Inferring the 3D Shapes of Extremely Metal-poor Galaxies from Sets of Projected Shapes

Simulations suggest that the main modes of galaxy growth over cosmic time involve merger (e.g., Conselice 2014) and cold gas streams (filaments of the cosmic web) that deeply penetrate the dark matter halo of galaxies and serve as the main driver of star formation (e.g., Dekel et al. 2009a). Although such cosmic accretion must become less intense as the universe ages, expands, and becomes less dense, evidence of cosmic accretion has been mounting in the local universe, as galaxies can be studied in greater detail nearby. Local extremely metal-poor galaxies (XMPs; defined as having an average gas-phase metallicity <0.1 solar; e.g., Kunth & Östlin 2000) typically have several properties consistent with the cosmic accretion scenario: off-center clumps (see Section 2.1; consistent with, e.g., Noguchi 1999; Elmegreen et al. 2008; Dekel et al. 2009b; Ceverino et al. 2016); high specific star formation rates (e.g., Morales-Luis et al. 2011); isolated (Filho et al. 2015); rich in H i, especially in the outer regions (Filho et al. 2013); and significant metallicity drops at the starburst regions (Sánchez Almeida et al. 2013, 2015), indicating recent accretion of metal-poor gas given that the timescale for gas mixing is short (e.g., de Avillez & Mac Low 2002). Thus, in the context of cosmic accretion, XMPs are leading local laboratories for studying galaxy formation and evolution.

About 500 XMPs have been identified thus far (e.g., Sánchez Almeida et al. 2017), and their 3D shape has yet to be explored. The 3D shape of a galaxy inevitably is linked to the past and/or current formation processes at work, so constraining the 3D shape of XMPs may lead to a better understanding of the nature of primitive galaxies. Specifically, it will provide insight regarding the roles of cosmic accretion and associated stellar feedback processes, the balance between external and internal processes, and the structural relation between the dark matter halo (e.g., Velliscig et al. 2015; Vega-Ferrero et al. 2017) and the stellar distribution, as dwarf galaxies, such as most XMPs, are likely dominated by dark matter (e.g., Behroozi et al. 2010).

Here we infer the 3D shape of XMPs using a simple and robust technique. It is simple in the sense that it requires only fitting ellipses to galaxy images, and it is robust because the distribution of projected shapes is a consequence of the 3D shape (and orientation) of the galaxies. Using an ellipsoid as our model for the 3D shape, the 2D projection in the plane of the sky is an ellipse. Thus, assuming the galaxies to have a common shape and to be oriented randomly, one can infer the parameters of the ellipsoid from the axial ratios of the ellipses fitted to their images. We refer to this general technique as the q technique, where q is the minor-to-major axial ratio of an ellipse fit to the image of a galaxy. The q technique has been used extensively starting with Hubble (1926). It is well suited for a hierarchical Bayesian inference model, and we use such an approach in this work (see Section 2.3). Our inference method was inspired by Sánchez-Janssen et al. (2016), who emphasize the power of a Bayesian approach to the q technique, namely that when using discrete q measurements instead of a histogram of them, a large observed sample is not necessary and the uncertainty of the individual q measurements can be used. But our Bayesian approach is significantly different from theirs; ours is hierarchical, whereby the likely shape and orientation of each galaxy is inferred while simultaneously inferring the average shape and dispersion.

The paper is organized as follows: Section 2 describes our sample selection, q measurements, and method for inferring 3D shapes. Section 3 presents our results and evaluates the potential bias caused by a surface brightness selection effect. In Section 4, we discuss our main results. Appendix A highlights the link between the distribution of q and 3D shape through simulations, Appendix B discusses the results of testing our inference model using a simulated galaxy population and small samples, and Appendix C shows how to modify the simple 3D model used for shape to account for the surface brightness variation with q.

2.1. Sample Selection

The galaxies used in our inference of 3D shape are from Sánchez Almeida et al. (2016), who mined the spectroscopic catalog of the Sloan Digital Sky Survey (SDSS) in producing the largest published sample (195) of XMPs from a single survey. To have a more homogeneous sample, we omitted four spiral galaxies (showing spiral arms and/or a bulge), six interacting galaxies with obvious tidal tails or bridges between separate nuclei, and five galaxies with a nonelliptical projected shape. The size of the remaining 180 galaxies relative to seeing is represented in Figure 1, top panel. It shows the observed axial ratio (computed in Section 2.2) versus R₉₀/R_seeing. R₉₀ is the equivalent radius enclosing 90% of the light ($\pi {R}_{90}^{2}$ is the area enclosing 90% of the light) whereas R_seeing stands for the half width at half maximum (HWHM) of the seeing point-spread function (PSF). Both R₉₀ and R_seeing were taken from the SDSS database and correspond to the r band. XMPs with R₉₀ ∼ R_seeing tend to be round (i.e., with axial ratio close to one), which can be an artifact produced by seeing. The trend disappears at R₉₀ > 5 R_seeing (Figure 1, top panel), therefore, we further omitted the 31 galaxies with R₉₀ < 5 R_seeing; this discarded compact objects appearing as clumps without a host, along with other poorly resolved galaxies. The final selected sample contains 149 XMPs. We note that the cut in size may potentially introduce an additional bias against physically small rounded galaxies (i.e., those to the left of the vertical line in Figure 2.2, top panel), however, the analysis carried out in Section 3.2 proves that it has insignificant impact on the inferred galaxy shapes.

