# 8. Milky Way-Like Galaxies¶

VICE’s milkyway object is designed handle multi-zone models of Milky Way-like galaxies in a flexible manner. Rather than requiring the user to construct them from scratch from the base multizone object, milkyway eases the burden by adopting a spatial configuration in which each zone represents an annulus of the Milky Way disk; these annuli are concentric from a radius $$R$$ = 0 to 20 kpc. The width of each annulus $$\Delta R$$ is a value the user may set upon construction of a milkyway object. As defaults, it adopts an observationally-motivated star formation law and a stellar migration prescription based on N-body simulations. For an in-depth example of an application of the milkyway object, we refer users to the models of Johnson et al. (2021) 1, for which these features were designed.

The default stellar migration model of the milkyway object is implemented in the vice.toolkit.hydrodisk.hydrodiskstars object. This object is built around data from the h277 simulation (Christensen et al. 2021) 2, a zoom-in hydrodynamical simulation ran from cosmological initial conditions which has made a number of appearances in the literature to date (e.g. Zolotov et al. 2012 3; Loebman et al. 2012 4, 2014 5; Brooks & Zolotov 6; Bird et al. 2021 7). Although h277 is currently the only simulation whose data is available to the hydrodiskstars object, it’s implementation could be extended to include others.

Note

The h277 star particle data is not included in VICE’s default distribution, but is available in its GitHub repository at vice/toolkit/hydrodisk/data. VICE will download these files automatically when a milkyway or hydrodiskstars object is created for the first time. With a decent internet connection, this process takes about one minute to complete, and does not need repeated. If this process fails, it may be due to not having administrator’s privileges; users in this situation should speak with their administrator, who would then be able to download these data with the following few lines in python:

>>> import vice


## 8.1. The Sample of Star Particles¶

The hydrodiskstars object, the default stellar migration prescription for the milkyway object, makes use of the birth radii, final radii, and birth times of star particles from hydrodynamical simulations, for which only the h277 simulation is currently available (see above). h277 did not record the birth radius of each star particle; however, each star particle does have an accurate age at each snapshot. The orbital radii of stars that are sufficiently young in their first snapshot should be good approximations of their birth radii since dynamical heating will have little effect in a short time interval. We have therefore restricted the sample of h277 star particles in the hydrodiskstars object to those with an age at first snapshot less than 150 Myr, adopting their galactocentric radius at first snapshot as their birth radius. The choice of 150 Myr makes no significant impact on the predictions of the milkyway object (see discussion in section 2.1 of Johnson et al. 2021).

Of the star particles that remain after imposing this cut, the oldest one has an age of 13.23 Gyr. Since h277 ran for ~13.7 Gyr, we have therefore subtracted 500 Myr from the birth times of all star particles, letting $$T$$ = 0 in the hydrodiskstars and milkyway objects correspond to $$T$$ = 500 Myr in h277, and placing the onset of star formation in these models at that time. As a consequence, these models support calculations of chemical evolution up to lookback times of 13.2 Gyr. Although this limit is not enforced in VICE, simulations on longer timescales using the milkyway object are highly likely to produce a segmentation fault.

We further restrict the sample of h277 star particles to only those with both formation and final radii of $$R \leq$$ 20 kpc, and to have formed within $$\left|z\right|\leq$$ 3 kpc of the disk midplane. These criteria ensure that our sample reflects only the star particles that formed in-situ, and can therefore be described by a disc GCE model. Although it’s possible some number of these star particles formed in a dwarf galaxy as it was being accreted by h277, these stars are few in number, and are only relevant at large radii and early times, where few stars form in nature anyway.

Based on a kinematic decomposition of these star particles, we exclude halo stars from the sample, but include those with bulge, pseudobulge, and disc-like kinematics. This ensures that all stars which can be attributed to the spatially confined regions reasonably defining a spiral galaxy disk can be modeled using the hydrodiskstars object. Altogether, these cuts yield a sample of 3,102,519 star particles from h277, accessible via the analog_data attribute of the hydrodiskstars object. For an analysis of the results of these cuts, we refer users to section 2.1 of Johnson et al. (2021).

## 8.2. Migration Models¶

As in many numerical models of galaxy evolution, stars in VICE are stand-ins for entire stellar populations. In the milkyway object (assuming the hydrodiskstars object is driving migration), they are said to be in a given zone if their radius is between the inner and outer edges of the annulus. At all times, VICE places their nucleosynthetic products and returned envelopes in the ISM of the annulus that they are in at that time.

The hydrodiskstars object assumes that star particles are born at the centers of their birth annuli. For a stellar population born at a time $$T$$ and galactocentric radius $$R$$, it first searches for star particles in the h277 sample (see above) which formed at $$T \pm$$ 250 Myr and $$R \pm$$ 250 pc. It then randomly selects a star particle from this subsample to act as an analog. The stellar population in the VICE model then adopts the change in orbital radius $$\Delta R$$, and moves their with an assumed time-dependence (see below). If no candidate analogs are found, the hydrodiskstars object widens the search to $$T \pm$$ 500 Myr and $$R \pm$$ 500 pc. If still no analog is found, it maintains the $$T \pm 500$$ Myr criterion, but finds the one with the smallest difference in birth radius, assigning that star particle as the analog. While this prescription allows stellar populations to be assigned analogs with significantly birth radii, this is only an issue for small $$T$$ and large $$R$$ where there are few star particles from h277, and where few stars form in nature anyway. When an h277 star particle is assigned as an analog, it is not thrown out of the sample of candidate analogs, in theory allowing a star particle to act as an analog for multiple stellar populations.

