THE RESOLVED STRUCTURE AND DYNAMICS OF AN ISOLATED DWARF GALAXY: A VLT AND KECK SPECTROSCOPIC SURVEY OF WLM

It is useful at this point to discuss the environment that WLM occupies within the Local Group, as the goal of this paper is to minimize the relative importance of external (tides and ram pressure) effects in order to learn more about the possible impact of internal feedback effects.

Derived and adopted properties for WLM are shown in Table 1 for clarity. WLM lies approximately 1 Mpc (Górski et al. 2011) from both the MW and M31, with its nearest neighbor being the small dSph Cetus (M_half  ∼  9 × 10⁷ M_☉; Walker et al. 2009), at 250 kpc away (Whiting et al. 1999), shown in Figure 1. Karachentsev (2005) calculate a tidal index for WLM of Θ = 0.3, where they define $\Theta \equiv {\rm max} [\log (M_{k}/D_{ik}^{3}){\rm ] + {\rm C}}$ to be the amount a galaxy is acted on by its largest tidal disturber.⁹ With the exception of Tucana (−0.2), Pegasus (−0.1), Aquarius (−0.1), and Leo A (0.2), WLM is one of the five least tidally disturbed and most isolated galaxies within a ∼1 Mpc sphere of the MW. Projection of WLM's heliocentric velocity (v_sys  ∼   − 130 km s⁻¹) toward the Local Group center of mass results in a velocity with respect to the barycenter of v_LG = −32 km s⁻¹, implying that it has just passed apocenter. With its current distance, that velocity, and an inferred Local Group mass of 5.6 × 10¹² M_☉ (based on M31-MW orbit timing arguments that assume an age of 13.7 Gyr for the universe; cf. Lynden-Bell 1981), the maximum apocenter that WLM could have is approximately 1.3 Mpc. Assuming a completely radial orbit, the implied orbital period for WLM to reach its pericenter with the MW is 11–17 Gyr. Thus, WLM, in addition to being currently quite isolated, has had at most one pericenter passage in its lifetime (which would have been at least 11 Gyr ago at z  ≃  2.5), meaning that its total evolution has been much less dominated by tidal and ram pressure effects from the MW than other Local Group galaxies.

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Table 1. WLM Properties

Quantity	Value	Description	Reference
(α, δ)	(00 01 58, −15 27 45)	J2000	Mateo (1998)
(l, b)	(75.85, −73.63)		Gallouet et al. (1975)
E(B − V)	0.035 (mag)	Line-of-sight reddening	McConnachie et al. (2005)
DMW	985 ± 33 kpc	Galactocentric distance	Average
e	0.45–0.6	Eccentricity	This work
q0	∼0.38–0.57	Intrinsic axial ratio	This work
z0	705 ± 28 pc	Vertical scale length	This work
rD	987 ± 30 pc	Radial scale length	This work
rh	1656 ± 49 pc	2D projected half-light radius	This work
hMA0	1209 ± 23 pc	Major-axis disk scale length	This work
PAkin	178° ± 1°	Kinematic position angle	This work
PAphot	179° ± 2°	Photometric position angle	This work
$V_{{\rm hel}}^{{\rm H\,\mathsc{i}}}$	−130 km s−1	Heliocentric velocity	Jackson et al. (2004)
$\left(\frac{V_{{\rm rot}}}{\sigma }\right)^{{\rm H\,\mathsc{i}}}$	∼6	H i rotational to pressure support	Kepley et al. (2007)
$\left(\frac{V_{{\rm rot}}}{\sigma }\right)^{\star }$	∼1	Stellar rotational to pressure support	This work
Mv	−14.1 (mag)	Integrated V-band absolute mag.	van den Bergh (1994)
$M_{{\rm H\,\mathsc{i}}}$	(6.3 ± 0.3) × 107 M☉	H i mass	Kepley et al. (2007)
M⋆	1.1 × 107 M☉	Stellar mass	Jackson et al. (2007)
$\frac{M_{g}}{M_{b}}$	0.86	Gas fraction ($M_{g} = M_{{\rm H\,\mathsc{i}}}/0.733$)	This work
$\frac{M_{b}}{M_{{\rm tot}}}$	0.05–0.10	Baryon fraction (Mb = M⋆ + Mg)	This work
$\frac{M_{{\rm tot}}}{L_{V}}$	17–30 M☉ L−1☉	Direct mass-to-light ratio (at rhalf to rlast)	This work
$\Upsilon = \frac{\eta 9 \sigma _{0}^{2}}{2 \pi G I_{0} r_{HB}}$	232 ± 31 M☉ L−1☉	Theoretical mass-to-light ratio	This work
[Fe/H]RGBphot	−1.45 ± 0.2 dex	Photometric RGB metallicity	Minniti & Zijlstra (1997)
[Fe/H]RGBspec	−1.28 ± 0.02 dex	Spectroscopic RGB metallicity	This work

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The paper will be presented as follows: in Section 2 we will present the data sets with a focus on the observations and reductions. Section 3 will discuss the spectral analysis techniques used to reduce and analyze the data in a homogeneous way to extract the chemodynamical properties for this study. Section 4 will present results on the structure and kinematics of the stellar and gaseous populations, Section 5 will present a discussion of the mass estimates, and Section 6 will focus on the evolution of the stellar dynamics.

Radial velocities were measured from the three strong calcium II triplet (CaT) lines (λ  ∼  8498, 8542, 8662 Å) that, in the case of the DEIMOS data, had a velocity correction applied from the sky OH lines in order to mitigate errors due to the instrument rotation. The average rms of the wavelength solution for the blue and red sides of the DEIMOS chips was 0.25 and 0.27 km s⁻¹, respectively. For the FORS2 spectra, the low S/N of the individual frames necessitated that we perform cross-correlation radial velocity calculations on the combined spectra, rather than on each individual image. As such, heliocentric velocity corrections were tailored to the individual exposures and applied prior to combining the spectra, due to the long temporal baseline (roughly four months) of the FORS2 observations. Once shifted and combined, the spectra were ready for radial velocity computation with the aid of template stars and a Fourier cross-correlation routine (fxcor). For the FORS2 data, a total of 23 template radial velocity stars observed with the same instrument setup were used with the cross-correlation routine. In the case of the DEIMOS data, the stellar spectra were cross correlated against a single synthetic template around the CaT region. Both computations provided in-line error estimates, with the velocity errors ranging from 1.0  ⩽  δV_hel  ⩽  8.0 km s⁻¹ with a mean of 〈δV_hel〉 = ±2.3 km s⁻¹ for the DEIMOS stars, and a range of 3.0  ⩽  δV_hel  ⩽  10.0 km s⁻¹ with a mean of 〈δV_hel〉 = ±5.0 km s⁻¹ for the FORS2 stars. For the FORS2 reductions systematic velocity errors due to a star's position in the slit were removed by centroiding the stars relative to the slit center. The fact that this procedure was done on combined spectra resulted in small absolute corrections, as the √n statistics meant that the individual slit errors were minimized in the combination and correction steps. The typical shift for the slit error on an individual exposure is approximately 6–9 km s⁻¹ for the FORS2 stars, and we note that this shift produces negligible uncertainties in the equivalent width error. The final average absolute corrections to the slit centering errors on the combined spectra were on the order of ⩽1.5 km s⁻¹ for the FORS2 stars.

Due to its distance off the plane of the MW (l, b = 75.85, −73.63; Gallouet et al. 1975), the contaminant fraction is expected to be very low in the direction of WLM. Nevertheless, the DEIMOS sample of WLM covers a much larger field of view than the FORS2 sample presented in Paper I, which means that the chance of foreground stars in the outer regions is increased. Additionally, as we are possibly sampling member stars at larger radii, they stand to be at larger velocities with respect to the central regions. Removing contaminants in a galaxy that is known to be rotating is not trivial, and in this case we culled the sample with an iterative maximum likelihood method. Rather than a global 3σ clipping, we adopted a spatially binned clipping routine that was much more robust, and by binning along the major axis we avoided artificially inflating the σ_v estimates. This procedure was based on the joint maximum likelihood method of Walker et al. (2006) (see also Gunn & Griffin 1979; Hargreaves et al. 1994), which we extended here to spatial bins of 25 stars each, so as to accurately throw out contaminants when the rotational profile enhances the spread of velocities at different positions in the galaxy. The algorithm simultaneously determines the dispersion (σ_v) and average velocity (〈u〉) in a given bin by maximizing the probability:

$$\begin{eqnarray} &&\vspace*{-2pt}\ln (p_{b}) = -\frac{1}{2}\sum _{i=1}^{N}\ln \big(\sigma _{i}^{2} + \sigma _{v}^{2}\big) {-} \frac{1}{2}\sum _{i=1}^{N}\frac{(v_{i}-\langle u \rangle)^{2}}{\big(\sigma _{i}^{2}+\sigma _{v}^{2}\big)} - \frac{N}{2}\ln (2\pi). \nonumber \\ \end{eqnarray} \tag{ 1 }$$

With this we culled the sample to 180 total member stars, with only one of the previous FORS2 stars removed. Figure 4 shows the distribution of probable member and contaminant stars in our catalog. In Figure 5, we plot the two-dimensional (2D) projected map of WLM with the H i gas contours and stellar sources determined to be members in our sample, both color coded by velocity.

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As in Paper I, [Fe/H] derivations were done through the empirically calibrated CaT method (Armandroff & Da Costa 1991) whereby the summed equivalent width of these lines correlates with [Fe/H] on a scale calibrated to galactic GCs (Cole et al. 2004). We here use the recently updated calibration given in Starkenburg et al. (2010). The details of the equivalent width measurements and placement onto the metallicity scale will be discussed in a forthcoming partner paper (Leaman et al. 2012), which will deal with the chemical analysis and interpretation of WLM. Of our 180 radial velocity members, we have derived [Fe/H] measurements for 126 of the stars that had high enough S/N (≳ 10 Å⁻¹). The uncertainties were estimated in the fashion described in Paper I, with the average internal error on metallicity being Δ_[Fe/H] = ±0.26 dex.

3.3.1. Repeat Observations

At the distance of WLM, deriving ages for individual stars is not possible with precision greater than ∼50% and in reality is simply providing a coarse correction to the assumption that metallicity and color are perfectly correlated. The reader is referred to Paper I for the discussion of the age derivations in detail, but relevant updates are briefly outlined here. The well-calibrated homogeneous photometry from McConnachie et al. (2005) is crucial in deriving accurate ages, but this method still suffers from difficulties (cf. Tolstoy 2003)—namely, the possibility of internal differential reddening and poorly constrained [α/Fe] values for the RGB stars as discussed in Paper I. Given the extended star formation history of WLM (Mateo 1998; Dolphin 2000) and the gas-rich nature of dIrr galaxies, we expect a significant range in red giant ages in the data set.

With our current spectroscopic sample, [Fe/H] values are now found ⩽  − 2.5 dex, which necessitated using the Yale-Yonsei evolutionary tracks (Demarque et al. 2004), which sample much lower metallicities. In Paper I, we provided age estimates based on both [α/Fe] =+0.3 and solar. Here, we use a compilation of high-dispersion spectroscopy of various stellar populations in WLM to constrain the mean [α/Fe] as a function of metallicity. Colucci et al. (2011) observed the lone GC of WLM and WLM-1 to have [Ca/Fe] = 0.25 and [Fe/H] = −1.71, which we adopt as the [α/Fe] values for all our targets with similar or lower metallicities. Of course GCs may undergo a separate formation and chemical evolution than field stars; however, in the MW, LMC, and Fornax the metal-poor clusters show similar [α/Fe] as the field stars (Pritzl et al. 2005; Hill 1997; Letarte et al. 2006), and Ca is not affected by GC mixing processes (e.g., Gratton et al. 2004). Stars more metal-rich than this are assigned an [α/Fe] based on a third-order polynomial, which is fit to the WLM-1 GC value, as well as the B and A supergiant studies of Venn et al. (2003) and Bresolin et al. (2006). The position of each star on the color–magnitude diagram was interpolated among a grid of Yale-Yonsei isochrones at appropriate values of [Fe/H] and [α/Fe] for the star.

