Dwarf Galaxy Mass Estimators vs. Cosmological Simulations

TL;DR

This study tests the accuracy of two widely-used mass estimators for dwarf galaxies using high-resolution cosmological simulations, finding they are highly accurate despite assumptions of spherical symmetry.

Contribution

It provides validation of the Walker and Wolf mass estimators against realistic, aspherical dwarf galaxy simulations, demonstrating their robustness and identifying key factors affecting their accuracy.

Findings

Estimators are accurate within ~20% across simulations.

Asphericity does not significantly impact estimator accuracy.

Using fitted surface density profiles improves mass estimates.

Abstract

We use a suite of high-resolution cosmological dwarf galaxy simulations to test the accuracy of commonly-used mass estimators from Walker et al.(2009) and Wolf et al.(2010), both of which depend on the observed line-of-sight velocity dispersion and the 2D half-light radius of the galaxy, $Re$. The simulations are part of the the Feedback in Realistic Environments (FIRE) project and include twelve systems with stellar masses spanning $10^{5} - 10^{7} M_{\odot}$ that have structural and kinematic properties similar to those of observed dispersion-supported dwarfs. Both estimators are found to be quite accurate: $M_{Wolf}/M_{true} = 0.98^{+0.19}_{-0.12}$ and $M_{Walker}/M_{true} =1.07^{+0.21}_{-0.15}$, with errors reflecting the 68% range over all simulations. The excellent performance of these estimators is remarkable given that they each assume spherical symmetry, a supposition that is broken in our simulated galaxies. Though our dwarfs have negligible rotation support, their 3D stellar distributions are flattened, with short-to-long axis ratios $ c/a \simeq 0.4-0.7$. The accuracy of the estimators shows no trend with asphericity. Our simulated galaxies have sphericalized stellar profiles in 3D that follow a nearly universal form, one that transitions from a core at small radius to a steep fall-off $\propto r^{-4.2}$ at large $r$, they are well fit by S\'ersic profiles in projection. We find that the most important empirical quantity affecting mass estimator accuracy is $Re$ . Determining $Re$ by an analytic fit to the surface density profile produces a better estimated mass than if the half-light radius is determined via direct summation.

Equations

$$\frac{1}{n_{\ast }}\frac{d}{dr}\left(n_{\ast }{\sigma }_r^2\right)+2\frac{\beta {\sigma }_r^2}{r}=-\frac{GM(r)}{r^2},$$

$$M(r)=\frac{r{\sigma }_r^2}{G}\left({\gamma }_{\ast }+{\gamma }_{\sigma }-2\beta \right).$$

$$M_{\rm Walker}(<\mbox{$R_{e}$})=\frac{5\mbox{$R_{e}$}\sigma^{2}_{\rm los}}{2\,G},$$

$$M_{\rm Wolf}^{\rm ideal}(<\mbox{$r_{3}$})=\frac{3\mbox{$r_{3}$}\sigma^{2}_{\rm los}}{G}.$$

$$M_{\rm Wolf}(<4/3\mbox{$R_{e}$})=\frac{4\mbox{$R_{e}$}\sigma^{2}_{\rm los}}{G}.$$

$$\rho (r)=\frac{{\rho }_s}{(r/r_s{)}^{\gamma }{\left[1+(r/r_s{)}^{\alpha }\right]}^{(\beta -\gamma )/\alpha }},$$

$${\rho }_{\ast }(r)=\frac{{\rho }_s}{(r/r_s{)}^{0.1}{\left[1+(r/r_s{)}^{2.25}\right]}^{1.82}}.$$

Full text

††pubyear: 2016††pagerange: Dwarf Galaxy Mass Estimators vs. Cosmological Simulations–A

1 Introduction

Dynamical measurements of galaxy masses are among the most important empirical quantities for testing theories of galaxy formation and cosmology (see Courteau et al. 2014 for a review). In rotationally-supported systems, for example, velocity field measurements can be used to trace the total mass enclosed as a function of radius, and this provides an important test of $\Lambda$CDM predictions for dark matter halo density profiles shapes (e.g., Flores & Primack, 1994; de Blok et al., 2001; Oh et al., 2011; Iorio et al., 2016). But for dispersion-supported galaxies, radial mass profiles are harder to extract from line-of-sight velocities, owing significantly to degeneracies associated with the unknown velocity dispersion anisotropy (e.g., Łokas, 2002; Binney & Tremaine, 2008). The inability to extract mass-density profiles in a straightforward manner from dispersion-supported galaxies is particularly unfortunate in the case of small (M_{*}$$\lesssim 10^{7} $M_{\odot}$) dwarf spheroidal/irregular galaxies as these systems are the primary focus of the Missing-Satellites problem (Klypin et al., 1999; Moore et al., 1999; Simon & Geha, 2007; Brooks et al., 2013) and the Too-Big-to-Fail problem (Boylan-Kolchin et al., 2011, 2012; Garrison-Kimmel et al., 2013; Tollerud et al., 2014).