The hydrodiskstars object provides four models for the time-dependence of a star’s radius between its birth and the present day. The first case is one in which stars remain at their birth radius until the present-day, at which time they instantly migrate; mixing is a post-processing prescription in this scenario. The second case is a generalization of this in which the sudden migration to the present-day radius occurs at some time randomly drawn between the birth time and the end of the simulation. The third is one in which the radius change with a $$\sqrt{\text{age}}$$ dependence, and the final is one with a linear dependence on time. These are the “post-processing”, “sudden”, “diffusion”, and “linear” migration models from Johnson et al. (2021, see section 2.2 therein for further details). The hydrodiskstars object adopts “diffusion” as the default.

Here we illustrate these four migration models in the $$R-T$$ plane. While VICE’s internal h277 data supply a stellar population in VICE’s models with $$\Delta R$$, given one of these four assumptions and its birth radius (assumed to be in the center of its zone of birth), its radius at all remaining times is known. We emphasize that there is no N-body integration that goes into VICE’s milkyway models. Although these are four built-in presets that users may choose from, they are not restricted to these options. A custom migration scheme based on the h277 data can be implemented by subclassing the hydrodiskstars object, overriding its __call__ function, and setting the attribute mode to None.

## 8.3. The Default Star Formation Law¶

As a default, the milkyway object adopts the vice.toolkit.J21_sf_law to describe the relation between the surface density of star formation $$\dot{\Sigma}_\star$$ and the surface density of the interstellar medium $$\Sigma_\text{g}$$. This is also the star formation law adopted in Johnson et al. (2021). This star formation law is a broken power-law with two breaks; below $$\Sigma_\text{g} = 5\times10^6 M_\odot~kpc^{-2}$$, the relation scales as $$\dot{\Sigma}_\star \propto \Sigma_\text{g}^{1.7}$$. Between $$5\times10^6 M_\odot~kpc^{-2}$$ and $$2\times10^7 M_\odot~kpc^{-2}$$, it scales as $$\dot{\Sigma}_\star \propto \Sigma_\text{g}^{3.6}$$. Above $$2\times10^7 M_\odot~kpc^{-2}$$, the relation becomes linear.

The J21_sf_law calculates the star formation efficiency timescale $$\tau_\star$$ (usually referred to as a “depletion time” in the star formation, feedback, and interstellar medium literatures) for use with the singlezone and multizone objects. This timescales is defined as the gas density per unit star formation: $$\tau_\star \equiv \dot{\Sigma}_\star / \Sigma_\text{g}$$. To set the normalization of the star formation law, the J21_sf_law object assumes that in the linear regime, $$\tau_\star = \tau_\text{mol}$$, the value of $$\tau_\star$$ for a star forming reservoir where hydrogen is entirely in the molecular phase. Below surface densities of $$2\times10^7 M_\odot kpc^{-2}$$, the timescale increases in a piece-wise continuous manner. The J21_sf_law object affords users the ability to modify the surface densities at which there are breaks in the power-law, as well as the power-law indeces themselves. For additional discussion, we refer users to section 2.5 of Johnson et al. (2021).

The milkyway object adopts a scaling of the mass loading factor $$\eta \equiv \dot{M}_\text{out} / \dot{M}_\star$$ with galactocentric radius. The scaling is tuned such that the equilibrium abundances as a function of radius reflect a reasonable metallicity gradient in agreement with observational results from APOGEE (see section 2.3 of Johnson et al. 2021). The default scaling is based on alpha-elements (e.g., O, Ne, Mg) under a constant star formation history. The slope of the gradient is assumed to be mode([$$\alpha$$/H]) $$\propto$$ -0.08 $$kpc^{-2}$$, with the normalization set by mode([$$\alpha$$/H]) = +0.3 at $$R$$ = 4 kpc. This default scaling is implemented via the function vice.milkyway.default_mass_loading, and like the star formation law and stellar migration prescription, is only a default and can be overridden by the user if they so choose.

The milkyway object does not have any treatment for vertical structure of the star forming disk; that is, the boundaries between zones are purely radial. There are no zones off the disk midplane. This implicitly assumes that the star forming reservoir is well-mixed in the azimuthal and vertical directions, and that significant abundance differences occur only in the radial direction. By default it also neglects gas migration, because the Johnson et al. (2021) models for which it was designed focused instead on the impact of varying assumptions about stellar migration.

For further discussion of the milkyway object, we refer users to section 2 of Johnson et al. (2021).

Relevant Source Code

• vice/milkyway/milkyway.py

• vice/toolkit/J21_sf_law.py

• vice/toolkit/hydrodisk/hydrodiskstars.py

• vice/toolkit/hydrodisk/_hydrodiskstars.pyx

• vice/src/toolkit/hydrodiskstars.c

1

Johnson et al. (2021), MNRAS, 508, 4484

2

Christensen et al. (2012), MNRAS, 425, 3058

3

Zolotov et al. (2012), ApJ, 761, 71

4

Loebman et al. (2012), ApJ, 758, L23

5

Loebman et al. (2014), ApJ, 794, 151

6

Brooks & Zolotov (2014), ApJ, 786, 87

7

Bird et al. (2020), arxiv:2005.12948

8

Johnson et al. (2021), MNRAS, 508, 4484