The random errors on ages for individual stars are ±50%. Systematic errors are present in these estimates due to the variable α-enhancement, extinction, and asymptotic giant branch (AGB) contamination. We have estimated the magnitude of these systematic uncertainties in our ages using artificial star tests, which will be described in more detail in the forthcoming paper (Leaman et al. 2012). In the relevant figures in this work we have overlaid both the individual random error in a star's age and the combined systematic uncertainties.

To derive the structural parameters of WLM, we used the photometric V, I, and JHK catalogs, as well as the H i data of Kepley et al. (2007). The color–magnitude diagrams are shown in Figure 6, including the INT V and I photometry, as well as the infrared UKIRT/WFCAM photometry of WLM. We have isolated several population tracers, as shown by the colored areas in the figures. For the V and I photometry a strict RGB locus is shown by the red box. The tip of the RGB location was taken from the analysis of McConnachie et al. (2005), which used the same photometric catalog as our present work. This selection represents the least contaminated RGB population in WLM. Outside of this lie primarily excessively young or metal-poor RGB and luminous AGB stars. In the JHK diagram, the oxygen-rich M stars (light blue) and the young (∼1 Gyr) carbon-rich C stars are identified, as those AGB stars above the TRGB and to the appropriate side of the metallicity-dependent color cut found for WLM in the work of Tatton et al. (2010). The stellar density profiles for the full V, I (black), strict RGB (red), and C-star (blue) populations are shown in Figure 7, constructed by binning stars in geometrical radii that have been referred to an average ellipticity of e = 0.55 at a position angle of 179°. The inner regions of the two INT populations suffer from crowding, and so the subsequent fits of the surface density profiles exclude these regions. Four profile fits have been applied: Plummer (Plummer 1911), King (King 1962), de Vaucouleurs, and exponential profiles. For the rest of this paper, we adopt the best-fitting exponential surface brightness profiles, $\Sigma (R) = \Sigma _{0}e^{-R/R_{d}}$, and convert the exponential scale length, R_d = 345 (for the full V and I samples), into the 2D projected half-light radius, r_h = 578, and where necessary the 3D deprojected half-light radius, r_h3 = 768, using the relations of Wolf et al. (2010). The background level in each population was simultaneously fit as a constant and was found to be consistent with selecting out by eye isolated regions of our field of view to estimate the background level.

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The 2D projected spatial distributions of these three populations and the H i surface density are shown in Figure 8. The mean and standard deviation of the background level were determined from regions at the outer limits of our field of view. Contour levels start at 2σ above the background, and each subsequent contour level is n = 1.2 times higher than the last. The H i radial extent is not significantly larger than the stellar component as reported in Kepley et al. (2007) and the recent single-dish observations of WLM by Hunter et al. (2011). The latter work also noted that the density profile of the H i shows a smooth drop-off to the detection limit, with no sharp truncation. The strict RGB population shows a slightly more spherical distribution than the H i, but otherwise the general shape of the evolved stellar populations is consistent with the H i population. As expected, the young C-star population occupies the central regions of the galaxy, where the most recent SF has occurred (Dolphin 2000). These centrally concentrated younger populations are similar to what has been found in other Local Group dwarf galaxies (e.g., Battaglia et al. 2006; Hidalgo et al. 2009; de Boer et al. 2011). However, we note that with characteristic ages of ∼1 Gyr, these stars would still have experienced ∼3 rotation periods. Ellipticity and photometric position angle (${\rm PA_{\rm phot}}$) are calculated as a function of radius by fitting ellipses using the task MPFITELLIPSE (Markwardt 2009) on the contours at various radii for each population.

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The resulting position angles and ellipticities are shown as a function of radius in Figure 9. There is excellent agreement between the H i and stellar structural properties, with only the evolved RGB population showing a slightly rounder ellipticity. This is consistent with literature values of past photometric studies of Ables & Ables (1977) and Minniti & Zijlstra (1997). The close agreement in position angle and ellipticity of the stellar populations in WLM is in contrast to some lower angular momentum dSphs such as Phoenix and Fornax (Martínez-Delgado et al. 1999; Battaglia et al. 2006). WLM clearly is flattened more than the nearby dSphs in the Local Group (Mateo 1998)—which may be due to the strong rotation, as we shall see in the next section.

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With nearly equal rotation and dispersion in the stellar component, care must be taken in interpreting WLM's velocity data. In rotating systems, classical mass estimates rely on the H i gas, which has negligible random motion, making for relatively simple analyses equating the enclosed mass to some circular velocity. The underlying mass profile can also be ascertained through velocity dispersion profile based methods—such as spherical Jeans modeling (e.g., Battaglia et al. 2008). In each of these cases, only one type of dominant kinematics is present in the tracer population—rotation or dispersion. The challenge in our case is to convert WLM's composite dynamical configuration into a singular rotating or dispersion-dominated data set. Without these corrections a number of systematic biases arise when interpreting rotational and pressure supported velocity distributions—such as encountered when analyzing an object in the configuration of WLM (cf. Łokas et al. 2010). Detailed dynamical modeling will be presented in a forthcoming paper (R. Leaman & G. Battaglia 2012, in preparation). We now proceed with analysis of the velocity dispersion and rotation profiles, with an emphasis on understanding the biases that arise when studying the kinematics of systems that simultaneously exhibit strong rotation and dispersion.

4.2.1. Stellar Velocity Field

As discussed in Paper I, the stellar velocities are markedly different in WLM from most of the dSph galaxies. The stars in WLM of all ages/metallicities appear to be rotating, in contrast to the purely random stellar motions in Local Group dSphs. We begin by looking at the velocity structure of the stellar and gaseous populations of WLM. As shown in Figure 10, WLM's stellar velocities, while showing rotation greater than seen in the dSphs of the Local Group, rotate with half the velocity of the H i gas and lag behind the H i by ∼15 km s⁻¹. Clearly the stellar rotation is decoupled from that of the H i, with the suppressed rotation velocity of the stars due to the effect of asymmetric drift, as we will demonstrate later. In Figure 11, we show the projection of the rotation velocity of the stars as a function of geometric radii (again referred to an ellipticity of e = 0.55). The binned data points are fit with an isothermal sphere rotation curve of the form V(r)² = 4πa₀a²₁(1 − a₁/r)(atan(r/a₁)) (Kepley et al. 2007), which slowly rises to a value of ∼17 km s⁻¹ at the last measured point.

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As later we wish to derive a line-of-sight velocity dispersion profile, it is important that we accurately characterize the velocity field of WLM. We therefore explored different models to characterize the rotation of the stars in WLM. A few methods that were tested were as follows: fitting the stellar velocity field simply along a strip of the major axis, describing the rotation field by iterating through fits at differing position angles, or treating the stars with the classical "spider diagram" velocity field:

$$\begin{equation} v(x,y) = v_{{\rm sys}} + v_{{\rm rot}}(R)\cos (\theta)\sin (i) + v_{{\rm rad}}(R)\sin (\theta)\sin (i), \end{equation} \tag{ 2 }$$

where θ is the azimuthal angle and i is the inclination of the galaxy.

The most consistent determination, as judged by the smallest deviation at all radii in the rotation-subtracted velocities, was a simple linear fit along the velocity pattern of the major axis. The rotation-subtracted velocities are shown in the second panel of Figure 11, with the mean values of the stars staying close to the systemic velocity at all radii. The velocity of the stars off the major axis shows little deviation from the model, suggesting that the stars are rotating cylindrically; however, due to the uncertainties in the inclination and disk thickness of WLM, we cannot examine this in detail.

4.2.2. Velocity Dispersion Profile

Information on the orientation and potential of WLM comes from comparing the position angle of the kinematic (${\rm PA_{\rm kin}}$) and photometric (${\rm PA_{\rm phot}}$) data. The kinematic position angle defines the major axis of rotation of a galaxy and, in cases where recent mergers have occurred or the observer is viewing a triaxial system, may not always lie along the same axis as the photometric position angle. To compute the ${\rm PA_{\rm kin}}$, we proceed as in, e.g., Walker et al. (2006). For every star, or interpolated H i velocity point in the tangent (ξ, η) plane, an angle of bisection is calculated based on the data point's position off the above-determined photometric minor axis. The galaxy is split along this bisection line, and velocity difference is computed by calculating the average velocities above and below the line and subtracting them. The bisection line is then rotated by 1° for the H i or to the next star and the procedure is repeated, building up the curves shown in Figure 12. A hyperbolic tangent function was fit to the data and the position angle determined based on the bisection angle where the velocity difference is minimized. With a position angle of zero northward in the tangent plane on the sky, the kinematic position angle is defined as 180° + or − the angle of bisection where the minimum velocity difference occurs. The uncertainty on the angle is determined by calculating the difference from the bisection line that would be produced if the velocity difference were changed by the average uncertainty in a star's velocity. We find consistent values for the kinematic position angle of the stellar and gaseous components of WLM, with the latter values in excellent agreement with the H i studies of Jackson et al. (2004) and Kepley et al. (2007), who used other methodology.

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With the photometric and kinematic position angles calculated, it is instructive to look at their relative alignment, typically designated as the parameter Ψ ≡ |PA_kin − PA_phot|. The quantity of Ψ is sometimes used as a signature of mergers, where the structural and dynamical misalignment would be high, or to give insight into non-circular gaseous motions due to infall or outflow (Fraternali et al. 2004). For galaxies in the ATLAS 3D study ∼90% of galaxies were found to have alignments ⩽15° (Krajnovic et al. 2011). For the stellar component of WLM we derive Ψ = 1°  ±  3°, and Ψ = 2°  ±  2° for the H i. The fact that the alignment of the kinematic and photometric position angles is close also constrains the viewing angle and shape. As discussed recently by Łokas et al. (2010), when rotation is significant in a triaxial system viewed perpendicular to the longest dimension, rotation other than the major axis is usually evident, as the likelihood of the rotation being aligned with either axis is small in a potential that permits more irregular orbits. In contrast, an axisymmetric system will show no rotation along the minor or intermediary position angles; thus, the kinematic and photometric position angles will show strong alignment like we observe in WLM. Additional constraints against WLM being triaxial come from its position on the V/σ versus e anisotropy diagram. As we shall see in the next section, WLM is flattened mostly by rotation rather than anisotropy, consistent with the interpretation that we are viewing a highly inclined oblate spheroid.

From Figure 9, the ellipticity and position angle of the gas and stars allow us to characterize the projected distribution of those populations. To untangle the intrinsic shape of WLM is more complicated. If we are observing an oblate spheroid that is rotationally flattened, it will have an intrinsic thickness, q₀ (for this orientation, q = c/a, with c being the short axis and a being the long axis)—which is difficult to disentangle without a constraint on the viewing inclination, i. The relation between the intrinsic and projected axial ratio is shown in Figure 13 for the range of inclinations determined in the tilted ring analysis of the H i data in Kepley et al. (2007). We note that because their inclinations were derived assuming a thin disk in the tilted ring analysis, they therefore provide lower limits on the possible inclination. Thus, given the high inclination for WLM, the range of measured projected axial ratios (b/a) for the stellar and gaseous components (horizontal lines) suggests that the intrinsic thickness ranges between 0.38  ⩽  q₀  ⩽  0.57. With the data presented here, we have evidence that WLM's vertical structure is quite extended in both its stellar and gaseous components. This is confirmed using the techniques described in van der Marel et al. (2002), whereby WLM can be approximated as a flattened spheroid with axial ratio q(R) = (2 + [2R_D/R(V_rot(R)/σ(R))²])^−1/2, assuming isotropic velocities for the stars (which are assumed to lie in a larger isothermal halo). Using the derived exponential scale length from Section 4.1, the intrinsic axial ratio (q) for WLM calculated using this method again lies between 0.38 and 0.56 at all radii—in good agreement with Figure 13.

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This value is consistent with the recent study of Roychowdhury et al. (2010), who showed that the average intrinsic gas disk thickness of dIrrs is a much higher 〈q₀〉 = 0.6, instead of the 0.2 that is canonically assumed for stellar and gaseous disks in higher mass galaxies. Our values for the stellar thickness are also in line with studies of distant dwarf galaxies that seem to show thicker stellar structure (Sánchez-Janssen et al. 2010).

We might ask, how does this thick configuration compare to other galaxies where direct measurements of the vertical height are possible? While we are unable to determine whether the extended vertical structure in WLM is due to just a single component flattened spheroid or multiple disks of varying thickness, we can compare the vertical extent to studies of edge-on galaxies that exhibit thick disks. In Figure 14, we plot the radial and vertical scale heights for WLM along with a sample of edge-on spiral galaxies from Yoachim & Dalcanton (2006) and the two large spirals in the Local Group (cf. Collins et al. 2011). The values for WLM were derived by fitting the major-axis stellar surface density with an exponential profile and the minor axis with a profile of the form Σ(z)∝sech²(z/z₀), for which we derive a vertical scale height of z₀ = 705 ± 28 pc. Assuming an inclination of i = 79, we show the corrected vertical scale height marked on the figure as the open circle. The best-fit lines are taken from Yoachim & Dalcanton (2006) and indicate that WLM has a similar relative vertical scale height as expected for its radial scale length. Direct confirmation of a distinct thick disk in WLM would only be possible after identifying a velocity lag or age difference in a statistical sense, which is beyond the scope of our spectroscopic sample. However, the vertical structure of WLM is clearly extended to an amount consistent with other galaxies that show increased vertical scale heights.

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We can use the inherent rotation of the stars to compute an estimate of the circular velocity of WLM, provided that we remove the component of random motions that are suppressing the rotation curve. As noted earlier, the stellar rotation curve differs in magnitude to the H i due to the effect of asymmetric drift. To directly compare the rotation curves of the stars and gas and allow for independent circular velocity estimates, we must correct for this effect. To analyze the stellar kinematics in a purely rotational sense, corrections due to asymmetric drift were applied:

$$\begin{eqnarray} v_{c}^{2} &=& v_{\phi }^{2} + \sigma ^{2}(r)\left[\frac{\partial \rho }{\partial R} + \frac{\partial \ln \sigma _{R}^{2}}{\partial \ln R} \right.\nonumber \\ &&+ \left. \left(1-\frac{\sigma _{\phi }^{2}}{\sigma _{R}^{2}}\right) + R\sigma _{Rz}^{2}\frac{\partial \big(\ln \rho \sigma _{Rz}^{2}\big)}{\partial z}\right]. \end{eqnarray} \tag{ 3 }$$

Here, v_ϕ is the inclination-corrected observed rotation velocity, ρ is the stellar density, σ is the velocity dispersion in a given component, and v_c is the drift-corrected rotation velocity. We follow the outline and assumptions of Hinz et al. (2001), namely, that the last term can be neglected, σ²_ϕ/σ_R² = 0.5, the density profile is of the form ρ∝exp (− R/R_d), and the slope of the radial component of the velocity dispersion is flat in the outer regions of the disk. The equation then reduces to v²_c = v_ϕ² + σ²_ϕ(2R/R_d − 1). We note that the assumed anisotropy is chosen primarily because it reproduces flat rotation curves in large galaxies; however, we have little in the way of constraints on it at this time. We use a nominal inclination of i = 79° for the inclination-corrected values of dispersion and rotation, allowing us to compute the corrected stellar circular velocity, v_c. As noted earlier, the tilted ring analysis of Kepley et al. (2007) assumed a thin disk; therefore, their inclination, which we adopt here, should provide a conservative lower limit. The results of this correction are seen in Figure 10. The final inclination and asymmetric drift-corrected curve shown by the blue binned points matches well with the H i gas—giving us confidence that our dispersion measurements are correct for the stars.

With this corrected stellar rotation curve we have estimates of the circular velocity at various projected radii in WLM, with which we can compute the associated rotationally derived mass as M(< r) = V²_cr/G. In Figure 15, we show these dynamical mass estimates as a function of radius as the black squares. The black solid and dashed lines show the enclosed mass curves based on the H i rotation analysis from the study of Kepley et al. (2007), assuming an isothermal sphere model. Agreement between the H i and stellar derived mass is roughly consistent when considering both the receding and approaching sides of the H i data. Differences most likely result from the uncertainty in assuming a velocity anisotropy for the stellar circular velocity reconstructions and possibly any non-circular motions not taken into account in the H i analysis. A weighted best-fit isothermal sphere model based on the six stellar bins is shown in magenta and a Navarro–Frenk–White (NFW) mass model in green. The rotationally derived mass at the half-light radii (1656 ± 49 pc) from the stellar data is M_half = (4.3  ±  0.3) × 10⁸ M_☉ and M_last  ≃  8.5 × 10⁸ M_☉ at the last measured stellar bin (2865 pc). Assuming the best-fitting enclosed mass profile from the H i study of Kepley et al. (2007), one finds masses of M_half ≃ 3 × 10⁸ M_☉ and M_last ≃ 7 × 10⁸ M_☉ from the gas rotation at those same radii—in close agreement with the values from the stellar rotation. The NFW and isothermal sphere models that are fit to the stellar data imply halo masses (taken to be the mass interior to a radius where the enclosed density equals 200 times the critical density) of 8.9 × 10⁹ M_☉ and 2.6  ×  10¹⁰ M_☉, respectively. We show these values plotted versus the known stellar mass of WLM (1.1 × 10⁷ M_☉; Jackson et al. 2007) in the inset of Figure 15, along with the M_*–M_halo relation of Guo et al. (2010). The isothermal sphere model and NFW model are offset from each other, but both are roughly consistent with the relation from Guo et al. (2010). However, the sensitivity of the derived halo mass to the rotation signature at small radii in WLM prevents us from discussing the implications of this in any further detail in this work. With the above values and assuming a V-band luminosity of 5.2 × 10⁷ L_☉ (Mateo 1998), we find a mass-to-light ratio of M/L = 8 and 29 M_☉ L⁻¹_☉ at the half-light radius and 4.5 kpc (roughly the extent of the stellar and gaseous body of WLM), respectively.

