# Three-Dimensional Structure and Dust Extinction in the Small Magellanic
Cloud

Petia Yanchulova Merica-Jones11affiliationmark:  , Karin M.
Sandstrom11affiliationmark:  , L. Clifton Johnson22affiliationmark:  , Andrew
E. Dolphin33affiliationmark:  , Julianne J. Dalcanton44affiliationmark:  ,
Karl Gordon55affiliationmark:  6 6affiliationmark:  , Julia Roman-
Duval55affiliationmark:  , Daniel R. Weisz77affiliationmark:  , Benjamin F.
Williams44affiliationmark:  1Center for Astrophysics and Space Sciences,
Department of Physics, University of California, 9500 Gilman Drive, La Jolla,
San Diego, CA 92093, USA  2Department of Physics and Astronomy, Northwestern
University, 2145 Sheridan Road, Evanston, IL 60208, USA  3Raytheon; 1151 E.
Hermans Road, Tucson, AZ 85756, USA  4Department of Astronomy, University of
Washington, Box 351580, Seattle, WA 98195, USA  5Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  6Sterrenkundig
Observatorium, Universiteit Gent, Gent, Belgium  7Department of Astronomy,
University of California, 501 Campbell Hall #3411, Berkeley, CA 94720-3411,
USA

###### Abstract

We examine the three-dimensional structure and dust extinction properties in a
~200 pc $\times$ 100 pc region in the southwest bar of the Small Magellanic
Cloud (SMC). We model a deep Hubble Space Telescope optical color-magnitude
diagram (CMD) of red clump and red giant branch stars to infer the dust
extinction and galactic structure. We model the distance distribution of the
stellar component with a Gaussian and find a centroid distance of 65.2 kpc
(distance modulus $\mu$ = 19.07 mag) with a FWHM $\approx$ 11.3 kpc. This
large extent along the line of sight reproduces results from previous studies
using variable stars and red clump stars. Additionally, we find an offset
between the stellar and dust distributions, with the dust on the near side
relative to the stars by 3.22 ${}^{+1.69}_{-1.44}$ kpc, resulting in a 73%
reddened fraction of stars. Modeling the dust layer with a log-normal $A_{V}$
distribution indicates a mean extinction $\langle A_{V}\rangle$ = 0.41 $\pm$
0.09 mag. We also calculate $A_{V}/N_{H}$ = 3.2 - 4.2 $\times 10^{-23}$ mag
cm2 H-1 which is significantly lower than the Milky Way value but is
comparable to previous SMC dust-to-gas ratio measurements. Our results yield
the first joint dust extinction and 3D geometry properties in a key region in
the SMC. This study demonstrates that CMD modeling can be a powerful tool to
simultaneously constrain dust extinction and geometry properties in nearby
galaxies.

###### Subject headings:

Interstellar dust extinction, Interstellar medium, Small Magellanic Cloud,
Star formation, Galaxy structure, Galaxy distances

′´

##  1\. Introduction

Understanding dust and dust extinction at low metallicity is important for a
number of reasons. Dust in galaxies attenuates starlight, and affects the star
formation, radiative transfer, thermodynamics, and chemistry in the
interstellar medium (ISM). The amount and type of dust is likely strongly
dependent on metallicity (e.g. Dwek, 1998; Zubko, 1999; Clayton et al., 2003;
Sofia et al., 2006; Zhukovska et al., 2008; Asano et al., 2013; Rémy-Ruyer et
al., 2014; Roman-Duval et al., 2014; Feldmann, 2015; Chastenet et al., 2019;
De Vis et al., 2019), making low-metallicity dust studies critical due to the
prevalence of these conditions in the early universe.

Fundamental dust extinction properties in low-metallicity environments show
distinct differences from the Milky Way. For example, the observed extinction
curve of the Small Magellanic Cloud (SMC) at 1/5 - 1/8 $Z_{\odot}$ (Dufour,
1984; Russell & Dopita, 1992; Kurt et al., 1999; Lee et al., 2005; Rolleston
et al., 1999, 2003) shows a steep UV rise and a weak or absent 2175 Å bump
(Lequeux et al., 1982; Prevot et al., 1984; Gordon & Clayton, 1998; Gordon et
al., 2003; Maíz Apellániz & Rubio, 2012). These differences may hold clues to
how the dust in such environments behaves and evolves, in turn affecting
galaxy evolution.

Accounting for dust extinction at high-redshift can be challenging (Somerville
et al., 2001), but fortunately a great deal can be learned from nearby low-
metallicity galaxies which can be considered analogues to the high-redshift
universe where galaxies must have formed at a very low metallicity (Madau &
Dickinson, 2014). The SMC in particular is an especially suitable target due
to its proximity, with its center at $\sim$ 62 kpc (de Grijs & Bono, 2015;
Scowcroft et al., 2016). Indeed, ”SMC-like” extinction is the standard used to
account for dust at high-redshift and/or low metallicity (Gordon et al.,
2003), thus it is a natural target for low-Z studies of dust properties.

The SMC’s proximity also allows us to study in detail its geometry, including
the distance to the stars and the ISM, and the relative positions of these two
galactic components. One of the goals of our study, named the Small Magellanic
Cloud Investigation of Dust and Gas Evolution (SMIDGE) (Yanchulova Merica-
Jones et al., 2017, hereafter YMJ17), is to examine dust extinction and 3D
geometry holistically and derive resolved maps of extinction ($A_{V}$ and
$R_{V}$) for low metallicity dust.

Traditionally, the shape of dust extinction curves are studied using the pair
method (Trumpler, 1930; Massa et al., 1983; Cardelli et al., 1992) which
compares the spectra of a reddened and an unreddened star of approximately the
same spectral type. However, this method relies on individual lines of sight
to a select sample of UV-bright stars with high signal-to-noise data. An
efficient and complementary technique to measure dust extinction and the
extinction curve is to use multiband photometric observations of resolved
stars to measure the average extinction properties in a galactic region. Using
resolved stellar photometry to measure dust extinction, however, may require
modeling the galactic geometry as well.

Figure 1.— The SMIDGE survey field (100 $\times$ 200 pc) in the SW bar of the
SMC is overlaid on a Spitzer Space Telescope composite image in 3.6, 8, and 24
$\mu$m from the SAGE-SMC survey (Gordon et al., 2011). The HST imaging
footprint is shown for ACS/WFC (green), WFC3/UVIS (magenta), and WFC3/IR
(blue). The cyan circle indicates the Weisz et al. (2013) SMC-2 field used in
our star formation history analysis.

Interstellar dust is expected to affect all color-magnitude diagram (CMD)
features. These effects depend on its distribution relative to the stars. If
the dust is in a foreground screen relative to the stars, the stars will be
reddened to various degrees depending on the ISM column density distribution,
which is often observed to be log-normal (Hill et al., 2008; Kainulainen et
al., 2009; Hennebelle & Falgarone, 2012). If it is mixed with the stars, some
stars will experience only a fraction of the overall reddening. If the dust is
in a thin layer and the stars are in a spatially extended distribution
(assuming they are not embedded in the dust layer), a part of the CMD will be
foreground to the dust and unreddened, while the rest will experience the full
reddening from the dust layer. In general, CMD features that are compact or
perpendicular to the reddening vector are the most useful for measuring the
effects of dust and geometry. CMD modelling of these effects can give clues
about the relative line of sight depths of the dust and the stars and the
reddened fraction of stars. The models can then be used to produce a map of
the extinction and/or the geometry. Extinction mapping of reddened stars using
color-color or color-magnitude diagrams has been the subject of numerous Milky
Way (Lombardi & Alves, 2001; Nataf et al., 2013; Schlafly et al., 2016), M31
(Dalcanton et al., 2015), LMC (Imara & Blitz, 2007; Dobashi et al., 2008; Choi
et al., 2018), and SMC studies (Gardiner & Hawkins 1991; Dobashi et al. 2009,
YMJ17).

Geometric effects, such as an elongation of the galaxy along the line of sight
spreads the stars in magnitude, but not color. Dust, on the other hand,
affects both the color and magnitude of stars in a CMD. Features that would
appear narrow in the absence of dust tend to get spread out when there is
dispersion in the amount of extinction due to the column density structure of
the ISM. If this effect can be modelled, then we can measure the mean and the
width of the extinction distribution. By finding the stars’ observed
displacement in a CMD from a predicted theoretical unreddened location, we can
calculate the dust extinction in magnitudes, $A_{\lambda}$, where $\lambda$ is
the effective wavelength of each photometric band. We can measure
$R_{\lambda}=A_{\lambda}/(A_{\lambda}\prime-A_{\lambda}\prime\prime)$ (related
to the dust grain size distribution) from a single CMD color. Measuring
$R_{\lambda}$ across photometric bands from IR to UV allows us to sample the
extinction curve shape.

The SMC, however, presents a unique challenge due to the combined effects of
its extinction, relative proximity and its appreciable elongation along the
line of sight (Subramanian & Subramaniam, 2009; Haschke et al., 2012;
Jacyszyn-Dobrzeniecka et al., 2016). For example, in YMJ17 we found that to
accurately measure the SMC extinction curve from the slope of reddened red
clump (RC) stars, we also needed to account for the galactic depth along the
line of sight. This effect, due to the appreciable spread in stellar
distances, could give the impression of a steeper RC slope since stars behind
the dust layer will have fainter magnitudes at increasing distances. This
observation would in turn seem to imply higher $R_{\lambda}$ values. We
accounted for the depth by employing a simple model for the RC which specified
a theoretical unreddened RC location based on the local star formation history
(SFH), and a set of parameters for the extinction distribution, the extinction
law, and the galactic depth along the line of sight.

In this paper, we build on our previous work and use the full RC and red giant
branch (RGB) CMD to model the dust extinction, 3D geometry, and the SFH
holistically. The effects of dust are ubiquitous in the SMIDGE data, and
beside spreading the reddened RC into a streak, dust also widens the reddened
RGB. The signature of the SMC’s geometry is most clearly seen in the double
RGB which is a result of the relative thicknesses along the line of sight and
the absolute positions of the dust and stellar components of the galaxy. The
SMC’s depth is not as immediately obvious from the CMD, but it nonetheless
results in a measurable and significant spread in the magnitude of both the RC
and the RGB. Analyzing the RC and the RGB simultaneously, versus only the RC,
allows us to better constrain the CMD stellar number counts statistically (and
thus the SFH), and to separately account for the effects of galactic geometry
and dust extinction. The reason for this is that the RC reddening vector slope
is a product of both the extinction curve and galactic geometry (where in
YMJ17 we showed these can be degenerate without additional information). At
the same time, while it would be challenging to constrain the slope of the
extinction vector using the RGB alone, the RGB very strongly constrains the
relative offset between the dust and the centroid of the stellar distribution.
Modeling the combined effects of the RC and the RGB in theory enables one to
separate these effects and better constrain the slope of the extinction
vector.

Accurately modeling the CMD requires a representative SFH as a starting point.
There have been numerous efforts to obtain the SFHs in the SMC by modeling
CMDs of resolved stellar populations (Tosi et al., 1989; Tolstoy & Saha, 1996;
Harris & Zaritsky, 2001; Dolphin, 2002; Williams, 2002; Zaritsky & Harris,
2004; Cignoni et al., 2009; Weisz et al., 2013; Rubele et al., 2015; Williams
et al., 2017; Rubele et al., 2018). These models commonly need to account for
the initial mass function (IMF), binary fraction, stellar evolution, time
resolution, metallicity, distance, extinction, and photometric noise.
Typically, the distance is a parameter specified by ancillary studies.
Although some approaches can also account for the extinction within a galaxy,
to simplify the solution most studies preferably examine regions with low
internal $A_{V}$ which is treated in a minimal way. Deriving the SFH itself
seems to be unaffected by the presence of a significant galactic depth (Harris
& Zaritsky, 2004; Rubele et al., 2018).

In this paper, we forward model the SMIDGE RC and RGB CMD with realistic noise
and a model which includes geometry, distance, and dust. We first create
synthetic model CMDs based on a set of input SFHs and stellar evolution
parameters. Then we model the line-of-sight depth of the stellar distribution,
the relative offset of the dust with respect to the stars, and the dust
extinction distribution. Using a realistic model of photometric uncertainties,
we compare model CMDs to SMIDGE observations to measure properties of the
SMC’s 3D distribution of gas and dust.

We briefly describe the SMIDGE data in Section 2. We discuss the CMD modeling
process in Section 3, including details about how we match CMDs and account
for the uncertainties. Our results are presented in Section 4 and discussed in
Section 5. We conclude with Section 6 and provide more relevant information in
the Appendix.

Figure 2.— The SMIDGE $\textit{F475W}-\textit{F814W}$ optical color-magnitude
shows the boundaries of the red clump and red giant branch region analyzed
here. Dust extinction is evident from the extended reddened RC (which would
theoretically be located within the red ellipse in the absence of dust) and
the extended and bimodal reddened RGB which would be represented by a single
vertical sequence if unreddened). The observed bimodal RGB indicates that the
dust layer is thin relative to the stellar distribution, with few embedded
stars. These combined effects cause a distinct unreddened and reddened stellar
populations.

##  2\. Data

The SMIDGE survey (GO-13659) consists of eight-band Hubble Space Telescope
(HST) imaging of a ~100 pc $\times$ 200 pc region in the SMC southwest bar.
Its location is shown in Figure 1. We refer to YMJ17 for the details of the
SMIDGE imaging footprint, HST camera and filter selections, photometry
processing, and data culling. Also see the SMIDGE survey paper (Sandstrom et
al., in preparation) for complete details. This work is based on the
$\textit{F475W}-\textit{F814W}$ vs F475W CMD shown in Figure 2 (where the
units of all magnitudes are Vega magnitudes).

##  3\. Modeling the SMIDGE Color-Magnitude Diagram

Our goal is to constrain the SMC dust extinction and 3D geometry parameters by
finding the best matching model CMDs to the SMIDGE observations. The best
match is determined by comparing the number of stars in bins of color and
magnitude between the modeled and the observed CMD. The optical CMD is ideal
for this purpose due to its high signal-to-noise (compared to other available
filter combinations), and broad wavelength baseline.

The location and number of stars in the CMD are mainly determined by these
factors:

  1. 1.

The SFH, i.e. the star formation rate SFR($t$) and metallicity evolution
Z($t$), which defines the density, unreddened position and morphology of the
stellar populations,

  2. 2.

The distribution of dust extinction, which sets the additional spread in color
and magnitude of the stars,

  3. 3.

The extinction law, which sets the slope of the reddening vector,

  4. 4.

The average distance modulus which determines the overall apparent magnitude
of the stars,

  5. 5.

The line-of-sight depth, which affects the spread in magnitude of the stellar
populations, and,

  6. 6.

The dust-stellar centroid offset (which we simply call the dust-stars offset),
which, along with an assumption of the relative thickness of the dust and
stellar distributions, determines the fraction of reddened stars and their
distance distribution.

To model these properties, we take a forward modeling approach to CMD matching
by creating synthetic CMDs which include the effects of the SFH, dust
extinction, galactic geometry, and observational noise, and compare the number
counts in the CMD to find the best match. Versions of CMD matching have been
used by several studies (see references in Sec. 1). The method we use is based
on the approach presented in Dolphin (2002, hereafter D02) who developed a
widely used technique for CMD modelling (e.g. Dolphin, 1997; Aparicio et al.,
1997) to derive the SFHs of nearby galaxies with spatially resolved stellar
populations (Tolstoy et al., 2009; Weisz et al., 2013, 2014). D02 created a
maximum likelihood CMD fitting package – MATCH – to recover the best SFH by
modelling the observed stellar density in a Hess diagram (a binned CMD showing
the relative density of stars). We rely on the D02 approach conceptually to
search combinations of dust extinction and geometry parameters for the best-
fit CMD.

Although MATCH can also model a dust extinction distribution, it is not
designed to model extinction and 3-D geometry (a spread of stellar distances
and a dust-stars offset) simultaneously. Since the SMC geometry necessitates
that we take into account all of these features at the same time, we do not
use MATCH per se to find the SMIDGE SFH or to fit the SMIDGE CMDs, but only
use it to generate single-distance, unreddened synthetic CMDs. We build on the
MATCH foundation by developing a fitting method which facilitates our use of
more complex models for distance and extinction distributions with the syncmd
111syncmd as of 2020 February 29 (C. Johnson):
https://github.com/lcjohnso/syncmd/tree/simple_attenuation code. syncmd
applies distance, reddening effects, and a noise model using functions
available from the Bayesian Extinction and Stellar Tool (BEAST) 222BEAST as of
2020 September 25: https://github.com/BEAST-Fitting/beast of Gordon et al.
(2016, hereafter G16) to create CMDs which we can directly compare to the
data.

In principle, finding the best fit model could involve searching through a
large grid of CMD models covering all possible variables. The dimensions of
this grid would be the SFH($t_{i}$) and Z($t_{i}$) for time bins $t_{i}$; the
dust extinction parameters; and the 3D geometry parameters. Depending on the
number of time bins in the SFH, such a calculation can become very
computationally expensive. We therefore simplify the procedure by making these
choices:

  1. 1.

We focus only on the RC/RGB region

These populations are intrinsically narrow and most strongly display the
effects of dust extinction and 3D geometry as seen in Figure 2, where the RC
is extended and the RGB is extended and doubled. We thus draw a selection box
around this region. To model the dust and geometry we do not need to include
the whole RGB, nor do we have to cut the selection box at fainter magnitudes
with an exquisite precision. On the other hand, if the selection box is
extended to fainter magnitudes, we would end up selecting main sequence stars
which is not optimal. Therefore, the results should not be affected by
expanding the box. We define the RC/RGB region as stars with
$\textit{F475W}-\textit{F814W}$ color of 1.2 - 2.2 mag and F475W magnitude of
18.5 - 22.75 mag and obtain a CMD subset of $\sim$12,000 stars. These ranges
encompass the unreddened RC, the streak of the reddened RC, and the tip and
base of the RGB.

  2. 2.

We simplify the SFH since we only model the RC/RGB

Since finding the SFH is not our main goal, and the extinction and geometry
parameter fitting only requires a broad knowledge of the SFH, the RC/RGB SFH
can be treated coarsely with limited time bins. See Section 3.2 for details.

  3. 3.

We separate the SFH and the dust/geometry grid search

We first measure the SFH by matching the stellar number counts in coarse
RC/RGB bins (see Fig. 3). After determining the approximate SFH, we proceed to
search the full grid of dust and geometry parameters.

Using the choices above, the conceptual outline of the CMD matching process is
the following, with details outlined in the rest of Section 3.

  1. I.

Fit a star formation history for the SMIDGE field. (Section 3.2).

    1. (a)

First generate unreddened, single-distance synthetic CMDs according to a range
of star formation rates and small metallicity offsets based on adjustments of
a known SFH of a nearby region.

Compare RC/RGB stellar number counts between the unreddened synthetic and the
observed CMDs in coarse bins designed to be insensitive to extinction and
geometry (left panel of Fig. 3). Select SFHs producing CMD statistics
consistent with the SMIDGE stellar number counts. (Sec. 3.1.1.)

    2. (b)

Refine the SFH selection by applying a fixed set of geometry, distance, and
dust extinction parameters (obtained in YMJ17) to the CMDs above. Model the
SMIDGE distance. Also include realistic observational noise (see Part II).
Recalculate the stellar number counts in the RC/RGB coarse bins. Perform a
second SFH refinement using a narrower range of the cuts. (Sec. 3.1.2 \-
3.1.5.)

    3. (c)

Select the SFH producing a best-fit coarse CMD and use it as a foundation for
the analysis in Part II.

  2. II.

Fit the dust extinction and geometry model combinations. (Section 3.1.4 \-
3.4).

    1. (a)

Generate model CMDs, with the SFH from step I, in a grid of combinations of
dust extinction and 3D galactic geometry parameters.

    2. (b)

Add observational noise to the stars in the model CMDs as characterized using
artificial star tests.

    3. (c)

Find the best fitting model and the uncertainty in the results by measuring
the probability density function of the model grid.

Table 1Grid of SFH parameters

Time [Gyr ago] | SFR [$10^{-4}M_{\odot}/yr$] | [M/H]
---|---|---
0.004 – 0.01 | 1.0 – 3.0 | -0.65 – -0.45
0.01 – 0.03 | 1.0 – 3.0 | -0.65 – -0.45
0.03 – 0.05 | 1.0 – 3.0 | -0.65 – -0.45
0.05 – 0.16 | 1.0 – 3.0 | -0.65 – -0.45
0.16 – 0.25 | 1.0 – 3.0 | -0.65 – -0.45
0.25 – 0.40 | 1.0 – 3.0 | -0.65 – -0.45
0.40 – 0.63 | 6.0 – 9.0 | -0.70 – -0.60
0.63 – 1.0 | 1.5 – 4.0 | -0.75 – -0.65
1.0 – 1.26 | 1.5 – 4.0 | -0.75 – -0.65
1.26 – 1.59 | 1.5 – 4.0 | -0.75 – -0.65
1.59 – 2.5 | 5.0 – 8.0 | -0.95 – -0.85
2.5 – 3.2 | 5.0 – 8.0 | -1.10 – -1.0
3.2 – 4.0 | 1.5 – 3.0 | -1.10 – -1.0
4.0 – 6.3 | 1.5 – 3.0 | -1.20
6.3 – 10.0 | 1.5 – 3.0 | -1.40
10.0 – 14.13 | 1.5 – 3.0 | -1.70

Note. — SFR and metallicity values in each age bin are equally spaced within
the range given: 4 - 400 Myr age bins assume 5 values; 400 Myr - 3.2 Gyr
assume 4 values, and bins with ages $\geq$ 3.2 Gyr assume 3 values. The
metallicity dispersion at each time bin is 0.25.

Table 2RC/RGB region stellar number counts for SFH selection

Region | All SFH Models | SMIDGE | Best-fit SFH
---|---|---|---
Red Clump | 3000-6000 | 4385 | 4332
Upper RGB | 800-1600 | 1350 | 1395
Mid RGB | 1000-2500 | 1469 | 1621
Low RGB | 1100-2400 | 1919 | 1899
Total RC + RGB | 6000-12000 | 9123 | 9247

Note. — The Region column corresponds to each of the RC/RGB regions in Figure
3. The “All SFH Models” column shows the full range of stellar number counts
in each region sampled by the combinations of SFH parameters in Table 1. The
SFH cuts we impose are based on the SMIDGE stellar number counts in a region
$\pm$ a percentage of the full range of this same region for all SFH models.
The cuts are, (a) $\pm$ 10 % for the initial unreddened CMD models, and (b)
$\pm$ 5% for the final reddened CMD models (there is a small deviation from
these cuts for the mid RGB region (red in Fig. 5) due to a horizontal branch
contribution noted towards the end of Section 3.2).

Figure 3.— (Left) The unreddened synthetic CMD generated by MATCH/fake
(Dolphin, 2002) based on the best-fit SMIDGE SFH chosen as described in
Section 3.2. (Right) The same synthetic CMD processed with observational
noise, dust extinction and geometry effects ( Sections 3.1.3 3.1.4). The CMD
is divided in the following subregions marked by diagonal bins: upper RGB
(URGB; light blue), RC & RGB (blue), mid RGB (MRGB; red), and low RGB (LRGB;
orange). A comparison of stellar number counts (in both the unreddened and the
reddened CMDs) is made with the same regions in the SMIDGE field. The diagonal
orientation of the bins follows the slope of the reddening vector (subject to
SMC Bar extinction (Gordon et al., 2003), as found in YMJ17). The boundaries
of the colored boxes coincide with the boundaries of the RC/RGB region chosen
here and in the rest of our analysis. The stellar number counts are shown in
the legend and are color coded correspondingly. The respective SMIDGE stellar
number counts for each region are in black. In the reddened CMD on the right
stars in CMD features outside of the defined RC/RGB region are reddened into
some of the subregions and increase the number of stars there. Other regions
(the RC) experience a negligible decrease in the number of stars due to a
low/no contribution from CMD features lying in the opposite direction of the
reddening vector and due to possible displacement outside of the defined
reddening vector boundaries. Figure 4.—  Distance modulus test for the SMIDGE
field using a Poisson likelihood ratio test (see Section 3.3). The SMC center
is measured to be at $\mu$=18.96 $\pm$ 0.01 mag by de Grijs & Bono (2015) and
Scowcroft et al. (2016), while we measure the center of the SMIDGE field to be
located at $\mu$=19.07 mag. The greater SMIDGE distance is expected from
models and measurements of the extended stellar distribution. Figure 5.— The
SMIDGE SFR derived as described in Section 3.2. Adjustments to the W13 SFR are
shown in dashed blue and were used in our YMJ17 work. We allow the SFR to vary
within the range in the gray band with the resolution specified in Table 1.
The SFR remains constant for ages beyond 7 Gyr. Our results for SMIDGE are in
red and show SFR enhancements to the W13 SFR at almost all ages, including at
times where our SFR is allowed to vary below their result. We see a SFR at
older ages ($\geq$ 3 Gyr) which is enhanced by a factor of 2.

###  3.1. Forward Modeling the CMD

####  3.1.1 Unreddened Single-Distance CMD

We can predict the number density of a population of stars in the CMD given a
SFR($t$) and a Z($t$). We use MATCH’s fake CMD simulation tool (D02) to
generate synthetic CMD models without noise, extinction or geometric effects
such as the one in the left panel of Figure 3. First we create a synthetic CMD
based on adjustments to the Weisz et al. (2013, hereafter W13) SMC SFH for a
nearby region. Our motivation for starting with this SFH is detailed in Sec.
3.2 where we discuss other details of the SFH analysis. We input the following
fixed parameters into fake: a Kroupa (2001) initial mass function, PARSEC
stellar evolution models (Bressan et al., 2012; Tang et al., 2014; Chen et
al., 2015), a binary fraction of 0.35, a foreground MW extinction of $A_{V}$ =
0.18 mag derived from MW HI foreground towards the SMIDGE field (Muller et
al., 2003; Welty et al., 2012), and a distance modulus $\mu$ = 18.96 mag for
the center of the SMC (de Grijs & Bono, 2015; Scowcroft et al., 2016).

####  3.1.2 Modeling the SMIDGE Distance

To properly model the CMD, we must also know the average distance to the stars
in the region. The SMC’s mean distance modulus varies across the galaxy due to
SMC’s three-dimensional structure. Scowcroft et al. (2016) measure a distance
modulus of $\mu$ = 18.96 $\pm$ 0.01 mag for the center of the SMC using
Cepheid variables, and de Grijs & Bono (2015) also find a mean SMC $\mu$ =
18.96 $\pm$ 0.02 mag. Haschke et al. (2012) find a distance modulus varying
between 18.94 mag $\leq\mu\leq$ 19.17 mag using RR Lyrae stars and Cepheids in
the bar and wing of the SMC. Nidever et al. (2013) use RC stars to measure the
SMC line-of-sight depth and find $\sim$23 kpc for the eastern side and
$\sim$10 kpc for the western side. Ripepi et al. (2017) use classical Cepheids
and find a distance spread in the SMC SW Bar of 62.5 - 65 kpc ( 18.98
$\leq\mu\leq$ 19.06 mag). Models of the SMC and the LMC (Besla et al., 2007,
2012, 2016), show that the Magellanic Clouds have experienced repeated
interactions with each other which cause asymmetry in the stellar structures
of the galaxies. Additionally, it is well-known that the SMC has a large depth
along the line of sight (Subramanian & Subramaniam, 2009, 2012; Subramanian et
al., 2017; Jacyszyn-Dobrzeniecka et al., 2016, 2017).

We investigate the distance modulus appropriate for the SMIDGE field in the SW
Bar by performing a preliminary Poisson likelihood ratio CMD matching
calculation using our previous knowledge of the SFH, extinction, and geometry
we obtained in YMJ17, namely: an extinction law consistent with SMC Bar Gordon
et al. (2003) extinction, a log-normal median $\widetilde{A_{V}}$ = 0.32 and
width $\langle{\sigma_{A_{V}}}\rangle$ = 0.3, a dust distance $D_{dust}$ = 60
kpc (dust-stars offset of $\sim$ 2 kpc), and a 10-kpc line-of-sight depth
(resulting in a 0.65 fraction of reddened stars). Using the adjusted W13 SMC
SFH, we proceed to shift the CMD in distance. The dust extinction and 3D
geometry parameters are held constant as the synthetic CMDs are shifted in
distance (magnitude) over the range 18.8 mag $\leq\mu\leq$ 19.3 mag. Since the
distance calculation relies only on shifting the CMD in magnitude, using a
preliminary version of these parameters does not affect the results in terms
of their sensitivity to a particular set of parameter values. Comparing to the
SMIDGE observations, we find a best match at $\mu$=19.07 mag as shown in
Figure 4. The larger distance is expected for the SMIDGE region based on
models of the extended stellar distribution placing SMC’s NE regions closer
than the SW regions (i.e. Haschke et al., 2012).

####  3.1.3 Modeling the 3D Geometry

We consider two aspects of the three-dimensional geometry which impact the
SMIDGE CMD: the depth along the line of sight of the stellar component and the
relative offset between the stars and a thin dust layer. We could assume that
the stellar density follows one of a number of distributions such as a
Gaussian, a log-normal, an exponential, etc. Subramanian & Subramaniam (2009)
find that the spread in color and magnitude of red clump stars in the central
region of the SMC (covering 2.5 square degrees, which includes the SMIDGE
region) is best fit with a Gaussian. Using Cepheid observations to study the
3D structure of the SMC, Jacyszyn-Dobrzeniecka et al. (2016) also find a
Gaussian-like distance distribution for the galaxy. Our model also assumes a
Gaussian stellar distribution along the line of sight (centered at a mean
distance of $\mu$=19.07 mag) with a FWHM between 3.5-19.8 kpc which spans the
range found by previous observations. We can adapt this model to explore a
variety of stellar density distributions in the future.

We apply distance and geometry offsets using the syncmd code in the following
way: We use the unreddened, single-distance CMD obtained as in Sec. 3.1.1 and
apply a random Gaussian distance distribution to the stars with a width
$\sigma_{DM}$. We then place a single thin dust layer at a specified position
and find which stars are foreground and background to the dust, and redden the
stars behind the dust as described in Section 3.1.4.

In our model, we assume the dust is in a thin layer relative to the stellar
distribution. We can test this assumption with a population of stars which is
distributed uniformly and independently of the dust layer. RGB stars can serve
this purpose since they are evolved stars and would be distributed regardless
of the location of the dust layer. Indeed, we do not see RGB stars covering
the full range of possible extinctions, but rather we see a bimodal RGB which
indicates that stars are essentially either unreddened or reddened, placing
them either in front of or behind a thin dust layer. If the dust layer had a
substantial thickness relative to the stellar distribution, this clear
bimodality would not be evident. From stellar counts in the reddened and
unreddened RGB, we conclude that the reddened fraction is $>$ 50%. We thus
consider only dust layer positions closer to us than the centroid of the
stellar distribution, as specified in Table 3.

Table 3Model Parameters

Parameter | Min | Max | Resolution
---|---|---|---
Galactic Line-of-Sight Depth1 | 0.05 mag (3.5 kpc) | 0.275 mag (21.3 kpc) | 0.025 mag
Dust-stars offset2 | 0.02 mag (0.6 kpc) | 0.2 mag (5.7 kpc) | 0.02 mag
Dust Extinction3, log-normal median $\widetilde{A_{V}}$ | 0.23 mag | 0.41 mag | 0.02 mag
Dust Extinction, width $\widetilde{\sigma_{A_{V}}}$ | 0.40 | 0.95 | 0.05

Note. — (1) The SMIDGE distance modulus is measured to be $\mu$=19.07 mag (see
Figure 4) indicating a distance of 65.16 kpc. Galactic depth range in kpc is
expressed as the FWHM of the Gaussian stellar distribution where $\mu$=19.07
$\pm$ 0.05 mag (lowest grid boundary for the galactic depth) corresponds to a
FWHM = 3.53 kpc (1 $\sigma$ depth of 1.50 kpc) and $\mu$=19.07 $\pm$ 0.275 mag
(highest grid boundary) corresponds to a FWHM = 19.81 kpc (1 $\sigma$ depth of
8.41 kpc). (2) The dust layer is offset on the near side of the stellar
distribution by the distance indicated. (3) Log-normal distribution of dust
extinctions where $\widetilde{A_{V}}$ is the log-normal median, and the
dimensionless $\widetilde{\sigma_{A_{V}}}$ is the distribution width (see
Section 3.1.4.)

####  3.1.4 Modeling the Extinction

The fake routine outputs the effective temperature, surface gravity, and
metallicity for each star in the CMD. We input these into syncmd which uses
the BEAST to compute the associated stellar spectra in order to correctly
apply extinction to the model spectra. (i.e., see Van De Putte et al. 2020;
Choi et al. 2020 for examples of using the BEAST). We apply a G03 SMC Bar
$R_{V}$ = 2.74 extinction law. The BEAST then extracts the integrated band
fluxes and computes synthetic HST photometry which we can compare to the
observations.

Following Dalcanton et al. (2015), we assume the extinction has a log-normal
distribution, with median $\widetilde{A_{V}}$ and a dimensionless width
$\widetilde{\sigma_{A_{V}}}$. The (normal) mean $A_{V}$ is related to the
median $\widetilde{A_{V}}$ and $\widetilde{\sigma_{A_{V}}}$ of the log-normal
by:

$\langle A_{V}\rangle=\widetilde{A_{V}}e^{\widetilde{\sigma_{A_{V}}^{2}}/2}$
(1)

The width of the log-normal, $\langle\sigma_{A_{V}}\rangle$, can be related to
$\widetilde{\sigma_{A_{V}}}$ by:

$\langle\sigma_{A_{V}}^{2}\rangle=\widetilde{A_{V}}^{2}e^{\widetilde{\sigma_{A_{V}}^{2}}}(e^{\widetilde{\sigma_{A_{V}}^{2}}}-1)$
(2)

or in an alternate way as in Sec. 3.1 of Dalcanton et al. (2015). The range of
extinction parameters is shown in Table 3.

It is also possible to vary the extinction curve, but since we are modeling
CMDs only with the optical color $\textit{F475W}-\textit{F814W}$, we are not
sensitive to the difference between an SMC and a MW extinction curve (see
discussion in the Appendix). Thus we model our CMDs with the extinction law
fixed to the G03 SMC Bar extinction curve, which is one of the reasons we
focus on this color combination. In future work we plan to expand this
technique to multiple color combinations and vary the extinction curve as
well.

####  3.1.5 Observational Noise Model

Accurate simulated noise is critical to our approach as it accounts for
photometric errors, crowding and incompleteness through artificial star tests
(ASTs) performed on the SMIDGE observations. For details on how ASTs are used
in our analysis, see Section 2.3.1 of Williams et al. (2014) who use the same
procedures for the Panchromatic Hubble Andromeda Treasury (PHAT) survey. We
employ a conservative noise model using the “toothpick” method of G16 which is
used for data that are known to be correlated between photometric bands by
treating the bands independently. The model is computed in equally spaced
logarithmic flux bins and assumes that each AST entry corresponds to one
artificial star. See Sections 4.4 and 5 in G16 for details on how ASTs are
used to determine the noise model.

###  3.2. Modeling the SMIDGE Star Formation History

The SFH affects the RC/RGB stellar density, position of the unreddened RC, and
morphology of CMD features. While finding the SFH for the SMIDGE field is not
the primary goal of our study, a representative SFH is key to successfully
reproducing these features in the unreddened CMD, and will ensure that our
model will be sensitive to potentially subtle variations in the dust
extinction and geometry properties.

The RC consists of low-mass K giants in their He-burning post-RGB phase. A K
giant has a nearly constant absolute magnitude during this phase of its
evolution, which places RC stars within a tight clump in the CMD. The RC is
thus less affected by age variations than the RGB in terms of CMD position.
However, both the RC and the RGB are sensitive to the metallicity and
metallicity dispersion at a given age. We model these effects simultaneously
by varying both the SFR($t$) and the metallicity, Z($t$), to the ranges we
specify in Table 1. We assume a constant metallicity dispersion for all ages.
To determine the value of the dispersion we compare the width of the
unreddened RGB in color to that of model CMDs in the following way: we define
a box around the unreddened RGB in a highly reddened region where the
unreddened and the reddened stars are well-separated from each other, and
compare the standard deviation in color between the models and the SMIDGE
data. We find that a metallicity dispersion of 0.25 dex best reproduces the
unreddened RGB width.

SFH studies exist for regions near SMIDGE (W13, Rubele et al. 2015, 2018).
Although those regions are close by, there are differences in the number
counts of RC/RGB stars between their fields and SMIDGE. Adjustments to
existing SFH results allow us to obtain a better fit to the observations since
stellar number counts of RC and RGB stars are directly related to the SFR(t)
over ages relevant for stars to evolve off the main sequence. For example,
when modeling the unreddened RC in YMJ17, we performed a rough refinement of
the SMC SFR($t$) and Z($t$) derived by W13 to match to first order the RC
morphology and stellar positions (we neglected the stellar number counts; see
Sec. 3.2 in YMJ17). Here we need an additional SFR refinement since our method
is sensitive to stellar number counts, the width of the unreddened RGB, and
the morphology of the red clump.

As a starting point we use the results of W13, who have used deep HST
photometry to study the SFH of a region in the SMC Bar near SMIDGE (shown in
Figure 1) with low internal dust extinction and an RGB stellar surface density
which is somewhat lower than that of the SMIDGE region. Rubele et al. (2015,
2018) have also examined the SFH of regions near and within SMIDGE. Upon a SFH
and age-metallicity comparison with W13 we find that although the Rubele et
al. (2015, 2018) results show higher metallicity and lower $A_{V}$, their
results for older dominant RC ages compensate for these differences and show a
consistency in the stellar number counts and morphology of the RC and the RGB.

The adjustments we made in YMJ17 to the W13 SFR are indicated by the dashed
blue line in Figure 5 and include star formation enhancements at t = 500 Myr
and 1.5 Gyr $\leq$ t $\leq$ 3 Gyr. We additionally enhance the SFR in this
work in order to match the total number of stars in the observed RC/RGB region
($\sim$ 12,300) where we increase the lower bound of the sampled SFR range for
ages $\geq$ 3 Gyr to 1.5 $\times 10^{-4}M_{\odot}$ per year. While we explore
a relatively narrow metallicity range, we find that Z($t$) offsets to lower
values by 0.2 - 0.3 dex produce a better fit. We use a set of 16 logarithmic
time bins (listed in Table 1) – re-binned from the original 40 time bins of
W13 – which range from 6.6 $\leq$ log($t$) $\leq$ 10.15. We make this
simplification since the RC/RGB populations consist of older ages and
smoothing over the W13 bins at younger ages results in a minimal variation in
the relevant areas of the CMD.

We search for a best match to SMIDGE by exploring a range around our initial
estimate for SFR($t$) and Z($t$) illustrated by the width of the gray band in
Figure 5. We define coarse diagonal CMD bins representing the upper RGB, RC &
RGB, mid RGB (below the reddened RC), and lower RGB (Figure 3), and compare
the stellar number counts per bin in each synthetic CMD to SMIDGE. First we
compare the unreddened, single-distance CMDs to SMIDGE, and then using
forward-modeling we compare CMDs with simulated reddening and geometry as
described in the next paragraph. We search for an initial set of SFHs
producing CMDs that fall within the cuts in Table 2 which are chosen to allow
for a small variation around the SMIDGE counts. The diagonal bins are parallel
to the extinction vector which we know from the approximate slope of the RC
reddening vector in YMJ17. They are designed to be relatively wide in color
and magnitude such that reddening, geometry and noise would minimally affect
the number counts.

To account for any subtle changes to the CMD stellar number counts due to dust
and geometry and due to the reddening of stars with unreddened positions bluer
than $\textit{F475W}-\textit{F814W}$ = 1.2 mag, we refine the initial
selection of models by transforming the unreddened CMDs with syncmd into
models incorporating geometry, dust extinction, and observational noise
effects. We use the dust extinction and geometry values we found for SMIDGE in
YMJ17 (listed in the last paragraph of Sec. 3.1.2) as approximate parameters
and follow the process in Sections 3.1.4 and 3.1.3. Our final SFH choice is
guided by a second round of narrower cuts around the SMIDGE number counts (see
Table 2) which improve the fit slightly. The resulting best-fit SFR is shown
in red in Figure 5.

The SFR we find to be suitable for the SMIDGE field in Figure 5 shows
enhancements to the W13 SFR at almost all ages, even those where we allowed
values to vary below their result. The most significant difference is at older
ages ($\geq$ 3 Gyr) where the SFR is enhanced by a factor of 2. The two star
formation peaks at more recent times are also slightly enhanced. These
differences are expected due to the higher stellar surface brightness of the
SMIDGE field compared to the W13 field.

We note that the model CMDs overpredict the contribution of red horizontal
branch (HB) stars due to known challenges in modeling the HB morphology
stemming from RGB mass-loss uncertainty (Catelan, 2009; Gratton et al., 2010).
Despite this contamination, we do not expect the HB contribution to be
significant or to dominate the statistics given that we find that the HB
accounts for only about 3% of the total number of RC/RGB stars.

Figure 6.— CMD matching of the SMIDGE color-magnitude diagram with a focus on
the red clump and the red giant branch. Top: The SMIDGE observations and the
best-fit model are binned in a Hess diagram to show the density of sources
according to the binning scheme used in our CMD fitting. Top left: the SMIDGE
data as in Figure 2. Top right: the best fit maximum probability model
obtained as in Section 3. Bottom left: Poisson Likelihood Ratio map shows the
PLR calculated for each CMD bin using Equation 6. Bottom right: Residual
significance map showing the difference in the number of sources between data
and model weighted by the uncertainty in the model measurement. The best-fit
model in the upper right has the following parameters: dust-stars offset of
0.10 mag, 1$\sigma$ distance modulus spread of 0.15 mag, dust extinction with
a log-normal median $A_{V}$ = 0.33 mag, and width $\sigma_{A_{V}}$ = 0.65.

###  3.3. Comparing Model and Observed CMDs

We generate a grid of model CMDs for each dust and geometry parameter
combination from Table 3 for a total of 12,000 models. To compare each model
CMD to the SMIDGE observations, we calculate the probability that a model
produces the observations by comparing the number of stars in the respective
CMDs in a way which is sensitive to CMD features (described below). This
calculation results in the probability density as a function of each dust
extinction and geometry parameter, or a combinations of parameters. The
probability density function (PDF) can in turn inform us about the confidence
interval for each parameter based on a set probability threshold.

To compare the number of stars in the model and data CMDs, we bin each model
CMD in an identical way to the observed CMD and calculate the probability,
from a Poisson distribution, that a model with a predicted number of stars
$n_{i}$ produces an observed $m_{i}$ stars in bin $i$:

$P_{i}=\frac{m_{i}^{n_{i}}}{e^{m_{i}}n_{i}!}$ (3)

We follow the recommendation of Dolphin (2002, 2013) who suggest that in the
case of binned CMD data, the Poisson probability distribution should be used
instead of the canonical Gaussian distribution, e.g. as in a $\chi^{2}$
minimization (Mighell, 1999). The results for all CMD bins are multiplied to
obtain the total probability for each model-data comparison.

The probability can be measured for any binning of the CMD in color and
magnitude. There are several considerations to take into account when defining
the binning. First, if there are features in the CMD which we would like to be
sensitive to, we must choose binning sufficiently small such that these
features are preserved. For our purposes, we want to be sensitive to features
such as the unreddened and the reddened RC, the tip of the RGB, and the
unreddened RGB, among others. The smallest (narrowest) of these features are
the unreddened RGB (which spans $\sim$ 0.1 mag in color) and the unreddened RC
(spanning $\sim$ 0.5 mag in magnitude). Second, we also make sure that our
bins are not too small that we end up with a significant number of bins
($\geq$ 5 %) containing no sources. Finally, we must choose either uniform
binning, where all CMD bins are of equal rectangular size, or irregular
binning, to fit parts of the CMD in a bin of a special size if we have a sense
for known errors or uncertainties in the data. Here we follow D02 in choosing
uniform binning. Our final bin size is 0.025 mag in color and 0.125 mag in
magnitude, which finely samples the CMD features of interest while minimizing
the number of empty bins. With our SFH analysis in Section 3.2 we ensure that
the difference in the total number of stars in the RC/RGB region between the
SMIDGE CMD and each model CMD is insignificant. The variation in this number
due to noise in each model CMD is addressed in Section 3.4.

