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Astron. Astrophys. 361, 379-387 (2000)

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4. Comparison with observational data and statistical analysis

Table 1 gives the periods P of the newly observed exoplanets. The masses M of the parent stars are taken from the compilation of Marcy et al. (1999). Only planetary companions are considered here, to the exclusion of brown dwarfs (but see also Sect. 6), using the criterion that their mass be smaller than 13 Jupiter masses (see e.g. Schneider 2000).


Table 1. Inner Solar System planets and extra-solar planets. The table gives, for each new exoplanet, the parent star mass M (in unit of [FORMULA]) and its uncertainty, the orbital period P(in years), the [FORMULA] ratio (in AU/[FORMULA]), given by [FORMULA] from Kepler's third law, and the effective quantum number [FORMULA]. Here a is the semi-major axis and v the average velocity, in km/s (see text).

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 P and of the star mass M for each planet and its parent star. The number [FORMULA] expresses, in terms of Solar System units (AU and [FORMULA]), the value [FORMULA] km/s which characterizes galactic and extragalactic systems (Tifft 1977) and also our own inner Solar System (Nottale 1996b; Nottale et al. 1997). Indeed the average Earth velocity is 29.79 km/s.

Therefore, our theoretical prediction can be summarized by the statement that the distribution of the values of [FORMULA] must cluster around integer numbers. These values are given in column 5 of Table 1 and their distribution is plotted in Fig. 1.

[FIGURE] Fig. 1. Observed distribution of [FORMULA] where the orbital period P and the star mass M are taken in Solar System units (AU and [FORMULA]), for the recently discovered exoplanet candidates (black dots) and for the planets of our inner Solar System (white dots). The planet around HR 7875, that remains unconfirmed, and the second planet around HD 114762, the existence of which is tentatively suggested here, are plotted as grey dots. The grey zone stands for the theoretically predicted low probability of presence of planets and the white zones for high probability. The error bars are typically of the order of [FORMULA].

Note that, since the main source of error is the uncertainty on the star mass (usually [FORMULA]), the relative uncertainty on [FORMULA] is [FORMULA]. Then our theoretical prediction according to which [FORMULA] must be close to an integer becomes more difficult to be checked beyond [FORMULA], since the error bar becomes too large ([FORMULA]).

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 [FORMULA] between [FORMULA] 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 [FORMULA] indeed cluster around integer values.

[FIGURE] Fig. 2. Histogram of the values of [FORMULA] for planets in the inner solar system and exoplanets. The mean velocity is computed from the original data using Kepler third law as [FORMULA], where M is the parent star mass and P is the planet period (in solar system units). Under the standard, no struturation, hypothesis, the distribution should be uniform in the interval [0,0.5]. On the contrary, it is found that 33 objects among 42 fall in the first half interval and only 9 in the second. The probability to obtain such a result by chance (which is as getting 9 heads while tossing a coin 42 times) is [FORMULA].

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 [FORMULA], where the value [FORMULA] km/s is taken from independent results (e.g., extragalactic data on binary galaxies) and where v is calculated from the observed star mass and planet orbital period. As a consequence the zero hypothesis corresponds to a uniform distribution of [FORMULA] values.

A Kolmogorov-Smirnov one-sample test yields a maximum difference [FORMULA] between the observed cumulative distribution and that of a uniform distribution. For n=42 points, this result has a probability [FORMULA] to be obtained by chance.

We can also make an independent test by separating the [FORMULA] 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 [FORMULA], where [FORMULA] denotes a binomial coefficient. Therefore we can exclude at better than the [FORMULA] 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].

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© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000