The mass-anisotropy degeneracy can in principle be broken by higher-order velocity moments (e.g. Łokas et al., 2005; Cappellari, 2008) and/or proper motion data (Strigari et al., 2007; Read & Steger, 2017). A common and easy-to-implement practice is to use the observed line-of-sight velocity dispersion to infer the mass within a single characteristic radius, close to the galaxy’s half-light radius, where mass measurements are most robust to degeneracies associated with velocity dispersion anisotropy. This approach was specifically advocated by Walker et al. (2009, Wa09) and Wolf et al. (2010, Wo10), who presented simple estimators designed for dwarf galaxies in particular. These estimators have become the go-to empirical measures of mass used in confronting the Too-Big-to-Fail problem (Fattahi et al., 2016; Wetzel et al., 2016) and in reporting mass-to-light ratios of newly discovered dwarfs to distinguish baryon-dominated star clusters from dark-matter dominated ultrafaint galaxies (Willman & Strader, 2012; Voggel et al., 2016; Caldwell et al., 2016).

In this paper, we will test these common mass estimators against fully self-consistent cosmological simulations of M_{*}$$\approx 10^{5}-10^{7} $M_{\odot}$  dwarf galaxies. Our simulations are part of a FIRE-2 simulation suite of the Feedback in Realistic Environments (FIRE) project (Hopkins et al., 2014; Fitts et al., 2016; Hopkins et al., 2017)111http://fire.northwestern.edu. We note that El-Badry et al. (2016a) have used FIRE simulations to test the underlying reliability of the Jeans equation in galaxies experiencing feedback and concluded that while larger dwarfs (M_{*}$$>10^{7} $M_{\odot}$) can experience feedback-driven potential fluctuations that induce $\sim 20\%$ errors in full Jeans modeled masses, the single small galaxy they considered (M_{*}$$\sim 2.5\times 10^{6} $M_{\odot}$) had only a $5\%$ offset in derived mass from Jeans modeling owing to the rather modest energy injection it had received from supernovae. This result provides hope that mass estimators we aim to test will prove reliable when compared against our larger suite of low-mass dwarfs.

Both the Walker and Wolf estimators were derived using idealized approximations, including the assumption that galaxies are spherically symmetric and non-rotating. While small dwarfs do tend to have little stellar rotation (Wheeler et al., 2017) , they can be quite aspherical (McConnachie, 2012), and this in itself motivates an exploration of the estimators’ reliability. Past tests of this kind have relied on analytic approaches (Sanders & Evans, 2016) and mock observations of galaxy-formation simulations (Campbell et al., 2016). Sanders & Evans (2016) showed that spherically-symmetric mass estimators can be applied to flattened stellar distributions, if $R_{e}$ is replaced by the so called “circularized’ analogue R_{e}$$\sqrt{1-\epsilon}, where $\epsilon$ is the ellipticity of the projected stellar distribution. Using the APOSTLE simulations that naturally included non-spherical galaxies, Campbell et al. (2016) found that both Wa09 and Wo10 provide accurate dynamical masses for their simulated galaxies with $\sim 25\%$ (unbiased) offsets at one-sigma. Our approach is similar, though complementary, to that of Campbell et al. (2016). While their simulations included a very large number of galaxies over a range of stellar masses ($M_{*}$= $10^{6.5}-10^{10}$ $M_{\odot}$) that are generally larger than most of the Milky Way satellites, ours includes $12$ systems with masses that are better matched to the range of classical Local Group dwarfs ($M_{*}$= $10^{5.6}-10^{7.2}$ $M_{\odot}$). Our simulations also have $\sim 15\times$ higher mass resolution and $\sim 3\times$ better force resolution222In GIZMO, we use the same definition of force softening $\epsilon$ and spatial resolution, corresponding to the inter-particle separation $h_{i}$; this is related to the Plummer equivalent softening ($\epsilon_{{\rm plummer}}$) by $\epsilon_{{\rm plummer}}\approx(2/3)\epsilon$. ($\epsilon_{{\rm DM}}=35$ pc) than the best-resolved systems in Campbell et al. (2016). High resolution is crucial for this kind of analysis, as the galaxies of concern are small ($R_{e}$ $\sim 500$ pc) and the mass estimators are proportional to galaxy size. Thus our analysis extends the mass-estimator test to the mass-scale of most relevance to the Missing Satellites and Too-Big-to-Fail problems and further allows us to perform a careful system-by-system analysis of each well-resolved galaxy. Finally, by testing these mass estimators with an entirely different simulation code and different modeling of galaxy formation physics, we provide an all-important cross check to ensure robustness to underlying model implementations and assumptions.

This article is organized as follows. In Section 2 we present our suite of simulations and discuss the methods to test mass estimators. In Section 3 we present our main results, which include a summary of the important properties of our simulated galaxies in 3.1 and the mass estimator performance on each galaxy in 3.2. Section 4 summarizes our findings, compares them to previous work, and discusses future directions.