The model with the highest probability is our best-fit model. To visualize the
goodness of fit per bin, it is convenient to calculate a maximum likelihood
statistic based on this probability. Similarly to D02 who applied this
statistic to find the best star formation history for nearby galaxies, we
calculate the Poisson likelihood ratio (PLR) for each model-data comparison.
Many studies use the PLR for CMD fitting, primarily to analyze SFHs (Dolphin,
1997; Aparicio et al., 1997; Dolphin, 2002; Harris & Zaritsky, 2009; Weisz et
al., 2011). The PLR is defined as the ratio of the probability of drawing
$n_{i}$ stars from model $m_{i}$ to the probability of drawing $n_{i}$ stars
from model $n_{i}$ and has the following form:

$PLR_{i}=\frac{m_{i}^{n_{i}}e^{n_{i}}}{n_{i}^{n_{i}}e^{m_{i}}}$ (4)

We simulate distance, dust extinction and observational noise simultaneously
on our unreddened synthetic CMDs and then measure the degree of match. In this
sense our synthetic model CMD is stochastically sampled, and to calculate the
likelihood – $L(n_{1},n_{2})$ – that two samples, $n_{1}$ and $n_{2}$, come
from the same underlying model, $m$, we use:

$L(n_{1},n_{2})=\int_{0}^{\infty}\frac{m^{n_{1}}}{e^{m}n_{1}!}\frac{m^{n_{2}}}{e^{m}n_{2}!}dm=\frac{0.5^{(n_{1}+n_{2}+1)}(n_{1}+n_{2})!}{n_{1}!n_{2}!}$
(5)

The equivalent PLR for this expression when $n_{1}$ is the observed data and
$n_{2}$ is the model is (see Sec. 2.3 of D02 for details):

$-2\ln PLR=-2\ln\frac{L(n_{1},n_{2})}{L(n_{1},n_{1})}$ (6)

We sum the log of the PLR for all bins to obtain a total PLR for each model-
data comparison.

The best-fit maximum probability model and the corresponding PLR results per
bin are shown in Figure 6. The residual significance in the lower right panel
measures the fit quality by subtracting the model from the observed CMD and
dividing this difference by the uncertainty in the model count,
$(n-m)/\sqrt{m}$. This result is simply the residual in units of standard
deviation (e.g., a value of $+1$ or $-1$ is a 1$\sigma$ outlier).

There is a very good agreement between the overall number of stars in the
RC/RGB CMD, but some weaknesses in the match are evident. The overestimated
contribution of the horizontal branch (HB) discussed in Section 3.2 can be
seen just below the red clump in both the CMD data-to-model comparison (as a
slight bulge), and in the PLR and residual significance maps (as a
green/yellow and a red patch, respectively). One consequence of this HB
overestimate is the appearance of a “dip” in the reddened model RC (near the
location of the unreddened RGB) even though the measured RC slopes of the data
and the best fit model are nearly identical. The steeper model RGB slope is
another potential consequence of this contribution. Both the PLR and the
residuals maps show a poor match (an overestimation) in the number of
(reddened and unreddened) RC stars which may have to do with the
overestimation of the number of stars in the HB. In the Appendix we show
examples of models which are bad fits to the data.

Figure 7.— 1D and 2D probability distribution functions based on CMD matching
results for the 3D geometry and dust extinction of the SMIDGE field. The 1D
68% and 95% confidence intervals fall between the red dashed and orange dot-
dashed lines, respectively. The 50% probability threshold shown in dotted blue
indicates the most likely value of the corresponding parameter. If the PDF
followed a normal distribution, then the 68% confidence interval would be
equivalent to $\pm 1\sigma$ uncertainty and the 50% blue line would indicate
the mean. 2D probability distribution functions show the correlation among the
geometry and extinction parameters. The black contours represent the 2D 68%
confidence interval threshold. Table 4SMIDGE 3D Structure and Dust Extinction
Results

Parameter | PDF 50% median and 68% confidence intervals
---|---
Dust-stars offset | 0.11 ${}^{+0.06}_{-0.05}$ mag; 3.22 ${}^{+1.69}_{-1.44}$ kpc
Distance modulus spread, $\sigma_{DM}$ | 0.16 ${}^{+0.07}_{-0.06}$ mag; FWHM $\approx$ 11.3 kpc
Reddened fraction, $f_{RED}$ | 0.73 ${}^{+0.13}_{-0.13}$
Extinction Log-normal, $A_{V}$ |  $\widetilde{A_{V}}$ = 0.33 ${}^{+0.05}_{-0.05}$ mag; mean $\langle A_{V}\rangle$ = 0.41 $\pm$ 0.09 mag
Extinction Log-normal width, $\sigma_{A_{V}}$ |  $\widetilde{\sigma_{A_{V}}}$ = 0.67 ${}^{+0.17}_{-0.15}$ mean $\langle\sigma_{A_{V}}\rangle$ = 0.097 $\pm$ 0.003

Note. — The dust layer is on the near side of the stars. The log-normal median
$\widetilde{A_{V}}$ and $\widetilde{\sigma_{A_{V}}}$ are converted into the
mean (normally-distributed) $\langle A_{V}\rangle$ and
$\langle\sigma_{A_{V}}\rangle$ using Equations. 1-2.

###  3.4. The Probability Density Function and Uncertainties in the Results

Each model CMD is subject to two main sources of uncertainty. One is a result
of number statistics due to the small number of stars in each CMD bin.
Additionally, each CMD contains an intrinsic uncertainty in the stellar
positions due to the random photon noise and crowding effects (simulated by
the BEAST). These effects result in an inherent variability in the model
predictions when a model is generated multiple times with the same set of
parameters.

We generate synthetic CMDs over a well-sampled evenly-spaced grid of models
based on the ranges of geometry and dust extinction parameters in Table 3. To
account for the sources of uncertainty above, we generate each CMD in the grid
50 times and calculate the average number of stars per CMD bin. Each model is
fit to the data by the process in Section 3.3. The repeated trials average out
the noise.

In addition to finding the best-fit model (see end of Section 3.3), we also
calculate the median and the confidence intervals of the 1-dimensional PDFs of
each parameter. We define the 50th percentile of the cumulative probability as
the PDF median, and the 16-84th percentile confidence interval threshold as a
measure of the PDF width (68$\%$ around the median), or the uncertainty in our
results, informing us of the range of parameter values which contains the
corresponding proportion of the probability. If the probability distribution
were Gaussian, then the 68% confidence interval would correspond to $\pm$ one
standard deviation, $\sigma$, and the 50th percentile would correspond to the
mean. The results of this analysis are shown in Figures 7 and 8.

To obtain a 1D PDF for each parameter, we marginalize over all other
parameters after one step of renormalizing. In this step, we assume that the
probability of a model outside the grid is $\sim$ 0, thus we cover a total
probability of 1 in the grid, where each model is normalized accordingly. We
marginalize over various dimensions of the grid to produce 1-D and 2-D PDFs
for each parameter and parameter pair. In this way we can find desired
confidence intervals for our parameters.

Figure 8.— The result of the SMC 3-dimensional structure on the reddened
fraction, $f_{RED}$, in terms of the relationship between the spread in the
stellar distribution and the location of the dust layer relative to the stars.
As expected, $f_{RED}$ increases as the dust is farther removed from the stars
(on the near side) while the stellar distribution remains compact. Lines of
constant reddened fractions align with the white 68% confidence interval
contours of the dust-stars offset and $\sigma_{DM}$ 2D PDF in Figure 7. The
observations indicate $f_{RED}$=0.73${}^{+0.13}_{-0.13}$. (See Section 4.)

##  4\. Results

The results for the 3D geometry and dust extinction parameters and their 68%
confidence interval thresholds are presented in Table 4 and Figures 7 and 8.
We find that the dust layer is offset on the near side of the centroid of the
stellar distribution by 3.22 ${}^{+1.69}_{-1.44}$ kpc, and that the CMD is
best fit with a stellar distribution along the line of sight with a 1 $\sigma$
width of 0.16 ${}^{+0.07}_{-0.06}$ mag, or an equivalent FWHM of 11.3 kpc. The
reddened fraction of stars is $f_{RED}$=0.73${}^{+0.13}_{-0.13}$ with the dust
on the near side of the stellar midplane. The log-normal of dust extinction
has a median of $\widetilde{A_{V}}$=0.33 $\pm$ 0.05 mag and width
$\widetilde{\sigma_{A_{V}}}$=0.67 ${}^{+0.17}_{-0.15}$. This corresponds to a
mean $\langle A_{V}\rangle$ = 0.41 mag and a width $\sigma_{A_{V}}$ = 0.097
(where the foreground $A_{V}$ = 0.18 mag applied to model the SFH is not a
part of this result, thus it is solely attributable to SMC dust.).

There is a lack of strong correlation between the geometry and the extinction
parameters in the fitting procedure. In particular, $\widetilde{A_{V}}$ and
$\widetilde{\sigma_{A_{V}}}$ do not appear correlated with the dust-stars
offset or the line-of-sight depth. The interpretation of this result is that
since stars lie either in front of or behind the thin dust layer, they
experience either all of the dust column or none of it (as discussed in
Section 3.1.4). The geometrical arrangement of the stars and the dust is
therefore independently constrained from the extinction properties of the dust
layer.

