The Optical Gravitational Lensing Experiment. Catalog of stellar proper motions in the OGLE

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The Optical Gravitational Lensing Experiment. Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

arXiv:astro-ph/0305315v2 28 Nov 2003

T. Sumi1, X. Wu1 , A. Udalski2, M. Szyma?ski2, M. Kubiak2, G. Pietrzy?ski2,3, n n 2 4 2 2 2 ˙ I. Soszy?ski , P. Wo?niak , K. Zebru? , O. Szewczyk & L. Wyrzykowski n z n
1 2 Warsaw 3 4

Princeton University Observatory, Princeton, NJ 08544-1001, USA; e-mail: (sumi, xawn)@astro.princeton.edu University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland; e-mail: (udalski,msz,mk,pietrzyn,soszynsk,zebrun,szewczyk,wyrzykow)@astrouw.edu.pl Universidad de Concepci?n, Departamento de Fisica, Casilla 160–C, Concepci?n, Chile o o Los Alamos National Laboratory, MS-D436, Los Alamos, NM 87545 USA; e-mail: wozniak@lanl.gov

Accepted Received in original form

ABSTRACT

We present a proper motion (?) catalogue of 5,080,236 stars in 49 Optical Gravitational Lensing Experiment II (OGLE-II) Galactic bulge (GB) ?elds, covering a range of ?11? < l < 11? and ?6? < b < 3? , the total area close to 11 square degrees. The proper motion measurements are based on 138 ? 555 I-band images taken during four observing seasons: 1997-2000. The catalogue stars are in the magnitude range 11 < I < 18 mag. In particular, the catalogue includes Red Clump Giants (RCGs) and Red Giants in the GB, and main sequence stars in the Galactic disc. The proper motions up to ? = 500 mas yr ?1 were measured with the mean accuracy of 0.8 ? 3.5 mas yr?1 , depending on the brightness of a star. This catalogue may be useful for studying the kinematic of stars in the GB and the Galactic disk. Key words: Galaxy:bulge – Galaxy:center – Galaxy:kinematics and dynamics– Galaxy:structure – astrometry

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INTRODUCTION

The Galactic Bulge (GB) is the nearest bulge in which individual stars can be studied in detail. A study of stellar populations and stellar dynamics in the Bulge may help us understand how bulges formed, what are their populations, gravitational potential and structure. The proper motions with precise photometry might make it possible to separate the observed populations based on their kinematics. Such study has ?rst been done by Spaenhauer, Jones & Whitford (1992) with photographic plates only for a few hundred of brightest red giants in Baade’s window. Recently deeper study has been done by Kuijken & Rich (2002) with Hubble Space Telescope (HST)/WFPC2 in Baade’s window. Several groups have carried out gravitational microlensing observations toward dense stellar ?elds, such as the Magellanic clouds, the Galactic center and disc. Until now, hundreds of events have been found (EROS: Aubourg et al. 1993; OGLE: Udalski et al. 2000; Wo?niak et al. 2001; z MACHO: Alcock et al. 2000; MOA: Bond et al. 2001; Sumi et al. 2003), and thousands are expected in the up-

coming years by MOA 1 , OGLE-III 2 and other collaborations. It is well known that the gravitational microlensing survey data is well suited for numerous other scienti?c projects (see Paczy?ski 1996; Gould 1996). The studies of n the Galactic structure certainly bene?t from this type of data. The microlensing optical depth probes the mass density of compact objects along the line of sight and the event time-scale distribution is related to the mass function and kinematics of the lensing objects. Observed high optical depth may be explained by the presence of the bar (Udalski et al. 1994; Alcock et al. 1997, 2000; Sumi et al. 2003; Afonso et al. 2003; Popowski et al 2003) There is substantial evidence that the Galaxy has a bar at its center (de Vaucouleurs 1964; Blitz & Spergel 1991; Stanek et al. 1994, 1997; Kiraga & Paczy?ski 1994; H¨fner et al. 2000). n a However, the parameters of the bar, e.g., its mass, size, and the motion of stars within it, still remain poorly constrained. Stanek et al. (1997) used the Red Clump Giants (RCGs) to constrain the axial ratios and orientation of

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Only about 70% of the area of the BUL SC1 ?eld overlaps with the extinction map made by Stanek (1996). The extinction map covering all OGLE-II ?elds has been constructed by Sumi (2003).

the Galactic bar. These stars are the equivalent of the horizontal branch stars for a metal-rich population, i.e., relatively low-mass core helium burning stars. RCGs in the GB occupy a distinct region in the colour magnitude diagram (Stanek et al. 2000 and references therein). The intrinsic width of the luminosity distribution of RCGs in the GB is small, about 0.2 mag (Stanek et al. 1997; Paczy?ski & Stanek 1998). Their observed peak and width n of the luminosity function are related to the distance and radial depth of the bar. Furthermore, Mao & Paczy?ski (2002) suggested that n there should be a di?erence in average proper motions of 1.6 mas yr?1 between the bright and faint RCG sub-samples, which are on average on the near and the far side of the bar, respectively, if their tangential streaming motion is 100 km s?1 . Following this suggestion, Sumi, Eyer & Wo?niak z (2003) measured mean proper motion of bright and faint RCGs in one OGLE-II ?eld in Baade’s Window, and they found a di?erence to be 1.5 ± 0.11 mas yr?1 . To expand this analysis we measured proper motions in all 49 GB ?elds observed by the Optical Gravitational Lensing Experiment 3 II (OGLE-II; Udalski et al. 2000) for stars down to I = 18 mag, which is su?ciently deep to include RCGs. There are several earlier proper motion catalogues of this general area (c.f. USNO-B:Monet et al. 2002, Improved NLTT:Salim & Gould 2003 and Tycho-2:H?g et al. 2000). Though the area covered by our catalogue is relatively small, it reaches deeper and covers a wide range of proper motion (? < 500 mas yr?1 ) with the accuracy as good as (? 1 mas yr?1 ). In § 2 we describe the data. We present the analysis method in § 3 and 4. In § 5, 6 and 7 we describe the property, zero point and problems in our catalogue. Discussion and conclusion are given in § 8.