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Among our selected sample, the majority have at least one clump appearing as a distinct object within a host (the clumps are off-center in all but two cases), a large fraction have an off-center bluer and/or brighter region (not appearing as a distinct object), and a small fraction show no signs of clumpiness but still appear to lack structure. We used the SDSS SkyServer⁵ RGB images for this visual assessment of clumpiness, but clumpiness was not part of our selection criteria. Galaxy redshifts go from 0 to 0.2, with most XMPs being closer than 0.05, and with no obvious trend between axial ratio and redshift once the galaxies with R₉₀ < 5R_seeing are discarded (see the blue symbols in Figure 1, bottom panel).

2.2. Axial Ratio Measurement

2.3. Inferring 3D Shape

Here, we explain our hierarchical Bayesian inference model to carry out the q technique; that is, to infer the 3D shape distribution of our sample from our q_obs measurements by modeling the galaxies as ellipsoids oriented randomly. Our inference model intertwines two stages, and Figure 2 shows the conditional dependencies involved (described below). The shape of each galaxy is inferred (stage 1) while inferring the global parameters characterizing the distribution of shapes (stage 2). The Bayesian solution to our statistical problem is known as the posterior probability,⁷ and it is proportional to some likelihood function⁸ times some prior probability.⁹

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To compute the posterior, one needs to define the likelihood function, which requires a generative model for modeling the probability of q_obs given q_syn, a synthetic q value. We assume the observed galaxies to have an ellipsoidal form with the mutually perpendicular axes length (A), width (B), and thickness (C) normalized to A, therefore leading to a ratio of axes 1:B:C, where A = 1 ≥ B ≥ C > 0. Thus, any ellipsoid shape is defined by B and C alone. The ellipsoid orientation depends on the inclination angle θ and the azimuthal angle ϕ. We used Simonneau et al. (1998) to compute q_syn for any {B, C, θ, ϕ} set of values; for the projected semimajor (a) and semiminor (b) axes, they provide the relations

$$\begin{eqnarray}&&\begin{array}{l}{a}^{2}{b}^{2}={f}^{2}\\ \ =\ {\left(C\sin \theta \cos \phi \right)}^{2}+{\left({BC}\sin \theta \sin \phi \right)}^{2}+{\left(B\cos \theta \right)}^{2}\end{array}\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{a}^{2}+{b}^{2}=g={\cos }^{2}\phi +{\cos }^{2}\theta {\sin }^{2}\phi \\ \ +\ {B}^{2}({\sin }^{2}\phi +{\cos }^{2}\theta {\cos }^{2}\phi )+{C}^{2}{\sin }^{2}\theta ,\end{array}\end{eqnarray} \tag{ 2 }$$

so that defining h as

$$\begin{eqnarray}&&h=\sqrt{\displaystyle \frac{g-2f}{g+2f}},\end{eqnarray} \tag{ 3 }$$

we can express q_syn as

$$\begin{eqnarray}&&{q}_{\mathrm{syn}}=\displaystyle \frac{b}{a}=\displaystyle \frac{1-h}{1+h}.\end{eqnarray} \tag{ 4 }$$

To highlight the link between q and 3D shape, Figure 7 in Appendix A shows histograms of q_syn for randomly oriented galaxies and a variety of 3D shapes.

Under the assumption of uncorrelated Gaussian noise, our generative model for sampling the posterior is q_syn(B, C, θ, ϕ) +  = q_obs, where the noise contribution is a Gaussian-distributed random variable, with mean zero and standard deviation ${\sigma }_{{q}_{\mathrm{obs}}}$, that is truncated and renormalized because q must be in the interval (0, 1]. We will refer to {B, C, θ, ϕ} as the set of galaxy parameters whose priors are further governed by the assumption that the observed galaxies make up a family of ellipsoids defined by the family parameters μ_B, μ_C, σ_B, and σ_C. Thus the galaxy parameters of a set of galaxies, ${\boldsymbol{B}}$ and ${\boldsymbol{C}}$,¹⁰ are assumed to follow from the same Gaussian distributions with means μ_B and μ_C, and standard deviations σ_B and σ_C.

