THE THREE-DIMENSIONAL STRUCTURE OF THE SMALL MAGELLANIC CLOUD

The OGLE III survey (Udalski et al. 2008) presented the V- and I-band photometric data of 16 deg² of the SMC consisting of about 35 million stars. We divided the observed region into 1280 regions with a bin size of 8.88 × 4.44 arcmin². The average photometric error of RC stars in the I and V bands is around 0.05 mag. Photometric data with errors less than 0.15 mag are considered for the analysis. For each sub-region the (V − I) versus I CMD is plotted and the RC stars are identified. A sample CMD is shown in Figure 1. For all the regions, RC stars are found to be located well within the box shown in the CMD, with boundaries 0.65–1.35 mag in (V − I) color and 17.5–19.5 mag in I magnitude. The number of RC stars identified in each sub-region ranges from 100 to 3000.

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The RC stars occupy a compact region in the CMD. Their number distribution profiles resemble a Gaussian. The peak values of their color and magnitude distributions are used to obtain the mean dereddened RC magnitude and hence the mean line-of-sight distance to each sub-region in the SMC. The dispersions in the magnitude and color distribution of the RC stars are used to obtain the line-of-sight depth of each sub-region in the SMC.

To obtain the number distribution of the RC stars, they are binned in color and magnitude with a bin size of 0.01 and 0.025 mag, respectively. These distributions are fitted with a Gaussian + quadratic polynomial. The Gaussian represents the RC stars and the other terms represent the red giants in the region. A nonlinear least-squares method is used to fit the profile and to obtain the parameters. The color and magnitude distributions along with the fitted profiles of a sub-region in the SMC are shown in the lower left panels of Figures 2 and 3, respectively. The parameters obtained are the coefficients of each term in the function used to fit the profile, error in the estimate of each parameter, and reduced χ² value. For all the sub-regions we estimated the peaks and the widths in the I mag and (V − I) mag of the distributions, associated errors with the parameters, and reduced χ² values. The errors associated with the parameters are obtained using the covariance matrix, where the square root of the diagonal elements of the matrix gives the error values.

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The number of RC stars identified within the box in the CMD is less (100–400) for 602 sub-regions. These sub-regions are located toward the edge of the survey and in the three isolated fields in the northwestern region of the SMC. For these 602 sub-regions we carefully analyzed the magnitude and color distributions. Because of the low number of stars the peaks of the distributions were not very clear and unique. It was difficult to fit those distributions with the profiles. As our methodology depends very much on the statistical analysis of the sample, we omitted these 602 sub-regions with RC stars less than 400 for the remaining analysis. Thus the number of regions used for the study reduced to 678. The magnitude and color distributions of most of these 678 regions were checked manually and we found that the fits are satisfactory.

The reduced χ² values and the fit errors of the peak and width of color and magnitude distributions are plotted against R.A. in the lower right, upper right, and upper left panels of Figures 2 and 3, respectively. The sub-regions with reduced χ² values greater than 2.0 for both magnitude and color distributions are omitted from the analysis. The regions with peak and width error greater than 0.05 mag in the magnitude distributions are also removed from the analysis. In the case of color distribution, regions with peak and width errors greater than 0.02 mag are not considered for the analysis. Thus we used only 553 out of 678 sub-regions for the final analysis.

The peak value of the color (V − I) mag at each location is used to estimate the reddening. The reddening is calculated using the relation E(V − I) = (V − I)_obs–0.89 mag. The intrinsic color of the RC stars in the SMC is chosen as 0.89 mag to produce a median reddening equal to that measured by Schlegel et al. (1998) toward the SMC. The interstellar extinction is estimated by A_I = 1.4 × E(V − I) (Subramaniam 2005). After correcting the mean I mag for interstellar extinction, I₀ mag for each region is estimated.

The variation in the I₀ mag between the sub-regions is assumed only due to the difference in the relative distances. The difference in I₀ mag is converted into relative distance, ΔD using the distance modulus formula,

$$\begin{eqnarray*} &&({I}_0 \ {\rm mean}\ - {I}_0\ {\rm of \ each \ region}) = 5 \log _{10}({D}_0/({D}_0 \pm \Delta {D}), \end{eqnarray*}$$

where D₀ is the mean distance to the SMC which is taken as 60 kpc. The average error in I₀ is calculated using the formula, δI₀² = (avg error in peak I)² + (1.4 × avg error in peak (V − I))², and the error is estimated as 0.013 mag which corresponds to ∼360 pc.

