The structure of the Galactic halo: SDSS versus SuperCOSMOS

Abstract

1 INTRODUCTION

The modern use of star counts in the study Galactic structure began with Bahcall & Soneira (1980). In Bahcall's standard model, the structure of the Galaxy is assumed to be an exponential disc and a de Vaucouleurs spheriodal halo. A lot of work has been done to constrain and examine this theoretical model, as summarized in Xu, Deng & Hu (2006) (XDH06 hereafter), most of them using only a small sky area. The global structure of the Galactic halo can only be inferred by different observations of small sky areas with different magnitude limits, photometric passbands and different original observational goals. Sloan Digital Sky Survey (SDSS) provides us with the opportunity to examine the large-scale structure of the Galaxy from optical photometry thanks to its deep photometry and large sky coverage. From SDSS observational data it is clear that the stellar halo of the Galaxy is asymmetric, contrary to what has been generally assumed. From colour star counts it is obvious that the asymmetric projected stellar number density is produced by halo stars. There are two possible explanations for such a halo structure. First, that there are some large-scale star streams embedded in the axisymmetric smooth structure of the Galactic halo (Jurić et al. 2005). Secondly, that the galactic stellar halo is intrinsically not axisymmetric (Newberg & Yanny 2005, XDH 2006). Based on the data we have so far, some combination of the two might also be possible. In Paper I, we tested the second option and fitted the observational data with triaxial halo. The triaxial halo model fits fairly well the projected number density near the northern cap of the Galactic stellar halo. However, in some sky areas, the triaxial halo model cannot reproduce the actual star counts. The multisolutions that are intrinsic in fitting the observational data with the triaxial halo model make the interpretation of the data somewhat difficult. On the other hand, the alternative option where the asymmetry of the halo is caused by large-scale star streams also has some problems, even if the overwhelmingly large Virgo overdensity that covers nearly a quarter of the Northern hemisphere can be explained by a large-scale star stream. The observed underdensity near Ursa Major with respect to the assumed axisymmetric halo still challenges such a picture. Nevertheless, the conservation of such a huge structure in the gravitational well of the Galaxy certainly needs to be verified. Martínez-Delgado et al. (2007) show that the Virgo overdensity can be reproduced by the dynamical evolution of the Sgr stream. Assuming a certain structure of the stellar halo (an oblate ellipse), their numerical simulation can predict an overdensity on a few hundred square degree scale.

Although it is the most advanced photometric sky survey in terms of depth and data quality, SDSS does not have good data coverage near the southern cap of the Galaxy which is, of course, crucial in understanding the overall structure of the stellar halo. Limited to the sky coverage of SDSS photometry data base, it is probably premature to draw a firm conclusion on the stellar halo structure. Assuming that the Galactic stellar halo is non-axisymmetric, and can be described by a triaxial model, there must be some corresponding evidence in the Southern hemisphere similar to what is found in XDH06 for the northern cap. In the axisymmetric halo model, the maximum star counts should be at longitude l= 0° (due to the location of the observer). In the case of a triaxial halo, however, the maximum projected number density also depends on a certain parameters of the halo including azimuth angle, axial ratios and the limiting magnitudes of the observations. In the simplest case, the plane defined by the primary and the middle axis of the triaxial halo stays in the Galactic disc, the azimuth angle is only related with the angle between the primary axis and the direction of the Galactic Centre from the Sun, therefore the expected star counts and asymmetric ratio of northern and southern sky ought to be mirror symmetric with respect to the Galactic plane, that is, what was found in the north cap should also be found in the south under such a halo model. If the two planes do not overlap, the situation will be more complicated, but similar results should still hold.

