Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge:
  Observations and Model Comparisons
Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 2 November 2018 (MN LATEX style file v2.2)
Proper Motion Dispersions of Red Clump Giants in the Galactic
Bulge: Observations and Model Comparisons
Nicholas J. Rattenbury1, Shude Mao1, Victor P. Debattista2, Takahiro Sumi3
Ortwin Gerhard4, Flavio De Lorenzi4 ⋆
1 University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, UK
2 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195-1580, USA
3 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
4 Max-Planck-Institut fuer extraterrestrische Physik, P.O. Box 1312,D-85741 Garching, Germany
Accepted ........ Received .......; in original form ......
ABSTRACT
Red clump giants in the Galactic bulge are approximate standard candles and hence they can
be used as distance indicators. We compute the proper motion dispersions of RCG stars in
the Galactic bulge using the proper motion catalogue from the second phase of the Optical
Gravitational Microlensing Experiment (OGLE-II, Sumi et al. 2004) for 45 fields. The proper
motion dispersions are measured to a few per cent accuracy due to the large number of stars
in the fields. The observational sample is comprised of 577736 stars. These observed data are
compared to a state-of-the-art particle simulation of the Galactic bulge region. The predictions
are in rough agreement with observations, but appear to be too anisotropic in the velocity
ellipsoid. We note that there is significant field-to-field variation in the observed proper motion
dispersions. This could either be a real feature, or due to some unknown systematic effect.
Key words: gravitational lensing - Galaxy: bulge - Galaxy: centre - Galaxy: kinematics and
dynamics - Galaxy: structure
1 INTRODUCTION
Many lines of evidence suggest the presence of a bar at the Galac-
tic centre, such as infrared maps (Dwek et al. 1995; Binney et al.
1997) and star counts (Stanek et al. 1997; Nikolaev & Weinberg
1997; Unavane & Gilmore 1998), see Gerhard (2002) for a review.
However, the bar parameters are not well determined. For example,
recent infra-red star counts collected by the Spitzer Space Tele-
scope are best explained assuming a bar at a ∼ 44◦ angle to the
Sun–Galactic centre line (Benjamin et al. 2005) while most previ-
ous studies prefer a bar at ∼ 20◦. In addition, there may be some
fine features, such as a ring in the Galactic bulge, that are not yet
firmly established (Babusiaux & Gilmore 2005). It is therefore cru-
cial to obtain as many constraints as possible in order to better un-
derstand the structure of the inner Galaxy.
Many microlensing groups monitor the Galactic bulge, in-
cluding the EROS (Aubourg et al. 1993), MACHO (Alcock et al.
2000), MOA (Bond et al. 2001; Sumi et al. 2003a) and OGLE
(Udalski et al. 2000) collaborations. In addition to discovering mi-
crolensing events, these groups have also accumulated a huge
amount of data about the stars in the Galactic bulge spanning sev-
eral years to a decade and a half.
⋆ e-mail: (njr, smao)@jb.man.ac.uk; debattis@astro.washington.edu;
gerhard@exgal.mpe.mpg.de; sumi@stelab.nagoya-u.ac.jp;
lorenzi@exgal.mpe.mpg.de
Eyer & Woźniak (2001) first demonstrated that the data can
be used to infer the proper motions of stars, down to ∼ mas yr−1.
Sumi et al. (2004) obtained the proper motions for millions of stars
in the OGLE-II database for a large area of the sky. In this pa-
per, we focus on the red clump giants. These stars are bright and
they are approximately standard candles, hence their magnitudes
can be taken as a crude measure of their distances. As the OGLE-II
proper motions are relative, in this paper we compute the proper
motion dispersions of bulge stars for all field data presented by
Sumi et al. (2004), as they are independent of the unknown proper
motion zero-points. These results could aid theoretical modelling
efforts for the central regions of the Galaxy.
The structure of the paper is as follows. In section 2, we
describe the OGLE-II proper motion catalogue and compute the
proper motion dispersions for bulge stars in 45 OGLE-II fields. In
section 3 we describe the stellar-dynamical model of the Galaxy
used in this work and detail how the model was used to generate
proper motion dispersions. These model predictions are compared
to the observational results in section 4 and in section 5 we discuss
the implications of the results.
2 OBSERVED PROPER MOTION DISPERSIONS
The second phase of the OGLE experiment observed the Galactic
Centre in 49 fields using the 1.3m Warsaw telescope at the Las
c© 2005 RAS
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2 Rattenbury et al.
−0.5 0 0.5 1 1.5 2
PSfrag replacements
(V − I)0
Figure 2. Extinction-corrected colour-magnitude diagram for stars in the
OGLE-II field 1. The ellipse defines the selection criteria for RCG stars
based on colour and magnitude, see text. Sample stars are also required to
have proper motion errors sl,b < 1 mas yr
−1 and total proper motion
µ < 10 mas yr−1.
Campanas Observatory, Chile. Data were collected over an interval
of almost four years, between 1997 and 2000. Each field is 0.24◦×
0.95◦ in size. Fig. 1 shows the position of the OGLE-II Galactic
Bulge fields which returned data used in this paper.
2.1 Red Clump Giants
The 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).
Below we describe the selection of RCG stars in more detail.
2.2 OGLE-II proper motion data
Bulge RCG stars are selected from the OGLE-II proper motion cat-
alogue by applying a cut in magnitude and colour to all stars in each
of the OGLE-II fields. We corrected for extinction and reddening
using the maps presented by Sumi (2004) for each field. Stars were
selected which are located in an ellipse with centre (V −I)0 = 1.0
, I0 = 14.6; and semi-major (magnitude) and semi-minor (colour)
axes of 0.9 and 0.4 respectively, see Fig. 2; a similar selection cri-
terion was used by Sumi (2004). Stars with errors in proper mo-
tion greater than 1 mas yr−1 in either the l or b directions were
excluded. Stars with total proper motion greater than 10 mas yr−1
where similarly excluded, as these are likely to be nearby disk stars,
see also section 3.2. Fields 44, 47-49 were not analysed due to the
low number of RCG stars appearing in these fields.
The proper motion dispersions for the longitude and latitude
directions (σl and σb) were computed for each field via a maxi-
mum likelihood analysis following Lupton et al. (1987). Assuming
a Gaussian distribution of proper motions with mean µ̄ and intrin-
sic proper motion dispersion σ, the probability of a single observed
proper motion µi with measurement error ξi is:
2π(σ2 + ξ2i )
(µi − µ̄)
2(σ2 + ξ2i )
Maximising the likelihood ln(L) = ln(
pi) for µ̄ and σ over all
observations we find:
∂ lnL
(µi − µ̄)
σ2 + ξ2i
= 0 (2)
⇒ µ̄ =
σ2 + ξ2i
−1 (3)
∂ lnL
σ2 + ξ2i
(µi − µ̄)
(σ2 + ξ2i )
= 0 (4)
which can be solved numerically to find σ2.