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ANALYSIS

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DATA

We use the data collected during the second phase of the OGLE experiment, between 1997 and 2000. All observations were made with the 1.3-m Warsaw telescope located at the Las Campanas Observatory, Chile, which is operated by the Carnegie Institution of Washington. The ”?rst generation” camera has a SITe 2048 × 2048 pixel CCD detector with pixel size of 24 ?m resulting in 0.417 arcsec/pixel scale. Images of the GB ?elds were taken in drift-scan mode at ”medium” readout speed with the gain 7.1 e? /ADU and readout noise of 6.3 e? . A single 2048 × 8192 pixel frame covers an area of 0.24 × 0.95 deg2 . Saturation level is about 55,000 ADU. Details of the instrumentation setup can be seen in Udalski, Kubiak & Szyma?ski (1997). n In this paper we use 138-555 I-band frames of the BUL SC1-49 ?elds. The centers of these ?elds are listed in Table 1. The time baseline is almost 4 years. There are gaps between the observing seasons when the GB cannot be observed from the Earth, each about 3 months long. The median seeing is ? 1.3′′ . We use the V I photometric maps of the standard OGLE template (Udalski et al. 2002) as the astrometric and photometric references.

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see http://www.astrouw.edu.pl/~ogle http://bulge.princeton.edu/~ogle

or

The analysis in this work follows Sumi, Eyer & Wo?niak z (2003), except our procedure makes it possible to detect high proper motions, extends to the limiting magnitude down to I = 18 mag and corrects the systematic e?ects. The standard OGLE template given by Udalski et al. (2002) serves as the ?xed astrometric reference in our analysis. In order to treat properly frame distortions in the y-axis (declination) due to drift-scan mode of observation each OGLE-II ?eld is divided into 64 subframes before processing. Subframes are 2048×128 pixels with a 14 pixel margin on each side. We compute the pixel positions of stars in the images using the DoPHOT package (Schechter, Mateo & Saha 1993). At the start of the processing for each exposure, the positions of stars in a single subframe are measured and cross-referenced with those in the template and the overall frame shift is obtained. Using this crude shift we can identify the same region of the sky (corresponding to a given subframe of the template) throughout the entire sequence of frames. To treat properly spatial PSF variations, each 2048×128 pixel subframe is divided into 4 smaller chunks with a size of 512×128 pixels with a 14 pixel margin on each side. Then the positions of stars in all chunks are computed by DoPHOT. We use all stars with I ≤ 18 mag (? 400 of them, depending on the stellar density in each ?eld) categorized by DoPHOT as isolated stars (marked as type=1) in the following analysis. We do not use the data points categorized by DoPHOT as a star blended with other stars (marked as type=3). The stars in each of the chunks are combined into the original 2048×128 subframes. We cross-reference the stars in the template and other frames with a search radius of 0.5 pixels and derive the local transformation between these pixel coordinate systems for each subframe. We use a ?rst order polynomial to ?t the transformation. The resulting piece-wise transformation adequately converts pixel positions to the reference frame of the template. Typical residuals are at the level of 0.08 pixels for bright stars (I < 16) and 0.2 pixels for all stars (I < 18). By using these transformation matrices, we crossreference the stars in the template and other frames with a search radius of 1.0 pixels instead of 0.5 pixels used in Sumi, Eyer & Wo?niak (2003) to increase the range of dez tectable high proper motion objects. We estimate that the probability of the mis-identi?cation in this search radius is negligible (0.26 %). We have found that there are systematic di?erences in the mean positional shifts of stars from the template position dx and dy depending on time and pixel coordinate in x. We have measured the dx and dy of the stars in 81 strips (X = 0 ? 80) centered at equal intervals in x coordinate between 0 ≤ x ≤ 2048 with a width of ±25 pixels. Each strip contains typically ? 2, 000 stars. In the upper panel of Fig. 1 we show dx as a function of time for the strip X = 40 (x = 1024±25 pixels) in BUL SC2. We can see

