The Optical–Mid-infrared Extinction Law of the l = 165° Sightline in the Galactic Plane: Diversity of the Extinction Law in the Diffuse Interstellar Medium

Three criteria can be independently used to characterize the ISM: visual extinction, IR dust emission, and CO line intensity. The extinction A_λ depends on the dust column density, ${N}_{{\rm{d}}}$, and the optical properties of the dust: ${A}_{\lambda }=1.086\,{N}_{{\rm{d}}}\,{C}_{\mathrm{ext}}(a,\lambda )$, where ${C}_{\mathrm{ext}}(a,\lambda )$ is the extinction cross-section of the dust with a typical size a at a wavelength λ. Hence, a high A_λ implies a dense cloud or a pile-up of many diffuse clouds along the line of sight. The dust-emission intensity I_λ is proportional to the dust column density, the absorption cross-section, and the specific emission intensity of the dust: ${I}_{\lambda }\propto {N}_{{\rm{d}}}\,{C}_{\mathrm{abs}}(a,\lambda ){B}_{\lambda }(T)$, where ${B}_{\lambda }(T)$ is the Planck function at the dust temperature T and wavelength λ. The high intensity of the CO emission line is often used to indicate dense interstellar environments. Generally speaking, with the intensity of the CO emission line, I(CO), the total mass of the molecular gas can be derived using an empirical CO-to-H₂ conversion factor ${X}_{\mathrm{CO}}\equiv {N}_{{{\rm{H}}}_{2}}/{I}_{\mathrm{CO}}$. If the gas is well mixed with the dust, with a constant gas-to-dust ratio, the total dust mass can be derived. Thus, the intensity of the CO emission line is also proportional to the dust column density. Indeed, Zasowski et al. (2009) used the ${}^{13}$CO (J = 1–0) line to trace dense interstellar clouds. Wang et al. (2013) used these criteria to distinguish complex environments in the Coalsack nebula region to investigate the variation of the mid-IR extinction law.

As described in the previous section, the CO (1–0) line intensity I(CO) is often used to characterize dense interstellar environments. Integrated CO line intensity contours reveal complex Galactic structures. Guided by this, we checked the CO emission intensity maps of the Galactic plane to find candidate diffuse ISM regions. A wealth of CO line intensity databases exists for the Galactic plane and some well-known molecular clouds. The Milky Way's CO emission map of Dame et al. (2001) covers the entire Galactic plane at Galactic latitudes $| b| \leqslant 30^\circ$ with an effective angular resolution of 05. The European Space Agency's Planck satellite observed the sky in nine bands covering frequencies of 30–857 GHz with high sensitivity and high spatial resolution. Planck CO maps have been extracted from the Planck HFI data. In this work, we adopt the Planck Type 3 CO (1–0) map because of its high resolution and sensitivity (Planck Collaboration XIII 2014). The angular resolution is $5\buildrel{\,\prime}\over{.} 5$, and the standard deviation is 0.16 K km s⁻¹ at an angular resolution of $15^{\prime}$. For comparison, the CO survey of Dame et al. (2001) has a typical uncertainty of 0.6 K km s⁻¹. Figure 1 displays integrated CO line intensity contours for $90^\circ \lt l\lt 180^\circ$, $| b| \leqslant 5^\circ$, where molecular clouds stand out clearly because of their intense CO line emission, while some regions are diffuse with low CO line intensities. Our candidate regions are restricted to the Galactic plane because the stars in the Galactic halo are usually metal-poor. We plan to use RC stars as tracers of the interstellar extinction. The intrinsic colors of RC stars need to be determined (Section 4). The optical intrinsic color of RC stars relates to their metallicity (e.g., Sarajedini 1999; Girardi & Salaris 2001; Nataf et al. 2010). Therefore, if we were to include stars in the Galactic halo, we should also consider the effect of metallicity in deriving the intrinsic color for RC stars. For this reason, we limit our candidate diffuse regions in the Galactic plane. In addition, RC stars will be selected based on the APOGEE survey; the diffuse region targeted must have been observed by it. The APOGEE survey targeted more than 100,000 red giant (RG) stars selected from the 2MASS database (Skrutskie et al. 2006), for which accurate stellar parameters were derived. However, APOGEE is not an all-sky survey. Therefore, the diffuse region targeted here was chosen by overlaying the APOGEE stars on the integrated CO emission intensity map. In Figure 1, the blue dots are giants from APOGEE.

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Based on Figure 1, we selected one candidate diffuse region, "l165°," centered on ($l=165\buildrel{\circ}\over{.} 0,b=0\buildrel{\circ}\over{.} 0$). It covers $4^\circ \times 4^\circ$ on the sky, and the mean I(CO) is about 2 K km s⁻¹; the region is indicated by the red square. The other two factors, the visual extinction and the dust IR emission intensity, are also used in selecting the diffuse region (see Section 1 for more details). The $l165^\circ$ region shows essentially very little extinction with ${A}_{V}\sim 1.0\,\mathrm{mag}$ in the visual extinction maps of Dobashi et al. (2005) and from Chen et al. (2015). This amount of extinction matches very well the characteristic extinction of diffuse clouds, i.e., the total visual extinction ${A}_{V}$ is ∼0–1 mag (Snow & McCall 2006; Draine 2011). In addition, the Spitzer/MIPS 24 μm image shows no detectable 24 μm emission for l165°. Therefore, the CO line intensity, visual extinction, and dust emission all indicate that the l165° region is very diffuse.

Broad-band photometric data from APASS, XSTPS-GAC, 2MASS, and WISE are used to derive the observed colors, while the stellar spectroscopic data set from APOGEE is used to determine the intrinsic colors.