The values of µ̄ and σ obtained using the above maximum-
likelihood analysis are virtually identical to those obtained via the
equations in Spaenhauer et al. (1992). The errors on the observed
proper motion dispersion values were determined from a bootstrap
analysis using 500 samplings of the observed dataset.
2.3 Extinction
In order to ensure the correction for extinction and reddening above
does not affect the kinematic measurements, σl and σb were recom-
puted for each OGLE-II field using reddening-independent mag-
nitudes. Following Stanek et al. (1997) we define the reddening-
independent magnitude IV−I:
IV−I = I − AI/(AV − AI) (V − I) (5)
where AI and AV are the extinctions in the I and V bands deter-
mined by Sumi (2004). The position of the red clump in the IV−I,
(V − I) CMD varies from field to field. The red clump stars were
extracted by iteratively applying a selection ellipse computed from
the moments of the data (Rocha et al. 2002) rather than centred on a
fixed colour and magnitude. The selection ellipse was recomputed
iteratively for each sample until convergence. The proper motion
dispersions σl and σb computed using RCG stars selected in this
way are consistent with those determined using the original selec-
tion criteria on corrected magnitudes and colours.
2.4 Results
Table 1 lists the observed proper motion dispersions along with
errors for each of the 45 OGLE-II fields considered in this paper.
Figures 3 and 4 show the proper motion dispersions σl and σb
as a function of Galactic longitude and latitude. A typical value of
σl or σb of 3.0 mas yr
−1 corresponds to ∼ 110 kms−1, assum-
ing a distance to the Galactic centre of 8 kpc. The proper motion
dispersion profiles as a function of Galactic longitude shows some
slight asymmetry about the Galactic centre. This asymmetry may
be related to the tri-axial Galactic bar structure (Stanek et al. 1997;
Nishiyama et al. 2005; Babusiaux & Gilmore 2005). The most dis-
crepant points in Fig. 3 correspond to the low-latitude fields num-
bers 6 and 7 (see Fig. 1). The varying field latitude accounts for
some of the scatter in Fig. 3, however we note below in section 4.1
that there are significant variations in the observed proper motion
dispersion between some pairs of adjacent fields. Owing to the the
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 3
−8−6−4−20246810
replacem
Galactic longitude (◦)
Figure 1. The position of the 45 OGLE-II fields used in this analysis. The field used in Spaenhauer et al. (1992) is shown, located within OGLE-II field 45
with (l, b) = (1.0245◦,−3.9253◦).
lack of fields at positive Galactic latitude, any asymmetry about the
Galactic centre in the proper motion dispersions as a function of
Galactic latitude is not obvious, see Fig. 4. Field-to-field variations
in the proper motion dispersions similarly contribute to the scat-
ter seen in Fig. 4, along with the wide range of field longitudes,
especially for fields with −4◦ < b < −3◦.
Table 2 lists the proper motion dispersions and cross-
correlation term Clb in the OGLE-II Baade’s Window fields 45
and 46 along with those found by Kozłowski et al. (2006) using
HST data in four BW fields. The two sets of proper motion dis-
persions results are consistent at the ∼ 2σ level. It is important to
note that the errors on the proper motion dispersions in Table 1 do
not include systematic errors. We also note that the selection cri-
teria applied to stars in the HST data are very different to those
for the ground-based data, in particular the magnitude limits ap-
plied in each case. The bulge kinematics from the HST data of
Kozłowski et al. (2006) were determined for stars with magnitudes
18.0 < IF814W < 21.5. The approximate reddening-independent
magnitude range for the OGLE-II data was 12.5 . IV−I . 14.6.
The effects of blending are also very different in the two datasets. It
is therefore very reassuring that our results are in general agreement
with those obtained by Kozłowski et al. (2006) using higher reso-
lution data from the HST. For more comparisons between ground
and HST RCG proper motion dispersions, see section 4.
Figure 5 shows the cross-correlation term Clb as a function of
Galactic co-ordinate. There is a clear sinusoidal structure in the Clb
data as a function of Galactic longitude, with the degree of corre-
lation between σl and σb changing most rapidly near l ≃ 0
◦. The
Clb data as a function of Galactic latitude may also show some ev-
idence of structure. It is possible however, that this apparent struc-
ture is due to the different number of fields at each latitude, rather
than some real physical cause.
3 GALACTIC MODEL
The stellar-dynamical model used in this work was produced us-
ing the made-to-measure method (Syer & Tremaine 1996). The
model is constrained to reproduce the density distribution con-
structed from the dust-corrected L-band COBE/DIRBE map of
Spergel et al. (1996). An earlier dynamical model was built to
match the total column density of the disk (Bissantz & Gerhard
2002). This dynamical model matched the radial velocity and
proper motion data in two fields (including Baade’s window) quite
well. No kinematic constraints were imposed during the construc-
tion of the model. We refer the readers to Bissantz et al. (2004)
for more detailed descriptions. The model used here is constructed
as in that case with the further refinement that the vertical den-
sity distribution is also included. This is necessary as the vertical
kinematics (σb) will also be compared with observations in this
paper. However the density distribution near the mid-plane is con-
siderably more uncertain, in part because of the dust extinction cor-
rection. Thus the model used in this paper can only be considered
illustrative, not final. Further efforts to model the vertical density
distribution are currently under way and will be reported elsewhere
(Debattista et al. 2007, in preparation).
In Fig. 6, we present the mean motion of stars in the mid-plane
of the Galaxy from this model. A bar position angle of θ = 20◦ is
shown here, as this is the orientation favoured both by optical depth
measurements (Evans & Belokurov 2002) and by the red clump gi-
ant brightness distribution (Stanek et al. 1997) and was the angle
c© 2005 RAS, MNRAS 000, 000–000
4 Rattenbury et al.
Table 1. Observed proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II
fields. High precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of
stars selected from each field. Fields 44, 47-49 were not analysed due to the low number of RCG stars appearing in these fields.
