A Method for Improving Galactic Cepheid Reddenings and Distances

In recent decades, the calibration of the Cepheid extragalactic distance scale has relied, in large measure, on observations of Cepheids in nearby galaxies (e.g., the LMC, M31, and NGC 4258; noting, for example, Freedman et al. 2001, 2012; Riess et al. 2016, and references therein). The Milky Way has historically been bypassed for the simple reason that very few long-period Cepheids are sufficiently close to measure their distances accurately, and none have been found in open clusters, near or far; therefore measurement of the slope of the period–luminosity (PL) relation (or Leavitt law) has been more robustly carried out using external galaxies where large, statistically complete samples of Cepheids at all periods are available. Progress on measurement of parallaxes for MW Cepheids has come from HST with the measurement of 10 Cepheids by Benedict et al. (2007) and two additional stars (both of which are long-period Cepheids) measured by Riess et al. (2014) and Casertano et al. (2016). Still, the sample of Benedict contains only a single Cepheid, l Car, with a period longer than 10 days. In contrast, most of the extragalactic searches have targeted Cepheids with periods mostly greater than 10 days.

Measurement of parallaxes, however, is only the first step in the PL calibration. Not only parallaxes but also wavelength-dependent corrections for interstellar extinction are required to obtain absolute magnitudes. Cepheids are young and rare, F and G-type supergiants. As such, they are sparsely distributed throughout the galaxy, and their apparent magnitudes are generally reddened and attenuated by intervening interstellar dust of varying amounts. Each Cepheid has a unique line of sight and reddening. These effects must be corrected for at each observed wavelength before intrinsic relations can be derived: both distances and extinctions must individually be derived independently for each star.

Many methods have been used for determining the total line-of-sight reddenings to individual Cepheids both in the Milky Way and in the nearest galaxies. In general, the quoted uncertainties in the reddenings range from ±0.03 to 0.05 mag in E(B − V). For our purposes in this study, we have adopted the Fouqué et al. (2007) sample of Galactic Cepheids, and we will use their absolute magnitudes and mean reddenings simply to illustrate our methodology designed to refine and produce high-precision reddenings. Making use of a reddening-independent magnitude to establish the relative positioning of individual stars within the PL relation, we use multiwavelength data to constrain the reddening estimates to individual Cepheids and to refine their distances. Initially, we began this investigation with the goal of improving reddenings to Cepheids. It later became clear that not only the reddenings could be improved, but the individual distance moduli themselves could be self-consistently updated as well.

We have used a published compilation of absolute magnitudes measured in seven bands, based on distances and reddenings independently determined for 59 Galactic Cepheids (Fouqué et al. 2007), with overtone and suspected overtone pulsators omitted. Where available, HST parallaxes have been adopted for these stars, followed by Infrared Surface Brightness (IRSB) determinations and then Interferometric Baade-Wesselink applications and, finally, revised Hipparcos parallaxes. A comprehensive discussion of these methods and all relevant references are given in Fouqué et al. Other data sets could have been considered, such as the more recent IRSB distances found in Storm et al. (2011) or those published by Groenewegen (2013); however, the purpose of this paper is not to give a final answer, but rather to introduce and test a technique in anticipation of the release of astrometric high-precision parallaxes from the Gaia mission. For this same reason there is no critiquing of the methods or implicit calibrations used in Fouqué et al. (2007) because our goal is to introduce and explain a new methodology, applied to an exemplary data set.

We start with a quick review of the basic attributes of the various PL relations relevant to the corrections detailed in the main body of this paper.

Figure 1 shows an expanded view of the B-band PL relation for the Milky Way Cepheids contained in Table 7 of Fouqué et al. (2007) and repeated for completeness and convenience of comparison in Table 1 of the present paper. We identify each of the stars for which there are HST parallaxes, using larger symbols. It can be seen immediately that this high-precision subset of the total sample does fairly outline the total width of the PL relation ($X\,{Sgr}$ being on the 2σ, bright (blue) edge of the instability strip and $l\,{Car}$ falling on the faint (red) edge). These stars also span almost the entire range of periods encompassed by the parent sample, falling short by about 0.1 in log P at the bright end. These same variables will be tracked into the revised B-band PL relation (Figure 15), to be presented later in this paper.

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Table 1.  Properties of 59 Galactic Cepheids

Cepheid	log(P)	E(B − V)	${\mu }_{o}$	π ($\sigma )$	MB	MV	MR	MI	MJ	MH	MK
	days	mag	mag	mas	mag	mag	mag	mag	mag	mag	mag
RT Aur	0.571489	0.059	8.099	2.400 (0.19)	−2.31	−2.84	−3.20	−3.40	−3.94	−4.15	−4.24
		0.202	8.169	2.324 (0.40)	−1.64	−2.31	−2.79	−3.11	−3.74	−3.99	−4.12
QZ Nor	0.578244	0.253	10.512	0.790 (0.10)	−1.83	−2.47	−2.85	−3.15	−3.67	−3.91	−4.01
		0.184	10.258	0.888 (0.98)	−2.37	−2.94	−3.27	−3.51	−3.99	−4.21	−4.29
SU Cyg	0.584952	0.098	9.622	1.190 (0.05)	−2.61	−3.08	−3.38	−3.63	−4.08	−4.28	−4.36
		0.163	9.625	1.188 (0.03)	−2.34	−2.87	−3.22	−3.53	−4.02	−4.24	−4.33
Y Lac	0.635863	0.207	11.783	0.440 (0.02)	−2.79	−3.31	−3.65	−3.90	−4.33	−4.58	−4.66
		0.263	11.979	0.402 (1.90)	−2.36	−2.94	−3.32	−3.62	−4.08	−4.35	−4.44
T Vul	0.646934	0.064	8.606	1.900 (0.23)	−2.48	−3.06	−3.41	−3.66	−4.12	−4.36	−4.45
		0.170	8.654	1.858 (0.18)	−1.99	−2.67	−3.11	−3.45	−3.98	−4.25	−4.36
FF Aql	0.650397	0.196	7.756	2.810 (0.18)	−2.46	−3.02	−3.35	−3.64	−4.09	−4.29	−4.37
		0.183	7.606	3.011 (1.12)	−2.66	−3.21	−3.53	−3.81	−4.25	−4.45	−4.52
T Vel	0.666501	0.289	10.022	0.990 (0.01)	−2.28	−2.93	−3.32	−3.64	−4.13	−4.41	−4.51
		0.189	9.922	1.037 (4.67)	−2.80	−3.35	−3.66	−3.89	−4.32	−4.57	−4.65
VZ Cyg	0.687034	0.266	11.338	0.540 (0.02)	−2.62	−3.23	−3.62	−3.88	−4.36	−4.60	−4.70
		0.291	11.476	0.507 (1.66)	−2.38	−3.01	−3.42	−3.70	−4.20	−4.45	−4.55
V350 Sgr	0.712165	0.299	9.853	1.070 (0.06)	−2.74	−3.34	−3.73	−4.02	−4.51	−4.78	−4.85
		0.279	9.984	1.007 (1.04)	−2.69	−3.27	−3.65	−3.92	−4.40	−4.66	−4.73
BG Lac	0.726883	0.300	11.146	0.590 (0.03)	−2.56	−3.22	−3.62	−3.91	−4.36	−4.64	−4.73
		0.282	11.196	0.577 (0.45)	−2.59	−3.23	−3.61	−3.89	−4.33	−4.60	−4.69
δ  Cep	0.729678	0.075	7.183	3.660 (0.15)	−2.88	−3.47	−3.87	−4.11	−4.55	−4.82	−4.91
		0.148	7.433	3.262 (2.65)	−2.32	−2.99	−3.44	−3.75	−4.23	−4.53	−4.63
CV Mon	0.730685	0.722	11.003	0.630 (0.05)	−2.46	−3.04	−3.51	−3.78	−4.35	−4.62	−4.72
		0.597	10.830	0.682 (1.04)	−3.15	−3.61	−3.98	−4.14	−4.64	−4.87	−4.94
V Cen	0.739882	0.292	8.913	1.650 (0.12)	−2.45	−3.03	−3.40	−3.68	−4.16	−4.43	−4.52
		0.307	8.578	1.925 (2.29)	−2.72	−3.32	−3.70	−3.99	−4.48	−4.76	−4.85
Y Sgr	0.761428	0.191	8.358	2.130 (0.29)	−2.57	−3.23	−3.63	−3.95	−4.45	−4.75	−4.83
		0.103	8.248	2.241 (0.38)	−3.05	−3.62	−3.95	−4.20	−4.64	−4.91	−4.97
CS Vel	0.771201	0.737	12.407	0.330 (0.03)	−2.48	−3.09	−3.54	−3.79	−4.33	−4.59	−4.69
		0.747	12.223	0.359 (0.97)	−2.62	−3.24	−3.70	−3.96	−4.50	−4.77	−4.87
BB Sgr	0.821971	0.281	9.481	1.270 (0.05)	−2.74	−3.45	−3.88	−4.19	−4.70	−4.99	−5.08
		0.226	9.521	1.247 (0.46)	−2.93	−3.58	−3.97	−4.23	−4.71	−4.98	−5.06
V Car	0.825860	0.169	10.089	0.960 (0.14)	−2.58	−3.29	−3.70	−4.01	−4.50	−4.79	−4.89
		0.157	9.945	1.026 (0.47)	−2.77	−3.47	−3.87	−4.17	−4.65	−4.94	−5.04
U Sgr	0.828997	0.403	8.760	1.770 (0.06)	−2.67	−3.36	−3.79	−4.10	−4.61	−4.89	−4.97
		0.369	8.685	1.832 (1.04)	−2.89	−3.55	−3.95	−4.23	−4.72	−4.99	−5.06
V496 Aql	0.832958	0.397	9.853	1.070 (0.11)	−2.63	−3.39	−3.85	−4.16	−4.67	−4.95	−5.02
		0.395	9.985	1.007 (0.57)	−2.50	−3.26	−3.72	−4.03	−4.54	−4.82	−4.89