The prior of our statistical problem is the product of the priors indicated in Table 1. We now have all the ingredients to write the expression for evaluating the posterior (Bayes' theorem) in terms of the parameters in our inference model:

$$\begin{eqnarray}&&\begin{array}{l}P({\boldsymbol{B}},{\boldsymbol{C}},{\boldsymbol{\theta }},{\boldsymbol{\phi }},{\mu }_{B},{\sigma }_{B},{\mu }_{C},{\sigma }_{C}| {{q}}_{\mathrm{obs}})\\ \ \propto \ { \mathcal L }({{q}}_{\mathrm{obs}}| {B},{C},{\boldsymbol{\theta }},{\boldsymbol{\phi }})P({\boldsymbol{B}}| {\mu }_{B},{\sigma }_{B})\,P({\boldsymbol{C}}| {\mu }_{C},{\sigma }_{C})\\ P({\boldsymbol{\theta }})P({\boldsymbol{\phi }})P({\mu }_{B})P({\mu }_{C})P({\sigma }_{B})P({\sigma }_{C}),\end{array}\end{eqnarray} \tag{ 5 }$$

where $P(y| x)$ stands for the probability of y given x, and ${ \mathcal L }$ is substituted for P to denote the likelihood. Note that the family parameters appear in the likelihood function implicitly.

Table 1.  Priors

Parameter	Probability Distributiona	Type
cos θ	U(−1, 1)	Prior
ϕ	U(0, 2π)	Prior
Bb	N(μB, σB)	Hierarchical prior
Cb	N(μC, σC)	Hierarchical prior
μB	U(0, 1)	Hyperprior
μC	U(0, 1)	Hyperprior
σB	Half-N(0.05)	Hyperprior
σC	Half-N(0.05)	Hyperprior

Notes.

^aU stands for uniform. N stands for normal. Given a normally distributed random variable X, the random variable $Y=| X|$ has a half-normal distribution Half-N. ^bB and C have the additional constraints 1 ≥ B ≥ C > 0.

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Our method to evaluate the posterior is computationally efficient for three reasons, and the first two are consequences of our hierarchical framework. First, we have circumvented computing q_syn histograms corresponding to different {μ_B, σ_B, μ_C, σ_C} sets. Second, only parameter estimates near the maximum likelihood will be sampled. Third, we implemented our inference model using Stan,¹¹ which uses the state-of-the-art sampling algorithms NUTS (Hoffman & Gelman 2014). We used Stan also because it makes relatively straightforward to account for the change of variables needed to ensure the condition B ≥ C.

We checked our inference model in three different ways. (1) Although the NUTS algorithm implemented in Stan is known to work properly even in problematic hierarchical inference models, we checked that the sampling of the posterior properly converged by checking standard convergence criteria.¹² (2) Posterior predictive checks (explained in Section 3.1) were carried out to be sure that our inference model is able to properly explain the observations. (3) We also fed model populations to our inference model to see how well it can infer the family parameters and how sample size plays a role. We found that indeed μ_B and μ_C can be accurately inferred from small samples and that σ_B can be particularly difficult to constrain. Appendix B presents the experiment and results in more detail, and Figure 8 is another way to argue that σ_B can be difficult to constrain.

Relative thickness C is directly signaled by the lowest q values, so we can already infer from Figure 3, showing q_obs versus semimajor axis, that smaller galaxies tend to be relatively thicker (larger C). We therefore divided our sample into Small, Medium, and Large, as shown in Figure 3. (To convert to physical size, we estimated distance using the SDSS redshift of each galaxy and a Hubble constant of 70 km s⁻¹ Mpc⁻¹. Only 4% of our sample is nearer than 10 Mpc—shortly beyond this point, proper motions with respect to the Hubble flow become negligible, so the vast majority of our size measurements are reasonable approximations.) We made the three cuts marked by the gray dashed lines in Figure 3. The cut positions were chosen at semimajor axis of 1.4 kpc (so that q_obs > 0.3 for all Small; see Figure 3), 3.9 kpc (so that q_obs > 0.2 for all Medium), and 8 kpc. Ten galaxies larger than 8 kpc (up to 23 kpc) were not included in Large to have a more homogeneous subsample, though we noted that this does not impact the results significantly. In addition to these three cuts, we also use the whole sample of 149 galaxies, naming it All.

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3.1. Observed q Histograms and Posterior Predictive Checks

The inferred 3D shapes to be presented in Section 3.2 are based on the discrete q_obs values and their uncertainties, but first we show our q_obs histograms in Figure 4 and their posterior-predicted fits to check that our inference model worked as expected. Our q_obs histograms and their 1σ uncertainty are represented by the red lines, generated via Monte Carlo simulation. The posterior-predicted histograms in gray show the q_syn values corresponding to the marginal posterior sampling of the galaxy parameters (see Section 2.3). The agreement between our observed distributions and the posterior-predicted histogram shows that our inference model is appropriate for explaining the observations.