Field Field centre PM Dispersions (mas yr−1) Clb N
l (◦) b (◦) Longitude σl Latitude σb
1 1.08 -3.62 3.10 ±0.02 2.83 ±0.02 -0.13 ±0.01 15434
2 2.23 -3.46 3.21 ±0.02 2.80 ±0.02 -0.14 ±0.01 16770
3 0.11 -1.93 3.40 ±0.01 3.30 ±0.02 -0.08 ±0.01 26763
4 0.43 -2.01 3.43 ±0.02 3.26 ±0.01 -0.11 ±0.01 26382
5 -0.23 -1.33 3.23 ±0.03 3.00 ±0.04 -0.04 ±0.02 3145
6 -0.25 -5.70 2.61 ±0.02 2.36 ±0.03 -0.06 ±0.01 7027
7 -0.14 -5.91 2.70 ±0.03 2.43 ±0.02 -0.05 ±0.01 6236
8 10.48 -3.78 2.80 ±0.03 2.29 ±0.02 -0.08 ±0.01 5136
9 10.59 -3.98 2.73 ±0.02 2.16 ±0.03 -0.06 ±0.01 5114
10 9.64 -3.44 2.77 ±0.02 2.27 ±0.02 -0.07 ±0.01 5568
11 9.74 -3.64 2.84 ±0.02 2.32 ±0.02 -0.10 ±0.01 5369
12 7.80 -3.37 2.66 ±0.03 2.31 ±0.03 -0.08 ±0.01 6035
13 7.91 -3.58 2.66 ±0.03 2.24 ±0.02 -0.07 ±0.01 5601
14 5.23 2.81 2.97 ±0.02 2.60 ±0.02 0.04 ±0.01 10427
15 5.38 2.63 3.02 ±0.02 2.64 ±0.03 -0.00 ±0.01 8989
16 5.10 -3.29 2.87 ±0.02 2.53 ±0.02 -0.12 ±0.01 9799
17 5.28 -3.45 2.81 ±0.02 2.42 ±0.01 -0.12 ±0.01 10268
18 3.97 -3.14 2.92 ±0.02 2.62 ±0.02 -0.13 ±0.01 14019
19 4.08 -3.35 2.90 ±0.02 2.60 ±0.02 -0.17 ±0.01 13256
20 1.68 -2.47 3.27 ±0.01 2.82 ±0.01 -0.12 ±0.01 17678
21 1.80 -2.66 3.31 ±0.02 2.90 ±0.02 -0.13 ±0.01 17577
22 -0.26 -2.95 3.17 ±0.02 2.84 ±0.02 -0.01 ±0.01 19787
23 -0.50 -3.36 3.15 ±0.01 2.84 ±0.02 -0.04 ±0.01 17996
24 -2.44 -3.36 2.96 ±0.01 2.48 ±0.01 0.02 ±0.01 16397
25 -2.32 -3.56 2.91 ±0.01 2.50 ±0.01 0.02 ±0.01 16386
26 -4.90 -3.37 2.68 ±0.02 2.17 ±0.01 0.02 ±0.01 13099
27 -4.92 -3.65 2.63 ±0.02 2.15 ±0.01 0.03 ±0.01 12728
28 -6.76 -4.42 2.63 ±0.03 2.12 ±0.02 -0.01 ±0.01 8367
29 -6.64 -4.62 2.66 ±0.03 2.09 ±0.02 -0.02 ±0.01 8108
30 1.94 -2.84 3.04 ±0.02 2.70 ±0.02 -0.12 ±0.01 17774
31 2.23 -2.94 3.11 ±0.02 2.74 ±0.01 -0.12 ±0.01 17273
32 2.34 -3.14 3.10 ±0.02 2.78 ±0.01 -0.13 ±0.01 15966
33 2.35 -3.66 3.08 ±0.02 2.77 ±0.02 -0.14 ±0.01 15450
34 1.35 -2.40 3.36 ±0.02 2.92 ±0.01 -0.11 ±0.01 16889
35 3.05 -3.00 3.09 ±0.02 2.72 ±0.02 -0.14 ±0.01 15973
36 3.16 -3.20 3.19 ±0.02 2.77 ±0.02 -0.16 ±0.01 14955
37 0.00 -1.74 3.29 ±0.02 3.04 ±0.01 -0.05 ±0.01 20233
38 0.97 -3.42 3.15 ±0.01 2.84 ±0.02 -0.12 ±0.01 15542
39 0.53 -2.21 3.21 ±0.01 3.00 ±0.01 -0.07 ±0.01 24820
40 -2.99 -3.14 2.84 ±0.01 2.47 ±0.02 0.05 ±0.01 13581
41 -2.78 -3.27 2.78 ±0.01 2.41 ±0.02 0.04 ±0.01 14070
42 4.48 -3.38 2.89 ±0.02 2.63 ±0.02 -0.15 ±0.01 10099
43 0.37 2.95 3.17 ±0.02 2.87 ±0.01 0.02 ±0.01 11467
45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2380
46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 1803
Table 2. Comparison between proper motion dispersions and cross-correlation term Clb in two of the OGLE-II fields (45 and 46) with proper motion
dispersions computed from four nearby HST fields (Kozłowski et al. 2006).
Field l (◦) b (◦) σl (mas yr
−1) σb (mas yr
−1) Clb Ref
119-A 1.32 -3.77 2.89 ±0.10 2.44 ±0.08 -0.14 ±0.04 1
119-C 0.85 -3.89 2.79 ±0.10 2.65 ±0.08 -0.14 ±0.04 1
OGLE-II 45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2
119-D 1.06 -4.12 2.75 ±0.10 2.56 ±0.09 -0.05 ±0.06 1
95-BLG-11 0.99 -4.21 2.82 ±0.09 2.62 ±0.09 -0.14 ±0.04 1
OGLE-II 46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 2
1Kozłowski et al. (2006) 2This work.
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 5
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PSfrag replacements
Galactic longitude (◦)
Figure 3. Proper motion dispersion in the Galactic longitude (σl) and lati-
tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of
field Galactic longitude. Open circles correspond to fields 6, 7, 14, 15 and
43 which have relatively extreme galactic latitudes, see Fig. 1.
used in deriving the model. Clearly one can see that the mean mo-
tion follows elliptical paths around the Galactic bar. The analysis of
OGLE-II proper motions by Sumi et al. (2003b) is consistent with
this streaming motion.
3.1 Model stellar magnitudes
The model has a four-fold symmetry, obtained by a rotation of π
radians around the vertical axis and by positioning the Sun above
or below the mid-plane. The kinematics of model particles falling
within the solid angle of each OGLE-II field were combined to
those from the three other equivalent lines-of-sight. This procedure
allows an increase in the number of model particles used for the
predictions of stellar kinematics.
