MNRAS 000,1{19 (2019) Simultaneous calibration of spectro-photometric distances and the Gaia DR2 parallax zero-point offset with deep learning Henry W. Leung1? & Jo Bovy1;2y 1Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada 2Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada ABSTRACT Gaia measures the five astrometric parameters for stars in the Milky Way, but only four of them (positions and proper motion, but not parallax) are well measured be- yond a few kpc from the Sun. Modern spectroscopic surveys such as APOGEE cover a large area of the Milky Way disk and we can use the relation between spectra and luminosity to determine distances to stars beyond Gaia's parallax reach. Here, we de- sign a deep neural network trained on stars in common between Gaia and APOGEE that determines spectro-photometric distances to APOGEE stars, while including a flexible model to calibrate parallax zero-point biases in Gaia DR2. We determine the zero-point offset to be −52:3 ± 2:0 µas when modeling it as a global constant, but also train a multivariate zero-point offset model that depends on G, GBP − GRP color, and Teff and that can be applied to all ≈ 139 million stars in Gaia DR2 within APOGEE's color{magnitude range. Our spectro-photometric distances are more pre- cise than Gaia at distances & 2 kpc from the Sun. We release a catalog of spectro- photometric distances for the entire APOGEE DR14 data set which covers Galacto- centric radii 2 kpc . R . 19 kpc; ≈ 150; 000 stars have < 10% uncertainty, making this a powerful sample to study the chemo-dynamical structure of the disk. We use this sample to map the mean »Fe/H¼ and 15 abundance ratios »X/Fe¼ from the Galactic center to the edge of the disk. Among many interesting trends, we find that the bulge and bar region at R . 5 kpc clearly stands out in »Fe/H¼ and most abundance ratios. Key words: astrometry | Galaxy: structure | methods: data analysis | stars: fundamental parameters | stars: distances | techniques: spectroscopic 1 INTRODUCTION (Gaia Collaboration, et al. 2016) are providing an unprece- dented view of the spatial and kinematic structure of the The Milky Way provides a unique opportunity for the study extended solar neighborhood (e.g., Gaia Collaboration et al. of galaxy formation and evolution, because we can determine 2018; Antoja et al. 2018; Bennett & Bovy 2019). However, the three-dimensional position and velocity (using astrom- even with Gaia's exquisite astrometric precision, it currently etry and spectroscopy), high-quality stellar parameters and only provides precise distances (and, thus, tangential mo- arXiv:1902.08634v1 [astro-ph.GA] 22 Feb 2019 elemental abundances (from high-resolution spectroscopy), tions) within about 2 kpc, even for relatively bright giants. and ages for large samples of individual stars (e.g., Freeman Thus, to take full advantage of the wide disk coverage of & Bland-Hawthorn 2002; Rix & Bovy 2013). Recently, great APOGEE, we require a method for obtaining precise dis- advances have been made in getting precise stellar param- tances and space velocities for all stars in APOGEE. The eters (e.g., Holtzman et al. 2015; Casey et al. 2016; Leung wealth of data from Gaia allows spectro-photometric dis- & Bovy 2019) and ages (e.g., Martig et al. 2016; Ness et tance methods to be calibrated using the large, nearby set al. 2016; Mackereth et al. 2019) for hundreds of thousands of Gaia parallaxes and then be applied to the full APOGEE of stars across the Milky Way from high-resolution spec- data set. This is what we set out to do in this paper, using troscopic surveys such as APOGEE (Majewski et al. 2017). the modern machine-learning technique deep learning. At the same time, astrometric data from the Gaia satellite Machine-learning techniques for spectro-photometric distances like deep learning are powerful, because they can ? E-mail: [email protected] be trained on stars with both high-resolution spectra from y Alfred P. Sloan Fellow APOGEE and parallaxes from Gaia to produce spectro- © 2019 The Authors 2 Leung & Bovy photometric distances. But applied blindly, such techniques for the spectro-photometric distances and one for the zero- propagate biases that are present in the training set to the point calibration, in such a way that the residual between model and the subsequently inferred distances. In particu- the spectro-photometric parallax and the calibrated Gaia lar, the Gaia DR2 parallaxes are known to suffer from a parallax has no information about the parallax itself. zero-point offset (Lindegren, et al. 2018; Zinn et al. 2018) The outline of this paper is as follows. Section2 de- that is known to be multivariate and may have a very com- scribes our methodology, with Section 2.1 focusing on a plex dependence on magnitude, color, sky position, or other general discussion of the method