---
abstract: 'We have used the latest HI observations of the Small Magellanic Cloud (SMC), obtained with the Australia Telescope Compact Array and the Parkes telescope, to re-examine the kinematics of this dwarf, irregular galaxy. A large velocity gradient is found in the HI velocity field with a significant symmetry in iso-velocity contours, suggestive of a differential rotation. A comparison of HI data with the predictions from tidal models for the SMC evolution suggests that the central region of the SMC corresponds to the central, disk- or bar-like, component left from the rotationally supported SMC disk prior to its last two encounters with the Large Magellanic Cloud. In this scenario, the velocity gradient is expected as a left-over from the original, pre-encounter, angular momentum. We have derived the HI rotation curve and the mass model for the SMC. This rotation curve rapidly rises to about 60  up to the turnover radius of $\sim3$ kpc. A stellar mass-to-light ratio of about unity is required to match the observed rotation curve, suggesting that a dark matter halo is not needed to explain the dynamics of the SMC. A set of derived kinematic parameters agrees well with the assumptions used in tidal theoretical models that led to a good reproduction of observational properties of the Magellanic System. The dynamical mass of the SMC, derived from the rotation curve, is $2.4\times10^{9}$ M$_{\odot}$.'
author:
- 'S. Stanimirović'
- 'L. Staveley-Smith'
- 'P. A. Jones'
title: A New Look at the Kinematics of Neutral Hydrogen in the Small Magellanic Cloud
---

Introduction
============

The Small Magellanic Cloud (SMC) is a nearby[^1], gas-rich, dwarf irregular galaxy. Its morphology, dynamics, and evolution are highly complex, and must have been heavily influenced by gravitational interactions with the nearby Large Magellanic Cloud (LMC) and the Galaxy [@Mary-nature]. As a dwarf irregular galaxy, the SMC is different from our own Galaxy in many respects, having a lower heavy element abundance, significantly lower dust content, and a consequently stronger interstellar radiation field [@Stanimirovic99; @Stanimirovic01].

Many recent high resolution studies have shown that dwarf galaxies in general have a very dynamic interstellar medium (ISM), structured mainly by star formation and its aftermath. The most obvious example of star-formation creativity are numerous expanding shells of gas [@Puche92; @Staveleyetal97; @Kim99; @Stanimirovic99; @Walter99]. Dwarf galaxies are also usually found to have large, and dynamically important, halos of dark matter [@Mateo98]. The relative roles of a galactic rotation, pressure support, dark matter, and magnetic fields on the 3-D structure and dynamics of these galaxies are still open questions. Furthermore, it is not clear yet whether different stages of galactic evolution are marked by fundamentally different phenomena, and whether these phenomena are selective with respect to spatial scales.

It has been known for some time that the morphology and kinematics of the SMC traced by different stellar populations show very different properties [@Gardiner92]. This has been one of the major reasons to start thinking about forces other than gravitational that may have played a significant role in the formation and evolution of the whole Magellanic System. The questions of morphology and kinematics are closely related to the long-standing and greatly controversial questions of the SMC’s 3-D structure and depth along the line-of-sight. As the results of several new optical and near-IR surveys are becoming available [@Zaritsky00; @Cioni00; @Maragoudaki01], revealing structural evolution of the SMC, as well as new N-body simulations of the dynamical evolution of the SMC [@Yoshizawa03], we find that it is timely to re-examine the morphology and kinematics of the SMC as traced by the neutral hydrogen (HI).

The mapping of HI in the SMC has a long and productive history. After the pioneering work by @KerrHindmanRobinson54 and @Hindmanetal63, @Hindman67 was the first to notice the velocity gradient in the SMC and to model its rotation curve, followed by @BajajaLoiseau82. However, the velocity field of the SMC is far from a simple text-book example. The HI profiles, often complex and with multiple peaks, have caused much controversy in the past, being interpreted as due to either expanding shells of gas or spatially separate systems [@Hindman67; @MathewsonFordVisvanathan88; @Martin89]. An improvement by a factor of ten in spatial resolution, relative to previous surveys with the Parkes telescope, was achieved in the HI survey by @Staveleyetal97, using the Australia Telescope Compact Array (ATCA). These new high resolution data showed that much of the HI profile complexity lies in the huge number ($\sim$ 500) of expanding shells. The new high resolution HI observations were complemented with new low resolution observations obtained with the Parkes telescope to provide information over a continuous range of spatial scales from 30 pc to 4 kpc [@Stanimirovic99].

The aim of this paper is to re-examine the HI kinematics of the SMC, as viewed from high resolution observations, and compare it with predictions from tidal theoretical models. We start by summarizing the HI observations in Section \[s:hi-data\]. The morphology and kinematics of the SMC from the HI distribution, as well as viewed using other tracers, are discussed in Section \[s:overview\]. The SMC 3-D structure and line-of-sight depth are reviewed briefly in Section \[s:3D-structure\]. We then compare HI data with predictions from several tidal models in Section \[s:theoretical-models\]. The rotation curve and mass model of the SMC are investigated in Section \[s:rotation-curve\]. Finally, a summary and concluding remarks are given in Section \[s:summary\].