We assign magnitudes to stars in the Galactic model described
above which appear in the same fields as that observed by the
OGLE collaboration. Number counts as a function of I-band ap-
parent magnitude, I , were used to compute the fraction of RCG
stars in each of the OGLE-II fields. Figure 7 shows an example of
the fitted number count function Nk(I) for one of the k = 1 . . . 49
OGLE-II fields, where Nk(I) is of the form of a power-law and a
Gaussian (Sumi 2004):
Nk(I) = ak10
(bkI) + ck exp
−(I − Ip,k)
−6 −4 −2 0 2 4
PSfrag replacements
Galactic latitude (◦)
Figure 4. Proper motion dispersion in the Galactic longitude (σl) and lati-
tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of
field Galactic latitude. Open circles correspond to fields 6, 7, 14, 15 and 43
which have relatively extreme galactic latitudes, see Fig. 1.
where the constants ak, bk, ck, Ip,k, σk are determined for each of
the k OGLE-II fields, see Table 3. The fraction Rk of RCG stars
is evaluated as the ratio of the area under the Gaussian component
of equation (6) to the area under the full expression. The integrals
are taken over ±3σk around the RCG peak in Nk(I) for each of
the k OGLE-II fields. Fields 44 and 47-49 are not included as there
are insufficient RCGs in the OGLE-II fields to fit equation (6). Fig-
ure 7 shows that the model number count function fails to fit the
observed number counts well for magnitudes I ≃ 15.4. In order to
convert stellar density to a distribution of apparent magnitude, the
relevant quantity is ρr3 (Bissantz & Gerhard 2002). Depending on
the line-of-sight, this quantity can give asymmetric magnitude dis-
tributions through the bulge. Using the best-fitting analytic tri-axial
density models for the bulge (Rattenbury et al. 2007, in prepara-
tion), this asymmetry is observed and may explain the excess of
stars in the number count histograms, compared to the best-fitting
two-component fit of equation (6). The inability of equation (6) to
model completely all features in the observed number counts in
some cases leads to an additional uncertainty in the magnitude lo-
cation of the fitted Gaussian peak. Computing the apparent magni-
tude distribution as ∝ ρr3 also produces a small shift in the peak of
the magnitude distribution. This shift is ∼ +0.04 mag for l = 0◦,
b = 0◦. The proper motion dispersions computed here are unlikely
to be sensitive to these small offsets.
c© 2005 RAS, MNRAS 000, 000–000
6 Rattenbury et al.
Table 3. Values of fitted parameters in equation (6) for all 45 OGLE-II fields used in this analysis. R is the ratio of observed RCG stars to the total number
of stars in each field, evaluated over ±3σ around the RCG peak magnitude, Ip, where σ is the fitted Gaussian spread in equation (6). The magnitudes of
the model RCG stars are shifted by ∆m to correspond with the observed mean RCG magnitude in each field. The total number of model stars in each field
assigned RCG magnitudes and colours is nrcg and the total number of model stars in each field is nall. The corresponding total model weight values for each
field are given by wrcg and wall respectively. The large values of σ for fields 8-11 might be related to their position at large positive longitudes, and could
indicate a structure such as the end of the bar, a ring or spiral arm. An analysis of the bar morphology based on these results is underway (Rattenbury et al.
2007, in preparation).
Field a b c Ip σ R ∆m nrcg nall wrcg wall
1 0.11 0.27 1735.70 14.62 0.29 0.40 0.43 585 1773 277.2 842.4
2 0.15 0.26 1876.47 14.54 -0.29 0.43 0.41 621 1802 298.1 853.9
3 0.16 0.28 4692.78 14.66 0.25 0.44 0.54 1264 3626 668.5 1911.1
4 0.17 0.28 4438.63 14.65 0.24 0.44 0.52 1298 3653 670.8 1922.2
5 0.05 0.33 4581.59 14.70 0.28 0.33 0.55 1342 4668 755.7 2685.7
6 0.04 0.27 519.71 14.57 0.37 0.34 0.36 152 583 69.5 270.8
7 0.03 0.28 457.42 14.55 0.39 0.32 0.36 143 527 71.9 243.8
8 0.04 0.27 259.65 14.37 -0.51 0.22 0.35 96 561 41.7 236.2
9 0.04 0.27 270.90 14.34 0.51 0.25 -0.05 96 497 46.1 230.9
10 0.08 0.26 321.32 14.44 0.52 0.22 0.40 131 654 49.1 260.1
11 0.04 0.28 316.25 14.45 0.50 0.23 0.28 128 695 57.5 339.4
12 0.12 0.25 546.85 14.43 0.38 0.28 0.41 238 908 100.7 393.1
13 0.10 0.25 520.45 14.45 0.37 0.29 0.15 190 863 83.9 392.4
14 0.09 0.28 1309.28 14.55 0.32 0.35 0.34 458 1587 216.0 767.4
15 0.05 0.29 1154.52 14.57 0.33 0.31 0.55 421 1661 185.2 761.8
16 0.12 0.27 1042.72 14.50 0.35 0.33 0.50 397 1383 172.8 601.1
17 0.12 0.26 1069.07 14.48 0.34 0.35 0.25 406 1443 212.4 753.4
18 0.17 0.26 1569.83 14.49 0.31 0.40 0.35 527 1564 234.7 702.4
19 0.17 0.26 1429.23 14.51 0.32 0.40 0.44 434 1365 184.4 608.5
20 0.20 0.27 3012.09 14.58 0.26 0.42 0.53 939 2728 480.3 1398.3
21 0.15 0.27 2793.36 14.58 0.26 0.43 0.45 900 2554 443.5 1260.0
22 0.12 0.28 2574.77 14.74 0.28 0.42 0.51 830 2419 382.5 1113.3
23 0.09 0.28 2147.71 14.73 0.29 0.42 0.47 767 2126 384.2 1060.6
24 0.12 0.27 2130.41 14.82 0.28 0.42 0.50 595 1864 269.6 905.4
25 0.07 0.28 2002.91 14.82 0.28 0.42 0.51 581 1782 289.5 885.1
26 0.09 0.27 1452.89 14.83 0.31 0.38 0.55 375 1325 159.7 570.5
27 0.07 0.27 1319.67 14.81 0.32 0.39 0.40 387 1238 172.5 578.9
28 0.04 0.28 563.00 14.79 0.31 0.31 0.62 162 649 72.3 293.5
29 0.05 0.27 559.86 14.78 0.31 0.32 0.44 156 607 70.7 267.5
30 0.18 0.27 2533.75 14.57 0.27 0.42 0.41 754 2195 362.4 1026.7
31 0.17 0.27 2354.64 14.53 0.28 0.43 0.32 763 2229 361.9 1122.1
32 0.17 0.26 2062.96 14.53 0.28 0.42 0.41 638 1962 291.8 938.5
33 0.13 0.27 1614.83 14.56 0.31 0.41 0.34 559 1586 265.5 760.7
34 0.18 0.27 3210.56 14.60 0.27 0.43 0.42 990 2936 503.0 1473.9
35 0.16 0.26 1963.53 14.53 0.29 0.41 0.45 663 1925 307.7 913.7
36 0.16 0.26 1773.62 14.51 0.30 0.41 0.47 574 1902 301.1 943.5
37 0.18 0.28 4901.22 14.64 0.25 0.42 0.43 1439 4077 794.9 2218.5
38 0.12 0.27 2091.19 14.64 0.28 0.43 0.46 662 1945 319.2 948.1
39 0.18 0.28 3919.30 14.69 0.26 0.44 0.65 1217 3456 631.8 1804.2
40 0.09 0.28 2181.18 14.87 0.29 0.41 0.62 668 1936 315.1 933.3
41 0.10 0.28 2180.49 14.87 0.28 0.42 0.55 626 1905 318.2 965.4
42 0.13 0.26 1215.38 14.52 0.35 0.37 0.40 425 1389 190.2 637.7
43 0.10 0.28 2659.91 14.84 0.27 0.41 0.79 777 2290 345.8 1074.6
45 0.11 0.27 1541.36 14.59 0.31 0.40 0.38 485 1568 228.3 767.7
46 0.09 0.27 1428.63 14.60 0.30 0.41 0.38 454 1400 221.6 669.5
Each star in the galactic model is assigned a RCG magnitude
with probability Rk for each field. The apparent magnitude is com-
puted using the model distance. Stars which are not assigned a RCG
magnitude are assigned a magnitude using the power-law compo-
nent of equation (6), defined over the same limits used to compute
Rk. Here we implicitly assume that the RCG stars trace the overall
Galactic disk and bulge populations.
