to obtain the potential at the surface.We now consider the integral Y ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) = (cid:90) + ∞ dk J ( kf ( z (cid:48) )) J ( kf ( z )) k e − k | z (cid:48) − z | . (8)This integral can be viewed as the Laplace transform of a combinationof Bessel and power functions, of the kind studied in (Kausel & Irfan Baig,2012). Specifically, using the notation from (Kausel & Irfan Baig, 2012), I λαβ ( a, b, s ) := (cid:90) + ∞ dx x λ J α ( ax ) J β ( bx ) e − sx , for α = 1 , β = 0 , λ = − , x = k, a = f ( z (cid:48) ) , b = f ( z ) , and s = | z (cid:48) − z | (9)we have that the right-hand side of equation (8) equals I − ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) . Using the formula (ENS-4.6) from (Kausel & Irfan Baig, 2012) we obtain I − ( a, b, s ) = 1 πa (cid:34) √ abκ E ( κ ) + ( a − b ) κ √ ab K ( κ ) (cid:35) + sπa Sgn ( a − b ) Λ ( ν, κ ) − sa Θ ( a − b ) , (10)where κ = 2 √ ab (cid:112) ( a + b ) + s , ν = 4 ab ( a + b ) , Λ ( ν, κ ) = | a − b | a + b s (cid:112) ( a + b ) + s Π ( ν, κ ) , (11)where E is the elliptic function of the first kind, K is the elliptic function ofthe second kind, Π is the elliptic function of the third kind, Sgn is the signfunction, and Θ is the unit step function.7or the convenience of the reader, we recall K ( κ ) = (cid:90) dt √ − t √ − κ t , E ( κ ) = (cid:90) √ − κ t √ − t dt, Π ( ν, κ ) = (cid:90) dt (1 − νt ) √ − t √ − κ t . (12)Substituting (9) in (10) we obtain the explicit expression Y ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) = (cid:112) ( z (cid:48) − z ) + ( f ( z (cid:48) ) + f ( z )) πf ( z (cid:48) ) · E (cid:32) (cid:112) f ( z (cid:48) ) f ( z ) (cid:112) ( z (cid:48) − z ) + ( f ( z (cid:48) ) + f ( z )) (cid:33) + f ( z (cid:48) ) − f ( z ) πf ( z (cid:48) ) (cid:112) ( z (cid:48) − z ) + ( f ( z ) + f ( z (cid:48) )) · K (cid:32) (cid:112) f ( z (cid:48) ) f ( z ) (cid:112) ( z (cid:48) − z ) + ( f ( z (cid:48) ) + f ( z )) (cid:33) + ( z (cid:48) − z ) ( f ( z (cid:48) ) − f ( z )) πf ( z (cid:48) )( f ( z (cid:48) ) + f ( z )) (cid:112) ( z (cid:48) − z ) + ( f ( z (cid:48) ) + f ( z )) · Π (cid:32) f ( z (cid:48) ) f ( z )( f ( z (cid:48) ) + f ( z )) , (cid:112) f ( z (cid:48) ) f ( z ) (cid:112) ( z (cid:48) − z ) + ( f ( z (cid:48) ) + f ( z )) (cid:33) − | z (cid:48) − z | f ( z (cid:48) ) Θ ( f ( z (cid:48) ) − f ( z )) . (13)Using equation (7), we obtain the following: Proposition 1
The gravitational potential at a point of cylindrical coordi-nates ( f ( z ) , φ, z ) on the surface of a body generated by revolving the graph of z (cid:55)→ f ( z ) ≥ , − z ≤ z ≤ z is given by U G = − πGρ (cid:90) z − z dz (cid:48) f ( z (cid:48) ) Y ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) . (14)8or a given body shape generated by the profile function f ( z ), equation(14) gives the gravitational potential U G as a function of z at any point ofthe surface.Equation (14) can be easily modified to obtain the exact gravitationalpotential at any point in space, as follows: Corollary 2
The gravitational potential at a point in space of cylindricalcoordinates ( s, φ, z ) , exerted by a body generated by revolving the graph of z (cid:55)→ f ( z ) ≥ , − z ≤ z ≤ z , is given by U G = − πGρ (cid:90) z − z dz (cid:48) f ( z (cid:48) ) Y ( f ( z (cid:48) ) , s, | z − z (cid:48) | ) . (15)The formulas (14) and (15) are very general as they apply to any solid ofrevolution. They give the gravitational potential in terms of a 1-dimensionalintegral of a combination elliptic functions. It is known that the ellipticfunctions have expansions in power series that are convergent, thus (14)and (15) can themselves be expanded in convergent power series (Byrd &Friedman, 2013). Also, elliptic functions are readily implemented in manynumerical computation software packages. In this section we