Abstract
We consider dynamics of a Sun–Jupiter–Asteroid system, and, under some simplifying assumptions, show the existence of instabilities in the motions of an asteroid. In particular, we show that an asteroid whose initial orbit is far from the orbit of Mars can be gradually perturbed into one that crosses Mars’ orbit. Properly formulated, the motion of the asteroid can be described as a Hamiltonian system with two degrees of freedom, with the dynamics restricted to a “large” open region of the phase space reduced to an exact area preserving map. Instabilities arise in regions where the map has no invariant curves. The method of MacKay and Percival is used to explicitly rule out the existence of these curves, and results of Mather abstractly guarantee the existence of diffusing orbits. We emphasize that finding such diffusing orbits numerically is quite difficult, and is outside the scope of this paper.
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Notes
To be precise, the perihelion is the point where the asteroid is at the closest point to the center of mass of the system, and the Sun is within \(\mu \) of the center of mass. However, in our Solar System, the radius of the Sun is approximately \(0.00089\) the Sun–Jupiter distance, so we allow this slight abuse in terminology for small \(\mu \).
For \(J_0\) near or less than \(1.52\) collisions with Jupiter are hard to exclude.
Notice that the angle \(\varphi \) enters into the perturbation \(\varDelta H\) (see (2)). As the Poincaré map \(\mathcal{F }_\mu \) is defined for approximately one revolution of the asteroid, then the change in the \(\varphi \) component should average out for higher order terms in the \(\mu \) expansion of \(\varDelta H\).
More precisely, it follows from the geometry of ellipses that \(e=\sqrt{1-\frac{G^2}{L^2}}\) in Delauney coordinates; a quick conversion to polar coordinates yields the formula \(e= \sqrt{1 - 2 J_0 P_{\varphi }^2 + 2 P_{\varphi }^3}\).
This construction actually localizes all Aubry–Mather sets with rotation symbol \(\omega \in \left[\frac{1}{n+1}+,\frac{1}{n}-\right]\). See Sect. 6 for definitions.
From hereon out, results may be stated using the \((\varphi ,P_\varphi )\) parameterization of annulus, as opposed to \((\varphi ,e)\). The former parameterization is easier to work with numerically; the later is better for intuition.
For \(v=(a,b)\), let \(\bar{v}=(1,\frac{b}{a})\) be a normalization of \(v\). A vector \(v_1\) lays above \(v_2\) iff \(x \ge y\), where \(\bar{v}_1=(1,x)\) and \(\bar{v}_2=(1,y)\).
For an introduction to interval arithmetic, refer to the paper by Wilczak and Zgliczynski (2007).
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Acknowledgments
The authors would like to acknowledge the guidance and direction given by Vadim Kaloshin. Without his input this paper certainly would not have been possible. The authors would also like to thank Anatoly Neishtadt for his remarks.
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Urschel, J.C., Galante, J.R. Instabilities in the Sun–Jupiter–Asteroid three body problem. Celest Mech Dyn Astr 115, 233–259 (2013). https://doi.org/10.1007/s10569-012-9461-8
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DOI: https://doi.org/10.1007/s10569-012-9461-8