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The Frequency of Giant Planets Around Solar-Type Stars
- The occurrence of Jovian planets and the habitability of planetary systems

Searches for extrasolar planetary companions to mature F-, G-, and K-type stars to date have yielded an occurrence of Jovian-mass companions of approximately 4% (4). Because other search techniques have either failed to definitively detect planetary-mass candidates (astrometry) or have not covered enough objects in a constrained volume of space to enable statistics to be accumulated (photometric transits, microlensing), this figure is the only statistically significant determination of planetary frequency.

There are two reasons, however, why the 4% number is probably not the actual frequency of extrasolar giant planets. First, the planetary mass determined by Doppler spectroscopy is a minimum mass, because only the velocity component of planet's orbit along the line of sight to Earth produces a Doppler shift. However, this is likely a small effect, because for a random distribution of planetary orbital inclinations to our line of site, the vast majority of detected objects should have masses within a factor of two of the Doppler spectroscopic mass (4). Second, the occurrence of planets beyond several AU around surveyed stars is effectively unknown, because the Doppler spectroscopic technique declines in sensitivity as planetary orbital semimajor axis increases. Therefore, in principle there could be a large number of giant planets around nearby stars in orbits equivalent to those of Jupiter or Saturn around our Sun still awaiting detection. Indeed, the process of giant planet formation hints at the possibility that the detected cohort of close-in giant planets may be derived from an initial population of more distant bodies. The remainder of the present section is devoted to a brief examination of this hypothesis.

It is generally agreed that the formation of giant planets occurs in disks of gas and dust spun out around newly forming stars. Disks are a product of the conservation of angular momentum during the collapse of a portion of the star-forming molecular cloud. They may range in mass during the planet-forming stage around a solar-type star from 0.001-0.3 solar masses, the lower limit driven by the mass required to form giant planets, the upper by fragmentation in more massive disks leading to multiple star formation (20, 21). Instabilities can be triggered in the gas either locally or by global processes in the disk, leading to direct collapse of the gas to form a giant planet (22). Alternatively, the collapse of the gas can be seeded by first forming a core accreted from solid materials. The core then attracts mostly gas but additional solids as well (23).

No compelling models have been offered by which giant planets form in place some 0.05 AU from their parent stars, either by nucleated accretion or by direct collapse. Various proposed mechanisms for formation of giant planets in close proximity to the parent star require a very large mass density of solids (difficult to sustain very close to a growing star) and some ad hoc assumptions regarding how a core might grow in such an environment (23). On the other hand, the extended radius of the transit planet HD209458 b requires that it be in place in its 0.05 AU orbit within tens of millions of years after formation (6). If the problems with in situ formation of close-in giant planets are physically real, then the case of HD209458 b argues for prompt inward evolution of giant planets---rapid migration---from more distant orbits where formation occurred to the orbits in which we observe them today.

There are three distinct environments within which giant planets might migrate. During formation, giant planets can interact gravitationally with and transfer angular momentum to the gaseous disk such that rapid inward evolution of their orbits takes place (24). After formation, giant planets gravitationally scatter icy and rocky debris; if this material is abundant, the angular momentum exchange produces significant inward orbital migration (25). Finally, giant planets can undergo mutual gravitational interactions, resulting in modification of orbits, ejection (26), and even merging to make larger planets (27). Migration through the gaseous disk arguably occurs before the other mechanisms because the gaseous disk is the earliest and most massive structure to form during planet growth. Giant planets can migrate rapidly enough to be consumed by the central star as the orbit reaches the point of Roche lobe overflow within a few stellar radii on timescales of 106 years or less (28, 29).

The existence of giant planets in extreme proximity to solar-type stars suggests ways of slowing or stopping the migration of some planets before they are consumed by the central star. Various mechanisms have been offered, and particularly intriguing is that the gaseous disk might be truncated late in its evolution on its inner edge by a magnetic cavity around the central star (24), which could act to slow or halt migration. Alternatively, the dissipation, or clearing out, of the disk toward the end of its lifetime might strand migrating planets at a range of semimajor axes (Fig. 1). In either case, the cohort of giant planets observed by Doppler spectroscopy must be a remnant of a larger population, some of which remain in larger undetected orbits, whereas others have been destroyed. To infer the original population of giant planets from that detected today requires an explicit model of migration and its termination.

