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# 2.1 From dust grains to rocks

Consider again the protoplanetary disc from Activity 1. The fact that the velocity of the gas in a protoplanetary disc is usually sub-Keplerian has important consequences for the evolution of solid particles embedded in it. A consequence of Equation 10 is that for geometrically thin discs () the radial pressure gradient makes a negligible contribution to the orbital speed of the gas. However, as seen in Activity 1, the difference in speed can be of the order of Δv ~ 100 m s-1 at ~1 au from the star and this turns out to be important in determining how the particles in a disc behave. In particular, a finite Δv can cause particles in the disc to slow down and drift inwards towards the star.

One of the most important parameters that determines how a particle of mass m interacts with the gas surrounding it is the stopping time, defined as

Equation label:(Equation 11)

where Fdrag is the magnitude of the drag force that acts in the opposite direction to Δv. This stopping time may be related to the Keplerian orbital speed by

Equation label:(Equation 12)

where τS is the Stokes number, which characterises how well particles follow fluid streamlines. Large particles will generally have large Stokes numbers (), and small particles will generally have small Stokes numbers ().

Small particles of radius s will be coupled with the gas; that is, they will move at almost the same speed as the gas. Such particles experience a drag force whose magnitude is given by

Equation label:(Equation 13)

where the thermal speed of the gas, vth, is roughly the same as its sound speed, cs. For spherical particles, the material density is . So, by combining this with Equations 11 and 13, we have

Equation label:(Equation 14)

We can now define the radial drift speed, vrad, as the speed with which particles in the disc move radially through it, drifting inwards towards the star. To derive a general expression for the radial drift speed we write the orbital speed as

Equation label:(Equation 15)

where, from Equation 10, we already know that . Remember that n is a dimensionless constant relating the pressure gradient to Pgas/r. After much algebra concerning the equations of motion and making appropriate approximations (the details of which are not important here), a general expression for the radial drift speed is found as:

Equation label:(Equation 16)

The radial drift speed will reach its maximum value when the Stokes number is τS = 1. This corresponds to the situation when the stopping time and Keplerian orbital speed are related by . In this case, Equation 16 shows that the radial drift speed is .

- Since η will be a small number for geometrically thin discs, make use of the approximation that to obtain a simple relation between Δv and vK. Hence write the maximum radial drift speed in terms of Δv.

- From Equation 10, the difference between the Keplerian speed and the orbital speed is

  Therefore, using the approximation for small values of η, we have

  and so . Hence, the maximum radial drift speed is

So the maximum radial drift speed is simply the difference between the Keplerian and orbital speeds.

## Activity 2

1. For large particles, with s > 1 m, the Stokes number is very large, . Obtain an expression for the radial drift speed in this case, in terms of τS and Δv only.

2. For small particles, with s < 1 cm, the Stokes number is very small, . Obtain an expression for the radial drift speed in this case, in terms of τS and Δv only.

### Discussion

1. For large particles so Equation 16 becomes

   Then, substituting for vK using the approximation , we have

2. For small particles so Equation 16 becomes

   Then, substituting for vK using the approximation , we have

Activity 2 showed that, in the limits of large or small particles, both values of the radial drift speed are independent of η. In each case, the radial drift speed is a small fraction of Δv.

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