Modelling the Galactic bar using OGLE-II Red Clump Giant Stars
Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 6 September 2021 (MN LATEX style file v2.2)
Modelling the Galactic bar using OGLE-II Red Clump Giant Stars
Nicholas J. Rattenbury1, Shude Mao1, Takahiro Sumi2, Martin C. Smith3 ⋆
1 University of Manchester, Jodrell Bank Observatory, Macclesfield, SK11 9DL, UK
2 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
3 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
Accepted ........ Received .......; in original form ......
ABSTRACT
Red clump giant stars can be used as distance indicators to trace the mass distribution of
the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing
collaboration database to constrain analytic tri-axial models for the Galactic bar. We find the
bar major axis is oriented at an angle of 24◦ – 27◦ to the Sun-Galactic centre line-of-sight.
The ratio of semi-major and semi-minor bar axis scale lengths in the Galactic plane x0, y0,
and vertical bar scale length z0, is x0 : y0 : z0 = 10 : 3.5 : 2.6, suggesting a slightly more
prolate bar structure than the working model of Gerhard (2002) which gives the scale length
ratios as x0 : y0 : z0 = 10 : 4 : 3.
Key words: Galaxy: bulge - Galaxy: centre - Galaxy: structure
1 INTRODUCTION
It is now generally accepted that the Galactic bulge is a tri-axial,
bar-like structure. Observational evidence for a bar has arisen
from several sources, such as the study of gas kinematics (e.g.
Binney et al. 1991), surface brightness (e.g. Blitz & Spergel 1991),
star counts (e.g. Nakada et al. 1991; Stanek et al. 1994) and mi-
crolensing (e.g. Udalski et al. 1994); see Gerhard (2002) for a re-
view.
Observational data have been used to constrain dynami-
cal models of the Galaxy. Dwek et al. (1995) used the COBE-
DIRBE multi-wavelength observations of the Galactic centre
(Weiland et al. 1994) to constrain several analytic bar models.
Stanek et al. (1997) used optical observations of red clump gi-
ant (RCG) stars to constrain theoretical bar models. Similarly,
Babusiaux & Gilmore (2005) and Nishiyama et al. (2005) traced
the bulge RCG population in the infrared. This work uses a sam-
ple of stars 30 times larger than that of Stanek et al. (1997), with
a greater number of fields distributed across a larger area of the
Galactic bulge, thus allowing finer constraints to be placed on the
bar parameters than those determined by Stanek et al. (1997).
Our current understanding of the Galactic bar is that it is ori-
entated at about 15− 40◦ to the Sun–Galactic centre line-of-sight,
with the near end in the first Galactic longitude quadrant. The
bar length is around 3.1 – 3.5 kpc with axis ratio approximately
10 : 4 : 3 (Gerhard 2002). The above bar parameters are gener-
ally accepted as a working model, however they are not well de-
termined. Our understanding of the complete structure of the inner
Galactic regions is similarly incomplete. For example, recent infra-
red star counts collected by the Spitzer Space Telescope for Galac-
⋆ e-mail: (njr, smao)@jb.man.ac.uk; sumi@stelab.nagoya-u.ac.jp;
msmith@astro.rug.nl
tic longitudes l = 10◦ – 30◦ are best explained assuming a long thin
bar oriented at an angle of ∼ 44◦ to the Sun–Galactic centre line
(Benjamin et al. 2005) while most previous studies (performed at
|l| . 12◦) prefer a short bar with an opening angle of ∼ 20◦. Re-
cently, Cabrera-Lavers et al. (2007) report that NIR observations of
RCGs support the hypothesis that a long thin bar oriented at ∼ 45◦
co-exists with a distinct short tri-axial bulge structure oriented at
∼ 13◦. In addition, there may be some fine features, such as a ring
in the Galactic bulge (Babusiaux & Gilmore 2005), or a secondary
bar (Nishiyama et al. 2005), that are not yet firmly established. It is
therefore crucial to obtain as many constraints as possible in order
to better understand the structure of the inner Galaxy.
In this paper we present an analysis of RCG stars observed
in the Galactic bulge fields during the second phase of the OGLE
microlensing project (Udalski et al. 2000). These stars are bright
and they are approximately standard candles, hence their magni-
tudes can be taken as an approximate measure of their distances.
Number counts in 34 central bulge fields with −4◦ 6 l 6 6◦
and −6◦ 6 b 6 3◦ are used to constrain analytic tri-axial bar
models, and thereby obtain estimates on bar parameters. We repeat
the analysis with 44 fields with −6.8◦ 6 l 6 10.6◦. We find the
fitted bar parameters support the general orientation and shape of
the bar reported by other groups. This paper is organised as fol-
lows: in Section 2 we describe the OGLE microlensing experiment
and photometry catalogue and we illustrate how RCG stars can be
used as approximate distance indicators; in Section 3 we detail how
RCGs in the OGLE-II proper motion catalogue are selected; in Sec-
tion 4 we compute the distance modulus to the red clump in 45
OGLE-II fields and thereby trace the central mass density of the
Galaxy; in Section 5 we describe how RCG star count histograms
for each field can be used to constrain analytic bar models of the
inner Galaxy; our results and their comparison to previous works is
c© 2005 RAS
http://arxiv.org/abs/0704.1614v1
2 Rattenbury et al.
given in Section 6 and in Section 7 we discuss the implications and
limitations of these results.
2 DATA
The OGLE (Udalski et al. 2000) and MOA (Bond et al. 2001;
Sumi et al. 2003) microlensing collaborations currently make rou-
tine observations of crowded stellar fields towards the Galactic
bulge, and issue alerts when a microlensing event is detected. A
result of this intense monitoring is the creation of massive photom-
etry databases for stars in the Galactic bulge fields. Such databases
are extremely useful for kinematic and population studies of the
central regions of the Galaxy.
Sumi et al. (2004) obtained the proper motions for millions of
stars in the OGLE-II database for a large area of the sky. Fig. 1
shows the OGLE-II fields towards the Galactic bulge. In this paper
we focus on the population of red clump giant stars at the Galac-
tic centre. Red clump giants are metal-rich horizontal branch stars
(Stanek et al. 2000 and references therein). Theoretically, one ex-
pects their magnitudes to have (small) variations with metallicity,
age and initial stellar mass (Girardi & Salaris 2001). Empirically
they appear to be reasonable standard candles in the I-band with
little dependence on metallicities (Udalski 2000; Zhao et al. 2001).
3 METHODS
Stanek et al. (1997) used RCG stars in 12 fields (see Fig. 1)
observed during the first phase of the OGLE microlensing ex-
periment, OGLE-I, to constrain several analytic models of the
Galactic bar density distribution. Babusiaux & Gilmore (2005),
Nishiyama et al. (2005) and Cabrera-Lavers et al. (2007) similarly
used IR observations of RCGs to trace the bulge stellar density. We
follow similar procedures to extract RCG stars from the OGLE-II
Galactic bulge fields and to constrain analytic models.
3.1 Sample selection
We compute the reddening-independent magnitude IV−I for all
stars in each of the 45 OGLE-II fields:
IV−I = I − AI/(AV − AI) (V − I)
where AI and AV are the extinctions in the I and V bands deter-
mined by Sumi (2004). We select stars which have I < 4(V −
I)+k, where k is a constant chosen for each field that excludes the
main-sequence dwarf stars, and IV−I < 14.66, which corresponds
to the magnitude of RCG stars closer than 15 kpc1. Fig. 2 shows the
sample of stars selected from the IV−I, (V −I) CMD for OGLE-II
field 1.
The selected stars are then collected in IV−I magnitude bins,
see Fig. 3. A function comprised of quadratic and Gaussian com-
ponents is used to model this number count histogram in each of
the OGLE-II fields:
N(x ≡ IV−I) = a+bx+cx2+
2πσRC
− (IV−I,RC − x)
2σRC2
1 We assume the fiducial RCG star at 15 kpc has an absolute magnitude and
colour of I0 = −0.26 and (V − I)0 = 1.0 respectively, with AI/(AV −
AI) = 0.96.
0.5 1 1.5 2 2.5 3
17PSfrag replacements
V − I
Figure 2. Reddening-independent magnitude vs colour diagram for OGLE-
II field 1. The red clump is clearly visible. Stars are selected (black dots)
using the criteria I < 4(V − I) + k, where k is a constant chosen for each
field, and IV−I < 14.66 (solid lines, see text).
12 12.5 13 13.5 14 14.5 15
PSfrag replacements
Figure 3. Number count histogram for selected stars in OGLE-II field 1.
The heavy solid line is the best-fitting model of the form given by equa-
tion (1). The last histogram bin is generally not included in the fitting due
to incompleteness effects near the limiting magnitude of IV−I < 14.66.
where σRC is the spread of the red clump giant magnitudes,
IV −I,RC is the mean apparent magnitude of the red clump, NRC
and a, b and c are coefficients for the Gaussian and quadratic com-
ponents respectively. These six parameters are allowed to vary for
each of the OGLE-II fields and the best-fitting values obtained by
minimising χ2 =
[(N − Nobs,i)/σi]2 where the sum is
taken over all i = 1 . . . 26 histogram bins which cover the range
12 6 IV−I 6 14.6. The error on histogram number counts is as-
sumed to be σi =
Nobs,i , i.e. Poissonian.
The errors on the mean magnitude and distribution width of
the red clump stars, ξIV−I,RC and ξσRC respectively, are deter-
mined using a maximum-likelihood analysis (see e.g. Lupton et al.
1987).
We determine the distance modulus to the red clump in each
field as:
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 3
−8−6−4−20246810
replacem
Galactic longitude (◦)
Figure 1. The position of the 45 OGLE-II fields used in this analysis. The solid black regions indicate the location of fields used in Stanek et al. (1997).
µ = 5 log(d)− 5 = I − I0,RC
= IV −I,RC +R (V − I)0,RC − I0,RC (2)
where d is the distance to the red clump measured in parsecs,
IV −I,RC is the fitted peak reddening-independent magnitude of the
red clump, R is the mean value of AI/(AV − AI) for each field.
I0,RC = −0.26 ± 0.03 (Alves et al. 2002) and (V − I)0,RC =
1.0± 0.08 (Paczynski & Stanek 1998) are the mean absolute mag-
nitude and colour of the population of local red clump giant stars.
We assume that the properties of the local population of red clump
giant stars are the same as the Galactic bulge population, however
it is likely that population effects are significant. We discuss the
effects of red clump population effects in Section 4.1.
4 DISTANCE MODULI OF RCGS
Table 1 lists the fitted parameters IV −I,RC ± ξIV−I,RC and σRC ±
ξσRC , along with the mean value of R for each field. The dis-
tance modulus, µ, computed from equation (2) showed that there
was a significant offset: the fitted mean red clump giant magni-
tudes were uniformly too faint compared to that expected of typ-
ical local RCG stars at 8 kpc, resulting in overly large distance
moduli. Sumi (2004) found that the OGLE-II RCGs are approxi-
mately 0.3 mag fainter than expected when assuming that the pop-
ulation of RCGs in the bulge is the same as local. For this reason,
we apply an offset of 0.3 mag to the distance moduli computed
above. The shifted distance moduli µ′ = µ − 0.3 are given in Ta-
ble 1. The true line-of-sight dispersion, σlos, can be approximated
by σlos = (σ
RC − σ2RC,0 − σ2e )1/2 where σRC is the Gaussian dis-
persion fitted using equation (1), σRC,0 is the intrinsic dispersion
of the RCG luminosity function and σe is an estimate of the pho-
tometric errors (Babusiaux & Gilmore 2005). We use σRC,0 = 0.2
(see Section 2) and σe = 0.02, along with the tabulated values of
σRC to determine σlos. Fig. 4 shows the mean distance to the red
clump stars in each of the 45 OGLE-II fields listed in Table 1. The
fields with Galactic longitude −4◦ 6 l 6 6◦ show clear evidence
of a bar, with a major axis oriented at ≃ 25◦ to the Sun-Galactic
centre line-of-sight. For fields with l > 6◦ and l < −4◦ the mean
position of the red clump stars do not continue to trace the major
axis of a linear structure. Babusiaux & Gilmore (2005) find similar
evidence that the position of the red clump stars is not predicted by
a bar for l = −9.7◦, suggesting that this is a detection of the end
of the bar, or the beginning of a ring-like structure. We investigate
these possibilities in Section 6.1.
The uncertainties on the mean position of the red clump are
large; the largest term in the error expression for ξIV−I,RC arises
from the relatively large uncertainty in the intrinsic colour of the
RCGs. The true line-of-sight dispersions, σlos, are consistent with
a wide range of bar position angles, but the mean position of the
red clump in each direction strongly suggest a bar oriented along
a direction consistent with that determined by the previous work
referred to in Section 1.
c© 2005 RAS, MNRAS 000, 000–000
4 Rattenbury et al.
Table 1. Fitted values of the red clump mean magnitude, IV −I,RC, and Gaussian dispersion, σRC for RCG stars selected from 45 OGLE-II fields. The
mean selective extinction R is also given. The shifted distance modulus µ′ = µ − 0.3 for each field is computed via equation (2), see text. σlos is the true
line-of-sight distance dispersion of the red clump giant stars. N is the total number of stars selected from each CMD, see Section 3.1
Field l b IV −I,RC σRC R µ
′ σlos N
1 1.08 -3.62 13.616 ±0.005 0.2936 ±0.0043 0.964 ±0.02 14.55 ±0.09 0.21 31002
2 2.23 -3.46 13.536 ±0.005 0.3130 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.24 33813
3 0.11 -1.93 13.664 ±0.003 0.2461 ±0.0026 0.964 ±0.04 14.60 ±0.09 0.14 66123
4 0.43 -2.01 13.655 ±0.003 0.2517 ±0.0026 0.964 ±0.04 14.59 ±0.09 0.15 65748
5 -0.23 -1.33 13.630 ±0.003 0.2809 ±0.0030 0.964 ±0.06 14.56 ±0.10 0.20 43493
6 -0.25 -5.70 13.606 ±0.011 0.4197 ±0.0088 0.964 ±0.03 14.54 ±0.09 0.37 12085
7 -0.14 -5.91 13.589 ±0.012 0.4255 ±0.0100 0.964 ±0.03 14.52 ±0.09 0.38 11328
8 10.48 -3.78 13.366 ±0.002 1.4277 ±0.0013 0.964 ±0.03 14.30 ±0.09 1.41 10248
9 10.59 -3.98 13.383 ±0.012 0.5855 ±0.0110 0.964 ±0.03 14.32 ±0.09 0.55 9971
10 9.64 -3.44 13.407 ±0.002 1.4178 ±0.0013 0.964 ±0.03 14.34 ±0.09 1.40 12068
11 9.74 -3.64 13.438 ±0.015 0.3902 ±0.0125 0.964 ±0.04 14.37 ±0.09 0.33 11345
12 7.80 -3.37 13.381 ±0.008 0.4692 ±0.0072 0.964 ±0.04 14.31 ±0.09 0.42 15936
13 7.91 -3.58 13.389 ±0.009 0.4234 ±0.0081 0.964 ±0.03 14.32 ±0.09 0.37 15698