The RCG luminosity function is approximated by a Gaussian
distribution with mean magnitude −0.26 and σ = 0.2. These as-
sumptions are mostly consistent with observations (Stanek et al.
1997) and the fitted distribution from Udalski (2000), but there may
be small offsets between local and bulge red clump giants. It was
noted in Sumi (2004) that there is some as-yet unexplained off-
set (0.3 mag) in the extinction-corrected mean RCG magnitudes in
the OGLE fields. A possible explanation for this offset is that the
RCG population effects are large: so that the absolute magnitude
of RCG stars is significantly different for RCGs in the bulge com-
pared to local RCGs, as claimed by Percival & Salaris (2003) and
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 7
−10−5051015
−0.15
−0.05
−6 −4 −2 0 2 4
−0.15
−0.05
PSfrag replacements
Galactic longitude (◦)
Galactic latitude (◦)
Figure 5. Cross-correlation term Clb for 45 OGLE-II Galactic bulge fields
as a function of field Galactic longitude (top) and latitude (bottom). Open
circles in the top plot of Clb vs. l correspond to fields 6, 7, 14, 15 and 43
which have relatively extreme galactic latitudes, see Fig. 1.
Salaris et al. (2003). A different value of the distance to the Galac-
tic centre to that assumed here (8 kpc) would in part account for
the discrepancy, however would not remove it completely. Using a
value of 7.6 kpc (Eisenhauer et al. 2005; Nishiyama et al. 2006) as
the distance to the Galactic centre would change the zero-point by
0.12 mag, resulting in an offset value of 0.18 mag. It is also possi-
ble that reddening toward the Galactic centre is more complicated
than assumed in Sumi (2004). In order to compare the model proper
motion results with the observed data, it was necessary to shift the
mean model RCG magnitudes to correspond with that observed in
each of the OGLE fields. The model RCG magnitudes were fitted
with a Gaussian curve. The mean of the model RCG magnitudes
was then shifted by a value ∆m, see Table 3, to correspond with
the observed mean RCG magnitude in each of the OGLE fields. No-
tice that we concentrate on second-order moments (proper motion
dispersions) of the proper motion, so a small shift in the zero-point
has little effect on our results.
Every model particle has an associated weight, wi. The par-
ticle weight can take values 0 < wi . 20. In order to account
for this weighting, ⌈wi⌉ stars are generated for each particle with
the same kinematics but magnitudes determined as above. ⌈wi⌉ is
the nearest integer toward +∞. Each model star is then assigned
a weight, γi = wi/⌈wi⌉. Notice this procedure allows us to in-
crease the effective number of particles to better sample the lumi-
nosity function. The total number of stars and the number of stars
assigned RCG magnitudes in each field are listed in Table 3 as nall
Figure 6. Galactic kinematics from the model of Debattista et al (2007, in
preparation). Bulk stellar motion in the mid-plane of the Galaxy is shown
super-imposed on the stellar density. The Sun is located at the origin (not
shown). An example line of sight is shown. The model can be rotated to four
equivalent positions for each line of sight due to symmetry (see section 3.1).
12 12.5 13 13.5 14 14.5 15 15.5 16 16.5
PSfrag replacements
stars
non-RCG stars
Figure 7. Number count as function of apparent magnitude, I , for OGLE-II
field 1. The number count histogram is shown along with the fitted function
equation (6). The fraction of RCG stars, Rk , is evaluated over the magni-
tude range Ip ±3σ for each of the (k = 1 . . . 49) OGLE-II fields. The ratio
Rk is assumed to be the same at all stellar distances for each field.
and nrcg respectively. 81806 stars from the model were used to
compare model kinematics to observed values.
3.2 Model kinematics
Stars with apparent magnitudes within the limits mmin = 13.7 and
mmax = 15.5, were selected from the model data. This magnitude
range corresponds to the selection criteria imposed on the observed
data sample, see section 2.2. Model stars with total proper motions
greater than 10 mas yr−1 (corresponding to > 380 kms−1 at a
c© 2005 RAS, MNRAS 000, 000–000
8 Rattenbury et al.
distance of the Galactic centre) were excluded on the basis that
such stars would be similarly excluded from any observed sample.
The fraction of weight removed and number of stars removed in
this way only amounted to a few per cent of the total weight and
number of stars in each field. Bulge model stars were selected by
requiring a distance d > 6 kpc.
The mean proper motion and proper motion dispersions in the
latitude and longitude directions were computed along with their
errors for all model stars in each field which obey the above selec-
tion criteria. The weights on model stars, γi, were used to compute
these values.
We then tested whether the finite and discrete nature of the
model data gives rise to uncertainties in the measured proper mo-
tion dispersion values. We measured the intrinsic noise in the model
by comparing the proper motion dispersions computed for four
equivalent lines-of-sight through the model for each field. The
spread of the proper motion dispersions for each field was then used
as the estimate of the intrinsic noise in the model. The mean (me-
dian) value of these errors in the longitude and latitude directions
are 0.08 (0.06) and 0.12 (0.097) mas yr−1 respectively.
