Vol.0 (20xx) No.0, 000–000

11institutetext: 1 Department of Physics and Astronomy, N283 ESC, Brigham
Young University, Provo, Utah 84602, USA<EMAIL_ADDRESS>
# Comparing the structural parameters of the Milky Way to other spiral
galaxies

Jacob A. Guerrette 11 Aleksandr V. Mosenkov 11 Dallin Spencer 11 Zacory D.
Shakespear 11

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The structural parameters of a galaxy can be used to gain insight into its
formation and evolution history. In this paper, we strive to compare the Milky
Way’s structural parameters to other, primarily edge-on, spiral galaxies in
order to determine how our Galaxy measures up to the Local Universe. For our
comparison, we use the galaxy structural parameters gathered from a variety of
literature sources in the optical and near-infrared wavebands. We compare the
scale length, scale height, and disk flatness for both the thin and thick
disks, the thick-to-thin disk mass ratio, the bulge-to-total luminosity ratio,
and the mean pitch angle of the Milky Way’s spiral arms to those in other
galaxies. We conclude that many of the Milky Way’s structural parameters are
largely ordinary and typical of spiral galaxies in the Local Universe, though
the Galaxy’s thick disk appears to be appreciably thinner and less extended
than expected from zoom-in cosmological simulations of Milky Way-mass galaxies
with a significant contribution of galaxy mergers involving satellite
galaxies.

###### keywords:

Galaxy: disc – galaxies: fundamental parameters – Galaxy: structure

##  1 Introduction

Galaxies are complex conglomerations of stars, gas, dust, and dark matter that
can have a variety of different shapes, sizes, and other general properties.
Due to this complexity, galaxies can contain a number of different structural
components, including active galactic nuclei, bulges, bars, rings, lenses,
disks, and stellar halos that determine their morphology (Buta 2011). By
studying the structural parameters of galaxies, we get valuable information on
their formation history through galaxy scaling relations, as different
groupings of galaxies on these scaling relations may point to different
scenarios of their formation and evolution (see Hohl 1978; Combes & Sanders
1981; Michel-Dansac & Wozniak 2004; Knapen et al. 2004; Knapen & Comerón
2013). Furthermore, by studying galaxies that are relatively similar in
morphology, we can begin to form an idea of the sequence of events required to
create that variety of galaxies (Muñoz-Mateos et al. 2015).

Due to our location within one of its spiral arms, the Milky Way, while being
our own Galaxy, remains somewhat of a mystery to us. The dust in the
interstellar medium (ISM) blocks our view of much of the Galactic plane in
addition to scattering what light we do receive (see Draine 2003 and
references therein). Because of this, the specific structural parameters
describing our Galaxy are difficult to study.

Luckily, due to infrared-submillimeter-radio observations where our Galaxy is
largely transparent, we can explore its stellar, dusty, and gaseous content in
great detail. In a review about our Galaxy’s parameters, Bland-Hawthorn &
Gerhard (2016) state, however, that there is much uncertainty about the
Galaxy’s structural parameters, especially parameters such as disk scale
length ($\sim$2$-$4 kpc; see e.g. Kent et al. 1991; Ruphy et al. 1996; Drimmel
& Spergel 2001; López-Corredoira et al. 2002; Benjamin et al. 2005; Reylé et
al. 2009) and scale height ($\sim$150$-$375 pc; see e.g. Kroupa 1992; Gould et
al. 1996; Ojha 2001; Siegel et al. 2002; Jurić et al. 2008). This uncertainty
in the Galaxy’s structural parameters makes it difficult to compare our Galaxy
with other similar spiral galaxies.

Recently, however, using unWISE 3.4 µm integrated photometry (Lang 2014),
Mosenkov et al. (2021, henceforth M21) fitted multiple disk models of the
Milky Way to determine which model best matches the observational data.
Importantly, they used the full treatment of the sky projection and dust
extinction effects in their 3D modelling of the Galaxy structure. Another
advantage of their work was that their approach of photometric decomposition
was essentially the same as has been used for external galaxies, making the
comparison of our Galaxy with other galaxies more reliable. They revealed that
the best fitting model contained two disks with exponential vertical density
profiles, outer flaring, warping, and two over-densities on either side of the
Galactic center represented by two blobs (see their table 1). Although this
was the best-fitting two-disk model, a similar model (with the only difference
being a sech2 vertical density profile instead of an exponential one,
henceforth called an isothermal disk) was a close second-best fit. Here, the
simple exponential model (Wainscoat et al. 1989; Xilouris et al. 1999;
McMillan 2011; Binney 2012) has a luminosity density profile $j(R,z)$ given by

$j(R,z)=j_{0}~{}\text{e}^{-R/h_{\mathrm{R}}-|z|/h_{\mathrm{z}}}$ (1)

where $j_{0}$ is the central luminosity density of the disk, $h_{\mathrm{R}}$
is the disk scale length, and $h_{\mathrm{z}}$ is the disk scale height.
Similarly, the isothermal model (Spitzer 1942; Camm 1950; van der Kruit &
Searle 1981a, b, 1982) has a luminosity density distribution given by

$j(R,z)=j_{0}~{}\text{e}^{-R/h_{\mathrm{R}}}~{}\text{sech}^{2}[|z|/(2h_{\mathrm{z}})].$
(2)

Both functional dependencies are often used in the literature to describe the
disk structure, but for our Galaxy none of the models provides a significantly
better fit. Therefore, throughout this paper, we will simultaneously use both
models for comparison purposes.

Figure 1: Semi-major axis at $\mu_{3.6}=25.5$ AB mag arcsec-2 for face-on
($i<40\degr$) disk galaxies from S10. Data points are colored based on the
galaxy’s numerical Hubble stage value. A trendline is included, along with a
$\pm 1\sigma$ spread. The Milky Way semi-major axis value of 15.1 kpc was
calculated using prescriptions from section 2.1 in Pilyugin et al. (2023)
adopting the surface stellar mass density $\Sigma_{*,R_{0}}=38$ M⊙ pc-2 at the
solar galactocentric distance $R_{0}=8.0$ kpc. The Milky Way mass value is
taken from Bland-Hawthorn & Gerhard (2016). The Galaxy is denoted by the blue
square, corresponding to a Type value of 5.

In this paper, our ambition is to compare some of the most recent Milky Way
structural models to those obtained for other spiral galaxies in the Local
Universe, helping us conclude how the Milky Way fits into the general picture.
This is done through studying the general statistics of spiral galaxies, for
which we have structural parameters from the literature, and by exploring
general galaxy scaling relations for a large range of stellar masses (see e.g.
Mosenkov et al. 2010, 2014) with the Milky Way superimposed. For example, in
Figure 1, one can see that the Milky Way perfectly follows the general trend
of the mass–size relation. We utilize one-dimensional, two-dimensional, and
three-dimensional modeling to compare the structural parameters of our Galaxy
to external edge-on and non-edge-on spiral galaxies. Our goal is to consider
models obtained with the same or similar decomposition methods for maximum
accuracy. By comparing the Milky Way structural models with those for other
galaxies, we can determine how normal or abnormal the Milky Way is, which, in
turn, can give us a better understanding of its evolutionary status.

The paper is organized as follows. In Section 2, we present galaxy samples
with available photometric decomposition taken from the literature for our
comparative analysis. In Section 3, we consider various models of our Galaxy
and external galaxies in the optical. In Section 4, we detail the single-disk
data comparisons in the near-infrared (NIR). In Section 5, we continue
detailing data comparisons in the NIR but for the two-disk models. In Section
6, we discuss how the bulge/bar in our Galaxy compares to the bulges in
external galaxies. In Section 7, we discuss how the mean pitch angle of the
Milky Way compares to external galaxies with similar bulge-to-total flux
ratios and similar morphological types. Finally, in Section 8, we summarize
our findings.

##  2 The samples

The structural parameters of a galaxy can vary with wavelength due to dust
attenuation and stellar population gradients (see de Jong 1996; MacArthur et
al. 2003; Taylor et al. 2005; Fathi et al. 2010). Thus, it is important to
analyze galaxy models in both optical and NIR wavelengths in order to best
determine how typical the Milky Way is among similar galaxies at wavelengths
where a mixed stellar population radiates (the optical part of the spectrum)
and where old stellar population makes up the bulk of the stellar mass (NIR).
For comparison purposes, we only use models obtained by means of the same or
similar methods and within similar wavebands.

