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Dust settling is the vertical drift of solid particles toward a disk's midplane under gravity and drag force. In the strong-drag limit, a particle with aerodynamic stopping time $\tau$ settles through a linear vertical gravitational field at rate ${\Omega }_z^2\tau$.

Let $v_x,v_y$ and $u_x,u_y$ be the dust and gas velocity perturbations after subtracting the common background shear, suppressing primes. Put $t_s>0$ for the aerodynamic stopping time. Linear axisymmetric dust perturbations about a Keplerian shearing sheet obey $D_{ϵ}{v}_x-2\Omega v_y=ϵ\Omega u_x$ and $D_{ϵ}{v}_y+\Omega v_x/2=ϵ\Omega u_y$, with $ϵ\Omega =t_s^{-1}>0$. Eliminating $v_y$ givesThus a gas epicycle $u_x=u\cos (kx-\Omega t)$, $u_y=u\sin (kx-\Omega t)/2$ supplies both $2ϵ{\Omega }^{2}u\sin (kx-\Omega t)$ and ${ϵ}^2{\Omega }^{2}u\cos (kx-\Omega t)$. Keeping the latter is necessary for an exact linear equation beyond first order in drag. The homogeneous roots are $-ϵ\Omega \pm i\Omega$, separating damping from epicyclic frequency.

Gas drag is a drag force on a solid body due to its motion relative to surrounding gas. With aerodynamic stopping time $t_s$, the linear approximation gives $\dot{\mathbf{v}}{|}_{\mathrm{drag}}=-(\mathbf{v}-{\mathbf{v}}_g)/t_s$.

For linear gas drag with aerodynamic stopping times $t_1,t_2$, masses $m_1,m_2$, total mass $m=m_1+m_2$, and relative velocity $\mathbf{v}$, defineIf $\mathbf{w}$ is the gas velocity relative to the centre of mass, the relative drag acceleration is $-\kappa \mathbf{v}+\Delta \mathbf{w}$. The first term dissipates internal specific orbital energy; the second is a differential headwind that vanishes for equal aerodynamic stopping times.

For a weakly perturbed circular binary planetesimal, the constant differential headwind does zero work averaged over the unperturbed orbit. Orbit-averaged drag work givesWith constant aerodynamic stopping times, $a(t)=a_0{e}^{-t/\tau }$. This averaging assumes drag is slow compared with the orbital period and does not appreciably distort the circular orbit within one period.

For a nearly circular stellar orbit of radius $a_b$ in gas moving at $(1-\eta )$ times the local circular speed, the averaged tangential gas drag on the centre of mass is $-\eta {\kappa }_b{v}_K$. Orbit-averaged drag work givesIf $m_2/m_1=ϵ\ll 1$ and aerodynamic stopping time scales as $m^k$, $0<k<1$, then $\tau /{\tau }_b\simeq \eta {ϵ}^k\ll 1$. Internal contraction can outpace stellar migration. Tightly gas-coupled drift requires solving radial and azimuthal motion together instead of this weak-drag approximation.

Put $m=m_1+m_2$, $\mu =Gm$, $n=\sqrt{\mu /a^3}$, $v=na$, and $\theta =nt$. Work instantaneously in the inertial frame in which the centre of mass is at rest. The unperturbed circular relative orbit hasThe individual velocities are ${\mathbf{v}}_1=-(m_2/m)\mathbf{v}$ and ${\mathbf{v}}_2=(m_1/m)\mathbf{v}$. Linear gas drag gives ${\mathbf{F}}_i/m_i=-({\mathbf{v}}_i-v_g\hat{\mathbf{y}})/t_i$, where $t_i=t_{si}$ are the aerodynamic stopping times. DefineThen the gas drag on a planetesimal binary isEquivalently, $\mathbf{f}=-\kappa \mathbf{v}+\Delta v_g\hat{\mathbf{y}}$. If the centre of mass moves, $v_g\hat{\mathbf{y}}$ here is its relative gas velocity, held approximately fixed during one short binary orbit.

For weak gas drag, evaluate the work on the unperturbed circular orbit and average over one orbital period. The gas velocity changes negligibly during that period and $⟨\cos \theta ⟩=0$. Hence gas-driven contraction of a planetesimal binary obeysWith constant aerodynamic stopping times,This is the secular, orbit-averaged separation. It assumes the perturbations are small enough for the orbit to remain nearly circular; a large differential headwind can excite eccentricity or disrupt a weakly bound pair instead.

Put $ϵ=m_2/m_1\ll 1$ and $t_{s2}/t_{s1}={ϵ}^k$, with $0<k<1$. The gas-driven contraction of a planetesimal binary and gas-driven migration of a binary centre of mass times obeyThereforeThe low-mass secondary has the shorter aerodynamic stopping time and chiefly controls internal damping, whereas the primary chiefly controls centre of mass drag. The pair can contract substantially, potentially merging, before its stellar orbital radius changes appreciably. This conclusion is restricted to a pair that remains bound and within the adopted linear-drag regime.

Gas in hydrostatic balance has zero vertical velocity. A dust grain subject to linear drag with aerodynamic stopping time $\tau$ therefore obeysThis is a damped harmonic oscillator. For ${\Omega }_z\tau >1/2$, the motion is underdamped, with angular frequencyand envelope $e^{-t/(2\tau )}$. Critical damping occurs at ${\Omega }_z\tau =1/2$. For ${\Omega }_z\tau <1/2$, the motion is overdamped. In the strong-drag limit ${\Omega }_z\tau \ll 1$, a rapid transient on timescale $\tau$ leaves slow dust settling in an astrophysical disk at rate ${\Omega }_z^2\tau$; in the weak-drag limit the grain makes many damped vertical oscillations.

A dilute dust component can be approximated as a pressureless fluid with velocity $\mathbf{v}$ and surface density of a disk ${\Sigma }_d$. In a gas velocity $\mathbf{u}$, linear drag force gives acceleration $-(\mathbf{v}-\mathbf{u})/t_s$ for positive aerodynamic stopping time $t_s$. Its equations include Coriolis acceleration, the shearing-sheet tidal potential, and ${\partial }_t{\Sigma }_d+\nabla \cdot ({\Sigma }_d\mathbf{v})=0$. Neglecting gas backreaction is justified only when the prescribed-gas approximation is appropriate. Strong orbit crossing or appreciable dust velocity dispersion can invalidate the single-valued pressureless-fluid description.