Quantum Signatures of Solar System Dynamics

Abstract

Let ω(i) be period of rotation of the i-th planet around the Sun (or ω j (i) be period of rotation of j-th satellite around the i-th planet). From empirical observations it is known that within margins of experimental errors i n i ω(i) = 0 (or j n j ω j (i) = 0) for some integers n i (or n j ), different for different satellite systems. These conditions, known as resonance conditions, make uses of theories such as KAM difficult to implement. The resonances in Solar System are similar to those encountered in old quantum mechanics where applications of methods of celestial mechanics to atomic and molecular physics were highly successful. With such a successes, the birth of new quantum mechanics is difficult to understand. In short, the rationale for its birth lies in simplicity with which the same type of calculations can be done using methods of quantum mechanics capable of taking care of resonances. The solution of quantization puzzle was found by Heisenberg. In this paper new uses of Heisenberg's ideas are found. When superimposed with the equivalence principle of general relativity, they lead to quantum mechanical treatment of observed resonances in the Solar System. To test correctness of theoretical predictions the number of allowed stable orbits for planets and for equatorial stable orbits of satellites of heavy planets is calculated resulting in good agreement with observational data. In addition, the paper briefly discusses quantum mechanical nature of rings of heavy planets and potential usefulness of the obtained results for cosmology. Key wordsHeisenberg honeycombs • Quantum and celestial mechanics • Group theory • Exactly solvable classical and quantum dynamical problems • Equivalence principle • Cosmological constant •(anti) de Sitter spaces

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