3.1.1. Optical to Infrared Photometric Data: APASS, XSTPS-GAC, 2MASS, WISE Surveys

3.1.2. Spectroscopic Data: The SDSS/APOGEE Survey

RG and RC stars are frequently used as IR interstellar extinction tracers. RG stars with a small scatter in the IR intrinsic color index (${C}_{{{JK}}_{s}}^{0};$ Gao et al. 2009; Wang et al. 2013) are usually selected based on mid-IR color restrictions: $[3.6]\mbox{--}[4.5]\lt 0.6$ mag and $[5.8]\mbox{--}[8.0]\lt 0.2$ mag (Flaherty et al. 2007). Although these criteria could effectively remove sources with intrinsic IR excesses, some asymptotic giant-branch (AGB) stars that suffer from circumstellar extinction may contaminate the RG sample. RC stars are a group of K2III-type stars in the core-helium-burning stage. Their absolute magnitude is around ${M}_{K}=-1.61\pm 0.03\,\mathrm{mag}$ (Alves 2000), and their near-IR intrinsic color index is $0.65\leqslant 0.75$ mag (Wainscoat et al. 1992; González-Fernández et al. 2014; Wang & Jiang 2014). Because of the constant IR luminosity and very small scatter in J − ${K}_{s}$, their distribution in the near-IR $(J-{K}_{s})$ versus ${K}_{s}$ CMD forms a narrow strip, which is commonly adopted to identify RCs. However, the observed $(J-{K}_{s})$ color depends only on interstellar extinction, while ${K}_{s}$ magnitudes depend on both interstellar extinction and distance, leading to a large dispersion of RC stars in the CMD. Therefore, selection of RC stars from the CMD may include some dwarf stars, specifically a fraction of ∼2.5%–5% for ${K}_{s}\lt 12.5\,\mathrm{mag}$, and up to ∼10%–40% for $13\,\mathrm{mag}\lt {K}_{s}\,\lt 14\,\mathrm{mag}$ (López-Corredoira et al. 2002; Cabrera-Laverset et al. 2007). In addition, selection of the RC strip in the CMD is not universal for all sightlines, and it is also somewhat subjective on the basis of empirical and visual inspection.

To avoid these uncertainties and contamination, we obtained a homogeneous RC sample with the current best available quality from a combination of photometric and spectroscopic data. First, the photometric quality must be $\sigma \lt 0.1$ mag in the B and V bands and $\sigma \lt 0.05$ mag in the $g,r,i,J,H,{K}_{s},W1$, and W2 bands. As the metallicity [Fe/H] would affect the intrinsic color at short wavelengths, we limit the [Fe/H] of giants ($\mathrm{log}g\leqslant 3.0$) to [Fe/H] > −0.5 dex. Next, likely RC candidates were selected based on their clumping in the ${T}_{\mathrm{eff}}$–$\mathrm{log}g$ contour map resulting from the entire APOGEE DR12 catalog, in the ranges $4550\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5050\,{\rm{K}}$ and $2.3\leqslant \mathrm{log}g\leqslant 3.0$. In addition, the selected candidates were cross-matched with the APOGEE–RC catalog. In fact, not all RC candidates are included in the APOGEE–RC catalog. Therefore, our final, homogeneous RC sample contains those RC candidates that are included in the APOGEE–RC catalog. For the diffuse l165° region, there are only 18 RC stars with the full ten-band data of high quality. Their names, locations, stellar parameters, and the $B,V$, and J-band photometric data are listed in Table 1, sorted by increasing ${T}_{\mathrm{eff}}$.

Table 1.  Eighteen RCs in the l165° Region Selected for Our Extinction Study

No.	Name	l	b	${T}_{\mathrm{eff}}$	$\mathrm{log}g$	[Fe/H]	V	B − V	V − J
		(°)	(°)	(K)			(mag)	(mag)	(mag)
1	2M05114361+4105251	166.05	0.97	4691.40	2.55	0.26	15.45	1.44	2.77
2	2M04564963+4125154	164.09	−1.05	4844.28	2.45	−0.19	14.61	1.53	2.83
3	2M05113045+4121184	165.82	1.10	4867.70	2.66	−0.04	14.60	1.47	2.41
4	2M05103457+4157164	165.23	1.31	4878.76	2.69	−0.45	13.99	1.24	2.19
5	2M05063114+4025047	166.01	−0.22	4907.07	2.75	0.10	14.45	1.41	2.62
6	2M05104317+4051399	166.13	0.68	4909.86	2.65	−0.10	14.19	1.26	2.37
7	2M05072939+4019254	166.19	−0.13	4910.46	2.66	−0.13	14.21	1.28	2.43
8	2M05014034+4045148	165.18	−0.75	4926.30	2.54	−0.28	15.11	1.30	2.60
9	2M05113657+4102440	166.08	0.93	4935.57	2.70	−0.42	15.42	1.09	2.30
10	2M05055643+4124051	165.16	0.29	4943.94	2.71	−0.15	14.36	1.38	2.51
11	2M05081159+4135433	165.25	0.74	4977.03	2.61	−0.23	14.48	1.48	2.79
12	2M05041714+4232248	164.06	0.73	4977.43	2.76	−0.33	13.95	1.21	2.37
13	2M05070978+4134192	165.16	0.57	4982.41	2.76	−0.27	14.25	1.26	2.38
14	2M05111337+4102241	166.04	0.87	4989.09	2.75	−0.03	13.77	1.32	2.41
15	2M05031839+4000169	165.96	−0.96	4992.78	2.63	−0.34	15.82	1.43	3.05
16	2M04592396+4023594	165.19	−1.30	4993.97	2.67	−0.39	15.14	1.49	2.95
17	2M05000136+4219417	163.75	−0.02	5007.93	2.93	−0.01	14.68	1.47	2.58
18	2M05102193+4121433	165.68	0.98	5027.16	2.79	−0.22	14.39	1.20	2.39