We employ single-disk and two-disk galaxy models (Salo et al. 2015; Comerón et
al. 2018) and galaxy pitch angles (Díaz-García et al. 2019) from the Spitzer
Survey of Stellar Structure in Galaxies (S4G, Sheth et al. 2010; Muñoz-Mateos
et al. 2013; Querejeta et al. 2015) and single-disk models from Bizyaev &
Mitronova (2009) and Mosenkov et al. (2010) based on the Two Micron All Sky
Survey (2MASS, Skrutskie et al. 2006) as our tools for comparing these NIR
models of the Milky Way to other galaxies. For these various sources, we take
orientation into account, separately comparing the Milky Way to edge-on
(inclinations greater than or equal to roughly 80 degrees; Salo et al. 2015)
and non-edge-on (inclinations less than 80 degrees) galaxies.

To obtain an accurate comparison in the optical, we also utilize the optical
Milky Way model from Natale et al. (2022). They created a radiative transfer
(RT) model of the Milky Way in the V band, from which we extracted the
structural parameters of a general stellar disk. For comparison, we gathered a
sample of RT models of galaxies in optical wavelengths from four papers:
Bianchi (2007), De Geyter et al. (2014), Peters et al. (2017), and Xilouris et
al. (1999). In addition, for our comparative analysis, we exploit $\sim$2800
3D models of edge-on galaxies from Bizyaev et al. (2014), who used $r$-band
data from the Sloan Digital Sky Survey (SDSS, York et al. 2000; Eisenstein et
al. 2011).

In Table 1, we provide brief information on the samples used. Once again, we
emphasize that in almost every case, we only compare the structure of our
Galaxy in a given waveband to the structure of other galaxies observed within
that same or a close waveband. This same restriction applies to the methods
used to model these galaxies, as well as to the complexity of the model. By
restricting these factors, we hope to isolate the structural parameters of the
galaxies we are analyzing. In the next sections, we make a comparison of the
Milky Way with other galaxies in the optical and NIR regions of the spectrum
using the aforementioned data sets.

Below we provide brief summaries of the data samples used, including details
on the modelling techniques employed.

Bianchi (2007) used RT modeling to retrieve the structural parameters of 7
galaxies in the V and K’ bands. They performed their own observations at the
3.5-m TNG (Telescopio Nazionale Galileo) telescope located at the Roque de Los
Muchachos Observatory in La Palma, Canary Islands. Their method involved the
iterative production of a model image of a galaxy which is then compared with
the observed image in an attempt to derive the structural parameters that
minimize $\chi^{2}$ (see their section 3 for an in-depth analysis of their
methods).

Bizyaev & Mitronova (2009) analyzed 139 galaxies from the Revised Flat
Galaxies Catalog (RFGC, Karachentsev et al. 1999) using 2MASS images
(Skrutskie et al. 2006) in the J, H, and Ks wavebands. To obtain the
structural parameters of these galaxies, they used the same method described
in section 3 of Bizyaev & Mitronova (2002) which is based on analyzing
photometric profiles drawn parallel to the major and minor axes of a galaxy at
a one-pixel interval (see their section 2 for further details).

Bizyaev et al. (2014) investigated the structural parameters of 5747 genuine
edge-on galaxies that were suitable for automatic analysis from SDSS (York et
al. 2000), RFGC (Karachentsev et al. 1999), the Third Reference Catalog of
Bright Galaxies (RC3, de Vaucouleurs et al. 1991), the EFIGI catalogue
(Baillard et al. 2011), and GalaxyZoo (Lintott et al. 2011) in the g, r, and i
wavebands. Of these 5747 galaxies, 3912 were fitted using a 3D model by
estimating a set of 1D galaxy photometric profiles and accounting for dust
absorption. We utilized 2789 galaxies from this sample that had radial
velocity data to perform redshift calculations and convert the structural
parameters to kiloparsecs.

Comerón et al. (2018) examined 141 galaxies using images from S4G and its
early-type extension (Sheth et al. 2010, 2013) in the 3.6 micron waveband.
Their complex fitting procedure involved steps the following steps: 1) fitting
the vertical surface-brightness profiles of galaxies (while accounting for the
gravitational effect of the gas disk), 2) fitting the axial surface-brightness
profiles, 3) refitting the vertical profiles while accounting for the presence
of central mass concentration (CMC) light, 4) producing an axial surface-
brightness profile for heights dominated by the thin and thick disks, and 5)
calculating the mass of each component (see their section 3 for an in-depth
breakdown of their methods).

De Geyter et al. (2014) analyzed 12 galaxies from the Calar Alto Legacy
Integral Field Area survey (CALIFA, Sánchez et al. 2012) in the g, r, i, and z
wavebands. To create RT models of these galaxies, they used the FitSKIRT
automated fitting routine (De Geyter et al. 2013) which couples the SKIRT
Monte Carlo RT code (Baes et al. 2011) with GAlib, the genetic algorithms-
based optimization library. FitSKIRT takes an arbitrary number of images and
fits a RT model with an arbitrary number of dust/stellar components (see their
section 3 for further information on their modeling).

Díaz-García et al. (2019) studied 391 spiral galaxies from S4G (Sheth et al.
2010) in the 3.6 micron waveband. Of these 391 galaxies, we use models for 233
of them that are classified as multi-armed or grand design. To model these
galaxies, Díaz-García et al. (2019) used deprojected images of the galaxies to
calculate global pitch angles from average measurements of individual
logarithmic spiral segments. Additionally, for a small subsample of galaxies
with measurements available in the literature, they performed a 2D Fourier
analysis by using Fourier transform spectral analyses of the deprojected
galaxy images (see their section 3 for more details on these methods).

Mosenkov et al. (2010) investigated the structural parameters of 175 edge-on
galaxies in the Ks waveband selected from the 2MASS-selected Flat Galaxy
Catalog (2MFGC, Mitronova et al. 2004). They used the BUDDA decomposition code
(de Souza et al. 2004) to fit the parameters of the Sérsic bulge and the edge-
on disk. BUDDA takes into account the entire 2D image of a galaxy to find the
optimal galaxy model by employing the multidimensional downhill simplex method
(Press et al. 1989).

Mosenkov et al. (2018) examined 7 large edge-on galaxies from the Herschel
(Pilbratt et al. 2010) Observations of Edge-on Spirals (HEROES) sample. They
used available databases for optical and NIR sky surveys to gather images of
these galaxies, then used a three step process to model the galaxies: first,
they fit a general model to each galaxy. Then, they used FitSKIRT to fit a
dust model to each galaxy. Finally, they performed SKIRT panchromatic
radiative transfer simulations from their FitSKIRT models to obtain the final
models.

Peters et al. (2017) used their own observations from 10 telescopes in their
modeling and analysis of 8 galaxies (with one galaxy being split into left and
right halves) in the B, V, R, I, J, H, K, Wise1, and Wise2 wavebands. Similar
to De Geyter et al. (2014), they utilized the FitSKIRT automated fitting
routine to model the structural parameters of their galaxies.

Pinna et al. (2023) studied 24 galaxies from the AURIGA
project111https://wwwmpa.mpa-garching.mpg.de/auriga/ (Grand et al. 2017),
which performs zoom-in magneto-hydrodynamical cosmological simulations of 30
Milky Way-mass galaxies. They projected these galaxies in an edge-on view,
then spatially binned them using Voronoi binning (Cappellari & Copin 2003).
Finally, they decomposed their edge-on galaxies into two disk components by
fitting vertical-brightness profiles to the galaxies to obtain their models.

Sheth et al. (2010) analyze 2331 galaxies observed with the Spitzer space
telescope (Werner et al. 2004) at 3.6 and 4.5µm. While their method for
obtaining models is complex and best described by their section 5, the summary
is as follows: for each galaxy, they use the initial observations to gather
parameters such as diameter and axial ratio. Then, they measure things like
total magnitude and the azimuthally-averaged radial profile of the surface
brightness. Finally, they perform a decomposition using galfit to obtain the
main constituent stellar components of each galaxy.

Salo et al. (2015) examined 2352 galaxies from S4G (Sheth et al. 2010) in the
3.6 and 4.5 micron wavebands. To model these galaxies, they performed 2D
surface brightness structural decompositions using galfit3.0 (Peng et al.
2010) which uses the Levenberg-Marquadr algorithm to minimize the weighted
residual $\chi_{\nu}^{2}$ between the observations and the models.
Additionally, they performed human-supervised multi-component decompositions
for each galaxy (see their section 2 for their decomposition pipeline).