Figure 7 and Figure 9 show that all parameters are well-constrained. Figure 9
shows a slice through the multidimensional grid, with all parameters held
fixed at their best-fit values from Section 3.3, aside from the value on the
x-axis. Both figures show that small values of the line-of-sight depth,
$\sigma_{DM}$, are strongly ruled out.

One of the most strongly constrained parameters from the observations is
$f_{RED}$, which can be found simply from counts of RGB stars in the
unreddened and reddened branches. $f_{RED}$ is a purely geometrical effect
which results from the relative arrangement of the dust-stars offset and the
stellar distribution along the line-of-sight ($\sigma_{DM}$). The correlation
between the line-of-sight depth and the dust-stars offset is the strongest
among all the parameters. $f_{RED}$ can be predicted theoretically from the
relationship between $\sigma_{DM}$ and the dust-stars offset as illustrated in
the right-hand panel of Figure 8. Regions of constant $f_{RED}$ indicate that
the closer the dust layer is to the centroid of the stellar distribution, the
smaller the line-of-sight depth required to preserve the same reddened
fraction. Indeed, the 68% confidence intervals of the 2D PDF of the dust-stars
offset – $\sigma_{DM}$ trace lines of constant $f_{RED}$. Our results for
$f_{RED}$ = 0.73${}^{+0.13}_{-0.13}$ are consistent with the 68% confidence
interval.

Figure 9.— Model grid slices in multidimensional space showing the
relationship between the best-fit CMD model parameters. Each point represents
the relative probability for a model corresponding to a parameter value shown
on the x-axis where the rest of the parameters for this model are fixed at
their best-fit values.

##  5\. Discussion

Our two main findings about the SMC galactic structure are that the SW bar of
the galaxy shows a significant spread in the distribution of stars along the
line of sight of more than 10kpc FWHM, and also that the dust layer, which is
clearly observed from its extinction effects, is offset from the centroid of
the stars on the near side of the SMC. We obtained a similar result for the
line-of-sight depth (10 $\pm$ 2 kpc) in YMJ17 employing a simple model for the
reddened RC which helped explain the observed offset in the extinction curve
in the SMC and the LMC.

Our line-of-sight depth finding is consistent with a number of other studies
of the SMC structure employing various methods which generally suggest that
the SMC is significantly elongated along the line of sight. Depending on the
region studied and the method used, results vary and indicate galactic depths
of less than 10 kpc and up to $\sim$ 23 kpc in the outer regions of the galaxy
(Hatzidimitriou & Hawkins, 1989; Gardiner & Hawkins, 1991; Harris & Zaritsky,
2004; Subramanian & Subramaniam, 2009, 2012; Nidever et al., 2013; Rubele et
al., 2018; Muraveva et al., 2018).

Distance tracers show a number of features in the SMC, such as a distance
gradient, galactic depth, and a distance bimodality. The distance gradient
observed generally suggests that the eastern regions are closer than the
western ones (Jacyszyn-Dobrzeniecka et al., 2016; Subramanian & Subramaniam,
2012; Ripepi et al., 2016; Subramanian et al., 2017; Muraveva et al., 2018).
The galactic depth has been observed by finding the distribution of RR Lyrae
and Cepheid variables (Harris & Zaritsky, 2004; Kapakos & Hatzidimitriou,
2012; Haschke et al., 2012; Jacyszyn-Dobrzeniecka et al., 2016; Muraveva et
al., 2018) and the luminosity dispersion of red clump stars (Gardiner &
Hawkins, 1991; Nidever et al., 2013; Subramanian et al., 2017). Using the
latter also shows a distance bimodality (Subramanian et al., 2017) with two
distinct bodies of stellar structures in the eastern regions of the galaxy.
Although we do not study the distance gradient here, we note that the distance
modulus we find via CMD matching discussed in Section 3.2 of $\mu$=19.07 mag
indicates that the SMIDGE region in the SMC’s SW Bar is offset from the center
of the SMC (found to be at a distance modulus of $\mu$=18.96 mag (Scowcroft et
al., 2016)), consistent with the studies above in that it indicates the
western region is farther away.

A significant line-of-sight depth is also consistent with an LMC/SMC
interaction. Numerical models for the SMC and the LMC indicate that the SMC
geometric features, as well as the internal kinematics of the Magellanic
Clouds, can be explained by repeated interactions between the two galaxies
(Besla et al., 2012, 2016). The models of Besla et al. (2012) show that ram
pressure decreases the velocity of the SMC’s ISM and plays a role in modifying
the properties of the ISM distribution, such as its location and velocity. Our
observations that the dust layer is thin and results that the dust is
positioned on the near side of the SMC relative to the stellar centroid
(therefore placing the dust nearer the LMC as well) may be an expected
consequence of such a ram pressure effect. Choi et al. (2018) measured the
dust reddening and 3D structure of the LMC using RC stars and discovered a new
stellar warp toward the SMC in the outer disk (and an LMC line-of-sight depth
of $\sim$7 kpc). It is clear from our study and others that the line-of-sight
geometry of both Magellanic Clouds is complicated, due to their interactions.

We compare our CMD fitting result for the average SMIDGE dust extinction with
$A_{V}$ based on dust mass surface densities, $\Sigma_{M_{d}}$, derived from
IR emission observations by the HERschel Inventory of The Agents of Galaxy
Evolution (Gordon et al., 2014). We fit the IR dust SED with a modified black
body dust emission model as in Chiang et al. (2018) to extract the average
dust mass surface density in the SMIDGE footprint, and find $\Sigma_{M_{d}}$ =
1 $\times$ 105 M⊙ kpc-2. Using $A_{V}/\Sigma_{M_{d}}$ = 0.7394
$mag/10^{5}M_{\odot}$ kpc-2 from the Draine & Li model and from Draine et al.
(2014), the IR-derived extinction is $A_{V}$ $\approx$ 0.75. This
overprediction of the observed extinction – here by a factor of $\sim$ 1.8 –
is consistent with other comparisons between IR-derived optical extinction
results using the Draine & Li dust model and alternative extinction methods.
Dalcanton et al. (2015) who use CMD fitting of RGB stars to derive the
distribution of dust in M31 find that the dust model $A_{V}$ estimates are too
high by a factor of $\sim$ 2.5 when comparing their results to those of Draine
et al. (2014). Similarly, Planck Collaboration et al. (2016) show a
discrepancy of a factor of 1.9 in $A_{V}$ estimates from all-sky Milky Way
optical photometry of quasi-stellar objects. This discrepancy can be
attributed to the Draine & Li model overestimating the dust mass from IR
emission, which would imply that dust grains are more emissive than the model
assumes. However, it is notable that M31 and the Milky Way are both at
relatively high metallicity. Our comparison is among the first of its kind for
a low-metallicity environment containing dust which potentially has very
different properties.

Further, an SMC $A_{V}$ comparison can be made with the work of Hagen et al.
(2017) who use UV, IR and optical observations integrated into 200′′ regions
to model the SMC SED and map $A_{V}$ for the galaxy. Half of their SMC regions
have a mean $\langle A_{V}\rangle<$ 0.25, with results for the SMIDGE region
varying between $A_{V}$ = 0.2 - 0.75 mag. Our results of mean $\langle
A_{V}\rangle$ = 0.41 are on the high end which is expected since the SMIDGE
field was specifically picked as one of the dustiest spots in the SMC. Our
upcoming study will resolve the SMIDGE field into smaller regions which will
allow for a better comparison with Hagen et al. (2017).

Dalcanton et al. (2015) studied the distribution of dust in M31 by modeling
the NIR CMD of red giant branch stars to obtain results for the same set of
dust and geometry parameters as the ones we explore here –
$\widetilde{A_{V}}$, $\widetilde{\sigma_{A_{V}}}$, and $f_{RED}$. Although
they approach modeling the RGB and RC quite differently for a number of
reasons, they also observe an RGB broadening and a bimodality due to the
combined effects of dust extinction and geometry, and rely on modeling these
observed effects to form the basis of their conclusions. Their study explores
regions in the star-forming disk of M31 which generally have a more widely-
varying and a higher $A_{V}$ than SMIDGE. They do not observe a correlation
between $A_{V}$ and $f_{RED}$ at high extinctions (median $A_{V}\gtrsim$ 1
mag). A correlation between the two at lower extinctions is interpreted not as
a physically-driven result, but rather as a result expected from their fitting
approach (which relates $f_{RED}$ to $A_{V}$ through a ”filling factor”
describing the areas of the gas cloud and the analyzed pixel such that an
increasingly smaller filling factor for dusty gas is due to a decreasing
median extinction and also contributes to a decreasing geometric reddened
fraction). At the lowest extinctions they observe, which are comparable to the
extinctions we find in SMIDGE, (median $A_{V}\lesssim$ 0.3 mag), the
broadening of the RGB in their NIR CMDs is not significant, thus their do not
have a strong constraint on $f_{RED}$.

The LMC, also known to have an appreciable depth along the line of sight
(Subramanian & Subramaniam, 2009) and a relatively low metallicity at 1/2
solar (Russell & Dopita, 1992), is another suitable target for this RC/RGB
dust and geometry modeling technique. Our goal is to apply this same approach
to data from the LMC METAL survey $HST$ (Roman-Duval et al., 2019) to study
dust extinction and geometry properties probing the LMC extinction curve and
extinction distribution, and 3D structure.

