Magnetocentrifugal launching criterion in a flattened power-law potential (${\alpha }_{\mathrm{crit}}=\arctan (|\lambda |/\sqrt{\beta +2})$)

In a flattened power-law gravitational potential $\Phi ={\Phi }_0{R}_0^{\beta }(R^2+{\lambda }^2{z}^2{)}^{-\beta /2}$, a magnetic surface corotating with its disk footpoint has an effective potential with meridional Hessian matrix ${\Omega }_f^2\mathrm{diag}[-(\beta +2),{\lambda }^2]$ there. A cold outward displacement along a field inclined by $\alpha$ to the vertical is downhill when ${\tan }^2\alpha >{\lambda }^2/(\beta +2)$. Equality is marginal and can depend on higher derivatives and field-line curvature; this is a local launching criterion, not a guarantee of a global escaping wind.