Finally, Xilouris et al. (1999) used their own observations to create RT
models for 5 galaxies in the B, V, I, J, and K wavebands. These RT models were
based on surface photometry (using the same process as Kylafis & Bahcall 1987)
and were used to input model parameters into their equations 1-6 to find the
parameters that best match the observed surface photometry of each galaxy.

Table 1: Details on the various data samples utilized. For subsamples from the
same source paper, a number in square brackets is added after their label to
differentiate between different subsamples. Details on why we only use a
specific number of galaxies from a paper are given in the caption.

Label | Method | Reference | # of Galaxies | Waveband
---|---|---|---|---
| of fitting |  |  | used
B07 | RT | Bianchi (2007) | 7 | V
BM09 | 1D | Bizyaev & Mitronova (2009) | 139 | Ks
B14 | 3D | Bizyaev et al. (2014) | 2789 | r
C181 [1] [2] | 1D | Comerón et al. (2018) | 96, 135 | 3.6 µm
DG14 | RT | De Geyter et al. (2014) | 12 | $g,r,i,z$
DG19 | 2D | Díaz-García et al. (2019) | 233 | 3.6 µm
M10 | 2D | Mosenkov et al. (2010) | 175 | Ks
M18 | RT | Mosenkov et al. (2018) | 7 | Several Optical-NIR bands
P17 | RT | Peters et al. (2017) | 9 | Several UV-MIR bands
P23 | 1D | Pinna et al. (2023) | 24 | V
S10 | 2D | Sheth et al. (2010) | 222 | 3.6 µm
S152 [1] [2] [3] | 2D | Salo et al. (2015) | 1287, 217, 128, | 3.6 µm
[4] [5] |  |  | 115, 576 |
X99 | RT | Xilouris et al. (1999) | 5 | V

1C18 [1] comprises edge-on two-disk galaxies; [2] consists of galaxies with
available disk mass data.
2S15 [1] comprises non-edge-on single-disk galaxies; [2] consists of edge-on
single-disk galaxies; [3] includes edge-on two-disk galaxies; [4] contains
galaxies with disk mass data; [5] is galaxies with B/T ratio data.

##  3 The single-disk models in the optical

Table 2: MW model structural parameter values from various sources. Scale
length, scale height, and radius values are given in kpc, the stellar masses
are given in solar masses, ratio values are unitless, magnitudes are given in
mag, and the absolute value of the mean pitch angle is given in degrees.

Model Type | Reference | Parameters | Values
---|---|---|---
RT (Optical) 1-disk | Natale et al. (2022) |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 3.10, 0.30, 10.33
NIR 1-disk Exp. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 3.19, 0.45, 7.02
NIR 1-disk Iso. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 3.02, 0.48, 6.36
NIR 2-disk Thick Exp. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 3.22, 0.71, 4.55
NIR 2-disk Thick Iso. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 2.79, 0.98, 2.85
NIR 2-disk Thin Exp. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 2.55, 0.25, 10.11
NIR 2-disk Thin Iso. | M21 |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio | 2.52, 0.24, 10.57
Optical | Pilyugin et al. (2023) |  $M_{*}$, $R_{\mathrm{opt}}$ |  $5.2\times 10^{10}$, 12.0
Optical | Bland-Hawthorn & Gerhard (2016) |  r-band Abs. Mag. | -21.64
NIR 2-disk | Bland-Hawthorn & Gerhard (2016) | $M_{*}$ | $4.1\times 10^{10}$
NIR 2-disk1 | M21 |  $B/T$ Max, $M_{\mathrm{T}}/M_{\mathrm{t}}$, $M_{\mathrm{T}}/M_{\mathrm{t}}$ | 0.12, 0.42, 1.34
Spiral Structure | Vallée (2015) | $|\phi|$ | 13.1

1Two values are included for the disk mass ratio due to uncertainty in the
local thick-to-thin disk surface density ratio (see Sect.

5.1 for further explanation).

Table 3: Quartiles for each model group. For model groups with multiple
parameters, the order of the quartile data matches the order of the listed
parameters. $h_{\mathrm{R}}$ and $h_{\mathrm{z}}$ values are in units of kpc,
while ratio values are unitless.

Model Type | Parameters | Quartiles
---|---|---
Optical 1-disk |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio |  $4.88^{+2.10}_{-1.62}$, $0.44^{+0.19}_{-0.15}$, $10.68^{+4.83}_{-3.09}$
NIR 1-disk Non-Edge-On | $h_{\mathrm{R}}$ | $2.38^{+0.88}_{-0.75}$
NIR 1-disk Edge-On |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio |  $3.75^{+1.43}_{-1.39}$, $0.43^{+0.17}_{-0.15}$, $8.52^{+2.28}_{-2.21}$
NIR 2-disk Thick |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio |  $3.07^{+0.84}_{-0.81}$, $0.60^{+0.46}_{-0.15}$, $4.83^{+2.23}_{-1.58}$
NIR 2-disk Thin |  $h_{\mathrm{R}}$, $h_{\mathrm{z}}$, Ratio |  $1.38^{+0.77}_{-0.49}$, $0.17^{+0.07}_{-0.04}$, $9.22^{+4.48}_{-3.28}$
NIR 2-disk | Mass Ratio | $0.95^{+1.07}_{-0.43}$
NIR Edge-On | $B/T$ | $0.47^{+0.15}_{-0.25}$
NIR Non-Edge-On | $B/T$ | $0.14^{+0.16}_{-0.08}$

We begin by comparing models obtained in optical wavelengths primarily using
RT modeling techniques. RT models simulate how photons within the galaxy
propagate and interact with the galactic interstellar medium. Natale et al.
(2022) (from which we gathered our RT Milky Way parameter values) created
images from such RT simulations to compare them to the observed surface
photometry of the Milky Way in order to find the most accurate RT model. We
gathered the RT model parameters for the stellar disks of other galaxies from
X99, B07, DG14, and P17 (see Table 1 for labels, information, and references
on any papers used in the making of this paper’s figures). In total, we have
33 RT models for external galaxies, which cannot by any means be a
representative sample of spiral galaxies. Unfortunately, detailed RT modeling
is very time consuming (see e.g. details in Mosenkov et al. 2018), so, to our
best knowledge, this is the largest combined sample of galaxies with available
RT models in the optical.

Here, we also use models from B14, who exploited optical observations from the
Edge-on Galaxies in the SDSS (EGIS) catalogue. They performed both 1D and 3D
photometric decomposition, but in this paper we only utilize their $r$-band 3D
models that account for dust attenuation.

Although we typically cannot study the Milky Way in optical wavelengths due to
the dust within the Galactic plane, the Natale et al. (2022) RT model allows
one to indirectly estimate our Galaxy’s radial scale length through a scaling
factor, making their model invaluable to our comparisons within optical
wavebands. Their scaling factor comes from table E.1 of Popescu et al. (2011)
who performed RT calculations for a number of spiral galaxies to find a
consistent model. They found that the disk scale length varies with
wavelength, so they normalized the scale lengths in their table E.1 to the
B-band scale length. Natale et al. (2022) then use these normalization factors
to estimate the V-band scale length of the Milky Way based on the K-band scale
length they constrained from their data, giving a value of 3.10 kpc. Popescu
et al. (2011) also found that the disk scale height does not vary with
wavelength, so Natale et al. (2022) utilize their own constrained K-band scale
height of 0.30 kpc for their V-band scale height.

In Figure 2, we depict the distributions of the selected galaxies by the disk
scale length, disk scale height, and the ratio of the two. We point out that
the stellar parameters in X99 and B07 are provided in different wavebands (we
use the $V$ band here), while DG14 and P17 provide the global properties of
the stellar fits based on all wavebands utilized. Despite this difference in
the output parameters, we see that the general distributions for the RT and
EGIS models are roughly consistent among the datasets under consideration,
although the low number of RT data points creates very spiky distributions.

Figure 2: Histograms of the scale length (left), scale height (middle), and
ratio between the two (right), with the respective Milky Way model parameter
values from Natale et al. (2022) highlighted. Edge-on single-disk galaxy
models in the optical are taken from ; ; ;  (‘RT Edge-On’) and EGIS (3D, B14).