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An alternative estimate of the dynamical mass can be taken from simple analytic expressions relating the mass to the line-of-sight velocity dispersion at a characteristic radius, as in the recent works of Walker et al. (2009), Wolf et al. (2010), and Amorisco & Evans (2012). Independently, the studies identified a characteristic radius where velocity anisotropy effects are minimized. With these formulae the enclosed mass is computed in accordance with simple virial estimates. The Wolf et al. (2010) relation requires a luminosity-weighted velocity dispersion, which may introduce bias when applied to composite populations that have spatially different luminosity distributions (cf. Amorisco & Evans 2012). Therefore, we use the relation of Walker et al. (2009), who found M(r_h) = μr_hσ², where μ ≡ 580 M_☉ pc⁻¹ km⁻² s². Being easily computed, this is used frequently to derive masses in the dSphs and ultra-faint dwarfs of the Local Group; however, there are several points of caution to be noted in this formula's application to WLM. Any of these dispersion-based mass estimators assume a system that is spherically symmetric, without rotation, and with a flat radial velocity dispersion profile—all conditions that are violated to some degree in WLM. Nevertheless, it is instructive for the time being to compare the dispersion-based mass to other estimators. We leave discussion of the biases that violation of these assumptions creates to our forthcoming paper on the dynamical modeling of WLM. With these caveats, we proceed and compute the average velocity dispersion within r_h using the rotation-subtracted velocity dispersion profile shown in Figure 11. With this the half mass from the Walker et al. (2009) relation is M_half = (2.1  ±  0.3) × 10⁸ M_☉. In Table 3, we summarize the mass estimates for WLM for ease of comparison.

Table 3. Mass Estimates

Description	Value	Notes
Mass within rh	(2.1 ± 0.3) × 108 M☉	Equation (1) of Walker et al. (2009)
Mass within rh from V⋆c	(4.3 ± 0.3) × 108 M☉
Mass within r⋆last from V⋆c	(8.5 ± 0.3) × 108 M☉	Last measured stellar bin = 2.9 kpc
Mass within $r_{{\rm last}}^{{\rm H\,\mathsc{i}}}$ from approaching side H i SIS Model	(9.4 ± 0.5) × 108 M☉	Last measured H i point = 4.5 kpc
Mass within $r_{{\rm last}}^{{\rm H\,\mathsc{i}}}$ from receding side H i SIS Model	(2.2 ± 0.1) × 109 M☉	Last measured H i point = 4.5 kpc
Mass within $r_{{\rm last}}^{{\rm H\,\mathsc{i}}}$ from stellar SIS Model	(1.2 ± 0.1) × 109 M☉	Last measured H i point = 4.5 kpc
Mass within $r_{{\rm last}}^{{\rm H\,\mathsc{i}}}$ from stellar NFW Model	(1.6 ± 0.1) × 109 M☉	Last measured H i point = 4.5 kpc
Mass within r200 from stellar SIS model	(2.6 ± 0.2) × 1010 M☉	r200(SIS) = 60.5 kpc
Mass within r200 from stellar NFW model	(8.9 ± 0.8) × 109 M☉	r200(NFW) = 42.0 kpc

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With the enclosed mass at the half-light radius, it is now instructive to compare WLM to the half-mass estimates of Local Group dSphs. In Figure 16, we show that WLM lies slightly on the more massive side of the sequence of M_half versus r_h shown in Walker et al. (2009). The other dSphs of the Local Group that lie off the relation in this direction are the isolated Cetus and Tucana dSphs. Tidal evolution tracks of Peñarrubia et al. (2008) are overlaid, showing the expected evolution in mass and radius for a galaxy that has lost various percentages of its dark and luminous mass due to tidal stripping. This may provide an order-of-magnitude estimate for the impact of stripping on WLM, but the models assume a dwarf galaxy with a King profile and minimal rotation, so only a qualitative comparison is possible. While WLM's location on this diagram is not evidence itself for an isolated history, we can say that if it were to go through significant tidal evolution it would likely transition down into the sequence of the diagram occupied by the majority of the dSphs.

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A more appropriate comparison may be to consider WLM with respect to the dwarf elliptical (dE) satellites of M31. Studies of NGC 147, NGC 185, and NGC 205 have shown them to be of similar total mass and metallicity as WLM (Bender et al. 1991; De Rijcke et al. 2006; Geha et al. 2010) although with higher luminosities and gas fractions more than two orders of magnitude smaller (Mateo 1998). Analyses of the velocity fields of NGC 147 and NGC 185 at large radii by Geha et al. (2010) suggest that these objects have similar kinematics as WLM, with all three exhibiting well-defined rotation. However, given the larger axial ratios of NGC 147 and NGC 185, their (V/σ)* values are 0.95 and 0.91, respectively (Geha et al. 2010), compared to (V/σ)*  ≃  1.1 for WLM. Therefore, while the dEs are more similar dynamically to WLM than the dSphs, the dEs likely have stronger anisotropy among their stars compared to WLM based on this ratio. Additionally, the mass-to-light ratio (M/L) of WLM is larger than the dEs despite having similar total masses at comparable radii. Specifically, while the M/L ratios for the three dEs are all consistent with M/L = 4 in solar units (De Rijcke et al. 2006), WLM's ranges from 16  ⩽  M/L  ⩽  30 over the half-light radius to the last measured point. From our circular velocity curve in Figure 15 we estimate WLM's dynamical mass at two effective radii to be M(2R_e)  ≃  1 × 10⁹ M_☉. This is very near to the mass of NGC 147 and NGC 185 at similar radius (5.6 × 10⁸ and 7.2 × 10⁸; Geha et al. 2010), but slightly lower than that of NGC 205 (2 × 10⁹; Geha et al. 2006). While WLM dynamically resembles NGC 147 in particular, it would seem that the relative dark matter content of WLM is somewhat higher than all of the Local Group dEs. However, it is interesting to note that a conversion of WLM's remaining gas mass to stellar mass would result in a M/L  ≃  4, consistent with the dEs.