It is also interesting to examine archived sky survey data that have the good coverage and reasonable quality in the southern Galactic halo: the photographic photometry of SuperCOSMOS is ideal for this purpose. As reviewed by Hambly et al. (2001a), photographic observations for the Galaxy started in the late nineteenth century. In the 1930s, the development of Schmidt telescopes with wide fields of view further advanced photographic surveys. The 1.2-m Palomar Oschin telescope, 1.0-m ESO telescope and 1.2-m United Kingdom Schmidt Telescope (UKST) finished the photographic whole sky survey in the last century; such surveys form a legacy library for examining the structure of the Galaxy. In the late twentieth century, the photographic plates were eventually digitized using microdensitometry and digital electronics machines. There are several major programmes to digitize the photographic plates, of which SuperCOSMOS is one. In Hambly et al. (2001a), a general overview of these programmes (APM, APS, COSMOS, DSS, PMM, SuperCOSMOS) is presented. The digitized photographic sky survey of SuperCOSMOS provides a catalogue of three bands, namely, blue (B_J), red (R_F) and near-infrared (I_VN), which have deeper detection limit for the same detection completeness compared to other similar survey programmes (see fig. 2 of Hambly et al. 2001a). We therefore adopt the SuperCOSMOS data archive for our present study.

Figure 2

The colour transformation between SuperCOSMOS B_J, R_F bands and SDSS g, r bands. The obvious offset between the two systems infers that a systematic correction is needed. See text for details.

Open in new tabDownload slide

In Section 2, the observational data are described and the stellar source cross-identification between the SuperCOSMOS data and the SDSS data is carried out, and the viability of using SuperCOSMOS data to study the structure of southern Galactic stellar halo is discussed. In Section 3, downgraded SDSS and SuperCOSMOS observational star count results are presented. In Section 4, the model fits to the SuperCOSMOS star counts are introduced. In Section 5, the SuperCOSMOS observational data and theoretical models are compared, and SuperCOSMOS southern sky star counts and SDSS downgraded northern sky star counts also compared and analysed. In Section 6, the result of star counts is summarized.

2 THE OBSERVATIONAL DATA

2.1 SuperCOSMOS photometric data

The SuperCOSMOS Sky Survey is a digitized photography sky survey. It is described in detail in a series of papers by Hambly and collaborators (Hambly et al. 2001a; Hambly, Irwin & MacGillivray 2001b; Hambly, Davenhall & Irwin 2001c).

The SuperCOSMOS photography atlas of the SuperCOSMOS sky survey includes blue (B_J), red (R_F) and near-infrared (I_VN) passband photometric surveys carried out by UK Schmidt survey for ⁠, ESO Red Survey of ⁠, and Palomar surveys including, POSS-I Red Survey for ⁠, POSS-II Blue Survey for ⁠, POSS-II Red Survey for ⁠. Data of B_J band has about 90 per cent detection completeness from 16.5 to 20.5 mag, and that of R_F band has same completeness from 16.5 to 19.5 mag. The photometric data have a magnitude error of 0.3 mag, but colour is accurate to about 0.16 mag (Hambly et al. 2001a).

There are two interface applications of SuperCOSMOS: the SuperCOSMOS Sky Survey (http://www-wfau.roe.ac.uk/sss, SSS hereafter) and the SuperCOSMOS Sky Archive (http://surveys.roe.ac.uk/ssa, SSA hereafter).

Images of small sky areas and catalogues from the SuperCOSMOS sky survey can be downloaded. We thank the SuperCOSMOS working group who made all the data available to the community. The SSA only includes photometric data from UKST and ESO. As made clear by Hambly et al. (2001a), although the entire sky is digitized, the data in this archive are released progressively. The total amount of data is enormous, only F-type stars (selected by 0.504 ≤B_J−R_F≤ 0.8236) from 20.4 to 20.415 mag are adopted to show the sky coverage which is in the upper panel of Fig. 1. The survey covers most of the high-latitude southern sky, and a little of the Northern hemisphere. The clump at ⁠) is the Small Magellanic Cloud, and the clump at ⁠) is the Large Magellanic Cloud.

Figure 1

Upper panel: the sky coverage of the SuperCOSMOS archive in Galactic coordinates as shown by F-type stars in the B_J band from 20.4 to 20.415 mag. Lower panel: Lambert projection of the sky coverage of SuperCOSMOS (shown by tiny dots of the same selection of stars in the upper panel), and the sky areas selected for this study (squares).

