The Initial Mass Function Based on the Full-sky 20-pc Census of $\sim$3,600 Stars and Brown Dwarfs

A complete accounting of nearby objects – from the highest-mass white dwarf progenitors down to low-mass brown dwarfs – is now possible, thanks to an almost complete set of trigonometric parallax determinations from Gaia, ground-based surveys, and Spitzer follow-up. We create a census of objects within a Sun-centered sphere of 20-pc radius and check published literature to decompose each binary or higher-order system into its separate components. The result is a volume-limited census of $\sim$3,600 individual star formation products useful in measuring the initial mass function across the stellar ($<8M_{\odot }$) and substellar ($\gtrsim 5M_{Jup}$) regimes. Comparing our resulting initial mass function to previous measurements shows good agreement above 0.8$M_{\odot }$ and a divergence at lower masses. Our 20-pc space densities are best fit with a quadripartite power law, $\xi (M)=dN\mathrm{/}dM\propto M^{-\alpha }$ with long-established values of $\alpha =2.3$ at high masses ($0.55<M<8.00M_{\odot }$) and $\alpha =1.3$ at intermediate masses ($0.22<M<0.55M_{\odot }$), but at lower masses we find $\alpha =0.25$ for $0.05<M<0.22M_{\odot }$ and $\alpha =0.6$ for $0.01<M<0.05M_{\odot }$. This implies that the rate of production as a function of decreasing mass diminishes in the low-mass star/high-mass brown dwarf regime before increasing again in the low-mass brown dwarf regime. Correcting for completeness, we find a star to brown dwarf number ratio of, currently, 4:1, and an average mass per object of 0.41 $M_{\odot }$.