14 5.23 2.81 13.550 ±0.006 0.3131 ±0.0053 0.964 ±0.04 14.48 ±0.09 0.24 27822
15 5.38 2.63 13.564 ±0.007 0.3027 ±0.0059 0.964 ±0.04 14.50 ±0.09 0.23 24473
16 5.10 -3.29 13.487 ±0.007 0.3200 ±0.0057 0.964 ±0.03 14.42 ±0.09 0.25 22055
17 5.28 -3.45 13.474 ±0.007 0.3270 ±0.0058 0.964 ±0.03 14.41 ±0.09 0.26 23132
18 3.97 -3.14 13.471 ±0.005 0.2983 ±0.0044 0.964 ±0.02 14.40 ±0.09 0.22 32457
19 4.08 -3.35 13.491 ±0.006 0.3026 ±0.0048 0.964 ±0.03 14.42 ±0.09 0.23 30410
20 1.68 -2.47 13.583 ±0.004 0.2726 ±0.0031 0.964 ±0.03 14.52 ±0.09 0.18 49900
21 1.80 -2.66 13.596 ±0.004 0.2886 ±0.0034 0.964 ±0.02 14.53 ±0.09 0.21 45578
22 -0.26 -2.95 13.741 ±0.004 0.2669 ±0.0034 0.964 ±0.04 14.67 ±0.09 0.18 42914
23 -0.50 -3.36 13.724 ±0.004 0.2778 ±0.0037 0.964 ±0.04 14.66 ±0.09 0.19 36030
24 -2.44 -3.36 13.817 ±0.004 0.2638 ±0.0037 0.964 ±0.04 14.75 ±0.09 0.17 35351
25 -2.32 -3.56 13.810 ±0.004 0.2695 ±0.0039 0.964 ±0.03 14.74 ±0.09 0.18 31801
26 -4.90 -3.37 13.815 ±0.005 0.2897 ±0.0046 0.964 ±0.02 14.75 ±0.09 0.21 26940
27 -4.92 -3.65 13.794 ±0.006 0.2782 ±0.0050 0.964 ±0.02 14.73 ±0.09 0.19 24603
28 -6.76 -4.42 13.785 ±0.010 0.3081 ±0.0082 0.964 ±0.02 14.72 ±0.09 0.23 13702
29 -6.64 -4.62 13.762 ±0.009 0.2792 ±0.0083 0.964 ±0.02 14.70 ±0.09 0.19 12893
30 1.94 -2.84 13.570 ±0.004 0.2746 ±0.0034 0.964 ±0.03 14.50 ±0.09 0.19 41748
31 2.23 -2.94 13.535 ±0.004 0.2892 ±0.0036 0.964 ±0.02 14.47 ±0.09 0.21 40623
32 2.34 -3.14 13.528 ±0.004 0.3001 ±0.0038 0.964 ±0.02 14.46 ±0.09 0.22 35954
33 2.35 -3.66 13.559 ±0.005 0.3206 ±0.0045 0.964 ±0.02 14.49 ±0.09 0.25 30882
34 1.35 -2.40 13.608 ±0.003 0.2711 ±0.0031 0.964 ±0.03 14.54 ±0.09 0.18 52216
35 3.05 -3.00 13.533 ±0.005 0.3049 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.23 36796
36 3.16 -3.20 13.508 ±0.005 0.3104 ±0.0042 0.964 ±0.02 14.44 ±0.09 0.24 34437
37 0.00 -1.74 13.636 ±0.003 0.2488 ±0.0025 0.964 ±0.05 14.57 ±0.10 0.15 72098
38 0.97 -3.42 13.637 ±0.005 0.2911 ±0.0038 0.964 ±0.02 14.57 ±0.09 0.21 34675
39 0.53 -2.21 13.687 ±0.003 0.2524 ±0.0027 0.964 ±0.04 14.62 ±0.09 0.15 60217
40 -2.99 -3.14 13.854 ±0.004 0.2459 ±0.0036 0.964 ±0.04 14.79 ±0.09 0.14 35426
41 -2.78 -3.27 13.857 ±0.004 0.2543 ±0.0035 0.964 ±0.04 14.79 ±0.09 0.16 34118
42 4.48 -3.38 13.494 ±0.006 0.3354 ±0.0051 0.964 ±0.03 14.43 ±0.09 0.27 27377
43 0.37 2.95 13.839 ±0.004 0.2654 ±0.0033 0.964 ±0.05 14.77 ±0.10 0.17 40730
45 0.98 -3.94 13.595 ±0.006 0.3302 ±0.0047 0.964 ±0.03 14.53 ±0.09 0.26 29009
46 1.09 -4.14 13.616 ±0.006 0.3189 ±0.0047 0.964 ±0.03 14.55 ±0.09 0.25 26027
4.1 RCG population effects
Sumi (2004) found that the extinction-corrected magnitudes of
RCG stars in the OGLE-II bulge fields were 0.3 mag higher than
that of a RCG with an intrinsic magnitude equal to those of lo-
cal RCGs, placed at a distance of 8kpc. Sumi (2004) notes that
the cause of this offset is uncertain but may be resolved when de-
tailed V-band OGLE-II photometry of RR Lyrae stars will allow
improved extinction zero-point estimations for all fields. There is
evidence that the properties of bulge RCGs are different to local
RCG stars, in particular age and metallicity, resulting in a differ-
ent average absolute magnitude (see e.g. Percival & Salaris (2003);
Salaris et al. (2003)). This could in part explain the observed off-
set of 0.3 mag between local and bulge RCG stars. The distance
modulus plotted in Fig. 4 is as for equation (2):
µ = IV −I,RC +R (V − I)0,RC − I0,RC + κ
where κ = −0.3 mag is the offset between the observed bulge
RCG population and a fiducial local RCG placed at 8 kpc. Includ-
ing the difference in intrinsic magnitude between local and bulge
RCG populations we have:
µ = IV −I,RC +R (V − I)0,RC − I0,RC +∆IRC + ν
where ∆IRC is the intrinsic RCG magnitude difference between lo-
cal and bulge populations and ν is a magnitude offset arising from
effects other than population differences. The theoretical popula-
tion models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 5
−2−1012
PSfrag replacements
x (kpc)
l = −10◦l = −5◦l = +5◦l = +10◦
Figure 4. Structure of the inner region of the Milky Way as traced by red
clump giant stars extracted from the OGLE-II microlensing survey data.
The mean position of the red clump for 45 OGLE-II fields is indicated by
black dots with errors shown by black lines. The grey lines along each line
of sight indicate the 1-σ spread in distances obtained from fitting equa-
tion (1) to the red clump data in each field and correcting for intrinsic red
clump luminosity dispersion and photometric errors; see text. A position
angle of 25◦ is shown by the thin solid line. Lines-of-sight at l = ±5◦ and
l = ±10◦ are indicated along the top axis.
The remaining magnitude offset ν = −0.2 can be accounted for
by decreasing the assumed value of the Galactocentric distance R0
from 8 kpc to 7.3 kpc. This assumption of R0 is in agreement
with the value of 7.52±0.10 (stat) ±0.32 (sys) kpc determined by
Nishiyama et al. (2006), and that of 7.63±0.32 kpc determined by
Eisenhauer et al. (2005). We note that as long as the stellar popula-
tion is uniform in the Galactic bulge (an assumption largely consis-
tent with the data), then this unknown offset only affects the zero
point (and hence the distance to the Galactic centre), and the abso-
lute bar scale lengths. However the ratios between bar scale lengths
are very robust, which we demonstrate in Section 6.2 where we
determine the bar parameters for several values of R0.
A metallicity gradient across the Galactic bulge would have an
effect on the intrinsic colour and absolute magnitude of bulge RCG
stars as a function of Galactic longitude. Pérez et al. (2006) find
that there is a connection between metallicity gradient and struc-
tural features in a sample of 6 barred galaxies. Minniti et al. (1995)
measured metallicities for K giants in two fields at 1.5 kpc and 1.7
kpc from the Galactic centre. The average metallicity was [Fe/H]
= −0.6, lower than that of K giants in Baade’s window. However,
no such metallicity gradient was reported for Galactocentric dis-
tance range 500 pc – 3.5 kpc by Ibata & Gilmore (1995). Similarly,
Santiago et al. (2006) determine that there is no metallicity gradient
within angles 2.2◦ to 6.0◦ of the Galactic centre (corresponding to
300 – 800 pc from the GC assuming a Sun-GC distance of 8 kpc).
