On the Link Between Energy Equipartition and Radial Variation in the
  Stellar Mass Function of Star Clusters
  We make use of $N$-body simulations to determine the relationship between two
observable parameters that are used to quantify mass segregation and energy
equipartition in star clusters. Mass segregation can be quantified by measuring
how the slope of a cluster's stellar mass function $\alpha$ changes with
clustercentric distance r, and then calculating $\delta_\alpha = \frac{d
\alpha(r)}{d ln(r/r_m)}$ where $r_m$ is the cluster's half-mass radius. The
degree of energy equipartition in a cluster is quantified by $\eta$, which is a
measure of how stellar velocity dispersion $\sigma$ depends on stellar mass m
via $\sigma(m) \propto m^{-\eta}$. Through a suite of $N$-body star cluster
simulations with a range of initial sizes, binary fractions, orbits, black hole
retention fractions, and initial mass functions, we present the co-evolution of
$\delta_\alpha$ and $\eta$. We find that measurements of the global $\eta$ are
strongly affected by the radial dependence of $\sigma$ and mean stellar mass
and the relationship between $\eta$ and $\delta_\alpha$ depends mainly on the
cluster's initial conditions and the tidal field. Within $r_m$, where these
effects are minimized, we find that $\eta$ and $\delta_\alpha$ initially share
a linear relationship. However, once the degree of mass segregation increases
such that the radial dependence of $\sigma$ and mean stellar mass become a
factor within $r_m$, or the cluster undergoes core collapse, the relationship
breaks down. We propose a method for determining $\eta$ within $r_m$ from an
observational measurement of $\delta_\alpha$. In cases where $\eta$ and
$\delta_\alpha$ can be measured independently, this new method offers a way of
measuring the cluster's dynamical state.
