A&A 443, 929-943 (2005)
DOI: 10.1051/0004-6361:20041393
S. Phleps^{1,2} - S. Drepper^{1} - K. Meisenheimer^{1} - B. Fuchs^{3}
1 - Max-Planck-Institut für Astronomie, Königstuhl 17,
69117 Heidelberg, Germany
2 - Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
3 - Astronomisches Rechen-Institut, Mönchhofstr. 12-14,
69115 Heidelberg, Germany
Received 2 June 2004 / Accepted 13 August 2005
Abstract
We used 1627 faint (
)
stars in five fields
of the Calar Alto Deep Imaging Survey (CADIS) to estimate the
structure parameters of the Galaxy. The results were derived by
applying two complementary methods: first by fitting the density
distribution function to the measured density of stars perpendicular
to the Galactic plane, and second by modelling the observed colors
and apparent magnitudes of the stars in the field, using Monte Carlo
simulations. The best-fitting model of the Galaxy is then determined
by minimising the C-statistic, a modified .
Our
model includes a double exponential for the
stellar disk with scaleheights h_{1} and h_{2} and a power law halo with
exponent .
24 480 different parameter
combinations have been simulated. Both methods yield consistent
results: the best fitting parameter combination is
(or
,
if we allow for a flattening of the halo with an axial
ratio of (c/a)=0.6),
h_{1}=300 pc, h_{2}=900 pc, and the contribution of thick disk
stars to the disk stars in the solar neighbourhood is found to be between 4 and 10%.
Key words: Galaxy: structure
In the standard model (Bahcall & Soneira 1980a,b) the Galaxy is of Hubble type Sbc, consisting of an exponential disk, a central bulge, and a spherical halo. The vertical structure of the disk follows an exponential law ( ) with scaleheight h_{z}=325 pc. However, as Gilmore & Reid (1983) showed, the data can be fitted much better by a superposition of two exponentials with scaleheights h_{1} of 90 to 325 pc and pc (Gilmore 1984). It is not clear whether this deviation from the single exponential is due to a distinct population of stars, although this is suggested by the different kinematics and lower metallicities (Freeman 1992), it is referred to as "thick disk'' in the literature.
In the last two decades highly efficient, large area surveys have been carried out, and large quantities of high-quality imaging data have been produced. The mere existance of a thick disk component is fairly established today, however, the size of its scaleheights are still under discussion (Siegel et al. 2002; Norris 1999), as several authors claim to have found a considerably smaller value for the scaleheight of the thick disk than the canonical one of kpc: Robin et al. (2000) found 750 pc, and Chen et al. (2001) found the scaleheight to be between 580 and 750 pc.
In general there are two different approaches to deduce Galactic Structure: the Baconian, and the Cartesian ansatz, as Gilmore & Wyse (1987) call it.
The Baconian ansatz tries to manage with a minimum of assumptions: by a pure measurement of the stellar distribution function by means of starcounts. Distances are estimated for each star and then their number density in dependence of distance from the Galactic plane is calculated (Gilmore & Reid 1983; Reid et al. 1996,1997; Gould et al. 1998). The parameters can then be determined by fitting distribution functions to the data. In a first paper on CADIS deep star counts (Phleps et al. 2000, Paper I in the following) we presented first results based on 300 faint stars ( ) in two CADIS fields covering an area of in total. From these data we deduced the density distribution of the stars up to a distance of about 20 kpc above the Galactic plane, using the Baconian ansatz. We found h_{1}= 280 pc, and unambigously the contribution of the thick disk with h_{2} on the order of 1300 pc.
On the other hand, as the main structural features of the Milky Way have been identified, it has been possible to design synthetic models of the Galactic stellar populations (Reid & Majewski 1993; Gilmore 1984; Mendez & van Altena 1998; Bahcall & Soneira 1981; Robin & Creze 1986; Chen et al. 1999,2001), assuming the forms of the density distribution function for the different components of the Galaxy. This method is what Gilmore & Wyse (1987) call the Cartesian ansatz.
