4 The frequency distribution $\Psi$(M $_\mathsf{2}$)

The cumulative frequency distribution of the  $M_2 \sin i$ values smaller than 17$M_{\rm J}$, where $M_{\rm J}$ is the mass of Jupiter (=$M_\odot$/1047.35), available in the literature (as of April 4, 2001) is presented in Fig. 1. It appears to be sufficiently well sampled to attempt the inversion procedure. The corresponding frequency distributions smoothed with two different smoothing lengths, locally self-adapting around $h_{\rm opt} = 1$$M_{\rm J}$ and 2 $h_{\rm opt}$ (see Appendix) are presented as well for comparison.

Figure 1: Bottom panel: the cumulative frequency distribution of the observed $M_2 \sin i$ values for the 67 exoplanets and very low-mass brown dwarfs (around 60 stars) known as of April 4, 2001. Top panel: the corresponding frequency distribution, represented either as an histogram, or smoothed using an Epanechnikov kernel (see Appendix) with an optimal smoothing length of $h_{\rm opt} = 1$ $M_{\rm J}$(dashed curve). The frequency distribution smoothed with $2 h_{\rm opt}$ is also shown for comparison (solid curve).

The sample includes 60 main-sequence stars hosting 67 companions with $M_2 \sin i < 17$ $M_{\rm J}$. Among those, 6 stars are orbited by more than one companion, namely HD74156 and HD82943 ( $M_2 \sin i = 1.50$, 7.40 $M_{\rm J}$, and  $M_2 \sin i =0.84$, 1.57 $M_{\rm J}$, respectively; Udry & Mayor 2001), HD83433 (0.16,0.38 $M_{\rm J}$; Mayor et al. 2000), HD168443 (7.22, 16.2 $M_{\rm J}$; Udry et al. 2000b), $\upsilon$ And (0.71, 2.20 and 4.45 $M_{\rm J}$; Butler et al. 1999) and Gliese 876 (0.56, 1.88 $M_{\rm J}$; Marcy et al. 2001). The inversion process is only able to treat these systems under the hypothesis that the orbits of the different planets in a given system are not coplanar, since Eq. (4) to hold requires random orbital inclinations. The case of coplanar and non-coplanar orbits are discussed separately in the remainder of this section.

4.1 Non-coplanar orbits

Figure 2 compares the solutions $\Psi (M_2)$ obtained from the Lucy-Richardson algorithm (after 2 and 20 iterations, denoted $\Psi _2$ and $\Psi_{20}$, respectively) and from the formal solution of Abel's integral equation with smoothing lengths  $h_{\rm opt}$ and 2 $h_{\rm opt}$ on $\Phi (Y)$ (the corresponding solutions are denoted $\Psi_{h_{\rm opt}}$ and $\Psi _{2h_{\rm opt}}$). The solutions from the two methods basically agree with each other, although solutions with different degrees of smoothness may be obtained with each method. On the one hand, $\Psi_{20}$ and $\Psi_{h_{\rm opt}}$ exhibit high-frequency fluctuations that may be traced back to the statistical fluctuations in the input data. This can be seen by noting that the peaks present in $\Psi_{h_{\rm opt}}$ correspond in fact to the high-frequency fluctuations already present in $\Phi_{h_{\rm opt}}$(Fig. 1). These fluctuations should thus not be given much credit. The same explanation holds true for $\Psi_{20}$, since it was argued in Sect. 3 that the solutions $\Psi_r$resulting from a large number of iterations tend to match $\Phi$ at increasingly small scales (i.e., higher frequencies) where statistical fluctuations become dominant. On the other hand, $\Psi _2$ and $\Psi _{2h_{\rm opt}}$ are much smoother, and are probably better matches to the actual distribution. The local maximum around  $M_2 \sim 1$ $M_{\rm J}$ is very likely, however, an artifact of the strong detection bias against low-mass companions.

Figure 2: Comparison of $\Psi (M_2)$ distributions obtained from the Lucy-Richardson algorithm (after 2 and 20 iterations, represented by the solid and dashed histograms, respectively), and from the formal solution of Abel's integral equation with smoothing lengths $2 h_{\rm opt}$ and $h_{\rm opt}$ applied on $\Phi (Y)$ (represented by solid and dashed curves, respectively). The supposedly best representation of the actual $\Psi (M_2)$ distribution is represented by the thick solid line. Planetary systems have been included in the inversion process with the assumption of non-coplanarity of the planetary orbits in a given system.