Figure 3: Left: Absolute r- or R-band magnitude versus scale length for our
optical datasets: B14 (’EGIS 3D’) and ; ; ;  (’RT’). A trendline is included,
along with a $\pm 1\sigma$ spread. Milky Way model magnitude is from Bland-
Hawthorn & Gerhard (2016) and its scale length is from Natale et al. (2022).
The Milky Way data point is denoted by the red star. Right: Optical radius
(semi-major axis of the 25 mag arcsec-2 isophote in the $B$ band taken from
HyperLeda) versus stellar mass for face-on ($i<40\degr$) disk galaxies from
S10 and 24 Milky Way-mass galaxies from the AURIGA zoom-in cosmological
simulations from Pinna et al. (2023). Milky Way values are from Pilyugin et
al. (2023), with the data point being marked by the red star.

Using the combined data in the optical, we can compare the structural
parameters of the Milky Way (see Table 2 for all main Milky Way parameter
values) to those for external disk galaxies. From our dataset’s quartiles Q1
(25%) and Q2 (median) scale length cutoff values (see Table 3 for this and all
future quartile cutoff values), we see that the Milky Way’s scale length is
lower than average, falling just below the Q1 cutoff. On the other hand, the
Q1 and median scale height cutoff values tell us that the Milky Way’s scale
height is average (close to Q2).

When we compare the flatness of the Milky Way to other galaxies (the right
plot in Fig. 2, the scale length divided by the scale height), the Milky Way’s
value of 10.33 seems to fall around the median value. Indeed, the dataset’s Q1
and median cutoffs indicate that the Milky Way’s ratio value is close to Q2,
making it quite average.

We thus conclude that the Milky Way’s scale length is slightly below average
in optical bands, although its scale height and flatness ratio are both
normal. This conclusion is also confirmed by the deviation of the Milky Way
from the general trend on the luminosity–scale length relation — our Galaxy’s
disk scale length appears to be appreciably smaller compared to those of disk
galaxies with similar luminosities (see Fig. 3, left plot). The optical radius
$R_{\mathrm{opt}}=12$ kpc of our Galaxy, estimated in Pilyugin et al. (2023),
is however, in good agreement with those of disk galaxies with similar stellar
masses. However, as one can see from Figure 3 (right plot), Milky-Way-like
galaxies with similar masses from AURIGA zoom-in cosmological simulations
(Grand et al. 2017) have significantly larger (0.3 dex) optical radii. We will
return to this fact in Section 8.

##  4 The single-disk models in the NIR

In this section, we present our analysis of single-disk galaxy models created
in NIR wavelengths, specifically the Ks and 3.6 $\mu$m bands. As a note, we
neglect the effects of dust attenuation and polycyclic aromatic hydrocarbon
emission in the case of the 3.6 $\mu$m waveband in these models (see M21). In
Section 3 where we explore edge-on disk models in the optical, all galaxy
models available in the literature consist of a single disk. Fitting two-disk
models in the optical is problematic because of the strong dust attenuation in
the plane of the galaxy where a thin disk may be present. In the NIR, the
observed vertical structure of an edge-on galaxy is much less distorted by the
internal extinction, so a large number of both single-disk and two-disk models
of edge-on galaxies have been fitted in the literature. Therefore, we
differentiate these one-disk and two-disk models and describe them in separate
sections. In this section, we focus on the NIR single-disk models (including
those fitted for non-edge-on galaxies for comparison), whereas the NIR two-
disk models are analyzed separately in Section 5.

As detailed in Section 1, for our Milky Way values, we make use of two single-
disk models from M21 obtained in 3.4 $\mu$m: one with a single exponential
disk and another with a single isothermal disk, both with outer flaring.
Having two points of reference for the Milky Way is beneficial in some
instances, as when both are out of the ordinary, it is indicative that the
Milky Way might have a non-typical structure. Also, it is known from the
literature that some vertical galaxy profiles are better fit with an
exponential disk, and some with an isothermal disk (see e.g. van der Kruit &
Freeman 2011, and references therein). For our Galaxy, M21 find that a single
isothermal disk slightly better agrees with the 3.4 $\mu$m Galaxy composite
image than a similar model but with an exponential disk. However, as noted in
Section 1, the difference between the exponential and isothermal disk models
is not significant enough to consider one of them more robust than the other.

For external galaxies, we make use of 2D galfit (Peng et al. 2002, 2010)
models obtained by S15 in the framework of the S4G project. We divide their
single-disk models into two groups (see Table 1): 1287 non-edge-on disk models
and 217 edge-on models. We do so because edge-on and non-edge-on galaxies have
different observed properties due to their inclination angle, such as the
levels of PAH + dust emission and attenuation (although at 3.6 $\mu$m the last
effect is drastically reduced), the visibility of different structural
features (such as spiral arms, rings, bars) which can be easily revealed at
specific inclination angles, and, of course, the resolution of the vertical
structure which can only be studied for edge-on or close to edge-on galaxies.

In addition, we also consider edge-on single-disk models from BM09 and M10
that were obtained in the 2MASS Ks waveband.

Figure 4 demonstrates a distribution of the non-edge-on single-disk scale
lengths for 1287 galaxy models from S15. Since these are non-edge-on galaxies,
we have no data values for their scale heights, so we only show a histogram
based on their scale lengths. We utilize the exponential and isothermal
single-disk Milky Way model scale length values from M21 as a comparison.

Figure 4: Distribution of the disk scale length for non-edge-on galaxies from
S15 [1] compared to the exponential (red solid line) and isothermal (red
dashed line) Milky Way model scale lengths from M21.

Let us now turn to edge-on galaxies for which NIR single-disk models are
available in the literature. Here we consider 139 galaxies from BM09, 175
galaxies from M10, and 217 edge-on single-disk models from S4G. The comparison
of these datasets for the disk scale length, disk scale height, and the ratio
of the two is displayed in Figure 5. As one can see, the edge-on S4G models
appear to have lower values, on average, than the other two datasets for both
scale length and scale height. The agreement between all M10 and BM09
distributions is good. Although the S4G models were obtained using Spitzer 3.6
$\mu$m observations while M10 and BM09 used 2MASS Ks data, the significant
difference between the 2MASS and Spitzer data cannot be explained by the
difference of the wavebands used. The Ks (2.16 $\mu$m) and 3.6 $\mu$m
wavebands are close and both map the old stellar population with little
effects from the galaxy ISM. Therefore, the main reason for this discrepancy
should be the morphological difference of the galaxies contained in the
samples. To confirm this, we present Figure 6, which compares the absolute
magnitudes and morphological types of the samples. We note here that while the
classification of morphologies of edge-on galaxies is more subjective than
that of face-on galaxies, there are methods that can be used to determine
these morphologies, such as calculating the contribution of the bulge mass to
the total mass (Obreschkow et al. 2009). For the galaxies in Figure 6, the
morphology data comes from HyperLeda222http://leda.univ-lyon1.fr/. As one can
see, the edge-on galaxies from S15 host systematically less luminous stellar
disks and are of later morphological types as compared to the BM09 and M10
samples. Due to this bias within the S15 sample, we do not include it within
our calculation of the quartiles for this comparison.

Figure 5: Histograms of the scale length (left), scale height (middle), and
ratio between the two (right) for the NIR edge-on single-disk galaxy models
from BM09, M10, and S15 [2]. The single-disk exponential and isothermal Milky
Way model parameter values from M21 are highlighted by the solid and dashed
red lines, respectively.

Figure 6: Comparison of the absolute magnitudes (left) and morphological types
(right) of galaxies from our NIR edge-on single-disk galaxy comparison (; ;
[2]). Milky Way values for both K-band absolute magnitude (Drimmel & Spergel
2001) and Type (Georgelin & Georgelin 1976) are highlighted.

Using the histograms presented in this section, we can draw the following
conclusions. Beginning with the non-edge-on galaxies in Figure 4, it is clear
that both values of the Milky Way’s scale length (the exponential and
isothermal model values) are slightly larger than average, but not by a
substantial amount. This is proven by the median and Q3 cutoff values for this
dataset, showing that both Milky Way model scale length values are only
slightly higher than the median.

Moving on to the edge-on single-disk models in the NIR (see Fig. 5), we see
more of the pattern of normality previously established by the non-edge-on
single-disk NIR galaxy models. Both scale length values are slightly below the
median but above the Q1 cutoff value. Thus, it is evident that the scale
length values are typical of spiral galaxies. Similarly, both scale height
values fall just above the median value.

The two ratios of the scale length and scale height values for the Milky Way
in Figure 5 are largely unremarkable. The exponential model value is slightly
larger than the isothermal model value, although both are close to Q2 and, as
such, can be considered normal.

