Collisions of Terrestrial Worlds: The Occurrence of Extreme Mid-Infrared
  Excesses around Low-Mass Field Stars
  We present the results of an investigation into the occurrence and properties
(stellar age and mass trends) of low-mass field stars exhibiting extreme
mid-infrared (MIR) excesses ($L_\mathrm{IR} / L_\ast \gtrsim 0.01$). Stars for
the analysis were initially selected from the Motion Verified Red Stars
(MoVeRS) catalog of photometric stars with SDSS, 2MASS, and $WISE$ photometry
and significant proper motions. We identify 584 stars exhibiting extreme MIR
excesses, selected based on an empirical relationship for main sequence $W1-W3$
colors. For a small subset of the sample, we show, using spectroscopic tracers
of stellar age (H$\alpha$ and Li ${\rm{\small I}}$) and luminosity class, that
the parent sample is likely comprised of field dwarfs ($\gtrsim$ 1 Gyr). We
also develop the Low-mass Kinematics ($LoKi$) galactic model to estimate the
completeness of the extreme MIR excess sample. Using Galactic height as a proxy
for stellar age, the completeness corrected analysis indicates a distinct age
dependence for field stars exhibiting extreme MIR excesses. We also find a
trend with stellar mass (using $r-z$ color as a proxy). Our findings are
consistent with the detected extreme MIR excesses originating from dust created
in a short-lived collisional cascade ($\lesssim$ 100,000 years) during a giant
impact between two large planetismals or terrestrial planets. These stars with
extreme MIR excesses also provide support for planetary collisions being the
dominant mechanism in creating the observed $Kepler$ dichotomy (the need for
more than a single mode, typically two, to explain the variety of planetary
system architectures $Kepler$ has observed), rather than different formation
mechanisms.
