Sample Completeness and Detection Thresholds

What we know today about extrasolar planets is the result of the selection criteria adopted for the stellar samples targeted, which are driven by the techniques adopted in the search, and their limiting capabilities.

2.2.1.1 Doppler Spectroscopy In the three panels of Figure 2.1 I summarize some of the general features of the planet host stars. If we do not include the handful of transiting planets discovered by transit photometry (Udalski et al. 2002a,b,2003; Alonso et al. 2004b) and confirmed by high-resolution spectroscopic measurements (Torres et al. 2004a;Bouchy et al. 2004;Moutou et al. 2004;Pont et al. 2004;Sozzetti et al. 2004b;Konacki et al. 2003,2004, 2005), belonging to very different stellar samples, the typical planet-harboring star has an average mass ¯M? = 1.1 M¯, is located at an average distance ¯D = 35.6 pc, and has an

average visual magnitude ¯mv = 6.9. This reflects the fact that the stellar reservoir utilized

by Doppler surveys is constituted by bright, very nearby, solar-type stars.

The groups leading the effort in radial-velocity planet searches (Korzennik et al. 1998; Queloz et al. 2000a;Butler et al. 2001;Endl et al. 2002;Tinney et al. 2002;Perrier et al. 2003; Marcy et al. 2004a;Cochran et al. 2004) are in fact targeting from both hemispheres a global sample of ∼ 2000 main-sequence and sub-giant (spectral subtype IV-V) late F, G, and early K stars, with typical selection criteria based on brightness (mv < 7.5 − 8), multiplicity (no

close binaries within a few arcsec), surface activity (adopting as a criterion the fractional Ca II H and K flux corrected for the photospheric flux (e.g., Noyes et al. 1984, Saar et al. 1998), typically log R0

HK . −4.5), metallicity ([Fe/H]≥ −0.5), rotation (Vrotsin i < 5 − 10

km/s), and the availability of Hipparcos distances (D < 100 pc) 2_{. Dedicated surveys of}

Figure 2.1 Left: mass histogram of the planet host stellar sample. Center: the histogram of apparent visual magnitudes. Right: the histogram of nominal distance estimates. Transiting planets lying at D > 100 pc are not included

stars in open clusters (Cochran et al. 2002), components of wide binaries (Gratton et al. 2004; Udry et al. 2004; Konacki 2005), metal-poor stars (Sozzetti et al. 2005), M dwarfs (Delfosse et al. 1998; Endl et al. 2003; K¨urster et al. 2003; K¨urster & Endl 2004; Bonfils et al. 2004), and K-G giants (Frink et al. 2002; Setiawan et al. 2003; Sato et al. 2003) bring the total to ∼ 3000.

For Doppler spectroscopy, the observable is the radial-velocity semi-amplitude K, which is a function of planet minimum mass Mpsin i, orbital period P , eccentricity e, and stellar

mass M? (expressed in units of solar masses M¯) through the following expression:

K = √28.4 1 − e2 µ Mpsin i MJ ¶ µ P 1yr ¶_−1/3µ M? M¯ ¶_−2/3 m/s (2.1)

The time coverage and achieved single-measurement precision σRV are highly variable,

ranging from a few up to 17 years and from σRV ' 10 − 15 m/s down to σRV ' 1 − 3 m/s,

respectively. In Figure 2.2 I show the histogram of the radial-velocity semi-amplitudes K reported for the extrasolar planet sample. As a result of variable measurement precision, time baseline, and detectability thresholds adopted by the different groups, the typical value of K is of order of at least a few tens of m/s, reflecting the fact that radial-velocity datasets still contain a significant fraction of lower precision measurements. In fact, for example, assuming σRV = 3 m/s, only ∼ 7% of the detected planets induces radial-velocity variations

on its parent star such that K/σRV . 4 − 5, and with one exception, they are all close-in

Figure 2.2 Log distribution of the radial-velocity semi-amplitudes for the sample of known extra- solar planets.

planets, with orbital periods not greater than several tens of days.

Statistical studies of the sensitivity of different radial-velocity surveys to planetary com- panions have been performed in the past by several authors (Walker et al. 1995;Cochran & Hatzes 1996; Nelson & Angel 1998; Cumming et al. 1999, 2004; Eisner & Kulkarni 2001a; Endl et al. 2002; Naef et al. 2004; Cumming 2004; Narayan et al. 2004). Such studies have utilized both numerical as well as analytical approaches to describe detection probabilities as a function of measurement errors, number and time sampling of the observations, and planet characteristics (orbital parameters, masses).