We can thus conclude that in NIR single-disk decompositions, the Milky Way’s
structural parameters are all well within the range of typical values. We also
demonstrate this in Figure 7 – the Milky Way perfectly obeys the correlation
between the disk scale length and stellar mass.

Figure 7: Stellar mass versus scale length for our NIR edge-on single-disk
data samples (S15 [2]; BM09; M10), though the stellar masses for S15 [2] come
from S10. The Milky Way mass value comes from Bland-Hawthorn & Gerhard (2016),
while the exponential (red circle) and isothermal (red star) scale length
values come from M21.

##  5 Two-disk NIR Waveband Galaxy Model

Figure 8: Scale length and scale height comparisons of the same 34 galaxies
from S15 [3] and C18 [1]. A dashed orange line representing $y=x$ is included
to aid in the comparison.

We now move on to the two-disk scale length and scale height comparisons. For
these comparisons, we utilize edge-on two-disk decomposition models created in
the NIR waveband, the only spectral region where we have a significant number
of galaxy models consisting of two photometrically-decoupled stellar disks.
The difference between the surface brightness profiles of the thin and thick
disks is that the scale height of the thick disk is significantly larger than
the scale height of the thin disk in a photometric rather than chemical
context (Burstein 1979; Walker et al. 1996; Schönrich & Binney 2009; Bournaud
et al. 2009; Beraldo e Silva et al. 2021). Such a two-disk model of the Milky
Way is also supported by a double exponential vertical number density profile
(Yoshii 1982; Gilmore & Reid 1983; Chen et al. 2001; Ojha 2001; Chrobáková et
al. 2020). There are other distinctive features of the thin and thick disks
such as the chemical composition and kinematics, but the surface brightness
profile of each is of the most interest to us in this study. Within M21, this
two-disk decomposition model best fits the observed data of the Milky Way,
with the exponential model being slightly more accurate than the isothermal
model (see Sect. 1). As such, the analysis of the scale length and scale
height in this section will be of the most worth for exploring the formation
scenario of the thin and thick disks in our Galaxy in comparison with other
spiral galaxies.

As with the single-disk models, we begin by gathering both the exponential and
isothermal models of the Milky Way from M21. These models also contain outer
flaring, warping, and two blobs on either side of the Galactic center
representing most apparent over-densities. In comparison to the other Milky
Way models we have used, the disk scale height and disk flatness values for
these two models are more different from each other, which provides for some
interesting situations in our analysis where one model’s value appears
abnormal and the other appears average.

For external galaxies, we gather our comparison data from four different
sources: S15, C18, M18, and P23. We note here that P23 only provide scale
height data and gather their data using optical wavebands, so the comparison
to their values is not entirely justified. However, we include them in order
to show the large discrepancy between the Milky Way and other Milky Way-mass
galaxies, so while the details might not be accurate, the overall comparison
still provides important information. As with the 2MASS data from the edge-on
single-disk NIR galaxies, both S15 and C18 provide models of S4G galaxies, but
because they utilize different fitting methods, their values are different and
thus we consider both datasets separately. Specifically, C18 utilize a more
extended point spread function (PSF) than S15 which likely causes the thin and
thick disk scale heights of galaxies contained in both datasets to be
systematically smaller in the C18 models (see Fig. 8 where we compare common
galaxies in both samples). The influence of the PSF on measured galaxy
observables was studied in detail in Sandin (2014) and Sandin (2015) which
indeed suggest that the vertical structure of edge-on galaxies may be
significantly affected by the instrumental scattered light. Also, the
difference in the results may be in the different methods used in these
studies. However, the full C18 dataset demonstrates higher scale heights, on
average, due to S15 being biased towards dimmer galaxies (similar to the left
panel in Fig. 6).

As part of their decomposition, C18 inspected the axial surface-brightness
profiles of the S4G edge-on galaxies in order to choose how many exponential
segments each disk contained. They were able to accomplish this because the
edge-on line-of-sight profiles of the galaxies were similar to piecewise
exponential profiles and thus could be separated into different segments, each
with its own scale length (see their sections 3.3.2 and 3.3.3 for further
details on the functions used and the fittings performed). Since the
exponential and isothermal Milky Way models created by M21 only have one scale
length for each of the thin and thick disks (the Galaxy radial surface
brightness profile does not demonstrate a significant brake or truncation), we
use the radial scale length from the first (closest to the center) exponential
section of each galaxy fitted by C18.

Figure 9: NIR waveband edge-on two-disk galaxy models (S15 [3]; ;  [1]; P23).
As a note, P23 use optical wavebands; see text for the reasoning of their
inclusion in this plot. Histograms for both disks are shown. Top Row: Thick
disk galaxy data. Histograms of the scale length (left, in kpc), scale height
(middle, in kpc), and ratio between the two (right) are shown. The thick disk
exponential and isothermal Milky Way model parameter values from M21 are
highlighted in each plot by the solid red line and the dashed red line,
respectively. Bottom Row: Thin disk galaxy data. The order of the histograms
is the same as the top row, with the respective thin disk exponential and
isothermal Milky Way model parameter values from M21 highlighted.

Using the combined data presented in Figure 9, we can compare the Milky Way’s
two-disk model to external galaxies. From an initial look at the thick disk
models, it appears that the Milky Way has relatively standard values. For the
thick disk scale length, scale height, and flatness ratio, the Milky Way’s
values are mostly within the second or third quarters for each parameter (with
the exception being the isothermal flatness ratio, which falls within the
first quarter). However, it is important to point out that our total data
sample is dominated by S4G galaxies, which we have shown to be of
sytematically lower mass (see Figs. 3, 6, and 7). The Milky Way-mass galaxies
from M18 and P23 (from the AURIGA zoom-in cosmological simulation) indicate
that the Milky Way’s thick disk is significantly thinner (smaller scale height
value) and much less extended (smaller scale length value) than would be
expected of a galaxy of its stellar mass. Thus, while at first glance the
Milky Way’s thick disk might appear normal, it is actually only normal for
galaxies of a much smaller mass, indicating it is far smaller than one would
anticipate.

Moving on to the thin disk data (the bottom row in Fig. 9), we see that the
scale length and scale height values of our Galaxy fall on the upper end of
the distribution of S4G galaxies, though, like the thick disk, both are below
the values of the galaxies from M18 and P23. At first glance, this would seem
to imply that the Milky Way has a relatively large thin disk; however, keeping
in mind the same points from the previous paragraph, this actually tells us
that, while the Milky Way’s thin disk is larger than lower-mass galaxies, it
is still thinner and less extended than other Milky Way-mass galaxies. While
less abnormal than the thick disk, the Galaxy’s thin disk still appears
slightly smaller compared to the HEROES and AURIGA galaxies with similar
stellar masses.

###  5.1 S4G Disk Mass Data and Analysis

Within this section we discuss models we gathered concerning the disk masses
of the S4G galaxies from S15 and C18. We analyze the thick-disk-to-thin-disk
mass ratio (see Fig. 10), the difference between the same galaxy’s mass values
in S15 and C18 (see Fig. 11), and the thick and thin disk masses of each
galaxy (see Fig. 12). As before, for the Milky Way we use the models from M21.
However, the only data they provide is the disk mass ratio, so we utilize a
total disk mass value of $4.1\times 10^{10}$ $M_{\sun}$ from Bland-Hawthorn &
Gerhard (2016) and use the two ratio values to calculate individual thick and
thin disk mass values (details on why there are two values are provided
later).

Figure 10: Disk mass ratios of galaxies from C18 [2] and S15 [4]. The minimum
and maximum values of the Milky Way’s disk mass ratio from M21 are
highlighted.

Figure 11: A comparison of the disk masses (in 109 solar masses) of the same
48 galaxies from C18 [2] and S15 [4]. Both the thick disk (left) and thin disk
(right) data are shown. On each plot, the line $y=x$ is shown to help better
visualize the difference between the two paper samples.