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4.1.1. Analytic ${T}_{\mathrm{eff}}$–Intrinsic Color ${C}_{{\lambda }_{1}{\lambda }_{2}}^{0}$ Relation

Wang & Jiang (2014) suggested that the intrinsic color index could be represented by the bluest observed color index under some circumstances for a given ${T}_{\mathrm{eff}}$. This means that the intrinsic color index ${C}_{{\lambda }_{1}{\lambda }_{2}}^{0}$ can be derived from their effective temperatures ${T}_{\mathrm{eff}}$ by considering the bluest star at the same ${T}_{\mathrm{eff}}$ not affected by reddening. They determined the ${T}_{\mathrm{eff}}$–near-IR intrinsic color index relation by means of a quadratic fit to the bluest stars in the ${T}_{\mathrm{eff}}$ versus observed color index diagram for APOGEE K-type giants ($3500\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 4800\,{\rm{K}}$). Xue et al. (2016) further applied this method to multiple mid-IR bands for the APOGEE G- and K-type giants ($3600\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5200\,{\rm{K}}$). In order to determine the blue envelope in the ${T}_{\mathrm{eff}}$ versus observed color index diagram, they adopted a mathematical definition for the blue edge. First, they chose the median color of the bluest 5% of stars in bins of ${\rm{\Delta }}{T}_{\mathrm{eff}}=100\,{\rm{K}}$ to represent the unreddened color. Then, they used an exponential or quadratic function to fit the bluest color. The original idea underlying this method was developed by Ducati et al. (2001).

Since RC stars in the range $4550\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5050\,{\rm{K}}$ are selected as tracers for investigating the interstellar extinction, we concentrate on the APOGEE K-type giants with $4500\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5100\,{\rm{K}}$ to determine the intrinsic colors in the optical bands, since the relation between IR color and ${T}_{\mathrm{eff}}$ has already been derived (Xue et al. 2016). The following constraints are used to obtain a K-giant sample: (1) $4500\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5100\,{\rm{K}};$ (2) $\mathrm{log}g\leqslant 3.0;$ (3) photometric uncertainties $\sigma \lt 0.1$ mag in the $B,V$ bands and $\sigma \lt 0.05$ mag in the $g,r,i,J,H,{K}_{s},W1,W2$ bands; (4) observed spectral signal-to-noise ratio $\gt 100$ and difference between the observed and synthetic model spectra ${\chi }_{\mathrm{ASPCAP}}^{2}\lt 30;$ (5) [Fe/H] $\gt \,-0.5$ dex. Note that this step employs stars from all fields, including from high Galactic latitude areas, and is unbiased as to any specific environment. A series of discrete median effective temperatures $\langle {T}_{\mathrm{eff}}\rangle$ and median observed colors $\langle {C}_{{\lambda }_{1}{\lambda }_{2}}\rangle$ in bins of ${\rm{\Delta }}{T}_{\mathrm{eff}}=50\,{\rm{K}}$ are selected from the bluest 5% of stars. A quadratic function is fitted to this $[\langle {T}_{\mathrm{eff}}\rangle$, $\langle {C}_{{\lambda }_{1}{\lambda }_{2}}\rangle ]$ series to determine the analytic expression for the ${T}_{\mathrm{eff}}$–intrinsic color relation. Figure 2 is the ${T}_{\mathrm{eff}}$ versus observed color $(B-V)$ diagram for the K-giant sample (black dots). The red asterisks are the median values for the bluest 5% of stars, and the red line is the quadratic best-fitting result. For comparison, we also plot the quadratic fit result for discrete intrinsic color data (blue asterisks) given by Johnson (1966; blue line). The two lines are highly consistent. The results in other bands are as follows:

$$\begin{eqnarray}&&{C}_{{BV}}^{0}=17.10-6.38\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}\,{\rm{K}}}\right)+0.628{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}{\rm{K}}}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{C}_{{gV}}^{0}=11.40-4.31\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}\,{\rm{K}}}\right)+0.424{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}{\rm{K}}}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{C}_{{Vr}}^{0}=12.93-5.03\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}\,{\rm{K}}}\right)+0.500{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}{\rm{K}}}\right)}^{2},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{C}_{{Vi}}^{0}=21.12-8.20\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}\,{\rm{K}}}\right)+0.818{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}{\rm{K}}}\right)}^{2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{C}_{{VJ}}^{0}=23.57-8.50\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}\,{\rm{K}}}\right)+0.823{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{3}{\rm{K}}}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where ${C}_{{BV}}^{0}$, ${C}_{{gV}}^{0}$, ${C}_{{Vr}}^{0}$, ${C}_{{Vi}}^{0}$, and ${C}_{{VJ}}^{0}$ represent the intrinsic color indices ${(B-V)}_{0},{(g-V)}_{0},{(V-r)}_{0}$, ${(V-i)}_{0}$, and ${(V-J)}_{0}$, respectively. Xue et al. (2016) already determined the multi-band intrinsic IR colors for APOGEE G- and K-type giants with $3600\,{\rm{K}}\leqslant {T}_{\mathrm{eff}}\leqslant 5200\,{\rm{K}}$. Therefore, we adopt the IR intrinsic colors ($J-H{)}_{0}$, ${(J-{K}_{s})}_{0}$, ${({K}_{s}-W1)}_{0}$, and ${({K}_{s}-W2)}_{0}$ from their work to determine ${(V-H)}_{0}$, ${(V-{K}_{s})}_{0}$, ${(V-W1)}_{0}$, and ${(V-W2)}_{0}$ (hereafter ${C}_{{VH}}^{0}$, ${C}_{{VK}}^{0}$, ${C}_{{VW}1}^{0}$, and ${C}_{{VW}2}^{0}$, respectively).

