Loi de titius-bode et formalisme ondulatoire |
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| Authors: | R Louise |
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| Institution: | 1. Observatoire de Marseille, Marseille, France
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| Abstract: | Several authors (Basano and Hughes, 1979; ter Haar and Cameron, 1963, Dermott, 1968; Prentice, 1976) give the revised Titius-Bode law in the form $$r_n = r_o C^n ,$$ wherer n stands for the distance of thenth planet from the Sun;r o andC are constant. They pointed out, in addition, that regular satellites systems around major planets obey also that law. It is now generally thought that the Kant-laplace primeval nebula accounts for the origin and evolution of the solar system (Reeves, 1976). Furthermore, it is shown (Prentice, 1976) that rings, which obey the Titius-Bode law, are formed through successive contractions of the solar nebula. Among difficulties encountered by Prentice's theory, the formation of regular satellites similar to the planatery system is the most important one. Indeed, the starting point of the planetary system is a rotating flattened circular solar nebula, whereas a gaseous ring must be the starting point of satellites systems. As far as the Titius-Bode law is concerned, we have the feeling that orbits of planets around the Sun and of satellites around their primaries do not depend on starting conditions. That law must be inherent to gravitation, in the same manner that electron orbits depend only on the atomic law instead of the starting conditions under which an electron is captured. If it is correct, then one may expect to formulate similarity between the T-B law and the Bohr law in the early quantum theory. Such a similarity is found (Louise, 1982) by using a postulate similar to the Bohr-Sommerfeld one — i.e., $$\int_{r_o }^{r_n } {U(r) dr = nk,}$$ whereU(r)=GM ⊙ /r is the potential created by the Sun,k is a constant, andn a positive integer. This similarity suggests the existence of an unknown were process in the solar system. The aim of the present paper is to investigate the possibility of such a process. The first approach is to study a steady wave encountered in special membrane, showing node rings similar to the Prentice's rings (1976) which obey the T-B law. In the second part, we try to apply the now classical Lindblad-Lin density wave theory of spiral galaxies to the solar nebula case. This theory was developed since 1940 (Lindblad, 1974) in order to account for the persistence of spiral structure of galaxies (Lin and Shu, 1964; Lin, 1966; Linet al., 1969; Contopoulos, 1973). Its basic assumption concerns the potential functionU expressed in the form $$U = U_0 + \tilde U,$$ whereU o stands for the background axisymmetric potential due to the disc population, and ?«U o is responsible of spiral density wave. Then, the corresponding mass-density distribution is \(\rho = \rho _o + \tilde \rho\) , with \(\tilde \rho \ll \rho _o\) . Both quantities ? and \(\tilde \rho\) must satisfy the Poisson's equation $$\nabla ^2 \tilde U + 4\pi G\tilde \rho = 0.$$ It is shown by direct observations that most spiral arms fit well with a logarithmic spiral curve (Danver, 1942; Considère, 1980; Mulliard mand Marcelin, 1981). From the physical point of view, they are represented by maxima of ? (or \(\tilde \rho\) ) which is of the form $$\tilde U = cte cos (q log_e r - m\theta ),$$ wherem is an integer (number of arms),q=cte, andr and θ are polar coordinates. The distancer is expressed in an arbitrary unit (r=d/do). In the case of an axisymmetric solar nebula (m=0), successive maxima of \(\tilde U\) are rings showing similar T-B law $$d = d_o C^n ,$$ withC=e 2 π/q constant, andn is a positive integer. It is noted, in addition, that the steady wave equation within the special membrane quoted above and the new expression of the Poisson's equation derived from (5) are quite similar and expressed in the form $$\nabla ^2 \tilde U + cte\tilde U/r^2 = 0.$$ This suggests that both spiral structure of galaxies and Prentice's rings system result from a wave process which is investigated in the last section. From Equation (2) it is possible to derive the wavelength of the assumed wave ‘χ’, by using a procedure similar to the one by L. De Broglie (1923). The velocity of the wave ‘χ’ process is discussed in two cases. Both cases lead to a similar Planck's relation (E=hv). |
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