X Sgr	0.845907	0.237	7.614	3.000 (0.18)	−3.31	−3.82	−4.17	−4.43	−4.87	−5.10	−5.17
		0.281	7.541	3.103 (0.57)	−3.20	−3.75	−4.14	−4.44	−4.90	−5.15	−5.23
U Aql	0.846591	0.360	9.178	1.460 (0.06)	−2.97	−3.65	−4.07	−4.35	−4.87	−5.12	−5.21
		0.416	9.342	1.354 (1.77)	−2.57	−3.31	−3.77	−4.10	−4.66	−4.92	−5.03
η  Aql	0.855930	0.130	6.910	4.150 (0.24)	−2.77	−3.43	−3.80	−4.14	−4.63	−4.91	−5.00
		0.070	6.612	4.760 (2.54)	−3.32	−3.92	−4.24	−4.53	−4.98	−5.24	−5.32
W Sgr	0.880522	0.108	8.210	2.280 (0.20)	−3.25	−3.89	−4.31	−4.58	−5.10	−5.40	−5.47
		0.177	8.446	2.045 (1.17)	−2.73	−3.44	−3.91	−4.24	−4.80	−5.12	−5.21
U Vul	0.902584	0.603	9.178	1.460 (0.06)	−3.32	−3.99	−4.47	−4.75	−5.17	−5.41	−5.46
		0.593	9.558	1.226 (3.91)	−2.98	−3.64	−4.11	−4.39	−4.80	−5.04	−5.08
S Sge	0.923352	0.100	9.149	1.480 (0.04)	−3.16	−3.86	−4.27	−4.57	−5.07	−5.35	−5.44
		0.141	9.237	1.421 (1.47)	−2.90	−3.64	−4.08	−4.42	−4.94	−5.24	−5.34
GH Lup	0.967448	0.335	10.253	0.890 (0.15)	−2.83	−3.71	−4.21	−4.56	−5.14	−5.47	−5.59
		0.275	10.417	0.825 (0.43)	−2.92	−3.74	−4.19	−4.49	−5.03	−5.34	−5.45
S Mus	0.984980	0.212	9.568	1.220 (0.08)	−3.50	−4.13	−4.50	−4.80	−5.27	−5.55	−5.66
		0.244	9.502	1.258 (0.47)	−3.43	−4.10	−4.49	−4.82	−5.31	−5.60	−5.71
S Nor	0.989194	0.178	9.873	1.060 (0.05)	−3.25	−4.02	−4.46	−4.80	−5.36	−5.68	−5.79
		0.137	9.992	1.003 (1.13)	−3.30	−4.03	−4.44	−4.74	−5.28	−5.59	−5.69
β  Dor	0.993131	0.052	7.515	3.140 (0.16)	−3.18	−3.93	−4.34	−4.68	−5.17	−5.49	−5.59
		0.042	7.460	3.221 (0.50)	−3.28	−4.02	−4.42	−4.75	−5.23	−5.55	−5.65
ζ  Gem	1.006507	0.014	7.780	2.780 (0.18)	−3.13	−3.93	−4.40	−4.70	−5.31	−5.60	−5.71
		0.075	7.919	2.608 (0.96)	−2.73	−3.60	−4.11	−4.47	−5.12	−5.42	−5.55
Z Lac	1.036854	0.370	11.379	0.530 (0.01)	−3.41	−4.14	−4.60	−4.89	−5.42	−5.73	−5.83
		0.417	11.482	0.505 (2.46)	−3.11	−3.89	−4.38	−4.72	−5.27	−5.60	−5.71
XX Cen	1.039548	0.266	10.902	0.660 (0.04)	−3.22	−3.93	−4.36	−4.68	−5.20	−5.50	−5.60
		0.265	10.666	0.736 (1.89)	−3.46	−4.17	−4.60	−4.92	−5.44	−5.74	−5.84
V340 Nor	1.052579	0.321	11.259	0.560 (0.11)	−3.10	−3.94	−4.42	−4.73	−5.36	−5.70	−5.82
		0.372	12.308	0.548 (0.11)	−2.83	−3.73	−4.25	−4.60	−5.26	−5.62	−5.75
UU Mus	1.065819	0.399	12.407	0.330 (0.04)	−3.14	−3.89	−4.36	−4.68	−5.29	−5.60	−5.72
		0.391	12.180	0.366 (0.91)	−3.40	−4.14	−4.60	−4.92	−5.52	−5.83	−5.95
U Nor	1.101875	0.862	10.458	0.810 (0.04)	−3.26	−4.01	−4.52	−4.80	−5.40	−5.70	−5.80
		0.929	10.356	0.849 (0.97)	−3.08	−3.89	−4.46	−4.80	−5.44	−5.76	−5.88
SU Cru	1.108800	0.942	10.038	0.620 (0.08)	−3.47	−4.30	−4.88	−5.23	−5.99	−6.53	−6.66
		0.779	11.321	0.544 (0.95)	−3.87	−4.54	−4.99	−5.20	−5.85	−6.35	−6.44
BN Pup	1.135867	0.416	12.925	0.260 (0.02)	−3.64	−4.42	−4.89	−5.23	−5.79	−6.11	−6.22
		0.395	13.004	0.251 (0.46)	−3.65	−4.41	−4.86	−5.18	−5.73	−6.04	−6.15
TT Aql	1.138459	0.438	10.022	0.990 (0.03)	−3.43	−4.30	−4.80	−5.15	−5.72	−6.05	−6.15
		0.397	10.142	0.937 (1.77)	−3.48	−4.31	−4.78	−5.09	−5.64	−5.95	−6.04
LS Pup	1.150646	0.461	13.389	0.210 (0.02)	−3.63	−4.40	−4.87	−5.20	−5.76	−6.09	−6.19
		0.456	13.387	0.210 (0.01)	−3.65	−4.42	−4.88	−5.21	−5.77	−6.10	−6.19