The Cartesian coordinates corresponding to each sub-region can be obtained using the R.A., decl., and ΔD. The x-axis is antiparallel to the R.A. axis, the y-axis is parallel to the declination axis, and the z-axis is toward the observer. The origin of the system is the centroid of our sample. The centroid estimated as the mean R.A. and decl. of the final sample of the RC stars in the 553 sub-regions of the SMC is α = 0^h52^m342, δ = −73°2'48''. The x, y, and z coordinates are obtained using the transformation equations given below (van der Marel & Cioni 2001, see also the Appendix of Weinberg & Nikolaev 2001; a similar transformation for the LMC is given in Subramanian & Subramaniam 2010):

$$\begin{eqnarray*} {x} &=& -{ D}\sin (\alpha - \alpha _0)\cos \delta,\\ \\ {y} &=& {D}\sin \delta \cos \delta _0 - {D}\sin \delta _0 \cos (\alpha - \alpha _0)\cos \delta,\\ \\ {z} &=& {D}_0 - {D}\sin \delta \sin \delta _0 - {D}\cos \delta _0 \cos (\alpha - \alpha _0)\cos \delta, \end{eqnarray*}$$

where D₀ is the distance to the SMC and D, the distance to each sub-region, is given by D = D₀ ± ΔD. The (α, δ) and (α₀, δ₀) represent the R.A. and decl. of each sub-region and the centroid of the sample, respectively.

The estimated dispersions of the color and magnitude distributions are used to obtain the line-of-sight depth of each region in the SMC. The total width of the Gaussian in the distribution of color is due to internal reddening apart from observational error and population effects. The width in the distribution of magnitude is due to population effects, observational error, internal extinction, and depth. By deconvolving the effects of observational error, extinction, and population effects from the distribution of magnitude, an estimate of the depth can be obtained. The contribution due to population effects on the observed dispersions discussed in Section 2 is corrected using the model predicted values given by Girardi & Salaris (2001). This analysis for the estimation of the depth is similar to that performed by Subramanian & Subramaniam (2009) for the OGLE II and the Magellanic Cloud Photometric Survey regions of the SMC:

$$\begin{eqnarray*} &&\sigma ^2_{{\rm col}} = \sigma ^2_{{\rm internal-reddening}} + \sigma ^2_{{\rm intrinsic}} + \sigma ^2_{{\rm error}}\\&&\ms \sigma _{{\rm internal-extinction}} = 1.4 \times \sigma _{{\rm internal-reddening}}\\&&\ms \sigma ^2_{{\rm mag}} = \sigma ^2_{{\rm depth}} + \sigma ^2_{{\rm internal-extinction}} + \sigma ^2_{{\rm intrinsic}} + \sigma ^2_{{\rm error}}. \end{eqnarray*}$$

Thus the width corresponding to the depth of the RC distribution in each sub-region is estimated. The error in the depth estimate which has contributions from the error in the widths of magnitude and color is also calculated. The average error in the depth estimate is ∼400 pc.

A catalog of the SMC RRLS from the OGLE III survey is presented by Soszyñski et al. (2010). To study the old stellar population in the SMC, we used 1933 fundamental mode RRLS present in the catalog. We removed 29 out of 1933 stars that are brighter than the SMC RRLS and are possible Galactic objects.

The ab-type RRLS can be considered to belong to a similar class and hence can be assumed to have similar properties. The mean magnitude of these stars in the I band, after correcting for extinction effects, can be used for to estimate distance, and the observed dispersion in their mean magnitude is a measure of the depth in their distribution. The reddening obtained using the RC stars (described in the previous sub-section) is used to estimate the extinction for individual RRLS. Stars within each bin of the reddening map are assigned a single reddening and it is assumed that reddening does not vary much within the bin. Thus the extinction corrected I₀ magnitudes for all the RRLS are estimated. As in the case of the RC stars, the difference in I₀ between each RRLS is assumed to be due only to the variation in their distances and the relative distance, ΔD, corresponding to each star is estimated. The error in the relative distance estimation of individual RRLS is basically the error in the extinction correction of I₀ mag. The average error in the extinction estimation, obtained from the RC stars, converts to a distance of ∼110 pc. The location of each RRLS in the Cartesian coordinate system is then calculated using the transformation equations given in the previous section.