2 Methods

2.1 Mass estimators and assumptions

The dynamical mass estimators we will test were derived from the spherically-symmetric Jeans equation:

[TABLE]

where $M(r)$ is the total mass within radius $r$. The quantities $n_{*}$, $\sigma_{r}$, and $\beta\equiv 1-\sigma_{t}^{2}/\sigma_{r}^{2}$ represent the stellar number density, the radial velocity dispersion, and the velocity dispersion anisotropy, respectively. The case of $\beta=0$ implies that the tangential velocity dispersion $\sigma_{t}$ ($=\sigma_{\theta}=\sigma_{\phi}$) is equal to the radial velocity dispersion. It is revealing to rewrite Equation 1 by solving for the mass profile and defining $\gamma_{*}\equiv-{\rm d}\ln n_{*}/{\rm d}\ln r$ and $\gamma_{\sigma}\equiv-{\rm d}\ln\sigma_{r}^{2}/{\rm d}\ln r$ to give

[TABLE]

We see that even at fixed radial velocity dispersion and radius, the associated mass is very sensitive to the local slopes of the tracer profile, the tracer’s velocity dispersion profile, and the velocity dispersion anisotropy. Each of these quantities is difficult to measure in a galaxy observed in projection.

Both Wa09 and Wo10 used Equation 1 as a starting point and derived mass estimators of the form $M(<r_{\star})=\alpha r_{\star}\,\sigma^{2}_{\rm los}/G$, where $r_{\star}$ is a characteristic stellar radius within which the mass is measured, $\sigma_{\rm los}$ is the tracer-weighted average line-of-sight velocity dispersion, and $\alpha$ is a constant of order unity.

Wa09 were guided by MCMC-enabled spherical Jeans modeling of their observed galaxies, which showed that masses within a radius r_{\star}=\mbox{R_{e}} (the projected half-light radius) were generally well constrained in posterior likelihoods after marginalizing over many other parameters. They therefore worked out analytically the mass within $R_{e}$ for spherical system simplified by three assumptions: i) that $n_{*}$ is given by a Plummer profile, ii) that $\beta=0$, and iii) that $\gamma_{\sigma}=0$. They obtained

[TABLE]

and showed that this estimator did well in comparison to their full Jeans/MCMC analysis.

Wo10 set out to find an ideal radius to use for $r_{\star}$. Assuming $\gamma_{\sigma}=0$, they were able to derive analytically that the mass within a radius r_{\star}=\mbox{r_{3}} is independent of $\beta$ for an observed $\sigma_{\rm los}$. Here, $r_{3}$ is the radius where $\gamma_{*}=3$ (the radius where the the log-slope of the tracer profile is equal to $-3$). The equation they derived is

[TABLE]

Wo10 also showed that for a wide range of commonly-adopted stellar profiles that \mbox{r_{3}}\simeq r_{1/2}, where $r_{1/2}$ is the 3D half light radius. The Wolf formula is then often quoted as an estimator for $M(<r_{1/2})$. Another good approximation that allows one to determine $r_{1/2}$ from the projected half-light radius $R_{e}$ is r_{1/2}\simeq 4/3\mbox{R_{e}} (see Wo10). Since $R_{e}$ can be measured on the sky, this is the most common form of the Wolf estimator used in the literature and we will adopt it as our fiducial comparison case:

[TABLE]

In what follows we will test Equations 3 and 5 using mock observations of our simulations.

2.2 Simulations

This work relies on a suite of twelve $\Lambda$CDM zoom-in hydrodynamical simulations of dwarf galaxies (Fitts et al., 2016) that were run using the GIZMO code333http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.html (Hopkins, 2015). The simulations are part of the Feedback In Realistic Environments (FIRE) project (Hopkins et al., 2014) and adopt the FIRE-2 model for galaxy formation physics (Hopkins et al., 2017). FIRE-2 implements the same main star formation and stellar feedback physics models as the original FIRE simulations, but with several numerical improvements, including a more accurate hydrodynamics solver (the “meshless finite mass” method). FIRE galaxies successfully reproduce several observables from real galaxies (see, e.g., Oñorbe et al., 2015; Chan et al., 2015; Wheeler et al., 2017; Muratov et al., 2015; Faucher-Giguère et al., 2015, 2016; Ma et al., 2016; El-Badry et al., 2016b). We refer the reader to Fitts et al. (2016) for a complete description of the simulations used in this work. Here we only briefly summarize the main characteristics.

The FIRE-2 simulations include cooling and heating rates tabulated from CLOUDY (Ferland et al., 2013). Star formation proceeds only in the densest regions ($\geq 1000\,{\rm cm}^{-3}$), mimicking the densities of molecular clouds. Each star particle is spawned as a single stellar population with a Kroupa (2001) initial mass function. The simulations model momentum and energy input from stellar feedback, including photo-ionization and photo-electric heating, radiation pressure, stellar winds, and supernovae (Types Ia and II).

Our simulated galaxies each form within a dark matter halo of virial mass $10^{10}M_{\odot}$ at $z=0$ with their properties summarized in Table 1 of Fitts et al. (2016). Specifically, we are studying simulations m10b to m10m. The initial conditions for these runs were chosen to span a variety of formation histories and associated halo concentrations and this results in a range of stellar masses in the final timestep: $M_{*}=5\times 10^{5}$ to 1.5\times 10^{7}\mbox{M_{\odot}}. The simulations use gas particle masses of 500 $M_{\odot}$ (with star particles having somewhat lower masses) and dark matter particles of mass 2500 $M_{\odot}$. The force resolution is $\epsilon_{DM}=35$ pc (corresponding to a “Plummer-equivalent” softening of 23 pc). The minimum gravitational and hydrodynamic resolution reached by the gas is $\epsilon_{g}=h_{g}=2$ pc, which corresponds to the inter-particle separation. Stars have a fixed force resolution at $\epsilon_{\star}=2$ pc. This level of mass and force resolution allows us to resolve the internal structure of simulated galaxies with thousands of star particles – and thus with tracers comparable in number to the largest kinematic stellar samples in local dwarf galaxies. As discussed in Fitts et al. (2016), these simulated dwarf galaxies have properties that broadly reproduce many observed properties of real galaxies of the same stellar mass and are highly dark-matter dominated. Their dark matter density profiles vary in shape from (feedback-driven) cores in systems with M_{*}$$\gtrsim 3\times 10^{6}\mbox{M_{\odot}} to cuspy profiles in galaxies that formed fewer stars than this threshold. This diversity in the underlying dark matter profile shape in our simulated systems adds to the generality of the tests of mass estimators.

