Pressureless dust fluid in a shearing sheet

A dilute dust component can be approximated as a pressureless fluid with velocity

v

and surface density of a disk

Σ_{d}

. In a gas velocity

u

, linear drag force gives acceleration

- (v - u) / t_{s}

for positive aerodynamic stopping time

t_{s}

. Its equations include Coriolis acceleration, the shearing-sheet tidal potential, and

\partial_{t} Σ_{d} + \nabla \cdot (Σ_{d} 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.

Table of contents
- Forced dust epicycle with aerodynamic drag Pressureless dust fluid in a shearing sheet
  - Resonant dust entrainment by an epicyclic gas wave Forced dust epicycle with aerodynamic drag
    - Density contrast of an entrained dust wave Resonant dust entrainment by an epicyclic gas wave

Forced dust epicycle with aerodynamic drag ( $D_{ϵ} = \partial_{t} + ϵ Ω$ )

 0  0

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} - 2Ω v_{y} = ϵ Ω u_{x}

and

D_{ϵ} v_{y} + Ω v_{x} /2 = ϵ Ω u_{y}

, with

ϵ Ω = t_{s}^{- 1} > 0

. Eliminating

v_{y}

gives

[\partial_{t}^{2} + 2 ϵ Ω \partial_{t} + (1 + ϵ^{2}) Ω^{2}] v_{x} = ϵ Ω \partial_{t} u_{x} + ϵ^{2} Ω^{2} u_{x} + 2 ϵ Ω^{2} u_{y} .

(1)

Thus a gas epicycle

u_{x} = u cos (k x - Ω t)

u_{y} = u sin (k x - Ω t) /2

supplies both

2 ϵ Ω^{2} u sin (k x - Ω t)

and

ϵ^{2} Ω^{2} u cos (k x - Ω t)

. Keeping the latter is necessary for an exact linear equation beyond first order in drag. The homogeneous roots are

- ϵ Ω \pm i Ω

, separating damping from epicyclic frequency.

Resonant dust entrainment by an epicyclic gas wave ( $v^{'} - u^{'} \propto e^{- ϵ Ω t}$ )

 0  0

If the prescribed gas velocity is itself a free epicyclic motion solution, subtract its two momentum equations from the forced dust epicycle with aerodynamic drag equations. The velocity difference obeys a damped epicycle and decays as

e^{- ϵ Ω t}

. Therefore dust approaches the gas velocity with identical phase and amplitude for every positive drag rate. At zero drag the difference does not decay, so the late-time limit is nonuniform as drag vanishes. A finite-wavelength inertial-acoustic wave is detuned from the free epicycle; matching its phase and amplitude in the weak-drag limit additionally requires

∣ ω - Ω∣ ≪ ϵ Ω

. For a gas density wave of frequency

ω

, the exact radial transfer amplitude is

\frac{V _{x}}{U _{x}} = \frac{ϵ Ω [ ϵ Ω - iω - i Ω ^{2} / ω ]}{( ϵ Ω - iω ) ^{2} + Ω ^{2}},

(1)

obtained by solving the two dust equations and using

U_{y} = - i Ω U_{x} / (2 ω)

. At

ω = Ω

it is exactly one.

Density contrast of an entrained dust wave ( $δ_{d} = (k u /Ω) cos (k x - Ω t) + C (x)$ )

 0  0

Let

Σ_{d 0}

be the uniform background dust surface density of a disk and

δ_{d} = (Σ_{d} - Σ_{d 0}) / Σ_{d 0}

its fractional perturbation. Here

k

is the radial wavenumber,

Ω

is the epicyclic frequency and

u

is a real velocity amplitude. For that uniform background and late velocity

v_{x} = u cos (k x - Ω t)

, linear dust continuity equation gives

\partial_{t} δ_{d} = - \partial_{x} v_{x} = k u sin (k x - Ω t)

. Integrating yields the displayed formula. The coherent travelling-wave fractional excess is

∣ k u ∣/Ω

, with validity requiring this number much smaller than one. The static term

C (x)

records initial density and transient displacement: drag damps velocity, but supplies no density diffusion. The long-wavelength gas wave has the same fractional density amplitude, so entrainment alone does not produce an additional dust-to-gas ratio enhancement at this order.