Using the dereddened I₀ magnitude of each RRLS, the dispersion in the surveyed region of the SMC can be found. The dispersion is a measure of the depth of the SMC. The data are binned in 05 in both x- and y-axes to estimate dispersion. Thus the observed region is divided into bins with an area of 0.25 deg². The number of stars in each bin has a range of 10–50. The bins with stars less than 10 are removed from our analysis. For each location we estimated the mean magnitude and the dispersion. The estimated dispersion has contributions from photometric errors, range in the metallicity of stars, intrinsic variation in the luminosity due to evolutionary effects within the sample, and the actual depth in the distribution of the stars (Clementini et al. 2003). We need to remove the contribution from the first three terms (σ_intrinsic) so that the value of the last term can be evaluated. Clementini et al. (2003) estimated a value of σ_intrinsic as 0.1 mag for their sample of 100 RRLS in the LMC. Subramaniam (2006) used a larger sample of RRLS in the LMC from the OGLE II catalog and estimated the value of σ_intrinsic as 0.15 mag. Both of these values are estimated using the globular cluster data in the observed field. In the case of the SMC, there is only one globular cluster and it is not in our observed field, so we used the intrinsic spread estimated for the RRLS in the LMC for the analysis of the RRLS in the SMC. Using both these values we estimated the corrected σ values which correspond only to depth. The relation used is

$$\begin{eqnarray*} &&\sigma ^2_{I_{0}{\rm mag}} = \sigma ^2_{{\rm depth}} + \sigma ^2_{{\rm intrinsic}}. \end{eqnarray*}$$

The mean dereddened I₀ magnitudes of the RC stars for the 553 sub-regions of the SMC are estimated. The mean magnitudes of different sub-regions are shown as different color points in the two-dimensional plot of X versus Y in Figure 5. The color code is given in the figure. The average dereddened magnitude I₀ of the 553 sub-regions is 18.43 ± 0.03 mag. The regions in the south (y < 0) are fainter compared to the regions in the north. The brighter regions (I₀ < 18.39) shown as blue points are located more in the northeastern side. This result indicates that the northeastern regions of the SMC are closer to us.

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The relative distance to each RRLS with respect to the mean distance to the SMC is estimated from the dereddened I₀ magnitude, assuming that the average distance to our sample of RRLS in the SMC is 60 kpc. The spatial distribution of RRLS in the XZ and YZ planes with overplotted density contours is shown in Figures 6 and 7, respectively. The convention of the +ve and −ve Z-axes is such that the +ve Z-axis is toward us and the −ve Z-axis is away from us. The distribution of RRLS in the SMC looks more or less symmetric and extends from −20 kpc to +20 kpc with respect to the mean. Most of the stars are located between ±10 kpc. An extension to large distances can be seen at x ∼ 0. The outer density contours shown in Figure 6 indicate that the eastern side of the SMC is on average closer to us than the western side. We calculated the average I₀ magnitude of RRLS in the eastern end (x < −1.5, 245 stars) and in the western end (x > 1.5, 221 stars). It was found that the eastern side is 0.034 mag brighter than the western side, which makes the eastern side on average 950 pc closer to us than the western side. Similarly, the outer density contours shown in Figure 7 suggest that the northern side is closer to us than the southern side. We calculated the average I₀ in the northern end (y > 1.5, 133 stars) and in the southern end (y < −1.5, 104 stars). It was found that the northern side is 0.019 mag brighter than the southern side, which makes the northern side on average 520 pc closer to us than the southern side. Thus the outer density contours of Figures 6 and 7 and the quantitative estimates mildly suggest that the northeastern part of the SMC is closer to us. This result is very similar to the result obtained from the RC stars. As this effect is significant in the outer regions and our study is limited to the inner regions, we need more data in the outer regions to securely say that the northeastern part of the SMC is closer to us. From the photometric study of stellar populations up to 11.1 kpc in the SMC, Nidever et al. (2011) found that the eastern part of the SMC is closer to us than the western side.

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In the case of the RC stars as well as RRLS, the variation in the I₀ magnitude is assumed to be due only to the difference in the distances. As discussed earlier in Section 2, there can be contributions in their mean magnitude from the population effects. Thus the line-of-sight distance estimates suggest that either the RC star and RRLS in the northeastern part of the SMC are different from those found in the other regions of the SMC and/or the northeastern regions of the SMC are closer to us. The previous studies mentioned in Section 2 suggest that there is no large variation in the age and metallicity among old stars in the inner SMC. Thus the contribution to the I₀ magnitude from the population effects across the SMC is expected to be minimum. The exact contribution can be understood only from the detailed spectroscopic studies of the old stellar populations.

The dispersions in the magnitude and color distributions of the RC stars are used to obtain the depth corresponding to the width (1σ) of the 553 sub-regions in the SMC. The width corresponding to depth is converted into depth in kiloparsecs using the distance modulus formula,

$$\begin{eqnarray*} &&\sigma _{{\rm depth}} = 5\log _{10}[({d}_0 + {d}/2)/({d}_0 - {d}/2), \end{eqnarray*}$$

where d is the line-of-sight depth in kiloparsecs.