The exact structure parameters can then be deduced by comparing model and star count data.
In this paper we use both approaches using the stellar component of the CADIS data. Deep (R<23) multi color data is now available for 1627 faint stars in five fields at high Galactic latitude and different Galactic longitudes.
In Sect. 2 the Calar Alto Deep Imaging Survey is briefly described. In Sect. 4 the results we deduced by applying the "classic'', Baconian method are shown, whereas in Sect. 5 we show the results given by the modelling approach. A summary and discussion is given in Sect. 6.
The seven CADIS fields measure each ( ) and are located at high Galactic latitudes to avoid dust absorption and reddening. In all fields the total flux on the IRAS 100 m maps is less than 2 MJy/sr which corresponds to E_{B-V} <0.07. Therefore we do not have to apply any colour corrections. As a second selection criterium the fields should not contain any star brighter than in the CADIS R band. In fact the brightest star in the five fields under consideration has an R magnitude of .
All observations were performed on Calar Alto, Spain. In the optical wavelength region the focal reducers CAFOS (Calar Alto Faint Object Spectrograph) at the 2.2 m telescope and MOSCA (Multi Object Spectrograph for Calar Alto) at the 3.5 m telescope were used. The NIR observations have been carried out using the Omega Prime camera at the 3.5 m telescope.
In each filter, a set of 5 to 15 individual exposures was taken. The images of one set were then bias subtracted, flatfielded and corrected for cosmic ray hits, and then coadded to one deep sumframe. This basic data reduction steps were done with the MIDAS software package in combination with the data reduction and photometry package MPIAPHOT (developed by H.-J. Röser and K. Meisenheimer).
The measured counts are translated into physical fluxes outside the terrestrial atmosphere by using a set of "tertiary'' spectrophotometric standard stars which were established in the CADIS fields, and which are calibrated with secondary standard stars (Walsh 1995; Oke 1990) in photometric nights.
From the physical fluxes, magnitudes and colour indices (an object's brightness ratio in any two filters, usually given in units of magnitudes) can be calculated. The CADIS magnitude system is described in detail in Wolf et al. (2001b) and Fried et al. (2001).
The CADIS color is defined by:
b-r can be calibrated to the Johnson-Cousins system (the CADIS
is
very close to the Cousins R) by using Vega as
a zero point:
(B-R)_{C} | = | ||
= | (b-r)+0.725. | (2) |
Using the minimum variance estimator (for details see Wolf et al. 2001b), each object is assigned a type (star - QSO - galaxy), a redshift (if it is not classified as star), and an .
Note that we do not apply any morphological star/galaxy separation or use other criteria. The classification is purely spectrophotometric.
Five CADIS fields have been fully analysed so far (for coordinates see Table 1). We identified 1627 stars with . The number of stars per field is given in Table 1.
Table 1: The Galactic coordinates of the five fields investigated so far, and the number of stars per field, .
(3) |
The radial change of the density can be taken into account by assuming an exponential decrease with scalelength h_{r}. The value of h_{r} is still poorly determined, measurements range from values as small as h_{r}=1.8 kpc (Bienaymé 1999), h_{r}=2.3 kpc (Ruphy et al. 1996; Ojha et al. 1996) and h_{r}=2.5 kpc (Robin et al. 1992), respectively, to h_{r}=3.5 kpc (Bahcall & Soneira 1980b,1981; Wainscoat et al. 1992).
For each
field with Galactic longitude l and latitude b, respectively, we
can describe the distribution of the stars in the disk by
(6) |
Several authors found that the stellar halo is significantly flattened, with an axial ratio of (c/a)around 0.6 (Lemon et al. 2004; Robin et al. 2000).
