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The formation of exoplanets
The formation of exoplanets

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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 ( cap h solidus r much less than one ) 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

tau sub stop equals m times normal cap delta times v divided by cap f sub drag comma
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

tau sub cap s equals tau sub stop times omega sub cap k comma
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 ( tau sub cap s much greater than one ), and small particles will generally have small Stokes numbers ( tau sub cap s much less than one ).

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

cap f sub drag equals four times pi divided by three times rho sub gas times s squared times v sub th times normal cap delta times v comma
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 rho sub m equals three times m solidus left parenthesis four times pi times s cubed right parenthesis . So, by combining this with Equations 11 and 13, we have

tau sub stop equals rho sub m divided by rho sub gas times s divided by c sub s full stop
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

v sub orb equals v sub cap k times left parenthesis one minus eta right parenthesis super one solidus two comma
Equation label: (Equation 15)

where, from Equation 10, we already know that eta equals n times left parenthesis cap h solidus r right parenthesis squared . 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:

v sub rad equals negative v sub cap k times eta divided by tau sub cap s plus tau sub cap s super negative one full stop
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 tau sub stop equals one solidus omega sub cap k . In this case, Equation 16 shows that the radial drift speed is v sub rad of max equals negative eta times v sub cap k solidus two .

  • Since η will be a small number for geometrically thin discs, make use of the approximation that left parenthesis one minus eta right parenthesis super one solidus two almost equals one minus left parenthesis eta solidus two right parenthesis 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

    equation sequence part 1 normal cap delta times v equals part 2 v sub cap k minus v sub orb equals part 3 v sub cap k times left square bracket one minus left parenthesis one minus eta right parenthesis super one solidus two right square bracket full stop

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

    equation sequence part 1 normal cap delta times v almost equals part 2 v sub cap k times left square bracket one minus one plus left parenthesis eta solidus two right parenthesis right square bracket almost equals part 3 eta times v sub cap k solidus two

    and so v sub cap k almost equals two times normal cap delta times v solidus eta . Hence, the maximum radial drift speed is

    equation sequence part 1 v sub rad of max almost equals part 2 negative left parenthesis eta solidus two right parenthesis multiplication two times normal cap delta times v solidus eta almost equals part 3 negative normal cap delta times v full stop

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

Activity 2

  • a.For large particles, with s > 1 m, the Stokes number is very large, tau sub cap s much greater than one . Obtain an expression for the radial drift speed in this case, in terms of τS and Δv only.

  • b.For small particles, with s < 1 cm, the Stokes number is very small, tau sub cap s much less than one . Obtain an expression for the radial drift speed in this case, in terms of τS and Δv only.

Comment

  • a.For large particles tau sub cap s much greater than one so Equation 16 becomes

    v sub rad almost equals negative eta times v sub cap k solidus tau sub cap s full stop

    Then, substituting for vK using the approximation v sub cap k almost equals two times normal cap delta times v solidus eta , we have

    v sub rad almost equals negative two times normal cap delta times v solidus tau sub cap s full stop

  • b.For small particles tau sub cap s much less than one so Equation 16 becomes

    v sub rad almost equals negative eta times v sub cap k times tau sub cap s full stop

    Then, substituting for vK using the approximation v sub cap k almost equals two times normal cap delta times v solidus eta , we have

    v sub rad almost equals negative two times normal cap delta times v times tau sub cap s full stop

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.