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4.1.2. Padova Stellar Models

4.1.3. Comparison

The intrinsic color indices of the 18 RC stars in the $l165^\circ$ region are included in Table 2. The top part displays the intrinsic values derived from the analytic ${T}_{\mathrm{eff}}$–${C}_{\lambda 1\lambda 2}^{0}$ relation; the bottom part displays the values derived from the Padova stellar models. The ${K}_{s}$-band absolute magnitudes from the Padova stellar models are also listed in the first column of Table 2. They range from ${K}_{s}=-0.97\,\mathrm{mag}$ to $-2.40\,\mathrm{mag}$. The average ${K}_{s}$-band absolute magnitude value, ${K}_{s}=-1.68\,\mathrm{mag}$, is slightly smaller than the typical RC value, $-1.54\,\mathrm{mag}$ to $-1.61\,\mathrm{mag}$, which results in a higher extinction value. We also compare the intrinsic color indices determined based on these two methods. Figure 3 shows the comparison. The vertical axis shows the intrinsic color indices derived from the analytic ${T}_{\mathrm{eff}}$–${C}_{\lambda 1\lambda 2}^{0}$ relation, and the horizontal axis shows the values derived from the Padova models. The intrinsic IR color indices, i.e., ${C}_{{VJ}}^{0}$, ${C}_{{VH}}^{0}$, ${C}_{{VK}}^{0}$, and ${C}_{{VW}1}^{0}$, are internally consistent. However, Figure 3 shows that the optical color indices exhibit notable differences: for ${C}_{{BV}}^{0}$ and ${C}_{{gV}}^{0}$, the analytic results are lower than the model results; for ${C}_{{Vr}}^{0}$ and ${C}_{{Vi}}^{0}$, the analytic results are higher than the model results. It is unclear whether these differences are caused by flaws in the stellar models or by the analytic method. Nevertheless, the difference is mostly smaller than 0.05 $\mathrm{mag}$ in color, comparable to the photometric uncertainties.

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Table 2.  Multi-band Intrinsic Color Indices for the 18 RCs in the l165° Region

No.	${M}_{{K}_{s}}$	${C}_{{BV}}^{0}$	${C}_{{gV}}^{0}$	${C}_{{Vr}}^{0}$	${C}_{{Vi}}^{0}$	${C}_{{VJ}}^{0}$	${C}_{{VH}}^{0}$	${C}_{{VK}}^{0}$	${C}_{{VW}1}^{0}$	${C}_{{VW}2}^{0}$
Analytic Results
1	⋯	0.992	0.505	0.337	0.644	1.81	2.35	2.45	2.52	2.43
2	⋯	0.932	0.463	0.297	0.582	1.71	2.21	2.30	2.37	2.29
3	⋯	0.925	0.459	0.293	0.576	1.70	2.19	2.28	2.35	2.27
4	⋯	0.922	0.457	0.291	0.573	1.70	2.19	2.27	2.34	2.26
5	⋯	0.916	0.452	0.287	0.568	1.68	2.17	2.25	2.32	2.24
6	⋯	0.915	0.452	0.287	0.567	1.68	2.16	2.25	2.32	2.24
7	⋯	0.915	0.451	0.287	0.567	1.68	2.16	2.25	2.32	2.24
8	⋯	0.912	0.449	0.285	0.565	1.68	2.15	2.24	2.31	2.23
9	⋯	0.910	0.448	0.284	0.563	1.67	2.15	2.23	2.30	2.22
10	⋯	0.908	0.447	0.283	0.562	1.67	2.14	2.22	2.30	2.22
11	⋯	0.903	0.443	0.281	0.559	1.66	2.12	2.20	2.28	2.20
12	⋯	0.903	0.443	0.281	0.559	1.66	2.12	2.20	2.28	2.20
13	⋯	0.903	0.443	0.281	0.559	1.66	2.12	2.20	2.27	2.20
14	⋯	0.902	0.442	0.280	0.559	1.65	2.12	2.20	2.27	2.19
15	⋯	0.901	0.442	0.280	0.559	1.65	2.12	2.19	2.27	2.19
16	⋯	0.901	0.442	0.280	0.559	1.65	2.11	2.19	2.27	2.19
17	⋯	0.900	0.441	0.280	0.558	1.65	2.11	2.19	2.26	2.18
18	⋯	0.898	0.440	0.279	0.558	1.64	2.10	2.18	2.25	2.18
Model Results
1	−2.17	1.14	0.59	0.32	0.58	1.85	2.39	2.48	2.52	2.44
2	−2.40	1.01	0.51	0.26	0.50	1.71	2.23	2.30	2.34	2.28
3	−1.83	1.01	0.52	0.27	0.50	1.70	2.20	2.28	2.32	2.25
4	−0.97	0.95	0.48	0.25	0.48	1.68	2.20	2.27	2.31	2.26
5	−1.74	1.01	0.52	0.27	0.49	1.68	2.17	2.25	2.28	2.21
6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
7	−1.81	0.98	0.5	0.26	0.48	1.67	2.16	2.24	2.27	2.21
8	−2.18	0.95	0.48	0.25	0.48	1.66	2.15	2.23	2.26	2.2
9	−1.19	0.92	0.47	0.24	0.46	1.64	2.14	2.21	2.24	2.2
10	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
11	−2.00	0.94	0.47	0.24	0.46	1.62	2.10	2.17	2.2	2.15
12	−1.34	0.92	0.46	0.24	0.46	1.62	2.11	2.18	2.21	2.16
13	−1.41	0.93	0.47	0.24	0.46	1.62	2.10	2.17	2.20	2.15
14	−1.74	0.96	0.49	0.25	0.46	1.61	2.09	2.16	2.19	2.13
15	−1.80	0.92	0.46	0.23	0.45	1.60	2.08	2.15	2.18	2.14
16	−1.65	0.91	0.46	0.23	0.45	1.60	2.09	2.15	2.18	2.14
17	−1.11	0.96	0.49	0.24	0.45	1.61	2.08	2.15	2.19	2.13
18	−1.48	0.91	0.46	0.24	0.45	1.59	2.06	2.13	2.16	2.11

