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Astron. Astrophys. 335, 281-286 (1998)

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2. Method

2.1. Determination of the rank of a body

We have built a software which automatically determines the best rank for each object of a given satellite system. The different steps of the process are the following, for a system of N bodies:

  1. we first normalize the values of [FORMULA], by dividing by the largest [FORMULA]

  2. we then set a largest possible rank m

  3. we take all slopes d of the lines [FORMULA] between 0 and 1 with steps of 10 (see Fig. 1)

  4. for each of our N bodies, we choose the rank n (less than m) so that s is minimum

  5. we compute [FORMULA], and finally we get a value of d which minimizes [FORMULA], and this value is associated with a given configuration of the rank n of the bodies

  6. we now repeat the steps 2 to 5 in order to get the best configurations for every value of the largest possible rank (in practice we ranged m from N to 3, see comment).


It is obvious that, if we take m larger and larger, it will be very easy to achieve a very good fit for our law. But, in practice, we see that generally at some value of m, the value of [FORMULA] suddenly decreases by a factor 10, and then stays stable for larger values of m. We call this phenomenon convergence, and we define M as the value of m achieving convergence.

Fig. 1 gives a graphical description of the method, applied to the Jupiter system.

[FIGURE] Fig. 1. Example of the technique to determine the rank applied to the satellite system of Jupiter, with N=5. The semi-major axis a is given in 103 km.

2.2. Computing the statistical significance of the result

It is important to know whether the results obtained by the previous method are statistically significant, or if we could fit any set of values, not involving any physics, with our law. In order to check this, we have made simulations with 10000 random systems having N bodies and for a maximum of m available positions. In Fig. 2 we show the distribution of [FORMULA] for random systems of N bodies as a function m. Analogous distributions were obtained for other values of N. For example, for Jupiter, N and M, and we see that only [FORMULA] of random systems have a better [FORMULA] that the [FORMULA] we have found for the Jupiter system.

[FIGURE] Fig. 2. Distribution of [FORMULA] giving percentage of random systems, for the number of bodies N, and the maximum rank m ranging from 5 to 9.

Table 1 gives the statistical significance for the different planetary systems. The significance is defined as [FORMULA]. We give also the correlation coefficient [FORMU__mark__LA] of the regression line fitting the data points as defined in step 3 of the software process, and the origin corresponding to x and y. The origin is given by equation 1for n.


Table 1. Statistical significance and correlation coefficient for the ranks attributed to the elements of the solar and the planetary systems

In the following pictures, the regression line has been calculated with the data represented by circles. The data represented by other symbols are added to the picture just for information, but are not taken into account for the computation.

Finally, we compute also a quantity called normalized standard deviation [FORMULA], which is the standard deviation divided by the mean value:


Ov is the observed value of the semi-major axis, Tv is the corresponding value given by the theory, and n is the number of values.

As a comparison, we have also considered the Titius-Bode law applied to the solar system (see for example (Nieto, 1972)):


It gives the distance r of a planet versus the rank n. The law does not fit well the observed data for Neptune and Pluto, and the normalized standard deviation looks very bad.

Finally, we give the statistical significance for the system made of 3 planets found around pulsar PSR B1257+12 (see (Nottale, 1996b)). The good results we obtain with our method are probably due to the fact that the law underlying this method originates in a real theory (Scale Relativity), whereas other methods, like the law of Titius-Bode, are only empirical approaches. (see also (Neuhauser and Feitzinger, 1986) and (Dubrulle B., 1996)).

Fig. 3 shows the solar system, which is divided in inner system and outer system. The objects for the inner solar system are: Mercury, Venus, the Earth, Mars, and the main mass peaks of the asteroid belts: Hungarias, Ceres, Hygeia and Hildas. The maximum of the mass distribution of the inner system, close to the position of the earth, fits well with the rank n of the outer system. This suggests that the inner system is a sub-system of the outer one, consistent with the fragmentation process proposed in (Nottale et al., 1997). It is important to note that, on this type of diagram, the horizontal correlation does not have any significance since it results from our construction. Only the vertical discrepancies to the straight line have a significance.

[FIGURE] Fig. 3. Solar system with the main asteroid belts. M is in Earth masses, an a in km.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998