The statistical error for the proper motion dispersions in the
longitude and latitude directions for each field were combined in
quadrature with the error arising from the finite discrete nature of
the model data to give the total error on the proper motion disper-
sions computed from the model.
4 COMPARISON BETWEEN THEORETICAL MODEL
AND OBSERVED DATA
The observed and predicted proper motion dispersions for each of
the OGLE-II fields are shown in Table 4. Fig. 8 shows the observed
proper motion dispersions for each of the analysed OGLE-II fields
plotted against the predicted model proper motion dispersions.
Fig. 8 shows that the model predictions are in general agree-
ment with observed proper motion dispersions for the OGLE-
II fields. The model has been used previously to predict the
proper motion dispersions of 427 stars1 entries observed by
Spaenhauer et al. (1992) in a single 6′× 6′ field toward the bulge
(Bissantz et al. 2004). The model value of σl in this previous anal-
ysis was in agreement with the observed value, yet the model and
observed values of σb were significantly different. The 6
′× 6′ field
used by Spaenhauer et al. (1992) falls within the OGLE-II field
number 45. The model prediction of σl for stars in OGLE field 45
is completely consistent with the measured value. The model pre-
diction of σb shows a similar discrepancy to the previous analysis
of Bissantz et al. (2004).
Fig. 9 shows the ratio R = σl/σb and cross-correlation term
Clb = σlb/(σlσb) computed using the model and observed data.
Typically the model predicts more anisotropic motion with R > 1
than what is observed.
The model predictions for stellar kinematics in the latitude di-
rection may be problematic. This is not surprising as the model is
not well constrained toward the plane due to a lack of observational
data because of the heavy dust extinction. The problem is currently
under investigation. Similarly, the model predictions for σl degrade
as l increases. This is because the model performance has been op-
timised for regions close to the Galactic centre.
1 There are two repeated entries in Table 2 of Spaenhauer et al. (1992).
The significant difference between the observed proper mo-
tion dispersions of adjacent fields (e.g. fields 1 and 45) might hint at
some fine-scale population effect, whereby a group of stars surviv-
ing the selection criteria have a significant and discrepant kinematic
signature. Higher-accuracy observations using the HST support this
evidence of such population effects (Kozłowski et al. 2006).
No attempt has been made to account for the blending of flux
inherent in the OGLE-II crowded-field photometry. It is certain
that a fraction of stars in each OGLE-II field suffers from some
degree of blending (Kozłowski et al. 2006). To investigate this ef-
fect, we checked one field covering the lens MACHO-95-BLG-
37 (l = 2.54◦, b = 3.33◦, Thomas et al. 2005) from the HST
proper motion survey of Kozłowski et al. (2006), which falls inside
OGLE-II field number 2. HST images suffer much less blending,
but the field of view is small, and so it has only a dozen or so clump
giants. We derive a proper motion of σl = 3.13 ± 0.57 mas yr
and σb = 2.17±0.40 mas yr
−1. These values agree with our kine-
matics in field 2 within 0.2σ for σl and 1.6σ for σb. The errors in
our proper motion dispersions are very small (∼ km s−1 at a dis-
tance of the Galactic centre), but it is likely that we underestimate
the error bars on the observed data due to systematic effects such
as blending.
4.1 Understanding the differences
We now seek to understand the cause of the differences between
the model and the Milky Way, at least at a qualitative level. We
first consider the possibility that the difference can be explained by
some systematic effect. We compute the differences between ob-
served proper motion dispersions of nearest fields for fields with
separations less than 0.25 degrees. No pair of fields is used twice,
and the difference ∆ = σi − σj is always plotted such that
|bi| ≥ |bj |. ∆l,obs and ∆b,obs denote the difference in observed
proper motion dispersions between adjacent fields in the longitude
and latitude directions respectively. The equivalent quantities pre-
dicted from the model are denoted ∆l,mod and ∆b,mod. In Fig. 10
we see that the deviations ∆l,obs and ∆b,obs scatter about 0, but
have a quite broad distribution in both the l and b directions, with
several fields inconsistent with zero difference at 1σ (defined as the
sum in quadrature of the uncertainties of the corresponding quanti-
ties of the two fields under comparison). Several deviations are as
large as 0.2 mas yr−1 (corresponding to ≃ 8 kms−1 at the Galac-
tic centre) and many σ’s away from zero. In view of the fact that
these differences have mean close to zero, it is possible that these
deviations are due to some systematic effect rather than to intrinsic
substructure in the Milky Way. We return to this point briefly in the
discussion.
In the case of the model uncertainties, however, Fig. 11 shows
that in most cases the differences ∆l,mod and ∆b,mod are consistent
with zero at the 1σ level, indicating that these error estimates are
robust.
We now seek to explore the correlations of the residuals with
properties of the model. We plot residuals δl,b = (σmod − σobs),
where σmod and σobs are the model and observed proper motion
dispersions in the corresponding Galactic co-ordinate. The errorbar
length is (u2mod+u
1/2 where umod and uobs are the uncertain-
ties in the model and observed proper motion dispersions, respec-
tively. Plotting these quantities as a function of l, we note that there
is no significant correlation, but that the largest deviations in the
latitude proper motion dispersion occur close to l = 0, see Fig. 12.
In plotting δl,b as a function of b, the reason which becomes evi-
dent is that the fields closest to the mid-plane have the largest δb,
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 9
2.6 2.8 3 3.2 3.4
27 28
PSfrag replacements
Model σl (mas yr
Model σb (mas yr
Observed σb (mas yr
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
11 12
24 25
35 36
46PSfrag replacements
Model σl (mas yr
Observed σl (mas yr
Model σb (mas yr
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
Figure 8. Comparison between observed and predicted proper motion dispersions for stars in the OGLE-II proper motion catalogue of Sumi et al. (2004). Left:
Proper motion dispersions in the galactic longitude direction, σl. The OGLE-II field number is indicated adjacent to each point, see also Fig. 1. Fields with
galactic longitude |l| > 5◦ are shown in magenta; fields within Baade’s window are shown in red; all other fields in blue. Right: Proper motion dispersions in
the galactic latitude direction, σb, shown with the same colour scheme.
0.8 1 1.2 1.4 1.6 1.8
PSfrag replacements
Model σl/σb
Model Clb
Observed Clb
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
−0.1 0 0.1
PSfrag replacements
Model σl/σb
Observed σl/σb
Model Clb
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
Figure 9. Left: Ratio of proper motion dispersions R = σl/σb for the observed OGLE-II proper motion data and model predictions. The model generally
predicts more anisotropic motion, i.e. R > 1 than is observed in the data. Right: The cross-correlation term Clb = σlb/σlσb.
see Fig. 13. The density distribution in this region is uncertain due
to presence of dust and the large extinction corrections required.