A color-coded, two-dimensional plot of depth is shown in Figure 8. The color code is explained in the plot. From the plot we can see that the depth distribution in the SMC is more or less uniform. The prominent feature in the plot is the enhanced depth (depth >8 kpc) seen near the central regions. An increased depth of around 6–8 kpc is also seen near the northeastern regions. The average depth obtained for the SMC observed region is 4.57 ± 1.03 kpc. These results match well with the previous depth estimates by Subramanian & Subramaniam (2009).

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The dispersion in the mean magnitude of RRab stars is also used to estimate the depth (1σ) of the SMC. The dispersion in the mean magnitude for 70 sub-regions of the SMC is estimated and converted into depth in kiloparsecs. The average value of depth estimated when the intrinsic spread was taken as 0.1 mag is 4.07 ± 1.68 kpc and the average depth is 3.43 ± 1.82 kpc when 0.15 mag was taken as the intrinsic spread. These estimates match with the depth estimate obtained from the RC stars within the error bars.

The depths estimated using the RC stars and RRLS are plotted together in Figure 9 against the X- and Y-axes. In both of the lower panels, the depth values are plotted against the X-axis and in both of the upper panels, they are plotted against the Y-axis. Basically, the depth values correspond to the extent (front to back distance) over which these stars are distributed in the SMC. The measured 1σ depth (front to back distance) is halved and plotted along the +ve and −ve depth axis, assuming that the depth is symmetric with respect to the SMC. The red crosses correspond to the depth estimated using the RC stars and the black points correspond to the depth estimated using the RRLS. The error bars for each point are not plotted to avoid crowding. In the left panels, the black dots correspond to the depth estimated using the RRLS when the σ_intrinsic is taken as 0.1 mag. Similarly, in the right panels the black dots correspond to the depth estimated using the RRLS when the σ_intrinsic is taken as 0.15 mag. The larger depth near the center (x ∼ 0 and y ∼ 0) is seen for both populations. The 18 regions in the right panels and the 3 regions in the left panels which are shown as open circles at zero depth are those for which the depth of the RRLS became less than zero when the correction for σ_intrinsic was applied. For a large number of regions the depth value becomes less than zero when σ_intrinsic is taken as 0.15 mag. Thus, it appears to be more appropriate to take σ_intrinsic as 0.1 mag for the RRLS in the SMC. The depth profiles of both populations are similar, suggesting that both the RC stars and RRLS occupy a similar volume in the SMC.

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It is important to see how the dispersion in the distribution of the RRLS is related to the real RRLS distribution, obtained from the individual RRLS distances. This comparison will help us to estimate the actual depth from the dispersion of the RRLS in the SMC. We halved the 1σ depth (front to back distance estimated after correcting for σ_intrinsic to be taken as 0.1 mag) obtained for each sub-region with respect to the mean distance and plotted in the −ve and +ve Z-axis as open circles against the X-axis in the lower left panel of Figure 10. Similarly, 2σ depth, 3σ depth, and 3.5σ depth are plotted in the lower right panel, upper right panel, and upper left panel of Figure 10, respectively. The individual RRLS distances with respect to the mean SMC distance are shown as black dots in all the panels of Figure 10. From the figure we can clearly see that the distribution of RRLS is at least extended up to 3.5σ dispersion, if we take an intrinsic spread of 0.1 mag. This 3.5σ width (σ_dep = 0.146 mag) translates to a front to back distance of around 14.12 kpc. In Figure 11, the depth estimated from RRLS after correcting for σ_intrinsic = 0.15 mag is plotted over the individual RRLS distances. The upper left panel shows the 4σ depth plotted over the individual RRLS distances. From the figure we can clearly see that the distribution of RRLS is at least extended to 4σ dispersion, if we take an intrinsic spread of 0.15 mag. This 4σ width (σ_dep = 0.124 mag) translates into a front to back distance of around 13.7 kpc. Thus the RRLS in the SMC are distributed over a distance of ∼14 kpc along the line of sight. In the case of the RC stars, we have estimated only the mean distances to the sub-regions and the depth of the sub-regions, so we cannot compare the real RC distribution with the width of the distribution to define the depth. Since the depth profiles of the RC stars and the RRLS are similar, we expect these two populations to occupy a similar volume in the SMC and the RC stars also to be distributed in a depth of 14 kpc. The large depth suggests a spheroidal/ellipsoidal distribution for the above populations.

