## 2. Method## 2.1. Determination of the rank of a bodyWe 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 -
we first normalize the values of , by dividing by the largest -
we then set a largest possible rank *m* -
we take all slopes *d*of the lines between*0*and*1*with steps of*10*(see Fig. 1) -
for each of our *N*bodies, we choose the rank*n*(less than*m*) so that*s*is minimum -
we compute , and finally we get a value of *d*which minimizes , and this value is associated with a given configuration of the rank*n*of the bodies -
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).
Comment: It is obvious that, if we take Fig. 1 gives a graphical description of the method, applied to the Jupiter system.
## 2.2. Computing the statistical significance of the resultIt 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
Table 1 gives the statistical significance for the different
planetary systems. The significance is defined as
. We give also the correlation coefficient
of the regression line fitting the data points
as defined in step 3 of the software process, and the origin
corresponding to
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 , which is the standard deviation divided by the mean value:
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 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
© European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |