1. Anthony R. Dobrovolskis, SETI Institute SPLIT SPHERES AS MODELS OF ASTEROIDS, COMETS, AND MOONS Objective & Approach The ”split sphere” model is used here to obtain simple criteria for the stability of asteroids and comets, modeled as uniform spheres, with respect to fracturing and failure. A split sphere is defined as a rotating, self-gravitating body with spherical symmetry, but with a single plane of weakness cutting the body into two rigid spherical caps; see the central sketch in FIGURE 1. Compare this with the image of asteroid (2867) Steins shown in FIGURE 2. The split sphere model is characterized by eight parameters, as described in TABLE 1 and as labeled in the center of FIGURE 1. The corners of FIGURE 1 sketch the four possible failure modes of a split sphere. The fault plane is ruptured by tension whenever the tensile stress T = (2R − h)h[ρω 2 sin2 β − 4πGρ 2 /3]/4 across it exceeds its tensile strength T ∗. Note that this tension is greatest when h = R and β = 90 ◦ ; that is, for faults through the sphere’s center and perpendicular to its equator. The left-hand panel of FIGURE 3 contours the corresponding tension T, divided by R 2, for spheres with a range of densities and rotation rates like those measured for asteroids. Likewise, the fault plane is ruptured by shear whenever the effective shear stress S = (2R−h)hρω 2 sinβ|cosβ|/4 +μ s T across it exceeds its shear strength S *. Like the tension, the effective shear stress is greatest when h = R, corresponding to a fault plane through the center of the sphere. However, S is not greatest for β = 90 ◦, but for β * =45 ◦ +θ/2 (or its supplement 180 ◦ − β * ), where θ = Arctan μ s is the ”friction angle”. FIGURE 4 shows the relations among μ s, θ, and β ∗ above. The right-hand panel of FIGURE 3 contours the effective shear S (also divided by R 2 ) over a fault plane through the center of the sphere, with a relatively low friction angle of θ = 15 ◦ and corresponding tilt of β * = 47 ◦.5. Conclusions Comparison of the left- and right-hand panels of Figure 4 reveals that the greatest shear S is comparable to the greatest tension T. But for most brittle materials like rock or ice, the tensile strength T ∗ is much less than the shear strength S ∗. Thus most split spheres are more likely to fracture by tension than by shear. Once the fault ruptures, the sphere fails by one of the four modes sketched in Figure 1. Because there is no net torque about the axis of symmetry of the spherical cap, Twisting cannot occur in this model. Separation occurs if and only if the tension T across the fault is positive. If T is negative or zero, Sliding occurs if and only if the effective shear S is positive. Whether Tipping occurs, either alone or together with Sliding, depends on the parameters of the problem in a complicated manner, not yet fully elucidated. This is a subject for further research. Also for future work, with the addition of tidal forces, split spheres can model failure of comets and asteroids on close approach to a planet, and of synchronously spinning satellites such as Phobos as well. FIGURE 5 sketches other possible generalizations of the simple split-sphere model: to certain kinds of inhomogeneities, such as concentric layers of different densities; to split ellipsoids; and to duplexes of two spheres joined together (Dobrovolskis and Korycansky, 2013). The duplex is a useful model for many bilobate comets (such as 67P/C-G) and asteroids (such as 216 Kleopatra), which are shaped like ”peanuts”, ”snowmen”, or ”dumbbells”. Reference A. R. Dobrovolskis and D. G. Korycansky 2013. The quadrupole model for rigid- body gravity simulations. Icarus 225, 623–635. Figure 1. Cross-section of a split sphere (along its spin axis and perpendicular to its fault plane), and its four possible failure modes. Figure 2. Diamond-shaped asteroid (2867) Steins (mean diameter 5.3 km) as imaged by the Rosetta spacecraft. The prominent chain of craters at the right is thought to be the surface expression of an internal fracture. Table 1. Parameters describing the split-sphere model. Figure 3: Relations among friction angle θ (dashed curve), static friction coefficient μ s = tan θ, and tilt angle β * = θ/2 + 45 ◦ (solid curve) for the fault plane with greatest shear. Note that θ is the same as the angle of repose for cohesionless granular materials, such as dry sand. Figure 4: Contours of the greatest tension and greatest shear stress in a split sphere, normalized by the square of its radius, as functions of the sphere’s density and rotation rate. The shear is plotted for a low friction angle of θ = 15 ◦ ; the corresponding plot for a high friction angle of θ = 45 ◦ is very similar to the plot of the greatest tension. Figure 5. Possible generalizations of the split-sphere model. Upper left: Concentrically layered internal structure. Upper right: Split ellipsoid. Bottom: ”Duplex” consisting of two overlapping spheres. Not shown: Tidal forces.