3 Results

3.1 Properties of simulated galaxies

As discussed in §2.1, the mass estimators we will be testing were derived under the simplifying assumptions of spherical symmetry and a constant stellar velocity dispersion profile. The Jeans equation is also sensitive to the shape of the underlying tracer density profile, and the Wo10 estimator in particular relies on the assumption that \mbox{r_{3}}\simeq r_{1/2}\simeq 4/3\mbox{R_{e}}. Each of these assumptions is broken to some degree by our simulated galaxies, and in this section we briefly present results on the stellar density distributions (both in 3D and in projection), the velocity dispersion profiles, and the ellipticity of our simulated galaxies before moving on to test the mass estimators applied against them. We note that our simulated galaxies display negligible stellar rotation, with very similar dispersion-supported properties to the simulated FIRE dwarfs discussed in Wheeler et al. (2017).

The left panel of Figure 1 presents the spherically-averaged stellar density profiles of each of our simulated galaxies. The color code corresponds to the total stellar mass of each galaxy, as shown by the color bar along the right. Arrows mark the radius where the local log-slope of the density profile equals -3 ($r_{3}$), and the 3D-half mass radius $r_{1/2}$, respectively. The size range spans $\sim 400-1100$ pc and is consistent with the sizes of real dwarf galaxies in this mass range, as discussed in Fitts et al. (2016). Recall that the Wo10 estimator assumes that \mbox{r_{3}}\approx r_{1/2} and this is, in fact, fairly accurate for almost all of the simulations. For nine of the twelve galaxies, the difference is smaller than $10\%$. The grey band marks the region where predictions are potentially unreliable due to numerical relaxation (Fitts et al., 2016).

As Figure 1 makes clear, the shapes of the $\rho_{*}(r)$ distributions in our simulated dwarf galaxies are quite similar. We find that all of them are reasonably well approximated by a single “universal" shape. Specifically, we have fit an $\alpha\beta\gamma$-profile (Zhao, 1996) of the form

[TABLE]

to each galaxy and find a good fit to the sample as a whole to be:

[TABLE]

This profile asymptotes to a core-like distribution at small radius ($\propto 1/r^{\gamma}$ with $\gamma=0.1$) and a steep fall-off at large radius ($\propto 1/r^{\beta}$ with $\beta=4.2$) and obeys \mbox{r_{3}}\simeq\,0.85\,r_{s}. In the right panel of Figure 1, we plot the density profiles for each simulated system normalized in radius by $r_{s}$ and in density by $\rho_{s}$. The black line shows Eq. 7, which goes dashed in the innermost radius of the galaxies, which may not be well resolved. Though this parameterization does a good job of characterizing the general shape of our galaxies, it is worth noting that the fit is not perfect: the smallest galaxies, for example, prefer somewhat steeper outer profiles.

The density profiles shown in Figure 1 enforce spherical symmetry, but in reality our simulated galaxies are non-spherical. We have computed the shape tensors $\mathbf{S}$ (as described in Zemp et al., 2011) for stars and dark matter within our galaxies and diagonalized them to obtain their eigenvalues, which are proportional to the square-root of the semi-principal axes of the 3D-ellipsoid $a\geq b\geq c$. While keeping the volume of an initial sphere of radius $r$ constant, we use an iterative method to construct ellipsoids until the axis ratio is stable to $1\%$. The points in Figure 2 show short-to-long axis ratios $c/a$ (where $c/a=1$ is perfectly spherical) of dark matter vs. stars calculated in volumes defined by spheres of radius $r=r_{1/2}$ for each simulated galaxy. Despite their lack of rotational support, our simulated galaxies are quite flattened ($c/a\approx 0.4-0.7$). The stellar and dark matter shapes are not identical, but the major axes of stars and dark matter align within 10 degrees for 11 of the 12 galaxies. The most misaligned system has a 20 degree offset between the major axes of dark matter and stars.