The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

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Table 1. Center positions of OGLE-II 49 ?elds. The number of frames Nf and stars Ns , and the mean of uncertainty in proper motion, < σ? > (mas yr?1 ), are given as a function of I (mag) for each ?eld. σ? is averaged over I ± 0.5 mag. ?eld BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10 SC11 SC12 SC13 SC14 SC15 SC16 SC17 SC18 SC19 SC20 SC21 SC22 SC23 SC24 SC25 SC26 SC27 SC28 SC29 SC30 SC31 SC32 SC33 SC34 SC35 SC36 SC37 SC38 SC39 SC40 SC41 SC42 SC43 SC44 SC45 SC46 SC47 SC48 SC49 α2000 18h 02m 32.5s 18h 04m 28.6s 17h 53m 34.4s 17h 54m 35.7s 17h 50m 21.7s 18h 08m 03.7s 18h 09m 10.6s 18h 23m 06.2s 18h 24m 02.5s 18h 20m 06.6s 18h 21m 06.5s 18h 16m 06.3s 18h 17m 02.6s 17h 47m 02.7s 17h 48m 06.9s 18h 10m 06.7s 18h 11m 03.6s 18h 07m 03.5s 18h 08m 02.4s 17h 59m 19.1s 18h 00m 22.3s 17h 56m 47.6s 17h 57m 54.5s 17h 53m 17.9s 17h 54m 26.1s 17h 47m 15.5s 17h 48m 23.6s 17h 47m 05.8s 17h 48m 10.8s 18h 01m 25.0s 18h 02m 22.6s 18h 03m 26.8s 18h 05m 30.9s 17h 58m 18.5s 18h 04m 28.6s 18h 05m 31.2s 17h 52m 32.2s 18h 01m 28.0s 17h 55m 39.1s 17h 51m 06.1s 17h 52m 07.2s 18h 09m 05.0s 17h 35m 13.5s 17h 49m 22.4s 18h 03m 36.5s 18h 04m 39.7s 17h 27m 03.7s 17h 28m 14.0s 17h 29m 25.1s δ2000 ?29? 57′ 41′′ ?28? 52′ 35′′ ?29? 57′ 56′′ ?29? 43′ 41′′ ?29? 56′ 49′′ ?32? 07′ 48′′ ?32? 07′ 40′′ ?21? 47′ 53′′ ?21? 47′ 55′′ ?22? 23′ 03′′ ?22? 23′ 05′′ ?23? 57′ 54′′ ?23? 57′ 44′′ ?23? 07′ 30′′ ?23? 06′ 09′′ ?26? 18′ 05′′ ?26? 12′ 35′′ ?27? 12′ 48′′ ?27? 12′ 45′′ ?28? 52′ 55′′ ?28? 51′ 45′′ ?30? 47′ 46′′ ?31? 12′ 36′′ ?32? 52′ 45′′ ?32? 52′ 49′′ ?34? 59′ 31′′ ?35? 09′ 32′′ ?37? 07′ 47′′ ?37? 07′ 21′′ ?28? 49′ 55′′ ?28? 37′ 21′′ ?28? 38′ 02′′ ?28? 52′ 50′′ ?29? 07′ 50′′ ?27? 56′ 56′′ ?27? 56′ 44′′ ?29? 57′ 44′′ ?29? 57′ 01′′ ?29? 44′ 52′′ ?33? 15′ 11′′ ?33? 07′ 41′′ ?26? 51′ 53′′ ?27? 11′ 00′′ ?30? 02′ 45′′ ?30? 05′ 00′′ ?30? 05′ 11′′ ?39? 47′ 16′′ ?39? 46′ 58′′ ?40? 16′ 21′′ l (deg.) 1.08 2.23 0.11 0.43 -0.23 -0.25 -0.14 10.48 10.59 9.64 9.74 7.80 7.91 5.23 5.38 5.10 5.28 3.97 4.08 1.68 1.80 -0.26 -0.50 -2.44 -2.32 -4.90 -4.92 -6.76 -6.64 1.94 2.23 2.34 2.35 1.35 3.05 3.16 0.00 0.97 0.53 -2.99 -2.78 4.48 0.37 -0.43 0.98 1.09 -11.19 -11.07 -11.36 b (deg.) -3.62 -3.46 -1.93 -2.01 -1.33 -5.70 -5.91 -3.78 -3.98 -3.44 -3.64 -3.37 -3.58 2.81 2.63 -3.29 -3.45 -3.14 -3.35 -2.47 -2.66 -2.95 -3.36 -3.36 -3.56 -3.37 -3.65 -4.42 -4.62 -2.84 -2.94 -3.14 -3.66 -2.40 -3.00 -3.20 -1.74 -3.42 -2.21 -3.14 -3.27 -3.38 2.95 -1.19 -3.94 -4.14 -2.60 -2.78 -3.25 Nf 259 264 555 514 392 306 323 289 288 291 280 294 270 290 285 277 284 275 273 316 321 414 350 359 342 346 334 321 313 323 334 313 273 329 260 290 406 268 415 325 312 273 382 343 140 138 242 237 234 Ns 11.5 120697 140398 167580 179906 113793 65578 62357 52549 51274 57064 51181 79162 79082 90091 84372 100885 101955 133282 112421 169423 161268 111768 94798 91733 90853 95233 92508 57501 57822 151877 155383 155100 121848 156281 139324 145376 152704 123428 155735 82152 87013 99152 76840 68457 107362 97197 47459 47673 43341 2.38 2.62 1.98 2.34 1.95 1.78 2.00 2.05 1.82 1.80 1.87 2.26 2.21 2.39 2.19 2.30 2.21 2.48 2.39 2.70 2.64 2.32 2.15 2.20 2.48 2.00 2.11 2.15 2.38 2.65 2.22 2.31 2.57 2.49 2.51 2.70 2.46 2.40 2.46 2.11 2.44 2.40 1.86 1.85 3.13 3.38 3.01 2.65 3.09 σ? (mas yr?1 ), for I (mag) = 12.5 13.5 14.5 15.5 16.5 0.92 0.94 0.66 0.69 0.78 0.92 0.93 0.90 0.90 0.99 1.01 0.98 1.04 1.01 0.98 1.01 0.95 0.97 0.97 0.99 0.99 0.79 0.80 0.87 0.84 0.85 0.88 0.83 0.81 0.91 0.95 0.93 0.92 0.99 0.96 0.99 0.75 0.93 0.79 0.89 0.90 0.98 0.80 0.86 1.67 1.86 1.32 1.39 1.29 0.88 0.89 0.63 0.69 0.74 0.85 0.90 0.86 0.85 0.93 0.93 0.93 1.02 0.97 0.94 0.93 0.92 0.91 0.93 0.98 0.96 0.77 0.73 0.80 0.78 0.81 0.84 0.75 0.76 0.87 0.92 0.91 0.85 0.95 0.89 0.95 0.72 0.90 0.77 0.82 0.86 0.94 0.82 0.86 1.65 1.78 1.29 1.29 1.26 0.99 1.04 0.78 0.89 0.84 0.94 1.00 0.95 0.91 0.97 0.96 1.02 1.07 1.05 1.03 1.03 1.02 1.08 1.02 1.15 1.13 0.83 0.79 0.85 0.86 0.94 0.94 0.79 0.78 1.05 1.11 1.10 1.00 1.16 1.05 1.10 0.87 1.05 0.91 0.88 0.91 1.03 0.84 0.90 1.85 1.94 1.33 1.28 1.25 1.34 1.40 1.22 1.44 1.12 1.25 1.30 1.20 1.17 1.27 1.24 1.36 1.47 1.44 1.39 1.42 1.40 1.47 1.38 1.67 1.63 1.19 1.10 1.19 1.14 1.31 1.29 0.98 1.00 1.51 1.54 1.55 1.34 1.70 1.45 1.55 1.27 1.43 1.38 1.19 1.24 1.45 1.14 1.17 2.45 2.46 1.59 1.56 1.50 2.53 2.83 2.02 2.50 2.03 2.23 2.27 1.99 1.96 2.13 2.01 2.37 2.50 2.41 2.26 2.57 2.58 2.91 2.53 3.36 3.34 1.92 1.74 1.88 1.85 2.28 2.22 1.64 1.66 2.94 3.11 3.21 2.56 3.04 2.87 3.27 2.13 2.66 2.26 1.85 1.96 2.50 1.91 1.98 4.62 4.45 2.47 2.44 2.32

