With improved statistics, in recent years a number of studies has been carried out to find evidence of correlations among orbital parameters and masses, and between planet charac- teristics and stellar host properties.
Figure 2.5 shows the e − P , Mpsin i − P , and e − Mpsin i diagrams for the extrasolar
planet sample. The visual impression is that all three parameters seem to be correlated. A Spearman rank-correlation test finds that the probability of the null hypothesis (no corre- lation) in the three cases is 1.44 × 10−10, 3.62 × 10−11, and 1.37 × 10−6, respectively. Each
Figure 2.5 Left: eccentricity versus orbital period for the extrasolar planet sample. Center: the mass-period diagram. Right: eccentricity versus minimum planet mass
diagram deserves a separate discussion.
If we limit the sample to objects with P > 10 days, no e − P correlation can be found. As a matter of fact, the few-days orbits of close-in planets are all likely to have been tidally circularized (Goldreich & Soter 1966). Beyond the circularization limit, extrasolar planets can be found on very eccentric orbits, regardless of the value of P .
The strong Mpsin i − P correlation can partly be explained in terms of an observational
selection effect. For periods greater than several hundred days, corresponding to separations of ∼ 2 AU, the sensitivity of Doppler surveys is now reaching the lower mass limit of 1 MJ (we recall in units of Mpsin i), while for shorter periods the availability of consistent
radial-velocity monitoring with precisions of 3-5 m/s allows for the detection of increasingly smaller masses. It is no surprise then that planets with minimum masses of order of the mass of Saturn and below have been found only with periods no greater than several tens of days.
However, as initially pointed out by Zucker & Mazeh (2002), Udry et al. (2002) and more recently by Eggenberger et al. (2004b), the paucity of high-mass planets on short- period orbits (the easiest to detect with the Doppler method) is real, and not due to any selection effects. The lack of massive, close-in planets deserves an explanation, which I shall address in Section 2.3.
The e − Mpsin i correlation arises, instead, as a consequence of the two main observed
e − P and Mpsin i − P trends. All low-mass planets are found on very short-period orbits,
Figure 2.6 Minimum mass of single planets versus mass of the host stars
In fact, with one exception, no planet with P < 30 days shows e > 0.3. If we remove this sample from the e − Mpsin i diagram, every hint of correlation disappears, and planets can
be found on orbits with a wide range of eccentricities, regardless of their mass.
The interplay between planets and their host stars at various stages of the formation and evolution processes could translate in possible correlations between the properties of the former and some of the characteristics of the latter. For example, in Figure 2.6 I show the distribution of projected masses for single planets orbiting single stars as a function of the mass of the parent star. A rank-correlation test gives a probability of no correlation of 6%, showing evidence for a marginal positive correlation.
The hint that more massive planets seem to orbit more massive parent stars might simply be due to observational selection effects. In fact, at a given period, for increasingly larger stellar masses the same radial-velocity amplitude is induced by an increasingly more massive planet, and increasingly hotter stars are at the same time more complicated targets for Doppler surveys (less, and weaker, spectral lines, jitter, and enhanced rotation being the major causes of degradation in the radial-velocity precision). The paucity of sub-Jupiter- mass planets around stars more massive than ∼ 1.2 M¯ (late F, or earlier), which have not
Figure 2.7 Mass versus metallicity for the planet-host stars
been considered in large numbers, should thus come as no surprise.
On the other hand, the paucity of planets orbiting late-K through early-M dwarfs (≈ 0.4 − 0.8M¯), and the hint for a decrease in planetary mass as we move toward smaller
planet-host masses, might not be accounted for only in terms of observational biases. In fact, at lower parent star masses, the typically lower achievable radial-velocity precision (due to intrinsic radial-velocity jitter and faintness) is at least partly counterbalanced by the fact that, at a given period, a planet of the same mass will induce a larger velocity variation. For the discovery of the Neptune-mass planet orbiting the M2.5V star GJ 436, Butler et al. (2004) report typical values of σRV ≈ 5 m/s, which would make objects with Mpsin i > 1
MJ detectable out to a few AU.
Although the sample size of late-K and M dwarfs monitored today is an order of mag- nitude smaller than that of solar-type stars, the trend appears to begin to be statistically significant. It is thus conceivable that a theoretical explanation for the paucity and smaller size of planets orbiting moderately late-type stars should be proposed. I will address this point in the next Section.
Figure 2.8 Left: minimum planet mass against [Fe/H] of the host stars. Center: the period- metallicity diagram. Right: eccentricity versus [Fe/H]
to study the probability of planet formation as a function of M? solely on the basis of
radial-velocity data. The adopted mass (or equivalently color, or Teff) cut-offs in Doppler
surveys show up when different planet-host properties are compared. In Figure2.7I show the distribution of masses of the planet hosts against their measured metallicity, and a marginal positive correlation can be found (probability of the null hypothesis of 10%). Santos et al. (2003) argue in fact that target lists are more likely to exclude massive metal-poor stars, or low-mass metal-rich stars. This bias does not prevent anyhow to draw conclusions on the dependence of planetary frequency on [Fe/H], as discussed previously.
Several authors have searched in the past for possible correlations between stellar metal- licity and planet properties. I show in the three panels of Figure 2.8 the distributions of planet masses, orbital periods, and eccentricities against [Fe/H] of the planet hosts. Udry et al. (2002), Santos et al. (2001,2003), and Fischer et al. (2002a) searched for correlations in the Mpsin i-[Fe/H] diagram, but concluded no statistically significant trend can be found. A
rank-correlation test on the present sample gives a probability of no correlation of 56%. No apparent correlation can be found in the e-[Fe/H] diagram as well, as already pointed out by Santos et al. (2003).
The P -[Fe/H] diagram deserves instead more attention. Gonzalez (1998b) and Queloz et al. (2000a) initially argued that metal-rich stars seem to possess an excess of very short- period planets with respect to other planet hosts. In later works (Santos et al. 2001, 2003; 2003 2003) no trend was found. However, after removing some potential sources of bias,
Sozzetti (2004) has shown how this trend is still present in the data, specifically when one restricts the analysis to single planets orbiting single stars. The correlation is significant at the 2-3 σ confidence level, and the latest planet announcements do not change the result: a rank correlation test on the less-biased sample of single stars orbited by only one detected planet used by Sozzetti (2004) gave a probability of no correlation 0.02.
It must be noted, however, that some bias sources, primarily small-number statistics, cannot be ruled out with high confidence. If real, the presence of a P -[Fe/H] correlation deserves an explanation, that theoretical models of formation and migration of giant planets could provide. I defer to Section 2.3 a further analysis of this point.