Self-consistent-field method for triaxial differentiated bodies in hydrostatic equilibrium

Abstract

Recent observations and models of Haumea and Quaoar suggest that both bodies are triaxial, but their shapes are inconsistent with Jacobi ellipsoids. To determine whether these objects can be at hydrostatic equilibrium, we propose a new numerical code, BALEINES, to study the hydrostatic shape of triaxial differentiated bodies. The fluid mass is assumed to be made of several homogeneous layers, which allowed us to rewrite the gravitational potential as a sum of proper surface integrals. In contrast to the classical self-consistent field method, we did not solve for the mass density, but for the shape of the boundary of all layers, meaning that only one point per layer is needed in the radial direction. The solution is still searched for iteratively. The code was benchmarked against analytical and numerical solutions. As a quick application, we studied the position of the axisymmetric-triaxial bifurcation point of two-layer systems. We show that the deviation from the Meyer bifurcation point in the single-layer case is below $10~\%$ in realistic cases. Based on this result, we conclude that the shape of Quaoar, as obtained in a recent work using a thermophysical model of the surface, is not compatible with a hydrostatic figure of equilibrium.

Figures (8)

• Figure 1: Possible axis ratio couples $(b/a,c/a)$ for ellipsoidal figures of equilibrium, with the Jacobi sequence $(a>b>c)$ as a black line and the Maclaurin sequence $(a=b>c)$ as a gray line. Haumea and Quaoar are placed in the diagram for illustration. The data and error bars for Haumea come from the observations of ortiz17, and the data for Quaoar come from the thermophysical model of kiss2024.
• Figure 2: Kernel function given by Eq. \ref{['eq:kernel']} for an ellipsoid with $a=1$, $b=0.9$, $c=0.6$ at $(\theta,\varphi)\approx(0.632,0.132)$, corresponding to a node of the Chebyshev grid, and for $r=1.001\, s(\theta,\varphi)$. The kernel, $\kappa_{\ell}$, is plotted as a function of $\varphi'$ for $\theta'=\theta$ and zoomed-in on the lobe. The Chebyshev grid has $N+1=17$ nodes. The dashed line corresponds to the Chebyshev approximation of the kernel, and the dashed-and-dotted line shows the kernel of the concentric and coincident sphere, corresponding to $\kappa_{\ell,0}$ in Eq. \ref{['eq:splitting']}.
• Figure 3: Comparison of numerical solutions from BALEINES ($N=16$) with analytical solutions from the Maclaurin-Jacobi sequence and numerical solutions reported by hachisu86III and descamps15.
• Figure 4: Evolution of the convergence indicator, $\epsilon$, and equatorial axis ratio, $b/a$, during the cycle for an initial ellipsoid with $c/a=0.40$, $b/a=1.00$, and $N=16$.
• Figure 5: Equilibrium figures of the free surface and the core-mantle boundary obtained with BALEINES for the configurations reported in Tables \ref{['tab:kyushu1']} (top) and \ref{['tab:kyushu2']} (bottom) in the $(x{\rm O}z)$ (left) and $(y{\rm O}z)$ (right) planes. The equivalent ellipsoids are also shown as dashed lines.