VW Cen	1.177138	0.428	12.764	0.280 (0.01)	−2.94	−3.87	−4.43	−4.83	−5.55	−5.95	−6.10
		0.257	12.460	0.322 (4.21)	−3.96	−4.72	−5.14	−5.39	−6.01	−6.36	−6.47
X Cyg	1.214482	0.228	10.328	0.860 (0.02)	−3.76	−4.66	−5.17	−5.53	−6.14	−6.48	−6.60
		0.208	10.614	0.754 (5.31)	−3.56	−4.44	−4.93	−5.27	−5.87	−6.21	−6.32
CD Cyg	1.232334	0.493	12.045	0.390 (0.01)	−3.87	−4.68	−5.19	−5.50	−6.13	−6.45	−6.55
		0.534	12.189	0.365 (2.50)	−3.56	−4.41	−4.95	−5.29	−5.95	−6.28	−6.39
Y Oph	1.233609	0.645	8.712	1.810 (0.13)	−3.89	−4.62	−5.16	−5.43	−5.96	−6.21	−6.28
		0.686	8.742	1.785 (0.19)	−3.69	−4.46	−5.03	−5.34	−5.89	−6.15	−6.23
SZ Aql	1.234029	0.537	11.686	0.460 (0.01)	−3.90	−4.79	−5.33	−5.68	−6.30	−6.63	−6.74
		0.523	12.095	0.381 (7.90)	−3.55	−4.42	−4.95	−5.29	−5.90	−6.23	−6.34
VY Car	1.276818	0.237	11.298	0.550 (0.03)	−3.69	−4.61	−5.13	−5.51	−6.13	−6.49	−6.61
		0.206	11.358	0.535 (0.50)	−3.76	−4.65	−5.14	−5.50	−6.10	−6.45	−6.56
RU Sct	1.294480	0.921	11.183	0.580 (0.05)	−3.94	−4.69	−5.26	−5.51	−6.09	−6.38	−6.46
		1.022	11.192	0.578 (0.05)	−3.51	−4.36	−5.01	−5.35	−5.99	−6.31	−6.41
RY Sco	1.307927	0.718	10.353	0.850 (0.04)	−3.91	−4.65	−5.19	−5.48	−6.09	−6.38	−6.49
		0.761	10.165	0.927 (1.92)	−3.92	−4.70	−5.27	−5.60	−6.24	−6.54	−6.66
RZ Vel	1.309564	0.299	10.775	0.700 (0.03)	−3.83	−4.66	−5.14	−5.51	−6.13	−6.47	−6.59
		0.267	10.541	0.780 (2.65)	−4.20	−5.00	−5.45	−5.79	−6.39	−6.72	−6.84
WZ Sgr	1.339443	0.431	11.221	0.570 (0.02)	−3.63	−4.60	−5.16	−5.55	−6.30	−6.70	−6.84
		0.343	11.153	0.588 (0.91)	−4.07	−4.95	−5.44	−5.75	−6.45	−6.82	−6.94
WZ Car	1.361977	0.370	12.688	0.290 (0.01)	−3.83	−4.61	−5.09	−5.43	−6.08	−6.42	−6.54
		0.389	12.172	0.368 (7.78)	−4.27	−5.07	−5.56	−5.92	−6.58	−6.92	−7.05
SW Vel	1.370016	0.344	11.884	0.420 (0.01)	−4.07	−4.88	−5.37	−5.73	−6.34	−6.68	−6.80
		0.313	11.640	0.470 (4.99)	−4.44	−5.22	−5.69	−6.02	−6.61	−6.94	−7.06
T Mon	1.431915	0.181	10.713	0.720 (0.03)	−4.18	−5.17	−5.68	−6.08	−6.75	−7.14	−7.27
		0.151	10.824	0.684 (1.20)	−4.19	−5.15	−5.64	−6.01	−6.67	−7.05	−7.17
RY Vel	1.449158	0.547	11.734	0.450 (0.03)	−4.32	−5.14	−5.65	−5.99	−6.62	−6.92	−7.04
		0.559	11.535	0.493 (1.44)	−4.47	−5.30	−5.82	−6.17	−6.81	−7.11	−7.23
AQ Pup	1.478624	0.518	12.407	0.330 (0.02)	−4.56	−5.39	−5.93	−6.28	−6.85	−7.20	−7.31
		0.483	12.406	0.330 (0.01)	−4.71	−5.50	−6.01	−6.33	−6.88	−7.22	−7.32
KN Cen	1.531857	0.797	12.843	0.270 (0.02)	−4.79	−5.59	−6.14	−6.43	−7.15	−7.53	−7.67
		0.914	12.917	0.261 (0.45)	−4.23	−5.14	−5.79	−6.18	−6.97	−7.38	−7.55
l Car	1.550816	0.147	8.484	2.010 (0.20)	−4.11	−5.22	−5.80	−6.21	−6.91	−7.33	−7.46
		0.170	8.582	1.921 (0.44)	−3.91	−5.05	−5.65	−6.08	−6.79	−7.22	−7.35
U Car	1.588970	0.265	10.870	0.670 (0.03)	−4.51	−5.43	−5.94	−6.32	−6.97	−7.32	−7.45
		0.285	10.596	0.760 (3.00)	−4.70	−5.64	−6.17	−6.56	−7.23	−7.58	−7.72
RS Pup	1.617420	0.457	11.298	0.550 (0.03)	−4.78	−5.76	−6.32	−6.72	−7.37	−7.74	−7.87
		0.412	11.405	0.524 (0.88)	−4.86	−5.80	−6.32	−6.68	−7.30	−7.66	−7.78
SV Vul	1.652569	0.461	11.686	0.460 (0.01)	−4.97	−5.97	−6.53	−6.90	−7.52	−7.86	−7.96
		0.536	11.981	0.402 (5.84)	−4.36	−5.44	−6.06	−6.49	−7.16	−7.52	−7.64

