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Protoplanetary discs comprise two species: gas and solids. The main difference between the two is that gas is partially supported against gravity by thermal pressure, so it orbits the host star slightly slower than the Keplerian velocity. Assuming the speed of sound is a fraction of the Keplerian velocity $ c = \alpha v_k $, the azimuthal velocity of the gas is

$ v_g \approx v_k \sqrt{1-\alpha^2} $

Small solid particles are carried along with the gas at the same velocity, but are not supported by thermal pressure, and therefore accelerate toward the host star. The acceleration stops when it is balanced by the drag force. The drag is given by Stokes' law

$ F_d \approx \rho_g s^2 v^2 $

where $ \rho_g $ is the density of the gas and $ v $ is the relative velocity between the gas and solids. The inward acceleration is given by

$ \Delta g \approx \alpha^2 \frac{v_k^2}{r} $

where $ r $ is the distance to the star. The terminal velocity is given by

$ F_d \approx \rho_s s^3 \Delta g \Rightarrow v_r \approx \alpha v_k \sqrt{\frac{s \rho_d}{r \rho_s}} $

where $ \rho_s $ is the density of an individual solid particle (as opposed to the average density of solids).

Bigger solid objects are only weakly coupled to the gas, and move at a velocity much closer to the Keplerian velocity than the gas velocity. The velocity of the solids is still a bit less than the Keplerian velocity because of drag. The inward velocity in this case is given by

$ v_r \approx v_k \frac{F_d}{\rho_s s^3 v_k^2/r} \approx v_k \frac{\rho_g s^2 \left(v_k-v_g\right)^2}{\rho_s s^3 v_k^2/r} \approx \alpha^4 v_k \frac{\rho_g a}{\rho_s s} $

The transition between the two regimes occurs when the size of the particles is $ s \approx \alpha^{8/3} r \frac{\rho_g}{\rho_s} $, where the velocity attains a maximum value $ v_r \approx \alpha v_k $.

Fot typical parameters of the protoplanetary disc of our solar system, the critical size for solid particles is about one metre. It has been argued that this could pose a problem for planet formation, as particles growing in size would drift into the host before they cross this threshold. However, in order to make this claim one has to show that the time it takes the solid particles to do so is smaller than the timescale for growth, and the timescale for the evaporation of the gas.

Epstein Drag Edit

When the solid particles are smaller than the average distance between gas particles, the drag is given by Epstein's law

$ F_e \approx \rho_g s^2 c v $

In this regime, the solids are strongly coupled to the gas, and so the terminal radial velocity is given by

$ \rho_g s^2 c v_r \approx \alpha^2 \frac{v_k^2}{r} \rho_s s^3 \Rightarrow v_r \approx \alpha v_k \frac{\rho_s s}{\rho_g r} $