Solar System Formation & Dynamics

The accepted theory for the formation of most giant planets, both in the Solar System and in extrasolar planetary systems, is core accretion. In one sentence, the basic idea of core accretion is that a massive solid core (probably with a mass of several to several tens of Earth masses) forms in the same way as do terrestrial planets, and only subsequently accretes a massive gaseous envelope from the surrounding disk.

The key to understanding core accretion lies in the behavior of the envelope as the core mass grows. A core of several Earth masses embedded with the gas of the protoplanetary disk has strong enough gravity that it is able to retain a dense atmosphere. This atmosphere or envelope is initially in hydrostatic equilibrium, meaning that the inward force of gravity on the envelope (coming primarily from the gravity of the core) is balanced by an outward pressure gradient. Somewhat surprisingly, however, there is a maximum core mass above which it is not possible to have an envelope in hydrostatic equilibrium. We can give a hand waving justification for the existence of such a critical core mass. As the core grows more massive, the mass of the envelope itself also increases. If the envelope is lower in mass than the core, this poses no problem: the extra gas is denser and can sustain the larger pressure gradient needed to support the larger envelope. Once the mass of the envelope approaches that of the core, however, a crisis develops. Adding extra gas still increases the pressure, but it also significantly increases the total gravity of the planet and makes the envelope harder to support. Above some limit hydrostatic equilibrium cannot be maintained, and the envelope contracts rapidly allowing more gas from the disk to be captured by the planet. Schematically,

A slightly more detailed calculation shows that hydrostatic equilibrium solutions for planets composed of a core plus a surrounding envelopes fall on to tracks in a plot of total (core plus envelope) planet mass versus core mass. For typical disk parameters, the maximum mass of a core that can maintain a hydrostatic envelope might be 5-10 Earth masses (though this depends on where in the disk the planet is growing, how much dust there is in the atmosphere, and various other hard-to-quantify effects). A representative plot looks like,

Based on these ideas, the general scenario for forming giant planets has four phases,

• Core formation – the solid core is formed by accretion of smaller bodies (in traditional models these are planetesimals, though recent work has also considered the possibility that substantial mass is gained by accretion of much smaller pebbles instead). Because of the boost in surface density outside the snow line, and because the Hill radius of a planet of fixed mass is larger at larger distance from the star, it is thought to be easiest to grow a massive core at radii between a few AU out to maybe 10 AU.
• Hydrostatic phase – once the core becomes massive enough it accretes an envelope, which is initially maintained in hydrostatic balance. Energy (provided either by bombardment of the planet by planetesimals, or by slow gravitational contraction of the envelope) steadily leaks out of the envelope, which contracts allowing more gas to be accreted. If the gas disk was dissipated during this phase, the resulting planet might resemble a Solar System ice giant.
• Rapid envelope growth – at some point the critical core mass is exceeded. Hydrostatic equilibrium is lost, and a much more rapid “runaway” phase of gas accretion ensues. This phase can see several hundred Earth masses of gas being captured, so that in the end the mass of the planet is totally dominated by gas rather than by the mass of the solid core that catalyzed the whole process.
• Gas starvation – accretion ceases either when the gas disk around the star is dissipated, or when the planet creates a local gap around its location that starves it of further gas supply.

Pollack et al. (1996) presented the first detailed calculations of the formation of the gas giants via core accretion. The plot below, based closely on their work, shows how the core, envelope and total mass of the planet increase over time.

The exact time scales and prerequisites for core accretion to work are debated. Current thinking is that it is possible to form a giant planet via core accretion on scales of a few AU within 1-3 Myr, consistent with the estimated life times of protoplanetary gas disks. It becomes harder to form a massive core at large orbital radii (because gravitational encounters between solid bodies lead more often to ejection from the shallower potential well of the star rather than accretion), and hence the formation channel of massive planets beyond a few tens of AU is especially contentious. It is not thought possible to form giant planets on very small scales, and hence standard explanations for extrasolar hot Jupiters appeal to orbital migration of these planets from formation sites further out in the disk.