Estimating accurate reddening values of LAMOST M dwarfs

In addition to the $E(G_{\rm BP}-G_{\rm RP})$ values, we also provide the mostly used $E(B-V)$ reddening values for the individual stars in our catalogue. In the current work, we have adopted a reddening coefficient of $R_{(G_{\rm BP}-G_{\rm RP})}$$=$ 1.33 from Chen et al. (2019a) to convert the Planck $E(B-V)$ to $E(G_{\rm BP}-G_{\rm RP})$ for the stars in the training data set. The reddening coefficient $R_{(G_{\rm BP}-G_{\rm RP})}$ should vary for the individual stars, due to: 1) the variation of the extinction curves in different environments (Jones et al., 2011; Schlafly et al., 2016), 2) the wide wavelength ranges of the Gaia $G_{\rm BP}$ and $G_{\rm RP}$ bands, and 3) the variation of the stellar spectral energy distributions (SEDs). Stars in the training sample are all located in very low extinction regions, with line-of-sight reddening $E(B-V)$$<$ 0.02 mag. The variation of the reddening coefficient $R_{(G_{\rm BP}-G_{\rm RP})}$ would not affect our results. However, to convert $E(G_{\rm BP}-G_{\rm RP})$ to $E(B-V)$ for all our catalogued stars accurately, we should consider the variation of the $R_{(G_{\rm BP}-G_{\rm RP})}$.

Physically, $R_{(G_{\rm BP}-G_{\rm RP})}$ is related to the stellar intrinsic colour $(G_{\rm BP}-G_{\rm RP})_{0}$ and the reddening $E(B-V)$ (Niu et al., 2021a, b; Sun et al., 2022). To explore the correlation between $R_{(G_{\rm BP}-G_{\rm RP})}$ and ($(G_{\rm BP}-G_{\rm RP})_{0}$, $E(B-V)$), we have selected stars with large distances to the Galactic plane ($Z$), where the stars are thought to be out of the Galactic dust disk and we assume their $E(B-V)$ values from the Planck Collaboration et al. (2014) dust map are reliable. We adopt the following criteria:

1. 1.

   LAMOST spectra S/N in the $i$-band snri $>$ 30 and photometric uncertainties of both the Gaia EDR3 $G_{\rm BP}$ and $G_{\rm RP}$ magnitudes $<$ 0.01 mag,

2. 2.

   reddening values $E(B-V)$ from the Planck two-dimensional dust maps (Planck Collaboration et al., 2014) between 0.02 and 0.5 mag,

3. 3.

   Galactic latitude $|b|$ $>$ 15°,

4. 4.

   Distance to the Galactic plane $|Z|$ $>$ 300 pc,

5. 5.

   matching distances between the LAMOST and Gaia positions $<$ 1^′′.

This leads to a sample of 62,905 stars. We note that stars with very red colours ($(G_{\rm BP}-G_{\rm RP})$$>$ 2.5 mag) are not included in this selected sample, since they are too faint to be observed at $|Z|>$ 300 pc. The reddening coefficients $R_{(G_{\rm BP}-G_{\rm RP})}$ of the selected stars are calculated by $R_{(G_{\rm BP}-G_{\rm RP})}$ $=$ $E(G_{\rm BP}-G_{\rm RP})$$/$$E(B-V)$, where $E(G_{\rm BP}-G_{\rm RP})$ are calculated in the current work and $E(B-V)$ are adopted from the Planck Collaboration et al. (2014) dust map. The selected stars are divided into 10 $\times$ 10 pixels in the $(G_{\rm BP}-G_{\rm RP})_{0}$ and $E(B-V)$ space. We obtain the averaged $R_{(G_{\rm BP}-G_{\rm RP})}$ value of each pixel and fit $R_{(G_{\rm BP}-G_{\rm RP})}$ as a function of $(G_{\rm BP}-G_{\rm RP})_{0}$ and $E(B-V)$. As a result, we obtain,

In Fig. 3 we show the fitting results. Our derived correlations between $R_{(G_{\rm BP}-G_{\rm RP})}$ and ($(G_{\rm BP}-G_{\rm RP})_{0}$, $E(B-V)$) are very similar as those derived from Sun et al. (2022, see their Fig. 1), which increase our confidence that both relations are robust. We have converted our resulted $E(G_{\rm BP}-G_{\rm RP})$ values to $E(B-V)$ for all the stars. For stars with colours redder than 2.325 mag, we adopt the reddening coefficients calculated with the colour $(G_{\rm BP}-G_{\rm RP})_{0}$= 2.325 mag. The derived $E(B-V)$ values are also listed in our catalogue.