Open in new tabDownload slide

Except for the difference in sky coverage, the two interface applications use different selection standards. The SSA SQL selection is much more configurable (private communication by email with Hambly) than that of the SSS. For example, there are four kinds of B magnitude in the SSA, namely, classMagB (B-band magnitude selected by B image class), gCorMagB (B-band magnitude assuming the object is a galaxy), sCorMagB (B-band magnitude assuming the object is a star), classB (image classification from B-band detection). The most appropriate attribute for point sources is sCorMag, while the most possible class of an object from all three bands is provided by parameter ‘meanclass’. The SSS only includes selection parameters applied to the primary passband, corresponding to classB of the SSA in the example. Our aim is to count the stars in each selected sky area, and using classB will lose some stars due to not synthesizing information of all the three bands. This will influence the result of star counts seriously. Thus, ‘sCorMagB’ of the SSA data is selected to carry out the study and the ‘meanclass’ is limited to 2 (star label). Because the SSA only covers limited sky areas of high-latitude Northern Galactic hemisphere (upper panel of Fig. 1) we cannot directly compare SuperCOSMOS star counts of northern sky with those of SDSS.

The SSA includes R_F-band data from both UKST and ESO. However, data from the R_F band of UKST is deeper than that of ESO. Therefore, only UKST is adopted. The detailed instrumental specifications of UKST can be found in Cannon (1984), the main parameters of the survey telescope and instruments are listed in Table 1.

As demonstrated in the upper panel of Fig. 1, the UKST atlas of SuperCOSMOS covers most of the high Galactic latitude Southern hemisphere. The structure of the Galactic halo near the southern cap can be studied using a stellar photometry catalogue selected in a similar way as we did for the northern sky in XDH06, shown here in the lower panel of Fig. 1. The selected sky area for this work is shown in lower panel of Fig. 1, the Lambert projection of Southern hemisphere. Each of the squares represents a rectangular sky area of about 2 × 2 deg². Some of the selected sky areas may be trimmed if sitting near the survey's edge, or the region is masked by contaminants such as saturated bright stars, or clumps such as the dwarf galaxy IC1613 in (130°, −60°). The first group of sky areas are along a circle of b=−60°, equally spaced by 10°. The other 12 groups are a selection of sky areas along longitudinal directions equally spaced by 30°. At a given longitude, the sky areas are selected by a step of 5°. This selection of sky areas can evenly cover the South Galactic Cap, so that the global structure of the halo near the southern pole can be examined.

2.2 Cross-checking of SuperCOSMOS and SDSS data sets

In our previous work (XDH06), SDSS data are used to study the structure of the Galactic stellar halo near the North Galactic Pole, The SDSS catalogue providing a uniform and accurate photometric data set. The five broad-band filters, u, g, r, i, z are 95 per cent complete to 22.0, 22.2, 22.2, 21.3, 20.5 mag, respectively, and the uncertainty in the photometry is about 3 per cent at g= 19 mag (Chen et al. 2001).

Compared to the high-quality photometry data of SDSS, the SuperCOSMOS data have a narrower dynamic range, lower magnitude limit and larger photometric error, due to photographic photometry. To evaluate any uncertainties due to misclassifications and the relatively less accurate photometry of SuperCOMOS, a comparison in areas common to both surveys is needed.

The photometric calibration between SDSS and SuperCOSMOS has been made available by the 2dF Galaxy Redshift Survey (2dFGRS) Final Data Release Photometric Calibration which defines a set of colour equations in its final data.1 The B_J band is correlated with SDSS g and r band, B_J= 0.15 + 0.13 × (g−r), while R_F is very similar to SDSS r band, R_F=r− 0.13. The results of such colour calibration are shown in Fig. 2.