Figure 1 shows the distribution of measured colors of the selected stars in the field. As in our first analysis, we use this bimodal distribution of colors, which comes from the interaction between luminosity function, density distribution and apparent magnitude limits ( ), to discriminate between disk and halo stars. The nearby disk stars are intrinsically faint and therefore red, whereas the halo stars have to be bright enough to be seen at large distances, and are therefore blue. According to the different directions of the fields the cut is located at slightly different colors, as indicated in Fig. 1. Unfortunately it is not possible to discriminate between thin and thick disk stars in the same way, so in the calculation of absolute magnitudes and thus distances we have to treat them as one sample.
Figure 1: The distribution of observed colors in the fields. The dotted line shows the color cut between halo and disk. Note that the ordinate has not the same scale in all cases, whereas the dotted line extends to 20 in each plot. The maxima are not located at the same colors, but depend on the direction of the field. | |
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We repeated the calculation of the photometric parallaxes
of the stars exactly as described in Paper I. Absolute magnitudes are inferred from a mean
main sequence relation in a color-magnitude diagram
(Lang 1992). The main sequence converted into a M_{R} vs. (b-r)relation (see Paper I) can be approximated by a fourth order
polynomial in the range
:
The mean main sequence relation is valid strictly only for stars with solar metallicities, whereas our sample may contain stars spread over a wide range of different metallicities. The influence of the varying metallicities on the main sequence color-magnitude relation is taken into account as described in Paper I: Halo stars are supposed to be metal poor, and are known to be fainter than disk stars with the same colors, so the relation is shifted towards fainter magnitudes by . This value is the mean deviation from the mean main sequence defined by the CNS4 stars (Jahreiß & Wielen 1997) of a subsample of 10 halo stars for which absolute Rmagnitudes were available (Jahreiß, priv. com)^{}.
The spread in a two color diagram (b-r) versus (r-i) (that is the CADIS color between and I_{815}, analog to Eq. (1)) becomes significant at , see Fig. 2. The maximal photometric error for the very faint stars is . Here metallicity effects will distort the relation between the measured (b-r) colors and the spectral type (temperature) and thus lead to wrong absolute magnitudes, so we have to correct for metallicity in order to avoid errors in the photometric parallaxes.
The filter is strongly affected by metallicity effects like absorption
bands of TiO_{2} and VO molecules in the stars' atmosphere, whereas
the B_{C} and the medium-band filter I_{815} (the
wavelength of which was chosen in order to avoid absorption bands in
cool stars) are not. So in a
first approximation
we can assume the "isophotes'' of varying metallicity in a (b-r)versus (r-i) two color diagram to be straight
lines with a slope of -1, along of which we project the measured colors with
onto the mean main
sequence track which in the interval
is defined by
(8) |
Figure 2: The stars in the five fields in a two color diagram ((b-r) versus (r-i)). The spread becomes significant at . The solid line is the mean main sequence track, onto which the (b-r) colours of the stars are projected along the R isophotes of varying metallicity (dotted lines). For some extremely red stars, photometric errors are plotted. | |
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Figure 3: Spatial distribution of metallicity corrected . The solid line represents the distance dependent upper color limit at , the dashed line is the lower limit due to the selection criterium of the fields (no star brighter than ). Since the metal-poor halo stars are intrinsically fainter, the color limits are shifted accordingly. The horizontal line at denotes the cut between halo and disk stars in the 13h field (which is why stars can be found "outside the boundaries'': the color cut is different in each field, see Fig. 1), the dashed-dotted line indicates the cutoff at due to the metallicity correction. The different symbols refer to the different fields: squares - 1 h field, triangles - 9 h field, dots - 13 h field, open circles - 16 h field, crosses - 23 h field. | |
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With the Poissonian errors
,
,
and
the
error of the corrected number counts becomes:
= | |||
(10) |
With the corrected number counts the density in the logarithmic spaced
volume bins (
)
can then be
calculated according to
(11) |
From the estimated density distribution in the five fields we now deduce the structure parameters by fitting the distribution function to the data.