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From the above section, we found that both the RC stars and RRLS have similar line-of-sight depth. The density distributions of both these populations will give a clue about the system in the XY plane. The surface density distribution and the radial density profile are studied to understand the structure of the SMC. The surface density is calculated by dividing the observed region into different sub-regions and obtaining the number of objects per unit area in each sub-region. The radial density profiles are obtained by finding the projected radial number density of objects in concentric rings around the centroid of the SMC. These profiles can be compared with the theoretical models. The two theoretical models which can be used for the comparison of spatial distribution of different stellar populations in the SMC are the exponential disk profile and the King's profile. The exponential disk profile is given by

$$\begin{eqnarray*} &&{f}({r}) = {f}_{0d}{\rm e}{^{-r/h}}, \end{eqnarray*}$$

where f_0d and the h represent the central density of the objects and the scale length, respectively, and r is the distance from the center of the distribution. The King's profile (King 1962), which is often used to describe the distribution of globular clusters, but also applies to dwarf spheroidal galaxies, is given by

$$\begin{eqnarray*} &&{f}(r) = {f}_{0k}\lbrace [1+({r/a})^2]^{-1/2} - [1+({r}_t/{a})^2]^{-1/2}\rbrace ^2, \end{eqnarray*}$$

where r_t and a are the tidal and core radii, respectively, f_0k is the central density of objects, and r is the distance from the center. Both these profiles are used to fit the observed distribution of the RC stars in the SMC.

The number of RC stars, identified from the CMD, in each sub-region of the OGLE III region is estimated. We used all 1280 sub-regions (each having an area of 32.6 kpc²) in the observed OGLE III region. We estimated the number density, number of the RC stars per unit area, for each sub-region. The number density distribution of the RC stars is shown in Figure 12. The number density of RC stars in each sub-region ranges from 7500 kpc⁻² to 200,000 kpc⁻². In this figure, discrete points represent each sub-region and the color denotes the number density. The plot clearly shows the smooth RC distribution with an elongation in the northeast to southwest (NE–SW) direction. The color code is given in the figure, where d denotes the RC number density. The position angle of the elongation of the RC distribution is around 54° for the eastern side. The plot also shows the shift in the RC density center, α = 0^h52^m342, δ = −73°2'48'' (shown as green hexagons in Figure 12) from the optical center, α = 0^h52^m45^s, δ = −72°49'43'' (shown as magenta hexagons in Figure 12) of the SMC. The center of the density distribution shown here is very similar to the centroid (given in Section 3.1) of our RC sample in 553 sub-regions which are used for the estimation of dereddened magnitude and depth.

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As the data coverage of the OGLE III is not symmetric with respect to the density center, the radial density distribution of the RC stars within a radius of 08 is obtained. The region within a 08 radius from the density center is divided into 16 equal area (28.3 arcmin²) annuli. The number of RC stars in each annuli is obtained and it is divided by the area of the annuli to obtain the number density. Thus the radial density distribution of the RC stars within a 08 radius from the density center is obtained and is shown as black squares in Figure 13. The best-fitted exponential disk profile and the King's profile are shown in the figure as green and blue solid lines, respectively. We can see that the radial density profile of the RC stars in the SMC is described marginally better by the King's profile than by the exponential profile. We obtained the surface density profile of the RC stars in the whole observed area of the SMC. The number density of the RC stars in the 1280 sub-regions of the SMC is plotted against the radial distance of each sub-region from the density center in Figure 13. The black points in the figure denote the surface density of the RC stars in each sub-region. The best-fit exponential disk profile and the King's profile are shown as green and blue dashed lines, respectively, in Figure 13. Here we can see that the surface density profile of the RC stars in the SMC is best described by the King's profile. The parameters obtained by fitting the radial density and surface density distributions by exponential and King's profiles are given in Table 1. The estimated tidal radius of the SMC system is ∼7–12 kpc. Thus the RC stars in the SMC are distributed in a spherical/ellipsoidal volume, which indicates that the SMC can be approximated as a spheroidal/ellipsoidal galaxy, but in the surface density profile we can see a spread in the points for each radii, which indicates that the volume in which the RC stars are distributed is not exactly spherical. Again, from the map of the surface density distribution shown in Figure 12, we can see the elongation in the NE–SW direction. Thus the RC stars in the SMC are distributed in an ellipsoidal system.

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Our sample of RRLS that are pulsating in the fundamental mode consists of 1904 stars. The density center of our sample is α = 0^h53^m31^s, δ = −72°59'157. The density center of our sample lies in between the two concentrations found by Soszyñski et al. (2010). To study the density profiles, a large number of samples is required. The surface density map of the RRLS identified from the OGLE III photometric maps is shown in the lower panel of Figure 7 in Soszyñski et al. (2010). They identified two concentrations in the RRLS spatial distribution and found that the RRLS in the SMC form a roughly circular structure in the sky which indicates that they are distributed in a spheroidal/ellipsoidal volume.

