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Astron. Astrophys. 357, 871-880 (2000)

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4. Structure of the Sagittarius dwarf galaxy

4.1. Surface density of Sgr

A surface density map is constructed from RRab with the amplitude cuts stated above. The spatial distribution of these RRab has been convolved with a Gaussian on a grid with a step of [FORMULA] and a variable filter size adapted to the local surface density [FORMULA], constrained between [FORMULA] and [FORMULA]. This map was then corrected for the different completeness in amplitude and crowding (see Sect. 3.2).

The resulting map is shown on Fig. 7 where the elongated shape of Sgr is clearly visible. This is the first map of Sgr in these regions, showing that Sgr extends far beyond the outer limit of the map previously published by IWGIS. One of the most striking features of this map is the slow decrease (if any?) of the density along the main axis of Sgr for [FORMULA].

[FIGURE] Fig. 7. Smoothed map of the Sagittarius dwarf galaxy. This map is based upon the spatial distribution of RRab with distance modulus greater than 16.3. Only those RRab with (amplitude in Bi)[FORMULA]0.60 in SAG and (amplitude in [FORMULA] in DUO have been used. Completeness corrections have also been applied (see text). Contours are labelled as number of RRab per square degree. The dotted line (labelled 5) is not equidistant from the other contours. 

The main source of uncertainty in the surface density is the Poissonian noise in the star counts, which is variable over the field and tends to increase towards lower [FORMULA]b[FORMULA]. To estimate this noise we simulated 1 000 maps by injecting 1 400 stars (corresponding to the number of RRab actually used to construct the final map) onto the field with a probability density matching the surface density of the real map. These spatial distributions were then processed exactly in the same manner as the real one and a 1[FORMULA] "noise map" has been deduced.

This map is shown on Fig. 8 where the contours are labelled in percent. The typical (relative) uncertainty is constrained between 10[FORMULA] and 15[FORMULA] over the main part of the field, but increases up to [FORMULA]40[FORMULA] towards the edges where the number of RRab drops.

[FIGURE] Fig. 8. Uncertainty of the density map of Sgr based on 1[FORMULA] Poissonian error in the star counts. The contours are labelled in percentage. The dotted line (labelled 15) is not equidistant with the other contours.

4.2. Surface density profile of the main axis

The position angle of Sgr has been determined by fitting an exponential to the surface density along various directions. The highest scale length was reached for an angle of [FORMULA], which we choosed as the direction of the major axis. Fig. 9 displays the density profile of Sgr along that axis. This figure is based on a map smoothed on a constant scale of [FORMULA]. The thick line is the density after correction for completeness whereas the dotted line is the density before that correction. The shaded region represents the 1[FORMULA] uncertainty issued from simulated maps. A discontinuity in the slope is clearly visible at [FORMULA] from the centre. After this point the surface density seems to be almost constant. It is however disconcerting that this discontinuity occurs near the limit between DUO and SAG and we may wonder if this is not an experimental effect. We have shown in Sect. 3 that the crowding correction in SAG and DUO were consistent with the completeness of each field in the overlap. Furthermore the break is also perceptible in the uncorrected density so it cannot be an effect of the crowding correction. Another possibility is that our amplitude cut in DUO is to low to be consistent with SAG . This is difficult to check and we can only rely on those 30 RRab in common in the overlap, from which we derived the relation in amplitude between DUO and SAG (Eq. 1). However, if we make the assumption that the RRab population is homogeneous over the field, it is possible to search for the relation [FORMULA] for which the amplitude distributions are the most similar (through Kolmogorov-Smirnov test). The resulting coefficients were a=0.95 and b=-0.08 and are within the error bars stated in Eq. 1. The corresponding cuts are [FORMULA]. These cuts would have reinforced the discontinuity, showing that our adopted amplitude cuts are not responsible for the break observed in Fig. 9. We conclude that the discontinuity in the slope of the density profile is probably real and not a consequence of the change of field.

[FIGURE] Fig. 9. Cross-section along the main axis of Sgr based on a density map smoothed with a constant filter size of [FORMULA]. The thick line represents the density after completeness corrections and the dotted line is the uncorrected density. The shaded region corresponds to the 1[FORMULA] uncertainty issued from simulated maps. Note the discontinuity in the density gradient at [FORMULA], also perceptible in the uncorrected density.