consider a solid of revolution as in Section 2, where the s -axis is chosen to pass through the center of mass. We now assume that thebody rotates around the s -axis with constant rotational speed ω . See Figure2. In the reference frame of the rotating object, that is, one rigidly attachedto the object, a centrifugal force ensues. The magnitude of this force can bederived from the rotational kinetic energy.To evaluate this rotational kinetic energy at any point on the surfacewe consider, as shown in Figure 2, a ring of surface points at coordinate z . A generic point on the ring has cylindrical coordinates ( f ( z ) , φ, z ), so itsdistance to the s -axis is f ( z ) sin ( φ ) + z . The rotational kinetic energy ofa point of unit mass at the surface, which rotates around the s -axis with9igure 2: Rotating dumbbell showing a circular region of surface points at z .rotational speed ω is: U Rot = − ω ( f ( z ) sin ( φ ) + z ) . (16) Corollary 3
The total energy U acting on a particle of unit mass at thesurface of the rotating body is the sum of the expressions in equations (14) and (16) : U = − πGρ (cid:90) z − z dz (cid:48) f ( z (cid:48) ) Y ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) − ω ( f ( z ) sin ( φ ) + z ) . (17)The negative sign in the last two terms of (17) accounts for the sign of therepulsive centrifugal force being opposite to that of the attractive gravitationalforce.Equation (17) can be used in two ways.First, if the explicit shape of a body is known, then the gravitationalacceleration g = −∇ U at the surface of the body can be computed.Second, one may ask, for a given family of shapes defined by a function f ( z )depending on parameters, to determine the parameters that (approximately)give the equilibrium shape. This can be re-phrased as a minimization problem,on determining the optimal parameters for the function describing the shape,10or which the total energy function at the surface has the lowest variability –expressed as normalized standard deviation (see Section 4).Thus equation (17) can be seen as a main result of this paper.In principle, for a given ω , functional variations δU should provide thefunction f ( z ) that give the equilibrium shape. However, in practice, that is avery challenging program, owing to the complicated form of the kernel of Y .Realistically one should explore the minimization problem in the parameterspace of a well suited family of functions.A problem that immediately becomes apparent from equation (17) is that,at the surface of the object, U G depends only on z , while U Rot depends, inaddition, on φ . Thus, as one walks along the ring of figure 2, U G remainsconstant while U Rot contributes with an additional sin φ dependence. It isclear that the total energy on a unit mass cannot be constant on the ring.However, the practical problem of a real celestial body must be interpretedin the context of rotating not with respect to a fixed axes, but secularly withrespect to all axes perpendicular to the z -axis. Under these conditions, andowing to the sin φ factor, outstretched shapes will develop perpendicularlyto z . But these shapes will subsequently develop in other directions, as theaxis of rotation itself wobbles. Hence, one should find among celestial bodies,those that after long times compared with the rotational period 2 π/ω , havecross sections averaged in φ . For this reason, we eliminate the dependenceon φ by replacing U Rot by its average with respect to φ . Thus, instead of thetotal energy (17) we consider the effective total energy U Eff = 12 π (cid:90) π dφ U ( z, φ ) , (18)which takes the following form U Eff = − πGρ (cid:90) z − z dz (cid:48) f ( z (cid:48) ) Y ( f ( z (cid:48) ) , f ( z ) , | z − z (cid:48) | ) − f ( z ) ω − z ω . (19)In the next section we give an example of a parametrized family ofdumbbell shapes, and investigate numerically the minimization problem offinding parameters for which the effective total energy function (19) at thesurface has the lowest normalized standard deviation.11 MINIMIZATION PROBLEM FOR APARAMETRIC FAMILY OFDUMBBELL SHAPES