5 MODELLING THE BAR
We continue to investigate the structure of the Galactic bar by fit-
ting analytic models of the stellar density to the observed RCG data
in the OGLE-II fields. Following Stanek et al. (1997) we use the an-
alytic models of Dwek et al. (1995) to fit the observed data. Three
model families (Gaussian, exponential and power-law) are used:
ρG1 = ρ0 exp(−r2/2) (3)
ρG2 = ρ0 exp(−r2s /2) (4)
ρG3 = ρ0r
exp(−r3) (5)
ρE1 = ρ0 exp(−re) (6)
ρE2 = ρ0 exp(−r) (7)
ρE3 = ρ0K0(rs) (8)
ρP1 = ρ0
1 + r
ρP2 = ρ0
r(1 + r)3
ρP3 = ρ0
1 + r2
where K0 is the modified Bessel function of the second kind and
The co-ordinate system has the origin at the Galactic centre,
with the xy plane defining the mid-plane of the Galaxy and the
z direction parallel to the direction of the Galactic poles. The x
direction defines the semi-major axis of the bar. The functions are
rotated by an angle α around the z-axis. An angle of α = ±π
corresponds to the major axis of the bar pointing towards the Sun.
The functions can also be rotated by an angle β around the y axis,
corresponding to the Sun’s position away from the mid-plane of the
Galaxy.
We aim to fit the observed number count histograms for each
field as a function of magnitude. Given a magnitude IV−I, the num-
ber of stars with this magnitude is (Stanek et al. 1997):
N(IV−I) = c1
Z smax
ρ(s)s
Φ(L)Lds (12)
where the integration is taken over distance smin = 3kpc < s <
smax = 13 kpc. We perform the integration over this range of R0±
5 kpc as we do not expect the tri-axial bulge density structure to
exceed these limits. The constant c1 is dependent on the solid angle
subtended around each line of sight. The luminosity L is given by
L = c2s
−0.4IV−I
c© 2005 RAS, MNRAS 000, 000–000
6 Rattenbury et al.
and c2 is a constant. The luminosity function Φ(L) is
Φ(L) = N0
2π σRC
− (L− LRC)
2σ2RC
where LRC is the luminosity of the red clump and σRC is the in-
trinsic spread in red clump giant luminosity and is held constant.
There are ten parameters to be determined in the above equa-
tions: the three bar scale lengths x0, y0, z0; the bar orientation and
tilt angles α and β; the luminosity function parameters N0, NRC ,
γ and LRC and the density function parameter ρ0.
There is evidence that the centroid of the bar is offset from
the centre of mass of the Galaxy, a feature commonly observed
in external galaxies (Stanek et al. 1997; Nishiyama et al. 2006 and
references therein). We include an additional parameter in the mod-
elling process, δl, which allows for this possible centroid offset.
Theoretical number counts are computed for the i = 1 . . .M fields
at longitudes li as Ni(li + δl) where the offset parameter δl is de-
termined over all fields for a given density model.
We apply an exponential cut-off to the density functions simi-
lar to that of Dwek et al. (1995):
f(r) =
1.0 r =
x2 + y2
6 rmax
− (r−rmax)
r > rmax
where r is in kpc, r0 = 0.5 kpc and rmax is a cut-off radius. The
theoretical constraint on the maximum radius of stable stellar or-
bits is the co-rotation radius, rmax. We adopt the co-rotation value
of rmax = 2.4 kpc determined by Binney et al. (1991) in equa-
tion (13). We later repeat the modelling process without this theo-
retical cut-off, see Section 6.2.
Stanek et al. (1997) included a further fitting parameter to ac-
count for a possible metallicity gradient across the bulge, but found
that this did not significantly affect the bar parameters. The discus-
sion in Section 4.1 suggests that there is no appreciable metallic-
ity gradient over the bulge region investigated here. We therefore
do not include an additional model parameter corresponding to a
metallicity gradient in the model fitting analysis.
The model fitting was performed using a standard non-linear
Neadler-Mead minimisation algorithm. For each of the nine density
profile models we minimise χ2:
(Nik(IV−I)− bNik(IV−I))2
where the summations are taken over each of the 26 IV−I histogram
bins in M OGLE-II fields, Nk(IV−I) is the observed histogram
data for field k and bNk(IV−I) is the model number count histogram
from equation (12). The error in Nik(IV−I) was taken to be σik =
Nik(IV−I).
6 RESULTS
The 11 parameters x0, y0, z0, α, β, A, N0, NRC, γ, LRC and
δl were fitted for each of the nine density profiles given in equa-
tions. (3 – 11) for the 34 OGLE-II fields2 which have −4◦ 6 l 6
6◦. A naive interpretation of Fig. 4 is that fitting all fields including
those with l > 6◦ and l < −4◦ with a single bar-like structure
2 Field 5 was excluded due to poor understanding of the dust extinction in
this field.
would not be successful and indeed initial modelling using data
from all fields resulted in bar angles of ≃ 45◦. As seen in Fig. 4
this result is consistent with the mean position angle of all fields,
but clearly does not describe the bar correctly. We therefore begin
modelling the bar using data from the central 34 fields which have
−4◦ 6 l 6 6◦. In Section 6.1 we proceed by including the data
from fields with l > 6◦ and l < −4◦ in the modelling process and
in Section 6.2 we consider the effect of changing R0.
The range of field latitudes is small and therefore the infor-
mation available in the latitude direction from the total dataset un-
likely to be able to constrain β strongly. Initial modelling runs held
β constant at 0◦. Once a χ2 minimum was determined for each
model holding β = 0◦, this parameter was allowed to vary with the
ten other parameters.
Figure 5 shows the number count histograms of observed red
clump giants in the 34 OGLE-II fields with −4◦ 6 l 6 6◦, along
with the best-fitting model (equation (12)) using the E2 density
profile given in equation (7) as an example3. Each set of axes is
arranged roughly in order of decreasing Galactic longitude. The
magnitude of the red clump peak increases with decreasing longi-
tude, consistent with a bar potential.
The observed number count histograms are reasonably well-
fitted by all models in most fields. The most obvious exceptions
are fields 6, 7, 14, 15 and 43. These fields are at the most extreme
latitudes represented in the data. In the case of fields 14 and 15
the algorithm fails to fit satisfactorily both observed histograms for
a given model. Upon closer inspection of the observed histogram
data for these two fields, it is noted that the total number of stars in
the field 15 histogram is significantly less than that for field 14. The
models preferred by all fields clearly cannot reproduce the decline
in total star count between fields 14 and 15. Upon inspection of the
CCD pixel position of stars for field 15 it was found that there are
no stars recorded in a region along one edge of the field. The lack
of stars in this strip is due to missing V -band data for these stars.
Similarly, there is a lack of stars in field 15 in the lower right corner,
due to heavy dust extinction in this area.
All models predict the peak of the red clump to be at a lower
magnitude than that observed for field 43. A similar offset is seen
in the other high latitude fields 14 and 15. These fields are all at
latitude b ≃ 3◦ and the observed offset in all three fields is likely
to be related to this common position in latitude. The cause of this
effect is currently unknown. The observed offset between the maxi-
mum density position predicted by the tri-axial bar models and that
observed for some fields may imply some asymmetry of the bar in
the latitude direction, as suggested by Nishiyama et al. (2006) and
references therein.
Fields 6 and 7, both at b ≃ −6◦, are also poorly fitted by
all models. The histograms of observed red clump stars in these
fields show a curious double peak. Clearly the density functions
equations (3 – 11) are inadequate for describing such a feature.
This double-peaked structure may be the result of another popu-
lation of stars lying along the line of sight, at a distance differ-
ent to the main bulge population. The best-fitting model curves
typically have a peak at magnitudes corresponding to the highest
peak in the observed histogram, thereby suggesting that the sec-
ond population, if real, and composed of a significant fraction of
RCGs, exists between us and the bulge population. A population
of asymptotic giant branch (AGB) stars (Alves & Sarajedini 1999;
3 Similar figures showing the best-fitting models to the data using all den-
sity profiles equations (3 – 11) appear in the on-line supplementary material.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 7
15 17 14 16 42
19 18 36 35 33
32 31 2 30 21
4000 20 L 34 L 46 R 1 R
200045 R
38 L 39 R 4 R 3 R
43 L 22 L 23 L 7 R
13 14 15
10006 R
12 13 14
  13 14
  13 14
13 14 15
PSfrag replacements
Figure 5. Red clump number count histograms and best-fitting profiles using density model E2 for 34 OGLE-II fields with −4◦ 6 l 6 6◦ . Fields are arranged
roughly in order of descending Galactic longitude. Field numbers are given in each set of axes. A ‘L’ or ‘R’ in the axes of rows 4 – 6 indicates whether the
vertical scale corresponds to left or right vertical axis of the row.
c© 2005 RAS, MNRAS 000, 000–000
8 Rattenbury et al.
Faria et al. 2007) may also be the cause of the secondary peak in
the number count histograms for these and similar fields (e.g. field
10, Fig. 9). Closer investigation of the stellar characteristics and
kinematics of stars in these fields may help in describing the origin
of the second number count peak.