17.5 6.01 6.76 4.66 5.73 3.54 4.47 4.40 3.79 3.71 4.09 3.83 4.71 4.93 5.23 4.88 5.49 5.50 6.64 5.84 8.10 7.77 4.59 4.10 4.22 4.30 5.03 4.90 3.43 3.41 7.05 7.42 7.56 6.01 7.54 6.80 7.92 4.49 6.41 5.53 3.99 4.34 5.31 3.65 3.75 10.57 10.43 4.74 4.70 4.51

the big jump at JD=2451041 (indicated by a vertical dashed line) where the exposure time of OGLE-II in the GB ?elds has been changed from 87 sec to 99 sec in the middle of 1998 season, on August 15. In the upper panel of Fig. 2 we show typical mean positional shifts in x, dx of stars in strips in BUL SC2 as a function of pixel coordinate x. The ?lled and open circles represent the dx of the frame taken at JD=2450887.822 (before the jump) and 2451336.769 (after the jump), respectively. There are also systematics in dy with the level of 0.04 pixels. We cannot see any such sys-

tematics as a function of y pixel coordinate. Because of the good coincidence between the jump and the change in the drift scan rate that determines the e?ective exposure time, the bulk of the systematics may be caused by the change in drift scan rate. However, we do not know the detail reasons behind this at the present time. Even within the period before and after the jump, the shapes of Fig. 2 di?er from time to time and from ?eld to ?eld at the level of 0.04 pixels. By interpolating these curves of dx and dy as a function of x for each frame (time) of

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Figure 1. Upper panel: The mean positional shift in x, dx of stars in a strip X = 40 (x = 1024 ± 25 pixels) in BUL SC2 as a function of time. We can see the big jump at JD=2451041 (indicated by a vertical dashed line) where the exposure time of OGLE-II in the GB ?elds has changed from 87 sec to 99 sec in the middle of 1998 season, on August 15. Lower panel: The same ?gure as above after the systematic correction.

Figure 3. Time variation of the position in αcosδ (upper) and δ (lower) for star ID=5935 (V-I=1.260) in BUL SC5. Filled circles represent actual positions, while open circles are positions corrected for di?erential refraction with an o?set of ?0.16 pixels. Solid lines indicate a model ?t for the proper motion (?α? , ?δ ) = (?1.19(0.41), 0.94(0.35)) (mas yr?1 ) with the 1 σ errors in bracket. The di?erential refraction coe?cient is a = ?22.42 mas.

An example of time dependence of the position for a star with a moderate detectable proper motion is shown in Fig. 3 with ?lled circles. To measure the proper motions in right ascension (?α? ≡ ?α cosδ) and in declination (?δ ), we ?t the positions as a function of time t with the following formula: α = α0 + ?α? t + a sin C tan z, δ = δ0 + ?δ t + a cos C tan z, (1) (2)

Figure 2. Upper panel: Typical mean positional shifts in x, dx of stars in strips in BUL SC2 as a function of pixel coordinate x. The ?lled and open circles represent the dx of the frame taken at JD=2450887.822 (before the jump) and 2451336.769 (after the jump), respectively. Lower panel: The same ?gure as above after the systematic correction. Note that vertical scale in the lower panel is very di?erent than in the upper panel.

each ?eld, we correct dx and dy for each star and frame. In the lower panel of Fig. 1 and Fig. 2, we show the same plots after this systematic correction. This procedure is based on the assumption that average proper motions of a large number of stars in separate groups of columns (i.e. di?erent values of X) should be the same. Note that the integral of the curves shown in Fig. 2 over all x-columns is unity, as this corresponds to the average position of all stars.

