SPar: estimating stellar parameters from multi-band photometries with empirical stellar libraries

Modern large-scale photometric surveys, such as the Sloan Digital Sky Survey (SDSS; York et al. 2000), the Pan-STARRS 1 Survey (PS1; Chambers et al. 2016), the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) and the Wide-Field Infrared Survey Explorer (WISE; Wright et al. 2010), have provided us board band photometry covering the wavelength from the optical to the infrared (IR) bands of billions of stars. The multi-band photometric measurements of stars contain the spectral energy distribution (SED) information of these stars, from which we can obtain the stellar atmosphere parameters (i.e. the effective temperature $T_{\rm eff}$, the metallicity $\rm[Fe/H]$ and the surface gravity log $g$), as well as the distance and extinction values of stars (Berry et al., 2012; Chen et al., 2014; Green et al., 2014, 2015).

Stellar atmospheric parameters are typically determined from their spectra. While large-scale multi-fibre spectroscopic surveys like the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST; Luo et al. 2015), the Apache Point Observatory Galactic Evolution Experiment (APOGEE; Abdurro’uf et al. 2022), the Galactic Archaeology with HERMES (GALAH, Buder et al. 2021), and the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2022) have provided spectra for tens of millions of stars, we can now determine the parameters of billions of stars with comparable precision using photometric data of exceptionally high accuracy or data obtained through specially designed filters. This approach allows us to obtain accurate stellar parameters without relying solely on spectroscopic data. For example, Xu et al. (2022) have measured metallicities of $\sim$ 27 million stars from the Gaia Early Data Release 3 (Gaia EDR3; Gaia Collaboration et al. 2021) photometric data of unprecedented millimagnitude precision. The typical metallicity precision is about $\delta$[Fe/H] = 0.2 dex. Based on the narrowband photometry from the SkyMapper Southern Survey (SMSS; Wolf et al. 2018; Onken et al. 2019), Huang et al. (2022) have determined stellar atmospheric parameters for $\sim$ 24 million stars. The precision of their metallicity estimates has typical values around 0.05 to 0.15 dex. Yang et al. (2022) obtained stellar parameters (effective temperature, $T_{\rm eff}$, surface gravity, ${\rm log}\,g$, and metallicity, $\rm[Fe/H]$) from the narrow-band photometries of the J-PLUS survey (Cenarro et al., 2019). They have achieved precisions of $\delta$$T_{\rm eff}$$\sim$ 55 K, $\delta$${\rm log}\,g$$\sim$ 0.15 dex, and $\delta$$\rm[Fe/H]$$\sim$ 0.07 dex, respectively. We note that these works are only for stars located at high Galactic latitudes, where the interstellar extinction is small. For stars at low Galactic latitudes, where the extinction effects are large, the stellar atmospheric parameters need to be estimated along with the stellar extinction values.

The future Multi-channel Photometric Survey Telescope (Mephisto; Yuan et al. 2020) photometric survey and the Chinese Space Station Telescope (CSST; Zhan 2011, 2018; Cao et al. 2018) optical survey will have both high precision and specially designed filters, which will provide great opportunities for us to obtain accurate stellar parameters of billions of stars. Mephisto is a wide-field survey telescope with a 1.6 m primary mirror. It is equipped with three CCD cameras and is capable to image the same patch of sky in three bands simultaneously, which will provide us with real-time colours of stars with unprecedented accuracy. The filters of Mephisto ($uvgriz$) are similar to those of SkyMapper (Bessell, 2001). The Mephisto-W Survey (Lei et al. 2021, 2022; Chen et al. in prep.) will target the northern sky of declination (Dec) between $-$21° and 75° with a coverage of over 27,000 deg². CSST is a 2 m space survey telescope, which shares the same orbit as the Chinese Space Station. The CSST optical survey (CSST-OS) will observe a large sky area of $\sim$ 18,000 deg² in seven photometric filters ($NUV,~{}u,~{}g,~{}r,~{}i,~{}z$ and $y$) covering the wavelengths from the near-ultraviolet (NUV) to near-infrared (NIR).

Deriving the stellar atmospheric parameters as well as the distance and extinction values is a fundamental task for the Mephisto and CSST surveys. In this work, we present a new algorithm，SPar (Stellar Parameters from multiband photometry)， to estimate stellar atmospheric parameters, such as the effective temperature ($T_{\rm eff}$) and metallicities ($\rm[Fe/H]$), absolute magnitudes ($M_{G}$), distances ($d$) and reddening values ($E(G_{\rm BP}-G_{\rm RP})$) of a large sample of stars from their multi-band photometries. Previous algorithms such as the Star-Horse (Anders et al., 2019, 2020) and the General stellar Parameterized from photometry (GSP-PHot; Andrae et al. 2022) rely on the theoretical stellar models, which may suffer systematic effects (Green et al., 2021). One method of dealing with these inaccuracies in theoretical models is to apply empirical corrections based on the observed photometry of stars of known type. Berry et al. (2012), Chen et al. (2014) and Green et al. (2015) measure stellar parameters and extinction based on the empirical stellar locus in the colour-colour space. Sun et al. (2018), Chen et al. (2019a) and Sun et al. (2021) calculated the intrinsic colours and reddening values of the individual stars based on a spectroscopic sample selected from the spectroscopic surveys. In this work, we will construct an empirical stellar library which maps the stellar parameters to multi-band photometries from a dataset with Gaia parallaxes, LAMOST atmospheric parameters, and optical to near-infrared photometry from several photometric surveys.

