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The aerodynamic stopping time is the characteristic time on which drag force damps a solid particle's velocity relative to the surrounding fluid. Linear drag gives relative acceleration $-{\mathbf{v}}_{\mathrm{rel}}/\tau$.

Consider a pressured dust layer in a prescribed Keplerian shearing sheet gas flow, neglecting gas backreaction. Let ${\sigma }_0$ be its unperturbed surface density, $c$ its constant effective random speed with pressure perturbation $c^2\delta \sigma$, and $\Omega >0$ the local angular velocity. Linear gas drag damps dust velocity at rate $\gamma =t_s^{-1}=ϵ\Omega >0$, where $t_s$ is the aerodynamic stopping time.

For axisymmetric perturbations proportional to $e^{st+ikx}$ with $k\ne 0$, write $\delta =\delta \sigma /{\sigma }_0$ and radial and azimuthal velocity amplitudes $u,v$. The linear mass and momentum balances areThe razor-thin disk Poisson kernel supplies the self-gravity term. The azimuthal coefficient $\Omega /2$ includes advection of the Keplerian shear. Taking the full amplitude determinant givesAt zero drag this factors as $s(s^2+{\omega }^2)=0$, with ${\omega }^2={\Omega }^2+c^2{k}^2-2\pi G{\sigma }_0|k|$. The zero root is the branch from which secular gravitational instability can develop when drag is restored. Dividing by $s$ before evaluating the determinant would discard that physical mode.

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.

A protoplanetary disk is a flattened, rotating reservoir of gas and solids around a young star. Its radial gravity is usually dominated by the host, making its rotation close to Keplerian rotation, ${\Omega }_K=(G{M}_{\star }/r^3{)}^{1/2}$. Gas pressure provides vertical support and a small radial correction to the orbital speed; it does not make the disk a nonrotating spherical atmosphere.

At height $|z|\ll r$, vertical hydrostatic equilibrium in the stellar gravitational field givesFor a locally vertically isothermal ideal gas, $P_g={\rho }_g{c}_s^2$ with $c_s^2=k_{B}T/(\mu m_H)$, where $\mu$ is mean molecular weight. Integration givesHere ${\Sigma }_g$ is the vertically integrated gas surface density of a disk, and $H$ is the isothermal disk scale height. The disk aspect ratio $H/r=c_s/v_K$ is small in a geometrically thin disk. For $T\propto r^{-q}$ it scales as $r^{(1-q)/2}$, so $q<1$ produces disk flaring. This illustrative local vertical model does not determine a unique radial surface-density law or temperature normalization from the information given.

The gas is predominantly molecular hydrogen and helium. A much smaller component consists of heavier elements, partitioned between vapor, solid refractory grains and condensed volatile ices. A gas-to-solid mass ratio of order $100$ is a common initial fiducial model, not a universal measurement applying at every disk radius or evolutionary stage. Solids can grow and settle while gas accretes or disperses, changing the local ratio substantially.^[ref]

Thermal structure organizes composition. In the hottest inner region, grains can sublimate and much of their material remains in the gas. Refractory silicate minerals and metallic grains survive at lower temperatures; farther out, water and then other sufficiently volatile species can condense when their local condensation conditions are met. The water snow line can raise the solid inventory available to planet formation. Its location depends on stellar irradiation, accretion heating, opacity and gas pressure, and evolves with time; it is not fixed merely by naming a star or adopting a solar-system distance.

Stellar radiation heats the exposed surface of a passive disk. Beneath it, shielding can leave a cooler, molecule-rich interior with a higher mass density. In the inner actively accreting disk, dissipation can instead make the optically thick midplane the hottest region. These are distinct heating regimes; a vertically isothermal profile is an analytic approximation rather than a claim that every real layer has the same temperature. Dust opacity controls how radiation is absorbed and re-emitted. Infrared and millimetre emission probe solid material, while spectral lines probe gas; neither observable alone supplies a model-free inventory of the whole disk.

Dust settling in an astrophysical disk makes larger, less tightly coupled particles more concentrated toward the midplane, while turbulence and mixing oppose that settling. For an aerodynamic stopping time $t_s$ short compared with ${\Omega }_K^{-1}$, terminal vertical balance gives $v_z\simeq -{\Omega }_K^2{t}_{s}z$. Small grains remain better coupled to the gas. Radial drift also changes the distribution of solids and their access to snow lines and embryos.

The disk evolves through accretion and angular momentum transport, potentially involving turbulent stresses, magnetorotational instability or magnetically driven winds where conditions permit. Protoplanetary disk dispersal eventually removes gas through accretion and mass loss, including photoevaporation. In standard evolutionary pictures this occurs on a scale of a few million years, with substantial system-to-system variation. That finite reservoir lifetime is central to whether an embryo acquires a massive gaseous envelope.^[ref]

The essential structure is a nearly Keplerian gas disk with vertical pressure support, radial and vertical thermal gradients, and an evolving solid population concentrated by settling, growth and drift. Its composition is set jointly by the original elemental inventory, local condensation conditions and transport.

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 $v_K=(G{M}_{\star }/r{)}^{1/2}$. Neglecting radial acceleration, radial force balance in the gas isAn outwardly decreasing pressure, $d{P}_g/dr<0$, supplies part of the inward-gravity balance through an outward pressure force. Less centrifugal support is needed, so $v_{g\varphi }<v_K$. Using $c_s^2=P_g/{\rho }_g$, define the pressure-gradient parameter of a protoplanetary disk byThenFor 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 $\mathrm{St}={\Omega }_K{t}_s$ 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 $u_{\varphi }=v_{p\varphi }-v_K$. The steady local radial and azimuthal balances areSolving givesThe sign is inward for the usual negative pressure gradient. Very small particles, $\mathrm{St}\ll 1$, closely follow the gas and have a small relative drift $v_{pr}\simeq -2\eta v_K\mathrm{St}$. Very weakly coupled large bodies, $\mathrm{St}\gg 1$, also drift slowly, with $v_{pr}\simeq -2\eta v_K/\mathrm{St}$. 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.

The drift timescale in this model isIt 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, $d{P}_g/dr=0$, 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. Pressure support creates both a drift obstacle to growth and opportunities for concentration where the pressure structure changes.

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.