We note that the derived mass for WLM is above the possible threshold for gas retention found in the dwarf galaxy simulations of Sawala et al. (2011). That paper, along with recent simulation and observation work in the last two years (Governato et al. 2010; Sánchez-Janssen et al. 2010), indicates that the global structural and dynamical properties of dwarf galaxies with gas are more strongly impacted by the initial halo mass, and their subsequent evolution driven by internal feedback or secular evolution, rather than tidal effects from a larger host galaxy. However, environment likely still plays some role given that WLM and the dEs (which both appear to have comparable total masses within two effective radii) have drastically different gas fractions and effective radii—presumably due to the fact that the dEs reside within the virial radius (⩽200 kpc) of M31, where ram pressure and tidal stripping is expected to be efficient. Differences such as these make it even more important to characterize the evolution of isolated galaxies, and the analysis in this work suggests that WLM represents one possible outcome of dwarf galaxy evolution in the absence of strong environmental forces.

In Figure 17, we show the velocity dispersion in stellar populations of different ages. There is a trend of increasing velocity dispersion with increasing stellar age, and at all ages the stellar dispersion is above that of the H i. It is useful now to discuss how exactly the stellar dispersions may have increased ∼10 km s⁻¹ higher than the neutral gas velocity dispersion.

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One interpretation of Figure 17 is that the stars of all ages are heated at some rate and originated with an initial, constant dispersion—which in our case agrees with the measured H i velocity dispersion from Kepley et al. (2007; dotted line).¹⁰ Alternatively, stars of different ages could have formed with different dispersions depending on the ISM kinematics at the time of their birth. Given that it is possible that both interpretations are working in combination, we now briefly discuss a few processes that might be expected to contribute to heating of the stellar populations in WLM.

6.1.1. Tidal Heating

6.1.2. ISM Pressure Floor

6.1.3. Giant Molecular Cloud and Dark Matter Subhalo Heating

The merger history of an object like WLM may also be important in dictating the evolution of its velocity dispersion and vertical thickness with time. Abadi et al. (2003) and Robertson et al. (2006) looked at the relative contribution of mergers of varying masses and gas fractions. The latter study noted the importance of gas-rich merging as a way to maintain rotation support in galaxies, while the work of Abadi et al. (2003) showed that a hot, thick stellar disk often forms early on, around the time of the last major merger. However, both studies could not include the high-resolution stellar feedback effects shown to be important in Governato et al. (2010) and were simulating galaxies that were several orders of magnitude more massive than WLM.

The case of heating by GMCs usually assumes that the clouds are occupying a relatively thin region of a galaxy and the encounter velocity is dictated by the peculiar velocity of the stars in a rotating frame. Therefore, ∂σ²/∂t∝1/σ (cf. Fuchs et al. 2001), and the heating is not particularly efficient as the rate drops as the random velocity component increases. This is due to the reduced encounter probability, as the stars spend more time outside the GMC "layer" as the dispersion increases. The blue solid line in Figure 17 shows a canonical relation of the form

$$\begin{equation} \sigma (t) \propto (t - t_{0})^{\alpha }. \end{equation} \tag{ 4 }$$

The maximum heating that is expected analytically and from simulations of GMC collisions alone is α_GMC  ⩽  0.25 (Lacey 1984). While the mass spectrum of GMCs in WLM is not precisely known, there is observational evidence to suggest that the properties are not strongly dependent on host galaxy (Bolatto et al. 2008). This maximum GMC contribution (indicated by the blue line) produces a curve that comes quite close to reproducing the observed stellar dispersion values.

An additional source of heating may be present due to interactions between dark matter subhalos and the stellar component of WLM (cf. Moore et al. 1999). These perturbations would have higher encounter velocities, and expected heating would be more or less isotropic (Fuchs et al. 2001). A simple calculation for the amount of heating that a host galaxy of halo mass M_h undergoes from minor mergers with dark matter subhalos of mass M_s can be derived assuming that the rate of heating for a subhalo to deposit energy into the disk component of the galaxy is

$$\begin{equation} \frac{\partial {E_{d}}}{\partial {t}} = \frac{\Delta {E_{d}}}{t_{{\rm df}}}, \end{equation} \tag{ 5 }$$

where t_df is the dynamical friction timescale:

$$\begin{equation} t_{{\rm df}} = \frac{1.17}{\ln (M_{h}/M_{s})} \left(\frac{M_{h}}{M_{s}}\right)\frac{r_{h}}{V_{c}}. \end{equation} \tag{ 6 }$$

For a given interaction, the amount of energy deposited into the halo component relative to the disk component of WLM scales as η∝2.3M_h/M_d  ≃  27 for WLM (Toth & Ostriker 1992). Thus, the actual energy input into the disk will not be a particularly large fraction of the satellite kinetic energy. Using the expected unevolved subhalo mass function, which is shown to be robust to halo size and over a range of redshifts (Giocoli et al. 2008),

$$\begin{equation} \frac{\partial {n}}{\partial \ln (M_{s}/M_{h})} = A\left (\frac{M_{s}}{f M_{h}}\right)^{-p} \exp \left [-\left (\frac{M_{s}}{f M_{h}}\right)^{q}\right ], \end{equation} \tag{ 7 }$$

Mo et al. (2010) show that the total heating rate by dark matter subhalos is

$$\begin{equation} \frac{\partial {E_{d}}}{\partial {t}} \propto \int _{m_{{\rm min}}}^{m_{{\rm max}}} n(M_{s}|M_{h})M_{s}^{2}dM_{s}, \end{equation} \tag{ 8 }$$

where the expected number of subhalos for a given primary halo can be further simplified to n(M_s|M_h)∝M^−1.8_s. In order to equate this heating rate by the dark matter halos to an observable, we make the assumption that the kinetic energy deposited in the disk by a single event is

$$\begin{equation} \Delta {E_{d}} = (1/2)M_{d}\sigma _{z}^{2}(R). \end{equation} \tag{ 9 }$$