This may explain why the residuals of σb seem to correlate more
with b than those of σl. We note that the σl residuals also seem to
have some dependence on b. A possible explanation is that there is
some additional effect due to dust which has not been accounted
5 DISCUSSION
Red clump giant stars in the dense fields observed by the OGLE-II
microlensing survey can be used as tracers of the bulge density and
motion over a large region toward the Galactic centre. The proper
motion dispersions of bulge RCG stars in the OGLE-II proper mo-
tion catalogue of Sumi et al. (2004) were calculated for 45 OGLE-
II fields. The kinematics derived from the ground-based OGLE-II
data were found to be in agreement with HST observations in two
fields from Kozłowski et al. (2006). It is reassuring that the results
presented here are consistent with those derived from the higher
resolution HST data, despite possible selection effects and blend-
The observed values of σl and σb were compared to predic-
tions from the made-to-measure stellar-dynamical model of De-
battista et al. (2007, in preparation). In general, the model gives
predictions qualitatively similar to observed values of σl and σb
for fields close to the Galactic centre. The model is in agree-
ment with observed OGLE-II data in the direction previously tested
by Bissantz et al. (2004). Using the definition of De Lorenzi et al.
(2007), the effective number of particles in the model used here
c© 2005 RAS, MNRAS 000, 000–000
10 Rattenbury et al.
Table 4. Proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II fields. High
precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of stars selected
from each field. Field 44 was not used due to the low number of RCGs in this field.
PM Dispersions (mas yr−1) Clb
Field Field centre Longitude σl Latitude σb
l (◦) b (◦) Model Observed Model Observed Model Observed N
1 1.08 -3.62 3.02 ±0.08 3.10 ±0.02 2.46 ±0.08 2.83 ±0.02 0.01 ±0.08 -0.13 ±0.01 15434
2 2.23 -3.46 3.02 ±0.06 3.21 ±0.02 2.68 ±0.15 2.80 ±0.02 0.12 ±0.03 -0.14 ±0.01 16770
3 0.11 -1.93 3.19 ±0.08 3.40 ±0.01 2.64 ±0.02 3.30 ±0.02 0.02 ±0.01 -0.08 ±0.01 26763
4 0.43 -2.01 3.26 ±0.05 3.43 ±0.02 2.80 ±0.06 3.26 ±0.01 0.01 ±0.03 -0.11 ±0.01 26382
5 -0.23 -1.33 3.22 ±0.15 3.23 ±0.03 2.30 ±0.07 3.00 ±0.04 -0.01 ±0.03 -0.04 ±0.02 3145
6 -0.25 -5.70 3.26 ±0.16 2.61 ±0.02 2.42 ±0.23 2.36 ±0.03 -0.02 ±0.13 -0.06 ±0.01 7027
7 -0.14 -5.91 2.95 ±0.15 2.70 ±0.03 2.49 ±0.12 2.43 ±0.02 -0.15 ±0.16 -0.05 ±0.01 6236
8 10.48 -3.78 3.07 ±0.09 2.80 ±0.03 1.98 ±0.14 2.29 ±0.02 0.00 ±0.09 -0.08 ±0.01 5136
9 10.59 -3.98 3.28 ±0.21 2.73 ±0.02 2.03 ±0.07 2.16 ±0.03 -0.03 ±0.07 -0.06 ±0.01 5114
10 9.64 -3.44 3.30 ±0.32 2.77 ±0.02 2.89 ±0.62 2.27 ±0.02 0.09 ±0.08 -0.07 ±0.01 5568
11 9.74 -3.64 3.01 ±0.20 2.84 ±0.02 2.22 ±0.29 2.32 ±0.02 -0.08 ±0.09 -0.10 ±0.01 5369
12 7.80 -3.37 3.31 ±0.10 2.66 ±0.03 2.29 ±0.06 2.31 ±0.03 -0.09 ±0.06 -0.08 ±0.01 6035
13 7.91 -3.58 3.26 ±0.18 2.66 ±0.03 2.29 ±0.12 2.24 ±0.02 0.05 ±0.02 -0.07 ±0.01 5601
14 5.23 2.81 3.21 ±0.05 2.97 ±0.02 2.62 ±0.13 2.60 ±0.02 0.06 ±0.04 0.04 ±0.01 10427
15 5.38 2.63 3.31 ±0.12 3.02 ±0.02 2.46 ±0.07 2.64 ±0.03 0.04 ±0.04 -0.00 ±0.01 8989
16 5.10 -3.29 3.19 ±0.07 2.87 ±0.02 2.23 ±0.08 2.53 ±0.02 0.03 ±0.08 -0.12 ±0.01 9799
17 5.28 -3.45 3.09 ±0.09 2.81 ±0.02 2.50 ±0.07 2.42 ±0.01 -0.01 ±0.11 -0.12 ±0.01 10268
18 3.97 -3.14 3.20 ±0.09 2.92 ±0.02 2.48 ±0.08 2.62 ±0.02 0.02 ±0.03 -0.13 ±0.01 14019
19 4.08 -3.35 3.06 ±0.13 2.90 ±0.02 2.49 ±0.26 2.60 ±0.02 0.01 ±0.03 -0.17 ±0.01 13256
20 1.68 -2.47 3.12 ±0.06 3.27 ±0.01 2.66 ±0.05 2.82 ±0.01 0.07 ±0.03 -0.12 ±0.01 17678
21 1.80 -2.66 3.12 ±0.06 3.31 ±0.02 2.57 ±0.08 2.90 ±0.02 -0.02 ±0.03 -0.13 ±0.01 17577
22 -0.26 -2.95 3.17 ±0.04 3.17 ±0.02 2.46 ±0.12 2.84 ±0.02 0.01 ±0.03 -0.01 ±0.01 19787
23 -0.50 -3.36 3.13 ±0.17 3.15 ±0.01 2.62 ±0.10 2.84 ±0.02 -0.02 ±0.14 -0.04 ±0.01 17996
24 -2.44 -3.36 2.77 ±0.04 2.96 ±0.01 2.32 ±0.10 2.48 ±0.01 -0.04 ±0.04 0.02 ±0.01 16397
25 -2.32 -3.56 2.76 ±0.07 2.91 ±0.01 2.47 ±0.15 2.50 ±0.01 -0.04 ±0.03 0.02 ±0.01 16386
26 -4.90 -3.37 2.80 ±0.17 2.68 ±0.02 2.22 ±0.04 2.17 ±0.01 -0.00 ±0.03 0.02 ±0.01 13099
27 -4.92 -3.65 2.78 ±0.07 2.63 ±0.02 2.19 ±0.04 2.15 ±0.01 -0.06 ±0.02 0.03 ±0.01 12728
28 -6.76 -4.42 3.02 ±0.11 2.63 ±0.03 2.44 ±0.36 2.12 ±0.02 0.05 ±0.06 -0.01 ±0.01 8367
29 -6.64 -4.62 3.02 ±0.21 2.66 ±0.03 1.79 ±0.14 2.09 ±0.02 -0.00 ±0.11 -0.02 ±0.01 8108
30 1.94 -2.84 3.13 ±0.07 3.04 ±0.02 2.59 ±0.11 2.70 ±0.02 -0.04 ±0.08 -0.12 ±0.01 17774
31 2.23 -2.94 3.08 ±0.05 3.11 ±0.02 2.68 ±0.11 2.74 ±0.01 0.08 ±0.05 -0.12 ±0.01 17273
32 2.34 -3.14 3.10 ±0.09 3.10 ±0.02 2.56 ±0.04 2.78 ±0.01 0.11 ±0.02 -0.13 ±0.01 15966