HI data {#s:hi-data}
=======

[lc]{} Property& Value\
\
[**Radio:**]{}&\
[*Measured:*]{}\
RA (J2000)$^{a}$ & 01$^{\rm h}$ 05$^{\rm m}$\
Dec (J2000)$^{a}$ & $-72^{\circ}$ $25'$\
Systemic velocity$^{b}$, $V_{\rm sys}$ (Gal.) & 24\
Systemic velocity$^{b}$, $V_{\rm sys}$ (Hel.) & 160\
HI mass$^{c}$, $M_{\rm HI}$ & $4.2 \times 10^{8}$ M$_{\odot} $\
[*Model-dependent:*]{}\
Inclination$^{d}$, $i$ & 40$\pm$ 20\
Kinematic Position angle$^{d}$, PA & 40\
V$_{\rm max}^{d}$ & 60\
R$_{\rm max}^{d}$ & 2.5 – 3 kpc\
M$_{\rm dyn}^{d}$ & $2.4\times10^{9}$ M$_{\odot}$\
\
[**Optical:** ]{}&\
Extinction$^{e}$, $E_{\rm B-V}$ & $\sim0.05$–0.25 mag\
Total visible magnitude$^{f}$, $V_{\rm T}$ & $+2.4$ mag\
Absolute visible magnitude$^{f}$, $M_{\rm V}$ & $-16.5$\
Visible luminosity$^{f}$, $L_{\rm V}$ & $3.1\times10^{8}$ L$_{\odot}$\
$M_{\rm H}/L_{\rm V}$ & 1.4 M$_{\odot}$/L$_{\odot}$\
$(B-V)_{\rm T}^{0}$ & 0.41\
$(U-B)_{\rm T}^{0}$ & $-0.23$\

The HI in the SMC was observed with the ATCA[^2], a radio interferometer, in a mosaicing mode (Staveley-Smith et al. 1997). Observations of the same area were obtained also with the 64-m Parkes telescope. The two sets of observations were then combined (Stanimirović et al. 1999), resulting in the final HI data cube with angular resolution of 98 arcsec, velocity resolution of 1.65 , and 1–$\sigma$ brightness-temperature sensitivity of 1.3 K to the full range of spatial scales between 30 pc and 4 kpc. The area covered with these observations is RA $00^{\rm h} 30^{\rm m}$ to $01^{\rm h} 30^{\rm m}$ and Dec $-71^{\circ}$ to $-75^{\circ}$ (J2000), over a velocity range of 90 to 215 . For details about the ATCA and Parkes observations, data processing, and data combination (short-spacings correction) see Staveley-Smith et al. (1997) and Stanimirović et al. (1999).

Structure and kinematics of the SMC {#s:overview}
===================================

HI distribution
---------------

Fig. \[f:HI-density\] shows the integrated HI column-density distribution. The large-scale HI morphology of the SMC is quite irregular and does not show obvious symmetry. The most prominent features are the elongation from the north-east to the south-west and the V-shaped concentration at the east (RA 01$^{\rm h}$ 14$^{\rm m}$, Dec $-73^{\circ}$ 15$'$, J2000). These are usually referred to as the ‘bar’ and the Eastern Wing, although their dynamical importance is still not well understood. A ‘bridge’ of gas appears to connect the ‘bar’ and the Wing (note that this is not the Magellanic Bridge that connects the SMC and the LMC), while the arm-like extension of the ‘bar’ towards the north-east is also prominent. On smaller scales and looking at different velocity channels (see Stanimirović et al. 1999), the HI distribution appears very complex and frothy, being dominated by numerous expanding shells, filaments and arcs. The total estimated mass of the HI, after correction for self-absorption, is $4.2 \times 10^{8}$ M$_{\odot} $ (Stanimirović et al. 1999). A summary of radio and optical properties of the SMC is given in Table \[t:summary\].

The HI column density distribution is super-imposed on the V-band optical image of the SMC, kindly provided to us by M. Bessell, in Fig. \[f:v-HI\]. In general, in the ‘bar’ and the Eastern Wing the stellar and HI distributions correlate well, however the HI is significantly more extended towards the south-west, the south-east, and the north-west.

HI kinematics {#s:profile-analysis}
-------------

The intensity-weighted mean velocity along each line-of-sight, or the first moment map, was derived from the full resolution HI data cube (corrected for short spacings), and is shown in Fig. \[f:1mom-mean\]. A large velocity gradient, from 91  in the south-west to 200  in the north-east, can be seen. The iso-velocity contours show some large-scale symmetry suggestive of differential rotation for the main gaseous body of the SMC. However, clear distortions are visible in the north-west, most likely corresponding to several shells and filamentary features aligned into a chimney-like structure at velocity $\sim$ 123  (Fig. 2 in Stanimirović et al. 1999), and in the south-east, towards the Eastern Wing region, where again a supergiant shell was found (494A; see Stanimirović et al. 1999). These perturbations form a S-shaped feature perpendicular to the direction of the main velocity gradient (as shown in Fig. \[f:1mom-mean\] with a dashed line).

The second moment map or the intensity-weighted velocity dispersion is shown in Fig. \[f:dispersion-mean\]. This velocity dispersion varies from $\sim$5 to 40  across the SMC. The regions with higher dispersion appear to be associated with the positions of the three largest supergiant shells (SGSs). The region around RA $01^{\rm h}$ $00^{\rm m}$, Dec $-71^{\circ} 30'$ (J2000) has the lowest dispersion with a mean value of $\sim$10 . As this region corresponds to the receding hemisphere of the SGS 304A (Stanimirović et al. 1999), the lower velocity dispersion may be explained with the fact that most of the approaching hemisphere of SGS 304A is missing at the northside.

