# Regularities in satellite systems

#### By Jean-Bernard DELOLY

Maurice Allais’s research on this subject was presented in a work published in 2005 by the Éditions Clément Juglar under the title “De très remarquables régularités dans les distributions des planètes et des satellites des planètes” [Highly remarkable regularities in the distributions of planets and of planets satellites].

Here is a summary of the research conducted:

Since the end of the 18th century it has been noticed that the distances between the Sun and the 6 planets then known (Mercury, Venus, Earth, Mars, Jupiter, Saturn) displayed remarkable relations to one another, which led to what is known as the Titius-Bode Law.

It later came to light (L. Gaussin, 1880) that relations of the same kind are found within the satellite system of planets having a significant number of satellites (Jupiter, Saturn and Uranus).

Using updated data, Maurice Allais reprised and completed this research, taking account in particular of the density of the central planet or star, which had not previously been done. This led to a single law applicable both to the Sun and to these three planets.

(1) $\log(d/r) \sim k d_e n^\prime$

with: $n^\prime = n+n^{*}_a$

where: $r$ : radius of the central star or planet under consideration $n$= number of the satellite under consideration (counting outwards) $d$ : distance from the satellite to the central celestial body $d_e$ : density of the central celestial body $k$ : coefficient having the value 0.4 (the various magnitudes being expressed in the CGS system) $n^{*}_a$ : a positive integer attached to the central celestial body

In the end, the only adjustment parameters in this model (i.e. its only parameters having no identified physical significance) are:

• the coefficient $k$
• for each celestial body , the integer $n^{*}_a$, which is no greater than a few units (the respective values for the Sun, Jupiter, Saturn and Uranus being: 7, 1, 3, 3).

Equation (1) serves to assign to the edge of the central star or planet $(d/r = 1 \Rightarrow n^\prime = 0)$ a number $(-n^{*}_a)$.

It should also be noted that Maurice Allais noticed that $d_e \sim 0,344 \lambda$, where $\lambda$ s a low integer (its respective values for the Sun, Jupiter, Saturn and Uranus being 4, 4, 2, 3).

In the end, in the above model, the satellites of each sun or central planet may be considered to be located at the nodes of a sinusoidal function of $n^\prime$ of period $T = 2$, the edge of the sun or central planet corresponding to $n^\prime = 0$.

All of this is of course located in isotropic, Euclidian space, which is the space in which the relevant distances were estimated.

The solar system considered as a whole therefore resembles a system of stationary waves.

To date, no theoretical explanation has been found for this formulation which involves the natural logarithms of distances in the fundamentally isotropic and Euclidian space of celestial mechanics.