EXTINCTION MAP OF THE SMALL MAGELLANIC CLOUD BASED ON THE SIRIUS AND 6X 2MASS POINT SOURCE CATALOGS

The 6X 2MASS catalog for the SMC contains ∼1.2×10⁶ stars and covers a ∼10° × 10° area around the SMC. Figure 3 shows a J-band star density map measured at m_J ⩽ 17.5 mag.

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The catalog is much deeper than the standard 2MASS catalog, and its limiting magnitudes vary from region to region in the SMC. For instance, a luminosity function of the stars indicates that the catalog is complete down to [m_J, m_H, m_K]≃[16.9,16.3,15.9] mag in the central region of the SMC, while on the outskirts the catalog is deeper and is complete down to [m_J, m_H, m_K] ≃ [17.5,16.9,16.3] mag.

In order to derive a color excess map, we selected stars from the catalog that satisfy the following three criteria: (1) magnitudes brighter than [m_J, m_H, m_K]=[18.0,17.5,17.0] mag; (2) high-quality detections, photometry, and astrometry (i.e., the read flag "rd-flg" in the catalog is either 1, 2, or 3, and the photometry flag "ph-qual" is either A, B, or C for the three bands); and (3) no counterparts among known minor planets (i.e., the minor planet flag "mp-flg" is 0).

The threshold magnitudes we used are higher than the limiting magnitude described above, but we used the above criterion (1) for constructing the color excess maps in order to increase the number of stars and to reduce noise.

For the selected stars, we applied the X percentile method described in Section 2: we placed circular cells with 26 diameter on a 1' grid along equatorial coordinates and calculated the colors J–H and H − K_S at every 5% for X₀ in the range 5 ⩽ X₀ ⩽ 90% with a common X₁ = 95%. The reason for adopting the 26 resolution is to compare our color excess map with the CO map newly obtained using the NANTEN telescope, which will be presented in a subsequent publication (A. Kawamura et al. 2009, in preparation).

We chose a small 15' × 25' region around the position α₂₀₀₀ = 0^h49^m40^s and δ₂₀₀₀ = −71°57'30'' as the reference field to derive the color excess maps. This region was selected as the reference field because it is located in the outskirts of the SMC, where the H i emission is weak. In addition, the region exhibits the bluest mean star colors in the SIRIUS color excess map (described in the next subsection, see Figure 4(b)). We assume E_bg = 0 mag for this region.

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As an example, we display the resulting E(J − H) map in Figure 4(a), which corresponds to $E_{X_m}$ in Equation (4) at X₀ = 80%. In the color excess map, a faint pattern of stripes along declination with constant right ascension can be seen. This is a known systematic error in the photometry of the individual scans taken over many different nights. The effect could be avoided by using the brighter threshold magnitudes of [m_J, m_H, m_K] ⩽ [16.8,16.1,15.3] mag (R. Cutri 2008, private communication).

The SIRIUS catalog covers a smaller area of the SMC than the 6X 2MASS catalog, but it has a higher sensitivity; within the ∼11 deg² coverage, ∼4.3 × 10⁵ stars are recorded in the catalog, and its 10σ limiting magnitudes are [m_J, m_H, m_K] = [18.8,17.8,16.6] mag.

For the SIRIUS catalog, we selected stars that satisfy the following criteria: (1) stars brighter than the above 10σ limiting magnitudes; (2) the "quality flag" representing the shape of the sources is better than "3" (c.f., "1" is the most point-like); and (3) the photometric uncertainties are smaller than 0.217 mag which corresponds to the "ph-qual" flag of C in the 6X 2MASS catalog.

We should note that there is a slight systematic error in photometry in the original SIRIUS catalog. A color map produced following Equation (1) shows a pointing-by-pointing noise pattern of a few times 0.01 mag, which is apparently due to an inappropriate flat field adopted to establish the photometry in one or more bands.

In general, photometric errors of this order are common in star catalogs, but it is not negligible in our study because dark clouds in the SMC are very faint with a color excess of only ∼0.1 mag in E(J − H) or E(H − K_S). We therefore attempted to apply a correction to the catalog. For all of the stars in the catalog (i.e., not only stars in the SMC but also those in the LMC and the Bridge), we checked their (X, Y) pixel coordinates in the detector of the SIRIUS camera which counts 1024 × 1024 pixels in total (Kato et al. 2007, see their Table 2). We then calculated star colors J–H and H − K_S for each star to derive the mean star color distribution as a function of (X, Y). As expected, a bumpy pattern of a few times 0.01 mag was found in the two color distributions, while a flat color distribution would be expected if the flat fields of the three bands were perfect: The mean star color J–H tends to be redder around the position (X, Y) ∼ (420, 500) and it gradually becomes bluer toward the edge of the detector. Similarly, the mean star color H − K_S around the positions (X, Y) ∼ (400, 150) and ∼(400, 1000) are redder than in the outer parts of the detector. The maximum color differences between these (X, Y) positions and the edge of the detector are Δ(J − H) ∼ 0.038 mag and Δ(H − K_S) ∼ 0.025 mag.