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With the observed color index (the difference between two observed magnitudes) and the intrinsic color index (derived from its dependence on ${T}_{\mathrm{eff}}$ or on isochrone sets), the color excess can be calculated easily. The color excesses $E(V-{\lambda }_{x})$ (${\lambda }_{x}:B,g,r,i,J,H,{K}_{s},W1,W2$) were derived for each sample star. In principle, the color-excess ratio, e.g., $E(V-{\lambda }_{x})/E(B-V)$, can be regarded as an indicator of the extinction law.

Using the method described above, the color-excess ratios $E(V-{\lambda }_{x})/E(B-V)$ were derived for the 18 RC stars in the $l165^\circ$ region. The results are tabulated in Table 3. Figure 4 displays the variations in the color-excess ratios with waveband, where the ratios were derived from the analytic ${T}_{\mathrm{eff}}$–${C}_{\lambda 1\lambda 2}^{0}$ relation.

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Table 3.  Multi-band Color-excess Ratios for the 18 RC Stars in the l165° Region

No.	${E}_{{BV}}$	${E}_{{gV}}$/${E}_{{BV}}$	${E}_{{Vr}}$/${E}_{{BV}}$	${E}_{{Vi}}$/${E}_{{BV}}$	${E}_{{VJ}}$/${E}_{{BV}}$	${E}_{{VH}}$/${E}_{{BV}}$	${E}_{{{VK}}_{S}}$/${E}_{{BV}}$	${E}_{{VW}1}$/${E}_{{BV}}$	${E}_{{VW}2}$/${E}_{{BV}}$
Analytic Results
1	0.448	0.398	0.731	1.22	2.13	2.45	2.62	2.73	2.72
2	0.598	0.599	0.455	0.980	1.87	2.12	2.33	2.44	2.45
3	0.540	0.501	0.314	0.690	1.31	1.59	1.63	1.79	1.78
4	0.319	0.459	0.428	0.952	1.55	1.76	2.02	2.18	2.25
5	0.493	0.441	0.563	1.04	1.91	2.13	2.31	2.44	2.44
6	0.347	0.387	0.692	1.12	1.97	2.28	2.47	2.66	2.67
7	0.362	0.481	0.605	1.03	2.07	2.39	2.66	2.64	2.73
8	0.386	0.583	0.701	1.33	2.40	2.78	3.03	3.21	3.27
9	0.184	0.448	1.309	2.17	3.43	4.16	4.54	4.72	4.78
10	0.476	0.610	0.433	0.942	1.76	2.03	2.22	2.29	2.28
11	0.575	0.554	0.472	0.991	1.96	2.28	2.48	2.59	2.62
12	0.308	0.482	0.716	1.30	2.33	2.62	2.79	3.05	3.07
13	0.361	0.596	0.501	1.01	1.99	2.32	2.58	2.69	2.67
14	0.416	0.636	0.489	0.992	1.81	2.11	2.23	2.35	2.36
15	0.533	0.561	0.650	1.31	2.62	3.03	3.31	3.41	3.49
16	0.586	0.591	0.512	1.08	2.21	2.59	2.75	2.89	2.93
17	0.573	0.572	0.384	0.828	1.62	1.90	2.06	2.15	2.17
18	0.303	0.544	0.698	1.40	2.47	2.90	3.07	3.20	3.25
Model Results
1	0.300	0.309	1.15	2.04	3.05	3.54	3.81	4.09	4.03
2	0.520	0.599	0.594	1.29	2.16	2.41	2.68	2.86	2.83
3	0.455	0.460	0.423	0.99	1.56	1.88	1.93	2.19	2.15
4	0.291	0.423	0.609	1.36	1.75	1.88	2.21	2.51	2.47
5	0.399	0.374	0.739	1.48	2.37	2.63	2.85	3.12	3.10
6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
7	0.297	0.423	0.828	1.56	2.56	2.93	3.26	3.38	3.42
8	0.348	0.559	0.878	1.72	2.71	3.10	3.38	3.70	3.71
9	0.174	0.348	1.64	2.90	3.81	4.45	4.92	5.36	5.20
10	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
11	0.538	0.542	0.580	1.24	2.17	2.48	2.71	2.91	2.89
12	0.291	0.452	0.897	1.72	2.59	2.82	3.03	3.45	3.38
13	0.334	0.563	0.664	1.38	2.26	2.57	2.88	3.13	3.02
14	0.358	0.606	0.653	1.43	2.23	2.53	2.70	2.96	2.92
15	0.514	0.546	0.771	1.57	2.82	3.21	3.52	3.70	3.72
16	0.577	0.569	0.606	1.28	2.34	2.67	2.87	3.08	3.07
17	0.513	0.543	0.506	1.14	1.88	2.18	2.37	2.54	2.53
18	0.291	0.496	0.861	1.83	2.75	3.15	3.35	3.64	3.60

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The derived extinction law is fitted with the CCM89 equations. Since CCM89 suggested that the extinction law can be described by ${R}_{V}$, this yields the best-fitting ${R}_{V}$ value for the prevailing extinction law. In practice, steps of 0.001 are adopted for $0.5\leqslant {R}_{V}\leqslant 7.0$. For each ${R}_{V}$, ${A}_{{\lambda }_{x}}/{A}_{V}$ is calculated based on the CCM89 equations in all ten bands, from which the color-excess ratio $E(V-{\lambda }_{x})/E(B-V)$ is derived. The best-fitting ${R}_{V}$ is determined by assessment of the minimum chi-squared value between the color-excess ratio derived using the CCM89 equations and this work.