A machine-readable version of the table is available.

Download table as:  DataTypeset images: 1 2

In Figure 2 we show the PL relations for the remaining six bands. What is immediately striking is the fact that the widths of these relations (as a function of increasing wavelength) are larger than those in other independent studies—for instance (as in Figure 3 of Freedman & Madore 2010), for the LMC. Indeed, this visual impression is confirmed and quantified by Table 8 in Fouqué et al. (2007), where it is seen that the dispersion in the B band is only ±0.20 mag, while at the other end of the wavelength range at J & K the dispersion has only dropped by 50% to ±0.15 mag. This is in contrast to the LMC values that start higher (±0.27 mag at B) and decrease much more (by nearly a factor of 2.5) to the red (to ±0.11 mag at K). Indeed the contrast is even more dramatic in the reddening-free Wesenheit functions (as introduced and defined by Madore 1976, 1982) where, again drawing the quantitative comparisons from the numbers quoted in their Table 8, the dispersions in W for the Galactic sample are around ±0.17 mag, while for the LMC they are around ±0.07 mag. This is despite the fact that for the LMC sample there is additional back-to-front scatter, which if taken into account would widen the disparity between the two samples even further.

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The Wesenheit Function, $W=V-{R}_{V}\times (B-V)$, was first introduced in this form by Madore (1976, 1982),³ implicitly accounting for reddening and extinction when applied to the PL relations for Classical Cepheids. By combining the apparent magnitude $V={V}_{o}+{A}_{v}={V}_{o}+{R}_{V}\times E(B-V)$ with the apparent color ${(B-V)=(B-V)}_{o}+E(B-V)$ scaled up by the ratio of total to selective absorption, ${R}_{V}\,={A}_{V}/E(B-V)$ mathematically assures that the numerical value of W is independent of reddening/extinction, where it can be shown that

$$\begin{eqnarray}&&W=V-{R}_{V}\times {(B-V)={W}_{o}={V}_{o}-{R}_{V}\times (B-V)}_{o}.\end{eqnarray} \tag{ 1 }$$

We begin with a full parameterization of the underlying period–luminosity–color (PLC) relation,

$$\begin{eqnarray}&&{M}_{{V}_{o}}=\alpha \,\mathrm{log}P+\beta \,{(B-V)}_{o}+\gamma ,\end{eqnarray} \tag{ 2 }$$

and the definition of the Wesenheit function,

$$\begin{eqnarray}&&W={M}_{V}-{R}_{v}\times (B-V).\end{eqnarray} \tag{ 3 }$$

From these functions, it follows that

$$\begin{eqnarray}&&W={M}_{{V}_{o}}+{A}_{v}-{R}_{v}\times [{(B-V)}_{o}+E(B-V)]\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&=\,{M}_{{V}_{o}}-{R}_{v}\times {(B-V)}_{o}+{A}_{v}-{R}_{V}\times E(B-V)\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&=\,{M}_{{V}_{o}}-{R}_{v}\times {(B-V)}_{o}={W}_{o},\end{eqnarray} \tag{ 6 }$$

since by definition,

$$\begin{eqnarray}&&{A}_{v}-{R}_{v}E(B-V)=0.00.\end{eqnarray} \tag{ 7 }$$

Therefore

$$\begin{eqnarray}&&W={W}_{o}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&={M}_{{V}_{o}}-{R}_{v}\times {(B-V)}_{o}=\alpha \,\mathrm{log}P\\ &&\quad +{(\beta -{R}_{v})\times (B-V)}_{o}+\gamma .\end{eqnarray} \tag{ 9 }$$

Had β been fortuitously equal to R_v then W would have been dispersionless; but clearly it is not. However, for our purposes, the intrinsic and measurable spread of the data points in W becomes an attribute to be taken advantage of: the scatter in W still contains information on the intrinsic colors of the Cepheids, individually free from the complicating effects of unknown amounts of absolute and differential reddening.

The two Wesenheit PL relations from Fouqué et al. (2007) are shown in Figure 3. Here it is interesting that the HST-calibrated Cepheids (again distinguished from the other stars with different distance determination methods being applied, using larger symbols) are readily seen to have a significantly smaller dispersion than the parent sample. This suggests that the larger dispersion in the general Galactic sample is at least in part due to distance modulus errors, given that W is, by definition, immune to reddening and, by extension, unaffected by errors in those reddening determinations.

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We begin by noting the obvious. From a comparison of Equations (2) and (9) as externally constrained by the blue and red edges of the instability strip, the magnitude widths of these equations, projected into their two (V and W) PL planes, scale like β and β − R, respectively. If β > R, then the magnitude width of the W − log P relation, at fixed period, will be less than the magnitude width of the M − log P relation. Furthermore, the relative ordering of the individual Cepheids across the strip will be identically represented in each of the PL relations—that is, the brightest, bluest stars marking one edge of the instability strip in V, say, will be the same brightest bluest stars in W, and so on across the strip, star by star. The decreased width of W with respect to PL relations at most other wavelengths is well established (e.g., Figure 3 in Freedman & Madore 2010). The synchronized ordering of points across the PL relations is obvious, but with one exception (Madore 1982), never before used for any practical purposes. We do that now. But having taken a first look at the equations, we next present a graphical view.