and assumptions and Sec- quantities. Simply training on the parallax data without any tion 2.2 discussing the specifics of the model and its imple- correction or using an inappropriate correction will result in mentation in more detail. Section3 provides information on a biased model. This would significantly bias distances ob- the data selection and processing from APOGEE DR14 and tained for distant stars such as those in the APOGEE sam- Gaia DR2 to construct training and test sets. Section4 and ple, because these have small parallaxes where even a small Section5 discuss our Gaia DR2 zero-point offset findings (tens of µas) zero-point offset has a large effect. and the precision of the spectro-photometric distances, re- Determinations of the Gaia DR2 parallax zero-point off- spectively. To illustrate the power of the derived data set of set have either been performed using quasars (Lindegren, et APOGEE stars with precise distances, Section6 shows maps al. 2018), which should have no measurable parallax on av- of the elemental abundances across a wide area of the Milky erage, or various types of stars. The determination using Way. Section7 discusses how our method compares to other quasars is precise, but likely not directly applicable to most methods for inferring the parallax zero-point and for deter- stars of interest, which are both brighter and bluer than mining spectro-photometric distances, and we look forward quasars and thus probably have a different zero-point off- to future applications of this methodology. Section8 gives a set due to its multivariate dependencies. Determinations us- brief overview of our conclusions. AppendixA describes how ing stars depend on trusting semi-empirical stellar-evolution to perform variational NN inference on arbitrary APOGEE models (Zinn et al. 2018) or use rigid (i.e., few parameter) spectra to determine their distances using the model used in models for standardizable candles (e.g., Sesar et al. 2017; this work. Riess, et al. 2018). These give precise determinations of Code to reproduce all of the plots in this paper as the zero-point, but have to assume that the zero-point is well as the FITS2 data file containing our neural net- a well-behaved (e.g., constant) function of observables such work's distance for the entire APOGEE DR14 data set as magnitude, parallax, and color. In this paper, we deter- is available at https://github.com/henrysky/astroNN_ mine the parallax zero-point offset using a diverse sample of gaia_dr2_paper. The data model for this FITS file is de- main-sequence and red-giant stars with a method that only cribed in Table1 at the end of this paper. rests on the assumption that continuum-normalized stellar spectra allow the intrinsic luminosity to be determined, but not the distance (or apparent magnitude). We represent the zero-point's multivariate dependence on magnitude, color, 2 METHODOLOGY and temperature using a flexible (i.e., many parameter) neu- ral network model using deep learning. 2.1 Basic idea and assumptions Deep learning is a subset of machine learning and the Our goal is to simultaneously calibrate spectro-photometric term refers to the usage of multi-layer (\deep") artificial neu- distances and the Gaia DR2 parallax zero-point offset by ral networks (NN) to do various kinds of machine learning training a model to predict the Gaia DR2 catalog value tasks in supervised and unsupervised learning, image recog- of the parallax from the near-infrared APOGEE spectra. nition, natural language processing, etc. NNs exist in various To do this, our supervised deep-learning algorithm requires architectures that mimic biological brains in order to repre- training labels for a set of stars observed by both Gaia and sent high-dimensional mappings in a versatile manner. On APOGEE and training proceeds by minimizing the predic- top of the versatility of the NN, we also employ in our work tion error for the training data set by adjusting the model a robust way of Bayesian deep learning that (a) takes data parameters. In the present application, these model param- uncertainties in the training data into account and (b) es- eters are the strengths of neural-network connections and timates uncertainty in predictions made with the NN using the optimization is performed using a gradient-descent op- an approximation to variational inference, drop out varia- timizer. tional inference. This methodology is mostly based on our Before explaining our method in detail, we briefly sum- previous work on deep learning of stellar abundances with marize the main ideas and assumptions that allow us to APOGEE spectra using astroNN1 (Leung & Bovy 2019). calibrate both the spectro-photometric distance model and Supervised learning requires trusted, labeled training data the Gaia DR2 zero-point offset at the same time: for the model to learn.