We begin by describing the process through which we obtained the stellar
masses of the disk models. C18 provide the masses of the thick and thin disks,
so we use their masses as they are. In S15, however, the stellar masses were
not computed; instead, there was data for each disk concerning the scale
length, scale height, edge-on central surface brightness, and distance (the
galaxy distances were originally provided in Sheth et al. 2010), which was
enough to allow us to calculate the disk masses. To compute the apparent face-
on central surface brightness of an isothermal disk

$I(r,z)=I(0,0)\,\mathrm{e}^{-r/h_{\mathrm{R}}}\mathrm{sech}^{2}(z/z_{0})\,,$
(3)

we need to integrate its profile along the $z$ axis at $r=0$, so

$I_{0}^{\mathrm{face-
on}}=I(0,0)\int_{-\infty}^{\infty}\mathrm{sech}^{2}(z/z_{0})\mathrm{dz}=2\,z_{0}I(0,0)\,.$
(4)

In a similar way, we can show that the apparent edge-on central surface
brightness is

$I_{0}^{\mathrm{edge-on}}=2\,h_{\mathrm{R}}I(0,0)\,.$ (5)

Hence, we can find the relationship between the apparent edge-on and face-on
central surface brightness through

$I_{0}^{\mathrm{edge-on}}=I_{0}^{\mathrm{face-on}}h_{\mathrm{R}}/z_{0}\,.$ (6)

Finally, we arrive at

$\mu_{0}^{\mathrm{face-on}}=\mu_{0}^{\mathrm{edge-
on}}+2.5\,\log{h_{\mathrm{R}}/z_{0}}\,.$ (7)

Similarly, it can be shown that for an exponential disk, the face-on central
surface brightness is related to the edge-on central surface brightness via

$\mu_{0}^{\mathrm{face-on}}=\mu_{0}^{\mathrm{edge-
on}}+2.5\,\log{h_{\mathrm{R}}/h_{\mathrm{z}}}\,.$ (8)

Then, using the newly-calculated face-on central surface brightness and the
scale length, we calculated the total apparent magnitude of each disk using

$m_{\mathrm{tot}}=\mu_{0}^{\mathrm{face-on}}-5\,\log(h_{\mathrm{R}})-1.955,$
(9)

where $m_{\mathrm{tot}}$ is the total apparent magnitude. Once we had
calculated the apparent magnitude of each disk, we used the distances to
calculate the absolute magnitudes of the galaxies. Finally, we used the
following equation from the literature (Muñoz-Mateos et al. 2013; Díaz-García
et al. 2016; see also Eskew et al. 2012) to calculate the mass of each disk in
solar masses:

$\log(M_{*}/M_{\odot})=-0.4\,M_{3.6\mathrm{{\mu}m}}+2.13$ (10)

where $M_{*}$ is the disk’s mass and $M_{3.6\mathrm{{\mu}m}}$ is the absolute
magnitude at 3.6 $\mu$m.

In Figure 10, we display a histogram showing the ratio of the thick disk mass
over the thin disk mass. By observation, we can see that the data from C18
(denoted by the blue bars in the histogram) contains slightly lower values, on
average, than the data from S15 (denoted by the orange bars). The Milky Way
data from M21, displayed in this histogram, contains two disk mass ratio
values, representing the minimum and maximum ratio values. The difference
between these minimum and maximum values is considerable, but even the minimum
$M_{\mathrm{T}}/M_{\mathrm{t}}=0.42$ implies that the contribution of the
thick disk to the Galaxy mass budget is substantial.

This range of values is due to the uncertainty in the local thick-to-thin disk
surface density ratio. Specifically, two values are given: $f_{\Sigma}=0.38$
from Snaith et al. (2014) and $f_{\Sigma}=0.12$ from Bland-Hawthorn & Gerhard
(2016). The reason for this uncertainty is because the local density
normalization of the Milky Way (the ratio of the thick disk density over the
thin disk density) is correlated to the scale height of the thick disk (higher
scale height estimates result in lower normalization estimates and vice versa,
Bland-Hawthorn & Gerhard 2016). Since the scale height values for both Milky
Way disks have a range of estimated values in the literature, the
normalization is poorly constrained, and thus the local surface density ratio
is also poorly constrained since they are proportional to each other.

The Milky Way’s low-end disk mass ratio value falls in the first quarter of
the distribution in Figure 10, while the high-end value lies amidst some of
the mid-sized values of our data samples within the third quarter. We see from
this comparison that the low Milky Way value, being smaller (though not an
outlier) than the average value for other galaxies, is still significant and
much higher than usually estimated with other methods (Bland-Hawthorn &
Gerhard 2016 give $M_{\mathrm{T}}/M_{\mathrm{t}}=0.17$ which lies well below
Q1). Therefore, our estimate of $M_{\mathrm{T}}/M_{\mathrm{t}}$ is more
realistic and signifies that the contribution of the thick disk to the total
mass budget in spiral galaxies is substantial, if not dominant.

Figure 12: Thin and thick disk masses for various galaxies from S15 [4], C18
[2], and M18. The masses are given in 109 solar masses. A trendline (green) is
included in the plot to show the trend of the distribution of data (with the
trendline equation being displayed above the plot). The Pearson correlation
coefficient for the trendline is 0.74. The two Milky Way datapoints included
represent the 0.42 disk mass ratio (red dot) and the 1.34 disk mass ratio (red
star).

Figure 11 shows the difference in disk mass values for the common 48 galaxies
in C18 and S15. For the thin disk, the S15 values are systematically higher
than those from C18, whereas for the thick disk, the general trend follows a
one-to-one relation. This difference can be explained by the different PSFs
and methods used in these studies.

Figure 12 displays a scatter plot of the thin disk’s and thick disk’s stellar
masses from S15 [4], C18 [2], and M18. The Milky Way data points we use in
these scatter plots come from a combination of sources: the total disk mass
value comes from Bland-Hawthorn & Gerhard (2016) while the individual thick
and thin disk mass values come from using the ratios discussed above (M21)
combined with the total disk mass value. Since we have two ratio values for
the disk masses, we added two Milky Way data points on the graph. A trendline
is included, with the equation being shown above the scatter plot. The Pearson
correlation coefficient for the trendline is 0.74, indicating a relatively
strong positive correlation. In the scatter plot, the 0.42 and 1.34 disk-mass-
ratio data points for the Milky Way lie relatively close the trendline (though
we note that the 0.42 data point lies almost directly on the line), making our
Galaxy quite typical in this context as well. We stress here that while the
Milky Way is certainly more massive than the majority of the S4G galaxies, the
correlation between mass and size for spiral galaxies is not nearly as
significant as it is for elliptical galaxies (Cebrián & Trujillo 2014), which
suggests that normal spiral galaxies are not so dramatically homologously
different. As such, the relationships considered in this section and the scale
length and height comparisons performed in previous sections are still
justified despite the difference in the stellar masses of the spiral galaxies
under study.

##  6 Bulge

We now analyze the bulge-to-total ($B/T$) flux ratio of the Milky Way against
other galaxies. M21 detail the morphology of the Milky Way’s bulge, confirming
the boxy/peanut/X-shape previously described in the literature (Dwek et al.
1995; McWilliam & Zoccali 2010; Saito et al. 2011) and conclude through an
N-body model of the bar that the asymmetry in the Galaxy bulge/bar’s shape is
likely caused by our viewing angle of the bulge/bar (see fig. 6 in their
paper). We utilize their value of the Milky Way’s $B/T$ ratio as a maximum
value for our analysis since they don’t consider the bulge and bar separately,
thus inflating the actual $B/T$ ratio with flux from the bar (technically, M21
give the bulge+bar-to-disk flux ratio, so we assumed a general decomposition
of only a disk and a bulge+bar and then calculated the $B/T$ ratio using the
bulge+bar-to-disk ratio). In an attempt to give a more accurate estimate of
our Galaxy’s $B/T$ ratio, we utilize a stellar mass ratio of the bulge and bar
from Portail et al. (2017) to account for and remove the excess bar flux from
the ratio, though this is just an estimate since, in principle, the mass-to-
light ratio of the bulge and bar can be different.

Figure 13: Bulge-to-total flux ratio of 55 edge-on (left) and 521 non-edge-on
(right) spiral galaxies from S15 [5]. 163 edge-on galaxies from M10 are also
included in the left plot, in addition to $\sim$20,000 edge-on galaxies from
Simard et al. (2011). In each plot, the Milky Way’s maximum $B/T$ value from
M21 is highlighted, as well as a lower estimate created using the stellar mass
ratio of the bulge and bar from Portail et al. (2017).