The best-fitting parameter values for all the nine density mod-
els are presented in Table 2. Two sets of parameter values are given,
one where the tilt angle, β, was held at 0.0◦, and the other where β
was allowed to vary. The values of χ2 are well in excess of the num-
ber of degrees of freedom: Ndof = 26 × 34 − 11 = 873, and can
therefore not be used to set reliable errors on the parameters values
in the usual way. It is likely that the errors on the histogram data
are underestimated, resulting in extreme values of χ2. We assumed
that the errors on the number count data would be Poissonian and
we have not added any error in quadrature that might arise from any
systematic effects. Furthermore, as mentioned above, the extreme
latitude fields 6, 7, 14, 15 and 43 are poorly fit by every model. The
cause for this is unknown, however the effect is to add a relatively
high contribution to the total value of χ2 compared to other fields.
The values of χ2 can only be used to differentiate between
the various models in a qualitative manner. In terms of relative per-
formance, model E3 best reproduces the observed number count
histograms for the fields tested here, and model G2 provides the
worst fit to the data.
The position angle of the bar semi-major axis with respect to
the Sun-Galactic centre line-of-sight is α′ and is related to the ro-
tation angle α by α′ = α− sgn(α)π/2. We find α′ = 20◦ – 26◦.
The addition of β as another variable parameter does not result in a
significant improvement in χ2. The lack of information in the lati-
tude direction means that the data have little power in constraining
parameters such as β.
The absolute values of the scale lengths in Table 2 cannot
be directly compared between models. The axis ratios x0/y0 and
x0/z0 can be compared however. Fig. 6 shows the axis ratios x0/y0
and x0/z0 for the nine models tested. The ratio of the major bar axis
scale length to the minor bar axis in the plane of the Galaxy, x0/y0,
has values 3.2 – 3.6, with the exception of that for model E1, for
which x0/y0 = 4.0. Stanek et al. (1997) found x0/y0 = 2.0 –
2.4, with the exception of model E1 for which x0/y0 = 2.9. It is
interesting to note that our values of x0/y0 have a similar range
to that of Stanek et al. (1997), when we exclude the outlying result
from the same model (E1). Our results do suggest a slimmer bar,
i.e. higher values of x0/y0 compared to Stanek et al. (1997). Both
these results and the results of Stanek et al. (1997) are consistent
with the value of x0/y0 ≃ 3± 1 reported by Dwek et al. (1995).
The ratio of the major bar axis scale length to the verti-
cal axis scale length x0/z0 was found to be 3.4 – 4.2, with the
same exception of model E1, which has an outlying value of
x0/z0 = 6.6. Again, the range of ratio values is comparable to
that of Stanek et al. (1997) who found x0/z0 = 2.8 – 3.8, with the
exception of model E1 again, which had x0/z0 = 5.6.
The mean axis ratios are x0 : y0 : z0 are 10 : 2.9 : 2.5.
Excluding the outlying results from the E1 model, the ratios are
10 : 3.0 : 2.6. These results suggest a bar more prolate than the
general working model with 10 : 4 : 3 (Gerhard 2002).
6.1 Evidence of non-bar structure? Including wide longitude
fields
Fig. 4 shows the mean position of the bulge red clump stars. At
longitudes −4◦ 6 l 6 6◦ the bulge red clump stars follow the main
axis of the bar. At greater angular distances, the mean positions of
G1 G2 G3 E1 E2 E3 P1 P2 P3
PSfrag replacements
Model
x0/y0, 34 fields
x0/z0, 34 fields
x0/y0, 44 fields
x0/z0, 44 fields
Figure 6. Scale length ratios x0/y0 (squares) and x0/z0 (triangles) for all
models. Solid and open symbols show the ratios for the best-fitting models
using the 34 fields with −4◦ 6 l 6 6◦, and all 44 fields (see Section 6.1)
respectively.
the red clump stars are clearly separated from the main bar axis.
Babusiaux & Gilmore (2005) find similar evidence and suggest that
this could indicate either the end of the bar or the existence of a
ring-like structure. We used the OGLE-II data to investigate these
possibilities.
The OGLE-II data from fields with longitude l < −4◦ and
l > 6◦ were excluded from the original bar modelling based on
the findings illustrated by Fig. 4, and because initial modelling ef-
forts using all data from all fields simultaneously failed to produce
satisfactory results. Using the best-fitting model (E3) determined
using the central OGLE-II fields, Fig. 7 shows the predicted num-
ber count densities for fields with l < −4◦ and l > 6◦, overlaid on
the observed number count histograms.
It is clear from the left hand column of Fig. 7 that the mean
position of the red clump in fields with l > 6◦ are significantly
removed from that predicted by the E3 linear bar model. However,
the right hand column of Fig. 7 shows that the observed position of
the red clump is in rough agreement with that predicted from the
model for fields with l < −4◦. We test this further, by taking lines
of sight through the best-fitting E3 model and computing the num-
ber count profile using equation (12). The mean magnitude of the
red clump is converted to a distance for each line of sight. Fig. 8
shows the density contours of the analytic E3 model overlaid with
the mean position of the red clump determined in this way. The fea-
tures of Fig. 4 at longitudes |l| & 5◦ are qualitatively reproduced in
Fig. 8. This suggests that the observed data are likely to be consis-
tent with a single bar-like structure, rather than requiring additional
structure elements. We note however, that the abrupt departure of
the location of the density peak in Fig. 4 at longitudes l . 5◦ is
best reproduced in Fig. 8 by the analytical density model when the
exponential cut-off of equation (13) is imposed.
Fields with l < −4◦ and l > 6◦ should be included in the
modelling of the bar, as they are likely to provide further constraints
on the final model. Initial attempts to model the bar using all fields
failed because the fitting algorithm found a model with a bar angle
of 45◦, consistent with the observed data, but not consistent with
the current understanding of the bar. Fig. 8 shows that the mor-
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 9
  13   14   15
  13   14   15
PSfrag replacements
Figure 7. Predicted number count profiles using the best-fitting E3 model (equation (12)) with the observed number count histograms for OGLE-II fields with
l > 6◦ (left column) and l < −4◦ (right column). The predicted number count profiles are significantly different for fields with l > 6◦, however the predicted
magnitude peak of the red clump roughly coincides with those observed for fields with l < −4◦ .
c© 2005 RAS, MNRAS 000, 000–000
10 Rattenbury et al.
Table 2. Best fitting parameter values for all density models (equations 3 – 11) fitted to the number count histograms from the 34 OGLE-II fields with
−4◦ 6 l 6 6◦. Two sets of parameter values are given, one where the tilt angle, β, was held at 0.0◦, and the other where β was allowed to vary. α′ is the
position angle of the bar semi-major axis with respect to the Sun-Galactic centre line-of-sight.