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3.2. Inferred 3D Shapes

The inferred 3D shape distributions for our 3 subsamples are shown in Figure 5. We plot the marginal posterior sampling of ${\boldsymbol{C}}$ versus that of ${\boldsymbol{B}}$ and translate the density of points to a color map where darker represents higher probability. In other words, each galaxy has its own cloud in the C (relative thickness) versus B (relative width) parameter space that is its inferred shape and the uncertainty in it, and the clouds in Figure 5 combine all of the individual galaxy clouds into a family cloud. The maximum a posteriori value of each family parameter is given in Table 2 for each of our subsamples. The shape parameters of the individual galaxies are included in Table 3, which is given online. The shape distributions in Figure 5 offer a more complete view because the ${\boldsymbol{C}}$ and ${\boldsymbol{B}}$ marginal posteriors inevitably are not perfectly Gaussian. The σ_B and σ_C results in Table 2 are poorly constrained (this is inherent to the inference model; see Appendix B), but we see that the average 3D shape of each subsample is nevertheless well constrained.

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Table 2.  Maximum A Posteriori Family Parameters^a

Sampleb	Sample Size	μC	σC	μB	σB
Small	54	${0.39}_{-0.02}^{+0.02}$	${0.03}_{-0.02}^{+0.02}$	${0.71}_{-0.05}^{+0.05}$	${0.04}_{-0.02}^{+0.03}$
Medium	54	${0.26}_{-0.04}^{+0.04}$	${0.06}_{-0.04}^{+0.03}$	${0.66}_{-0.03}^{+0.03}$	${0.03}_{-0.02}^{+0.02}$
Large	31	${0.16}_{-0.03}^{+0.03}$	${0.03}_{-0.02}^{+0.02}$	${0.71}_{-0.08}^{+0.08}$	${0.04}_{-0.02}^{+0.03}$
All	149	${0.28}_{-0.02}^{+0.02}$	${0.09}_{-0.02}^{+0.02}$	${0.67}_{-0.03}^{+0.03}$	${0.03}_{-0.02}^{+0.02}$

Notes.

^aWe report the median values of the marginal posteriors and express the uncertainty using the 16th and 84th percentiles (the probability that the parameter is within this range is 68%). ^bThe subsamples are divided as shown in Figure 3. The subsample All also includes the 10 galaxies excluded in Large.

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Table 3.  3D Shape Parameters for Individual Galaxies^a

R.A.b	Decl.b	C	B	θc	Subsampled	IDe
(deg)	(deg)	(A)	(A)	(deg)	⋯	⋯
9.42130	0.55561	${0.39}_{-0.04}^{+0.04}$	${0.72}_{-0.07}^{+0.07}$	${43.0}_{-20.1}^{+20.8}$	S	6
19.80951	−9.59617	${0.37}_{-0.04}^{+0.03}$	${0.71}_{-0.07}^{+0.07}$	${81.8}_{-8.5}^{+5.6}$	S	9
23.46897	13.70264	${0.39}_{-0.04}^{+0.04}$	${0.71}_{-0.07}^{+0.07}$	${49.2}_{-22.5}^{+22.1}$	S	10
45.45427	−0.88260	${0.39}_{-0.04}^{+0.04}$	${0.71}_{-0.07}^{+0.07}$	${72.5}_{-13.9}^{+12.0}$	S	19
130.65241	10.55387	${0.39}_{-0.04}^{+0.04}$	${0.71}_{-0.08}^{+0.07}$	${69.6}_{-13.3}^{+13.8}$	S	36
⋮	⋮	⋮	⋮	⋮	⋮	⋮

Notes.

^aThe full table, available only online, contains 139 galaxies divided among our three subsamples. It is sorted by subsample and then by R.A. The corresponding family parameters are described in Section 2.3 and Table 2. We report the median values of the marginal posteriors and express the uncertainty using the 16th and 84th percentiles (the probability that the parameter is within this range is 68%). ^bR.A. and decl. in J2000 coordinates. ^cInclination of the major axis with respect to the line of sight. ^dFigure 3 shows how subsamples are divided. S, M, and L stand for Small, Medium, and Large, respectively. ^eIdentification number from Sánchez Almeida et al. (2016).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In Figure 5 we see the relation that relative thickness increases with decreasing galaxy size. We see no trend in the results for relative width, but we see that for all three subsamples, the galaxies are inferred to be notably far from axisymmetric disks. In Figure 5 we have included a variation of the intuitive-shape framework (elongated versus disky versus spheroidal) proposed by van der Wel et al. (2014).