The ${R}_{V}$ values and best-fitting CCM89 lines are shown in Figure 4. The CCM89 extinction curve can fit all cases. Table 3 shows that the ${E}_{{BV}}$ and ${E}_{{gV}}$ of RC star No. 9 are rather small and the color-excess ratios ${E}_{V{\lambda }_{x}}$/${E}_{{BV}}$ in the $r,i,J,H,{K}_{s},W1$, and W2 bands are all apparently larger than the values for the other 17 RC stars. This is caused by the extremely small ${E}_{{BV}}$ value, which may be due to unreliable calibrations in the $B,V$, and g bands. This star was removed from further analysis.

The mean ${R}_{V}$ value of the remaining 17 RCs is 2.8, which is close to ${R}_{V}=3.1$, commonly adopted for the average extinction law toward Galactic diffuse clouds. Schlafly & Finkbeiner (2011) measured reddening values for the diffuse ISM based on a large sample of SDSS sources, and found an average extinction law consistent with ${R}_{V}=3.1$. On the other hand, as can be seen from Figure 4, there is a clear diversity of ${R}_{V}$ values among stars even in such a small region, which was carefully selected to be very diffuse along any of its sightlines. The lowest ${R}_{V}$ value is ${R}_{V}=1.72$ (No. 3); the highest value reaches ${R}_{V}=3.79$ (No. 15). This diversity significantly exceeds the intrinsic errors (see the next section), and so it is likely real. Moreover, the lowest value of ${R}_{V}=1.72$ is smaller than the previously published lowest value of ${R}_{V}=2.1$ toward the HD 210101 sightline. The reddening toward Type Ia supernovae commonly shows a low ${R}_{V}$ value <2.0 (Howell 2011, and references therein), in some cases smaller than 1.0, which means the possibility for ${R}_{V}$ being smaller than 1.7. If we search in larger sample sizes, even lower ${R}_{V}$ values may be found in the Milky Way. As for the IR bands, there is little variation in $E(J-H)/E(J-{K}_{s})$, with a mean around 0.64, which agrees with the result of Wang & Jiang (2014). The W1 and W2 bands seem to show some diversity, but with less confidence because of their low sensitivity.

The error in the color-excess ratios ${[E(V-{\lambda }_{x})/E(B-V)]}_{\mathrm{err}}$ originates from a few contributors, including the observed and intrinsic color indices. By constraining the photometric quality of our sample stars to $\sigma \lt 0.1$ mag in the $B,V$ bands and $\sigma \lt 0.05$ mag in the $g,r,i,J,H,{K}_{s},W1$, and W2 bands, the average photometric error is ∼0.06 mag in B, ∼0.04 mag in V, and ∼0.02 mag in $g,r,i,J,H,{K}_{s},W1$, and W2.⁸ Consequently, the average error in the observed color index is ∼0.1 mag for ${C}_{{BV}}$ and ∼0.06 mag for ${C}_{{gV}}$, ${C}_{{Vr}}$, ${C}_{{Vi}}$, ${C}_{{VJ}}$, ${C}_{{VH}}$, ${C}_{{{VK}}_{S}}$, ${C}_{{K}_{S}W1}$, and ${C}_{{K}_{S}W2}$. The average error in the APOGEE ${T}_{\mathrm{eff}}$ is $\sim 50{\rm{K}}$. Using the ${T}_{\mathrm{eff}}$–intrinsic color index relation to derive the intrinsic colors from Equations (1)–(5), the ${T}_{\mathrm{eff}}$ error causes an average error of ∼0.006 mag for ${C}_{{BV}}^{0}$, ∼0.004 mag for ${C}_{{gV}}^{0}$, ∼0.003 mag for ${C}_{{Vr}}^{0}$, ∼0.004 mag for ${C}_{{Vi}}^{0}$, ∼0.013 mag for ${C}_{{VJ}}^{0}$, ∼0.022 mag for ${C}_{{VH}}^{0}$, ∼0.023 mag for ${C}_{{VK}}^{0}$, ∼0.023 mag for ${C}_{{VW}1}^{0}$, and ∼0.025 mag for ${C}_{{VW}2}^{0}$. Combining the photometric and intrinsic color errors, the uncertainties in the color excesses are ${({E}_{{BV}})}_{\mathrm{err}}\sim 0.11$ mag, ${({E}_{{gV}})}_{\mathrm{err}}\sim 0.064$ mag, ${({E}_{{Vr}})}_{\mathrm{err}}\sim 0.063$ mag, ${({E}_{{Vi}})}_{\mathrm{err}}\sim 0.064$ mag, ${({E}_{{VJ}})}_{\mathrm{err}}\sim 0.073$ mag, ${({E}_{{VH}})}_{\mathrm{err}}\sim 0.082$ mag, ${({E}_{{{VK}}_{S}})}_{\mathrm{err}}\sim 0.083$ mag, ${({E}_{{VW}1})}_{\mathrm{err}}\sim 0.083$ mag, and ${({E}_{{VW}2})}_{\mathrm{err}}\sim 0.085$ mag.

