Sur un formalisme explicitant la loi des distances des planetes |
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| Authors: | R Louise |
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| Institution: | 1. Observatoire de Marseille, Marseille, France
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| Abstract: | The well-known Titius-Bode law (T-B) giving distances of planets from the Sun was improved by Basano and Hughes (1979) who found: $$a_n = 0.285 \times 1.523^n ;$$ a n being the semi-major axis expressed in astronomical units, of then-th planet. The integern is equal to 1 for Mercury, 2 for Venus etc. The new law (B-H) is more natural than the (T-B) one, because the valuen=?∞ for Mercury is avoided. Furthermore, it accounts for distances of all planets, including Neptune and Pluto. It is striking to note that this law: - does not depend on physical parameters of planets (mass, density, temperature, spin, number of satellites and their nature etc.).
- shows integers suggesting an unknown, obscure wave process in the formation of the solar system.
In this paper, we try to find a formalism accounting for the B-H law. It is based on the turbulence, assumed to be responsible of accretion of matter within the primeval nebula. We consider the function $$\psi ^2 (r,t) = |u^2 (r,t) - u_0^2 |$$ , whereu 2(r, t) stands for the turbulence, i.e., the mean-square deviation velocities of particles at the pointr and the timet; andu 0 2 is the value of turbulence for which the accretion process of matter is optimum. It is obvious that Ψ2(r n,t0) = 0 forr n=0.285×1.523 n at the birth timet 0 of proto-planets. Under these conditions, it is easily found that $$\psi ^2 (r,t_0 ) = \frac{{A^2 }}{r}\sin ^2 \alpha log r - \Phi (t_0 )]$$ With α=7.47 and Φ(t 0)=217.24 in the CGS system, the above function accounts for the B-H law. Another approach of the problem is made by considering fluctuations of the potentialU(r, t) and of the density of matter ρ(r, t). For very small fluctuations, it may be written down the Poisson equation $$\Delta \tilde U(r,t_0 ) + 4\pi G\tilde \rho (r,t_0 ) = 0$$ , withU(r, t)=U 0(r)+?(r, t 0 ) and \(\tilde \rho (r,t_0 )\) . It suffices to postulate \(\tilde \rho (r,t_0 ) = k\tilde U(r,t_0 )/r^2 ](k = cte)\) for finding the solution $$\tilde U(r,t_0 ) = \frac{{cte}}{{r^{1/2} }}\cos a\log r - \zeta (t_0 )]$$ . Fora=14.94 and ζ(t 0)=434.48 in CGS system, the successive maxima of ?(r,t 0) account again for the B-H law. In the last approach we try to write Ψ(r, t) under a wave function form $$\Psi ^2 (r,t) = \frac{{A^2 }}{r}\sin ^2 \left {\omega \log \left( {\frac{r}{v} - t} \right)} \right].$$ It is emphasized that all calculations are made under mathematical considerations. |
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