Figure 4 illustrates in graphical terms the underpinnings of the methodology explored here, designed originally to adjust the individual reddenings to our Cepheid sample. Our ingoing position was that the distance moduli (parallaxes) were of high precision, but that the reddenings were still in need of star-by-star refinements. Later we relaxed the assumption that the individual moduli are not in need of corrections themselves, and thereby expanded the method to allow for refinements in both parameters.

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Figure 4 is a color–magnitude diagram crossed by the Cepheid instability strip. The bright (blue) edge of the strip is traced by the broken thick (blue) line going diagonally to redder colors and brighter magnitudes. Similarly, the red edge is traced by the slanting thick, broken red line. A line of constant period traverses the instability strip in the middle of the plot, slanting down (to fainter magnitudes) and to the right (to redder intrinsic colors). The slope of this line of fixed period is $\beta ={\rm{\Delta }}V/{\rm{\Delta }}(B-V){| }_{P}$. Five evenly separated Cepheids (along the line of constant period) are marked across the instability strip as numbered, circled dots. Cepheid No. 1 marks the bright, blue limit of the instability strip; No. 5 is at the red edge; and No. 3 sits on the central ridge line. Additional lines of increasing period (if plotted) would be found parallel to these points, but displaced to higher luminosities and redder colors, and so on.

Marginalizations of the instability strip (i.e., orthogonal projections) can be thought of as resulting in PL and/or period–color (PC) relations. Marginalization of the color–magnitude diagram to the left axis gives rise to the Cepheid PL relation; marginalizing over magnitude and projecting down on to the color axis gives rise to the PC relation. One slice of each of these "projections" (shown at fixed period) is graphically illustrated by the horizontal displacement of our numbered points out of the instability strip to the left and onto the magnitude axis. This new distribution of points now shows the full width of the PL relation at a fixed period. The downward projection of those same Cepheids out of the instability strip and onto the color axis gives the full width of the PC relation, again at this fixed period.

The relative magnitude and/or the color ordering of the individual stars should be preserved (and identical in their order and separation) within the instability strip and for any of its projections, when intrinsic colors and magnitudes are being considered. In Figure 5 (right panel) we show that this expectation is not fulfilled for this Galactic Cepheid data set.

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However, we now introduce the complicating factor of reddening (and extinction), which was the (original) focus of this paper. The shallower but still downward-sloping line (still within the instability strip, in this case) connecting Cepheid No. 2 to the point at 2' is a reddening trajectory taking the Cepheid to redder colors and fainter magnitudes, quantified by a reddening $E{(B-V)=(B-V)-(B-V)}_{o}$ and an extinction ${A}_{V}=V-{V}_{o}$. These are conventionally related to each other by the ratio of total to selective absorption ${R}_{V}={A}_{V}/E(B-V)$, which is shown in our color–magnitude diagram by the arrow passing through 2 and 2' and having a slope equal to R_V.

It is important to note that R_V and β are closely parallel, but they are not equal, and so their trajectories do diverge; but nevertheless the two lines are heading qualitatively in the same direction. That is, both effects (i.e., crossing the instability strip at fixed period, or being reddened) move stars to fainter magnitudes and redder colors; but in one case the effect is intrinsic to the star (and contingent on the internal structure of the Cepheid instability strip), while in the other case it is an extrinsic effect, contingent upon the amount of dust along a particular line of sight. Over the years decoupling these two effects has been problematic, but as we will now show in detail, it is not insoluble.

If we plot the magnitude residuals from the mean PL relation at each wavelength (B through K in this case) versus the corresponding (star-by-star) residuals taken from the W − log P plot,⁴ we should expect a one-to-one mapping of the residuals with a slope of ${\rm{\Delta }}{M}_{\lambda }/{\rm{\Delta }}{W}_{{VI}}={\beta }_{\lambda }/({\beta }_{\lambda }-R)$. If there are random errors in the adopted reddenings, they will contribute to the scatter in the fit, but only in the vertical ${M}_{\lambda }$ direction, given that W and ΔW are both (by design) totally independent of reddening, whether those reddenings are correctly estimated or not. Within the photometric precision of the magnitudes and colors (and, for the moment, also assuming that the adopted distance moduli also have negligible random errors; but see Section 4), we can then attribute vertical deviations away from the ΔM/ΔW plot to systematic errors in the individually adopted extinctions. To gain some perspective on how these Δ–Δ plots (both Figures 5 and 6) relate back to their parent PL relations, we offer a graphical explanation in Figure 7.

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If the previous assertion is the dominantly correct one, then the deviations in magnitude in the various ${\rm{\Delta }}{M}_{\lambda }/{\rm{\Delta }}W$ plots⁵ should all be deterministically correlated through their commonly shared, but as yet undetermined residual errors, ${\rm{\Delta }}E(B-V)$ in their published color excesses, and they each should scale with ${R}_{\lambda }:{\rm{\Delta }}{M}_{B}={R}_{B}\times {\rm{\Delta }}E(B-V)$, ${\rm{\Delta }}{M}_{V}\,={R}_{V}\times {\rm{\Delta }}E(B-V),$ and so on. We test this assertion in Figures 8 and 9.