33 2.35 -3.66 2.82 ±0.11 3.08 ±0.02 2.57 ±0.12 2.77 ±0.02 0.07 ±0.06 -0.14 ±0.01 15450
34 1.35 -2.40 3.18 ±0.06 3.36 ±0.02 2.62 ±0.03 2.92 ±0.01 0.04 ±0.02 -0.11 ±0.01 16889
35 3.05 -3.00 3.05 ±0.05 3.09 ±0.02 2.59 ±0.07 2.72 ±0.02 0.08 ±0.03 -0.14 ±0.01 15973
36 3.16 -3.20 3.00 ±0.06 3.19 ±0.02 2.95 ±0.40 2.77 ±0.02 -0.05 ±0.08 -0.16 ±0.01 14955
37 0.00 -1.74 3.29 ±0.04 3.29 ±0.02 2.70 ±0.04 3.04 ±0.01 -0.01 ±0.01 -0.05 ±0.01 20233
38 0.97 -3.42 3.01 ±0.07 3.15 ±0.01 2.60 ±0.14 2.84 ±0.02 0.07 ±0.07 -0.12 ±0.01 15542
39 0.53 -2.21 3.22 ±0.03 3.21 ±0.01 2.69 ±0.06 3.00 ±0.01 0.01 ±0.04 -0.07 ±0.01 24820
40 -2.99 -3.14 2.84 ±0.04 2.84 ±0.01 2.28 ±0.07 2.47 ±0.02 -0.09 ±0.07 0.05 ±0.01 13581
41 -2.78 -3.27 2.86 ±0.06 2.78 ±0.01 2.60 ±0.19 2.41 ±0.02 -0.16 ±0.07 0.04 ±0.01 14070
42 4.48 -3.38 3.07 ±0.05 2.89 ±0.02 2.44 ±0.15 2.63 ±0.02 0.02 ±0.02 -0.15 ±0.01 10099
43 0.37 2.95 3.13 ±0.06 3.17 ±0.02 2.72 ±0.10 2.87 ±0.01 0.04 ±0.07 0.02 ±0.01 11467
45 0.98 -3.94 3.02 ±0.05 2.97 ±0.04 2.42 ±0.14 2.61 ±0.04 0.06 ±0.11 -0.13 ±0.02 2380
46 1.09 -4.14 2.87 ±0.08 2.90 ±0.04 2.53 ±0.21 2.67 ±0.04 -0.03 ±0.06 -0.16 ±0.03 1803
is 3986. This relatively low number results in large errors on the
model proper motion dispersions and we therefore recommend re-
garding interpretations based on this model with some caution. An
improved model with a larger number of particles (the recent study
by De Lorenzi et al. (2007) has an effective particle number ∼ 106)
will decrease the errors on the model predictions and allow a more
useful comparison between model and observed proper motion dis-
persions.
The OGLE-II fields mostly extend over ∼ 17◦ in longi-
tude and about 5◦ in latitude across the Galactic bulge region
and can therefore provide a more powerful set of constraints on
stellar motions predicted by galactic models. The high-accuracy
proper motion data for the 45 fields and those obtained with HST
(Kozłowski et al. 2006) can be used as direct input in the made-to-
measure method to construct a better constrained dynamical model
of the Milky Way.
The statistical errors of our proper motion dispersions are
small (∼ km s−1), but systematic uncertainties (for example due
to incorrect dust extinction treatment) which were not included in
the analysis may be significant. Nevertheless, it is interesting to
note that there appears to be significant difference between the ob-
served proper motion dispersions of adjacent fields (e.g. fields 1
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 11
0.22 0.23 0.24 0.25
PSfrag replacements
Separation (◦)
Figure 10. Difference between observed proper motion dispersions for
pairs of fields with separations less than 0.25 degrees (corresponding to
≃ 40 pc at the Galactic centre).
and 45). This might hint at some fine-scale population effect,where
the kinematics of the bulge may be not in total equilibrium (e.g. due
to a small accretion event). Higher-accuracy observations using the
HST may provide further evidence of such population effects. We
note that Rich et al. (2006) suggest the possible existence of cold
structures using data from a radial velocity survey of Galactic bulge
M giant stars although their conclusion could be strengthened by a
larger sample of stars.
The OGLE-II proper motion catalogue (Sumi et al. 2004) for
millions of bulge stars is still somewhat under-explored. For exam-
ple, it will be interesting to explore the nature of the high proper
motion stars (µ > 10 mas yr−1) and search for wide binaries in
the catalogue. Some exploration along these lines is under way.
ACKNOWLEDGEMENTS
We thank Drs. Vasily Belokurov, Wyn Evans and Martin Smith for
helpful discussions, and the anonymous referee for their helpful
suggestions.
NJR acknowledges financial support by a PPARC PDRA
fellowship. This work was partially supported by the European
Community’s Sixth Framework Marie Curie Research Training
Network Programme, Contract No. MRTN-CT-2004-505183 ‘AN-
GLES’. VPD is supported by a Brooks Prize Fellowship at the Uni-
0.22 0.23 0.24 0.25
PSfrag replacements
Separation (◦)
Figure 11. Difference between model proper motion dispersions for pairs
of fields with separations less than 0.25 degrees.
versity of Washington and receives partial support from NSF ITR
grant PHY-0205413.
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	Introduction
	Observed Proper Motion Dispersions
	Red Clump Giants
	OGLE-II proper motion data
	Extinction
	Results
	Galactic Model
	Model stellar magnitudes
	Model kinematics
	Comparison between theoretical model and observed data
	Understanding the differences
	Discussion
