10-4 10-3 10-2 10-1 10° 101 102 103 104 105 106 107 Diameter, cm
Fig. 5.24 Evolution of the size distribution of particles with a disk as a function of time. Initially, the grains have a diameter of 1 |m. The conditions in the protoplanetary disk are those in the primordial solar nebula, for a distance of 1 AU from the Sun (After Weidenschilling, 2000)
Kilometre-sized objects move in Keplerian orbits and are no longer susceptible to drag forces exerted by the surrounding gas. The orbits are almost circular and copla-nar, but the differential between the Keplerian velocities creates collisions and gravitational interactions. These interactions have the effect of increasing the eccentricity and inclination of the bodies, whereas collisions and friction with the gas tends to circularize the orbits. Because the relative velocities of the bodies are related to their velocities and their masses, the size distributions and the velocities evolve together and in a non-linear fashion.
The following section summarizes the discussion by de Pater and Lissauer (2001) on the growth rate of planetesimals.
Collisions between solid bodies may result in accretion, fragmentation or inelastic rebound. The impact velocity at which two solid bodies collide is:
where v is the relative velocity of one body with respect to the other far from encounter, and ve is the escape velocity at the point of impact:
where m1 and m2 are the respective masses of the two bodies, and R1 and R2 are their respective radii. The rebound velocity is evi, with e < 1. Accretion may occur if evi is smaller than ve. For a 10-km rocky object, the escape velocity is about 6 m/s. This is larger than the typical relative velocities of planetesimals, so that a 10-km body is likely to accrete the surrounding planetesimals, whereas fragmentation will preferentially occur for very small planetesimals.
The mean growth of a planetary embryo of mass M is dM 2
where ps is the density of the planetesimals, and v is the average relative velocity between the embryo and the small bodies (assumed to be much smaller than the embryo). F is the gravitational enhancement factor, given by
in the 2-body approximation.
It is possible to express the growth rate of the embryo's radius as a function of the surface mass density op, the embryo density pP and the Keplerian orbital angular velocity n:
dt y n 4pp
This equation leads to a growth time of about 2 x 107y for the Earth, and more than 108 yr for Jupiter. In the latter case, we know that other more efficient factors have been involved, because Jupiter and Saturn must have formed within 107 years, before the T-Tauri phase and the dissipation of the gas.
As the embryo grows, its escape velocity increases. If the relative velocity of the embryo versus the swarm of planetesimals remains small, the F factor may increase by large factors, leading to the runaway growth of the embryo. The embryo's feeding zone is limited to the annulus of planetesimals which the embryo may perturb gravi-tationally. Thus, rapid runaway will stop when the embryo has consumed the matter contained in the annulus. This mechanism thus leads to the formation of gaps inside protoplanetary disks. The size of the object that is formed depends on the material available within the gap that it creates.
The size of the gap depends primarily on the mass of the object that is forming. For a disk with no gas, the size of the gap is equal to a few times rH, where rH is the Hill radius, the latter being defined as the distance beyond which the gravitational force exerted by the star exceeds that of the protoplanet:
where a is the semi-major axis of the protoplanet's orbit, m is the mass of the proto-planet, and M the mass of the star. In the case of a disk with a high gas content that is strongly viscous, the width of the gap is a function of the Reynolds number where A is the width of the gap (Varniére et al., 2004; Beust, 2006).
Another situation occurs when the relative velocity of the swarm of planetesimals is comparable with or higher than the escape velocity of the embryo. In this case, the F factor remains close to unity and the evolutionary path of the planetesimals exhibits an orderly growth in the entire size distribution. This is why numerical simulations of planetary formation typically lead to two types of solutions:
• a burst of growth, which results in the rapid growth of one body, which sweeps up material from the surrounding space;
• orderly growth, resulting in several large bodies of similar mass, with a power distribution of smaller bodies.
In the latter case, the small bodies could have a very long lifetime. Such evolution could possibly explain the existence of debris disks around evolved stars (Table 5.1).
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