To avoid the effect of seeing on the inferred shapes, we removed from the original sample 31 galaxies with size R₉₀ < 5 R_seeing (Section 2.1). This cut in size may potentially introduce a bias against physically small round galaxies. To discard it, we repeated the computation of 3D shape including all the small filtered out galaxies for which q could be estimated. The new values for B and C are very close to those in Table 2, namely, μ_C and μ_B are ${0.39}_{-0.02}^{+0.02}$ and ${0.73}_{-0.07}^{+0.05}$, ${0.29}_{-0.04}^{+0.04}$ and ${0.67}_{-0.04}^{+0.04}$, and ${0.17}_{-0.03}^{+0.03}$ and ${0.79}_{-0.05}^{+0.05}$, for the subsamples Small, Medium, and Large, respectively.

3.3. Surface Brightness Selection Effect

Disk-like galaxies tend to have fainter surface brightness when observed face-on. A drawback of our implementation of the q technique is that it does not account for the bias introduced by this effect, i.e., for the fact that near the surface brightness limit of a survey different orientations have different probabilities of being observed. To assess the potential impact of this effect on our results, we estimate the expected surface brightness bias. As we will show, the predicted decrease in surface brightness when q increases is not present in the observed data set. Thus, the surface brightness bias does not seem to be important for the XMP galaxies analyzed in our work, supporting the reliability of the shapes inferred using our implementation of the q technique.

In this estimate, we crudely model the surface brightness of a galaxy (SB) assuming no internal extinction. (Extinction is known to be low in XMPs, and will be treated in Appendix C.) Then the total flux emitted by the galaxy is independent of its orientation; therefore, the flux per unit surface scales as the inverse of the area projected by the galaxy on the plane of the sky. Assuming the 3D model used in the paper, this area is proportional to the product ab given in Equation (1), so that

$$\begin{eqnarray}&&\mathrm{SB}={\mathrm{SB}}_{0}-2.5\,\mathrm{log}\left[B\,/\,({ab})\right],\end{eqnarray} \tag{ 6 }$$

where SB₀ stands for the largest SB considering all possible orientations of the galaxy, and B is the maximum value of ab.¹³ Given B, C, θ, ϕ and SB₀, SB(q) follows implicitly from Equations (1), (4), and (6). Figure 6, bottom row, shows scatter plots of SB versus q for collections of 100 identical galaxies randomly oriented. (Their actual B, C and SB₀ values are given in the figure.) To account for the limited completeness of our survey, the 100 model galaxies were included in Figure 6 at random, with the probability depending on their SB. The probability of being included was given by the completeness function of the SDSS spectroscopic survey worked out by Blanton et al. (2005), who added mock disk galaxies to the raw SDSS images, which then went through the standard pipeline to be selected as member of the SDSS spectroscopic catalog. The completeness function thus estimated has 50% completeness at 23.4 mag arcsec⁻² and falloff width of around 1 mag arcsec⁻². Thus, the selection effect is not noticeable until SB₀ = 24 mag arcsec⁻², but it is severe for SB₀ = 25 mag arcsec⁻² and larger.

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The model distribution SB(q) in Figure 6 (bottom row) have been chosen to have B and C as the means μ_B and μ_C observed in the samples Small, Medium, and Large (Table 2). They have to be compared with the observed distribution in Figure 6, top row. For our observed SB distributions in Figure 6 we plot the half-light surface brightness in the r band, as this is the quantity employed by Blanton et al. (2005) to quantify the SDSS completeness function used in our modeling. Comparing our model with observed half-light surface brightness implicitly assumes the galaxies to have a single shape, from the isophotes traced by the half-light radius to those where q is measured. We used SExtractor (Bertin & Arnouts 1996) to measure the integrated magnitudes of our galaxies and their half-light radii. (Magnitudes from the task to determine q_obs, i.e., IRAF ellipse, are not employed because they exclude the contribution of bright clumps, which disable their use in flux determinations.) We argue that the surface brightness selection effect modeled in Figure 6 is not significantly affecting us for three reasons. (1) Each of our observed SB distributions shows a rough upward trend as q increases, whereas the model populations, which are affected by the selection effect, have a downward trend. (2) Figure 6 shows that only a small percentage of the observed galaxies allows for a selection effect against face-on disks. The selection effect predicts the existence of many galaxies below the 50% completeness line, which are not observed (compare top and bottom rows in Figure 6). (3) Nearly all of our galaxies have some degree of clumpiness, so face-on counterparts may in fact appear brighter than when edge-on due to less extinction of the light from the clumps. This effect may partially account for the upward trend in each of our observed SB distributions even though the dust content tends to be low in XMPs (Sánchez Almeida et al. 2016). Differences between the central parts of the galaxies, contributing to SB, and the outskirts, contributing to the q, could also explain the observed increase of surface brightness with increasing q. These two possibilities are discussed in more detail in Appendix C.