For the uncertainties of our derived reddening $E(G_{\rm BP}-G_{\rm RP})$, we first consider the random errors, which are estimated from the dispersions of reddening differences (divided by $\sqrt{2}$) yielded by duplicate observations. We have found 96,118 unique stars in our catalogue that have more than one spectra. We select stars with duplicate spectra that have similar $i$-band S/N and compare their resulted reddening values. Fig. 4 shows the difference dispersions divided by $\sqrt{2}$, i.e. the random errors, as a function of the $i$-band S/N of the LAMOST spectra and the photometric errors of the Gaia $(G_{\rm BP}-G_{\rm RP})$ colours $\sigma$$(G_{\rm BP}-G_{\rm RP})$ computed by adding the photometric errors in quadrature. For stars of LAMOST S/N of 10, the random errors are about 0.10 mag. The random errors decrease to $\sim$ 0.03 mag for stars with S/N of about 50 and becomes better than $\sim$ 0.02 mag at S/N values larger than 100. We fit the random errors as a function of the S/N and $\sigma$$(G_{\rm BP}-G_{\rm RP})$ values. As a result, we obtain,

which are used to calculate the random errors of the individual stars in our catalogue. For stars of S/N better than 200, their random errors are assigned to be values of those at S/N $=$ 200. The typical random error of our catalogued stars is about 0.03 mag in $E(G_{\rm BP}-G_{\rm RP})$.

In addition to the random errors described above, the uncertainties of our derived reddening values also contain systematic errors which arise from the fact that our training data set are not from perfectly extinction-free lines of sight. Our training sample stars were chosen from regions of integrated reddening of up to 0.02 mag in $E(B-V)$. The median line-of-sight reddening of our training sample stars is $E(B-V)$$=$ 0.016 mag. Since most of the training sample stars are located at high Galactic latitudes ($b>15$°) and large distances to the Galactic plane ($|Z|>300$ pc), the deviation between their actual reddening values and those from the Planck Collaboration et al. (2014) dust map should be negligible. Because of this, we have ignored the systematical uncertainties in the current work and adopted only the random errors as the final reddening uncertainties. An alternative method is to obtain the intrinsic reddening values of the training sample star from a synthetic spectral library. However, it would result in additional errors from the uncertainties of the synthetic stellar models (Yuan et al., 2015).

The typical error of our resultant reddening values is estimated to be $\sim$ 0.03 mag. To verify the robustness of our reddening values, we have compared our results with those from previous works. The comparisons are shown in Fig. 5. Chen et al. (2014) obtained $r$-band extinction values for over 13 million stars in the Galactic anticentre from the SED fits to the multi-band photometric measurements of the Xuyi Schmidt Telescope Photometric Survey of the Galactic Anticentre (XSTPS-GAC), the Two Micron All Sky Survey (2MASS) and Wide-Field Infrared Survey Explorer (WISE). We cross-matched our catalogue with that from Chen et al. (2014) with a radius of 3^′′, which yields 112,621 common stars. The Chen et al. extinction values are converted to reddening values by the extinction coefficients from Yuan et al. (2013): $E(B-V)$= 0.43$A_{r}$. Our resulted reddening values are in good agreement with those from Chen et al. (2014). The mean difference is 0.014 mag and the dispersion is 0.061 mag.

Based on Gaia parallaxes and stellar photometry from Pan-STARRS 1 (PS1) and 2MASS, Green et al. (2019) obtained reddening values of 799 million stars in the northern sky. To compare the Green et al. results with our values, we have selected stars in the Green et al. catalogue by the following criteria: 1) stars been detected in all the eight PS1 and 2MASS passbands ($g_{\rm P1}$, $r_{\rm P1}$, $i_{\rm P1}$, $z_{\rm P1}$, $y_{\rm P1}$, $J$, $H$ and $K_{\rm S}$) and the photometric errors in all the eight bands smaller than 0.05 mag; 2) the goodness-of-fit (maximum-likelihood $\chi^{2}/$passband) smaller than 5; and 3) the uncertainties of reddening smaller than 0.08 mag. The selected stars are cross-matched with our catalogue and we have obtained 173,603 common stars. Our results agree well with those selected from the Green et al. catalogue, with a negligible averaged difference and a small dispersion of only 0.058 mag.

Anders et al. (2022) presented a catalogue of 362 million stars. Their reddening values are derived from the photometric catalogues of Gaia EDR3, PS1, SkyMapper, 2MASS, and WISE. We select stars from the Anders et al. catalogue by the following criteria: 1) stars with good astrometric measurements (astrometric fidelity flag larger than 0.5; Rybizki et al. 2022) and good $G_{\rm BP}$$G_{\rm RP}$ photometries ($\lvert C_{*}\rvert/{\sigma_{C^{*}}}<0.5$, where $C_{*}$ and ${\sigma_{C^{*}}}$ are respectively the bp_rp_excess_corr value and a simple function of the Gaia $G$ magnitude defined in Riello et al. 2021) ; 2) stars been detected in all the Gaia $G$$G_{\rm BP}$$G_{\rm RP}$, the 2MASS $JHK_{\rm S}$, the WISE $W1W2$ and the PS1/SkyMapper $griz$ bands; 3) stars with good fitting models (the first number of the flag ‘sh_outflag’ $=$ 0) and small fitting errors (the third number of ‘sh_outflag’ $=$ 0). The selected stars are then cross-matched with our catalogue, which yields 99,273 common stars. The agreement of the comparison between our results and those from Anders et al. (2022) is good. The average difference is negligible and the dispersion is 0.075 mag in $E(G_{\rm BP}-G_{\rm RP})$.