The method described in the Appendix can be applied to our sample of the RRLS to estimate the parameters of the ellipsoidal component of the SMC. First, we applied this method only to the (x,y) system and found that the SMC RR Lyrae distribution is elongated with an axes ratio of 1:1.3 and the major axis has a position angle of 74° (NE–SW). We repeated the procedure for the (x, y, z) coordinates of the RRLS. The axes ratio obtained was 1:1.3:6.47 and the longest axis is inclined with the line-of-sight direction with an angle (i) of 04. The position angle of the projection of the ellipsoid (ϕ) on the plane of the sky is given by 744. The data of the RRLS used here contain the RRLS in the isolated northwestern OGLE III fields. As they are discrete points in the density distribution, we removed the RRLS in those fields and repeated the procedure. When we removed the RRLS in those fields, the density center changed to α = 0^h54^m386, δ = −73°4'522. The axes ratio obtained is 1:1.57:7.71 with i = 04 and ϕ = 660. Here we can see that the coverage of the data plays an important role in the estimation of the structural parameters of the ellipsoid. This may also suggest that there may be a variation in the inner and outer structures of the SMC. Nidever et al. (2011) found that the inner SMC (R < 3°) is more elliptical than the outer component (3 °<R < 75).

In order to understand the radial variation of the inner structure, we estimated the parameters using the data within different radii. At first, the analysis is done excluding the RRLS in the northwestern fields. The axes ratio i and ϕ of the RRLS distribution are obtained for the data within different radii, starting from 075. The values obtained are given in Table 2. The values show that i is more or less constant with a value of around 05, but the axes ratio and ϕ have a range of values. The axes ratio has a range from 1:1.05:19.84 to 1:1.57:7.7. ϕ ranges from 60° to 78°. The data within 075 of the density center are symmetric and the parameters obtained confirm that the RRLS distribution in the SMC is slightly elongated in the NE–SW direction. As the radius increases the data coverage is not symmetric and circular, and the axis ratio of x and y shows an increasing trend. The surface density map of RRLS shown in Figure 7 of Soszyñski et al. (2010) does not show large elongation; rather the map looks nearly circular with a mild elongation in the NE–SW direction. Thus the elongation obtained for the RRLS distribution in the plane of the sky, at a larger radius where the data coverage is not even, may not be real. We conducted a similar analysis to understand the radial variation including the fields in the northwestern regions. Here the concentric circles with different radii are centered on the density center (given in the last paragraph) estimated including the RRLS in the northwestern fields. These values are also given in Table 2. From this analysis it is also evident that there is a mild elongation in the NE–SW direction in the distribution of the RRLS in the SMC. Here we have to also keep in mind that the density center estimation of the SMC RRLS is not accurate as there are two concentrations found in the density distribution shown in Figure 7 of Soszyñski et al. (2010). The variation in the density center also modifies the structural parameters of the distribution.

Table 2. Orientation Measurements of the Ellipsoidal Component of the SMC Estimated Using the RR Lyrae Stars

Data	No. of RRLS	Axes Ratio	i	ϕ
Excluding the three northwestern fields,
Center: α = 0h54m386, δ = −73°4'522
r < 075	421	1:1.05:19.84	04	7883
r < 100	676	1:1.03:14.59	04	7220
r < 125	924	1:1.04:11.35	02	7568
r < 150	1187	1:1.10:9.43	01	6341
r < 175	1407	1:1.23:8.66	01	6168
r < 200	1563	1:1.34:8.21	01	6600
r < 225	1711	1:1.47:7.87	03	6762
r < 250	1780	1:1.54:7.73	04	6748
r < 275	1798	1:1.57:7.71	04	6630
r < 300	1803	1:1.57:7.71	04	6596
Including the northwestern fields,
Center: α = 0h53m31s, δ = −72°59'157
r < 075	428	1:1.07:20.01	05	4884
r < 100	671	1:1.03:14.01	04	6488
r < 125	927	1:1.03:11.19	04	6784
r < 150	1177	1:1.13:9.59	02	5954
r < 175	1399	1:1.23:8.70	02	6010
r < 200	1568	1:1.30:8.00	01	6487
r < 225	1755	1:1.34:7.22	03	6878
r < 250	1845	1:1.36:6.89	04	7080
r < 275	1892	1:1.34:6.57	04	7370
r < 300	1904	1:1.33:6.47	03	7440