We have investigated two effects that could significantly influence the observed velocity field shown in Fig. \[f:1mom-mean\]: (i) the effect of the proper motion of the SMC, and (ii) the effect of multiple-peak velocity profiles.\
(i) The binary motion of the SMC around the Galaxy, combined with its large angular extent, could have a significant contribution to the observed velocity field, as was shown for the case of the LMC by @LuksRohlfs92 and @Kim98. This contribution consists of the projection of the transverse velocity of the SMC’s center of gravity on the line-of-sight at each position. In the case of a large angular extent on the sky, this projection can vary greatly across the field of observation and cause a significant change in the observed velocity field. To correct for this effect, we used values for the SMC’s proper motion and the heliocentric transversal velocity predicted by numerical simulations by @Gardiner94, which are in agreement with the estimates based on a combination of proper motion measurements by @Kroupa94 and @KroupaBastian97. At any position across the SMC the proper-motion corrected heliocentric velocity was obtained by subtracting the projection of the heliocentric transverse velocity of the center of the SMC on the line-of-sight from the observed radial heliocentric velocity. The values for the conversion between the observed and the proper-motion corrected heliocentric velocity range between $-23$ , in the north-west, and 28 , in the south-east. The correction for the proper motion, however, did not make as big difference as in the case of the LMC [@LuksRohlfs92]. The reason is that the angular extent of the SMC is less than half that of the LMC. In addition, the direction of the SMC’s motion is perpendicular to the observed velocity gradient resulting in the gradient being almost unaffected by the proper motion.

\(ii) Velocity profiles in the SMC are usually very complex, having double or even multiple-peak components. Since the intensity-weighted mean velocity (Fig. \[f:1mom-mean\]) is biased towards the velocity component with higher intensity, it is not necessarily the best representation of complex velocity profiles. To avoid this bias, we found, for each line-of-sight, the minimum and maximum velocity for which the intensity is 5% of the peak value for that line-of-sight. The mean of the minimum and maximum velocity, determined in this way, was then taken to be our new velocity estimate. Since such an analysis requires a high signal-to-noise ratio, we only used the low resolution Parkes data cube. The resultant velocity field has a more regular ‘spider’ pattern than the mean one, however irregularities are still present in the north and close to the Eastern Wing region. We also investigated the existence of a bimodal velocity field, by modeling each velocity profile in the Parkes data cube with one or a superposition of two independent Gaussian functions. As a result, two separate velocity structures were depicted, with central velocities corresponding to the low and high velocity entities of 137  and 174 , respectively. The position-velocity diagrams show almost parallel velocity fields of these two components. However, in many cases the velocity fields intersect suggesting that two separate velocity components may be a consequence of the statistical data handling of complex HI profiles with no substantial physical meaning.

The proper-motion corrected velocity field in the Galactocentric velocity reference frame, derived from the Parkes HI data cube using the 5% cut-off described in (ii), is shown in Fig. \[f:gal\_velfield\]. A clear velocity gradient is noticeable from $-15$ , in the south-west, to 53 , in the north-east. The positions of the three largest shells in the SMC are also overlaid in Fig. \[f:gal\_velfield\], suggesting that supershells 304A and 494A may still be responsible for some of the perturbations in the velocity field. Some perturbations are also visible in the north-west.

Comparison with other tracers {#s:other-tracers}
-----------------------------

### SMC morphology from stellar populations

It has been known for some time that young and old stellar populations in the SMC have significantly different spatial distributions (e.g. @Gardiner92). Several recent stellar surveys have especially enhanced our knowledge about the different morphological properties of different stellar populations. The recent stellar survey of the central SMC (4$\times$ 4) by @Zaritsky00 showed that while young stars (with ages $\la 200$ Myr) exhibit an irregular distribution, similar to that seen in HI, the old stellar population (with ages $\ga 1$ Gyr) shows regular, undisturbed, and almost spheroidal distribution. Similar results were reached from the [DENIS]{} survey by @Cioni00 based on $IJK_{s}$ IR bands. These studies also pointed out that the ‘bar’ and the Eastern Wing region may not be distinct dynamical entities, but could be a result of recent hydrodynamic interactions between the LMC and SMC’s gaseous components. @Maragoudaki01 further investigated the dynamical origin of the ‘bar’ by using isodensity contour maps of stars with different ages. They found similar results concerning old stellar populations. In addition, their data show that the ‘bar’ and the Eastern Wing region were already prominent features some 0.3–0.4 Gyr ago; however, it is not clear whether this is a temporary extended region of star formation or a genuine, dynamical feature.

### SMC kinematics from stellar populations {#s:other-tracers-kinematics}

The intermediate age and old stellar populations, traced by planetary nebulae [@Dopita85b] and carbon stars [@Hardy89; @Hatzidimitriou97; @Hatzidimitriou99; @Kunkel00], do not show signs of rotation. The red horizontal branch clump stars in the north-east show a velocity gradient of 70  which was interpreted as due to a distance spread of 10 kpc along the line-of-sight (Hazidimitriou et al. 1993). The young stellar population, traced by HI shells, shows a clear velocity gradient from the south-west to the north-east [@Staveleyetal97], while Cepheids may have a similar trend with velocity [@MathewsonFordVisvanathan88]. All populations show similar mean heliocentric velocity and velocity dispersion regardless of age and location [@Hatzidimitriou99; @Staveleyetal97].

From an analysis of 150 carbon stars, @Kunkel00 estimated the inclination of the SMC orbital plane to $i=73^{\circ}\pm4^{\circ}$ relative to the plane of the sky. They found that carbon stars are associated with more negative velocity and denser HI, while low density HI seems to be devoid of carbon stars. This suggests that non-gravitational forces must be acting on the gaseous component to provide its separation from stellar systems. Similar estimates for the SMC’s inclination were reached using Cepheids by @Caldwell86, $i=70^{\circ} \pm 3^{\circ}$, and by @Groenewegen00, $i=68^{\circ} \pm 2^{\circ}$.