We have constrained the 3D shapes of XMPs from their distribution of projected shapes using a hierarchical Bayesian inference model. Our main findings (Figure 5) are that (1) XMPs tend to be triaxial (A > B > C) with an intermediate axis ≈0.7 times the longest axis and that (2) the shortest axis of XMPs ranges between ≈0.4 and ≈0.15 times the longest axis, with smaller galaxies tending to be relatively thicker. Our subsample cuts based on physical size (Figure 3) are somewhat arbitrary, but small changes would not significantly affect the inferred range for relative thickness.

To put these results into the proper context, we have to be aware that galaxies of nearly all types have a 3D shape with some elongation (e.g., Binggeli & Popescu 1995), defined as having B < 1. Spirals are among the least elongated, with B ≈ 0.9 on average (e.g., Padilla & Strauss 2008). However, we note that the Milky Way has a stellar halo that is much more elongated than this (Iorio et al. 2018). More elongated 3D shapes (compared with spirals) are common in irregular galaxies (e.g., Sung et al. 1998; Hunter & Elmegreen 2006; Roychowdhury et al. 2013), including ultra diffuse galaxies,¹⁴ and in high-redshift, star-forming galaxies (e.g., Elmegreen et al. 2005; Ravindranath et al. 2006; Law et al. 2012). Simulations have found dark matter halos (e.g., Bekki & Freeman 2002; Velliscig et al. 2015; Vega-Ferrero et al. 2017) and their stellar counterparts (e.g., Ceverino et al. 2015; Velliscig et al. 2015) also to be notably elongated.

Our results for C are similar to the findings of Roychowdhury et al. (2013), who observed relative thickness to increase with decreasing luminosity for dwarf irregular galaxies. Our average C for subsample Large (0.16) is consistent with the relative thickness of the disk component of spiral galaxies (e.g., Kregel et al. 2002), and our average C for sample All (0.28) agrees closely with Law et al. (2012) for high-redshift, star-forming galaxies. Sung et al. (1998) found an average C of 0.55 for blue compact dwarfs (BCDs); this is much greater than our average, yet many XMPs are BCDs (e.g., Kunth & Östlin 2000; Morales-Luis et al. 2011). This discrepancy may reflect in part that XMPs are not a single morphological class (which was evident to us during our sample selection and from the change of surface brightness with q). This issue has been mitigated by omitting obvious spirals and other poorly resolved XMPs and by forming subsamples based on physical size, but a natural extension of this work would be to evaluate the different morphological types among XMPs.

The trend that smaller galaxies are relatively thicker is understood dynamically, as thickness is proportional to the square of the velocity dispersion divided by the disk surface density, which tends to be low for dwarf galaxies like XMPs. If our galaxies all have a similar velocity dispersion and if the smaller ones rotate more slowly, then we would expect the smaller XMPs to be relatively thicker. The ratio of the clump size to the galaxy size also scales with relative thickness because the clump size is usually comparable to the turbulent Jeans length, which scales with velocity dispersion and column density in the same way as the thickness. These scalings make smaller galaxies relatively thicker and more clumpy, which is a morphology that also applies, for the same reasons, to more massive galaxies at high redshift (for further details, see Elmegreen et al. 2009). This tendency for more dynamically primitive systems to be thicker is also found among the faint galaxies in the Virgo and Fornax clusters (Sánchez-Janssen et al. 2019).

In addition to slow rotation, large relative thickness may result from feedback ejecting gas and causing the stellar body to expand in response or from stellar scattering, which could be more important in a low mass galaxy with more clumps. It may also result from the accretion of external gas that induces random motions in the gas that forms stars, which seems to be the case for primitive, high-redshift galaxies (e.g., Elmegreen et al. 2005, 2012; Olmo-García et al. 2017).

Isolated galaxies dominated by their own rotation evolve to become axisymmetric disks. Thus the elongated 3D shapes of XMPs have to be due to a variety of factors that make them deviate from the ideal case. According to simulations, larger dark matter halos tend to be more elongated in part because they formed more recently (Vega-Ferrero et al. 2017). Thus, an elongated shape appears to be normal for dynamically young systems in which rotation is probably less dominant. Lending observational support to this, Lelli et al. (2014) found that starburst dwarfs with a younger starburst tend to have a more asymmetric H i morphology compared with those with an older burst. In this sense, XMPs often have large H i reservoirs with asymmetric morphology (Filho et al. 2013). Simulations have also revealed elongated stellar distributions within elongated dark matter halos (Velliscig et al. 2015), so that the shape of the stellar distribution is inherited in part from the dark matter halo shape (e.g., Zaritsky et al. 2013; Laine et al. 2014). The presence of filaments (Vega-Ferrero et al. 2017), cosmic accretion (Ravindranath et al. 2006), gas clumps, and radiative feedback (Ceverino et al. 2014) may also all contribute to an elongated shape, and such factors are all expected for typical XMPs. Tidal forces could also cause an elongated shape, but during our sample selection we found a tiny fraction of interacting galaxies with a distorted shape.