As the nearest stars in our fields have still distances of about 200 pc the normalization at z=0 has to be established by other means: we take stars from the CNS4 (Fourth Catalogue of Nearby Stars, Jahreiß & Wielen 1997), which are located in a sphere with radius 20 pc around the sun. The stars in our normalization sample are selected from the CNS4 by their absolute visual magnitudes, according to the distribution of absolute magnitudes of the CADIS disk stars ( ).
Since the thin disk completely dominates the first kpc and there is no indication of a change of slope within this distance (Fuchs & Wielen 1993), it is justified to fit the density distribution in this range with a single exponential component (taking of course the radial decrease into account). The sum of thick and thin disk stars in the solar neighbourhood, kpc^{-3} with an uncertainty of 10%, is given by the CNS4 stars. We estimate the scaleheight h_{1} from this fit, assuming two extreme values of the scalelength: h_{r}=2.0 kpc, and h_{r}=3.5 kpc, respectively (see Table 2). For h_{r}=2.0 kpc we find that the weighted mean of the scaleheight of the thin disk is , for h_{r}=3.5 kpc we find . These values are indistinguishable from each other, and also the values of indicate that the value adopted for h_{r} does not influence the measurement of the scaleheight h_{1}.
Table 2: The scaleheights of the thin disk, assuming a scalelength of and , respectively. The numbers in brackets are the corresponding values of .
We then fitted Eq. (5) to the data over the range 0< z < 4 kpc, again for h_{r}=2.0 kpc, and h_{r}=3.5 kpc, respectively, keeping the corresponding values of h_{1} fixed. With h_{1} and (n_{1}+n_{2}) known, the free parameters are now the scaleheight h_{2} of the thick disk, and the ratio n_{2}/(n_{1}+n_{2}). Again the measurement is insensible to the scalelength assumed in Eq. (5). This is due to the high Galactic latitude of the CADIS fields ( ) - the halo becomes dominant already at a projected radial distance of only about 7 kpc.
Although the number of degrees of freedom is already kept to a minimum, this fit is highly degenerated, as can be seen from the distribution (see Fig. 4), where we asumed h_{r}=3.5 kpc. The contour lines show the 1, 2 and confidence levels. Neither the ratio n_{2}/(n_{1}+n_{2}) nor the value of the scaleheight of the thick disk can be estimated with high accuracy. Except for the 9 h field, where the distribution suggests a very large value of h_{2}, the minimum can be found at around kpc, and . The fit of the scaleheight of the thick disk in the 9 h field is mainly influenced by the one data point at kpc, where the density seems to be extraordinarily high. Because this value appears to be an exception, we disregard it in the calculation of the mean. The mean , excluding the 9 h field, is , . Including the 9 h field, the mean (lower right corner) is approximately at , .
Figure 4: plane for the parameters h_{2} and n_{2}/(n_{1}+n_{2}) for the fit of the measured density distribution in the five fields, and the mean of all five ones ( bottom right). The scalelength is assumed to be h_{r}=3.5 kpc, and the values of h_{1} are given in Table 2. The contour lines show the , and confidence levels. | |
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Figure 5 shows the density distribution of the disk stars in the five fields, and the corresponding best fits. The value for the local normalisation is given by the CNS4, and the values for h1 can be found in Table 2 (we assumed h_{r}=3.5 kpc). The fraction of thick disk stars in the local normalisation is assumed to be n_{2}/(n_{1}+n_{2})=0.04, and the mean value of h_{2}=1.0 kpc was adopted for all fields except for the 9 h field, where a formal value of h_{2}=2.6 kpc is required to achieve a satisfying fit.
For the local normalisation we take stars from the CNS4, which have been discriminated against disk stars by their metallicities and kinematics (Fuchs & Jahreiß 1998). From these we selected stars with (which corresponds to ). All these stars have very low metallicities ( dex) and large space velocities (see Table 1 in Fuchs & Jahreiß 1998), which clearly identifies them as halo stars. The following six stars in a radius of pc around the sun satisfy the criteria: Gl 53A, Gl 451A, Gl 158A, GJ 1064A, GJ1064B, LHS 2815. Thus we find stars/kpc^{3}.
