Global semi-analytic model of the Galactic disk based on Gaia and APOGEE

Introduction

We develop a semi-analytic model of the Milky Way (MW) disk with a fine time resolution of 25 Myr (Just and Jahreiß 2010, Paper I). The Just-Jahreiß (JJ) model is based on an iterative solving of the Poisson equation and reconstructs a self-consistent pair of the total vertical gravitational potential and density. The thin disk is modeled in a presence of gas, thick disk, stellar halo, and dark matter (DM). The disk evolution is governed by four input functions: star formation rate (SFR), initial mass function (IMF), age-velocity dispersion relation (AVR), and age-metallicity relation (AMR).

Local calibration based on Gaia (and other data)

The JJ model has been many times tested and calibrated against different data in the Solar neighborhood:

• In Just et al. (2011) (Paper II), we used star counts towards the north Galactic pole from the SDSS DR7 to optimize the SFR and thick-disk parameters. 
• In Rybizki and Just (2015) (Paper III), we improved the IMF with the help of Hipparcos data complemented by an updated version of the CNS4 (Catalog of Nearby Stars). 
• In Sysoliatina et al. (2018), we demonstrated a good model-to-data consistency (with a ~7.3-% discrepancy in terms of star counts) in the local 1-kpc height cylinder using stars from the RAVE DR5 and Gaia DR1 stars (TGAS). 
• Most recently, we used Gaia DR2 stars within 600 pc from the Sun and simultaneously optimized 22 model parameters within the Bayesian framework (Sysoliatina and Just 2021, Paper IV). We also calibrated the AMR against the local Red Clump (RC) sample from the APOGEE survey. 

In Paper IV, we have identified two recent star formation (SF) burst episodes in the thin-disk evolutionary history, which happened ~0.5 Gyr and ~3 Gyr ago and left behind dynamically heated fossil populations. This interesting result is consistent with Mor et al. (2019) who found a single SF burst centered at the age of 2-3 Gyr. Also, Ruiz-Lara et al. (2020) report three SF enhancements 5.7 Gyr, 1.9 Gyr, and 1 Gyr ago, which were presumably triggered by recurrent passages of the Sagittarius dwarf galaxy. 

More details on the local calibration of the JJ model can be found in our EAS2021 K-poster, "Dynamically hot recent star bursts in the Galactic disc".

We take the first step towards calibrating the JJ model at Galacocentric distances other than Solar. We start with bringing our AMR into consistency with the APOGEE data.

Following Paper IV, we constrain the thin- and thick-disk AMR using the generalized JJ model with the radially extended APOGEE RC sample. We then propose a new analytic form for the thin-disk AMR, which is able to comprise the variation of its shape across the studied distances 4 kpc < R < 14 kpc. 

AMR reconstruction procedure

1. We select the sample at < 1kpc in |z| within 4-14 kpc in R; for low-alpha stars, we radially bin the sample with a 1-kpc step. We calculate the observed cumulative metallicity distribution functions (CMDFs) of the low- and high-alpha RC stars. 
2. Assuming AMR for the thick disk (same at all radii) and for the thin disk (radially dependent), we calculate the total vertical potential and density of the RC population at each radius. The density profile is then used to construct the cumulative age distribution functions (CADFs) of the thin- and thick-disk RC stars within the selected R-z limits. Here we also mimic the APOGEE pencil-beam-like geometry of the sample. 
3. We put in correspondence stellar metallicities and ages by comparing the observed CMDFs to the predicted CADFs, and thus, arrive at the updated AMR of the thin- and thick-disk. To achieve self-consistency, we iterate over steps (2) and (3).
4. We fit the obtained thin- and thick-disk AMR with analytic functions. In the case of the thin disk, we investigate the radial trends of the AMR best-fit parameters and also fit them with simple laws (Figure 5). As a result, we obtain a 'generalized' thin-disk AMR fit, which is well-consistent with the originally reconstructed 'raw' AMR (Figure 4).
5. Finally, we perform a consistency test: calculate the thin- and thick-disk metallicity distributions of the RC stars using the updated AMR and compare these predictions to the data (Figure 6). 

For the fitting, we use the following equation: 

We found that most of the parameters of thin-disk AMR equation are simple functions of Galactocentric distance (can be fitted with a constant or linear profile). Only parameters t₀₁ and t₀₂ have complicated radial trends, so we describe them as three-slope broken linear laws. 

In our new work, Sysoliatina and Just (2022) (submitted to A&A), we generalize our local JJ model for the range of Galactocentric distances 4 kpc < R <14 kpc by building the disk out of a set of independent radial annuli. We assume radially exponential gaseous, thin and thick disks, a spherical stellar halo, and a cored isothermal DM halo. We fix the disk thickness by assuming a constant-thickness thick disk and a constant-thickness or flaring thin disk. Then, we allow such radial variations of the input functions SFR, AVR, and AMR, which correspond to the inside-out disk growth scenario (Figure 2).

In the generalized JJ model, thin-disk SFR is defined as a smooth continuum (blue curve in Figure 1), which can be modified by an arbitrary number of additional Gaussian peaks. This allows testing of very different and complicated SFR shapes. Also, by analogy to our approach in Paper IV,  subpopulations associated with these extra SF bursts can have special vertical kinematics, different from the thin-disk AVR (see Figure 2). 

Figure 3 shows predicted vertical and radial density profiles of the thin-disk mono-age populations (top) and radial variation of the scale lengths in age and metallicity bins (bottom). Parameters of the model which define its properties at the non-solar radii need further fine-tuning (see the next section on the AMR).

We present a publicly available tool, python package jjmodel, which can be used to perform stellar population synthesis within the JJ model framework. The package is complemented by three sets of isochrone grids generated by PARSEC, MIST, and BaSTI codes.