The average SB of the XMPs measured in the r band is found to increase as q increases (Figure 6, top row). This is unexpected for disk-like galaxies. Unfortunately, we do not have a good explanation for this behavior. The ellipse fit employed to infer shapes traces the outer parts of the galaxies, whereas the contribution to the surface brightness is more centrally concentrated. Thus the clumps in XMPs, which we have avoided in the measure of q, may bias the trend observed in the SB. The unexpected behavior probably indicates that XMPs are not homologous, being more prolate in the centers and more oblate in the outskirts. This nonhomology plus the small contribution of extinction might eventually explain the observed trend.

This work has been partly funded by the Spanish Ministry of Economy and Competitiveness (MINECO), projects AYA2016-79724-C4-2-P (ESTALLIDOS) and AYA2014-60476-P and by the La Caixa Foundation. We thank Nicola Caon, Benjamin Alan Weaver, Ignacio Trujillo, Rubén Sánchez-Janssen, and Benne Holwerda for assisting with different aspects of the paper. We also thank the anonymous referee for suggestions to analyze and clear out the surface brightness bias. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III website is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.

To highlight the link between q and our model for 3D shape (see Section 2.3), Figure 7 shows a q_syn histogram for a perfect disk (in green), for a highly triaxial shape (in blue), for a highly elongated shape (in gray), and for a highly spheroidal shape (in red). Each histogram is from a model population of one million randomly oriented galaxies all having the same shape, defined in the legend of Figure 7. Gaussian noise (with the standard deviation set to 4% of a) was added to a and b, which were then reordered to ensure q_syn ≤ 1.

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We experimented with q_syn data of a model population to see how well our inference model can retrieve the set of family parameters {μ_B, σ_B, μ_C, σ_C}. We tested 10 subsamples all randomly extracted from the sample of one million galaxies that make up the blue histogram in Figure 7. Five of the samples have a sample size of 25, the other five samples have a sample size of 200, and all samples have C = 0.3, B = 0.7, and noise (added as described in Appendix A). If the family parameters are correctly inferred, we should find that the maximum A Posterior solution is consistent with μ_C = 0.3, σ_C = 0, μ_B = 0.7, and σ_B = 0. The inferred family parameter results are summarized in Table 4. We can conclude that μ_B and μ_C can be accurately inferred from small samples even though σ_B and σ_C are difficult to constrain, and inferring σ_B remains particularly difficult even as sample size increases. Another way to see this is in Figure 8, where the blue histogram in Figure 7 is shown again along with the population when σ_B and σ_C are set to 0.05 and with noise in q_syn (as described in Appendix A). Comparing the two histograms, we see that they are virtually identical in the falloff toward higher q_syn, which is where most of the information on B is contained (see Figure 7). Thus, it makes sense that σ_B remains inherently difficult to constrain even as sample size increases.

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Table 4.  Maximum A Posteriori Family Parameters: Model Population Tests^a

Test No.	Sample Size	${\mu }_{{\rm{C}}}$	σC	μB	σB
1	25	${0.28}_{-0.04}^{+0.03}$	${0.03}_{-0.02}^{+0.03}$	${0.68}_{-0.05}^{+0.06}$	${0.04}_{-0.02}^{+0.03}$
2	25	${0.28}_{-0.03}^{+0.03}$	${0.02}_{-0.01}^{+0.02}$	${0.71}_{-0.21}^{+0.12}$	${0.03}_{-0.02}^{+0.03}$
3	25	${0.27}_{-0.03}^{+0.03}$	${0.02}_{-0.01}^{+0.02}$	${0.76}_{-0.06}^{+0.06}$	${0.03}_{-0.02}^{+0.03}$
4	25	${0.30}_{-0.03}^{+0.03}$	${0.03}_{-0.02}^{+0.02}$	${0.53}_{-0.07}^{+0.13}$	${0.04}_{-0.02}^{+0.03}$
5	25	${0.33}_{-0.04}^{+0.03}$	${0.03}_{-0.02}^{+0.02}$	${0.77}_{-0.10}^{+0.08}$	${0.04}_{-0.02}^{+0.03}$
6	200	${0.32}_{-0.01}^{+0.01}$	${0.02}_{-0.01}^{+0.02}$	${0.71}_{-0.02}^{+0.02}$	${0.03}_{-0.01}^{+0.02}$
7	200	${0.31}_{-0.01}^{+0.01}$	${0.01}_{-0.01}^{+0.02}$	${0.73}_{-0.03}^{+0.03}$	${0.04}_{-0.02}^{+0.03}$
8	200	${0.30}_{-0.01}^{+0.01}$	${0.02}_{-0.01}^{+0.02}$	${0.69}_{-0.03}^{+0.03}$	${0.05}_{-0.03}^{+0.03}$
9	200	${0.29}_{-0.01}^{+0.01}$	${0.01}_{-0.01}^{+0.01}$	${0.74}_{-0.03}^{+0.04}$	${0.04}_{-0.02}^{+0.03}$
10	200	${0.30}_{-0.01}^{+0.01}$	${0.02}_{-0.01}^{+0.01}$	${0.67}_{-0.03}^{+0.03}$	${0.06}_{-0.03}^{+0.03}$
11	1000	${0.30}_{-0.00}^{+0.00}$	${0.01}_{-0.00}^{+0.01}$	${0.98}_{-0.01}^{+0.01}$	${0.00}_{-0.00}^{+0.00}$

