## 4. Comparison with observational data and statistical analysisTable 1 gives the periods
The references for the observed orbital periods and for the star masses are as follows: (1) Lang (1992); (2) Mayor & Queloz (1995); (3) Butler et al. (1998); (4) Butler et al. (1997); (5) Noyes et al. (1997); (6) Fischer et al. (1998); (7) Marcy et al. (1999); (8) Cochran et al. (1997); (9) Marcy et al. (1998); (10) Butler & Marcy (1996); (11) Queloz et al. (2000); (12) Marcy & Butler (1996); (13) Mazeh et al. (1996); (14) Mayor et al. (1999); (15) Butler et al. (1999); (16) Santos et al. (2000); (17) Udry et al. (2000); (18) Mayor et al. (1998), Marcy (2000); (19) Charbonneau et al. (1999), Henry et al. (2000); (20) Vogt et al. (2000); (21) Kurster et al. (1998); (22) Marcy (2000). We define an effective "quantum number" which is computed directly from the observational data, i.e., the
values of the orbital period Therefore, our theoretical prediction can be summarized by the statement that the distribution of the values of must cluster around integer numbers. These values are given in column 5 of Table 1 and their distribution is plotted in Fig. 1.
Note that, since the main source of error is the uncertainty on the star mass (usually ), the relative uncertainty on is . Then our theoretical prediction according to which must be close to an integer becomes more difficult to be checked beyond , since the error bar becomes too large (). The values of the masses of the parent stars have been taken from the compilation of Marcy et al. (1999), and for more recently discovered planets, from Mayor et al. (1999: HD 75289), Charboneau et al. (1999: HD 209458), Vogt et al. (2000), Marcy (2000). For two of these planets, HD 177830 and HD 10697, the masses being badly determined and in contradiction with the mass expected from their spectral type, we have taken an average value (1.0 solar mass). We have plotted in Fig. 2 the histogram of the differences between and the nearest integer, for the data of Table 1. As can be checked in this figure, as well as in Table 1 and Fig. 1, we verify that the observed values of indeed cluster around integer values.
Let us make a statistical analysis of this result. We recall that
we have performed no fit of the data. Indeed, we look for clustering
around integer values of the ratio ,
where the value km/s is taken from
independent results (e.g., extragalactic data on binary galaxies) and
where A Kolmogorov-Smirnov one-sample test yields a maximum difference between the observed cumulative distribution and that of a uniform distribution. For n=42 points, this result has a probability to be obtained by chance. We can also make an independent test by separating the domain into two equal intervals respectively of high probability, [-0.25,+0.25] and low probability, [0.25,0.75]. It can be seen in Fig. 2 that among 42 points, 33 fall in the interval [-0.25,+0.25] and only 9 in [0.25,0.75]. The probability that, for 42 trials, 9 events (or less) fall in a 1/2 large interval and 33 in the complementary one is , where denotes a binomial coefficient. Therefore we can exclude at better than the level of statistical significance that such a result be obtained by chance. Finally, we have performed the same analysis by taking star masses deduced from their spectral type (from Allen 1973). One finds essentially the same statistical result. Among 42 points, 33 fall in the interval [-0.25,+0.25] and 9 in [0.25,0.75]. © European Southern Observatory (ESO) 2000 Online publication: September 5, 2000 |