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
>>> vice.toolkit.hydrodisk.data.download()
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).
8.4. Additional Parameters¶
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), arxiv:2103.09838
- 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), arxiv:2103.09838