The two small sky areas with superpositions of SDSS and SuperCOSMOS surveys in both the northern and southern sky are chosen to examine the colour equations and the classification of SuperCOSMOS objects. The northern area is located at (l, b) = (280°, 60°) with 2 × 2-deg² field of view (FOV), the southern area is at (l, b) = (62°, −59°) with 1 × 4-deg² FOV. The equinox of SuperCOSMOS data associated with the photometric image library is J2000.0. The position accuracy of SuperCOSMOS is ±0.2 arcsec at B_J= 19 mag, R_F= 18 mag, ±0.3 arcsec at B_J= 22 mag,R_F= 21 mag (Hambly et al. 2001c). Taking into account proper motion, cross-identification is carried out between SDSS and SuperCOSMOS in the two superimposed sky areas in an identification criterion box of 0.3 mag and 10 arcsec. In such a box, multiple sources can be present, the pair of stars with the nearest coordinates and magnitudes are identified as the same source. We take the SDSS data as the ‘true’ values of both position and magnitude. Based on the matched star list in the two areas, uncertainties in the magnitude of SuperCOSMOS photometry for each object can be measured. The systematic error calculated this way infers the error of the colour equations from 2dFGRS calibration; while the scatter can be used to measure the error in SuperCOSMOS photometric data. Fitting the systematic error with a second-order polynomial, the colour equations are refined. Using the modified colour equations, B_J and R_F magnitude of SDSS data is defined as B_JSDSS=g+ 0.15 + 0.13 × (g−r) +▵ mod, R_FSDSS=r− 0.13 +▵mod. Iterating the cross-identification procedure reduces the systematic error. The error in the SuperCOSMOS data in the B_J band is found to be ⁠, and that in ⁠. The variance of the errors as functions of magnitude is obtained from fitting the scatter with a Gaussian.

After such modification, and repeating the cross-identification, the source matching ratios between the two surveys are improved. In the end, the SuperCOSMOS data match that of SDSS in the B_J-band magnitude limits by 92–93 per cent in a 10-arcsec and 0.3-mag box. For the R_F band, the matching ratio can be raised to 85 per cent or larger in the 16.5 –19.5 mag range. The matching ratio of the R_F band is not as good as that of the B_J band, this is likely due to the lower sensitivity in the R_F band. A 85 per cent is still lower than the intrinsic completeness estimated for different surveys in the SuperCOSMOS atlas (see fig. 12b of Hambly et al. 2001b). This is possibly caused by the brighter magnitude limit of the SSA compared to that of SDSS as the bright stars are saturated, therefore influencing more neighbours.

Figure 12

The difference of projected surface number densities between the sky areas along the circles in the north (b= 60°) and south (b=−60°). Stars are selected with 16.5 mag < B_JSDSS, B_J < 20.5 mag, 16.5 mag < R_FSDSS, R_F < 19.5 mag. Upper panel: data points are constrained by 0 < B_JSDSS−R_FSDSS, B_J−R_F < 1.6 which roughly represents the halo population. Lower panel: data points are constrained with 1.6 < B_JSDSS−R_FSDSS, B_J−R_F < 3.0 which roughly represents the disc population.

Open in new tabDownload slide

In the upper and lower panel of Fig. 3, the contours in the colour–colour diagram represent the SuperCOSMOS data in the three bands that are cross-identified in the SDSS data. Black points overplotted on the contours are the matched stellar sources, while the crosses represent SuperCOSMOS sources which are unmatched.

Figure 3

The cross-identification between the SuperCOSMOS and SDSS data sets in colour–colour space. The contour and the black points on the colour–colour diagram represent the B_J-band matched sources in the same overlapping sky areas, limited in 0.3 mag in magnitude and 10 arcsec in angular distance box and a magnitude interval of 16.5 mag < B_J, B_JSDSS < 20.5 mag. The crosses represent the unmatched sources. Upper panel: the northern overlapped sky area around (280°, 60°); lower panel: the southern one around (62°, −59°).

Open in new tabDownload slide

3 OBSERVATIONAL STAR COUNTS

3.1 Star counts from downgraded SDSS data

The examination of halo structure through star counts depends critically on the depth of the photometry. The SSA has a narrower dynamic range and shallower detection limit than SDSS. A test is carried out to check if the asymmetric structure found in XDH06 is still present with the shallower limit of SSA data. The data used in XDH06 are downgraded by applying the SuperCOSMOS magnitude limits, photometric errors of SuperCOSMOS are also added to the SDSS data. A Monte Carlo method is used to reproduce the photometric errors as of SuperCOSMOS (Rockosi, private communication). Gaussian errors similar in size to those of the SuperCOSMOS data are added to the magnitude of each star before measuring the star counts. We find that the results of XDN06 are recovered, with the average fluctuation raised only by about 3.7 per cent. After transforming into the SuperCOSMOS system, the errors are 16.5 < B_JSDSS < 20.5 mag and 16.5 < R_FSDSS < 19.5 mag, respectively.