To cancel these pixel-dependent systematic errors, we attributed the errors in the J–H pixel map to the J-band photometry and those in the H − K_S map to the K_S band photometry, and modified the J and K_S magnitudes of stars in the original SIRIUS catalog according to the flat-field correction derived above and their detector pixel coordinates (X, Y). Note that our correction should work properly only for the colors (J–H and H − K_S), but not for the magnitudes (m_J, m_H, and m_K) of the individual stars. A new star catalog corrected for the flat-field problem is now being prepared by the authors of the original catalog, which will be available to the public in the near future (D. Kato et al. 2009, private communication).

Using the stars with thus modified photometry, we applied our X percentile method to derive the color excess maps of E(J − H) and E(H − K_S) in the same way as for the 6X 2MASS data. An example of the resulting color excess map is shown in Figure 4(b). We also show the central region of the SMC in Figure 5 on a finer scale.

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In order to estimate the noise level of the color excess maps, we performed a Monte Carlo simulation. First, we composed a histogram of star colors (J–H and H − K_S) found within a 15' radius around individual observed positions, and generated N stars randomly whose color distribution follows the composed color histogram. Here, N is the number of stars actually found in the 26 cells. Following Equations (1) and (2), we calculated the colors using the N stars in the same way as for the color excess maps. We repeated this simulation 100 times for each cell and regarded the standard deviation of the resulting colors as the 1σ noise level of the cells.

The noise level inferred in this manner varies from region to region mainly depending on the observed number of stars N as well as on the X₀ values. In general, the noise level is lower in the central region of the SMC with larger N, and is higher in the outer regions with smaller N. For instance, in the center of the SMC, N ∼ 300 and ∼100 stars are observed for the colors J–H and H − K_S, resulting in the noise levels of σ ∼ 0.01 and ∼0.008 mag over the range 30 ⩽ X₀ ⩽ 80%, respectively. In external regions having N ∼ 50 and ∼15 in J–H and H − K_S, respectively, the noise level for the two colors is σ ∼ 0.025 mag over the same X₀ range.

In addition to the random noise described above, there may be a systematic error arising from the selection of the reference field which we set at the region indicated by the small box in Figure 4. The error would result in an offset in our final color excess maps. In Section 5.1, we attempt to analyze dark clouds detected in the color excess map by separating them into the background component (E⁸⁰_bg in the subsection) and the cloud component (E⁸⁰₀). In the background component, the error should be comparable to the possible offset, while the cloud component should be much less affected. In order to determine the magnitude of the offset, we use some other regions where the mean star colors are relatively blue (e.g., top left and bottom left corners of Figure 5) as the reference field, finding that the star colors in these regions vary within a maximum uncertainty of Δ(J − H) ∼ 0.02 mag and Δ(H − K_S) ∼ 0.01 mag. Our color excess maps may therefore contain an offset of this order.

We further note that there may be an additional systematic error due to the extinction by the Galactic dust lying in the foreground of the SMC. Such errors may not be removed perfectly by subtracting a simple mean star color in the reference field, if the foreground extinction varies significantly over the SMC. The Galactic component of the H i emission observed by Brüns et al. (2005) does vary over the SMC. The error, however, is likely to be much smaller than the ambiguity arising from the choice of the reference field: assuming that the standard Galactic dust-to-gas ratio A_V/N_H (Bohlin et al. 1978) and reddening law (Cardelli et al. 1989) of R_V = 3.1 are valid for the Galactic H i emission, we estimate the error in our color excess maps to be ΔE(J − H) ∼ ΔE(H − K_S) ∼ 0.003 mag at most over the region shown in Figure 5.

In this subsection, we compare the E⁸⁰(J − H) map with our model calculation to infer the cloud locations in the star distribution along the LOS.

As illustrated in Figure 1, the model parameters are: E_bg, the background color excess; ϕ(c), the distribution function of the intrinsic star colors; E₀, the color excess by the cloud; Z, the cloud position; and T, the cloud thickness. Among these, we first estimate E_bg and ϕ(c) independently, and then fit the observed X₀ versus $E_{X_0}$ and X₀ versus $E_{X_m}$ diagrams (see Figure 2) with the rest of the parameters.

