Solar System Formation & Dynamics

As we discussed previously, how planetesimals form remains a substantial mystery of planet formation. If we allow ourselves to stipulate that they do form, however, we can develop a theory for how planetesimals collide and grow into larger objects. In the simplest case, we make three assumptions,

• Planetesimals form with a smooth radial distribution across some relatively wide range of disk radii. We denote the surface density of the planetesimals (smoothed out over the area of the disk) as . As with other surface densities, it has units of .
• Planetesimals form with sizes that are large enough that gravity, rather than aerodynamic forces, dominates their subsequent evolution. This is often taken as a definition of “planetesimal”, though one should note that the dominance of gravity does not mean that aerodynamic forces suddenly become zero – to the contrary aerodynamics continues to play an important role in damping the random motions of planetesimals which are continually being excited by gravitational perturbations from other planetesimals and from growing planets.
• Collisions between planetesimals, and between planetesimals and protoplanets, lead to accretion. As we showed in class, this is generally a good assumption as long as the random velocities of the bodies are small compared to the escape speed from their surfaces. Provided this is the case, the energy of collisions is controlled primarily by the acceleration the two bodies experience as they fall into their gravitational potential wells. In this limit, even a modest amount of dissipation during the impact renders the two bodies gravitationally bound, and even if pieces fly off during the collision they will tend to reimpact later on (forming a rubble pile).

We can develop a theory for the early phases of planet formation that is quite analogous to the statistical theory of a gas. If we consider a gas, the molecules are moving around “randomly”, in the sense that the motions of neighboring molecules are un-correlated. For a gas with number density , collision cross-section , and random velocities , the collision rate of one molecule with others is,

.

Here , the collision cross-section, can be thought of as a measure of the physical size of the colliding molecules.

To apply the same ideas to planet formation there is one crucial difference. Planets and planetesimals are gravitating objects, which attract each other when close approaches happen. If we sketch the geometry of a close encounter, it is clear that the effect of gravity is to bring the bodies closer together at closest approach than they would have been if gravity had been negligible. This effect, called gravitational focusing, increases the collision cross-section to a value that is larger than the physical area of the colliding bodies.

To compute the enhancement to the collision rate caused by gravitational focusing, we set up the problem as shown above. We consider two equal mass bodies , moving with relative velocity at infinity on trajectories that would result in an impact parameter (this is the distance of closest approach ignoring gravity). Suppose that the distance of closest approach (with gravity) is , and the velocity of each body at closest approach is . We can solve for these two unknowns using conservation of energy and angular momentum. Equating the total energy in the initial state (just kinetic) to that when the bodies are at the moment of closest approach (kinetic plus potential) we have,

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Noting that there is no radial velocity component to the relative motion at closest approach, angular momentum conservation gives,

.

With two equations in two unknowns, we can solve for the distance and velocity at closest approach. If we then note that a physical collision happens if , where is the sum of the physical radii of the bodies, we obtain that physical collisions occur provided that,

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In terms of the escape velocity from the point of contact, , we have,

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Equivalently, the cross-section for collisions is,

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The first term is here the usual physical cross-section, the second term is the boost to the collision cross-section due to gravitational focusing. Gravitational focusing dominates collisions if the random velocity of the colliding bodies is small compared to the escape velocities from their surfaces. if this condition is met, it can result in a much higher rate of collisions than would be the case for non-self-gravitating bodies.

For this analysis, we assumed that the two colliding bodies were the only objects that matter. It is generally safe to ignore the serendipitous presence of a third planetesimal (three-body “collisions” are rare events), but the effects of the Sun need more careful attention. If the two planetesimals are at different orbital radii, there will be a differential (“tidal”) gravitational force from the Sun acting on each of them. The Sun will only be negligible for the dynamics of collisions if the tidal forces are small compared to the pairwise gravitational forces between the bodies.

We can estimate the role of tidal forces with a simple time scale argument. Consider a planet, mass , orbiting the star at distance $a$. A nearby planetesimal has a radial separation from the planet of distance . If the planetesimal orbits the star, its angular velocity is,

,

whereas if it orbits the planet the angular velocity is,

.

Setting these frequencies equal defines the distance from the planet within which the planet’s gravity dominates over the Solar tidal forces. This is known as the Hill radius (after George Hill). A more precise version of the argument given above gives a formula for the Hill radius,

,

that depends only on the one third power of the planet mass. This means that the gravitational “reach” of planets – the region of the disk within which the dynamics is controlled by the planet – increases more slowly with mass than one might expect. If we compare Jupiter to the Earth, for example, the former is 300 times more massive. The size of the Hill sphere as a fraction of the orbit, , on the other hand, is only larger by a factor of 6.7.

The size of the Hill sphere defines the region of space within which a planet can hold on to permanently bound satellites. However, the concept has much more general importance. The stability of a planetary system, for example, can be roughly measured in terms of the planetary separation measured in units of the Hill radii of the planets involved.