where a is the coe?cient of di?erential refraction, z is the zenith angle, and C is the angle between the line joining the star and Zenith and the line joining the star with the South Pole, and α0 and δ0 are constants. The parameter a is a function of the apparent star colour. We neglect the parallactic motion due to the Earth’s orbit because its e?ect is strongly degenerate with the e?ect of di?erential refraction for stars in the direction of the GB (Eyer & Wo?niak 2001). z In Fig. 3 we present with solid lines and open circles the best ?t model for the proper motion (?α? , ?δ ) and positions allowing for the di?erential refraction respectively. As is written in this ?gure, the parameters for this object are: ?eld SC5-1-6, which means that this object is in the OGLE-II ?eld BUL SC5, and chunk (Xchunk , Ychunk )=(1, 6), OGLE ID=5935, I = 14.093, V ? I = 1.260, the number of data points N = 392, proper motion (?α? , ?δ ) = (1.19(0.41), 0.94(0.35)) (mas yr?1 ) with 1 σ errors in brackets. The di?erential refraction coe?cient is a = ?22.42 mas. We computed α0 , δ0 , ?α? , ?δ and a for all stars used to transform coordinate systems (approximately the number of ?elds times the number of chunks times the typical number of stars per chunk, i.e. 49 × 256 × 400). In cases where the star is measured in the overlap region of more than one chunk of a given ?eld, the data set with the largest

The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

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Figure 4. Colour Magnitude Diagram of stars with σ? < 2.5 mas yr?1 in BUL SC2. I0 and (V ? I)0 are extinction corrected I-band magnitude and V ? I colour. The stars in GB are de?ned within the ellipse centered at the center of RCGs plus 0.4 mag in I.

Figure 5. Upper panel: The mean proper motions in x (?lled circle) and in y (open circle), for stars in 50 pixels strips in BUL SC2 as a function of pixel coordinate x. Lower panel: The same ?gure as above after the systematic correction.

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HIGH PROPER MOTION OBJECTS

number of points is selected. Stars with the fewer than 20 data points are rejected. The catalogue of whole BUL SC1 was recomputed in this analysis because the catalogue presented by Sumi, Eyer & Wo?niak (2003) contains only 70 % z of this ?eld. Our catalogue contains 5,080,236 stars, which is 79.8 % of all objects with I ≤ 18 mag in the original OGLE template (Udalski et al. 2002). 7% of these stars do not have any V -band photmetry. The missing V -band photometry of stars can be estimated from the di?erential refraction coe?cient because there is good correlation between the di?erential refraction coe?cient a and the apparent V ? I colour for stars (see Fig. 8), provided they belong to the same population as that of the majority. We have measured the mean proper motions of stars in the Galactic bulge (GB) de?ned in the ellipse in the Colour Magnitude Diagram (CMD) in Fig. 4, where the extinction and reddening are corrected by using the extinction map of Sumi (2003). The ellipse is located at the center of the RCGs estimated in Sumi (2003) plus 0.4 mag in I and have semi-major and minor axises of 0.9 mag and 0.4 mag, respectively. This ellipse includes RCGs and Red Giants in the GB. Here we chose only objects whose proper motion accuracy is better than 2.5 mas yr?1 . These mean proper motions of stars in the GB are assumed to be constant within each ?eld. In Fig. 5 we show these measured mean proper motions of stars in the GB as a function of x (upper panel) and after the systematic correction (lower panel). We can see how well the systematic distortions are corrected. Note that our instrumental reference frame is de?ned by all stars, and this is why the average proper motion of stars in the GB are not zero.

The detectability of high proper motions is limited by the search radius used for cross – identi?cation of stars on all images. The search radius, 1 pix yr?1 , corresponds to ? 400 mas yr?1 . Objects with ? ≥ 100 mas yr?1 cannot be identi?ed over the full four year long observing interval of OGLEII, as they move out of the search radius. In order to be able to follow fast moving stars over four observing seasons we made additional astrometry for objects for which preliminary estimates gave a proper motion ? ≥ 100 mas yr?1 . Whenever a star moved more than 0.4 pixels from the previous search center we moved that center to the median location of the last 3 data points. This procedure was adopted when it allowed us to locate the star in a larger number of CCD images. We show the positional movement of one of the highest proper motion objects in Fig. 6 and images at 1997 and at 2000 in Fig. 7. This star has relatively fewer data points because this star is overexposed on the good seeing frames in the I-band. This star dose not have any V -band measurements because of the failure of cross-identi?cation in the V -band template image due to its high proper motion. I-band measurements of such objects may also be unreliable because the OGLE photometric maps are based on the measurements by the ”?xed position mode” over years (Udalski et al. 2002). The photometries of this star obtained by hand relative to the neighboring stars are: I = 11.70, V ? I = 2.86. This colour is very red as expected from the relation between di?erential refraction coe?cient a from the ?t and colour in Fig.8, but its a and V ? I do not match the relation exactly. The relation between a and V ? I for BUL SC42 in which the extinctions are relatively small, is slightly di?erent from that for BUL SC5 plotted in Fig. 8. The zero-point of a is 15 mas higher than that for BUL SC5, but the expected colour of this star from this relation is still redder (V ? I = 4 ? 5). In these ?gures the slopes di?er between ?elds for

