The Dependence of the Extinction Coefficient on Reddening for Galactic Cepheids

TL;DR

This work demonstrates a clear anti-correlation between the Gaia $G$-band extinction coefficient $R_G$ and reddening $E(G_{ m BP}-G_{ m RP})$, quantified as $R_G = 1.921 \pm 0.060 - (0.107 \pm 0.022)\,E(G_{ m BP} - G_{ m RP})$. By combining a convolution-based assessment of non-linear band extinction with MCMC fits to Gaia data and infrared distances, the authors show that roughly half of the reddening dependence arises from the broad Gaia bands and half from variations in $R_V$ along different sightlines. Ignoring this reddening dependence biases Cepheid metallicity calibrations and systematically underestimates distances for highly reddened Cepheids, while Wesenheit magnitudes built with $R_G$ become unreliable in high-extinction regimes. As a practical alternative, infrared-based distances—less sensitive to $R_V$ variations and non-linear effects—are recommended, with two viable approaches: infrared Wesenheit indices or the infrared multi-band optimal distance method; future work will aim to determine star-by-star $R_V$ values to further refine Cepheid distances.

Abstract

Cepheids are fundamental distance indicators, playing a crucial role not only in the cosmic distance ladder but also in mapping the structure, kinematics, and extinction properties of the Milky Way. Using high-precision photometry and parallaxes from $Gaia$, we identify a significant anti-correlation between the $G$-band extinction coefficient and reddening for Galactic Cepheids, quantified as $R_G = 1.921 \pm 0.060 - (0.107 \pm 0.022)\,E(G_{\mathrm{BP}} - G_{\mathrm{RP}})$. We propose that this anti-correlation is partly driven by the pronounced non-linear effects inherent to the broad $Gaia$ bands, while the remaining part arise from the $R_V$ variations caused by diverse interstellar medium. Adopting a fixed $R_G$ would not only lead to an overestimation of the metallicity dependence of Cepheid luminosities, but also systematically underestimate the distances to highly reddened Cepheids. Moreover, the strong reddening dependence of $R_G$ makes Wesenheit function based on it unsuitable for highly reddened Cepheids, since the definition of Wesenheit magnitudes requires a fixed extinction coefficient. In contrast, infrared-based distances, being less affected by non-linear effects and insensitive to $R_V$, provide the most reliable Cepheid distances at present. This work emphasizes the importance of accurately determining $R_V$ for Galactic Cepheids and accounting for non-linear effects in distance measurements, particularly in the optical bands.

Figures (4)

• Figure 1: The left panel shows the non-linear effect of the effective wavelength in the Gaia bands. The middle panel shows the non-linear effects of $R_G$ as a function of $E(G_{\mathrm{BP}}-G_{\mathrm{RP}})$ for five different extinction laws, where CCM89 represents cardelli1989, F99 represents Fitzpatrick1999, M14 represents maiz2014, WC19 represents wang2019, and G23 represents Gordon2023. The black curve represents the average of the five curves, while the dashed gray line denotes a linear fit to this average curve. The right panel shows the distribution of $R_G$ as a function of $E(G_{\mathrm{BP}}-G_{\mathrm{RP}})$ for different values of $R_V$.
• Figure 2: The left panel shows the posterior distribution for Model 2, while the right panel shows that for Model 3.
• Figure 3: The left panel shows the distribution of parallax residuals as a function of $E(G_{\mathrm{BP}} - G_{\mathrm{RP}})$ for Model 1, while the right panel shows that for Model 2.
• Figure 4: The left panel shows the distribution of $R_G$ versus $E(G_{\mathrm{BP}} - G_{\mathrm{RP}})$ for 2,846 Cepheids, with the green dashed line representing the MCMC fitting result. The right panel presents the distribution of $R_V$ versus $E(G_{\mathrm{BP}} - G_{\mathrm{RP}})$ for the same sample, where the red points indicate the median $R_V$ values within bins of width 0.5 mag for $E(G_{\mathrm{BP}} - G_{\mathrm{RP}}) > 0.5$ mag.