In this section we consider the family of curves f ( z ) = γ (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) − (cid:18) zz (cid:19) (cid:33) (cid:32) β − β (cid:18) zz (cid:19) (cid:33) , (20)for − z ≤ z ≤ z . The functions in this family are symmetric relative to the s -axis, so the solid of revolution generated by these functions have a mirrorsymmetry with respect to the plane through the origin orthogonal to the z -axis. Different values of the parameters β and γ give different shapes. Theparameter γ gives the value f (0) of the radius of the section at the origin,and β controls the convexity. We note that some values of β the resultingbody is dumbell shaped while for some others it is not. The parameter z can be used as the unit of length and, without loss of generality, it can be setto unity. Examples of graphs of f ( z ) obtained for some choices of parameters( β, γ ) are depicted in Figure 3a and Figure 3b. (a) The function f ( z ) at z equals 10, γ equals 0 . β equals 0 .
9. (b) The function f ( z ) at z and γ bothequal 1 (i.e. dimensionless) and β equals0 . Figure 3: Examples of shapes given by the function f ( z ), which is defined asin equation (20). 12eplacing f ( z ) from equation (20) into equation (19) and computing theintegral numerically, we obtain numerical values of the function U Eff ( z, β, γ ).We then search, among all pairs ( β, γ ), the ones which produce, for a given ω , an effective total energy U Eff with the least normalized standard deviationin z . The variant of the normalized standard deviation that we use is givenby the quantity σ/ | µ | , that is, the standard deviation divided by the mean.This quantity measures the extent of variability in relation to the mean. Itis known as the coefficient of variation, and it has been used extensivelyin the optimization literature (Abdi, 2010). It is a dimensionless quantitythat is very practical in comparing the variability of different data sets. Oneparticularity is that it emphasizes deviations for smaller means.In our case, we compute the coefficient of variation σ/ | µ | as a measureof the variability of the effective total energy U Eff computed for all values of z , for different choices of parameters ( β, γ ) which control the shape of thebody. The optimization problem is to find those parameters for which thecoefficient of variation attains the lowest values.Practically, for each value of the rotational speed ω in a grid, we computethe effective total energy for each coordinate z , for fixed ( β, γ ). The goal isto find dumbbell shapes that yield a nearly constant effective total energy atthe surface, in practice one with a relatively small coefficient of variation of U Eff over the surface of the body. For each value of ω – which we increaseat each step by an increment of δω = 0 .
1, we record the local minimumvalues of σ/ | µ | and the parameters γ and β for which the local minimum isattained. The obtained results suggest different interesting dumbbell shapes,as in Figures 4, 5, 6. 13 =0.1 ω =0.3 σ/|μ|=0.0748821 σ/|μ|=0.0861589γ = 0.095 γ = 0.650β = 0.97 β = 0.40σ/|μ|=0.0657014 σ/|μ|=0.0614073γ = 0.6 γ = 0.75β = 0.4 β = 0.2σ/|μ|=0.0568046 σ/|μ|=0.0498435γ = 0.65 γ = 0.800β = 0.2 β = 0.050σ/|μ|=0.03282420.7850.050 ω =0.2 ω =0.4 σ/|μ|=0.0675384 σ/|μ|=0.0980369γ = 0.65 γ = 0.700β = 0.4 β = 0.400σ/|μ|=0.0570862 σ/|μ|=0.0828665γ = 0.750 γ = 0.750β = 0.200 β = 0.2σ/|μ|=0.0402647 σ/|μ|=0.0683985γ = 0.785 γ = 0.800β = 0.05 β = 0.125 σ/|μ|≈0.03-0.07 σ/|μ|≈0.06-0.09σ/|μ|≈0.05-0.07 σ/|μ|≈0.07-0.10 Figure 4: Approximate equilibrium shapes for ω = 0 . ω = 0 . ω = 0 . ω = 0 .