While the 3D shape of the underlying stellar density distributions have important implications for full Jeans modeling, in practice it is the 2D (projected) profile that is observed and that must be used when inferring masses. Specifically, the 2D half-light radius $R_{e}$  is the only stellar distribution parameter that is explicitly included in the Walker and Wolf formulas (Eq. 3 and 5). Figure 3 shows the surface density profiles for three of our simulated dwarfs plotted as a function of 2D (circularly-averaged) radius. The shaded bands show the $1\,\sigma$ range of density profiles observed over 1,000 random projections covering the unit sphere around each galaxy. The three colored lines show average Sérsic profile fits over all projections for each galaxy (color coded by galaxy $M_{*}$). We perform these fits within $R=4\,r_{1/2}$. In general, our simulated surface density profiles are well-characterized by Sérsic fits with $0.7\lesssim n\lesssim 1.3$ in their inner regions ($r\lesssim 2-4$ $R_{e}$). In this sense, our galaxies are realistic analogs for testing mass estimators. Interestingly, while a single galaxy can yield a variable Sérsic index $n$ when viewed from different projections, the resultant 2D half-light $R_{e}$ values derived from the fits are quite robust from projection-to-projection. The color-matched vertical dotted lines show $\pm 1\sigma$ range of $R_{e}$ values for each galaxy over all projections. This narrow range of inferred $R_{e}$ for each galaxy contributes to the accuracy of the mass estimators (as we discuss below).

Note that the two most massive systems shown in Figure 3 demonstrate an outer flattening in their projected light profiles. We see this behavior – an excess above the Sérsic extrapolation at large radius – in most of the more massive systems in our suite. Similar features have been observed in the outer regions of larger disk galaxies (see e.g., Herrmann et al., 2016), and it is possible that behavior of this kind exists in the outskirts of smaller dwarf galaxies such as the ones simulated here. However detecting such a flattening in a diffuse stellar component could be challenging. With resolved stars, such a break might be hard to distinguish from foreground or background stars. In diffuse light, the fairly low surface brightness of the expected upturn will make it difficult to detect. In Figure 3 the equivalent surface brightness $\mu_{r^{{}^{\prime}}}$ in the SDSS $r^{\prime}$ band is shown in the right-hand axis. Observational searches for behavior of this kind in the outer regions of dwarf galaxies could enable an interesting constraint on simulations of the type presented here.

The top panel of Figure 4 shows the radial velocity dispersion profiles of each simulated dwarf galaxy. The bottom panel shows the velocity anisotropy profiles for the same systems. The radii have been normalized by $r_{1/2}$ in each case. The radial velocity dispersion profiles rise slowly towards larger radius. The velocity dispersion anisotropy also rises from nearly isotropic $\beta\sim 0$ at small $r$ to a radially biased velocity dispersion at large $r$. The fact that $\sigma_{r}$ is not constant for our galaxies provides an interesting test for both the Wa09 and Wo10 estimators, given that both assume $\gamma_{\sigma}\equiv$ - d$\ln\sigma_{r}^{2}$/d$\ln r=0$. Our galaxies, on the other hand, have $\gamma_{\sigma}\sim-0.5$ near r\sim\mbox{r_{1/2}}.

Figure 5 shows that while the $\sigma_{r}$ tends to rise with radius, our galaxies produce flat line-of-sight velocity dispersion profiles when observed in projection, similar to those seen in real dwarfs (see, e.g., Wa09). In a similar approach as used in Figure 3, Figure 5 presents line-of-sight velocity dispersion profiles of two simulated dwarfs plotted as a function of 2D (circularly-averaged) radius (normalized by $R_{e}$). The shaded bands show the $1\,\sigma$ range of velocity dispersion profiles observed over 1,000 random projections covering the unit sphere around each galaxy. The two colored lines show the median over all projections for each galaxy (color coded by galaxy $M_{*}$).

3.2 Testing the Walker and Wolf Mass Estimators

We perform mock observations of each of our dwarf galaxy simulations viewed along 1,000 randomly-sampled line-of-sight projections. For each projection, we compute the stellar-mass-weighted velocity dispersion, $\sigma_{\rm los}$, and measure an $R_{e}$ value using Sérsic fits to the circularly-binned stellar surface density profiles. Figure 6 shows the resultant predicted mass obtained using both the Wo10 formula (black) and the Wa09 formula (grey) versus the true dynamical mass measured within the corresponding radii, 4/3\mbox{R_{e}} and $R_{e}$, respectively. Each symbol represent the median value over all projections. There are two symbols for each galaxy, with the Wo10 mass always the larger of the two owing to the fact that it measures the mass within a larger radius. The error bars span the 16th and 84th percentiles of the distributions over all projections. Note that each galaxy is plotted at a single ‘true’ mass for either estimator. This does not necessarily have to be the case, as the value of $R_{e}$ inferred from different projections could shift enough to force the true mass to shift accordingly. However, as we discussed in §3.1, when we determine $R_{e}$ from fitting Sérsic profiles, the inferred distribution of $R_{e}$ values over all projections is extremely narrow, and this results in true total mass ranges (corresponding to ranges of $R_{e}$ value) that are narrower than the widths of the points plotted.

It is clear from Figure 6 that both estimators track the one-to-one line fairly well. However, there are a few individual galaxies that tend to produce biased masses, with medians falling either above or below the line (slightly). Figure 7 refines the comparison by showing the ratio of the estimated mass (Wolf, top; Walker, bottom) to true mass for each galaxy, again with points and error bars reflecting the median and one sigma range of the ratio, respectively. Each point is plotted against the $3D$ short-to-long axis ratio of the stars $c/a$ (as introduced for Fig. 3) measured within the radius r_{\star}=4/3\mbox{R_{e}} (top) and $R_{e}$ (bottom) to reflect the radii of interest for the estimators. The dotted line in each panel shows the average of the median data points for all galaxies. We see no strong trend in the median points with $c/a$.