The paper is structured as follows: Section 2 presents the relevant dataset. Section 3 describes in detail the empirical stellar library we constructed. Section 4 describes our algorithm and Section 5 tests it. Finally, Section 6 summarizes the algorithm.

As the Mephisto and CSST surveys have not yet started, in the current work we have used photometric data from the SMSS survey for the experiment. This work is based on the Gaia Data Release 3, the broad-band photometry from SMSS, Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) and the Wide-field Infrared Survey Explorer survey (WISE; Wright et al. 2010), and the spectroscopic data from LAMOST, APOGEE and GALAH.

The SMSS is an ongoing photometric survey of the Southern sky (Wolf et al., 2018; Onken et al., 2019). The survey depth is between 19.7 and 21.7 mag in six optical bands: $u,~{}v,~{}g,~{}r,~{}i$ and $z$. In this work, we adopt the data from its second data release (SMSS DR2; Onken et al. 2019). The photometry has a internal precision of 1 per cent in the $u$ and $v$ bands, and 0.7 per cent in the other four bands ($g,~{}r,~{}i$ and $z$).

To break the degeneracy of effective temperature (or intrinsic colours) and extinction for the individual stars, we combine the SMSS photometry with the IR photometry of 2MASS and WISE. The 2MASS survey is a full-sky survey undertaken in three filters: $J,~{}H$ and $K_{\rm S}$. The systematic errors of the 2MASS photometry are estimated to be less than 0.03 mag (Skrutskie et al., 1997). The WISE survey is a full-sky survey undertaken in four bands: $W1,~{}W2,~{}W3$ and $W4$. In the current work, we adopt the AllWISE Source Catalog (Kirkpatrick et al., 2014). We use only the data in the $W1$ and $W2$ bands, as the $W3$ and $W4$ measurements have lower sensitivities and poorer angular resolutions.

The Gaia DR3 (Gaia Collaboration et al., 2022) photometric data and parallax measurements are also adopted to derive the stellar parameters when available. Gaia DR3 was released by the Gaia mission (Gaia Collaboration et al., 2016). It contains more than a billion sources with five astrometric parameters (position, parallaxes, and proper motions) and three-band photometry ($G$, $G_{\rm BP}$ and $G_{\rm RP}$). The median uncertainty of the parallax is $\sim$ 0.02 - 0.03 mas for G $<$ 15 mag sources, 0.07 mas at G = 17 mag, and 0.5 mas at G = 20 mag (Gaia Collaboration et al., 2021). At G = 20 mag, the typical uncertainties for the Gaia DR3 photometries are 6, 108, and 52 mmag for the Gaia $G$, $G_{\rm BP}$ and $G_{\rm RP}$ bands, respectively (Gaia Collaboration et al., 2021).

The LAMOST, APOGEE and GALAH spectroscopic data are adopted in the current work for two aims: to construct the empirical stellar library and to validate the resulting stellar properties. In this work, we use the ‘LAMOST LRS Stellar Parameter Catalog of A, F, G and K Stars’ from the LAMOST data release 8 (LAMOST DR8; Luo et al. 2015). The catalogue contains stellar atmospheric parameters ($T_{\rm eff}$, ${\rm log}\,g$ and $\rm[Fe/H]$) derived from over 6 million low-resolution spectra. For the APOGEE data, we use the APOGEE stellar parameters catalogue from the SDSS data release 17 (SDSS DR17; Abdurro’uf et al. 2022). The catalogue contains stellar atmospheric parameters ($T_{\rm eff}$, ${\rm log}\,g$ and [M/H]) derived from over 0.6 million near-IR spectra. For the GALAH data, we adopt its DR3 catalogue (Buder et al., 2021), which contains stellar parameters ($T_{\rm eff}$, ${\rm log}\,g$ and [Fe/H]) of over 0.5 million nearby stars.

An empirical stellar library is first created based on a sample of stars selected from the LAMOST spectroscopic data and Gaia DR3, for which atmospheric parameters, distances, and extinction values can be well measured. We cross-match the LAMOST and Gaia stars with the optical and near-IR photometric data, i.e. the SMSS, 2MASS and WISE, using a radius of 1 arcsec. To exclude the stars with bad observations, a sample of stars are selected by the criteria as below:

1. 1.

   LAMOST spectral signal-to-noise ratio (SNR) larger than 20 and effective temperature $T_{\rm eff}$ between 4000 and 8000 K,

2. 2.