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galactic coordinate ?l and ?b with their errors, di?erential refraction coe?cient a, standard deviation (Sdev) of data points in the ?tting, equatorial coordinates α2000 , δ2000 in 2000, apparent I-band magnitude and V ? I colour, pixel coordinates on CCD x and y, and position of chunk Xc and Yc to which this object belongs. ID, 2000 coordinates, I and V ? I for each object are identical to those in Udalski et al. (2002). If the object doesn’t have V -band photometry the V ? I colour is written as 9.999. Note that the number of data points N di?ers from star to star even if they have similar brightness. This is because some are near the edge of the CCD image or CCD defects, and others are a?ected by blending. We use the positions which are categorized as a single isolated star by DoPHOT. Hence, blended stars can not be measured when the seeing is poor. In Fig. 8 we present a colour – magnitude diagram (CMD) for stars in OGLE ?eld BUL SC5, which is one of the most reddened OGLE-II ?elds. We also show the correlation between the di?erential refraction coe?cient a and the apparent V ? I colour for stars with I < 16 in this ?eld. This correlation for other ?elds is similar, but it has a slight dependence on the population of the majority of stars in each ?eld. In Fig. 9, we plot the uncertainties in ?α? (σ?α? , upper panel), in ?δ (σ?δ , middle panel) and di?erence between them (σ?α? ?σ?δ , lower panel) in BUL SC3 as a function of the apparent I-band magnitude. In the lower panel, where open circles represent mean values for each 1 magnitude bin, we can see that σ?α? is systematically larger than σ?δ at ? 0.1 mas yr?1 level. We see the same trend in all our ?elds. This trend is expected from the residual scattering due to the systematic correction and the di?erential refraction which is large in the direction of α. The mean 2 2 σ?α? + σ?δ with 2σ clipping as a uncertainties σ? ≡ function of I are listed in Table 1. Note: that the accuracy of proper motions in our catalogue is better than 1 mas yr?1 for 12 < I < 14. Fig. 10 shows the histogram of our proper motion measurements ? for the whole catalogue, with the lines with increasing thickness corresponding to all stars, and to those with the proper motion detected with con?dence better than 3σ, 5σ and 10σ, respectively. The total number of stars with proper motions measured to 3σ, 5σ and 10σ accuracy is N3 = 1, 469, 838, N5 = 568, 665, N10 = 61, 231, respectively. The inclined solid line: (log(N ) = ?3log(?) + const.), has a slope corresponding to the expectation for a uniform distribution and kinematics of stars in space. The distribution of accurate (5σ and 10σ) proper motions seems to be roughly consistent with the uniform distribution. However, the number of very high proper motion stars, (? > 200 mas yr?1 ), appears to be smaller than expected from a uniform distribution. This may be due to saturation of images of nearby, and therefore apparently bright, stars in OGLE images. The distribution of less accurate (≤ 3σ) proper motions has an apparent cut-o? at ? ? 100 mas yr?1 . Such stars are apparently faint, their positions have large errors, and they may be di?cult to identify in a search radius of 1 pixel, which corresponds to ? ? 100 mas yr?1 . A more thorough analysis of various selection e?ects is beyond the scope of this paper. Readers are advised to use caution in a statistical analysis of our catalogue.

Figure 6. Same plots as Fig. 3 for ID=133638 (V-I=none) in OGLE-II ?eld BUL SC42, one of the highest proper motion stars. Filled circles represent the actual positions and open circles are the positions corrected for di?erential refraction, with an o?set of +1.5 pixels. Solid lines indicate a model ?t for the proper motion (?α? , ?δ ) = (?261.19(2.51), ?436.90(1.2)) (mas yr?1 ) with 1 σ errors in bracket. The di?erential refraction coe?cient is a = 26.04 mas, which indicates the star is very red. This star dose not have V -band magnitude because of the high proper motion. The photometries of the star obtained by hand relatively to the neighboring stars are: I = 11.70, V ? I = 2.86.

19-4-1997

10"

30-9-2000

Figure 7. Images of the high proper motion star in Fig. 6, i.e. ID=133638 in OGLE-II ?eld BUL SC42, at 19 April 1997 (left) and at 30 September 2000 (right).

the region V ? I > 3. This is because of the di?erence of the population. In BUL SC5, the majority in this colour region are RCGs and Red Giant Branch stars, but Red Super Giants in BUL SC42. So this disagreement might be because this object is a nearby very red dwarf of M4-5 spectral type, not typical for this catalogue.

5

CATALOGUE

A sample for our catalogue of proper motions is shown in Table 2. The complete list of all 5,080,236 stars is available in electronic format via anonymous ftp from the server ftp://ftp.astrouw.edu.pl/ogle/ogle2/proper motion/ and ftp://bulge.princeton.edu/ogle/ogle2/proper motion/. The list contains star ID, the number ?α? and ?δ and in

Table 2: Sample of proper motion catalogue for BUL SC2.

ID 13213 13214 13215 13216 13217 13218 13219 13220 13222 13223 13224 13225 13227 13228 13229 13230 13231 13232

N 203 258 255 247 250 264 263 247 262 245 189 131 248 262 245 264 249 200

?α? 2.96 1.09 -1.71 -0.11 -3.44 1.49 -3.31 -1.11 -3.98 2.25 -2.72 0.83 0.32 0.45 -1.94 3.89 0.82 2.53

σ?α? 0.90 0.66 0.68 0.67 0.83 1.01 0.91 0.78 0.55 1.11 0.87 0.88 0.53 0.60 0.92 0.57 0.76 1.72

?δ -0.44 -2.03 -4.75 -1.46 -0.74 -0.30 -1.75 -4.67 -6.50 -7.76 0.31 -3.41 -5.23 1.12 -6.90 3.85 -0.58 11.04

?l σ?δ (mas yr?1 ) 0.63 0.41 1.01 0.51 0.76 0.66 0.39 1.41 0.50 1.14 0.64 1.01 0.68 0.62 0.98 0.60 0.68 1.50 1.06 -1.24 -4.98 -1.33 -2.32 0.46 -3.14 -4.62 -7.62 -5.68 -1.06 -2.57 -4.41 1.20 -6.97 5.26 -0.11 10.87