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From the detailed analysis described in the last two paragraphs, it is clear that the quantitative estimates of the structural parameters of the SMC are very much dependent on the data coverage. Though there are differences in the values, we can say that the RR Lyrae distribution in the inner SMC is slightly elongated in the NE–SW direction. The longest axis Z', which is perpendicular to the X'Y' plane (a plane obtained by the counterclockwise rotation of the XY plane with an angle ϕ with respect to the Z-axis), is aligned almost parallel to the line of sight Z-axis. The x, y, and z values of the whole sample of the RRLS are plotted in Figure 14 to get a three-dimensional visualization of the SMC structure. The figure clearly shows the elongation in the Z' ∼ Z axis.

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From the previous sections, we found that the RC stars and RRLS occupy a similar volume of the SMC. Hence we can apply the method of inertia tensor to the RC stars for the estimation of the parameters of the ellipsoidal component of the SMC. First, we apply this method to the (x, y) coordinates of the 553 sub-regions of the SMC. Here each (x, y) pair represents the coordinates of a sub-region. We weighted it using the number of RC stars identified in that region to obtain the axes ratio. We find that the RC distribution in the SMC is elongated with an axes ratio of 1:1.48 and the position angle of the major axis is 553. Now we applied the same procedure to the (x, y, z) system of the RC stars. In the case of RRLS, we took the distance of the individual RRLS with respect to the mean distance as the z coordinate, but in the case of RC stars we only have the mean magnitudes corresponding to the mean distances to the sub-regions. The mean magnitude of a region in the ellipsoidal or nearly spheroidal system is the average of the magnitude of symmetrically distributed RC stars. In order to obtain the real RC distribution we did the following. After subtracting the average extinction, we obtained the magnitude distribution of the RC stars for each sub-region. Using the method explained earlier in Section 3.1, we estimated the peak and the width of the magnitude distribution for each sub-region and eventually the width corresponding to depth. The depth (front to back distance) of the RC stars in a region is the measure of the extent to which the RC stars in that region are distributed. Most of the RC stars in a sub-region are at the mean distance obtained for that sub-region and the remaining are distributed within the depth of the region. We assume the distribution to be symmetric with respect to the mean distance to that sub-region. Initially, we took the I₀ values of the bins within 1σ width and converted them into z-distances. As we have the number of stars in each bin we weighted and applied the inertia tensor analysis to the x, y, and z coordinates.

Earlier we found that the RC and RR Lyrae depth distributions are similar. Also, from Figures 10 and 11 we can see that the RR Lyrae distribution extends up to 3σ–4σ depth. Hence we can take the z component to extend up to higher sigma levels (2σ, 3σ, 3.5σ, 4σ) to understand the real RC distribution in the SMC. We applied the inertia tensor analysis using z components which extend up to 2σ, 3σ, 3.5σ, and 4σ. The parameters obtained are given in Table 3. From the table we can see that the orientation measurements of the ellipsoidal component of the SMC using RC stars, where the z component extends up to 3.5σ, are similar to the values obtained for the RR Lyrae distribution excluding the northwestern fields. As we removed the regions with RC stars less than 400, the northwestern fields are not included in the analysis. The axes ratio and the angle i match well with the ellipsoidal parameters of the SMC estimated using the distribution of the RRLS. The position angle, ϕ, estimated using the RC stars and RRLS is slightly different.

Table 3. Orientation Measurements of the Ellipsoidal Component of the SMC Using the RC Stars

Data	Axes Ratio	i	ϕ
RC stars within 1σ depth	1:1.49:3.08	422	557
RC stars within 2σ depth	1:1.48:5.26	123	554
RC stars within 3σ depth	1:1.48:7.10	068	556
RC stars within 3.5σ depth	1:1.48:7.88	058	555
RC stars within 4σ depth	1:1.48:8.58	049	554

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Based on the analysis described above, we can say that the RC stars in the SMC are distributed in an ellipsoidal system with an axes ratio of 1:1.48:7.88. The position angle, ϕ,is 555. The longest axis Z' which is perpendicular to the X'Y' plane (obtained by the counterclockwise rotation of the XY plane with an angle 555 with respect to the Z-axis) is inclined with the Z-axis with an angle of 058. The inclination of the Z'-axis with the line of sight is very small and we can assume the longest axis to be almost along the line of sight.