It is also possible to derive an upper limit for the extension of Sgr along the line of sight: the distance modulus histogram can be roughly fitted by a Gaussian with a width of 0.2 mag, corresponding to a depth of [FORMULA]4.5 kpc for an assumed distance of 24 kpc.

4.3. Model fitting

We define the following analytical functions to fit to the density profile:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Where r represents the distance from the Centre of Sgr, and all other parameters are variables to be fitted. Eq. 2 refers to the empirical King model (K) with a core radius rc and tidal radius rt (King 1962). Eq. 3 refers to an exponential model (E) with a radius re. Eq. 4 refers to a Gaussian (G) with a width of [FORMULA]. Finally, Eq. 5 refers to a linear model (L) where the density profile is modeled by a straight line. These models have been fitted along three segments. Two of these segments are located on the main axis: one corresponding to the main axis over its entire length [FORMULA], referred to as Maj10 ; and another one corresponding to the portion of the main axis contained within SAG ([FORMULA] from the center), referred to as Maj6 . The latter segment has been chosen in order to avoid fitting the stars past the break and also because it is more consistent since it is entirely contained within SAG . Finally the minor axis is not present within our field and instead we fitted an axis making a large angle relative to the main axis [FORMULA], referred to as Min . The fit of model L on Maj10 has been performed by fitting the density before [FORMULA] and after [FORMULA] separately. In order to get uncorrelated points for the fit, we took the densities (after correction for completeness) of RRab inside boxes with a size of [FORMULA] located along each axis (see Fig. 10). The results of the fit are shown in Table 2 and in Fig. 11.

[FIGURE] Fig. 10. Location of the axes on which various models have been fitted. Each point is represented by the number of RRab (corrected for completeness) within a box of size [FORMULA]. The vertical dashed line at [FORMULA]18.55 indicates the limit of the Maj6 axis (see text).


[TABLE]

Table 2. Results of the model fitting to the surface density of Sgr. The three first columns give respectively the axis, model and parameter considered. Column 4 gives the result of the fit. Column 5 gives the uncertainty on the parameter value as given by the covariance matrix issued from the fitting procedure. Column 6 gives the [FORMULA] of the fit normalized by the number of degree of freedom. All values given in kpc assume a distance to Sgr of 24 kpc.


[FIGURE] Fig. 11. Results of the fit to the surface density of Sgr. Left panels: density profile along the main axis. The solid line is the fit to the whole main axis (Maj10 ) while the dotted line corresponds to the fit to the inner part of the main axis (Maj6 ). The dotted line is prolonged beyond [FORMULA] to allow visual comparison between the data and the inner fit. Right panel: density profile along an axis rotated [FORMULA] relative to the main axis (Min ). The errors bars represent the Poissonian noise. The fitted model is indicated in each panel.

The single function model that best fits Maj10 is model E ([FORMULA]=3.08). However, the fit is significantly improved if we consider the model L ([FORMULA]=1.77) which reproduces the break already observed in Fig. 9. A Fisher test shows that the probability for the ratio of the [FORMULA] of these two fits to be lower than the observed value by chance is only [FORMULA]13[FORMULA]. Note that we were unable to fit any convergent two-component model to Maj10 : this is due to the almost constant density of the external region which causes one of the components to increase as we move away from the centre in order to compensate the decrease of the other component.

Concerning the core of Sgr (Maj6 ), the density profile is equally well fitted by model E and L ([FORMULA]2.1). The scale length derived from model E is [FORMULA] (1.7[FORMULA]0.2 kpc). This value is slightly lower to the one derived by MOM who find an inner scale length of [FORMULA] in the Southern part of Sgr. Model K and G also give an acceptable fit to the core of Sgr but they fail to reproduce the high density in the first bin. Furthermore, the uncertainties on the parameters of the empirical King model are quite large and the infinite tidal radius is rather unrealistic.

Finally, the best fit on Min is achieved by model G ([FORMULA]=1.75), but again it fails to reproduce the high density of the first bin. The only model that reproduces the high central density is model E but the [FORMULA] of this model is worsened by the poor fit on the three last bins. However these bins contain only very few points (between 1 and 5), introducing uncertainties induced by small-numbers statistics.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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