The community of astronomical strawmen says that \({R(V)}\) should correlate with ISM density – dust grows and/or agglomerates in dense structures. We can look for this correlation in the intersection of the Valencic+ (year) and Jenkins (2009) samples. Valencic+ provides extinction information, including \({R(V)}\), and Jenkins provides ISM density and dust-to-gas ratio information along the same lines of sight. However, while there is a clear correlation between density and the dust-to-gas ratio (Figure \ref{fig:nH_F}), there does not appear to be a correlation between density and \({R(V)}\) (Figure \ref{fig:lognH_RV}).

By looking at Eddie’s map of \({R(V)}\) over a large area, we can generate the hypothesis that there’s not a clear density-\({R(V)}\) correlation because much of the \({R(V)}\) variation in Eddie’s map happens on much larger spatial scales than the angular size of a dense ISM structure. There could be a density-\({R(V)}\) relationship on top of this large-scale variation, but there isn’t a clear \({E(B-V)}\) vs. \({R(V)}\) correlation because the large-scale variation has a higher magnitude.

So, if we want to look for an \({E(B-V)}\) vs. \({R(V)}\) correlation, we need to filter out the large scale structure. One way to do this is to look at differences in \({E(B-V)}\) vs. differences in \({R(V)}\) between pairs of sightlines as a function of the sightlines’ separations.

\label{fig:nH_F} The mean dust-to-gas ratio along a sightline (y-axis) as encoded by the \(F*\) parameter from Jenkins (2009) vs. the mean volume density of atomic and molecular hydrogen along the same sightline. \(F*=0\) and \(F*=1\) correspond to dust-to-gas mass ratios of around 1/250 and 1/100, respectively. This is a recreation of Figure 16 from Jenkins (2009).

\label{fig:lognH_RV} \({R(V)}\) from Valencic+ (y-axis) vs. log of the mean density along the same sightline from Jenkins 2009 (x-axis). While the dust-to-gas ratio is clearly directly correlated with volume density (see previous figure), the dust properties traced by \(R(V)\) apparently are not.

The sample

Start with Eddie’s full \({E(B-V)}\) catalog. Throw out stars towards which \({R(V)}\) has not been measured and stars which fall outside of \(130^\circ < {\ell}< 250^\circ\) and \(-10^\circ < {b}< 10^\circ\). We restrict ourselves to \(130^\circ < {\ell}\) because we will want the distribution of \({\mathrm{T}_\mathrm{eff}}\) to not depend on \({\ell}\) later on, and there’s a dependence of \({\mathrm{T}_\mathrm{eff}}\) on \({\ell}\) when \({\ell}< 130^\circ\); this is shown in figure \ref{fig:glon_teff}.

\({R(V)}\) uncertainties

The amount of scatter in \({R(V)}\) at fixed \({E(B-V)}\) decreases as \({E(B-V)}\) increases from 0 to around 0.5. These low-\({E(B-V)}\) \({R(V)}\) measurements are clearly noisier than \({R(V)}\) measurements at higher \({E(B-V)}\). If we want to do any sort of regression with these clearly heteroskedastic measurements, we’re going to need to estimate these uncertainties. We can estimate these uncertainties using STUPID IMPORTANCE SAMPLING TRICKS and they behave pretty much the way we expect them to – sharp decline from 0 to 0.5, gradual decline from there, uptick at very high reddening, as can be seen in figure \ref{fig:EBV_Runc}.

Regressing out large scale effects

\label{sec:deLSS} There’s some pretty clearly large scale structure in the \({R(V)}\) map which, on small spatial scales, is mostly a DC offset. However, boundaries between these LSSs fall on a lot of our fields and are pretty narrow, so LSS is going to leak into our attempts to get at small-separation signals if we don’t regress the LSS effect out. Luckily, there’s a very good variable to regress on – the fraction of \({E(B-V)}\) at distances smaller than 1.2 kpc in the skewer containing each APOGEE star in Eddie’s 3D map, which we’ll call \(f_n\) (fraction near). To compute \(f_n\), we take each APOGEE star’s on-sky position, take the \({E(B-V)}\) profile as a function of distance at that position in Eddie’s 3D map, and just divide the total \({E(B-V)}\) up to 1.2 kpc by the total \({E(B-V)}\) along the profile. We show \({R(V)}\) as a function of \(f_n\) in figure \ref{fig:fn_rv}. The \(f_n\) we derive from the 3D map seems to be a pretty decent estimate of the actual \(f_n\). This does not obviously have to be true, since the 3D map extends out past many of the APOGEE stars.

The simplest model for the \(f_n\) dependence of \({R(V)}\) is that the near and far dust each have their own \({R(V)}\) values which get superimposed along the line of sight. The \({R(V)}\) that comes out of this admixture is given by the expression \[\label{eqn:geometric_RV} {R(V)}= 1.2 \frac{R_{0,g} - R_{0,W2} + \left({\frac{\rm{d}{R_g}} {\rm{d} {x} }} - {\frac{\rm{d}{R_{W2}}} {\rm{d} {x} }} \right) (f_n x_n + (1 - f_n) x_f)} {R_{0,g} - R_{0,r} + \left({\frac{\rm{d}{R_g}} {\rm{d} {x} }} - {\frac{\rm{d}{R_{r}}} {\rm{d} {x} }} \right) (f_n x_n + (1 - f_n) x_f)} - 1.18,\] where \(x_n\) and \(x_f\) are the value of the \({R(V)}\)-like principal component strengths of the near and far dust, respectively.

There’s also some obvious structure in \({E(B-V)}\). Most, though not all of this structure, is determined by \({b}\). T