Similar to previous sections, the models of galaxies are split between edge-on
and non-edge-on galaxies. All of our $B/T$ ratio comparison data for this
section comes from S15 and M10. However, we also compare the $B/T$ ratios of
our galaxies with their numerical Hubble stage (Type). For the S15 data, these
Type values were gathered from Sheth et al. (2010), while for the M10 data,
they were included in the original paper. Using an Sc classification for the
Milky Way (Georgelin & Georgelin 1976), we convert that to a value of 5 for
its numerical Type. While Figure 13 compares the Milky Way’s $B/T$ value to
all the galaxies in our sample, Figure 14 allows us to compare it to other Sc-
type galaxies, which is a much better comparison to determine if the Milky Way
is abnormal in its $B/T$ value. Nonetheless, we include the comparison to all
the galaxies for thoroughness.

We begin our analysis with Figure 13 which displays two histograms of our
galaxy data (the first histogram shows the edge-on galaxies while the second
one shows the non-edge-on galaxies). Unfortunately, the size of the sample of
edge-on galaxies is significantly smaller than that of non-edge-on galaxies.
This caused the seemingly random pattern for the S15 and M10 galaxies in the
left plot. In an attempt to show the true pattern for edge-on galaxies, we
include $\sim$20,000 galaxies from Simard et al. (2011), which we magnitude-
restricted to only contain edge-on galaxies of similar absolute magnitude to
the Milky Way. From this, we see that edge-on galaxies have seemingly larger
bulges than non-edge-on galaxies, with the distribution being much more spread
out. While the Simard et al. (2011) galaxies are useful for this
visualization, they were obtained through optical observations, so we omit
them from the rest of this NIR $B/T$ analysis. Additionally, due to the small
size of the sample of edge-on galaxies in the NIR, we will not give much
weight to the conclusion in our overall analysis of the Milky Way’s bulge.

Figure 14: Bulge-to-total flux ratio data with the galaxy’s numerical Hubble
stage from Sheth et al. (2010) (the numerical Hubble stage values are rounded
to the nearest integer). In both plots, the line passes through the mean $B/T$
value of each numerical Hubble stage value, with the vertical lines
representing one STD above and below the mean. Numerical Hubble stage values
with fewer than four data points have no STD calculated. Top: 55 edge-on
galaxies from S15 [5] and 163 edge-on galaxies from M10. We utilize a maximum
value of 0.12 for the Milky Way’s $B/T$ ratio from M21. We estimate a lower
value of 0.09 by using the stellar mass ratio of the bulge and bar from
Portail et al. (2017). We also estimate the Milky Way’s numerical Hubble stage
as 5, which corresponds to an Sc-type galaxy. Its maximum value data point is
highlighted by the red triangle, while the its lower estimate is highlighted
by the red star. Bottom: 521 non-edge-on galaxies from S15 [5] with the same
Milky Way data highlighted as in the edge-on galaxy plot.

Looking at the edge-on galaxy histograms in Figure 13 (left plot), we see
high-frequency bars occurring throughout the $B/T$ ratio for the samples in
the NIR, likely due to the presence of bars inflating the $B/T$ value in edge-
on galaxies (although we point out that for the Milky Way, the $B/T$ ratio
also includes the presence of the bar, hence our effort to correct it) and
selection effects. It is clear that the Milky Way’s $B/T$ value is far below
the average value for these edge-on galaxies. The Q1 cutoff value indicates
that the Milky Way’s maximum and estimated $B/T$ values are well within the
first quarter. Therefore, the Milky Way appears to have an abnormally small
$B/T$ value compared to edge-on galaxies in the samples used. If we compare
the Milky Way with the sample of $\sim$20,000 edge-on galaxies from Simard et
al. (2011), both of the Milky Way’s $B/T$ values appear to be somewhat small.
However, as previously stated, this comparison is not fair due to the Simard
et al. (2011) values being fitted in the optical waveband. Nevertheless, since
the $B/T$ ratio is higher in the NIR than in the optical due to the stellar
population gradient and differential dust attenuation (Tortora et al. 2010;
Sánchez-Blázquez et al. 2014; García-Benito et al. 2017; Zhuang et al. 2019;
Dale et al. 2020; Parikh et al. 2021), we can conclude that the Milky Way’s
$B/T$ should be quite average among the edge-on galaxies with similar stellar
mass.

The histogram for non-edge-on galaxies (right plot in Fig. 13), however,
appears very much like how we would expect for late-type galaxies: the
frequency rate of galaxies dramatically decreases with increasing $B/T$. The
Milky Way’s values are well within the range of normality, being between the
Q1 cutoff value and the median.

In Figure 14, we compare the Milky Way’s $B/T$ to morphological type, together
with other galaxies in our sample. The Type values given for our dataset
originally had various decimal point values, so we rounded these values to the
nearest integer for easier visualization. Since our main desire was to view
how the $B/T$ values of galaxies with similar Type values compare with one
another, this rounding seems acceptable. In each scatter plot, we also
represented the mean and standard deviation (STD) values for each grouping. If
a grouping had fewer than four galaxies, we disregarded the STD value.

Similar to the histogram in Figure 13 for edge-on galaxies, the upper scatter
plot in Figure 14 also shows no pattern: the $B/T$ ratio does not correlate
with Hubble stage, in contrast to what we see on the bottom plot for non-edge-
on galaxies. In the bottom plot, the $-$5-Type-value grouping, representing
elliptical galaxies, has some of the highest $B/T$ values with the trend then
decreasing as the Type values increase, until the $B/T$ values taper off
around the 7-8-Type-value groupings333We note here that despite the negative
morphological type for some of the selected galaxies in our samples, their
$B/T<1$ values indicate that these galaxies were probably misclassified as
elliptical galaxies and should instead class as lenticular galaxies.. In the
upper plot, the average $B/T$ ratios are typically higher for the same
morphological type as compared to the same distribution but for non-edge-on
galaxies. The reason for this may be that all morphologies we use in our study
are taken from the HyperLeda database (Makarov et al. 2014). Since HyperLeda
utilizes optical observations, the morphology of galaxies (especially edge-on
galaxies) is heavily affected by dust attenuation, and, thus, may be
influenced by the projection effect. Moreover, the bulge contribution, not
speaking of the galaxy spiral pattern, is hard to visually determine. At a
Type value of 5, both of the Milky Way’s $B/T$ values lie within one STD from
the mean value for edge-on galaxies, indicating an average $B/T$ value for our
Galaxy for its morphological type. Similarly, the Milky Way’s maximum $B/T$
value lies just over one STD above the mean value for non-edge-on galaxies of
the same morphological type, with our lower estimate falling within one STD of
the mean. Since our Galaxy’s maximum $B/T$ value is barely one STD above the
mean, we can safely make the assumption that the true value of the Milky Way’s
$B/T$ ratio is average when compared to non-edge-on galaxies of the same
morphology and mass.

##  7 Pitch Angle

Figure 15: 127 galaxies with both mean pitch angle data from DG19 and $B/T$
data from S15. The data was split into five equally-sized bins of $B/T$,
whereupon means and STDs of the dataset were calculated. The Milky Way’s mean
pitch angle value and max $B/T$ value were taken from Vallée (2015) and M21,
respectively, with a lower estimate of the $B/T$ value included (as in Fig.
14).
Figure 16: Mean pitch angle data from DG19 and rounded numerical Hubble stage
data from Sheth et al. (2010). The line passes through the mean pitch angle
value of each numerical Hubble stage value, with one STD being shown. The mean
and STD data begins at Type $=2$ due to previous values having too few
datapoints. The Milky Way’s mean pitch angle value and numerical Hubble stage
value were taken from Vallée (2015) and Georgelin & Georgelin (1976),
respectively.

Finally, within this section, we provide a brief analysis of the mean pitch
angle of the Milky Way compared to galaxies within the S4G survey. For these
comparisons, we collected a Milky Way pitch angle from Vallée (2015), who
utilized four different statistical methods to find the mean pitch angle of
our Galaxy. Each method produced a result between $12\degree$ and $14\degree$,
with the mean of the results being $13.1\degree$. The exact number of the
spiral arms in the Milky Way remains uncertain: a two-arm and four-arm spiral
pattern has been proposed (see e.g. Efremov 2011 and references therein). In a
recent study, Xu et al. (2023) conclude that our Galaxy is probably a
multiarmed galaxy. Despite this uncertainty on the number of the spiral arms,
it is evident that the Milky Way’s spiral structure is not flocculent (patchy)
because individual arms have been identified, which is an attribute of grand-
design and multiarmed galaxies.

