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Chaos and Stability in Planetary Systems by Rudolf Dvorak, F. Freistetter, Jürgen Kurths

By Rudolf Dvorak, F. Freistetter, Jürgen Kurths

This e-book is meant as an advent to the sphere of planetary platforms on the postgraduate point. It includes 4 wide lectures on Hamiltonian dynamics, celestial mechanics, the constitution of extrasolar planetary structures and the formation of planets. As such, this quantity is especially appropriate if you have to comprehend the great connections among those varied topics.

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RN , t) = 0 with i = 1, 2, . . , R and R ≤ 3N − 1. The constraints determine the direction of the constraint force C. Using this property, equation (71) can be solved. A constraint restricts the motion of a particle on a surface – but not inside the surface. Thus, the constraint force has only a component orthogonal to the surface: h (r, t) = 0 → C || ∇h (r, t) (77) where ∇ is the Nabla operator. Now we can make the following ansatz for the constraint force C: C (r, t) = α (t) · ∇h (r, t) (78) where α (t) is a unknown function.

His main contribution to the R3BP was the introduction of a synodic coordinate system, where the two massive bodies have fixed positions. 30 28 29 30 The actual value for a given date has to be computed from the respective formula given in the Nautical Almanac. Leonhard Euler (1707 – 1783) Later in this chapter we will see, that this problem can be regarded as a generalization of the MacMillan problem, where a massless body moves up and down in between two equally massive bodies. Stability and Chaos in Planetary Systems 51 Fig.

23: r2 ∆f = abπ N (156) where abπ is the area of the ellipse and N the number of sections. By introducing the orbital period U one obtains with (151) abπ 1 1 U = c∆t = c N 2 2 N and thus c= 2πab nap = nab = √ U 1 − e2 (157) (158) where 2π (159) U is the mean motion of the body which is related to the mean anomaly M (see Fig. 24) by M = nt (160) n= Equation (158) can be transformed by using (148): c = nab = n pa3 = na2 nap 1 − e2 = √ 1 − e2 (161) Stability and Chaos in Planetary Systems 41 Fig. 24.

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