   All Gaia $G$$G_{\rm BP}$$G_{\rm RP}$  photometric errors smaller than 0.1 mag and phot_bp_rp_excess_factor $>$ 1.3 + 0.06($G_{\rm BP}$ - $G_{\rm RP}$)²,

3. 3.

   Photometric errors in any SMSS $uvgriz$, 2MASS $JH$$K_{\rm S}$ and WISE $W1W2$ bands less than 0.1 mag.

To obtain the absolute magnitude of our selected stars in each filter, the extinction values in the individual filters and the distance of each star need to be calculated.

This section introduces how we derive the stellar parameters from the multi-band photometries. To allow our algorithm SPar to be applied to large samples of billions of stars, we need to minimise the computational cost. Our algorithm fits only four stellar parameters, $T_{\rm eff}$, $\rm[Fe/H]$, $M_{G}$ and $E(G_{\rm BP}-G_{\rm RP})$. The distances $d$ of stars can be further derived from the fitting results. SPar uses an ensemble Markov-chain Monto-Carlo (MCMC) method to obtain the best parameters of the individual stars by adopting a set of initial values derived from a minimum $\chi^{2}$ method.

In this section, we will test the SPar algorithm. We cross-match the SMSS data with the LAMOST, APOGEE and GALAH spectroscopic data, and have obtained a test sample of stars with parameter measurements from the spectroscopic data. The SPar algorithm is then applied to the sample stars to obtain their parameters, which are then compared with those obtained from the spectra. The test sample contains 1,046,722 stars, of which 408,671, 200,902 and 437,149 stars are from the LAMOST, APOGEE and GALAH surveys, respectively.

Fig. 4 displays the distribution of $\chi^{2}$ values for all sources included in the test sample. A typical $\chi^{2}$ distribution is observed, with a prominent peak at 2 and a long tail. Some stars in the sample exhibit a large $\chi^{2}$ value, indicating that our empirical models are unable to effectively match their observed data. This may be due to the atmospheric parameters of these stars lying outside the range of our models. To optimize computational efficiency, we have excluded these stars from subsequent MCMC analysis. Consequently, by adopting a conservative $\chi^{2}$ threshold of less than 10 in our study, we have retained 96% of LAMOST sources, 85% of APOGEE sources, and 97% of GALAH sources.

After conducting the MCMC analysis for the remaining 982,927 stars, we obtained their final parameters. To evaluate the accuracy of our algorithm, we first compared the differences between our predictions and the observations. We simulated the observed magnitudes of the individual stars at various wavelengths and their parallaxes based on the MCMC results, which were compared with the observed data presented in Fig. 5. Our simulations show good agreement with the observational results. The mean values of the differences are negligible. The dispersions of the differences are about 0.020-0.024 mag for the optical magnitudes ($G$ $G_{\rm BP}$$G_{\rm RP}$$griz$), 0.034 - 0.039 mag for the UV and near-IR magnitudes ($uvJH$$K_{\rm S}$$W1W2$), and 0.042 mas for parallax, respectively.

In this work, we have developed a new algorithm called SPar to derive stellar parameters from multi-band photometries, which can be applied to large samples of stars. The algorithm takes advantage of empirical stellar libraries constructed from Gaia, LAMOST, and other photometric surveys. It leverages the minimum $\chi^{2}$ fit of the stellar SEDs to obtain the initial values for the MCMC analysis, which results in the stellar parameters, including $T_{\rm eff}$, $\rm[Fe/H]$, $E(G_{\rm BP}-G_{\rm RP})$ and $M_{G}$, of the individual stars. Our algorithm is tested on the LAMOST, APOGEE, and GALAH stars. The typical dispersion values of the differences between our results and literature values were 170 K for $T_{\rm eff}$, 0.23 dex for $\rm[Fe/H]$, 0.13 mag for $M_{G}$ and 0.05 mag for $E(G_{\rm BP}-G_{\rm RP})$.

In the future, our new SPar algorithm will be implemented on large samples of stars obtained from the Mephisto and CSST surveys to derive atmospheric parameters, distances, and extinction values for billions of stars. This will give us crucial insights into the structure, chemistry, and other properties of the Milky Way.

We would like to thank the referee for providing us with detailed and constructive feedback that has significantly enhanced the quality of the manuscript. We are grateful to Professor Biwei Jiang for help and discussion. This work is partially supported by the National Key R&D Program of China No. 2019YFA0405500, National Natural Science Foundation of China 12173034, 11833006, 12203016 and 12173013, Natural Science Foundation of Hebei Province No. A2022205018, A2021205006, 226Z7604G, and Yunnan University grant No. C619300A034, and Science Foundation of Hebei Normal University No. L2022B33. We acknowledge the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-A09, CMS-CSST-2021-A08 and CMS-CSST-2021-B03. We are grateful for the support of the Postdoctoral Research Station in Physics at Hebei Normal University.

This research made use of the cross-match service provided by CDS, Strasbourg.

Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.

This work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia.

The national facility capability for SkyMapper has been funded through ARC LIEF grant LE130100104 from the Australian Research Council, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University’s Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth’s Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS).

This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

This publication makes use of data products from the Widefield Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.