The 3-D structure of the SMC {#s:3D-structure}
============================

Previous observational models
-----------------------------

The 3-D structure of the SMC has been a matter of great controversy in the past. Since early mapping of HI, complex HI profiles pointed to the existence of unusual motions of the gas [@Hindman67]. [@Hindman67] also suggested the first model of the SMC, as a flattened disk with three supergiant shells in the main body.

From the analysis of the radial velocity distributions of HI, stars, HII regions, and planetary nebulae, [@MathewsonFord84] and Mathewson, Ford & Visvanathan (1986, 1988) suggested the ‘two separate entities’ model for the SMC, whereby the SMC is, along most of its angular extent, broken into two velocity subsystems (the Small Magellanic Cloud Remnant, SMCR, and the Mini Magellanic Cloud, MMC). In this model, the SMCR and MMC are separated in velocity by $\sim$ 40 , and have their own nebular and stellar populations. The reason proposed for this disruption of the SMC was its close encounter with the LMC some 2$\times$10$^{8}$ yr ago. A slight modification of the ‘two separate entities’ model was suggested by [@Torres87], and [@Martin89], who found four, instead of two, different velocity components.

[@Caldwell86] also used Cepheids to suggest the ‘bar and three arms’ geometrical model of the SMC. The central bar is very elongated (5-to-1) and is seen edge-on. A distant arm of material is pulled from the center of the SMC to the west, while two sides of the bar located in the north-east and the south-west are considered as two near arms.

Depth along the line-of-sight {#s:smc-depth}
-----------------------------

The question of the 3-D structure of the SMC is closely related to the long-standing and greatly controversial issue of the SMC’s depth along the line-of-sight. By measuring distances and radial velocities of 161 Cepheids, [@MathewsonFordVisvanathan86] found a great depth along the line-of-sight of $\sim$30 kpc, and a distance gradient from the north-east to the south-west. The large depth of the SMC was supported later by [@MathewsonFordVisvanathan88], using Cepheids again, and by [@Hatzidimitriou89] and [@Hatzidimitriou93] from a study of the intermediate-age (halo) population.

[@Welch87] have pointed out that the determination of the Cepheid distances in [@MathewsonFordVisvanathan86] suffers from a few possible problems: an inconsistent correction to mean magnitude, the assumption of zero intrinsic scatter for the period-luminosity (P-L) relation, and sample inhomogeneity. [@Welch87] concluded that the SMC does not extend in depth beyond its tidal radius (4 – 9 kpc). [@Martin89] confirmed that the young stars in the SMC lie within a depth of $<10$ kpc. On the other hand, Groenewegen (2000) used near-infrared observations of Cepheids, which are less affected by reddening and metallicity, and derived the depth of the SMC of 14 kpc, assuming that the P-L relation has the same scatter for both the LMC and the SMC.

Although Cepheids are used as the main distance indicators, the P-L relation requires an independent zero-point calibration, which is very difficult to achieve, and could suffer from a significant metallicity dependence. Another source of error in the distance determination, that has not been previously fully appreciated, is differential extinction toward the SMC. While reddening of Cepheids in the SMC has been extensively studied (see [@Welch87] for a summary), a constant value has been often applied for distance determination. For example, [@MathewsonFordVisvanathan86] found no sign of differential reddening and assumed a uniform correction of 0.06 mag for their sample of 161 Cepheids.

Recently, [@Zaritsky02] determined the extinction map across the SMC, using the Magellanic Clouds Photometric Survey, and showed that extinction varies both spatially across the SMC and with stellar population. They found that young, hot stars have an average extinction 0.3 mag higher than old, cool stars, and that young stars show a significant increase in extinction along the main SMC ridge, from the north-east to the south-west. [@Zaritsky02] found $E_{\rm B-V} \sim$ 0.05 – 0.25 mag. For comparison, the mean value for the interstellar reddening previously measured by [@Caldwell85] was $E_{\rm B-V}=(0.054 \pm 0.021)$ mag, while [@Sasselov97] found a slightly higher value of $E_{\rm B-V} = (0.125 \pm 0.009)$ mag.

The differential extinction across the SMC has a significant influence on the distance estimate. Differential reddening toward Cepheids in the south-west part of the SMC is $\sim0.2$ mag higher than toward in the north-east. This results in the interstellar infrared absorption being larger in the south-west by $\sim$0.4 mag. Hence, the Cepheid distances estimated by [@MathewsonFordVisvanathan88] could be over-estimated by up to 18%. This corresponds to about 10 kpc at a distance of 60 kpc, and suggests that the correction for the interstellar absorption can significantly influence the distance determination, and easily bring the depth of the SMC within its tidal radius (4 – 9 kpc).

Comparison with theoretical models {#s:theoretical-models}
==================================

There are two families of theoretical models, based on either a tidal or a ram pressure scenario, that try to reproduce observational features in the Magellanic System, caused by interactions between the SMC, LMC, and the Galaxy. The 3-D structure and kinematics of the SMC were particularly addressed in tidal models by @Gardiner94, @Gardiner96, and @Yoshizawa03. The recent work by @Yoshizawa03 provides, to date, the most detailed model for the evolution of the SMC. We summarize here results from these tidal simulations and their major predictions concerning the structure and kinematics of the SMC.

