this is a necessary but not sufficient condition; a would-be satellite approaching a planet too fast will simply pass through the Hill sphere with some deflection, rather than be captured.

If the relative velocities between neighbouring planetesimals are dispersiondominated - i.e., the main contribution to their relative velocities is their mutual perturbations, rather than the difference in Keplerian speed between neighbouring orbits - their growth rate can be estimated using a simple 'particle in a box' approach (Lissauer (1993) and references therein):

where pplsmls is the density with which planetesimals are distributed in the planetesimal disk, a is the collisional cross-section of a planetesimal, and vrel is the average relative velocity between nearby planetesimals. The collisional cross-section of a planetesimal is given by

where R is the planetesimal's physical radius, vesc = -J2GM /R is the escape velocity from its surface, and fg is the enhancement of the cross-section by gravitational focusing. With a few substitutions and simplifications, we can gain more insight from Equation (3.2). To begin with, we will assume that gravitational focusing is effective, so that fg » 1. Also, Pplsmls = pplsmls/2HplsmlS, where Splsmls is the surface density of the planetesimal disk, and Hplsmls is the disk's scale height. If the planetesimals have a characteristic random velocity v, then the vertical component is ~ v/\/3,andso H ~ (rv)/(V3vKep). Finally, vKep a r-1/2, R a M1/3 and vesc a M1/3 , and so the growth rate has the form dM XplsmlsM4/3

As shown by Wetherill and Stewart (1989), planetesimal accretion is subject to positive feedback, which results in runaway growth: the largest bodies grow the fastest, rapidly detaching themselves from the size distribution of the planetesimals. The reason for this can be seen from Equation (3.4): dM/dt a M4/3, and so the growth timescale is

thus larger bodies grow faster than smaller ones.

Ida and Makino (1993) showed that runaway growth only operates temporarily; once the largest protoplanets are massive enough to dominate the gravitational stirring of the nearby planetesimals, the mode of accretion changes. From this point on, near a protoplanet of mass M, the planetesimal random velocity is approximately proportional to the surface escape speed from the protoplanet, and so v a M1/3 . Therefore, we now have dM SplsmlsM2/3

Thus growth becomes orderly; larger bodies grow more slowly, and we have the onset of oligarchic growth (Kokubo and Ida, 1998). In the terrestrial region, the transition from runaway to oligarchic growth already happens when the largest protoplanets are still many orders of magnitude below an Earth mass (Thommes et al., 2003). As a result, protoplanets spend almost all of their time growing oligarchically.

As long as nebular gas is present, the random velocity v of the planetesimals is set by two competing effects: gravitational stirring acts to increase v, while aerodynamic gas drag acts to reduce it. One can estimate the equilibrium random velocity by equating the two rates (Kokubo and Ida, 1998). In this way, one can eliminate v and write the protoplanet growth rate (recall that there is essentially no more planetesimal-planetesimal growth by this time) in terms of planetesimal and gas disk properties. Also, for simplicity we approximate the planetesimal population as having a single characteristic mass m. The result is (for details see Thommes et al. (2003))

where b5/2CD/5G1/2M+1/6 pg2a/5

Pp4CPm r1/Wm2/15

CD is a dimensionless drag coefficient ~1 for kilometre-sized or larger planetesimals, M* is the mass of the central star, pgas is the density of the gas disk, and PM and Pm are the material densities of the protoplanet and the planetesimals, respectively. The parameter b is the spacing between adjacent protoplanets in units of their Hill radii (3.1). An equilibrium between mutual gravitational scattering on the one hand and recircularization by dynamical friction on the other keeps b ~ 10 (Kokubo and Ida, 1998).

Protoplanet growth by sweep-up of planetesimals ceases when all planetesimals are gone. This leads to the notion of the isolation mass, which is the mass at which a protoplanet has consumed all planetesimals within an annulus of width brH centred on its orbital radius. The isolation mass is given by

V3M*

Note that when Spismis a r-2, Miso is constant with radius from the star; the increasing gravitational reach at larger distance from the star exactly balances the falloff in the density of planetesimals. For any shallower surface density profile, Miso increases with r.

In order to obtain values for protoplanet accretion times and final masses, a useful starting point is the 'minimum-mass Solar Nebula' (MMSN) model (Hayashi, 1981), which is obtained by 'smearing out' the refractory elements contained in the Solar System planets into a power-law planetesimal disk, then adding gas to obtain solar abundance:

is the 'snow line' solids enhancement factor: beyond rSN (= 2.7 AU in the Hayashi model), water freezes out, thus adding to the surface density of solids.

Since the exponent of the surface density power law is greater than -2, we know from Equation (3.10) that the isolation mass will increase with distance from the star (though from Equation (3.8), the time to finish growing also increases with r). Using Equation (3.10) with b = 10, the MMSN yields an isolation mass of 0.07 M© at 1 AU. If we assume a material density of 2 g cm-3 for planetesimals and protoplanets, and a characteristic planetesimal size of ~ 1 km (with a corresponding

3.4 The growth of planets Protoplanet mass (Earth masses) 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Orbital radius (AU)

Fig. 3.1. Protoplanet accretion during the 'oligarchic growth' phase in an MMSN. The average protoplanet mass as a function of distance from the star is plotted at 104 (dashed), 105 (dotted), and 106 (solid) years, showing how growth proceeds as a 'wave' from inside out through the disk. The upper limit, or isolation mass, is also plotted (thick). By a million years, growth has finished, i.e., essentially all planetesimals have been agglomerated into protoplanets, well out into the region of the present-day asteroid belt.

mass of 10-12 Me), the growth timescale for an isolation-mass body at 1 AU is tgrow = M/(dM/dt) ~ 6 x 104 y. Thus, the sweep-up of planetesimals into protoplanets proceeds very rapidly in the terrestrial region, concluding long before the nebular gas dissipates, which takes a few million years (Haisch et al., 2001). Figure 3.1 plots the solution to Equation (3.8) at different times, showing the progression of protoplanet growth through the terrestrial region during the first million years of the disk's lifetime.

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