 Forced dust epicycle with aerodynamic drag

ID: forced-dust-epicycle-with-aerodynamic-drag

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} .

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

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