Trilling et al. (31) constructed such a model, placing Jupiters with a range of masses at or beyond 5 AU and allowing them to migrate through disks with varying lifetimes, masses, and levels of turbulence (which affect migration times). For simplicity, they assumed no special stopping mechanism except dissipation of the disk itself. The result of their study was that, to match the observed incidence of giant planets as seen by Doppler spectroscopic studies, roughly 10% of solar-type stars must possess giant planets today, but most are in orbits with semimajor axes beyond the reach of Doppler spectroscopic studies. Further, they conclude that most giant planets formed are lost to consumption by the parent star during the gaseous disk phase, and hence giant planet formation must be a common phenomenon among solar-type stars to account for the statistical occurrence of planets. Of course, variants of the model can be envisioned that would decrease or increase the mortality of young giant planets. Adding stopping mechanisms at the inner edge of the disk, for example, would reduce the loss rate. Including migration of giant planet cores during formation (28) as well as later migration of planets in particulate disks and by multiplanet interactions would have the opposite effect. Ultimately, completing the observational census of giant planets in all possible orbits around solar-type stars would allow the problem to be inverted, placing constraints on what happens to giant planets (and by implication, terrestrial planets) during formation.

Does observational evidence exist supporting the notion that stars sometimes do consume young giant planets? Some solar-type stars show an enrichment of metals (defined in astronomical parlance simply as all elements heavier than helium) in their atmospheres. It is commonly accepted that the Sun's observed complement of metals is close to, but slightly enriched, relative to the average value for nearby G-type stars (32, 33). There is a statistically significant relationship between stars with enhanced metallicity in their observable atmospheres and the occurrence of planets around those stars. Giant planets, which during formation tend to build up high metallicity associated with their sweep up of large amounts of solid material, are a potential source of stellar enrichment.

The stellar interior consists of an inner region in which radiative transport of photons carries the energy of thermonuclear fusion outward, without turbulent mixing. Atop this zone is a region of turbulent mixing---the stellar convection zone---which ranges from 30% of the Sun's volume to 100% of the volume of the very lowest mass stars (34). Because of the steeply decreasing density of the interior as one approaches the surface, the Sun's convection zone is of order only a percent of the total mass of the star. The significance of a convection zone of restricted extent is that limited mixing of material occurs through and below the base of the zone. Therefore, material of different composition---e.g., higher metallically---injected into the zone will tend to remain there and have an effect on the observed composition well out of proportion to its contribution in terms of the total mass of the Sun. The same applies for other solar-type stars.

Fig. 2 shows the enrichment in metallicity in a star of one solar mass, as a function of the mass fraction of the star in the well-mixed layer. I calculate the enrichment for four cases, corresponding to the introduction into the star of 1, 5, 10, and 15 Jupiter mass planets, respectively. From mapping Jupiter's gravitational field and modeling its interior, we know that it contains perhaps 15-30 Earth masses of heavy elements (18). That is an enrichment of 5-10 relative to solar and clear evidence for the significance of core and planetesimal accretion. However, we do not know well the relative enrichment among the various different elements, except for the lightest elements carbon, nitrogen, oxygen, and the noble gases (35, 36). This is a problem for comparison with stellar metallicity in which iron is the standard---and for which we know nothing for Jupiter apart from the bulk metallicity. The major heavy element---oxygen---is enriched in Jupiter by a factor of 2-10 times solar. We take a value of five times solar for oxygen, and we use it as the proxy for metallicity in Fig. 2. The results seem consistent with the detailed numerical modeling of Sandquist et al. (37).

I have shown in Fig. 2 a range of masses for the mixing zone much larger than expected for F-, G-, and K-type stars to emphasize an important caveat. During the formation of stars, when planets are migrating, the zone of mixing may be much deeper. Indeed, I deliberately do not call it the convection zone, because several processes may lead to mixing of planetary material well below the traditionally regarded convection zone. This includes penetration by the planetary core (37), temporary increases in accretion rate onto the star forcing advective mixing, and transient changes in stellar structure during accretion. Although these are difficult to quantify in a uniquely defined historical model of a particular star-planet system, it is important to recognize that there may not be a direct one-to-one correlation between the metallicity of a star today and the number of companions it swallowed. For this reason, the very modest enrichment of heavy elements in the Sun relative to the average for stars of comparable vintage in the galactic neighborhood need not necessarily imply that the Sun consumed few or no giant planets during the formation of our own planetary system.

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