In most cases, detectability thresholds (using K/σRV as a proxy) were defined on the basis

of χ2 _{and Lomb-Scargle (Lomb 1976; Scargle 1982) periodogram tests. The relative roles of}

short and long orbital periods (compared to the time-span T of the observing campaign), high and low eccentricity, sparse and dense, even and uneven sampling have been quantified. As a rule of thumb, assuming circular orbits, and σRV = 3 m/s, the present, robust (5σ

criterion) lower limit to the detectable minimum planet mass by radial-velocity surveys of solar-like stars (0.8 ≤ M? ≤ 1.2M¯) would be of order of 0.1 MJ, 0.5 MJ, and 1.5 MJ at

0.05 AU, 1 AU, and 5 AU, respectively. Below these limits in mass, radial-velocity surveys are largely incomplete.

However, when the time-span of observations exceeds a few years, the contribution from lower-precision measurements becomes important, and direct extrapolation of the mass de- tection threshold to 4-5 AU is likely to be incorrect. Furthermore, even at very short periods the situation is already changing, as new instrumentation begins to attain higher-level per- formances, toward the 1 m/s precision.

The improved sensitivity has recently opened the door for the detection of planets close to the terrestrial planet mass regime, as confirmed by the recent discoveries of Neptune-class objects on few-day orbits (Butler et al. 2004; McArthur et al. 2004; Santos et al. 2004b). The almost simultaneous announcements have sparked a heated debate on who announced the first one first (Irion 2004).

2.2.1.2 Transit Photometry Except for HD 209458b (Charbonneau et al. 2000;Henry et al. 2000), all transiting planets have estimated distances exceeding 100 pc. This reflects the two main choices made by ground-based photometric transit surveys regarding the targeted stellar samples. About two dozen groups worldwide are now targeting hundreds of thousands of stars, trying to detect transiting Jupiter-sized planets in very close-in orbits (for a review see Horne (2004) and Charbonneau (2003), and references therein). While wide-angle transit surveys target large samples of moderately bright stars (9 ≤ mv ≤ 13), deep galactic-plane

surveys and surveys in stellar clusters focus instead on faint objects (14 ≤ mv ≤ 19.5). As a

consequence, the former primarily investigate a sphere of a few hundred pc centered on the Sun, while the typical target of the latter lies at D > 1 kpc.

I recall that the main observable in this case is the transit depth ∆F , where F is defined as the total observed stellar flux:

∆F = Fno transit− Ftransit Fno transit = µ Rp R? ¶₂ (2.2)

Assuming the stellar radius R? = 1R¯, and the planet radius Rp = 1RJ, then ∆F ' 0.01.

The requisite photometric precision (a few times better than 1%) and phase coverage for large stellar samples are attained by most of the surveys, so completeness in the limiting

detectable mass (about 1 MJ) is reached in most cases, in the typical period range 1-4 days.

Completeness boundaries in the stellar samples targeted are however harder to assess, and useful constraints on the occurrence rate of hot Jupiters as a function of the environment and properties of the central star will be in turn difficult to establish. In fact, transit surveys are essentially magnitude-limited, thus the selection function of spectral types, ages, and metallicities is unknown (except for stars in clusters).

As recently pointed out (Moutou et al. 2004; Gaudi et al. 2005), the apparent inconsis- tency between the presence of a 50% of transiting objects with periods of order of 1.5 days in the OGLE surveys (Udalski et al. 2002a,b,2003) and the fact that among the ∼ 20 planets with P ≤ 5 days detected by Doppler surveys the shortest period is rather of order of 3 days can be reconciled by observing that, once a variety of selection biases and small-number statistics are taken into account, only one in a few thousands stars is likely to harbor a transiting close-in planet on a 1.5 day period orbit. These objects are at least a few times less common than hot Jupiters with 3 . P . 10 days, thus very close-in planets (P < 3 days) are not out of reach of future radial velocity surveys, such as that described by Fischer et al. (2005).

Furthermore, the lack of OGLE transiting planets with 3 . P . 10 days, as opposed to the ∼ 50% of such objects being present in the sample with P < 10 days detected by Doppler surveys, can also be explained in terms of selection biases such as uneven sampling and finite duration of the photometric transit campaigns (Charbonneau 2003;Gaudi et al. 2005), resulting in a strong inverse dependence of the probability of detection of transiting planets on P which significantly limits the sensitivity of transit surveys for P & 3.5−4 days. Indeed, such limitations constitute the primary reason for which the number of transiting planets is not significantly larger than what is observed, as naive considerations based on the large stellar samples, achieved sensitivity, and frequency of close-in planets, would have suggested (Horne 2004).

Finally, additional complications arise due to the fact that transit searches are prone to a very high rate of false alarms. As extensively discussed by e.g, Latham (2004), Brown (2003) and Charbonneau et al. (2004), a variety of astrophysical false positives (small stars eclipsing large stars, grazing eclipsing binaries, and most importantly faint eclipsing binaries

blended with a physically associated or unassociated brighter star) can outnumber true planetary transits by well over an order of magnitude. To this end, very careful analysis of the light curves and of spectral lines profiles of the targets must be conducted (Torres et al. 2004b, 2005; Mandushev et al. 2005), in order to ascertain or rule out the existence of a blend.