Having thus determined the errors in the color excesses, we use Monte Carlo simulations to calculate the statistical uncertainties in the color-excess ratios ${[E(V-{\lambda }_{x})/E(B-V)]}_{\mathrm{err}}$ and in the total-to-selective ratio ${({R}_{V})}_{\mathrm{err}}$. Taking the error in the color excess into account, we performed 20,000 simulations. A Gaussian function was used to fit the distributions of $E(V-{\lambda }_{x})/E(B-V)$ and ${R}_{V}$. The widths of the Gaussian distributions were considered to represent the errors in the color-excess ratios ${[E(V-{\lambda }_{x})/E(B-V)]}_{\mathrm{err}}$ and the error in the reddening ${({R}_{V})}_{\mathrm{err}}$.

The distributions of $E(V-J)/E(B-V)$ and ${R}_{V}$ resulting from the Monte Carlo resampling for RC star No. 1 are shown in Figure 5 as an example. The distributions of $E(V-J)/E(B-V)$ and ${R}_{V}$ are both well-fitted by a Gaussian function, with a peak at 2.13 and a width of 0.07, and with a peak at 3.19 and a width of 0.10, respectively. In comparison, our best-fitting results $E(V-J)/E(B-V)\,=2.13$ and ${R}_{V}=3.20$ are highly consistent with the Monte Carlo simulation results, so the latter method reconfirms our fits. The results of our error analysis for the 18 RC stars are presented in Table 4. Except for RC star No. 9, the average error in ${R}_{V}$ is about 13.4%. In comparison, the range of the derived ${R}_{V}$ values, from 1.7 to 3.8, is much larger than the typical error, and thus the variation in ${R}_{V}$ is real and cannot be attributed merely to errors.

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Table 4.  Results of Our Error Analysis for the 18 RC Stars in the l165° Region

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
${R}_{V}$	3.19	2.47	1.72	2.18	2.68	2.97	2.98	3.45	5.68	2.41	2.77	3.45	2.90	2.52	3.79	3.11	2.25	3.75
${({R}_{V})}_{\mathrm{err}}$	0.10	0.20	0.16	0.29	0.23	0.54	0.34	0.56	1.58	0.28	0.20	0.88	0.44	0.38	0.33	0.36	0.24	0.95
Peak(${R}_{V}$)	3.19	2.47	1.74	2.17	2.68	2.90	2.95	3.41	5.10	2.31	2.68	3.22	2.71	2.37	3.73	2.99	2.15	3.36

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Variations in ${R}_{V}$ are usually related to the interstellar environment, whether diffuse or dense. Although the sightlines in the $l165^\circ$ region are essentially diffuse, the extinction law still exhibits significant variations. A possible reason may be that the degree of diffusivity varies along different sightlines. In order to quantify the interstellar environment, we derived the distances to the sample stars so that the specific extinction per kiloparsec (kpc) can be measured.

RC stars are good standard candles for estimating astronomical distances, since their absolute luminosities are fairly independent of stellar composition or age. In particular, the near-IR I- and ${K}_{s}$-band luminosities have been widely used to retrieve RC distances. Alves (2000) was the first to consider using the RC ${K}_{s}$-band magnitude as a distance indicator. He found ${M}_{{K}_{s}}=-1.61\pm 0.03\,\mathrm{mag}$, with a weak dependence on metallicity. Groenewegen (2008) reinvestigated the absolute magnitude of the RC stars based on revised Hipparcos parallaxes. He obtained ${M}_{{K}_{s}}=-1.54\pm 0.04\,\mathrm{mag}$, which is slightly fainter than previously published values. In addition, Nishiyama et al. (2006) used ${M}_{{K}_{s}}=-1.59\,\mathrm{mag}$ based on Bonatto et al. (2004) to measure the distance to the Galactic center. This distance thus measured is in agreement with results based on other methods (e.g., de Grijs & Bono 2016). Therefore, we adopt ${M}_{{K}_{s}}=-1.59\,\mathrm{mag}$ to estimate the distances to our RC stars.

In order to calculate the distance d using $\mathrm{log}d=1+0.2({m}_{{K}_{s}}-{M}_{{K}_{s}}-{A}_{{K}_{s}})$, we need to know ${A}_{{K}_{s}}$, the ${K}_{s}$-band extinction. As we have derived a series of color excesses $E(V-{\lambda }_{x})$ and the relative extinction ${A}_{{\lambda }_{x}}$/${A}_{V}$ using the CCM89 equations, a series of corresponding ${A}_{V}$ values can be obtained: ${A}_{V}=E(V-{\lambda }_{x})/(1-{A}_{{\lambda }_{x}}/{A}_{V})$. The median $\langle {A}_{V}\rangle$ of this series is taken as the absolute V-band extinction. Then, ${A}_{{K}_{S}}={A}_{{K}_{S}}/{A}_{V}\times \langle {A}_{V}\rangle$ is obtained. The derived distances range from 2.67 kpc to 4.49 kpc. The average ${A}_{V}$ per kiloparsec is 0.37 mag kpc⁻¹, based on taking all 17 RC stars in this region. Considering that the average rate of interstellar extinction in the V band is usually taken as 0.7–1.0 mag kpc⁻¹, simply based on observations in the solar neighborhood (Gottlieb & Upson 1969; Milne & Aller 1980), a gradient of 0.37 mag kpc⁻¹ means that we are dealing with a really diffuse medium. The highest specific extinction in our region of interest is approximately 0.56 mag kpc⁻¹, still significantly below the average rate.