Figs 4 and 5 show the star counts from the SDSS data with same sky areas as in XDH06 but downgraded to the SuperCOSMOS magnitude limits, for B_JSDSS and R_FSDSS, respectively. From the present SDSS public data release, the sky area l= 210° is now added. Panel (a) is for star counts in sky areas along the b= 60° circle. Panels (b)–(f) are for star counts of sky areas along the selected longitudinal directions paired by mirror symmetry on the both sides of the l= 0° meridian. The asymmetric structure still appears clearly with the magnitude limit of the downgraded SDSS data (especially in Fig. 4a). The asymmetric structure is not so prominent as with the original SDSS magnitude limits, but we can still see that the star counts in l∈[180°, 360°] are systematically higher than in l∈[0°, 180°]. The largest asymmetry of star counts appear in panels (b), (c) and (d). In panels (e) and (f), the errors are so large at the downgraded limits that the asymmetric differences between sky areas found in XDH06 are only marginally visible. As in Fig. 4, Fig. 5 shows the results of star counts from the R_FSDSS data. Again, the most prominent excess over mirror symmetry is found in panels (b), (c) and (d). However, the R_FSDSS-band magnitude limit is fainter than that of the B_JSDSS band, which leads to weaker features of asymmetry than Fig. 4. Tables 2 and 3 describe the asymmetric ratio and its uncertainty in the downgraded SDSS data. Columns 1–4 are the Galactic coordinates (l and b), counted numbers and the corresponding errors for sky areas with l≤ 180°, and Columns 5–8 are the same quantities for sky areas on the other side of the l= 0° meridian. Comparison is between sky areas paired with mirror symmetry with respect to the l= 0° meridian. The asymmetric ratios are defined by: asymmetric ratio = number density₂− number density₁)/(number density₁) × 100  per cent which are given in Column 9; Column 10 gives the uncertainties in the ratios which are inferred from the error of the number densities (Tables 4 and 5 all have the same entries, but for different data). The asymmetric ratios measured from downgraded SDSS data are all positive with one exception which is very near to zero, this means that all the sky areas in l∈[180°, 360°] have higher projected number densities than those in l∈[0°, 180°]. The largest asymmetric ratio is 16.9 ± 6.3  per cent in the B_JSDSS band and 13.5 ± 6.7  per cent in the R_FSDSS band.

Figure 4

Projected surface number density of SDSS data in the same selected sky areas as in XDH06, but downgraded to 16.5 < B_JSDSS < 20.5 mag. Panel (a) is for the selected areas along a circle of b= 60°, the horizontal axis is the Galactic longitude in degrees; while the others are for the ones along different paired longitudinal directions, with the longitudes indicated in the inlet of each panel, all the horizontal axis in panels (b)–(f) are the Galactic latitude in degrees.

Open in new tabDownload slide

Figure 5

The same as Fig. 4 but for the projected surface number density of SDSS data in R_JSDSS downgraded to 16.5 < R_FSDSS < 19.5 mag.

Open in new tabDownload slide

Therefore if there are similar levels of asymmetric structure in the southern sky, they should be visible even with the SuperCOSMOS magnitude limit.

3.2 South Galactic Cap: star counts from SuperCOSMOS data

In XDH06, star counts from SDSS data show a prominent asymmetric structure in the Northern Galactic hemisphere through comparing the projected number densities of sky area pairs with mirror symmetry on both sides of the l= 0° meridian. We will use the same method to examine the structure of stellar halo in the southern sky, in particular to check whether the halo structure has the same features or is different from its northern counterpart.

Star counts for southern sky from the SuperCOSMOS data are shown in Figs 6 and 7. Star counts in each sky area are plotted using triangles and squares. Panel (a) shows the results of star counts for the selected sky areas along a circle of b=−60°. Panels (b)–(f) are for sky areas along the longitudinal directions, also paired with mirror symmetry on the both side of the l= 0° meridian. Each of the sky areas is divided into four subfields to account for the fluctuation of star counts over the average value of the area. The fluctuations calculated for all sky areas this way are used as error bars in the plots. The error bars actually measure the intrinsic fluctuations of the projected number density, and the uncertainties in classification and photometry. The average error of star counts will be discussed in Section 6.

