= Solution
Take every finite positive <drag coefficient> $\gamma>0$ and define
$$
K=2\Omega(2\Omega-S),\qquad L=\Omega(\alpha+\beta-S),\qquad P=\alpha\beta.
$$
The coefficients of the <characteristic polynomial> are $c_3=2\gamma$, $c_2=K+\gamma^2$, $c_1=2L\gamma$ and $c_0=P\gamma^2$. The nontrivial <Routh-Hurwitz criterion> reduces exactly to
$$
c_3c_2c_1-c_3^2c_0-c_1^2=4\gamma^2\left[L(K-L)+\gamma^2(L-P)\right]>0.
$$
The printed strict chain $K>L>P>0$ makes every coefficient positive and both bracket terms positive, so it proves attraction for every positive <drag coefficient>.
There is, however, a boundary error in the asserted necessity. To have the bracket positive for every $\gamma>0$, its constant and slope as a function of $\gamma^2$ must be nonnegative, and cannot both vanish. Together with coefficient positivity this gives the exact <drag-polynomial stability for all positive stopping rates>:
$$
\boxed{K\geq L\geq P>0,\qquad K>P.}
$$
Necessity follows by taking arbitrarily small and arbitrarily large positive $\gamma$; if one bracket coefficient is negative, the inequality fails in the corresponding limit. Sufficiency follows because at least one coefficient is strictly positive. Thus $K=L>P$ and $K>L=P$ still give strict decay at each finite positive $\gamma$. They need not give a uniform stability margin in the zero- or infinite-drag limit.
In a <Keplerian shearing sheet>, $S=3\Omega/2$ and
$$
\frac K{\Omega^2}=1,\quad\frac L{\Omega^2}=\frac{3(r+1)}{2r(r-1)},\quad\frac P{\Omega^2}=\frac9{4(r-1)^2}.
$$
The differences are
$$
\frac{K-L}{\Omega^2}=\frac{(r-3)(2r+1)}{2r(r-1)},\qquad\frac{L-P}{\Omega^2}=\frac{3(2r+1)(r-2)}{4r(r-1)^2}.
$$
Hence $r>3$ establishes the requested strict chain and trapping conclusion. At $r=3$, however, $K=L=\Omega^2$ and $P=9\Omega^2/16$, so
$$
c_3c_2c_1-c_3^2c_0-c_1^2=\frac74\Omega^2\gamma^4>0.
$$
All coefficients remain positive. This is an explicit counterexample within the stated vortex model: \b[$r=3$ also attracts particles for every finite positive <drag coefficient>, so the literal strict “if and only if” is false.] The exact Keplerian all-positive-drag condition is $r\geq3$. If $\gamma=0$ is included, the polynomial has zero and purely imaginary roots and there is no asymptotic attraction; “all drag coefficients” must mean positive drag. The local attraction claim presumes the particle remains in the patch, as is assured for sufficiently small perturbations of its centre.
Back to article page