Note.

^aWe report the median values of the marginal posteriors and express the uncertainty using the 16th and 84th percentiles (the probability that the parameter is within this range is 68%). The true values are μ_C = 0.3, σ_C = 0, μ_B = 0.7, and σ_B = 0 for test number 1 to 10. Trial 11 has been run with all other parameters the same but μ_B = 1.0.

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Our simple 3D model is used in Section 3.3 to argue that the bias against face-on disk-like galaxies is not present in the analyzed XMP data set. Rather, galaxies become slightly brighter as q increases (Figure 6). The 3D model used to represent galaxy shapes is simply too elementary to account for the observed effect, however, small modifications of the original model allows us to reproduce the observed trend. The issue is discussed here for consistency, to show that there is no fundamental limitation that prevents our 3D model from reproducing the trend in Figure 6.

If dust extinction is included in the model, objects observed edge-on may be more obscured and so should present lower surface brightness. The effect of extinction on SB can be easily incorporated into Equation (6) adding an extra term, namely

$$\begin{eqnarray}&&\mathrm{SB}={\mathrm{SB}}_{0}-2.5\,\mathrm{log}\left[B/({ab})\right]+2.5\,\kappa \,({q}_{\mathrm{syn}}^{-1}-1),\end{eqnarray} \tag{ 7 }$$

where κ stands for the extinction coefficient, and the dependence (q_syn⁻¹ − 1) is an ansatz to provide a mathematical representation of the attenuation. It was chosen because $({q}_{\mathrm{syn}}^{-1}-1)$ is zero for face-on disks (q_syn = 1) and increases as q⁻¹ with decreasing q_syn, which is the behavior expected for thin disks, for which q_syn is just the cosine of the inclination with respect to the line of sight. Figure (9), center panel, shows the same type of Monte Carlo simulation worked out in Section 3.3 but now based on Equation (7). We begin with 100 randomly oriented galaxies, for which ab and q_syn are computed from Equations (1) to (4). These galaxies are then selected or not with a probability set by completeness function (see Section 3.3). We also make two assumptions regarding SB₀, i.e., it is the same for all galaxies (the magenta symbols in Figure 9), or they have a random spread of values of ±0.7 mag (the yellow symbols in Figure 9). We use κ = 0.3, which is a reasonable value for XMPs (see, Section 4.1 in Sánchez Almeida et al. 2016, where κ has been ascribed to the extinction coefficient in Hβ). As Figure 9 evidences, including extinction produces an increase SB as q_syn decreases, in qualitative agreement with the observed trend (compare Figures 9, left and center panels).

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Another alternative explanation relies on the inside-out change in galaxy shape. The increase of surface brightness with increasing q_syn is to be expected in 3D prolate ellipsoids, which have A > B, C and B ≈ C. According to Equation (6), the galaxies are brightest when their projected area is smallest. In the case of a prolate ellipsoid, the minimum area on the sky results when the galaxy is observed along its major axis, and because B ≈ C, this maximum brightness corresponds to q ≃ 1. The behavior is shown in Figure 9, right panel, which follows from ellipsoids having A:B:C = 1:0.30:0.25. This prolate shape disagrees with the oblate shape inferred from the q technique. Thus, for this to be an explanation, the 3D shape of the galaxies should be non-homologous, being more prolate in the centers and more oblate in the outskirts. The ellipse fit used to infer shapes traces the outer parts of the galaxies, whereas the contribution to the surface brightness is more centrally concentrated. This nonhomology might explain the trend observed in SB(q). The points in Figure 9, right panel, make two assumptions regarding SB₀: it is the same for all galaxies (the magenta symbols), or they have a random spread of values of ±0.5 mag (the yellow symbols). The actual values of SB₀ have been tuned to match the level and spread of the observed SB (Figure 9, left panel).