It is important to note that none of the theoretical models so far predicts the bimodal velocity distribution throughout the main gaseous body of the SMC. This gives support to the idea that most, if not all, of the observed line-splitting comes from the combined effects of numerous expanding shells (Staveley-Smith et al. 1997).

Model predictions
-----------------

The above three models come from N-body simulations of the gravitational interactions in the Galaxy-LMC-SMC system to reproduce the observed gas distribution in the Magellanic System, primarily those of the Magellanic Stream and Bridge. While @Gardiner94 modeled the SMC as a single-component disk-like system, in simulations by @Gardiner96 the SMC was represented by a two-component particle system consisting of a nearly spherical halo and a rotationally supported disk, with a disk-to-halo mass ratio of 1:1. This high mass ratio resulted in the original disk quickly becoming unstable, and being transformed into a bar-like structure. The tidal model by @Yoshizawa03 included, for the first time, gas dynamics and star formation processes, while representing the SMC also as a rotationally supported exponential disk with a nearly spherical halo, having the disk-to-halo mass ratio of 3:7. The significantly lower disk mass resulted in the disk being stable against bar instabilities. Following typical findings for the Magellanic-type galaxies, a slowly rising rotation curve was assumed in both models, with a turnover radius of 3.5 kpc and maximum rotation velocity of 50 . The best spatial orientation of the original SMC disk was determined in @Gardiner96 to have an inclination $i=45^{\circ}$ and major axis $\theta=230^{\circ}$ in order to match the observed gas and stellar distributions. These values were adopted by @Yoshizawa03 and simulations were repeated for different star formation parameters.

The major and common result of all three models is that the current 3-D structure of the SMC is composed of a central, disk- or bar-like, component and two tidal, spiral-arm-like tails. These tidal tails extend into the Magellanic Bridge, were formed 200 Myr ago, and are seen in both gas and stellar components. In particular, simulations by @Yoshizawa03 show great morphological change of the initial SMC’s gas disk – from its original size of about 10 kpc in diameter the disk shrunk forming first the Magellanic Stream and the Leading Arm (about 1.5 Gyr ago), and then later two tidal arms that form the Magellanic Bridge (about 200 Myr ago). At the end of the simulation the gas disk is still present but is now significantly smaller in size, approximately by a factor of 2.5. The current SMC consists of the following components.

1.  The left-over [**disk-like component**]{}, located at 0$^{\rm h} 15^{\rm m}<$RA$<1^{\rm h}$, is not greatly elongated along the line-of-sight, with the stellar disk having a higher distance dispersion than the gas disk. Kinematically, the gas disk has a significant velocity gradient, from $\sim$80 to 180 . The left-over stellar disk, on the other hand, shows high dispersion in the velocity field but no significant velocity gradient. In simulations by @Gardiner96, this is a bar-like feature slightly bigger and centered more westward, covering 0$^{\rm h} 30^{\rm m}<$RA$<1^{\rm h} 30^{\rm m}$, and a slightly wider velocity range.

2.  The [**eastern tail**]{}, starting around RA $\sim1^{\rm h}$, extending into the Magellanic Bridge, and covering a distance range from 55 to 40 kpc. This feature starts at a heliocentric velocity of about 170  and gradually decreases to about 120  at RA $>2^{\rm h}$. It is seen in both gas and stars but the gas tail is more extended.

3.  The [**western tail**]{}, starting around RA $\sim0.5^{\rm h}$ and covering a distance range from 55 to almost 80 kpc. Kinematically, this feature starts westward from the main disk-like component at a velocity of about 80 , but then turns east passing the main component with increasing velocity all the way to about $>250$  around RA $>2^{\rm h}$. The stellar western tail follows the gas one and is less extended.

Hence, the two tidal tails contribute mostly to the SMC’s elongation along the line-of-sight, while the SMC disk-like component is not significantly elongated ( $\sim$5 kpc).

The best simulation by @Yoshizawa03 reproduces a large number of observational properties in the Magellanic System, suggesting that most of them may be predominantly of tidal origin. Very importantly, a good morphological and kinematic reproduction of the Magellanic Stream with almost no stars was achieved for the first time, by using a very compact initial configuration of the SMC’s stellar disk. The major shortcoming of the model is its failure to reproduce gas masses and mass ratios in the Magellanic Stream, Bridge, and the SMC. This may be related with the need for a more massive initial disk of the SMC, or a different halo-to-disk mass ratio, and requires further investigation. In addition, as pointed out by @Putman03, a number of detailed features related to the Magellanic Stream (e.g. its double-helix appearance and numerous small clouds surrounding its main filaments) need to be explained. It has also been shown that forces other than gravity must play a significant role in the kinematics of the SMC [@Kunkel00; @Zaritsky00].

The results of tidal numerical simulations show that the current central gaseous body of the SMC ($\sim4$ degrees in extent) has a significant velocity gradient. @Gardiner96 explained this gradient as being due to the elongation of the SMC’s bar-like component along the line-of-sight. However, a significant angular momentum left from before the last two close encounters with the LMC is expected to be present, if the pre-encounter SMC disk had an internal spin. The N-body simulations by @Mayer01 showed that in general disk-like dwarf irregular galaxies undergo a great morphological transformation due to close encounters with the more massive Milky Way. While dwarfs lose their mass through tidal stripping, their angular momentum gets gradually removed over a period of 6-10 Gyrs. This transformation is gradual, and on short timescales ($\sim2$ Gyrs) a significant angular momentum ($>$80%) is still preserved. Focusing especially on the SMC case, a similar conclusion was reached by @Kunkel00 who found that it is extremely hard for the SMC to lose its pre-encounter angular momentum during an encounter with the LMC.