σ?l 0.70 0.48 0.94 0.55 0.78 0.76 0.56 1.29 0.51 1.13 0.70 0.98 0.65 0.62 0.97 0.59 0.70 1.56

?b -2.80 -1.94 -0.82 -0.62 2.64 -1.45 2.04 -1.31 0.31 -5.75 2.53 -2.39 -2.83 0.15 -1.67 -1.52 -1.00 3.17

σ?b 0.84 0.61 0.77 0.64 0.81 0.94 0.82 0.97 0.54 1.12 0.82 0.91 0.57 0.61 0.93 0.58 0.74 1.67

a

Sdev (mas) 13.02 10.16 16.10 10.95 14.72 16.19 13.20 20.88 9.87 20.22 12.55 13.38 11.08 11.51 17.20 11.09 13.26 26.09

α2000 (deg) 270.99213 271.03837 271.04729 271.00104 271.00654 271.04246 271.02271 270.99650 271.01058 271.04275 270.99100 270.98617 271.03579 271.00475 270.99608 271.02446 271.03533 271.00042

δ2000 (deg.) -29.24292 -29.24297 -29.24297 -29.24281 -29.24286 -29.24286 -29.24231 -29.24206 -29.24183 -29.24197 -29.24144 -29.24125 -29.24092 -29.24047 -29.24036 -29.24042 -29.24047 -29.24025

I

V ?I (mag) 1.522 1.716 1.599 1.990 1.746 1.711 1.931 1.644 1.853 1.903 1.915 1.795 1.690 1.909 1.876 1.757 1.850 1.658

x (pixel) 62.52 413.60 481.45 130.10 171.88 444.59 294.82 95.64 202.60 446.73 53.69 17.09 394.10 158.16 92.42 307.89 390.64 125.33

y 904.72 904.92 904.94 905.70 905.41 905.97 910.42 912.22 914.42 913.63 917.54 919.06 922.69 926.03 926.87 926.93 926.65 928.07

Xc 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Yc 7 8 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8

0.69 -0.88 1.18 -0.57 -2.72 -3.97 -2.80 2.30 0.29 5.67 -3.43 1.48 0.41 0.31 0.10 -0.14 -1.17 0.86

15.282 15.507 15.572 15.635 15.768 14.960 15.132 15.436 15.310 15.369 15.874 15.460 15.015 15.208 15.402 15.082 15.090 15.608

The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

The proper motions in our catalogue are relative values. We need QSOs behind our ?elds to de?ne the zero point of our proper motions in the inertial frame. However we can get rough information of the zero point of our proper motions by measuring the average proper motion of stars located in the GB, which is presumably close to absolute proper motion of the Galactic Center (GC). We select stars in the GB de?ned in Fig. 4. Here we choose only objects whose proper motion accuracy is better than 2.5 mas yr?1 . We divide them into bins with a width of ?I = 0.1 mag in I-band magnitude and take mean of their proper motions for each bin. We plot these mean proper motions ?l,b as a function of I for BUL SC2 in Fig. 11. In this ?gure we can see the streaming motion of bright (I ? 14.1) and faint (I ? 14.7) RCGs in ?l . The proper motion of Red Giants (I > 15) is the same as the average motion of the RCGs, as expected since Red Giants are on average in the GC. To measure the proper motion of the GC, ?l,GC and ?b,GC in our reference frame, without any bias due to the incompleteness for fainter stars, we take a mean of these ?l,b without weighting by their errors. This provides an estimate that is more reliable than taking the mean proper motion of all individual stars. The measured ?l,GC and ?b,GC are shown as a solid and dashed line for BUL SC2 in Fig. 11 and listed for other ?elds in Table 3 along with those in equatorial coordinates (?α?,GC , ?δ,GC ). Table 3 is also avail-

6 ZERO POINT

Figure 8. Upper: Colour – magnitude diagram of a quarter of the analyzed stars in OGLE-II ?eld BUL SC5. Lower: correlation between the di?erential refraction coe?cient a and the apparent colour for stars with I < 16 in this ?eld.

7

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Figure 11. Mean proper motions of stars at GC ?l (?lled circle) and ?b (open circle) within ?I = 0.1 mag bin as a function of I-band magnitude. Solid and dashed lines represent the mean of these bins without weighting by errors. We can see the streaming motion of bright (I ? 14.1) and faint (? 14.7) RCGs. The proper motion of Red Giants (I > 15) is the mid of them as expected from that they are on average on the GC.

Figure 9. Uncertainties in ?α? (σ?α? , upper panel), ?δ (σ?δ , middle panel) and the di?erence between them (σ?α? ?σ?δ , lower panel) in OGLE-II ?eld BUL SC3 as a function of the I-band magnitude. In the lower panel, where open circles represent mean values for each 1 magnitude bin, we can see that σ?α? is systematically larger than σ?δ by ? 0.1 mas yr?1 level.

able in electronic format via anonymous ftp with the main catalogue (see § 5). The proper motion measurements in our catalogue can be transformed to the inertial frame by formula; ?α?OGLE ?δOGLE = = ?α? ? ?α?,GC + ?α?,GC,inert ?δ ? ?δ,GC + ?δ,GC,inert . (3) (4)

Figure 10. Histograms are shown for ? for all N = 5, 080, 236 stars, and for measurements with con?dence better than 3σ, 5σ and 10σ (from the thinnest to the thickest line, respectively) for all 49 OGLE-II ?elds. The total number of stars with proper motion with 3σ, 5σ and 10σ accuracy are N3 = 1, 469, 838, N5 = 568, 665, N10 = 61, 231. The inclined solid line: (log(N ) = ?3log(?) + const.) has the slope expected from stars that have uniform distribution and kinematics.

