The next attempt to model the relative distances of planets in the Solar System is known today as the Titius–Bode law. This empirical law in its original form states that the mean distance d from the Sun to each of

the six (known to Titius) planets can be approximated by the relation $$ d=0.4+0.3\times 2^i, $$ (1)where i = − ∞ , 0, 1, 2, 3, 4, 5 and d is given in astronomical units (AU). Modern observations show however that the structure of our Solar System is much more complex than what can be predicted from these simplified models. An enormous influence on the planetary system dynamical structure is exerted by an apparently small gravitational effect caused by the resonance phenomenon. The resonances can easily form due to the orbital migration and they are a central theme of this article. Resonances In most general terms, a resonance Cetuximab datasheet occurs when some

frequencies ω i of the system are commensurable with each other. This means that there is a linear relation between these frequencies of the kind: $$ \sum\limits_i k_i\omega_i=0, $$ (2)where the k i are integers, and the index i spans over a set of consecutive natural numbers. The frequencies ω i can refer to a single object. This is for instance the case of a spin-orbit coupling, where i = 1,2 and ω 1 is the rotational frequency RG7422 cell line while ω 2 is the orbital frequency. Nevertheless, they can also be related to two or more bodies as in the case of orbit-orbit interactions, where i ≥ 2 and ω i is the orbital frequency of the i-th body. There are also other more complicated relations as for example the secular resonances, which are connected with the orbital precession. Here we will concentrate on the orbit-orbit resonances, in particular, the mean-motion resonances. The name “mean motion” derives from the fact, that the frequency under consideration is the mean motion n i defined through the orbital period P i in the following way \(\omega_i= n_i =\frac2\piP_i\). Let us denote the mean motion of the inner Methocarbamol planet as n 2 and that of the outer planet by n 1. The “exact”

resonance occurs when $$ (p+q) n_1 – p n_2 \approx 0, $$ (3)where p and q are positive integers and q is the order of the resonance. Therefore, if q = 1 then the resonance under consideration is called the first order resonance, if q = 2 then it is the second order, and so on. The nominal resonance location can be found from the relation $$a_2 \over a_1 = \left(p \over p+q \right)^2/3, $$ (4)where a 1 and a 2 are the semi-major axes of the outer and inner planets, respectively. One of the most interesting examples of the commensurabilities in our Solar System is the resonance 4:2:1 between the orbital periods of the Galilean satellites of Jupiter: Io, Europa and Ganymede. Io is in the 2:1 resonance with Europa and Europa is in the 2:1 resonance with Ganymede. This commensurability is called the Laplace resonance.