Figure 6 shows the completeness corrected density distribution of all stars. We fitted Eq. (5) to the last four data points, while keeping the normalisation at z=0 fixed to the value given by the stars from the CNS4. We estimated the power law index for a flattened halo with an axis ratio of c/a=0.6 (dashed line in Fig. 6), and no flattening (c/a=1.0, solid line). The values of we deduced from the fit to the data are listed in Table 3.
Figure 5: Density distribution of the stars in the disk. The solid line is the best fit for a sum of two exponentials. | |
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Figure 6: Density distribution of all stars. The solid lines are the best fits for the disk (as in Fig. 5) and a spherical halo, the dashed line is the fit for a flattened halo with axial ratio (c/a)=0.6. The corresponding exponents are given in Table 3). The dotted line is the sum of disk and spherical halo (which is slightly favoured by the fit). | |
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Clearly the fit in the 9 h field is dominated by one exceptionally high data point at kpc, and does not give a reliable result. The weighted mean, disregarding the 9 h field, is for (c/a)=0.6, and for (c/a)=1.0, where the values favour a halo without strong flattening: The sum of the values of the fits in the four single fields which enter in the determination are for the flattened, and for the spherical halo.
The axis ratio assumed for the estimation of the power law index influences the fitted value: the flatter the halo, the smaller . But not only the axial ratio influences the measurement of , also the local normalisation adopted for the fit. In order to estimate the effect we fitted the data for and , respectively, and found weighted means of (with ) and ( ) for (c/a)=0.6, and ( ) and ( ) for (c/a)=1.0, respectively (again leaving out the 9 h field).
The values of are getting larger for a larger value of , and smaller for smaller . However, the the summed indicate that the normalisation given by the CNS4 stars is not too far away from the "true'' normalisation. The true value is definitely not smaller. If it is slightly larger, then a flattened halo is favoured. However, it is not possible to measure the flattening of the halo accurately with the current data set.
(12) |
Table 3: The exponent from equation (5), for a flattened halo with an axis ratio of c/a=0.6, and for c/a=1.0 (no flattening). The numbers in brackets are the corresponding values.
The weighted mean of the SLFs of the five CADIS fields is shown in Fig. 7. For each single field we also calculated the effective volume by using the measured values of h_{1}. Also shown in Fig. 7 are the SLFs one would measure for h_{1}=0.250 kpc and h_{1}=0.350 kpc (dotted and dashed lines in the upper panel), in order to demonstrate how the SLF changes with the scaleheight assumed for the calculation of the effective volume. The dash-dotted line in the middle shows our old determination as published in Paper I (Fig. 10), the dashed line is the local SLF (Jahreiß & Wielen 1997) of stars inside a radius of 20 pc around the sun, which is based on HIPPARCOS parallaxes. In the range where the SLFs overlap, they are consistent with each other. As in Paper I we calculated the weighted mean of CADIS and local SLF, which is shown in the lower panel of Fig. 7, in comparison with a photometric SLF which is based on HST observations (Gould et al. 1998).
Figure 7: Upper panel: the mean SLF of the five CADIS fields (solid line). For the calculation of the effective volumes we used the fitted values of h_{1} (see Table 2). The dotted and dashed lines are the determination for h_{1}=0.250 kpc, and for h_{1}=0.350 kpc, respectively. Middle panel: The solid line is our current measurement, as in the upper panel. The dotted line is our first measurement as published in Paper I, the dashed line is the local SLF (Jahreiß & Wielen 1997), which is based on HIPPARCOS parallaxes. Lower panel: weighted mean of CADIS and local SLF, in comparison with a photometric SLF which is based on HST observations (Gould et al. 1998). | |
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We will use this luminosity function as input for the simulation, which we describe in the following section.