The most striking feature of the $\Psi (M_2)$ distribution displayed in Fig. 2 is its decreasing character, reaching zero for the first time around M₂ = 10 $M_{\rm J}$, and in any case well before 13.6 $M_{\rm J}$. The latter value, corresponding to the minimum stellar mass for igniting deuterium, does not in any way mark the transition between giant planets and brown dwarfs, as sometimes proposed. That transition, which is thus likely to occur at smaller masses, must rely instead on the different mechanisms governing the formation of planets and brown dwarfs. Another argument favouring a giant-planet/brown-dwarf transition mass smaller than 13.6 $M_{\rm J}$is provided e.g., by the observation of free-floating (and thus most likely stellar) objects with masses probably smaller than 10 $M_{\rm J}$ in the $\sigma$ Orionis star cluster (Zapatero Osorio et al. 2000). The $\Psi (M_2)$ distribution nevertheless clearly exhibits a tail of objects clustering around  $M_2 \sim 15$ $M_{\rm J}$, due to HD114762 ( $M_2 \sin i = 11.5$ $M_{\rm J}$), HD162020 (14.3 $M_{\rm J}$), HD202206 (15.0 $M_{\rm J}$) and HD168443c (16.2 $M_{\rm J}$). It would be interesting to investigate whether these systems differ from those with smaller masses in some identifiable way (periods, eccentricities, metallicities, ...), so as to assess whether or not they form a distinct class (Udry et al., in preparation).

The jackknife method (e.g., Lupton 1993) has been used to estimate the uncertainty on the $\Psi _{2h_{\rm opt}}$ solution. In a first step, 67 input $\Phi _{2h_{\rm opt}}$ distributions are computed, corresponding to all 67 possible sets with one data point removed from the original set. Equation (8) is then applied to these 67 different input distributions. The resulting distributions are displayed in Fig. 3, which shows that the threshold observed at 10 $M_{\rm J}$ is a robust result not affected by the uncertainty on the solution.

Figure 3: Comparison of the input $\Phi _{2h_{\rm opt}}$ distribution (thick dashed line) and the 67 $\Psi _{2h_{\rm opt}}$ distributions (thin dotted lines) resulting from the application of the jackknife method (see text), which illustrates the uncertainty on the solution $\Psi _{2h_{\rm opt}}$ (thick solid line). To guide the eye, a power-law of index -1.6 has been plotted as well (dot-dashed line).

4.2 Coplanar orbits in multi-planets systems

All the results discussed so far are obtained under the assumption that orbits of planets belonging to a planetary system are not coplanar. To evaluate the impact of this hypothesis, the following procedure has been applied. In a first step, the Lucy-Richardson algorithm is applied on the data set excluding the 13 planets belonging to planetary systems. That mass distribution obtained after 2 iterations is then completed by mass estimates for the remaining 13 planets. For each of the 6 different systems, an inclination i is drawn from a $\sin i$ distribution. This is done through the expression $i = {\rm arccos} \; x$, where x is a random number with uniform deviation. The same value of i is then applied to all planets in a given system to extract M₂ from the observed  $M_2 \sin i$value. The distributions of exoplanet masses obtained with and without the hypothesis of coplanarity are compared in Fig. 4, and it is seen that planetary systems are not yet numerous enough for the coplanarity hypothesis to alter significantly the resulting $\Psi (M_2)$ distribution.

Figure 4: Evaluation of the impact of the coplanarity hypothesis on the resulting $\Psi (M_2)$ distribution. The dashed histogram corresponds to the mass distribution obtained assuming coplanar orbits in planetary systems (see text), as compared to the $\Psi _2$ (solid histogram) and $\Psi _{2h_{\rm opt}}$ solutions (see Fig. 2).

In any case, the main result of the present paper is that the statistical properties of the observed  $M_2 \sin i$ distribution coupled with the hypothesis of randomly oriented orbital planes confine the vast majority of planetary companion masses below about 10 $M_{\rm J}$. Zucker & Mazeh (2001) reach the same conclusion.

It should be remarked that the above conclusion cannot be due to detection biases, since the high-mass tail of the M₂ distribution is not affected by the difficulty of finding low-amplitude, long-period orbits.