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}.
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}.
\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 \({b}\). This isn’t all however – the residuals from just regressing out \({b}\) are extremely correlated with \(f_n\). We can improve on this by regressing \({b}\) out the near and far \({E(B-V)}\) maps separately. Even this procedure leaves some LSS in due to the fact that (1) the dense ISM falls off non-linearly in \({b}\) and (2) the \({b}\) centroid of the MW disk has pronounced warps. These two effects can be removed by regressing \({b}\) out of the near and far \({E(B-V)}\) maps one field at a time.
Maps of \({E(B-V)}\) and \({R(V)}\) with these simple models subtracted are shown in figure \ref{fig:deLSSd}.
Take a pair of stars, \(A\) and \(B\). Compute the difference between the values of variable \(X\) for the two stars, \(\Delta X_{AB} = X_A - X_B\), and between the values of variable \(Y\), \(\Delta Y_{AB} = Y_A - Y_B\). Repeat for every star in the sample.
Take every pair of stars whose angular separation \(s\) on the sky is in some interval. Compute the slope, \({\frac{\rm{d}{\Delta X}} {\rm{d} {\Delta Y} }}\), of the relation between \(\Delta X\) and \(\Delta Y\). Repeat for every interval present in the sample to get a function, \({\frac{\rm{d}{\Delta X}} {\rm{d} {\Delta Y} }} (s) \), that tells you how the correlation between pairwise differences in \(X\) and \(Y\) depends on separation \(s\).
We compute these pairwise-difference correlation functions (PDCFs) for \({R(V)}\) residuals as a function of \({E(B-V)}\) residuals and for stellar temperature as a function of \({E(B-V)}\) residuals. These PDCFs are shown in figure \ref{fig:dRdE_dTdE}, along with PDCFs derived from scrambled catalogs.
We make these scrambled catalogs by shuffling some properties between different stars while keeping others fixed. The scrambled catalogs shown in figure \ref{fig:dRdE_dTdE} are generated by fixing the \({\ell}\) and \({b}\) of each star and replacing its \({E(B-V)}\) residuals, \({R(V)}\) residuals, and effective temperature with those of a different star. We require that the separation between the second star and the center of its field be similar to the separation between the first star and the center of its field; this seems to be necessary in order for the shape of the scrambled data PDCFs to resemble that of the actual data PDCFs. Requiring the second star to instead have a similar Galactic longitude, Galactic latitude, effective temperature, or \({E(B-V)}\) (original or residual) to the first star produces PDCFs that look quite different from the actual ones.
A note on \({E(B-V)}\), \({R(V)}\), and their various residuals – all of the different ways of subtracting off large-scale trends in \({E(B-V)}\) and \({R(V)}\) produce basically identical PDCFs. At pairwise separations shorter than a degree or so, all of these differently computed residuals are also basically indistinguishable from those of just the raw \({E(B-V)}\) and \({R(V)}\) maps. The purpose of removing the large-scale trends is to allow us to make useful scrambled catalogs. For example, if we just regress out a single, global \(\vert {b}\vert\) dependence, we can see the distance-dependence of the angular scale height of the ISM disk. Because our scrambling procedure shuffles stars between widely separated parts of the sky, the range of \({E(B-V)}\) residual values in a scrambled field will be much higher than in an actual field. This greater range drives down the strength of the \({E(B-V)}\)-\({R(V)}\) correlation in the scrambled data PDCFs. Since we use the scrambled data PDCFs to estimate the significance of the actual data PDCFs, it’s important to filter out as much large-scale power as possible.