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We study the effect of variations in dust size distribution and compositionon the correlation between the spectral shape of extinction (parameterized by$R_{\textrm{V}}$) and far-infrared dust emissivity (parameterized by thepower-law index $\beta$). Starting from the size distribution models proposedby Weingartner & Draine (2001), using the dust absorption and emissionproperties derived by Laor & Draine (1993) for carbonaceous and silicategrains, and by Li & Draine (2001) for PAH grains, we calculate the extinctionand compare it with the reddening vector derived by Schlafly et al. (2016). Anoptimizer and an MCMC are used to explore the space of available parameters forthe size distributions. We find that larger grains are correlated with high$R_{\textrm{V}}$. However, this trend is not enough to explain theemission-extinction correlation observed by Schlafly et al. (2016). For the$R_{\textrm{V}}-\beta$ correlation to arise, we need to impose explicit priorsfor the carbonaceous and silicate volume priors as functions of$R_{\textrm{V}}$. The results show that a composition with higher ratio ofcarbonaceous to silicate grains leads to higher $R_{\textrm{V}}$ and lower$\beta$. A relation between $E(\textrm{B}-\textrm{V})/\tau_{353}$ and$R_{\textrm{V}}$ is apparent, with possible consequences for the recalibrationof emission-based dust maps as a function of $R_{\textrm{V}}$.

Implications of Grain Size Distribution and Composition for the Correlation Between Dust Extinction and Emissivity