A Pilot-Wave Gravity and the Titius-Bode Law J.R. Croca 1,2 , P. Castro 2 , M. Gatta 2,3 , A. Cardoso 2 and R. Moreira 2 1 University of Lisbon, Faculty of Sciences, Department of Physics 2 Center of Philosophy of Sciences of the University of Lisbon 3 CINAV and Escola Naval (Portuguese Naval Academy) Email: jncroca@fc.ul.pt; jpcastro@fc.ul.pt; mariogatta@gmail.com; amb.cardoso@gmail.com; ranmoreira@gmail.com Abstract. Since its initial proposal in 1766, Titius-Bode empirical law has remained a puzzling source of discomfort as it predicts the average distances from the planets to the Sun for no apparent reason. Using a framework analogous to de Broglie’s pilot wave theory and the self-organizing Principle of Eurhythmy, we claim that several main physical quantities describing the Solar System are quantified. Hence the Titius-Bode Law is a direct manifestation of gravitational pilot-waves at work in the Solar System. Keywords: Titius-Bode Law, Gravity, Pilot wave theory, planet-star interactions. 1 Titius-Bode Law History and Formulation The occurrence of simple numerical relations between observable quantities in natural phenomena has been a constant source of interest and even fascination, as has happened with the Pythagoreans. In some occasions such relations go beyond the purely mystical and lead to important scientific insights. Among many possible examples, one may recall de Broglie’s fundamental intuition that the occurrence of integer numbers in Bohr’s model, for the hydrogen atom, should have something to do with wave phenomena, for which that occurrence was well established and easily understood. This was undoubtedly one of the major impulses leading to the birth of present-day quantum mechanics. Less successful has been the hypothesis put forward in 1766 by Johann Daniel Titius von Wittenberg for the Solar System in his German translation of the French book “Contemplation de la nature” by Charles Bonnet, and first published in 1764, in Amsterdam. In a paragraph added by himself, Titius proposes that the distances N r of the planets to the Sun are described approximately by a simple algebraic relation that reads, 4 3 2 N N r = + × (1) taking the integer N to be –∞ for Mercury, 0 for Venus, 1 for the Earth and so on, and where at this scale the size of Earth’s orbit is 10. In 1772, the renowned German astronomer Johann Elert Bode paid further attention to this curious relation by including a footnote in his second edition of “Anleitung zur Kentniss gestirnten des Himmels”, praising the same sequence of planetary distances he had read in Titius book, although not crediting the original author for the remark. History nevertheless was generous enough to call this empirical relation the “Titius-Bode Law”. At the time of the first publication of Titius-Bode Law only six planets, up to Saturn, were known. The resulting distances would fit quite well, provided one skipped the case N = 3 where the dwarf planet Ceres was eventually discovered in 1801. Uranus also fit well the expected distance, but there were serious disagreements for Neptune and Pluto. Various attempts to improve the applicability of the Titius-Bode Law were made over the years. The most important of these are due to the astronomers Mary Adela Blagg in 1913 [1] and E. Richardson, in 1945 [2]. A salient fact of Blagg’s formula is that the base is no longer the number 2 but 1.7275: ( ) α β = + + ⎡ ⎤ ⎣ ⎦ (1.7275) N N r A B f N (2) New Horizons in Mathematical Physics, Vol. 1, No. 3, December 2017 https://dx.doi.org/10.22606/nhmp.2017.13001 75 Copyright © 2017 Isaac Scientific Publishing NHMP