###  5.1. Calculating $A_{V}/N_{H}$

We measure the average H column density in the SMIDGE field using HI
observations from Stanimirovic et al. (1999) using the ATCA and Parkes
telescopes and 12CO J=(2$-$1) observations of the SW Bar with the APEX
telescope (Rubio et al. in prep). We extract the average integrated
intensities of the HI and CO lines over the full coverage of the ACS imaging
from SMIDGE. We convert the HI line integrated intensity to column density
assuming no opacity correction. For H2, we must adopt a CO-to-H2 conversion
factor. The behavior of the conversion factor as a function of metallicity is
the subject of much investigation (Bolatto et al., 2013). For the SMC, XCO has
been found to be larger than the MW value of X${}_{\rm CO}=2\times 10^{20}$
cm-2 (K km s-1) by a variety of studies (Leroy et al., 2011; Roman-Duval et
al., 2014). The SMIDGE field is atomic gas dominated, so for a wide range of
conversion factors NH does not change dramatically depending on the conversion
factor choice. We convert the CO (2-1) line to (1-0) using a
$R_{21}=(2-1)/(1-0)=0.7$ and we use XCO values between the MW and 10 times the
MW value to obtain NH2, which we multiply by 2 to obtain NH. We calculate a
mean $\langle A_{V}\rangle$ = 0.41 $\pm$ 0.09 mag. The final range of NH for
SMIDGE is $9.7\times 10^{21}$ to $1.3\times 10^{22}$ cm-2. This yields
$\langle A_{V}\rangle/N_{H}=3.2-4.2\times 10^{-23}$ mag cm2 H-1. These values
are substantially lower than the canonical MW $\langle
A_{V}\rangle/N_{H}=5.3\times 10^{-22}$ (Bohlin et al., 1978; Rachford et al.,
2009) by more than an order of magnitude. G03 measure the $HI$ column density
from the UV Ly$\alpha$ absorption profile for four sightlines in the SMC Bar
and find $A_{V}/N_{H}=7.7\times 10^{-23}$ mag cm-2 H-1, which is only a factor
of $\sim$ 2 higher than our result. Roman-Duval et al. (2014) measure a
comparable ratio in the SMC using Herschel IR observations where they find the
gas-to-dust ratio to be between 4 and $\sim$ 10 times the MW value in the
diffuse and dense ISM, respectively, depending on the methodology used.

##  6\. Conclusions and Further Work

We model the dust extinction and 3D geometry properties of the SMIDGE field in
the SW Bar of the SMC using the optical $\textit{F475W}-\textit{F814W}$ CMD of
RC and RGB stars. We find the following:

  1. 1.

The distance modulus for the midpoint of the stellar distribution within the
SMIDGE region is 65.2 kpc ($\mu$ = 19.07 mag).

  2. 2.

The CMD is best fit with a 1$\sigma$ line-of-sight depth $\sigma_{DM}$ = 0.16
${}^{+0.07}_{-0.06}$ mag, or an equivalent FWHM of 11.3 kpc.

  3. 3.

The dust layer is offset on the near side of the stars by 3.22
${}^{+1.69}_{-1.44}$ kpc and is located at a distance modulus of $\mu$ = 18.96
mag (61.94 kpc).

  4. 4.

The combination of dust position and stellar distribution results in a
73${}^{+0.13}_{-0.13}$ % reddened fraction of stars.

  5. 5.

The distribution of dust extinction has a mean $\langle A_{V}\rangle$ = 0.41
$\pm$ 0.09 mag and width $\langle\sigma_{A_{V}}\rangle$ = 0.097 $\pm$ 0.003
mag (log-normal median $\widetilde{A_{V}}$ = 0.33${}^{+0.05}_{-0.05}$ mag and
dimensionless width $\widetilde{\sigma_{A_{V}}}$ = 0.67 ${}^{+0.17}_{-0.15}$).

The SMC’s dust extinction curve is notably different from that of the Milky
Way in terms of the lack of a strong 2175 Å bump and a steep rise in the UV
(G03). Here we base our model grid on the G03 SMC Bar extinction law
($R_{V}$=2.74) as discussed in Section 3.1.4 since our 2017 work (YMJ17)
showed consistency with the G03 results. To probe different extinction laws,
including a potential extinction law mixture of SMC and Milky Way extinction
curves (see Appendix), in subsequent work we will use multiple color
combinations for a similar analysis, fixing the geometric parameters of the
SMC at the values found here.

We thank the anonymous referee for a very thorough and helpful report that
substantially improved the paper. This work was supported in part by NASA
through grant number HST-GO-13659 from the Space Telescope Science Institute,
which is operated by AURA, Inc., under NASA contract NAS5-26555. D.R.W.
acknowledges support from an Alfred P. Sloan Fellowship and an Alexander von
Humboldt Foundation Fellowship. These observations are associated with program
# GO-13659 and are based on observations made with the NASA/ESA Hubble Space
Telescope. We made extensive use of NASA’s Astrophysics Data System
bibliographic services. This research made use of Astropy, a community-
developed core Python package for astronomy (Astropy Collaboration et al.
2013), NumPy (Van Der Walt et al. 2011), Matplotlib (Hunter 2007), the syncmd
tool for BEAST-based post-processing of Synthetic Stellar Populations (C.
Johnson), the BEAST (Bayesian Extinction and Stellar Tool; K. Gordon), and the
MATCH CMD fitting software package (A. Dolphin).

## Appendix

We perform tests to determine the sensitivity of our technique to a number
physical- and technique-specific factors in our attempt to measure quantities
of interest.

Our sensitivity to the extinction law is limited since extinction curve
differences in the optical part of the spectrum are small, and our analysis is
based only on optical CMDs. We test how sensitive we are to differentiating
between an SMC and a MW extinction curve by simulating CMDs with a mixture of
an SMC-like G03 $R_{V}$=2.74 law and a Milky Way-like Fitzpatrick (1999)
$R_{V}$=3.1 law. We do this by varying $f_{A}$, which is the fraction
corresponding to the mixture between the two extinction laws (see Sec. 3.2 of
G16). Although we see a trend of increasing PLR (worse match) as the model
moves away from an SMC-like extinction, a pure Milky Way-like law is within
the SMC noise model uncertainties. Thus we are not sensitive to the extinction
law to a sufficient degree using the optical CMD only.

While Figure 6 shows our best fit model, here we show examples of bad model
fits where all parameters are held fixed at their minimum-PLR values, except
for one parameter which is fixed at a bad-fit value. The PLR and the residual
significance plots show the mismatch clearly. The top panel of Figure 10 shows
the result of placing the dust layer in the middle of the stellar distribution
causing a $\sim$ 50% reddened fraction. We can see that both the RC and the
RGB have an overdensitiy of stars which we do not see in the data.

The middle panel shows a model which underestimates the spread of the stars
along the line of sight (FWHM = 5.3 kpc while our results point to a FWHM
$\approx$ 11.3 kpc). We clearly see that the model lacks a sufficient number
of unreddened stars extending to brighter magnitudes (which would result from
a higher line-of-sight depth). Correspondingly, the model overestimates the
number of reddened stars (calculated to be $f_{RED}$) since the smaller spread
in the stellar distribution places a higher fraction of stars behind the dust
(see Fig. 8).

It can be also seen that the slope of the reddened portion of the red clump is
incorrect since a smaller galactic depth produces a shallower RC slope, and in
our case a mismatch to the data. Indeed, the main measurement of our previous
work (YMJ17), which used the slope of the reddened RC across multiple-color
CMD combinations to measure the shape of the SMIDGE extinction curve, showed
an offset from previous extinction curve measurements. This result was due to
the measured steepness of the RC reddening vector which resulted in higher
extinction values across all CMD combinations. Using a toy RC model to
simulate an SMC depth, we showed that the offset was consistent with a stellar
distribution with a FWHM = 10 $\pm$ 2 kpc along the line of sight.

The bottom panel of Figure 10 shows a model which underestimates the amount of
dust by incorporating a distribution of dust extinctions with
$\widetilde{A_{V}}$=0.1 (contrasted with our result of
$\widetilde{A_{V}}$=0.3). This model also results in a reddened fraction
$f_{RED}$=0.7, since the extinction has no bearing on the dust-stars geometry
(and vice-versa) as discussed in Section 5. However, since $\widetilde{A_{V}}$
changes both the color and magnitude of stars, we can see the effect in both
CMD dimensions for the RC and the RGB as they tend to be closer to their
unreddened positions (see the left panel of Figure 3).

Figure 10.— Examples of bad model fits where all but one parameters are at
their best fit value listed in Section 4. Top panel: Dust layer is aligned
with the center of the stellar distribution causing a 50% reddened fraction.
Mid-panel: The spread of the stellar distribution is 1/2 of its best-fit value
($\sigma_{DM}$=2.25 kpc which results in a distribution with a FWHM = 5.3 kpc.
Bottom panel: The mean of the log-normal distribution of dust extinctions is
$A_{V}$=0.1 mag. )

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