For the Wolf estimator (top), the median values are only $\sim 2\%$ low on average, with the worst outliers biased low by $\sim 10\%$. The one-sigma ranges are more important than the bias, with some galaxies having errors as large as $\pm 25\%$ with respect to the true mass, though $\pm 15\%$ is more typical. The Walker estimator (bottom) does only slightly worse, with median values averaging about $\sim 10\%$ high compared to the true mass. The most extreme outliers are biased high in the median by $\sim 25\%$ and the one-sigma range about the median is as large as $\pm 30\%$. While these small discrepancies in the median do not correlate with $c/a$, many of the most aspherical galaxies are also the ones with the largest scatter. This is not surprising, since they will display the most pronounced variance in shape with viewing angle. We have explored whether the offset in mass estimator correlates with the observed projected shape on the sky and find no systematic trend (see Appendix  A). Unfortunately, based on our simulations, it is difficult to infer the expected variance based on the observed shape on the sky.

While asphericity in the stellar distribution does not seem to be the primary culprit in biasing estimators in the median, we do find that the mapping between 2D and 3D density – specifically the relationship between $R_{e}$ and $r_{3}$ – is a source of bias. Specifically, the Wo10 mass estimator assumes r_{3}\approx 4/3\mbox{R_{e}} because this is a good approximation for a variety of commonly-adopted analytic profiles (Sérsic, King, and Plummer). However, this approximation does not have to hold in general. In Figure 8 we plot the ratio of estimated Wolf mass to true dynamical mass against the \mbox{r_{3}}/(4/3\mbox{R_{e}}) ratio derived for each system. As before, each symbol is the median over 1,000 lines of sight for the twelve dwarfs. The error bars account for the 16th and 84th percentiles in the distribution. We see that the two cases where the Wo10 estimator is biased ($\sim 10\%$ low in the median) are also the two cases where the r_{3}\approx 4/3\mbox{R_{e}} approximation fails by more than 20 percent. It therefore appears that the mapping between 2D stellar distribution and 3D distribution represents an underlying uncertainty in the mass measure, one that is difficult to overcome empirically.

In Figure 9 we show the probability distribution functions (PDF) obtained by stacking the ratios between the estimated mass and the true dynamical mass over all projections in all twelve simulated dwarf galaxies, with the Wolf estimator plotted in the upper panel and the Walker estimator plotted in the lower panel. The solid vertical lines show the median of the distributions and the dotted lines show the 16th-84th percentiles: $M_{\rm Wolf}/M_{\rm true}=0.98^{+0.19}_{-0.12}$ and $M_{\rm Walker}/M_{\rm true}=1.07^{+0.21}_{-0.15}$.

3.3 Dependence on methods for inferring $R_{e}$

In the previous subsection (Figure 8), we showed that the relationship between the inferred $R_{e}$ of a galaxy and its underlying $r_{3}$ value is one important source of mass estimator error. In this subsection we investigate alternative methods for measuring $R_{e}$ and determine which of these provides the most robust variable to use in the Walker and Wolf estimators. The fiducial approach we have adopted in proceeding sections – to measure $R_{e}$ via a Sérsic fit to circularly-symmetric bins – does best, and this is the reason we have used it above.

When attempting to measure $R_{e}$ one, first needs to decide how to assign a radius to a distribution that is not necessarily circularly symmetric on the sky. One common choice is to simply enforce circular symmetry. Another is to measure the axis ratio on the sky and circularize elliptical bins via $R\rightarrow R\sqrt{1-\epsilon}$  (e.g., Koposov et al., 2015). Once we have a definition of radius, we must then chose a method for measuring the 2D half-light radius $R_{e}$. Two common choices are 1) to directly count half the light using concentric radial bins or 2) to perform an analytic fit to the projected distribution and to infer $R_{e}$ based on fit parameters. We have explored all four of these methods to determine $R_{e}$ for our galaxies: both circular and ellipsoidal bins with direct counts and Sérsic fits. The results for our stacked samples over all projections are shown in Figure 10 for each of the four methods. These results indicate that measurement of $R_{e}$ via an analytic fit is preferred over direct measurement of the half-light radius.444For the Wolf estimator, the fitting method does better in finding an $R_{e}$ that obeys \mbox{r_{3}}=4/3\mbox{R_{e}}. The $1\,\sigma$ scatter about the median falls from $0.25$ to $0.16$ ($0.29$ to $0.18$) for the Wolf (Walker) estimator as one moves from direct counts to fitted $R_{e}$.

Figure 10 also shows that the choice of circular vs. ellipsoidal bins is not very important when $R_{e}$ values are determined via fit. The circular approach does slightly better when $R_{e}$ is inferred via direct counts. This is somewhat surprising given the work of Sanders & Evans (2016), who showed analytically that “circularized” half-mass radii R_{e}$$\sqrt{1-\epsilon} are preferred for flattened stellar distributions. Particularly, because these authors worked under the assumption that stars and dark matter were stratified along the same self-similar concentric ellipsoids. As shown in Figure 2, our simulated dwarfs seem to obey this assumption on average, though individual systems are not perfectly self-similar, with axis ratios in stars to dark matter $(c/a)_{*}/(c/a)_{DM}$ that range from $\sim 0.8$ to 1.2.

