Solar System Formation & Dynamics

Although there is vastly more gas than solid material in protoplanetary disks (typically by a factor of the order of 100), it is the evolution of the solid component that is critical to the formation of both terrestrial planets and the cores of gas giants. As long as the solid particles are relatively small (anywhere between microns in size up to tens or hundreds of meters) their dynamics is closely coupled to that of the gas via aerodynamic forces. These forces comes in two flavors, depending upon the particle size,

Protoplanetary disks are extremely tenuous gases. The mean free path at the mid plane of the disk may be of the order of tens of cm or meters in size, depending upon the distance from the star and the surface density of the disk. As a result, the earliest phases of solid growth are always securely in the Epstein regime of drag. Consider a solid particle modeled (probably not very well) as a sphere of radius . The particle has a velocity relative to the local gas of (note that this is not the orbital velocity of either the gas or the particle, but generally a much smaller number). The gas has density , and a mean thermal speed . Under these conditions, the Epstein drag is given by,

.

As you might guess, the drag depends upon the frontal area of the particle (i.e. it scales with the square of the particle radius), but as you might not guess the force is only linear in the relative velocity. It is this feature of Epstein drag which is distinct from the Stokes drag that we’re more familiar with in terrestrial situations.

The aerodynamic interactions between solid particles and the gas are particularly important for the vertical and radial mobility of solids with the disk. Let us first consider a dust particle suspended in the disk above (or equivalently below) the mid plane. The gas within the disk is maintained in hydrostatic equilibrium by a balance of forces: the vertical component of stellar gravity, acting downward, is balanced by an upward-acting vertical pressure gradient. A particle, however, does not feel the pressure force within the gas. With the only force acting on the particle being gravity, there is a tendency for dust to settle toward the mid plane, potentially forming a dense particle layer.

We can calculate how quickly dust settling occurs in a laminar (non-turbulent) disk quite easily. We begin by defining a quantity known as the friction time (or stopping time),

.

Since is the linear momentum associated with the relative motion of the particle against the gas, this time scale is the time on which aerodynamic drag tries to slow down the particle to match the local gas speed. Often, it is useful to use a dimensionless version of the same quantity, which we accomplish by multiplying by the local Keplerian orbital frequency,

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Substituting for the drag in the Epstein regime (and remembering that , with the material density of the particle) we find,

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For small particles this time scale is very small indeed. For micron sized particles at 1 AU, for example, a few seconds is a representative estimate. As a consequence, it is generally reasonable to assume that small dust particles attain terminal velocity as they settle toward the mid plane, in much the same way as a skydiver jumping from a plane accelerates downward only up to the point where air resistance matches gravity. In our situation, we have the same two forces acting, gravity, acting vertically downward toward the mid plane,

,

and aerodynamic drag, given by the Epstein formula as before,

.

Equating these forces defines the settling speed,

,

from which we could also compute a rough estimate of the settling time scale,

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Even for quite small particles (micron sized dust grains, which are coupled the gas very strongly) this time scale is generally shorter than the lifetime of the disk. At 1 AU, the settling time scale, ignoring both collisions with other particles (which may lead to coagulation and faster settling) and turbulence (which by stirring up particles opposes settling), typically may be of the order of 100,000 yr.

What about the effects of aerodynamic drag on the radial motion of solid particles? It is not immediately obvious that there are any effects, as a solid body such as a planet orbits a star in a balance between a radial gravitational force and an outward centrifugal force. There are thus no “unbalanced” forces of the kind that we appealed to when discussing settling. This is too simple, however, The gas in a protoplanetary disk is subject to additional non-gravitational forces, which cause it to orbit the star at a very slightly different speed from the Keplerian value, . This difference means that solid bodies do have a velocity differential with respect to the gas, and suffer aerodynamic effects as a consequence.

Quantitatively, we consider a gas disk that is circular and static (for the curious, the terms we are about to neglect that arise from radial gas flows are, indeed, generally negligible). Working in the disk mid plane, the orbital velocity is determined by the sum of the radial gravitational force and the pressure gradient,

.

If we neglect pressure, then this simplifies to the usual formula for the Keplerian orbital velocity. The pressure term, however, contributes a small but important correction. If – as is normally the case – the disk mid plane pressure is high at small radius and lower further out, then the pressure gradient term is negative. The pressure gradient provides an outward “push” that partially opposes gravity, and as a result the gas does not have to rotate as fast to maintain radial force balance.

The key result is thus: gas in protoplanetary disks typically rotates at less than the local Keplerian velocity.

For many purposes, this effect is small and can be safely ignored. If we write the pressure as a power law in radius,

,

then substituting we find,

.

Since , the gas velocity differs from the Keplerian value by a small amount, normally less than 1%. This is still, however, quite a large absolute velocity – at 1 AU of the order of 100 meters per second – and it is has a profound effect on the dynamics of solid bodies. Since the gas rotates slower than Keplerian velocity, while the solid particles want to orbit at the Keplerian velocity, the solids feel a headwind as they orbit faster than the local gas. The aerodynamic effect of this headwind causes the solids to lose angular momentum, and spiral in toward the star.

The striking fact is how fast this radial drift is. Weidenschilling (1977) showed that the rate of aerodynamic radial drift (not to be confused with the gravitational migration we will discuss later) is a function of the particle size. The effects of drift slow down for very massive bodies (obviously, as those have a lot of inertia relative to their surface area), but they are also small for extremely small particles that are effectively frozen into the gas by tight aerodynamic coupling. The strongest effects occur for a dimensionless friction time . Plotted against the friction time, the radial velocity has the following form,

The values on the y-axis depend upon the disk properties (especially on , which sets the velocity differential that drives radial drift) – here I have assumed . The peak drift speeds in this case exceed 0.1% of the Keplerian orbital velocity. This implies very short decay times,

For the same disk model (the middle line in the plot), particles with are predicted to be lost into the star on a time scale that is only of the order of 100 or 1000 years in the planet forming region! What size these most vulnerable particles are again depends upon the disk model, but in the terrestrial planet forming region values of tens of centimeters to around a meter are typical.

Radial drift, due to aerodynamic forces against a sub-Keplerian gas disk, is the cause of the radial drift problem or meter-sized barrier in planet formation. Clearly, if meter-sized bodies only survive for a brief period within the disk, either growth through the adjacent size scales must proceed very quickly, or the disk must possess “traps” within which radial drift is halted or slowed. In class we discussed a number of possible ways to get around the radial drift problem, but this is still an open research problem. Indeed, how to form planetesimals (the km-sized bodies that are the smallest objects immune to radial drift) out of smaller solid particles is often considered to be the most serious unsolved problem of planet formation.