Bar orientation (◦) Scale lengths (pc) Axis ratios
Model β α′ x0 y0 z0 χ2 x0 : y0 : z0
G1 0.00 21.82 1469.00 449.28 391.80 15302.31
-0.99 21.82 1469.41 448.78 391.62 15282.57 10.0 : 3.1 : 2.7
G2 0.00 19.76 1206.92 375.94 353.81 16558.28
-0.28 19.79 1203.37 377.22 354.24 16547.35 10.0 : 3.1 : 2.9
G3 0.00 25.57 5289.91 1512.32 1277.52 14306.20
-0.30 25.55 5288.64 1511.72 1277.98 14306.06 10.0 : 2.9 : 2.4
E1 0.00 21.63 2143.51 540.08 325.49 15766.77
0.06 21.62 2135.11 539.86 325.48 15753.98 10.0 : 2.5 : 1.5
E2 0.00 23.68 1034.30 306.39 261.43 11930.61
-0.01 23.68 1034.24 306.38 261.47 11930.58 10.0 : 3.0 : 2.5
E3 0.00 21.92 1039.86 323.80 299.08 10722.69
0.47 21.97 1039.62 323.75 298.75 10711.13 10.0 : 3.1 : 2.9
P1 0.00 24.66 1988.33 562.49 478.93 13952.47
0.08 24.66 1988.10 562.39 478.84 13952.09 10.0 : 2.8 : 2.4
P2 0.00 25.07 3906.98 1088.90 927.00 15183.47
0.76 25.03 3907.42 1089.36 927.61 15180.89 10.0 : 2.8 : 2.4
P3 0.00 23.81 1992.91 580.32 491.30 12123.07
0.10 23.82 1993.85 580.33 491.09 12122.82 10.0 : 2.9 : 2.5
−3 −2 −1 0 1 2 3 
PSfrag replacements
x (kpc)
Figure 8. Mean position of the red clump determined using the best-fitting
analytic bar model (E3). Gray lines indicate 11 lines-of-sight through the
E3 model, with l = −10◦,−9.8◦, . . . , 10◦ and b = 0.0◦ and contours in-
dicating the density profile of the model. The dashed line shows the orienta-
tion of the bar major axis, with the Sun positioned at (0,−8) kpc. Number
count profiles are generated using equation (12). The mean red clump mag-
nitude for each line-of-sight was converted to a distance, and plotted as solid
circles. Black circles indicate the mean position of the red clump when the
exponential cut-off, equation (13) is applied. Grey circles indicate the mean
RCG position when this cut-off is not imposed. The observed features in
Fig. 4 at longitudes |l| & 5◦ are more clearly reproduced when the cut-off
is applied to the analytic model.
phology indicated by the observed data in Fig. 4 can be explained
using a bar structure oriented at ∼ 20◦ to the Sun-GC line-of-sight.
Cabrera-Lavers et al. (2007) also note that the line-of-sight density
for tri-axial bulges reaches a maximum where the line-of-sight is
tangential to the ellipsoidal density contours. These authors also
quantify the difference between the positions of maximum density
and the intersection of the line-of-sight with the major axis of the
The fitting procedure for all nine tri-axial models was re-
peated, including now the 10 OGLE-II fields with l < −4◦ or
l > 6◦. Figure 9 shows the best-fitting number count profiles for the
ten non-central fields for the ‘G’ type of analytic tri-axial model4.
The best-fitting number count profiles for fields with−4◦ 6 l 6 6◦
are not significantly different to those shown in Figure 5. The main
features of the observed number count profiles are reproduced by
the best-fitting analytic models. There are however instances where
details of the observed number counts are not reproduced. Magni-
tude bins with IV−I & 13 are poorly fitted by the G-type models in
positive longitude fields. Two of the E-type models (E2, E3) sim-
ilarly fail to trace these data. The number count profile for model
E1 shows a flattened peak, resulting from the pronounced box-like
nature of the density profile. All the P-type models appear to repro-
duce the data at these magnitudes with similar profiles. All models
predict a RCG peak at magnitudes greater than that observed in
fields 28 and 29. This offset between predicted and observed RCG
peak location is also seen in fields 26 and 27 but to a lesser degree.
The best-fitting parameters are shown in Table 3. The model
which results in the lowest value of χ2 is still E3. Model G2 also
remains the worst-fitting model to the data.
We now consider the change to the best-fitting model param-
eters when data from all fields are used in the analysis. The po-
sition angle of the bar semi-major axis with respect to the Sun-
Galactic centre line-of-sight is now α′ = 24◦ – 27◦. The bar
4 Similar figures showing the best-fitting models using the ‘E’ and ‘P’ type
density profiles appear in the on-line supplementary material.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 11
111111
101010
131313
  13   14   15
1000 121212
2000262626
272727
292929
  13   14   15
282828
PSfrag replacements
Figure 9. Best-fitting number count profiles using the Gaussian ‘G’ type tri-axial models with the observed number count histograms for OGLE-II fields with
l > 6◦ (left column) and l < −4◦ (right column). Solid, dashed and dot-dashed lines correspond to model subtype 1, 2 and 3 respectively.
c© 2005 RAS, MNRAS 000, 000–000
12 Rattenbury et al.
Table 3. Best fitting parameter values for all density models (equations 3-11) fitted to the number count histograms from all 44 OGLE-II fields.
Bar orientation (◦) Scale lengths (pc) Axis ratios
Model β α′ x0 y0 z0 χ2 x0 : y0 : z0
G1 0.75 27.06 1505.20 568.49 392.19 23808.97 10.0 : 3.8 : 2.6
G2 -7.68 24.49 1276.56 473.28 359.91 26571.42 10.0 : 3.7 : 2.8
G3 -0.72 26.56 4787.17 1680.64 1279.94 17397.76 10.0 : 3.5 : 2.7
E1 1.95 23.82 1901.43 626.89 324.93 19170.60 10.0 : 3.3 : 1.7
E2 0.97 26.43 986.60 356.65 260.88 15674.39 10.0 : 3.6 : 2.6
E3 2.50 25.54 1045.67 378.20 294.89 15110.75 10.0 : 3.6 : 2.8
P1 2.19 25.30 1810.98 609.40 478.93 16720.85 10.0 : 3.4 : 2.6
P2 3.67 25.16 3513.34 1160.72 927.54 18069.53 10.0 : 3.3 : 2.6
P3 -0.84 26.06 1876.75 658.13 487.76 15228.11 10.0 : 3.5 : 2.6
G1 G2 G3 E1 E2 E3 P1 P2 P3
PSfrag replacements
Model
x0,M=44/x0,M=34
y0,M=44/y0,M=34
z0,M=44/z0,M=34
Figure 10. Ratio of bar scale lengths from best-fitting models using data
from all 44 OGLE-II fields to those using data from the central 34 OGLE-II
fields.
scale lengths x0, y0 and z0 also changed; Fig. 10 shows the ra-
tio of the best-fitting scale lengths determined using all 44 fields
to those found using only the central 34 fields. On average, upon
including the data from wide longitude fields, the semi-major axis
scale length, x0, decreased by ∼ 5%; the semi-minor axis scale
length, y0, increased by ∼ 16% and the vertical scale length, z0,
remained essentially unchanged. The relatively large change in the
semi-minor axis scale length is intuitively understandable, as a bar
position angle of ∼ 25◦ with respect to the Sun-GC line-of-sight
mean the semi-minor axis direction has a large vector component
in the Galactic longitude direction. The additional constraints of
the data from fields at extended Galactic longitudes can therefore
have a pronounced effect on the fitted values of y0. Similarly, we
expect some weak correlation between the semi-major and semi-
minor axis scale lengths, evidenced by the slight decrease on av-
erage for the x0 values upon adding the data from wide longitude
fields. It is unsurprising that the values of z0 are unchanged, as
adding data from the wide longitude fields has little power in fur-
ther constraining model elements in the direction perpendicular to
the Galactic midplane.
As above, we consider the ratio of the axis scale lengths for
each model (see Fig. 6). The ratio x0/y0 for all nine models using
all field data has values 2.7 – 3.0. This range is now completely
consistent with the range 2.5 – 3.3 reported by Bissantz & Gerhard
(2002). The ratio x0/z0 now lies in the range 3.6 – 3.9, with the
exception of that for model G1 which has x0/z0 = 5.9. The range
x0/z0 = 3.6 – 3.9 is narrower than that found previously, 3.4 – 4.2,
yet still higher on average than 2.8 – 3.8 reported by Stanek et al.