Figure 6

The same as Fig. 4, but for the fitting of the surface number density counted from SuperCOSMOS B_J-band data. The solid and dashed lines are the theoretical predictions, while the diamonds and triangles show the observational data.

Open in new tabDownload slide

Figure 7

The same as Fig. 6, but for R_F-band results.

Open in new tabDownload slide

Dividing the southern cap into two halves by the l= 0°, 180° meridian, the data for both B_J and R_F bands show that the structures of the two halves are basically symmetric within statistical errors. This is clearly shown in panels (b)–(f) of Figs 6 and 7.

The B_J-band data show smaller error bars and obvious smoother structure in the projected number density distribution than the R_F band data do. Sizable fluctuations over an axisymmetric structure do exist in the B_J-band data. In two pairs of data, that is, (150°, −60°) and (210°, −60°) and (90°, −60°) and (270°, −60°), the projected number density at l > 180° side is higher than the other side (l < 180°). While the pair of (60°, −70°) and (300°, −70°) shows a reversed excess.

The star counts from the R_F band have a larger scatter than that of the B_J band. The R_F-band data also have less coincidence in classification with SDSS data than the B_J-band data. Moreover, its limiting magnitude is shallower than the B_J band by about 1 mag. For example, for F0-type stars, the distance limits given by the B_J band are from 5.23 to 32.98 kpc, while those defined by the R_F band are from 6.46 to 25.72 kpc; and for F8-type stars, they are 2.68–16.89 kpc for the B_J band, and 3.79–15.09 kpc for the R_F band. Selecting redder stars from a shallower box in the R_F band, star counts show larger deviations.

In both the B_J and R_F bands, there is an odd data point at (130°, −60°), which has a projected number density obviously lower than its neighbour sky areas. Because this sky area is near the edge of the survey, it is very likely that this is a boundary effect.

Table 4 (for B_J) and Table 5 (for R_F) list the projected number densities and their corresponding errors, and the asymmetric ratio measured in the SuperCOSMOS data. Comparing the uniform positive asymmetric ratios in the downgraded SDSS data (Tables 2 and 3), the values given by SuperCOSMOS are quite irregular, with apparently random positive and negative values.

4 THE THEORETICAL MODEL

From SuperCOSMOS star counts there is no obvious asymmetric structure in the southern halo. A theoretical axisymmetric halo model is therefore adopted here. Because the R_F-band data are shallower and less consistent with SDSS data, only B_J-band data are used to constrain the model parameters.

From the Bahcall standard model (Bahcall & Soneira 1980), the projected number density in a certain apparent magnitude interval along a fixed direction can be described by the integral of a density profile and a luminosity function.

1

where m₁, m₂ are the limits of the given magnitude interval, R is the heliocentric distance of a star, ρ is the density profile of each stellar population and φ(M) is the luminosity function of the population.

For the thin and thick disc components, the density profile is assumed to be exponential,

2

where |z| is absolute value of height of a star above the Galactic plane, x is distance between the Galactic Centre and the projected point of that star on the Galactic plane, H is the scaleheight and h is the scalelength of the exponential disc.

The power-law halo density profile in Reid (1993) is adopted,

3

where r₀ is the distance from the Sun to the Galactic Centre, and a₀= 1000 is a normalization constant. r is the distance from the star to the Galactic Centre. as in XDH06. x, y, z define the position vector in three axes of coordinate frame adopted in Paper I. p, q are the axial ratios of the middle axis and shortest axis to the major axis, respectively. The triaxial halo model naturally degenerates to asymmetric halo model when p= 1. The luminosity functions of the halo, thick and thin disc components in B_J, R_F bands are transformed from the luminosity functions of Robin & Crézé (1986), with B_J=B− 0.304 × (B−V) and R_F=R+ 0.163 × (V−R) as provided by the photometric calibration of the final data release of 2dFGRS. Given the R_F luminosity function, star counts in the R_F band can also be obtained, but these are not used for model fitting for the reasons given above.

