Viscously Stirring Particle Disks into Lorentzians and Gaussians to Infer Dynamical and Collisional Masses (ARKS XIII)
Disks (Keplerian or otherwise, particulate or fluid) are often assumed to have densities that drop off vertically as Gaussians. Recent mm-wave imaging of circumstellar debris disks contradicts this assumption, revealing vertical profiles in dust that resemble Lorentzians. As part of the ARKS ALMA Large Program, we show how Lorentzians and Gaussians define an evolutionary sequence for disks of gravitationally scattering (viscously stirring) particles. When orbits are crossing and eccentricities $e \gg$ inclinations $i$, each scattering can change a particle's inclination by $\pm \,\Delta i \propto i$. A random walk with fixed steps in $\Delta i/i = \Delta \ln i$ produces a thick, log normal tail at large $i$ that leads to Lorentzian tails in density. This result holds independent of the origin of the large eccentricities, which may characterize either the stirrers or the objects being stirred; what matters is that relative motions parallel to the disk midplane are faster than perpendicular motions, and that vertical displacements during encounters are smaller than horizontal displacements. After enough scatterings, $i$ comes into equipartition with $e$, $\Delta i$ stops exponentiating, and the vertical density relaxes to a Gaussian. We identify four regimes of dispersion-dominated viscous stirring, three of which are out-of-equipartition and where $i$ is stirred faster than $e$. The stirrers for ARKS debris disks may range from Pluto to a few times Mars in size, and be sufficiently few as to be collisionless. Or the stirrers may be even smaller, and be so numerous and collide so frequently that they source the collisional cascades that produce observable dust.