4 Conclusions and Discussion

We have used a suite of twelve state-of-the-art cosmological simulations of dwarf galaxies to test the mass estimators of Walker et al. (2009) and Wolf et al. (2010), which are frequently applied to dispersion-supported dwarf galaxies in the literature. The simulated dwarfs used in this comparison were presented first in Fitts et al. (2016). They have stellar masses \mbox{M_{*}}=5\times 10^{5}\mbox{M_{\odot}} to 1.5\times 10^{7}\mbox{M_{\odot}} and are each resolved with more than one thousand star particles and up to tens of thousands of star particles. Our dwarfs broadly reproduce the observed properties of low-mass galaxies seen in the local universe, including sizes, velocity dispersions, flat line-of-sight velocity dispersion profiles, and negligible rotation support. The 3D stellar density profiles for these galaxies is well-represented by a single universal form (Eq.  7, Fig. 1), with a core-like distribution $\rho_{*}\propto 1/r^{0.1}$ at small $r$ a steep fall-off $\rho_{*}\propto 1/r^{4.2}$ at large $r$. The stellar distributions are aspherical, with an approximately constant short-to-long axis ratios $c/a\sim 0.5$ within twice the half-light radius (Fig. 2).

When applied to our simulations in mock projections, both the Walker (Eq. 3) and Wolf (Eq. 5) estimators successfully recover the total mass for dispersion supported dwarf galaxies with medians and 16th-84th percentiles given by $M_{\rm Walker}/M_{\rm true}=1.07^{+0.21}_{-0.15}$ and $M_{\rm Wolf}/M_{\rm true}=0.98^{+0.19}_{-0.12}$ when averaged over projection angle for all of our simulated dwarfs. In a dwarf-by-dwarf comparison, we found that the Walker estimator is biased slightly high ($\sim 10\%$) in the median (Fig. 7) and has a slightly higher dispersion about the median than the Wolf estimator. We found that it is important to measure the half-light radius $R_{e}$ using an analytic fit to the projected density profile rather than cumulative binning in order to achieve an accuracy better than $\sim 25\%$ (Fig. 10). The reason is that part of the uncertainty in the mass estimators comes from the mapping between the projected and 3D stellar distributions (Section 3.2), and the analytical fits to the projected profiles provide more robust determinations of $R_{e}$ (and $r_{3}$).

We find no correlation between the median accuracy of these mass estimators and the triaxial distribution of stars. However, for individual galaxies viewed over many projection angles, the dispersion about the median increases as galaxies become more flattened (Fig. 7). For one of the most flattened dwarfs ($c/a\approx 0.4$) the $1\,\sigma$ dispersion is $\pm 25\%$ ($\pm 30\%$) for the Wolf (Walker) estimator, while it decreases to $\pm 5\%$ ($\pm 7$) for the most spherical of our galaxies ($c/a\approx 0.7$). Unfortunately, we find that the observed axis ratio on the sky is not a good indicator of the underlying dispersion about the mean (see Appendix A), so it will be difficult to estimate the expected uncertainty owing to projection by observing the shape on the sky.

Recently, Campbell et al. (2016) made use of the APOSTLE simulations of the Local Group to test the same mass estimators explored here against both dispersion-supported galaxies and galaxies with significant rotational support. They also found that the Walker and Wolf estimators were accurate, especially for their dispersion-supported galaxies. The dispersion over their entire sample was 25 and 23 percent, respectively. We have found similar qualitative results here, but we find a scatter about the median that is significantly lower: 18 and 16 percent. The lower dispersion in our results could be due to differences in the physical processes included in our respective simulations, as well as our focus on dwarf galaxies with M_{*}$$\lesssim 10^{7} $M_{\odot}$. However, it may have to do with the methods used to measure $R_{e}$. Our higher resolution simulations have allowed us to construct and measure $R_{e}$ values directly by profile fits in each projection. Campbell et al. determined $R_{e}$ by measuring cumulative half-mass radii. When we use a similar method to the one in Campbell et al. (2016), we find larger dispersions: 29 and 25 percent, respectively. As a final point of comparison we mention that Campbell et al. (2016) provided a new mass estimator that did better in their simulations than either the Walker or Wolf estimator. When we tested their estimator, we found it to be biased low and with larger scatter than either of the two primary estimators we considered. We conclude that the mass estimates presented in Wa09 and Wo10 (1) are accurate and (2) do not have errors that are substantially underestimated.

Sanders & Evans (2016) explored mass estimators for dispersion-supported galaxies with $R_{e}$ replaced by the “circularized” half-mass radius R_{e}$$\sqrt{1-\epsilon}. Using an analytic approach, they showed that this replacement provided a better mass estimator than the un-circularized approach. To do this, they assumed that “stars and dark matter are stratified on the same self-similar concentric ellipsoids." As shown in Figure 2, our simulated dwarfs only roughly conform to this assumption, though in detail the shape of the dark matter and stars are not identical, with ratios $(c/a)_{\rm*}/(c/a)_{\rm DM}\simeq 0.8-1.2$. We tested the estimator using the “circularized half-mass radius” and we found that $r_{1/2}$  is not well represented by 4/3\;\mbox{R_{e}}\sqrt{1-\epsilon} and, consequently, the scatter around the median (above 20 percent) is larger than the estimators we tested in this work. Sanders & Evans (2016) also pointed out that the uncertainty in the estimate might depend on the ellipticity, which is qualitatively similar to our result that dispersion in mass estimator increases with decreasing axis ratio (see Fig. 7).