Since the color excess $E(B-V)$ represents in general the density of dust, larger ${R}_{V}$ is expected at larger $E(B-V)$, as dense clouds usually show high ${R}_{V}$ values. We investigate the variation in ${R}_{V}$ with $E(B-V)$ shown in Figure 6(a). The Pearson correlation coefficient is −0.34, indicating no correlation between ${R}_{V}$ and $E(B-V)$ in the $E(B-V)$ range from 0.2 to 0.6 mag. Schlafly et al. (2016) measured optical–IR extinction curve spatial variations to tens of thousands of APOGEE stars. They also found that the variation in ${R}_{V}$ is uncorrelated with dust column density up to $E(B-V)\approx 2$ mag. This non-correlation seems to contradict with normal expectation. However, dense molecular clouds exhibit high dust extinction with ${A}_{V}\gt 5\mbox{--}10$ mag (Snow & McCall 2006), corresponding to $E(B-V)\gt 1.6$ mag (for ${R}_{V}=3.1$). Therefore, these RC sightlines are still diffuse regions. Dense regions are needed to investigate a possible correlation between ${R}_{V}$ and $E(B-V)$. On the other hand, a large $E(B-V)$ may result from a pile of diffuse clouds. The specific extinction per distance would be a better measure of the ISM environment. Figure 6(b) displays the variation in ${R}_{V}$ with specific visual extinction per kiloparsec, ${A}_{V}/d$. The error bars were determined using error-propagation theory and based on Monte Carlo simulations. Since the Pearson correlation coefficient between ${R}_{V}$ and ${A}_{V}/d$ is 0.14, There is no apparent trend for ${R}_{V}$ with ${A}_{V}/d$. If we were to adopt different ${M}_{{K}_{s}}$ values, ${M}_{{K}_{s}}=-1.54\,\mathrm{mag}$ or $-1.61\,\mathrm{mag}$, the distance would change by about 3%, but this change is systematic and would therefore not change our conclusion, i.e., that the variation in ${R}_{V}$ is independent of both $E(B-V)$ and ${A}_{V}/d$ within this small range of $E(B-V)(\lt 0.6)$ mag.

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The extinction toward the l165° region is relatively small, with ${A}_{V}$ ranging from about 0.7 to 2.0 mag, i.e., $E(B-V)$ ranges from about 0.2 to 0.6 mag. For such a low extinction, a small variation in color excess would lead to a significant difference in the color-excess ratio $E(V-{\lambda }_{x})/E(B-V)$. Metallicity is a secondary parameter affecting the intrinsic color index, in addition to the effective temperature. In our derivation of the intrinsic colors (Section 4.1.1), metallicity was not taken into account. Therefore, here we discuss the effect of metallicity on the intrinsic colors of RC stars. For each of the 18 target RC stars, stars were selected that had similar stellar parameters (including metallicities) to the target stars and were located "close to" the analytic best-fitting ${T}_{\mathrm{eff}}$–${C}_{{BV}}^{0}$ intrinsic color line. Instead of reading the intrinsic color index from the best-fitting line, the average values of the observed color indices of these similar stars were taken as the intrinsic color indices for each target RC star. The "similar" stellar parameters are defined as those parameters that are found within the ranges ${T}_{\mathrm{eff}}(\mathrm{target})\pm 100\,{\rm{K}}$, $\mathrm{log}g(\mathrm{target})\pm 0.2$, and $[\mathrm{Fe}/{\rm{H}}](\mathrm{target})\pm 0.4$ dex.⁹ The "close" distance is defined in the sense that the star deviates from the best-fitting ${T}_{\mathrm{eff}}$–${C}_{{BV}}^{0}$ line by less than the uncertainty of 0.06 $\mathrm{mag}$ in the observed color index ${({C}_{{BV}})}_{\mathrm{err}}$.¹⁰ These stars are considered reddening-free, since the deviation from the line of intrinsic color is not larger than the photometric uncertainty. Therefore, the observed color indices of these stars can represent the intrinsic colors. This method was used by Yuan et al. (2013) to derive the empirical extinction law.

Figure 7(a) exhibits the results for the similar stars selected for the 18 target RC stars in the effective temperature ${T}_{\mathrm{eff}}$ versus observed color index (B − V) diagram. The yellow asterisks represent the 18 target RC stars in the $l165^\circ$ region, the blue solid line denotes the ${T}_{\mathrm{eff}}$–${C}_{{BV}}^{0}$ relation, with the cyan dashed line representing the 1σ envelope. The red asterisks are the average observed color index values $[{C}_{{BV}}]$ of the similar stars, and these average values turn out to be the intrinsic color indices of the 18 RC stars. This method is affected by a systematic bias, as shown in Figure 7(a): all reference stars lie below the best-fitting line, which means that they suffer from some extinction. Consequently, the derived $(B-V{)}_{0}^{\mathrm{metals}}$ is redder than the ${(B-V)}_{0}$ from the ${T}_{\mathrm{eff}}$–${C}_{{BV}}^{0}$ relation, as shown in Figure 7(b). Except for ${C}_{{gV}}^{0}$, this method results in redder intrinsic color indices ${(V-{\lambda }_{x})}_{0}$ in the $r,i,J,H,{K}_{s},W1$, and W2 bands. The corresponding color excesses $E(B-V)$ and $E(V-{\lambda }_{x})$ become smaller, and $E(g-V)$ becomes larger. We also investigated the effect on the ${R}_{V}$ value. The slightly different intrinsic color indices cause the ${R}_{V}$ value to become smaller, as shown in Figure 7(c): $-0.42\,\lt {({R}_{V})}_{\mathrm{metals}}-{({R}_{V})}_{0}\lt -0.14$, while the diversity is still there.

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