The total heating rate due to dark matter subhalos from masses m_min to m_max can then be rewritten as

$$\begin{equation} \frac{\partial \sigma ^{2}}{\partial {t}} \propto \frac{2\sqrt{(}3)M_{s}^{1.2}}{1.2M_{d}}\left|^{M_{s}=m_{{\rm max}}}_{M_{s}=m_{{\rm min}}}\right.. \end{equation} \tag{ 10 }$$

Therefore, the expected total heating rate by dark matter substructure can be roughly estimated by fitting our velocity-dispersion–age data with a function of two variables: the maximum subhalo mass (m_max) and the WLM disk mass (M_d). We assume equipartition of energy in all velocity directions, which has been shown to be appropriate for satellites with circular orbits of various inclinations, and an isotropic Maxwellian distribution of orbits (Toth & Ostriker 1992). Similarly, assuming an average dispersion within 1r_e for an age bin is justified, as the scale affected by the energy injection of satellites has been shown to be at least that large (Sellwood et al. 1998). Therefore, it is a reasonable assumption for this simple model that disk stars at nearly all radii are heated. In red we overlay two examples for a heating model based solely on dark matter (DM) interactions. Both curves have a maximum perturber corresponding to m_max = 5 × 10⁸ M_☉, or ∼5% of the total halo mass of WLM. The dashed line uses a disk mass of 1 × 10⁸ M_☉, close to the present-day value, while the solid line assumes a disk mass of 8 × 10⁵, which is approximately the disk mass of WLM 11 Gyr ago using the scaling relations of Dutton & van den Bosch (2009). Given the assumptions discussed in this paragraph, both curves are strong upper limits on the amount of heating substructure would predict. Despite this, the fact that the energy deposited in the disk scales as the mass ratio of WLM's disk to halo prevents strong contribution from this effect. Even computing substructure models with an unphysical value of η = 1 does not match the observed dispersion evolution. This is in good agreement with semianalytic and n-body simulations of disk heating such as Benson et al. (2004), which found similar inefficiencies for substructure heating.

In that study the authors showed that for lower mass primary galaxies, the contribution of heating from GMC encounters will dominate relative to substructure interactions. As an additional independent check, we compute the heating/thickening predicted by Equations (11) and (12) of Benson et al. (2004), expressed as a ratio of vertical to radial scale lengths, h = z_x/r_d. With modest assumptions for the parameters of those equations, we find the same trend—GMC heating can produce a system with 0.32 ⩽ h ⩽ 0.57, close to the observed inclination-corrected axial ratio of WLM (see Section 4.4). Using the same range of disk and perturber masses in our Equation (10), we again find that the substructure mechanism is not enough to produce the observed vertical scale height—with only 0.02 ⩽ h ⩽ 0.09 found at the end of 12 Gyr in those models. However, recent simulations have shown that disk heating by dark subhalos may be more effective if higher mass ratio accretion events on plunging radial orbits are considered; however, the probability of such events occurring is still thought to be low (T. Starkenburg 2012, private communication). While there are obvious degeneracies that prevent us from accurately knowing the exact portion of the heating that came from GMC or DM substructure encounters (especially given the uncertain merger rate as a function of time), combined they are clearly of the right order of magnitude to explain the increased velocity dispersion and thickening in WLM. Most importantly, these processes can occur in a galaxy in isolation and do not require influence from the MW or M31 to modify a galaxy.

Additional secular processes may work to heat the disk in the vertical direction, including long-scale length global disk instabilities, as well as transient spiral or overdensity features (cf. Sotnikova & Rodionov 2003). However, as those features may only last 1 Gyr, we cannot comment on their impact directly. Although we have not ruled out a scenario in which gas dispersion decreases with time (Section 6.1.2), the dispersion evolution presented in Figure 17, when coupled with WLM's isolation, strongly suggests that the gaseous and stellar populations became heated and thickened through continual internal effects such as GMC and substructure interactions, rather than tidal transformations triggered by the MW.

How do these observations fit in with the larger picture of dwarf galaxy formation and evolution? The structure and dynamical evolution of this isolated dwarf galaxy can be examined in perhaps one scenario that is consistent with the findings presented in this paper.

From Sawala et al. (2011) we see that the ability of a dwarf galaxy to retain gas until the present day may be due in most part to the total halo mass of the dwarf—with a threshold of approximately 1 × 10⁹ M_☉ below which the galaxies are less likely to retain gas. Gas-rich galaxies in that simulation also tended to have a late infall (z  ⩽  0.2) to the group and pericenter distances that were greater than 300 kpc. Because of this, dIrrs could have gas over a larger percentage of their history and experience ongoing SF and SNe—internal feedback that contributes to heating the dwarf, creating a slightly puffier system. Additionally, simulations by Schroyen et al. (2011) show that dwarf galaxies with high angular momentum tend to have SF continuously over the full radial extent of the galaxy—compared to non-rotating systems in which the gas can efficiently fall to the center and undergo extreme centralized SF and blowout events. As discussed in Robertson & Kravtsov (2008), if SF is regulated by the ability of the gas to settle into the molecular phase, a thick, hot ISM would naturally result in low-efficiency SF, in line with the high gas fraction seen for WLM today. Therefore, higher mass, rotating dwarf galaxies would more likely be the type where we would expect to see signatures of extended internal feedback effects—namely, a thick stellar configuration. These systems, if isolated, will be solely subject to internal feedback effects such as SF, GMC interactions, and secular global disk instabilities (cf. Sotnikova & Rodionov 2003), which only produce mild transformations in their structure and dynamics.

Thus, we could have a scenario where the initial mass dictates the probability of gas retention (i.e., Sawala et al. 2011), which, in combination with angular momentum (Schroyen et al. 2011; Koleva et al. 2009), sets the level of internal-feedback-driven heating. In this picture, WLM could be an example of an isolated dIrr at high enough mass and isolation to allow for ongoing SF-driven feedback that maintains a dynamically hot stellar structure, low SF rate, and high gas content.