For comparison, we utilize the pitch angles for a large sample of multiarmed
and grand-design S4G galaxies from DG19. To aid in this comparison, we used
two other parameters: $B/T$ ratios and Type values. For both the Milky Way and
external galaxy data, we retrieved these parameter values from the same
sources described in Section 6, although we note that in our $B/T$-pitch-angle
comparison, we only use galaxies that have both pitch angle values from DG19
and $B/T$ values from S15. Additionally, similar to our data in Section 6, we
round the Type value for each galaxy to the nearest integer to better see the
trend between earlier- and later-type disk galaxies.

Beginning with Figure 15, we grouped the data into five equally-sized bins of
$B/T$ values before calculating the mean and STD data. It is evident that
spiral galaxies are far more likely to have $B/T$ values between 0 and 0.2, as
shown by the large number of data points within that range, and so those
groupings are more indicative of the actual trend between galactic $B/T$ ratio
and pitch angle. Upon grouping the data, we calculated both the mean $B/T$
value and the mean pitch angle for each grouping, along with a standard
deviation for each parameter. From these values we see that for galaxies with
$B/T$ values between 0.1 and 0.2, the maximum Milky Way value is within the
average range of $B/T$ values, though it lies a little more than one STD below
the mean pitch angle value. Our lower $B/T$ value estimate falls in the 0.0 to
0.1 $B/T$ category, and is more than one STD above the mean $B/T$ value and
more than one STD below the mean pitch angle value for this category. This
would imply that the Milky Way’s spiral arms are more tightly wound than other
galaxies with similar $B/T$ ratios.

Figure 16 shows the pitch angle of the S4G galaxies compared to their Type,
similar to Figure 14. Likely due to the same issues with HyperLeda galaxy
classification discussed previously (i.e. the galaxies being heavily
influenced by dust attenuation because of the observations being performed in
the optical waveband), there are a few early-type galaxies in this dataset,
though the vast majority are spirals. From this data, we see an interesting
trend: as the galaxies progress to later and later Types, the mean pitch angle
steadily increases as expected, indicating more loosely wound spiral arms.
When we focus on the 5-Type-value grouping, we again see that the Milky Way
falls a little more than one STD below the mean, similar to Figure 15. This
would imply that the Milky Way’s spiral arms are also more tightly wound than
in other galaxies of a similar morphology. To summarize, it appears that when
compared to galaxies with similar $B/T$ ratio values and when compared to
galaxies with similar morphological types, the Milky Way has more tightly
wound spiral arms.

##  8 Conclusion

Studying the structural parameters of our Galaxy is crucial to understanding
how the Milky Way fits in with the rest of the Universe. To that end, we have
compared various parameters of our Galaxy with other spiral galaxies from a
variety of sources. We compared the scale length and scale height of the Milky
Way to a large sample of observed disk galaxies using a radiative transfer
model in the optical. Also, we performed a similar comparison in the NIR with
either a general single-disk or a two-disk (thin+thick disk) model, comparing
these models to observed single-disk and two-disk galaxies in a varied sample
of edge-on and non-edge-on galaxies. For both optical and NIR comparisons,
we’ve included mass/magnitude versus scale length comparisons to better see
the scaling relation of our Galaxy in context. Then, we compared the Milky
Way’s bulge-to-total luminosity ratio $B/T$ to the entire sample of edge-on
and non-edge on galaxies, and also considered the $B/T$ ratio versus Hubble
type. Finally, we compared our Galaxy’s mean pitch angle to other galaxies
with similar $B/T$ values and other galaxies of a similar morphological type.
We summarize our findings from these various analyses as follows:

  1. 1.

In the optical, the Milky Way’s scale length appears to be below average
(falling in the first quarter), while the scale height and flatness ratio for
the general stellar disk are very typical of spiral galaxies (falling in the
second quarter).

  2. 2.

In the NIR, where the comparison is more straightforward due to a much lower
internal extinction, those same structural parameters are all normal for the
single-disk models of the Milky Way. Compared to edge-on and non-edge-on
galaxies, the Milky Way’s values are within either the second or third
quarters, depending on the parameter.

  3. 3.

For the two-disk Milky Way models in the NIR, we found that the thick disk has
normal scale values when compared to smaller-mass galaxies, though when
compared to galaxies of similar mass, both its scale length and scale height
are very small, indicating it is thinner and less extended than expected. The
thin disk follows a similar, though not quite as extreme, trend: its scale
length and scale height are larger than the smaller-mass galaxies, though they
still fall short of the Milky Way-mass galaxies. This would imply that while
the thick disk of the Milky Way is abnormally small for a galaxy of its mass,
the thin disk, while still small, is more how one would expect.

  4. 4.

As for the thick-to-thin disk mass ratio, we considered low (0.43) and high
(1.34) estimates, which fall in the first quarter and in the third quarter for
the reference sample, respectively. Since we do not know which of these
estimates is accurate and only suggest that the true value lies between the
two, we cannot state with precision where the true Milky Way disk mass ratio
lies. The fact that the low mass ratio estimate falls almost directly on the
trendline of Figure 12 would seem to imply that, despite the evidence from the
linear relationship in Figure 10, a logarithmic relationship shows that the
Milky Way’s disk mass ratio is larger than normal.

  5. 5.

When compared with the full dataset of edge-on galaxies in the NIR, the Milky
Way’s bulge-to-total flux ratio value falls easily within the first quarter.
However, the small size of our sample could have influenced this, and
therefore this result should be considered with caution. Compared to non-edge-
on galaxies, the Milky Way’s bulge-to-total flux ratio value is completely
normal, falling comfortably within the second quarter. When only compared to
other galaxies of its Hubble type, we received contradicting results. Among
edge-on Sc-type galaxies, the Milky Way is within a STD of the mean value.
When viewed against non-edge-on Sc-type galaxies, the Milky Way’s maximum
$B/T$ value is just over one STD above the mean, with our lower estimated
value falling within one STD. Given the fact that our sample of non-edge-on
galaxies contains far more data points, we consider that comparison more
valid. Thus, by attempting to correct for bar corruption in the $B/T$ ratio,
we find that the Milky Way’s value is well within the range of normal values.

  6. 6.

Finally, we found that the absolute value of the mean pitch angle of our
Galaxy is below average when compared to galaxies with similar bulge-to-total
flux ratio values as well as when compared to galaxies of similar
morphological type.

Combining all of these results, we find that the scale length and scale height
of the Milky Way in the optical and NIR single-disk models generally lie
within the typical range of an average spiral galaxy in the Local Universe.
The disagreement between the isothermal and exponential models of the Milky
Way in fitting the thick disk indicates that further research may need to be
done to determine the correct model for the Milky Way disk. We found that the
Milky Way’s thick disk and optical radius are unusually small when compared to
galaxies of similar mass. This indicates that mergers probably played a less
significant role in the formation of the thick disk than in other similar-mass
galaxies. For example, the galaxies in the AURIGA cosmological simulations
(P23) had, on average, 22% of the thick disk mass come from accretion, which
thickened the disk and caused the larger scale height values seen in Figure 9.
Since the thick disk in our Galaxy is several times thinner than in the HEROES
(M18) and AURIGA galaxies, we can conclude that the Milky Way’s thick disk
formed in situ without a significant influence from satellite mergers.
Additionally, the Milky Way’s mean pitch angle indicates that the spiral arms
of the Milky Way are more tightly wound than in other similar galaxies.

Further study in this area could include comparing other structural parameters
of the Milky Way, such as the dust disk mass, to other galaxies. Additionally,
as further galaxy data is gathered from databases such as
RCSEDv2444https://rcsed2.voxastro.org (Chilingarian & Zolotukhin 2012;
Chilingarian et al. 2017), larger sample sizes (for example, using the recent
EGIPS catalog of edge-on galaxies, Makarov et al. 2022) would help give more
accurate comparisons for all of the analyses performed in this paper.

## Acknowledgements

We acknowledge the usage of the HyperLeda database (http://leda.univ-
lyon1.fr).

The Spitzer Survey of Stellar Structure in Galaxies (S4G) is a volume-,
magnitude-, and size-limited survey of over 2300 nearby galaxies at 3.6 and
4.5µm. This is an extremely deep survey reaching an unprecedented 1$\sigma$
surface brightness limit of 3.6µm(AB) = 27 mag arcsec-2. This translates to a
stellar surface density of $<<$ 1 M⊙ pc-2 ! S4G can thus probe the stellar
structure in galaxies in a regime where the gas dominates the stars (typical
HI surface density $\sim$ a few M⊙ pc-2).

This dataset or service is made available by the Infrared Science Archive
(IRSA) at IPAC, which is operated by the California Institute of Technology
under contract with the National Aeronautics and Space Administration.

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