4. 14 =0.5 ω=0.7 σ/|μ|=0.152336 σ/|μ|=0.242303γ = 0.650 γ = 0.700β = 0.600 β = 0.650σ/|μ|=0.11306 σ/|μ|=0.179832γ = 0.700 γ = 0.800β = 0.400 β = 0.400σ/|μ|=0.0945129 σ/|μ|=0.150703γ = 0.800 γ = 0.850β = 0.200 β = 0.200 ω=0.6 ω=0.8 σ/|μ|=0.178958 σ/|μ|=0.288899 γ = 0.700 γ = 0.750β = 0.550 β = 0.600σ/|μ|=0.154153 σ/|μ|=0.248736 γ = 0.750 γ = 0.800β = 0.450 β = 0.475σ/|μ|=0.108252 σ/|μ|=0.174104 γ = 0.850 γ = 0.900β = 0.200 β = 0.200σ/|μ|≈0.09-0.15 σ/|μ|≈0.15-0.24σ/|μ|≈0.11-0.18 σ/|μ|≈0.18-0.29
Figure 5: Approximate equilibrium shapes for ω = 0 . ω = 0 . ω = 0 . ω = 0 .
8. 15 =0.9 ω =1 σ/|μ|=0.400951 σ/|μ|=0.428434γ = 0.750 γ = 0.850β = 0.700 β = 0.550σ/|μ|=0.348109 σ/|μ|=0.359681γ = 0.800 γ = 0.900β = 0.575 β = 0.400σ/|μ|=0.247292 σ/|μ|=0.294837γ = 0.900 γ = 0.950β = 0.200 β = 0.200 σ/|μ|≈0.25-0.40 σ/|μ|≈0.29-0.43 Figure 6: Approximate equilibrium shapes for ω = 0 . ω = 1 . ω , there may be several different choicesof parameters γ and β – hence different shapes – for which σ/ | µ | attains alocal minimum. Since at a practical level we do not look for exact solutionsof the optimization problem, but for approximate ones for which σ/ | µ | is‘relatively small’, we do not impose a precise threshold on what ‘relativelysmall’ means. However, we speculate that for a given ω , dumbbell shapeswith relatively large values of σ/ | µ | are less likely to occur in real life thanthose with relatively small values of σ/ | µ | . We also notice that some of theshapes that we obtain appear to be similar to the observed shapes of someasteroids and comets, such as 624 Hektor, 103P/Hartley, and 8P/Tuttle. First, we have derived a formula in terms of elliptic integrals for the gravita-tional potential at any point on the surface of an axisymmetric body, as wellas at any point outside the body. Second, we have derived a formula for thetotal energy of an unit mass particle on the surface of an axisymmetric body16hat rotates around an axis perpendicular to the symmetry axis. Third, wehave formulated an optimization problem of finding approximate equilibriumshapes, based on the principle of minimizing the coefficient of variation ofthe effective total energy at the surface. As an application, we have consid-ered a two-parameter family of dumbbells, and computed numerically theirapproximate equilibrium solutions, i.e., the choices of parameters, dependingon the rotational speed, for which the effective total energy at the surface isapproximately constant.We note that there also exist exact equilibrium solutions of dumbbellshapes (Eriguchi et al., 1982). Such dumbbell shapes are not given by closedform equations. In contrast, we provide a family of dumbbell shapes thatare given by simple, explicit formulas, and depend only on two parameters.However these only correspond to approximate energy level sets. Our familyof dumbbell shapes could be potentially utilized to find first approximationsfor irregularly shaped asteroids and comets. Our approach can be extendedto other families of shapes (depending on more parameters), as well as toshapes that are not generated as solids of revolution.
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