A contour map of the rotating-frame effective potential for a Sun-Jupiter type system…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch Perturbed orbits, precession and the three-body problem

Problem

A contour map of the rotating-frame effective potential Φeff\Phi_{\rm eff} for a Sun-Jupiter type system (μ2=9.54×10−4\mu_2=9.54\times10^{-4}) has been drawn, and the values of Φeff\Phi_{\rm eff} at the five Lagrange points, in units of (Ωa12)2(\Omega a_{12})^2, are read off the contours as Φeff(L1)=−1.5194,Φeff(L2)=−1.5187,Φeff(L3)=−1.5005,Φeff(L4)=Φeff(L5)=−1.4995.\begin{aligned}\Phi_{\rm eff}(L_1)&=-1.5194,\\\Phi_{\rm eff}(L_2)&=-1.5187,\\\Phi_{\rm eff}(L_3)&=-1.5005,\\\Phi_{\rm eff}(L_4)=\Phi_{\rm eff}(L_5)&=-1.4995.\end{aligned} A test body has Jacobi constant CJ=−1.5030 (Ωa12)2C_{\rm J}=-1.5030\,(\Omega a_{12})^2. ((a)) State, for each of L1L_1, L2L_2, L3L_3, whether the corresponding neck in the zero-velocity curve Φeff=CJ\Phi_{\rm eff}=C_{\rm J} is open or closed for this body, and whether the triangular (L4L_4, L5L_5) region is accessible. ((b)) Describe in words the shape of the resulting allowed region: which of the Sun's neighbourhood, Jupiter's neighbourhood and the region beyond Jupiter's orbit are connected to each other, whether the body can be found on the far side of the Sun, and where it can cross Jupiter's orbital radius.

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