The High Mass Slope of the IMF

Recent papers have found that the inferred slope of the high-mass ($>1.5$ M$_\odot$) IMF for field stars in the solar vicinity has a larger value ($\sim 1.7-2.1$) than the slopes ($\sim 1.2-1.7$; Salpeter= 1.35) inferred from numerous studies of young clusters. We attempt to reconcile this apparent contradiction. Stars mostly form in Giant Molecular Clouds, and the more massive stars ($\gtrsim 3$ M$_\odot$) may have insufficient time before their deaths to uniformly populate the solar circle of the Galaxy. We examine the effect of small sample volumes on the {\it apparent} slope, $\Gamma_{\rm app}$, of the high-mass IMF by modeling the present day mass function (PDMF) over the mass range $1.5-6$ M$_\odot$. Depending on the location of the observer along the solar circle and the size of the sample volume, the apparent slope of the IMF can show a wide variance, with typical values steeper than the underlying universal value $\Gamma$. We show, for example, that the PDMFs observed in a small (radius $\sim 200$ pc) volume randomly placed at the solar circle have a $\sim 15-30$\% likelihood of resulting in $\Gamma_{\rm app} \gtrsim \Gamma+ 0.35$ because of inhomogeneities in the surface densities of more massive stars. If we add the a priori knowledge that the Sun currently lies in an interarm region, where the star formation rate is lower than the average at the solar circle, we find an even higher likelihood ($\sim 50-60\%$ ) of $\Gamma_{\rm app} \gtrsim \Gamma+0.35$, corresponding to $\Gamma_{\rm app} \gtrsim 1.7$ when the underlying $\Gamma= 1.35$.