Comparison with HI
------------------

In Section \[s:profile-analysis\] we showed that the HI distribution contains a large velocity gradient and some signatures for the existence of ordered, systematic motions. This agrees with theoretical simulations which show that the current central gaseous body of the SMC should still posses angular momentum left over from before the last two close encounters with the LMC. This angular momentum should be less significant in the case of older stellar populations, as indeed has been found from observations (see Section \[s:other-tracers\] for a summary). We now compare RA-velocity slices through the HI data cube, shown in Fig. \[f:ra\_vel1\], with the theoretical predictions, shown in Fig. 17 of @Yoshizawa03 and Fig. 10 of @Gardiner96. The area enclosed in our observations contains only the central region of the SMC ($4.5^{\circ}\times4.5^{\circ}$).

Images in Fig. \[f:ra\_vel1\] show coherent large-scale features with a wealth of small-scale structure. Many of the features have already been interpreted as expanding shells of gas (Staveley-Smith et al. 1997; Stanimirović et al. 1999). At a low Dec of $\sim-74$, the two most dominant features are supergiant shells 37A (RA 01$^{\rm h}$ 25$^{\rm m}$) and 494A (00$^{\rm h}$ 35$^{\rm m}$). These shells were investigated in Stanimirović et al. (1999). Around Dec $\sim-73$20$'$, Fig. \[f:ra\_vel1\] shows an HI emission, bar-like, feature stretching over the RA range 00$^{\rm h}$ 40$^{\rm m}$ to $\sim$00$^{\rm h}$ 55$^{\rm m}$, and having a velocity gradient from $\sim$120 to $\sim$170 . This feature continues eastward as a complex network of filaments and shell-like features, culminating around Dec $\sim-72$37$'$ with two long filaments centered at heliocentric velocities of 130 and 170 . The coherent appearance of these features traced throughout the data cube was interpreted as the low-Dec hemisphere of the supergiant shell 304A (see Stanimirović et al. 1999). At a very high Dec of $\sim-71$55$'$, a high-velocity component is apparent, at about 200 .

We do not find obvious features that could correspond to the predicted eastern and western tails within the region covered by our observations. To search for potential tidal tails running out of the central SMC main body an investigation of a larger area is essential; this will be possible in the near future with new observations by @Bruns02 and @Muller03, but is beyond the scope of this paper. Concerning the position and velocity range of the central component, our observations are in better agreement with the @Gardiner96 model predictions: the coherent structure seen from RA 00$^{\rm h}$ 30$^{\rm m}$ to 01$^{\rm h}$ 30$^{\rm m}$ corresponds to the area where the central bar-like feature predicted by the model should be found, and its velocity span from 100 to 200  is very close to the predicted velocity range for the bar-like component. The @Yoshizawa03 model predictions for the left-over disk component cover a smaller area and velocity range than what is found in the observations.

Rotation curve and mass modeling of the SMC {#s:rotation-curve}
===========================================

As discussed in the previous section, the central $4^{\circ} \times 4^{\circ}$ of the SMC, covered by the HI observations, most likely corresponds to the left-over part of the original SMC’s gaseous rotationally supported disk. This disk should contain significant signatures of angular momentum left from the pre-encounter disk. Motivated by this, we proceed further to derive the rotation curve from the observed HI velocity field, and the mass model, assuming that the entire velocity gradient is due to rotation.

Tilted ring analysis
--------------------

The velocity field derived in Section \[s:profile-analysis\], and shown in Fig. \[f:gal\_velfield\], was used to define the major and minor kinematic axes, and the apparent kinematic center at $\sim$ RA $01^{\rm h} 05^{\rm m}$, Dec $-72^{\circ} 25'$ (J2000), with a systemic velocity of $\sim$ 20  in the galactocentric reference frame. The position angle (PA) of the major kinematic axis is around 50. The tilted ring algorithm [rocur]{} in the [aips]{} package was then used to derive the HI rotation curve at different radii from the center. A fairly standard procedure [@Meurer98; @Bureau02] was applied to estimate a set of kinematic parameters that represents well the observed velocity field at all radii. It was found that solutions for the inclination vary greatly especially for the outer radii. In the end, the value $i=40$$\pm$20 was adopted. The newly determined least-squares solution for the systemic velocity was found to be 24 . Finally, the solutions for the rotational velocity and position angle were determined and are shown in Fig. \[f:pa-vrot\]. The position angle varies systematically from 30 to 50 (as shown in the top panel of Fig. \[f:pa-vrot\]). Using the mean PA, the global and the separate rotation curves for the receding and approaching sides of the velocity field were derived, as shown in Fig. \[f:pa-vrot\] (bottom panel).

The global rotation curve shows a slow rise up to $R\sim3$ kpc, where it reaches the maximum velocity of $\sim$40 . The receding and approaching curves are significantly different, which is not surprising since the velocity field is quite asymmetric (Fig. \[f:gal\_velfield\]). The receding curve is less smooth and could have a local maximum around 1 kpc from the center of rotation. The maximum rotation velocity found in previous studies [@Hindman67; @LoiseauBajaja81] was $\sim$ 36 .

The line-of-sight HI velocity dispersion distribution shown in Fig. \[f:dispersion-mean\] pointed to a high velocity dispersion across most of the SMC. The mean value, $\sigma_{\rm HI}=(22\pm2)$ , is significantly higher than what is found for spiral galaxies, or even other dwarf galaxies [@Meurer98; @Cote00]. It is also large compared to the rotational velocity (Fig. \[f:pa-vrot\]). If interpreted as being all due to random motions, rather than bulk motions along the line-of-sight, such high values for $\sigma_{\rm HI}$ suggest that the turbulence in the ISM of the SMC has an important influence on the system dynamics.

