A&A 443, 929-943 (2005)
DOI: 10.1051/0004-6361:20041393
S. Phleps1,2 - S. Drepper1 - K. Meisenheimer1 - B. Fuchs3
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 h1 and h2 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),
h1=300 pc, h2=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 hz=325 pc. However, as Gilmore & Reid (1983)
showed, the data can be fitted much better by a superposition of two
exponentials with scaleheights h1 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 h1= 280 pc, and
unambigously the contribution of the thick disk with h2 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
EB-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, .
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The radial change of the density can be taken into account by assuming an exponential decrease with scalelength hr. The value of hr is still poorly determined, measurements range from values as small as hr=1.8 kpc (Bienaymé 1999), hr=2.3 kpc (Ruphy et al. 1996; Ojha et al. 1996) and hr=2.5 kpc (Robin et al. 1992), respectively, to hr=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
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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.
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Figure 1:
The distribution of observed
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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 MR 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 I815, 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 TiO2 and VO molecules in the stars' atmosphere, whereas
the BC and the medium-band filter I815 (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
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Figure 2:
The stars in the five fields in a two color diagram
((b-r) versus (r-i)). The spread becomes significant at
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Figure 3:
Spatial distribution of metallicity corrected
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With the Poissonian errors
,
,
and
the
error of the corrected number counts becomes:
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= | ![]() |
|
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With the corrected number counts the density in the logarithmic spaced
volume bins (
)
can then be
calculated according to
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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 h1 from this fit, assuming
two extreme values of the
scalelength: hr=2.0 kpc, and hr=3.5 kpc, respectively (see
Table 2). For hr=2.0 kpc we find that the
weighted mean of the scaleheight of the thin disk is
,
for hr=3.5 kpc we find
.
These values are indistinguishable from
each other, and also the values of
indicate that the value adopted for hr does not influence the
measurement of the scaleheight h1.
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 hr=2.0 kpc,
and hr=3.5 kpc, respectively, keeping the corresponding values of h1 fixed.
With h1 and (n1+n2)
known, the free parameters are
now the scaleheight h2 of the thick disk, and the ratio
n2/(n1+n2). 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
hr=3.5 kpc. The contour lines show the 1, 2 and
confidence levels. Neither the ratio
n2/(n1+n2) 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 h2, 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
,
.
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Figure 4:
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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 hr=3.5 kpc). The fraction of thick disk stars in the local normalisation is assumed to be n2/(n1+n2)=0.04, and the mean value of h2=1.0 kpc was adopted for all fields except for the 9 h field, where a formal value of h2=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/kpc3.
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.
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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 ![]() |
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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.
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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 h1. Also shown in Fig. 7 are the SLFs one would measure for h1=0.250 kpc and h1=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).
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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 h1 (see Table 2). The dotted and dashed lines are the determination for h1=0.250 kpc, and for h1=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 h1 < h2, 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,
n2/(n1+n2), is left as
a free parameter, and is varied from 2% to 18% in steps of 1%. The scaleheights
h1 and h2 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 hr=2000 pc, and for hr=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:
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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 MV 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 ni is the number of stars in the bin, ei the expectation
value predicted by the model and N the number of bins, then
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The overall C is lowest for
,
(c/a)=0.6,
hr=3500 pc, and h1=300 pc, while C is only marginally larger
if (c/a)=1.0 (and
). There is no significant difference
between models with hr=2000 pc or hr=3500 pc. This is
consistent with the results from the previous section. Figure 8 shows the C planes for h2 and
n2/(n1+n2) (with hr=3500 pc, h1=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.
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Figure 8:
The C planes of the five fields for the
parameter h2 and
n2/(n1+n2) (with hr=3500 pc, h1=300 pc,
(c/a)=0.6,and
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From the mean of the five fields we find the minimum of the Cstatistic to be at h2=900 pc and
.
While the
value of h2 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
n2/(n1+n2)-axis. Thus the relative
normalisation is only measurable with extremely large errors, and the
rather high value of
n2/(n1+n2)=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 (hr=3500 pc, h1=300 pc,
h2=900 pc,
n2/(n1+n2)=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.
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Figure 9:
The number density of stars in the (b-r)C-![]() ![]() ![]() |
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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
hr=2000 pc, and
pc if hr=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.
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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 h1=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 h1, h2, n1, n2,
,
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.