The three-dimensional extinction model of the Milky Way derived from COBE observations is adopted four our model. Directly correcting the observational data for extinction is not possible due to the lack of distance information for individual stars (Drimmel, Cabrera-Lavers & López-Corredoira 2003); however, we solve this problem by applying COBE extinction data to the theoretical model.

Using equation (1), the projected number density of each sky area can be obtained. To reveal the distribution of star counts in apparent magnitude, the B_J-band magnitude is divided into eight bins (16.5 –20.5 mag, in steps of 0.5 mag), the number density in each magnitude bin is then calculated. Constraining star counts in B_J-band magnitude limits of 16.5 –20.5 mag, a non-negligible number of stars have no corresponding R_F-band data, therefore colour counts need to be treated with special care, and that will be discussed later in Section 5.2.

As a continuation of XDH06, the present work is focused on halo structure near the southern cap of the Galaxy. For a better comparison between the present work and XDH06, the parameters of the thin and thick discs are fixed with the values used in XDH06, which were taken from Chen et al. (2001). Only the halo parameters are adjusted to fit the SuperCOSMOS observations, the scope of parameters is listed in Table 6. In an axisymmetric halo model there are only two parameters: n is power-law index of halo density profile, and q is the axial ratio z/x.

A χ² minimization is adopted to compare the theoretical results and the observational data sets (Press et al. 1992). For a non-Gaussian distribution of discrete data, Pearson's χ² is used,

4

The meanings of all the symbols are described in XDH06. χ² is calculated in order to evaluate the similarity between the theoretical projected number density and the observational data. χ²_bin describes the difference between the distribution of the theoretical star counts in apparent magnitude bins and that of observations for each sky area. is the average value of χ²_bin in all sky areas.

5 RESULTS AND ANALYSIS

5.1 Fitting SuperCOSMOS star counts with the axisymmetric model

The theoretical projected surface number densities are calculated using the axisymmetric model described in Section 5, with extinction included. The theoretical model that best fits the SuperCOSMOS observational data in both B_J and R_F bands is shown in Figs 6 and 7 as the solid and dashed lines, respectively . The model parameters are n= 2.8, q= 0.7. This is one of the best-fitting models in the provided parameter space. An axisymmetric model can fit SuperCOSMOS data reasonably well within the statistical error bars. The best-fitting theoretical model (solid line) and the observational data (diamonds with error bars) for l=−60° fields are shown in Fig. 6(a) in which the data show an irregular pattern of deviations from the symmetric model. The projected number densities at (40°, −60°), (60°, −60°), (270°, −60°), (290°, −60°) are higher than the model, while the observational data at (110°, −60°), (340°, −60°), (350°, −60°) are lower than the model. It is clear from Figs 6(b)–(f) that the two theoretical lines do not overlap perfectly due to different extinctions. In Fig. 6(e) of b=−70°, l= 120°, the value of the theoretical curve has a dip because that sky area (120°, −70°) has a large extinction from COBE, such that when the distance is larger than 550PC, the extinction takes Av= 0.174 mag. While in other areas around this, the extinction ranges only 0.07–0.08 mag.

The above model parameter set is used to calculate the theoretical R_F-band star count, and the results are plotted in Fig. 7. In Fig. 7(a), star counts from (40°, −60°) to (110°, −60°), (270°, −60°) to (340°, −60°) also fluctuate around the theoretical value. In Figs 7(b)–(f), the pairs of (120°, −75°) and (240°, −75°) and (150°, −75°) and (210°, −75°) show counts higher than the theoretical line, while counts in all other areas are random around the theoretical prediction.

As well as calculating the stellar projected number density, we also compare the theoretical and the observational star counts in apparent magnitude bin for each sky area. Fig. 8 shows, as an example, the observational (grey diamond) and the theoretical (dark line) star counts in 12 sky areas of b=−60°, the Galactic coordinates of each sky area are indicated in the corresponding panel. As shown in these plots, the theoretical model can fit observation data fairly well, but with a few exceptions. In (90°, −60°) and (330°, −60°) at the bin of 20 –20.5 mag, the theoretical value is higher than the observational one. Similar to what is shown in Figs 6 and 7, the distribution of star counts in apparent magnitude also fits fairly well a homogeneous axisymmetric structure.