In order to estimate E_bg, we attempted to separate the observed color excess E⁸⁰(J − H) into background and cloud components. For this, we tentatively assumed that the regions with E⁸⁰ < 0.05 mag (Figure 5) contain only the background component and the regions with E⁸⁰ ⩾ 0.05 mag comprise both cloud and background components. We applied a mask to the regions with E⁸⁰ ⩾ 0.05 mag and then estimated the background level in the masked regions using the IDL routine TRIGRID, and subtracted them from the original map to deduce the cloud components. In Table 2, we list the background and cloud components derived at the peak positions of the selected clouds, denoted as E⁸⁰_bg and E⁸⁰₀, respectively, in the table. Note that E⁸⁰_bg + E⁸⁰₀ corresponds to E⁸⁰ in Table 1. The relation between E⁸⁰_bg and the total background E_bg is almost linear, depending on the distribution function of the intrinsic star color ϕ(c) (c = J − H in this case, see Section 3). Using ϕ(c) observed in the reference field, where we assume E_bg = 0 (see Figure 4(b)), we deduce E_bg from E⁸⁰_bg. The resulting E_bg values are listed in the fourth column of Table 2. Note that the inferred total background E_bg is approximately twice E⁸⁰_bg.

The distribution function ϕ(c) should vary slightly from region to region in the SMC, mainly depending on the population of giants and dwarfs (e.g., Bessell & Brett 1988), and can also be different inside the SMC to the reference field located in the outskirts. We therefore also attempted to derive ϕ(c) at Cloud B, the densest part in Region 1 (see Figure 6). We first measured the color distribution of stars falling within 15' of the peak position of Cloud B and then deduced ϕ(c), assuming that the distribution is reddened by the background E_bg estimated above. The derived ϕ(c) is similar to that observed in the reference field. In the following analysis, we tested both forms of ϕ(c), measured in the reference field and at Cloud B, and found that the results are in good agreement.

Using E_bg and ϕ(c) estimated in this manner, we fitted the behavior of $E_{X_0}$ and $E_{X_m}$ as a function of X₀ as observed at the peak positions of Clouds A–J. In the model described in Figure 1, we set the estimated E_bg and randomly distributed 10⁵ stars whose colors follow the ϕ(c) at Cloud B. We then calculated $E_{X_0}$ and $E_{X_m}$ for different sets of parameters [E₀, T, Z], varying E₀ by 0.001 mag and T and Z by 2.5%, and compared the resulting $E_{X_0}$ and $E_{X_m}$ with those observed to search for the set of parameters that minimize the χ². Results are shown in Figure 9. We summarize the best model parameters [E₀, T, Z] in Table 2.

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In Figure 9, the $E_{X_0}$ and $E_{X_m}$ diagrams at Clouds A and G are flat, indicating that the clouds at these positions are located almost in front of the star distribution in the SMC and should be completely detected in our E⁸⁰ map. On the other hand, the diagrams at Clouds I and J in Region 3 are rather complex. In particular, the diagram measured at Cloud I cannot be fitted by our simple model. This is most likely due to two or more cloud components lying on the same LOS, which is not taken into account in our model. In fact, it is interesting to note that recent CO observations made with the NANTEN telescope have revealed two distinct velocity components toward Cloud I: one is a component at V_LSR = 160 km s⁻¹ extending over the cloud surface, and the other is a weak component at V_LSR = 150 km s⁻¹ detected only around the peak position of Cloud I (A. Kawamura et al. 2009, in preparation).

For the other clouds, the diagrams increase with X₀ and some of them show concave $E_{X_m}$ curves, which is a characteristic feature of clouds whose far sides are located toward the back of the star distribution (i.e., Z + T>50%, see Figure 2(c)). The color excess of such clouds can be highly underestimated in our E⁸⁰ map. In fact, a comparison with the model calculations indicates that our values of E⁸⁰₀ underestimate E₀ by a factor of ∼2 (see Table 2).

We applied the above fitting procedure to the regions with E⁸⁰ ⩾ 0.05 mag in Region 1 to estimate how much we may underestimate the true color excess in our E⁸⁰ map. The resulting E₀ map is shown in Figure 10, together with Z, T, and E_bg maps. Although the overall distribution of the E₀ is similar to that of E⁸⁰ map, we found that E₀ is larger than E⁸⁰₀ by a factor of ∼2 on average, as expected. In addition, contrasts of the clouds are different in the E₀ map, depending on the cloud parameters Z and T. For example, the color excess at Cloud A is unchanged and remains E⁸⁰₀ = E₀=0.07 mag because it is located at Z ∼ 0 (Figure 10(b) and (c)). On the other hand, the color excess measured at Cloud F, which has larger Z, increases by a factor of 1.8 from E⁸⁰₀=0.121 mag to E₀=0.220 mag (see Table 2), indicating that Cloud F is about twice as dense as observed in the E⁸⁰ map.