The quantities estimated in this analysis strongly depend on the data coverage. In order to compare the axes ratio and the orientation measurements obtained from both populations, it is important to take the sample of both population of stars from the same region. In the case of the RC stars, regions that include the northwestern fields and also regions in the edges of the data set are omitted based on the selection criteria. Such a truncation of the data is not done in the case of the RRLS. Only the RRLS which are possible Galactic objects are removed from the analysis. These stars are not concentrated on a particular region but are scattered. If we remove the RRLS in the northwestern fields and the edges of the data, then the sample of RRLS covers nearly the same region as the RC stars' sample. We used the RRLS within the box of size −1.8 to 1.8 in both the X- and Y-axes. Thus the RRLS in the edges of the data set and in the northwestern fields are nearly removed. Then we estimated the parameters using this sample of the RRLS and the values are given in Table 4. From the table we can see that when the sample of both populations is taken from similar regions of the SMC the estimated parameters match well.

Table 4. Orientation Measurements of the Ellipsoidal Component of the SMC Using the RC Stars and the RRLS in the Same Region

Data	Axes Ratio	i	ϕ
1654 RRLS	1:1.46:8.04	050	583
RC stars within 3.5σ depth	1:1.48:7.88	058	555

Note. Excluding the northwestern fields and the edges of the data.

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We find that the Z' ∼ Z axis is the longest axis in the analysis of both the RRLS and the RC stars. The reason for such a result is mainly the coverage of the SMC. The studied data cover only the central regions thus restricting the coverage in the XY plane. On the other hand, the full Z-direction is sampled. Hence the present view of the SMC based on our axes ratio is like viewing only the central part of a sphere along the Z-axis. Such a perspective will give an elongated Z-axis. This is clear from Figure 14. In order to test whether the z' ∼ z axis is due to the viewing perspective, we have sampled the data similarly along the three axes. The extent of the sample along all three axes can be made equal by taking the data only within spherical radii (ρ) of  2°, 25, and 3° from the center. We took the RRLS stars within these different radii and estimated the structural parameters. The estimated parameters are given in Table 5. An elongation in the NE–SW axis is seen as suggesting that the elongation of the RRLS in the NE–SW is a real feature. The decrease in the relative length of the z'-axis is expected as the selection mentioned above removes relatively more stars along the z'-direction than in the x' and y' directions. From Table 5 we can see that the inclination of the longest axis with the line-of-sight axis decreases from inner to outer radii. The finite XY coverage further restricts analysis of this trend. The very small value of i (given in Tables 2 and 4) estimated from the analysis of all the RRLS, including those at larger z-distances from the center, indicates that the decreasing trend of i is also continued to outer radii. In Table 5, we can also see an increase in ϕ and the relative length of the y'-axis from inner to outer regions. From Section 6.1 and Table 2, we can understand that these two quantities are dependent on the choice of the centroid of the system. So we cannot definitely say anything more about this trend. Another important result to be noted from Table 5 is that even though the z'-axis is the longest of the three, the values of the longest and the second longest (y') axes are comparable. This suggests that when the coverage along all three axes becomes comparable, the structure of the SMC is spheroidal or slightly ellipsoidal.

The above analysis indicates that the XY extent of the SMC up to which stellar populations are studied/detected plays an important role in understanding the actual structure of the SMC and hence the estimation of the structural parameters. The tidal radius estimates give the radial extent to which the stellar populations are bound to the SMC or to which they are expected to be found. Gardiner & Noguchi (1996) estimated the tidal radius of the SMC to be ∼5 kpc. Later, Stanimirović et al. (2004) suggested the tidal radius of the SMC to be 4–9 kpc. The estimations obtained in our study from the radial density and surface density profiles of the RC stars, given in Table 1, suggest the tidal radius to be 7–12 kpc. There are many recent studies which found old and intermediate stellar populations in the outer regions. Noël et al. (2007) found that up to 6.5 kpc from the SMC center, the galaxy is composed of both intermediate-age and old populations without an extended halo. De Propris et al. (2010) estimated the SMC edge in the eastern direction to be around 6 kpc from the survey of red giant stars in 10 fields in the SMC. A very recent photometric survey of the stellar periphery of the SMC by Nidever et al. (2011) found the presence of old and intermediate-age populations up to at least 9 kpc. The above-mentioned observational studies and tidal radius estimates suggest that the full extent of the SMC in the XY plane (2×9 = 18 kpc) is of the order of the front to back distance estimated (14 kpc) along the Z-axis. We found in earlier sections that the inner SMC is slightly elongated in the NE–SW direction. Thus we suggest that the actual structure of the SMC is spheroidal or slightly ellipsoidal. Better estimates of the structural parameters can be obtained with a data set of larger sky coverage. In the following section, we combine previous studies and suggest an evolutionary model for the SMC.