(1997). The scale length ratios are now x0 : y0 : z0 = 10 : 3.5 :
2.6, which compared to those determined using only the central
OGLE-II fields (x0 : y0 : z0 = 10 : 3.0 : 2.6) are closer to the
working model proposed by Gerhard (2002) which has x0 : y0 :
z0 = 10 : 4 : 3.
There are features in the number count histograms that are not
faithfully reproduced by the analytic tri-axial bar models for wide
longitude fields. Specifically, the predicted number count disper-
sions around the RCG peak magnitudes for fields with l > 6◦ are
too small compared to that observed, see Fig. 9. While the location
of the observed maximum line-of-sight density can be reproduced
using just a bar, it is possible that the reason why the analytic bar
models underestimate the observed number count dispersions for
fields with l > 6◦ is because these models do not include elements
which correspond to extended aggregations of stars at or near the
ends of the bar. Clumps of stars at the ends of the bar would in-
crease the line-of-sight density dispersion at wide longitudes. Such
extra aggregations of stars have been observed in galaxies with
boxy or peanut-shaped bulges (Bureau et al. 2006). Their origin
may be due to the superposition of members of the x1 family of
orbits (Patsis et al. 2002; Binney & Tremaine 1987) many of which
have loops near the end of the bar. Alternatively, the aggregations
of stars near the ends of the bar may be the edge-on projection
of an inner ring (Bureau et al. 2006 and references therein). The
presence of another density structure such as a long thin bar ori-
ented at ∼ 45◦ (Cabrera-Lavers et al. 2007) might also contribute
to the relatively large line-of-sight density dispersion. The effects
of such a structure would be most pronounced at wide longitudes.
The presence of a spiral arm might similarly contribute to the large
line-of-sight density dispersion for these fields.
6.2 Varying R0
We consider in this section the effect on the fitted model parameters
of changing the Galactocentric distance R0. It was noted in Sec-
tion 4.1 that there is an offset between the peak magnitude of RCGs
observed in the OGLE-II data and that expected for a RCG, with ab-
solute magnitude similar to that of local RCGs, placed at a distance
of 8.0 kpc. The observed RCGs are systematically 0.3 mag fainter
than the fiducial local RCG at 8.0 kpc. There are two possible rea-
sons for this magnitude offset. Firstly, the adopted Galactocentric
distance of R0 = 8.0 kpc may be incorrect; secondly, there may be
population variations between local and bulge RCGs, resulting in
different intrinsic magnitudes in the two populations. In Section 4.1
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 13
we noted that the offset predicted from the theoretical population
models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1. We
postulated that if the remaining magnitude offset is completely ac-
counted for by a change in the Galactocentric distance, this would
mean that R0 = 7.3 kpc. Conversely, if we adopted the value
of R0 = 7.6 ± 0.32 kpc determined by Eisenhauer et al. (2005),
the magnitude offset would become −0.18 mag, where including
the population effect predicted by Girardi & Salaris (2001) would
leave an unaccounted-for offset of −0.08 mag. In this section we
present the results of further modelling where we apply magnitude
offsets which correspond to different values of R0.
In Section 6, we note that for the OGLE-II fields 6, 7, 14, 15
and 43, which have latitudes removed from the majority of OGLE-
II fields, there are systematic offsets between the RCG peak in ob-
served number count histograms and the predictions of all nine tri-
axial models tested. For this reason, we limited the fields used to
those which have Galactic latitude −5◦ 6 b 6 −2◦. We also noted
in Section 6 that due to the lack of field coverage in the latitude
direction the data are not effective at constraining the bar tilt an-
gle β. For this reason, β was held constant at 0.0◦ in the following
modelling analysis. During the previous modelling efforts of Sec-
tion 6 it was also found that applying the exponential cut-off of
equation (13) had little effect on determining the best-fitting bar
parameters. This cut-off was not applied in the following analysis.
The best-fitting bar parameters were determined for the nine
tri-axial bar models for the OGLE-II fields occupying the latitude
strip−5◦ 6 b 6 −2◦ using magnitude offsets corresponding to R0
values of 8.0 kpc, 7.6 kpc and 7.3 kpc. The resulting best-fitting
parameters are listed in Table 4.
From the results in Table 4 we see that the fitted bar opening
angle α increases for all models as R0 decreases. The fitted scale
lengths x0, y0 and z0 all decrease linearly as R0 decreases. The val-
ues of χ2 in Table 4 indicate that a smaller value of R0 is favoured
by all models. The ratio between the scale heights remains remark-
ably constant with varying R0, indicating that while the orientation
of the bar changes slightly with varying R0, the overall shape of
the bar does not change.
The mean values of the scale length ratios for all models, and
all values of R0 are x0 : y0 : z0 = 10 : 3.6 : 2.7. This result
is very close to that determined using data from all 44 OGLE-II
fields, see Section 6 above.
7 DISCUSSION
Red clump giant stars in the OGLE-II microlensing survey cata-
logue can be used as tracers of the bulge density over a large re-
gion towards the Galactic centre. Nine analytic tri-axial bar mod-
els were initially fitted to the number count histograms of red
clump stars observed in 34 OGLE-II fields which have −4◦ 6
l 6 6◦. The models all have the major axis of the bar oriented
at 20◦ – 26◦ to the Sun-Galactic centre line-of-sight. This orien-
tation is in agreement with the results of Stanek et al. (1997) and
Nikolaev & Weinberg (1997) which give a bar angle of 20◦ – 30◦,
and is marginally in agreement with the value of 12 ± 6◦ from
López-Corredoira et al. (2000). Bissantz & Gerhard (2002) obtain
best-fitting non-parametric models to the COBE/DIRBE L band
map of the inner Galaxy with bar angles of 20◦ – 25◦.
We find the ratio of the bar major axis scale length to minor
axis scale length in the plane of the Galaxy to be x0/y0 = 3.2
– 3.6, higher than the value of 2.0 – 2.4 reported by Stanek et al.
(1997), but consistent with the upper end of the range 2.5 – 3.3
found by Bissantz & Gerhard (2002). The ratio of bar major axis
scale length to vertical axis scale length was found to be x0/z0 =
3.4 – 4.2, again higher on average than that reported by Stanek et al.
(1997) who found x0/z0 = 2.8 – 3.8, and higher than the value
of ≃ 3.3 generally adopted (Gerhard 2002). The working model
proposed by Gerhard (2002) gives the scale length ratios as x0 :
y0 : z0 = 10 : 4 : 3. Our results suggest a more prolate model with
x0 : y0 : z0 = 10 : 3.0 : 2.6.
The observed separation of the mean position of red clump
giants from the bar major axis at |l| & 5◦ was shown to be a ge-
ometric effect, rather than evidence of a more complicated struc-
ture such as a ring. The observed data from these fields were used
to further constrain the bar models. The resulting bar position an-
gles was found to be 24◦ – 27◦. This narrower range is consis-
tent with the several values of the bar position angle found by pre-
vious studies. The bar scale length ratios were determined to be
x0 : y0 : z0 = 10 : 3.5 : 2.6, which are closer to the working
model proposed by Gerhard (2002) than those values found using
data from only the central 34 OGLE-II fields.
Reasons for the difference between the bar axis ratios deter-
mined here and the general working model of Gerhard (2002) may
include RCG population effects noted above in Section 4.1. The in-
trinsic luminosity of bulge RCG stars was assumed to be the same
as the local population, however it was found that an offset of −0.3
mag had to be applied to the computed distance moduli in order
to obtain results consistent with standard bar models. Sumi (2004)
found the mean magnitude of observed bulge clump stars is found
to be 0.3 mag fainter to that expected assuming (i) the properties
of bulge RCGs are the same as the local population and (ii) the
distance to the Galactic centre is 8 kpc. A possible implication is
that the adopted distance to the Galactic centre of 8 kpc may be in-
correct. The bar modelling procedure was repeated for three values
of the Sun-Galactic centre distance R0: 7.3 kpc, 7.6 kpc and 8.0
kpc, using data from OGLE-II fields which have −5◦ 6 b 6 −2◦.