Based on the stars for which we measured accurate reddening values, we have plotted the Hertzsprung–Russell diagram (HRD) of the LAMOST M dwarfs (Fig. 6). We have selected 528,368 stars in our catalogue that have Gaia EDR3 parallax errors smaller than 20 per cent. The distances of the selected stars $d$ are calculated from the Gaia EDR3 parallaxes $\varpi$ by a simple Bayesian algorithm (Bailer-Jones et al., 2018; Chen et al., 2019b). We adopt a posterior probability given by,

where $\sigma_{\varpi}$ is the Gaia parallaxes uncertainties, $\varpi_{\rm zp}$ the global parallax zero point and $p(d)$ the space density distribution prior for the M dwarfs. In the current work, we adopt $\varpi_{\rm zp}$ $=$ $-$0.026 mas from Huang et al. (2021) and the Galactic disk model derived from the late type stars (1.0 $<~{}r-i~{}<$ 1.4 mag) by Jurić et al. (2008).

With the resultant distances, the Gaia $G$-band absolute magnitudes are then calculated by $M_{G}=G-5\lg\,d+5-A_{G}$. In the current work, we adopted the $G$-band extinction coefficient from Chen et al. (2019a) to obtain the $G$-band extinction, by $A_{G}=1.91E(B-V)$. We show the dust-corrected colour and absolute magnitude diagram for the selected 528,368 stars in Fig. 6. A PARSEC isochrone with the Solar metallicity and age of 10 Gyr (Bressan et al., 2012) is overplotted in the diagram. The distribution of our catalogued stars is in good agreement with the theoretic isochrone, which suggests the robustness of our resulted reddening values. Most of our catalogued stars have intrinsic colours $(G_{\rm BP}-G_{\rm RP})_{0}$ between 1.6 and 3.2 mag and absolute magnitudes $M_{G}$ between 7 and 12 mag. According to the PARSEC isochrone, our catalogued stars have masses between 0.2 and 0.65 $M_{\odot}$.

Finally, we plot the $E(G_{\rm BP}-G_{\rm RP})$  reddening maps in the Galactic longitude $l$ and latitude $b$ coordinates and the Galactocentric coordinates based on our catalogued M dwarfs. In Fig. 7, we show the reddening map in the Galactic longitude $l$ and latitude $b$. We use the HEALPix pixelization scheme (Górski et al., 2005) to bin the sample stars into individual pixels. We adopt a resolution of 27 arcmin (with HEALPix nside = 128) and average the reddening values of stars in the individual pixels. Fig. 7 presents a nice local dust reddening map, which show very similar features to the nearest distance slice of Galactic three-dimensional (3D) dust maps, such as the Green et al. (2019) 3D extinction map with distance 0$~{}<~{}d~{}<~{}$0.5 kpc (their Fig. 1) and the Guo et al. (2021) map with 0$~{}<~{}d~{}<~{}$0.4 kpc (their Fig. 14). The famous nearby giant molecular clouds, such as the Orion ($l,~{}b$ $\sim$ $-$160°, $-$15°), the Taurus ($l,~{}b$ $\sim$ 170°, $-$15°), the Perseus ($l,~{}b$ $\sim$ 155°, $-$20°), the Aries ($l,~{}b$ $\sim$ 160°, $-$30°), the Eridanus ($l,~{}b$ $\sim$ 190°, $-$35°), the Aquila Rift ($l,~{}b$ $\sim$ 30°, 10°) and the Pegasus ($l,~{}b$ $\sim$ 90°, $-$35°), are clearly visible in the map.

We have also created averaged $E(G_{\rm BP}-G_{\rm RP})$ maps in the Galactocentric coordinates based on the 528,368 stars with Gaia EDR3 parallax errors smaller than 20 per cent. The maps are presented in Fig. 8. In the left panel of Fig. 8, we show the distribution of reddening $E(G_{\rm BP}-G_{\rm RP})$ values in the Galactic $X$-$Y$ plane for stars with distances to the Galactic plane within 50 pc. The most significant feature in the Figure is the Local Bubble, which are marked by the $E(G_{\rm BP}-G_{\rm RP})$= 0.15 mag contour lines. The shape of the Local Bubble is in good agreement with those derived from the previous 3D extinction maps (e.g. Fig. 12 from Chen et al. 2019a and Fig. 4 from Lallement et al. 2022). In the right panel of Fig. 8 we show the averaged reddening $E(G_{\rm BP}-G_{\rm RP})$ maps in the Galactic cylindrical coordinates $R$ and $Z$. The Figure displays a remarkable shape of the Galactic dust disc. Based on our catalogued M dwarfs with accurate reddening values, we will be able to investigate the large and small scales dust structure in the solar neighbourhood. More quantitative analysis will be presented in future works.