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To orient the reader in both anticipating and then navigating the plots appearing in Figures 11 and 12, we offer in Figure 10 a graphical illustration of the reddening and distance modulus correction procedure used here. In this figure we take the reader through one of seven such ($\delta B...{VRIJHK}$ versus ΔW) plots considered for each of the 59 Cepheids studied here. Figure 10 plots ΔB versus ΔW. Crossing the plot from the lower left to the upper right is a thick, solid black line representing the intrinsic relation between the deterministically correlated and rank-ordered pairs of residuals from the two chosen PL relations, B − log P and W − log P, in this case. In the absence of reddening and/or distance errors (and assuminh high-precision photometry), all stars should fall somewhere along this line. However, star B, in the upper middle of the diagram, does not fall on the line, from which we must conclude that either the reddening adopted is wrong or the distance modulus is in error, or both. The broken arrow pointing straight down from B reaches the intrinsic line at point B^o. In this case, all of the error is assumed to be in the extinction, and the only correction would then be $\delta {A}_{B}^{o}$, as annotated to the left of the arrow. That option, however is far from unique. The arrow from B going horizontally to the left gives a second path, using a combination of distance modulus and reddening error to get back to the intrinsic line. The arrows across and down from B (both marked as $\delta {\mu }^{o}$) show the effect of a distance error. Unlike the reddening vector in this plot, a distance error affects both axes equally. The thin solid line passing diagonally and downward to the left through star B is a line of unit slope; the two distance error arrows bring the star across and down that line. But this action does not (yet) get our star back to its intrinsic line. Indeed, it is now further from it. The addition of a new (larger) extinction correction ($\delta {A}_{B}^{1}$) can then bring the star down to the intrinsic line. Similar (non-unique) trajectories are shown for star C, which also deviates from expectations, and likewise has an array of distance/extinction pairs of corrections that can return it to the line.

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Had there only been the B-band data to work with this degeneracy in the two corrections, they would be unbroken. However, we have seven such plots at our disposal, each of which give extinction/distance correction trajectories, and each of those trajectories has a different slope because of the differing relationships that ${\rm{\Delta }}B,{\rm{\Delta }}V,{\rm{\Delta }}R,$ and so on have with ΔW. The solution for a unique value of the distance correction and its corresponding extinction correction fast becomes overdetermined by the data. And that solution resides in an eight-dimensional covariance matrix, which brings us to Figure 12. In those panels for individual Cepheids, we have collapsed six of the dimensions into one by projecting each of the multiwavelength extinction/distance trajectories into a single plane, shown by dotted straight lines of systematically decreasing slopes as a function of increasing wavelength. Under ideal circumstances (given that there is physically only one solution for the extinction and distance modulus for a given star), all of these lines should cross at a single point. In the presence of photometric noise and varying sensitivity of each of the bandpasses to extinction, the result is more likely to be a minimum dispersion region in their intersections. That expectation is realized in practice and is captured in the run of dispersion in the reddening correction, with the distance modulus correction shown in the adjacent right-hand panels for each of the stars in Figure 12.

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In the panels of Figure 11 we have plotted for each of the Cepheids their vertical scatter, taken from the ${\rm{\Delta }}{M}_{\lambda }/{\rm{\Delta }}W$ plots, divided by their corresponding wavelength-dependent values of ${R}_{\lambda }$. If the aforementioned vertical deviations are due to individual errors in the adopted extinctions/reddenings, then each of the "normalized residuals" (scaled by their respective wavelength-dependent values of R) should be constant as a function of wavelength; that is, their means should correspond to the $\delta E(B-V)$ needed to fine-tune their total color excesses. Over the seven available bands, the "normalized residual" plots for the 59 Cepheids are remarkably flat (especially for the most reddening-sensitive shorter-wavelength data; more on this as follows). This strongly supports the idea that the reddenings, as initially adopted here, are in need of adjustment and that the precision of the incoming photometry supports the determination of those adjustments.

Given the excellent agreement of these reddening estimates across wavelengths for a given Cepheid (as shown for a representative sample of Cepheids in Figure 12), and between the two independent Wesenheit determinations, we apply these "second-order corrections" to the absolute magnitudes; and perhaps more interestingly at this juncture, we apply them to the intrinsic colors. In doing this we can investigate the micro-structure of the instability strip, probing in the first instance, the slope of lines of constant period crossing the CMD. This, of course, is the color term, β, in the PLC, an example of which is given in Equation (2). Figure 13 shows the dramatic improvement in the mapping of relative positions of the Cepheids in the various PL relations originally seen to be scrambled in Figure 5.

In Table 1 we give two lines for each of the 59 Galactic Cepheids in Fouqué et al. (2007). At the beginning of the first line is the name of the Cepheid followed by the period, the $E(B-V)$ color excess, their adopted distance modulus, parallax (and quoted uncertainty), followed by their mean absolute magnitudes in the BVRI (optical), and JHK (near-infrared) bandpasses. The second line gives the updated values for all of the previously listed quantities, excluding the periods that were retained in all cases. The revised PL relations and their uncertainties, based on the corrected data in Table 1, are given in Table 2, and plotted in Figures 14–16. So as to decouple the errors quoted in the zero point from errors in the slopes, all quantities were calculated with respect to the approximate pivot point of log P = 1.0. The corrected run of slope with wavelength is shown in Figure 17. The adopted changes in reddening and distance modulus, expressed as a one-sigma dispersion, are ±0.06 and ±0.19 mag, respectively, and are shown in Figures 18 and 19 as a check on covariance and systematics.