The two basic incredients are the stellar density distribution function and the stellar luminosity function. We will keep the luminosity function fixed, while varying the structure parameters of the density distribution.
We use the same model of the Milky Way as described in Sect. 3: a sum of two exponential disks with scaleheights h_{1} < h_{2}, and a power law halo.
The local density is again given by the CNS4 stars (Fuchs & Jahreiß 1998; Jahreiß & Wielen 1997). The contribution of thick disk stars to the local normalisation, n_{2}/(n_{1}+n_{2}), is left as a free parameter, and is varied from 2% to 18% in steps of 1%. The scaleheights h_{1} and h_{2} are varied from 150 pc to 400 pc in steps of 50 pc, and from 200 pc to 1800 pc in steps of 100 pc, respectively. The simulation was run for a radial scalelength h_{r}=2000 pc, and for h_{r}=3500 pc. The power law index of the stellar halo was increased from to in steps of 0.5, for a spherical as well as for a flattened halo with an axis ratio (c/a)=0.6.
For each field, the expected number of stars N in the model under consideration is
calculated by integrating the distribution function along the line of sight:
(15) |
In the range
the mean of the measured CADIS and the local SLF from
Jahreiß & Wielen (1997) can be described quite precisely by a third-order
polynomial in log-space:
Each star is assigned an absolute V band luminosity, randomly chosen from a distribution following Eq. (16). Since there is no V or comparable filter available in the CADIS filter set, we have to convert the M_{V} magnitudes into magnitudes (see Paper I), i.e. . CADIS (b-r)_{C} colors can be calculated from the absolute band magnitudes using the main sequence relation described in Paper I. The same relation was used in Sect. 4 to deduce absolute magnitudes from the (b-r)_{C} colors of the stars.
In order to take the spread in metallicities into account, the simulated colors are then smeared out by adding a value taken from a uniform distribution of numbers in the range .
The simulated stars are selected by their apparent magnitudes ( ). We bin the data, observed and simulated, into a grid with steps of in magnitude and in color.
Due to our survey selection criteria the bins at bright apparent magnitudes contain only very few stars, which makes a fit of the model to the data extremely difficult. Instead of using as an estimate of the goodness of fit, we use the C-statistic for sparse sampling, which was developed by Cash (1979). It is based on the likelihood ratio, and is suited for small number statistics. In the limit of large numbers it converges to .
If n_{i} is the number of stars in the bin, e_{i} the expectation
value predicted by the model and N the number of bins, then
(17) |
The overall C is lowest for , (c/a)=0.6, h_{r}=3500 pc, and h_{1}=300 pc, while C is only marginally larger if (c/a)=1.0 (and ). There is no significant difference between models with h_{r}=2000 pc or h_{r}=3500 pc. This is consistent with the results from the previous section. Figure 8 shows the C planes for h_{2} and n_{2}/(n_{1}+n_{2}) (with h_{r}=3500 pc, h_{1}=300 pc, (c/a)=0.6, and fixed at 2.5), for each of the five individual fields and the mean of all of them.
Figure 8: The C planes of the five fields for the parameter h_{2} and n_{2}/(n_{1}+n_{2}) (with h_{r}=3500 pc, h_{1}=300 pc, (c/a)=0.6,and ). The plot in the lower right corner is the mean of the five indidual fields. The contour lines show the , , and confidence levels. | |
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From the mean of the five fields we find the minimum of the Cstatistic to be at h_{2}=900 pc and . While the value of h_{2} is consistent with the value derived from the direct measurement, the relative normalisation is higher. However, the fit is highly degenerated, and the minimum in the distribution of C is streched along the n_{2}/(n_{1}+n_{2})-axis. Thus the relative normalisation is only measurable with extremely large errors, and the rather high value of n_{2}/(n_{1}+n_{2})=0.09 should be regarded as an upper limit.
