= Solution
<Sub-Keplerian rotation> means that the gas rotates more slowly than a pressure-free particle on a <circular orbit> at the same cylindrical radius. Let $v_K=(GM_\star/r)^{1/2}$. Neglecting radial acceleration, radial <force> balance in the gas is
$$
\frac{v_{g\phi}^2}{r}=\frac{GM_\star}{r^2}
+\frac1{\rho_g}\frac{dP_g}{dr}.
$$
An outwardly decreasing <pressure>, $dP_g/dr<0$, supplies part of the inward-gravity balance through an outward pressure force. Less centrifugal support is needed, so $v_{g\phi}<v_K$. Using $c_s^2=P_g/\rho_g$, define the <pressure-gradient parameter of a protoplanetary disk> by
$$
\eta\equiv-\frac{r}{2\rho_gv_K^2}\frac{dP_g}{dr}
=-\frac12\left(\frac{c_s}{v_K}\right)^2\frac{d\ln P_g}{d\ln r}.
$$
Then
$$
\boxed{v_{g\phi}=v_K\sqrt{1-2\eta}\simeq v_K(1-\eta).}
$$
For a thin disk and a smooth radial pressure profile, $\eta$ is small, of order $(H/r)^2$. The factor of two is part of this explicit convention; changing the convention for $\eta$ changes the drift formula correspondingly.
A solid particle feels essentially no continuum gas-pressure support. Left alone, it would orbit at $v_K$, so in the gas it feels a headwind. <Drag> removes <angular momentum>, causing <radial drift of solids in a gas disk>. Define <Stokes number> $\mathrm{St}=\Omega_Kt_s$ using the <aerodynamic stopping time> $t_s$. In a dilute test-particle limit with linear drag, negligible radial gas flow and small departures from a circular Keplerian orbit, let $u_\phi=v_{p\phi}-v_K$. The steady local radial and azimuthal balances are
$$
0=2\Omega_Ku_\phi-\frac{v_{pr}}{t_s},\qquad
0=-\frac{\Omega_Kv_{pr}}2-\frac{u_\phi+\eta v_K}{t_s}.
$$
Solving gives
$$
\boxed{v_{pr}=-\frac{2\eta v_K\mathrm{St}}{1+\mathrm{St}^2},\qquad
v_{p\phi}-v_K=-\frac{\eta v_K}{1+\mathrm{St}^2}.}
$$
The sign is inward for the usual negative pressure gradient. Very small particles, $\mathrm{St}\ll1$, closely follow the gas and have a small relative drift $v_{pr}\simeq-2\eta v_K\mathrm{St}$. Very weakly coupled large bodies, $\mathrm{St}\gg1$, also drift slowly, with $v_{pr}\simeq-2\eta v_K/\mathrm{St}$. \b[The fastest pressure-driven inward drift occurs at $\mathrm{St}=1$, where $|v_{pr}|=\eta v_K$.] The corresponding physical grain or aggregate size depends on gas density, orbital radius and the drag regime; “metre-size barrier” is a conventional description of some disk models, not a universal size.
\Image[/past-exam-of-the-exoplanets-and-planetary-systems-course-of-the-university-of-cambridge/2016/paper-1-dust-drift.png]
{title=Pressure support creates a headwind, with maximum inward solid drift at unit <Stokes number>}
The drift timescale in this model is
$$
t_{\rm drift}=\frac r{|v_{pr}|}
=\frac{1+\mathrm{St}^2}{2\eta\Omega_K\mathrm{St}}.
$$
It can be much shorter than the disk gas lifetime, threatening loss of the material needed to make <planetesimals>. Growth through the vulnerable coupling range must compete with this loss, while differential drift generates collisions that may stick, bounce or fragment. Radial redistribution also changes solid-to-gas ratios and moves material across condensation fronts. These are consequences for <planet formation>, rather than merely a small correction to an orbital frequency.
The same mechanism can help concentrate solids. At a local radial pressure maximum, $dP_g/dr=0$, so this component of drift vanishes. On the inside, a positive pressure gradient gives outward drift; on the outside, a negative gradient gives inward drift. Particles therefore converge toward a <dust trap>. Turbulent diffusion and inward gas advection can limit retention. With a nonzero radial gas speed $v_{gr}$, the test-particle result generalizes to $v_{pr}=[v_{gr}-2\eta v_K\mathrm{St}]/(1+\mathrm{St}^2)$, so gas advection need not vanish even at a pressure maximum.
Finally, if local solid loading becomes large, backreaction on the gas can no longer be neglected. Coupled gas-particle drift can support <streaming instability> and strong solid concentrations, potentially enabling gravitational collapse into <planetesimals>. The simple test-particle drift formula is a useful starting point, but not a description of that nonlinear high-loading regime. \b[Pressure support creates both a drift obstacle to growth and opportunities for concentration where the pressure structure changes.]
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