To account for this dynamical support we estimated the asymmetric drift correction. The usual prescription for determining this correction [@Meurer96] was followed: we derived the azimuthally averaged HI surface brightness and velocity dispersion profiles (for deprojected circular annuli with a mean position angle of 40 and an inclination of 40), and assumed a constant vertical scale height. In general, the derived asymmetric drift correction is quite significant and range from 0, for the inner radii, to $\sim 40$  for the outer radii. The observed rotation curve corrected for the pressure support is presented in Fig. \[f:rotmod-final\] (crosses). The difference is significant for $R>0.5$ kpc with the corrected curve being higher by $\sim$ 10  than the observed one.

The mass model
--------------

A two-component mass model was fitted to the corrected rotation curve.

\(a) To derive the deprojected rotation curve due to the potential resulting from the neutral gas alone, the radial HI surface density profile was used in the [gipsy]{}’s [rotmod]{} task [@gipsy]. An exponential density law was assumed for the vertical disk distribution, with a scale height of 1 kpc. An indication of the large scale height was given by the typical size of large HI shells in the SMC, as well as from previous estimates based on the velocity dispersion and an average surface density of matter for the case of an isothermal disk. The resultant rotation curve ($V_{\rm g}$) is shown in Fig. \[f:rotmod-final\] with dotted line.

\(b) To derive the deprojected rotation curve due to the optical surface density distribution, the V-band image (shown in Fig. \[f:v-HI\]) was used. This image was first smoothed to 20 arcmin resolution and scaled to match the total luminosity of the SMC in the V-band of $3.1\times10^{8}$ L$_{\odot}$ [@RC3]. The rotation curve arising from the stellar potential alone ($V_{\ast}$) was estimated from the V-band stellar density profile assuming, in first approximation, that $M_{\ast}/L_{\rm V}=1$ (see the curve shown as a dash-dotted line in Fig. \[f:rotmod-final\]). The best fit of the total predicted rotational velocity, $\sqrt{V_{\rm g}^{2}+V_{\ast}^{2}}$, to the observed rotational velocity corrected for the pressure support was obtained for $M_{\ast}/L_{\rm V}=1.02$ (solid line in Fig. \[f:rotmod-final\]). The two component mass model appears to fit the observed rotational velocities quite well, suggesting that no additional component, such as a dark halo, is needed to explain the rotation of the SMC.

The rotation velocity of only the stellar component implies a total stellar mass of the SMC of $1.8\times10^{9}$ M$_{\odot}$, within a radius of 3 kpc. From Section \[s:hi-data\] and after correction for neutral He, the mass of HI$+$He is $5.6\times10^{8}$ M$_{\odot}$, being almost a third of the stellar mass within the same radius. The total mass of the SMC, implied from the rotation curve, is thus $2.4\times10^{9}$ M$_{\odot}$. This is almost twice that of $1.5\times10^{9}$ M$_{\odot}$, derived by @Hindman67 within the slightly smaller radius of 2.6 kpc, and is similar to the inital mass of the SMC assumed by @Gardiner94 (about $2\times10^{9}$ M$_{\odot}$) and by @Yoshizawa03 ($3\times10^{9}$ M$_{\odot}$).

### The stellar $M/L$ from population synthesis models

Stellar population synthesis (SPS) models provide an independent way to estimate the stellar mass-to-light ratio. @Bell01 used simplified spiral galaxy evolution models, based on several different SPS models, to investigate trends of the stellar $M/L$ with galaxy properties, such as colors and gas fraction. They found good agreement with observational data and showed that the trend of $M/L$ with color, $\log_{10} (M/L)=a_{\lambda}+b_{\lambda}$Color, is robust with respect to the SPS models but is dependent on the type of the initial mass function (IMF) used. They tabulate coefficients $a_{\lambda}$ and $a_{\lambda}$ for different colors, stellar metallicity, SPS models, and IMFs.

We have used coefficients for the case of a stellar metallicity appropriate for the SMC, which is about 25% of the solar abundance, based on the SPS model by [@Bruzual03], for a Salpeter IMF scaled by a factor of 0.7 [@Bell01], and the [@Schmidt59] star formation law. The stellar $M/L_{\rm V}$ was then estimated using the broad band colors given in Table 1, $B-V=0.41$ and $B-R=0.7$. We get $M/L_{\rm V}=0.8\pm0.2$ which agrees with the value obtained from the HI rotation curve.

Critical density for star formation
-----------------------------------

We now go a step further and address the gravitational stability of the current gaseous disk of the SMC and its possible consequences for star formation. We estimated the disk stability parameter, $Q$, [@Toomre64] using the mean velocity dispersion, azimuthally averaged HI surface brightness, and the angular velocity derived using the rotation velocity. According to the Toomre’s stability criterion, the disk is stable for $Q>1$, while regions with $Q \la 1$ are unstable and under possible fragmentation which can lead to new star formation [@BinneyTremaine]. The estimated values of $Q$, as a function of radius, are shown in Fig. \[f:Q\]. The plot shows that most of the gaseous disk, for $R<2.1$ kpc, is unstable and, most likely, under current star formation. The unstable region appears to be quite extended, reaching almost the cut-off H$\alpha$ radius, $R({\rm H}_{\alpha})\approx2.6$ kpc (based on H$\alpha$ observations by [@Kennicuttetal95]), which delineates the parts of the gaseous disk under the most recent star formation.