Figure 9 shows the color-coded representation of the density of the stars in the magnitude-color diagram for the five fields. The images have been rebinned in steps of 0.1 in both color and magnitude. The contour plots overlaid are the simulated best-fit cases of the model (h_{r}=3500 pc, h_{1}=300 pc, h_{2}=900 pc, n_{2}/(n_{1}+n_{2})=0.09, (c/a)=0.6and ).
The prominent structure at the left of each image consists of disk stars, whereas the halo stars are mainly distributed at the right. The simulation matches the data very well except for the 13 h field, where the gap between halo and disk is not modelled. In all cases the halo does not match the data very well, a possible explanation might be that the real halo luminosity function, which we do not know, is in fact significantly different from the disk luminosity function which we use for the simulation.
Figure 9: The number density of stars in the (b-r)_{C}- plane, for the five fields. The densities are normalised to the total number of stars in each field. The images have been rebinned in steps of 0.1 in both color and magnitude. The contours overlaid are the simulated data for h_{r}=3500 pc, h_{1}=300 pc, h_{2}=900 pc, n_{2}/(n_{1}+n_{2})=0.09, (c/a)=0.6and , also normalised to the number of simulated stars. The contours are plotted for in steps of 0.001. | |
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Figure 10 shows the distribution of absolute V band magnitudes of the CADIS data (calculated from the colors as described in Sect. 5.2) in comparison with the simulated data, for the best fit cases. Both histograms have been normalised to the total number of stars, in order that they can be compared directly.
With exception of the CADIS 23 h field the distribution of absolute V band magnitudes is matched very well by the simulation. In the 9 h field the model can not account for the large peak at . It is not improbable that the Galactic halo is clumpy (Newberg et al. 2002), so the the feature may well be an overdensity of stars with respect to the smooth powerlaw halo.
In the 23 h field the number of bright halo stars is overpredicted. Since the 16 h and the 23 h fields are located at the same Galactic longitude and , both fields should show the same distribution of color or absolute magnitudes, respectively. However, as both distributions differ from each other at the bright end, it seems more likely that the 23 h field is "missing'' some stars, rather than that the model (which matches the 16 h field more closely) overpredicts stars which are not supposed to be there.
We have estimated the structure parameters of the stellar density distribution of the Milky Way using two different, complementary methods: from a direct measurement of the density distribution of stars perpendicular to the Galactic plane (Baconian ansatz), and by modelling the observed colors and apparent magnitudes of the stars and comparing them to the data (Kartesian ansatz).
Both methods have their advantages and limitations: the direct determination of the density involves completeness corrections because faint stars are only traced to small distances. These rely on the assumption that the stellar luminosity function (SLF) does not change with distance and is the same for disk and halo stars. The same assumption of course is used in the simulation. However, the simulation avoids completeness corrections and thus the errors do not depend on an iterative correction. A fitting of a simulated color-magnitude distribution of stars works very well for a large number of stars; with the CADIS data the method is at the edge of feasibility. We therefore used the C statistic developed by Cash (1979) for sparse sampling to find the best fitting parameter combination.
From the
fit to the direct measurement of the density
distribution of the stars, we find
pc if we assume
h_{r}=2000 pc, and
pc if h_{r}=3500 pc. Obviously,
due to the high Galactic latitude of the fields (
),
we are not able to measure the scalelength (the halo becomes dominant
already at a projected radial distance of only about 7 kpc). The fit
of the scaleheight of the thick disk and the local percentage of thick
disk stars is highly degenerate. The best estimate is
kpc, and
.
We also fitted
the density distribution in the halo and find
for
(c/a)=1.0, and
if we allow for a flattening of
the stellar halo with an axial ratio of (c/a)=0.6. The values of
formally favour the spherical halo, but this result is not
significant, since the power law slope and the axial ratio are highly
correlated. Furthermore the measured values of the exponent ,
and the axial ratio c/a depend on the local normalisation assumed
for the fit: slightly higher (lower) values of
yield slightly
higher (lower) values of ,
and a flattened (spherical) halo is favoured.