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To conclude, both the background and cloud components observed in our E⁸⁰(J − H) map are likely to underestimate the true color excess by a factor of ∼2 on average. However, the error (∼2) is rather uncertain, because the cloud parameter maps in Figure 10 are still tentative and may be erroneous. This is because it is difficult to determine the model parameters E₀, Z, and T that minimize χ² in the case of clouds with Z + T>50% showing concave $E_{\rm X_m}$ curves. In general, E₀ tends to be overestimated for such $E_{\rm X_m}$ curves, and the error in E₀ is larger for clouds with fainter color excess (e.g., E⁸⁰ < 0.05 mag). Our E₀ map can therefore be regarded as the upper limit to the actual E₀ values, while E⁸⁰₀ should be considered as a lower limit.

As seen in Figure 9, clouds in Region 1 have rather simple $E_{X_0}$ and $E_{X_m}$ curves and are fitted well by our model. Four of them, Clouds B, D, E, and F, are also well detected in CO using the NANTEN telescope and their virial masses M_VIR have been estimated by Mizuno et al. (2001, see their Figure 4) as M_VIR ∝ R(ΔV)², where R and ΔV are the cloud radius and the observed CO line width. We compare the virial masses with our E(J − H) data to derive the dust-to-gas ratio A_V/N_H of these clouds, which can be expressed as,

$$\begin{equation} \frac{{A_V }}{{N_{\rm H}}} = \alpha \left({\frac{{\int {E(J - H)ds} }}{{\rm mag\; arcmin ^2 }}} \right)\left({\frac{{M_{\rm VIR} }}{{10^{5}\; M_\odot }}} \right)^{ - 1}, \end{equation} \tag{ 5 }$$

where ∫E(J − H)ds is the color excess integrated over the cloud surface, and α is a coefficient. We assume the cloud surface to be the area defined by the lowest contours (0.45 K km s⁻¹) in the velocity-integrated CO map presented by Mizuno et al. (2001, see their Figure 3). The coefficient α is taken to be 2.67 × 10⁻²²mag H^-1 cm², assuming the relation A_V = 10.9E(J − H) that is derived from the reddening law determined by Cardelli et al. (1989) for R_V = 3.1. We note that the reddening law in the SMC is known to be similar to that in the MW in the VIS to near-infrared wavelengths, despite their significant difference in the ultraviolet (Gordon et al. 2003). Moreover, it was recently found by Zagury (2007) that the extinction curves in the SMC, MW, and LMC obey the same reddening law, similar to that found by Cardelli et al. (1989).

For E(J − H) in Equation (5), we used the values of the cloud components (E⁸⁰₀ and E₀) described in the previous subsection, since CO emission is expected to arise from the dense clouds but not from the diffuse background components (E⁸⁰_bg and E_bg). Resulting values of A_V/N_H for the individual clouds are summarized in Table 3, as well as the surface areas and virial masses measured by Mizuno et al. (2001). The minimum estimate of A_V/N_H based on E⁸⁰₀ varies around ∼1 × 10⁻²²mag H^-1 cm², and the maximum estimate using E₀ ranges from ∼2 ×10⁻²² to ∼ 3 ×10⁻²² mag H^-1 cm². The mean value of the four clouds, weighted by M_VIR, is ∼1.5 ×10⁻²², which is about a half the value of A_V/N_H measured using the same method for the clouds near the 30 Dor region in the LMC (e.g., ∼3×10⁻²² for LMC-154; Dobashi et al. 2008), and about one third of the standard Galactic value (5.34 × 10⁻²² for R_V = 3.1; Bohlin et al. 1978). Our average value of A_V/N_H is about twice the value measured for a limited number of sightlines in the SMC by Gordon et al. (2003, 7.59 × 10⁻²³), which is closer to our lower limit (∼1×10⁻²²). We should note that the A_V/N_H value measured by Gordon et al. takes into account only the neutral hydrogen (H i) but not hydrogen molecules (H₂), which may increase the difference between their and our estimates for A_V/N_H. The effect should be small, however, since the total extinction toward their sample stars is rather low (E(B − V) ≲ 0.2 mag) and therefore H₂ may not be abundant along the LOS toward their sample stars.

Instead, we think that the difference between our value of A_V/N_H and the value determined by Gordon et al. is probably due to the variations of A_V/N_H within the SMC. Variations in A_V/N_H of a factor of ∼2 are observed also among clouds in the LMC (Dobashi et al. 2008; Bernard et al. 2008) and among the Galactic clouds within 1 kpc from the Sun (F. Egusa et al. 2009, in preparation). In addition, the rather large uncertainty in our analysis (see Section 5.1) may also be responsible for the difference. However, it is also worth noting that the virial mass derived as M_VIR ∝ R(ΔV)² (Mizuno et al. 2001) may underestimate the true cloud mass by a factor of ∼2 in the case of giant molecular clouds in the SMC, not only because of the ambiguity in determining the cloud radius but also because of possible cloud support by the magnetic field (Bot et al. 2007). If this is the case for the M_VIR values that we adopted in our analysis, then our estimate for A_V/N_H is very close to that found by Gordon et al.