Figure 18 shows the run of color excess corrections as a function of period of the Cepheids involved. No trend is seen, indicating that the corrections are not biasing the slope of the resulting PL relations. Similarly, Figure 19 shows a plot of the color excess corrections against the true distance modulus corrections. We expect no correlation, given the fact that the original, full reddening corrections were taken from averages over the literature and were thereby completely decoupled from the methods used in determining the parallaxes, and so any errors in the two quantities should also be decoupled, which apparently they are.

Intrinsic PL relations, based on the corrected data in Table 1, are given in equation form below. A revised and updated version of the B-band PL relation is given in Figure 14, again, explicitly marking the HST parallax sample as in Figure 1. The individual PL relations are given in the six panels of Figure 15, which are to be compared with the original plots given in Figure 2.

$$\begin{eqnarray*}{M}_{B} & = & -\,2.277\,(\mathrm{log}\,P-1.0)-3.214\,\,(\sigma =0.27),\\ {M}_{V} & = & -\,2.670\,(\mathrm{log}\,P-1.0)-3.944\,\,(\sigma =0.21),\\ {M}_{R} & = & -\,2.874\,(\mathrm{log}\,P-1.0)-4.396\,\,(\sigma =0.17),\\ {M}_{I} & = & -2.983\,(\mathrm{log}\,P-1.0)-4.706\,\,(\sigma =0.14),\\ {M}_{J} & = & -\,3.198\,(\mathrm{log}\,P-1.0)-5.258\,\,(\sigma =0.12),\\ {M}_{H} & = & -\,3.333\,(\mathrm{log}\,P-1.0)-5.558\,\,(\sigma =0.12),\\ {M}_{K} & = & -\,3.377\,(\mathrm{log}\,P-1.0)-5.659\,\,(\sigma =0.12),\end{eqnarray*}$$

$$\begin{eqnarray*}{W}_{{VI}} & = & -\,3.476\,(\mathrm{log}\,P-1.0)-5.889\,\,(\sigma =0.08),\\ {W}_{{BV}} & = & -\,3.600\,(\mathrm{log}\,P-1.0)-5.997\,\,(\sigma =0.07).\end{eqnarray*}$$

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Below we give the PLC relations for M_V and all six of the independent colors. The quoted sigmas are consistent with photometric errors on the photometry, amounting to ${\sigma }_{V},{\sigma }_{R},{\sigma }_{I}=0.02$ mag, ${\sigma }_{B},{\sigma }_{J}=0.03$ mag, and ${\sigma }_{H},{\sigma }_{K}=0.05$ mag. However, it is interesting to note that the V-band PLC has a sigma of only 0.06 mag, meaning that if independent reddenings can be obtained, this formula is capable of delivering distances to Cepheids that are individually good to ±3%. Magnitude versus color residual plots leading to the color terms in these equations are given in Figure 20.

$$\begin{eqnarray*}{M}_{V} & = & -3.60\,(\mathrm{log}\,P-1.0)+4.55\,{(V-R)}_{o}-6.00(\sigma =0.10),\\ {M}_{V} & = & -4.18\,(\mathrm{log}\,P-1.0)+3.84\,{(B-V)}_{o}-6.75(\sigma =0.15),\\ {M}_{V} & = & -3.62\,(\mathrm{log}\,P-1.0)+3.03\,{(V-I)}_{o}-6.25(\sigma =0.06),\\ {M}_{V} & = & -3.80\,(\mathrm{log}\,P-1.0)+2.13\,{(V-J)}_{o}-6.74(\sigma =0.10),\\ {M}_{V} & = & -4.00\,(\mathrm{log}\,P-1.0)+2.00\,{(V-H)}_{o}-7.17(\sigma =0.13),\\ {M}_{V} & = & -3.98\,(\mathrm{log}\,P-1.0)+1.85\,{(V-K)}_{o}-7.12(\sigma =0.14).\end{eqnarray*}$$

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We have introduced and applied a novel technique for refining the individual distances and line-of-sight reddenings to Galactic Cepheids. Starting with the reddening-independent Wesenheit function, it is possible to order individual Cepheids by their position within and across the instability strip. Demanding the same relative ordering across the corresponding projections of the instability strip into the multiwavelength PL relations reveals discrepancies in the ordering that can be corrected by applying differential corrections to their true distance moduli and their published reddenings. By minimizing the scatter in the adopted reddening as a function of the reddening-independent, true distance modulus, updated parallaxes, and reddenings can be obtained on a star-by-star basis, self-consistently determined across seven optical and near-infrared bands. The resulting PL relations, updated in this way, show the expected monotonic increase of slope and corresponding monotonic decrease in width as a function of increasing wavelength of the observations. Residuals measured from the ridge lines of the multiwavelength PL relations are highly correlated (again as expected), revealing the intrinsic PLC relation underwriting the individual PL relations.

We thank the Carnegie Institution for Science and the University of Chicago for their continuing generous support of our long-term research into the expansion rate of the universe. We also thank Dylan Hatt for careful and insightful reading of an early draft of this paper. Finally, we are indebted to the referee for their careful reading of the paper and the many detailed suggestions offered that made the final paper more readable, and for strongly urging us to describe Figure 11, which is at the heart of the process by which our finally adopted values of extinctions and distances were self-consistently derived.