Figure 10: Comparison of absolute V band magnitudes of the simulation (dotted line) with the CADIS data (solid line), for the best fit case. | |
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From the Cartesian method we derive kpc, . We also find that the overall or C, respectively, is smallest for h_{1}=300 pc and an exponent . If we allow for a flattening of the halo with an axial ratio of (c/a)=0.6, we find that a slightly smaller value of fits the data better. The corresponding values of or C, respectively, are only marginally smaller for a flattened halo. The values of are generally slightly larger than in the direct measurement. A possible explanation for this difference is that when we fit the simulated distribution of stars in the color-magnitude diagram, we fit all components of the galaxy simultaneously (whereas in the direct measurement we fit disk and halo separately), and there might be a degeneracy between halo and thick disk parameters.
However, in general both methods yield essentially the same result, which we regard as a corroboration that both work correctly within their limitations.
Chen et al. (2001) carried out an investigation of Galactic structure using stars brighter than from the Sloan Digital Sky Survey (SDSS), covering a total area of 279 . They presented models of the Milky Way for a large number of free parameters, that is, they determine by means of a maximum likelyhood analysis the scaleheights of the thin and thick disk, respectively, the relative normalisation of both disk components in the solar neighbourhood, the offset of the sun above the Galactic plane, the exponent of the power law halo and the flattening of the halo.
They find pc, pc, , and , and . Lemon et al. (2004) also found a significant flattening of the halo ( ).
While our results lie well within the range found by most authors, our measurements for the scaleheights of the disk are quite dissimilar from the Sloan results.
A possible explanation might be that their determination, the results of which differ significantly from all other determinations, suffers from the degeneracy due to the large number of degrees of freedom in their simulation: their fitting routine may have found a secondary minimum.
Robin et al. (2000) found that there is a significant degeneracy between the power law index and the axis ratio of the halo. Their best fit to the data yields and c/a=0.76, but a flatter spheroid with c/a=0.6 with is not excluded either. We find the same degeneracy in our data: If we assume (c/a)=0.6, we find (or in the simulation), whereas for (c/a)=1.0 we find that ( ) fits the data better.
Therefore it certainly depends on the assumptions and the number of the degrees of freedom which values of h_{1}, h_{2}, n_{1}, n_{2}, , and (c/a) are determined in different analyses. The more structure parameters are known and kept fixed, the narrower is the (or C) distribution. Thus it is important to combine results from different methods.
The scaleheight of the thick disk for example will be determinable with high accuracy if the local normalisation of thick disk stars is known better. The local density of stars belonging to different components of the Galaxy has to be determined by means of kinematics and metallicities of the stars in order to break the degeneracy of the fit of a double exponential.
The degeneracy between the power law slope and the flattening of the halo can be broken by simultaneously fitting density distributions estimated from deep star counts in a large number of fields at different Galactic latitudes and longitudes, respectively.
When the "size'' and exact shape of the stellar halo is known, it will become much easier to determine the structure parameters of the thick disk.
CADIS, although designed as an extragalactic survey, provided a sufficient number of stars (and fields) to investigate the structure of the Milky Way. However, not only larger statistics, but also a much better knowledge of the luminosity function of halo stars is required to increase the accuracy of the results.
Acknowledgements
We thank all those involved in the Calar Alto Deep Imaging Survey, especially H.-J. Röser and C. Wolf, without whom carrying out the whole project would have been impossible.
We also thank M. Alises and A. Aguirre for their help and support during many nights at Calar Alto Observatory, and for carefully carrying out observations in service mode.
We are greatly indepted to our referee, Annie. C. Robin, who pointed out several points which had not received sufficient attention in the original manuscript. This led to a substantial improvement of the paper.
S. Phleps acknowledges financial support by the SISCO Network provided through the European Community's Human Potential Programme under contract HPRN-CT-2002-00316.