Our overall conclusion is that the mass estimators proposed by both Walker et al. (2009) and Wolf et al. (2010) do a better-than-expected job at reproducing the dynamical masses of simulated galaxies, which have stellar masses of 5\times 10^{5}\mbox{M_{\odot}} – 1.5\times 10^{7}\mbox{M_{\odot}}. We caution that errors may be larger for more massive galaxies, which often show more rotational support. It is remarkable that both the Walker and Wolf mass estimators are accurate to better than $\sim 20\%$ considering that our simulated systems break the key assumption of spherical symmetry. Errors at this level are not large enough to explain the inferred mass discrepancies that are at the origin of the Too-Big-to-Fail problem, and furthermore, the unbiased nature of the estimators makes it unlikely that misestimation of dynamical masses is the explanation of the problem. Going forward it will be useful to extend this comparison to the question of cusp/core measurement. Specifically, one of the most interesting uses of these estimators was proposed by Walker & Peñarrubia (2011), who used the existence of multiple populations of stars in individual dwarfs to measure masses at two distinct radii and thus constrain the log-slope of the underlying density profile. The potential to constrain the density profile slope in very low-mass dwarf galaxies could be a key diagnostic of the CDM paradigm (Bullock & Boylan-Kolchin 2017, in preparation), and thus testing this method against simulated galaxies with both cusps and cores presents an important next step. This will be the subject of a forthcoming work.

Acknowledgements

A.G-S. was supported by a UC-MEXUS Fellowship. JSB and OE were supported by NSF grant AST-1518291. JSB was also supported by HST theory programs AR-13921, AR-13888, and AR-14282.001 and program number HST-GO-13343. These HST programs were provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. MBK and AF acknowledge support from the National Science Foundation (grant AST-1517226). MBK was also partially supported by NASA through grant NNX17AG29G and HST theory grants (programs AR-12836, AR-13888, AR-13896, and AR-14282) awarded by the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS5-26555. DK was supported by NSF grant AST-1412153 and the Cottrell Scholar Award from the Research Corporation for Science Advancement. CAFG was supported by NSF through grants AST-1412836 and AST-1517491, and by NASA through grant NNX15AB22G.

This work used computational resources of the The University of Texas at Austin and the Texas Advanced Computing Center (TACC; http://www.tacc.utexas. edu), the NASA Advanced Supercomputing (NAS) Division and the NASA Center for Climate Simulation (NCCS) through allocation SMD-16-7760, and the Extreme Science and Engineering Discovery Environment (XSEDE, via allocation TG-AST140080), which is supported by National Science Foundation grant number OCI-1053575. This research made use of astropy (Astropy Collaboration et al., 2013).

Appendix A Projected Shape

We compute the axis ratios555Note that that the projected axis values ($a$,$b$) are different from the ones ($a$,$b$,$c$) used in the main text, since the last are calculated by analyzing the 3D distribution. ($b/a$) for each of the projected distributions in each simulated galaxy. As we found that both estimators gave “similar” (see Figure 9) results, in the following we will focus our analysis in the Wo10 estimator. We use an iteratively method (Dubinski & Carlberg, 1991) for all the stellar particles within 4/3\mbox{R_{e}} (Wo10). In Figure 11 we plot the ratio between the estimated mass by Wo10 estimator and the actual dynamical mass within 4/3$$R_{e} against the axis-ratio of the projected 2D stellar distribution for all the galaxies. We stacked the results for all the dwarf galaxies in our suite of simulations (purple shaded area). In order to have a more clear understanding of how the axis-ratios correlate with the mass estimator, we have evenly-binned the whole distribution according to their axis-ratio value, the mean in each bin is showed in red points and the error bars represent the $1\,\sigma$ scatter. It is clear that for 2D-ellipse axis-ratios larger than 0.7 the Wo10 estimator tend to overestimate the true dynamical mass within r=4/3\mbox{R_{e}}, and the larger the $b/a$ values, the more inaccurate become the estimator, reaching a 25 percent offset in the limit where $b/a\approx 1$. For $b/a$ values smaller than 0.65, the opposite behavior occurs, but in this case Wo10 result in a $\approx$ 10 percent constant underestimation of the true total mass.

We would be tempted to think that $b/a$ values closer to one, reflecting a projected spherical distribution would give better results, taking into account that mass estimators assume this kind of symmetry. Certainly that would be the case if the stellar particles in our simulated galaxies were distributed spherically. However, we found that estimators with $b/a$ values larger than 0.7 almost always overestimate the true dynamical mass in the dwarfs; what this is telling us is that the intrinsic 3D-distribution of the stellar particles is triaxial, and l.o.s. in which $b/a$ is close to one, represent projections that are far form the real stellar distribution. Indeed, the lowest 3D-ellipsoid semi-axes ratios, $c/a$, that gives an idea of how spherical is the 3D-distribution, have values around 0.7 for most of our runs.