Figure 8

Model fitting to B_J star counts in selected sky areas whose Galactic coordinates are indicated in each panel. The grey dots are observational star counts, and the solid lines are the theoretical predictions.

Open in new tabDownload slide

Using equation (4), χ² and for each parameter grid can be obtained. The contour plots of χ² and in the n–q plane are presented in Fig. 9. The minimum value of χ² is 1.53, and the maximum is 11.915, while the values for are 0.939 and 3.424, respectively, 20 levels of contours are used. The innermost (smallest values of χ² and ⁠) contour indicates the best-fitting combinations n and q. The open diamonds in both panels of Fig. 9 indicate the most favourable parameters given by both χ² and minimizations, which are listed in Table 7. The fitting of the observed projected number density using one of the best combinations (n= 2.8, q= 0.7) is shown in Fig. 6.

Figure 9

Upper panel: contours of χ² of the theoretical models in power-law index (horizontal) and axial ratio (vertical) plane. Lower panel: contours of of the models in the same plane. The overlapped grids of the minimum contour level are labelled by diamonds, which define the best-fitting model parameters.

Open in new tabDownload slide

Table 7

Open in new tab

The best-fitting parameters.

n	q	χ2
2.5	0.6	1.714	0.990
2.6	0.6	1.658	0.964
2.7	0.6	1.615	0.948
2.8	0.6	1.583	0.940
2.8	0.7	1.561	1.064
2.9	0.6	1.559	0.939
2.9	0.7	1.587	1.044
3.0	0.6	1.543	0.944
3.0	0.7	1.621	1.032
3.1	0.6	1.534	0.955
3.1	0.7	1.662	1.026
3.2	0.6	1.530	0.971
3.2	0.7	1.708	1.025
3.3	0.7	1.760	1.029
3.4	0.7	1.815	1.037
3.5	0.7	1.874	1.050

n	q	χ2
2.5	0.6	1.714	0.990
2.6	0.6	1.658	0.964
2.7	0.6	1.615	0.948
2.8	0.6	1.583	0.940
2.8	0.7	1.561	1.064
2.9	0.6	1.559	0.939
2.9	0.7	1.587	1.044
3.0	0.6	1.543	0.944
3.0	0.7	1.621	1.032
3.1	0.6	1.534	0.955
3.1	0.7	1.662	1.026
3.2	0.6	1.530	0.971
3.2	0.7	1.708	1.025
3.3	0.7	1.760	1.029
3.4	0.7	1.815	1.037
3.5	0.7	1.874	1.050

Table 7

Open in new tab

The best-fitting parameters.

n	q	χ2
2.5	0.6	1.714	0.990
2.6	0.6	1.658	0.964
2.7	0.6	1.615	0.948
2.8	0.6	1.583	0.940
2.8	0.7	1.561	1.064
2.9	0.6	1.559	0.939
2.9	0.7	1.587	1.044
3.0	0.6	1.543	0.944
3.0	0.7	1.621	1.032
3.1	0.6	1.534	0.955
3.1	0.7	1.662	1.026
3.2	0.6	1.530	0.971
3.2	0.7	1.708	1.025
3.3	0.7	1.760	1.029
3.4	0.7	1.815	1.037
3.5	0.7	1.874	1.050

n	q	χ2
2.5	0.6	1.714	0.990
2.6	0.6	1.658	0.964
2.7	0.6	1.615	0.948
2.8	0.6	1.583	0.940
2.8	0.7	1.561	1.064
2.9	0.6	1.559	0.939
2.9	0.7	1.587	1.044
3.0	0.6	1.543	0.944
3.0	0.7	1.621	1.032
3.1	0.6	1.534	0.955
3.1	0.7	1.662	1.026
3.2	0.6	1.530	0.971
3.2	0.7	1.708	1.025
3.3	0.7	1.760	1.029
3.4	0.7	1.815	1.037
3.5	0.7	1.874	1.050