Some fields with latitudes outside this range were found to have
number count histograms which were not reproducible by any lin-
ear tri-axial model of the bar tested in this work, and were there-
fore excluded. In addition, most of the OGLE-II fields latitudes
−4◦ < b < −2◦ were excluded. The low amount of informa-
tion in the latitude direction means that the data have little leverage
in determining some of the model parameters. Without more data
in these regions, model fitting algorithms may not be able to refine
some model parameters such as the tilt angle β. The three values
of R0 used are the ‘default’ value of 8.0 kpc; 7.6 kpc, as deter-
mined by Eisenhauer et al. (2005); and 7.3 kpc, which corresponds
to the value of R0 consistent with the observed mean RCG magni-
tudes in the OGLE-II bulge fields assuming the population effects
of Girardi & Salaris (2001). The fitted scale lengths x0, y0 and z0
were found to increase linearly with increasing values of R0, while
the bar opening angle α decreased slightly with increasing R0. The
shape of the bar, as quantified by the ratio of axis scale lengths,
was found to be insensitive to different values of R0. The mean ra-
tio of scale lengths over all models and values of R0 was found to
be x0 : y0 : z0 = 10 : 3.6 : 2.7, which is slightly closer to the
working model of Gerhard (2002) for the Galactic bar than that de-
termined using all 44 OGLE-II fields. The goodness-of-fit measure
χ2 decreased for all models as the value of R0 was lowered.
Improved modelling for the Galactic bar may be possible
through the addition of further elements to the methods presented
here. The possibility of a metallicity gradient across the bulge was
not accounted for in this work. Including spiral terms (see e.g.
Evans & Belokurov 2002) in the analytic density profiles may sim-
c© 2005 RAS, MNRAS 000, 000–000
14 Rattenbury et al.
Table 4. Best-fitting parameters for all density models (equations 3-11) fitted to the number count histograms for OGLE-II fields with Galactic latitude
−5◦ 6 b 6−2◦ . Three values of Galactocentric distance R0 were used. The bar tilt angle, β, was held at 0.0◦.
Scale lengths (pc) Axis ratios
Model R0 (kpc) α (◦) x0 y0 z0 χ2 x0 : y0 : z0
8.0 26.74 1525.19 569.35 382.73 14622.23 10.0 : 3.7 : 2.5
G1 7.6 27.67 1430.76 528.21 363.45 14384.62 10.0 : 3.7 : 2.5
7.3 28.63 1357.09 494.16 349.04 14165.15 10.0 : 3.6 : 2.6
8.0 25.91 1313.82 467.27 337.77 15676.58 10.0 : 3.6 : 2.6
G2 7.6 26.65 1235.00 435.18 320.48 15303.12 10.0 : 3.5 : 2.6
7.3 27.42 1173.14 408.84 307.58 14993.34 10.0 : 3.5 : 2.6
8.0 24.16 4587.68 1658.73 1330.48 9712.92 10.0 : 3.6 : 2.9
G3 7.6 25.32 4266.68 1528.53 1272.76 9502.48 10.0 : 3.6 : 3.0
7.3 26.52 4011.04 1418.58 1226.61 9353.45 10.0 : 3.5 : 3.1
8.0 21.41 1710.52 626.99 343.73 11530.12 10.0 : 3.7 : 2.0
E1 7.6 22.48 1595.52 575.62 327.43 11028.20 10.0 : 3.6 : 2.1
7.3 23.70 1509.95 532.02 316.25 10707.93 10.0 : 3.5 : 2.1
8.0 24.56 974.73 351.07 264.40 9135.87 10.0 : 3.6 : 2.7
E2 7.6 25.63 907.63 323.75 250.86 9085.08 10.0 : 3.6 : 2.8
7.3 26.75 855.05 301.02 240.66 9043.39 10.0 : 3.5 : 2.8
8.0 23.87 1023.09 365.74 297.10 9341.10 10.0 : 3.6 : 2.9
E3 7.6 24.75 954.56 338.68 281.02 9254.02 10.0 : 3.5 : 2.9
7.3 25.69 900.61 316.34 268.93 9180.48 10.0 : 3.5 : 3.0
8.0 22.05 1698.22 609.30 485.75 9418.38 10.0 : 3.6 : 2.9
P1 7.6 23.34 1550.61 554.41 456.64 9288.41 10.0 : 3.6 : 2.9
7.3 24.68 1435.21 509.34 434.79 9191.40 10.0 : 3.5 : 3.0
8.0 21.69 3192.12 1144.07 919.70 10148.09 10.0 : 3.6 : 2.9
P2 7.6 23.03 2894.57 1035.28 861.54 9957.52 10.0 : 3.6 : 3.0
7.3 24.43 2663.48 946.72 817.88 9813.78 10.0 : 3.6 : 3.1
8.0 23.61 1827.34 651.79 487.70 8606.41 10.0 : 3.6 : 2.7
P3 7.6 24.77 1686.69 599.37 460.45 8566.67 10.0 : 3.6 : 2.7
7.3 25.88 1584.40 555.15 444.66 8479.57 10.0 : 3.5 : 2.8
ilarly result in a closer reproduction of the observed number count
profiles.
The observed maximum line-of-sight density can be repro-
duced using just a bar, without requiring additional structure ele-
ments. However, the finer details of the number count histograms,
especially at wide longitudes, were not reproduced by the tri-axial
bar models used here. The predicted number count dispersions
around the peak RCG magnitude were underestimated by the ana-
lytical models, particularly for fields with l > 6◦. It is possible that
these finer features can be reproduced using models which include
extra stellar agglomerations at the ends of the bar. The additional
stellar densities could arise from specific stellar orbits aligned with
the bar, or due to the projection effect of an inner ring. A long thin
bar as postulated by Cabrera-Lavers et al. (2007) might similarly
increase the line-of-sight density dispersion, particularly for wide
longitude fields, as might the presence of a spiral arm. Modelling
the bar using non-parametric methods (see e.g. Efstathiou et al.
1988) may provide valuable insight into the existence and nature
of such additional features. Some preliminary work applying these
methods has begun.
Data from current and future surveys of the Galactic bulge
region will be useful for refining the constraints on the bar param-
eters. The third evolution of the OGLE microlensing experiment,
OGLE-III, is currently in progress, covering a larger region of the
central Galactic region than OGLE-II.
Infra-red observations of the bulge have the advantage of
lower extinction effects due to dust, compared to optical observa-
tions. Current catalogues which would be suitable for investigat-
ing the structure of the Galactic bar include the point source cata-
logue from the 2MASS All Sky data release (Skrutskie et al. 2006),
the Galactic Plane Survey from the UKIRT Infra-Red Deep Sky
Survey (Lawrence et al. 2006; Dye et al. 2006) and data from the
ISOGAL (Omont et al. 2003), Spitzer/GLIMPSE (Benjamin et al.
2005) and AKARI (Ishihara & Onaka 2006) space telescope sur-
veys. Proposed infra-red surveys towards the Galactic centre in-
clude the Galactic Bar Infra-red Time-domain (GABARIT) survey
which aims to monitor the Galactic bar region in the K band for mi-
crolensing events (Kerins 2006, private communication). The anal-
ysis of red clump giant star counts in these surveys may result in
tighter constraints on the properties of the bar, and can be combined
with constrains on stellar kinematics from proper motion surveys
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 15
(Rattenbury et al. 2006) in order to develop dynamical models of
the inner Galaxy.
ACKNOWLEDGEMENTS
We thank D. Faria, Ł. Wyrzykowski, R. James and W. Evans
for helpful discussions. NJR acknowledges financial support by a
PPARC PDRA fellowship and the LKBF. NJR thanks the Kapteyn
Astronomical Institute, RuG, for visitor support. This work was
partially supported by the European Community’s Sixth Frame-
work Marie Curie Research Training Network Programme, Con-
tract No. MRTN-CT-2004-505183 ‘ANGLES’.
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	Introduction
	Data
	Methods
	Sample selection
	Distance Moduli of RCGs
	RCG population effects
	Modelling the bar
	Results
	Evidence of non-bar structure? Including wide longitude fields
	Varying R0
	Discussion