We also determined the so called critical surface density, defined as the surface density needed to stabilize the disk (or for $Q=1$). Both the observed ($\Sigma_{\rm HI}$) and critical ($\Sigma_{\rm c}$) surface densities are shown in Fig. \[f:stability\] (top panel). The ratio of the observed to critical surface density ($\alpha=\Sigma_{\rm HI}/\Sigma_{\rm c}=1/Q$) is shown in the same figure (bottom panel). For the unstable part of the disk, $\alpha$ varies between 1.0 and 2.0, while it has a constant value of $\alpha=0.74\pm0.04$ for the stable part of the disk. This agrees with the empirical relationship between the star formation threshold and the disk stability found by @Kennicutt89 for a sample of normal galaxies, whereby most of the star formation is taking place for $\alpha>0.67$ (and hence for $Q<1.5$).

Summary and concluding remarks {#s:summary}
==============================

We have used the latest HI observations, obtained with the ATCA and the Parkes telescope, to re-examine the kinematics of the SMC. The HI velocity field, derived in Section \[s:profile-analysis\], shows a large velocity gradient from the south-west to the north-east. The iso-velocity contours of this velocity field show some symmetry, suggestive of a differential rotation. Some large-scale distortions in this velocity field are easily visible, but could be related to positions of several supergiant shells. In contrast to the HI distribution, the old stellar populations appear to have a spheroidal spatial distribution and a total absence of rotation. In Section \[s:3D-structure\], we summarized previous observational models of the SMC and the greatly controversial issue of its possibly large depth along the line-of-sight. We cautioned that the differential extinction across the SMC, revealed recently by @Zaritsky02, could be a significant source, unappreciated previously, for an overestimate of stellar distances, and therefore the SMC’s depth.

In Section \[s:theoretical-models\], we summarized several tidal models that are concerned with the 3-D structure and kinematics of the SMC. In order to reproduce observational characteristics of the Magellanic System these models needed to assume the existence of angular momentum in the pre-encounter SMC (about 1.5–2 Gyr ago). Models predict that the current, left-over material from the pre-encounter SMC disk is a disk-like or a bar-like feature, about 4 kpc in extent, with a velocity gradient of about 100 . This velocity gradient is at least partially from the original angular momentum, as it is very difficult to lose the original spin during a galaxy encounter [@Mayer01; @Kunkel00]. The elongation along the line-of-sight of the pre-encounter SMC disk may also be partially responsible. It is, however, not clear from the models what are the relative contributions of these two effects to the predicted velocity gradient. In addition, theoretical models predict higher velocity dispersion for the post-encounter stellar disk without a significant velocity gradient; this agrees with observations of older stellar populations summarized in Section \[s:other-tracers-kinematics\].

While we do not find evidence for the existence of tidal tails predicted by the models, the central region of the SMC covered in these observations most likely corresponds to the central component left-over from the pre-encounter SMC gaseous (rotationally supported) disk. The observed HI velocity gradient agrees well with the model predictions. We then proceeded further to derive its rotation curve and mass model in Section \[s:rotation-curve\]. A set of kinematic parameters derived from the tilted ring analyses agrees extremely well with the assumptions used in theoretical models that led to a good reproduction of observational properties of the Magellanic System. The derived HI rotation curve in Section \[s:rotation-curve\] rapidly rises to about 60  up to the turnover radius of $\sim3$ kpc. A stellar mass-to-light ratio of about unity was required to scale the stellar component of the SMC’s rotation curve to match the observed rotation curve. This suggests that a dark matter halo is not needed to explain the dynamics of the SMC. This is a surprising result, as dwarf irregular galaxies are often found to be dark matter dominated, and could be related to the mechanism of tidal stripping, as indicated by [@Mayer01]. The total dynamical mass of the SMC derived from the rotation curve is $2.4\times10^{9}$ M$_{\odot}$, three quarters of which are due to the stellar mass only. We also derived Toomre’s disk stability parameter $Q$, which shows that almost all of the SMC disk is in the unstable regime. The gravitationally stable part, on the other hand, has a ratio of the observed to critical surface density $\alpha=0.74$ which is in excellent agreement with the empirical stability threshold found by @Kennicutt89.

All of the above suggest that the HI distribution is capable of providing valuable information about the SMC dynamics. Although very disturbed, it still contains imprints of the original system prior to the latest encounters with the LMC and the Galaxy. The HI data discussed here comprise only the central $4.5^{\circ}\times4.5^{\circ}$ of the SMC. It is very important to compare tidal theoretical models with data comprising a larger area around the SMC to search for signatures of the proposed tidal tails. This will be possible in the near future using the Parkes Multibeam HI observations of the whole Magellanic System [@Bruns00], as well as the ATCA and Parkes observations of the Magellanic Bridge [@Muller03].

The HI SMC data set is available from the ATNF SMC web page\
(http://www.atnf.csiro.au/research/smc\_h1/).

Acknowledgments {#acknowledgments .unnumbered}
===============

We are greatly thankful to Mike Bessell for providing us with the V-band optical image of the SMC prior to publication. We thank Greg Bothun for sending us the H$\alpha$ image. We thank John Dickey, Jacco van Loon, and Mary Putman for stimulating and fruitful discussions, and an anonymous referee for insightful comments. This work was supported in part by NSF grants AST-0097417 and AST-9981308.

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\[lastpage\]

[^1]: Throughout this paper we assume a distance to the SMC of 60 kpc [@Westerlund97].

[^2]: The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