Here (?α?,GC,inert , ?δ,GC,inert ) = (?2.93, ?5.17) mas yr?1 is the expected proper motion of the GC relative to the inertial frame, assuming a ?at rotation curve of vr ? 220 km s?1 , the distance between the GC and the Sun of R0 = 8.0 kpc (Eisenhauer et al. 2003) and Solar velocity of (v⊙l , v⊙b )= (5.25 km s?1 , 7.17 km s?1 ) relative to the Local Standard of Rest (RSL) (Dehnen & Binney 1998). This transformation gives us crude absolute proper motions, while this gives us very good relative zero points from ?eld to ?eld as we show later. The reader can apply this transformation formula to our catalogue to get value in the inertial frame. We didn’t apply this transformation to our catalog because the transformation to the inertial frame can be improved with the QSOs to be discovered behind our ?elds in the future. To check our measurements we cross-identi?ed stars in our catalogue with the Tycho-2 catalogue (H?g et al. 2000). We selected from our catalogue objects with proper motions higher than 10 mas yr?1 and measured with a signi?cance above 3σ, to avoid mis-identi?cation. Most high proper motion stars in Tycho-2 are saturated in OGLE images. We found 65 Tycho-2 stars in our catalogue and their proper motions: ?α? (thin) and ?δ (thick) are presented in Fig. 12. Here OGLE proper motions have been transformed into the inertial frame by equations (3) and (4). We can see a very good correlation between OGLE and Tycho-2 measurements. Dashed lines indicate ?OGLE = ?T ycho?2 , and solid lines represent best ?t with ?xed unit slope and a possible

The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

9

Table 3. Proper motions of the GC for all 49 OGLE-II ?elds. The number of stars used in measurements Nstar , proper motions of the GC in our reference frame in equatorial coordinate ?α,GC and ?δ,GC and galactic coordinate ?l,GC and ?b,GC with their errors. ?eld BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL BUL SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10 SC11 SC12 SC13 SC14 SC15 SC16 SC17 SC18 SC19 SC20 SC21 SC22 SC23 SC24 SC25 SC26 SC27 SC28 SC29 SC30 SC31 SC32 SC33 SC34 SC35 SC36 SC37 SC38 SC39 SC40 SC41 SC42 SC43 SC44 SC45 SC46 SC47 SC48 SC49 Nstar 23938 25738 43597 42847 12704 10167 9431 7397 7364 8476 8067 9913 9899 19398 16454 15801 16665 23064 21441 31933 30486 31278 27679 27355 25782 21577 20250 12159 11755 29140 28389 25750 23709 32041 25997 24278 38711 25633 41129 25729 25665 18649 25012 – 13776 12608 6294 6965 6299 ?α,GC -0.57 -0.61 -0.66 -0.57 -0.72 -0.61 -0.55 -0.98 -0.85 -0.84 -0.80 -0.95 -0.84 -0.57 -0.53 -0.71 -0.68 -0.60 -0.60 -0.42 -0.55 -0.57 -0.59 -0.55 -0.52 -0.62 -0.60 -0.73 -0.75 -0.54 -0.44 -0.53 -0.61 -0.53 -0.56 -0.59 -0.68 -0.46 -0.47 -0.52 -0.50 -0.79 -0.36 – -0.43 -0.40 -1.18 -1.09 -0.99 σ?α,GC 0.07 0.05 0.05 0.06 0.08 0.05 0.06 0.05 0.05 0.06 0.06 0.05 0.06 0.05 0.05 0.06 0.05 0.05 0.05 0.06 0.06 0.05 0.06 0.06 0.05 0.05 0.04 0.05 0.04 0.05 0.06 0.06 0.06 0.06 0.05 0.05 0.06 0.06 0.06 0.05 0.05 0.06 0.07 – 0.08 0.08 0.07 0.06 0.06 ?δ,GC -0.55 -0.64 -0.68 -0.62 -0.94 -0.80 -0.68 -0.97 -1.02 -0.98 -1.03 -1.33 -1.13 -0.70 -0.79 -1.00 -0.97 -0.73 -0.77 -0.61 -0.71 -0.60 -0.70 -0.59 -0.66 -0.66 -0.72 -0.78 -0.78 -0.64 -0.67 -0.65 -0.69 -0.59 -0.66 -0.64 -0.75 -0.64 -0.64 -0.57 -0.56 -0.88 -0.36 – -0.32 -0.23 -1.20 -1.00 -0.86 ?l,GC σ?δ,GC (mas yr?1 ) 0.11 -0.76 0.09 -0.86 0.08 -0.92 0.10 -0.82 0.12 -1.18 0.08 -0.99 0.10 -0.86 0.08 -1.31 0.08 -1.30 0.11 -1.26 0.10 -1.28 0.08 -1.62 0.10 -1.39 0.08 -0.89 0.08 -0.95 0.09 -1.22 0.07 -1.18 0.09 -0.93 0.08 -0.96 0.10 -0.74 0.09 -0.89 0.08 -0.80 0.10 -0.91 0.08 -0.79 0.09 -0.83 0.07 -0.88 0.07 -0.92 0.07 -1.04 0.05 -1.06 0.08 -0.82 0.10 -0.80 0.09 -0.82 0.10 -0.90 0.09 -0.78 0.09 -0.85 0.08 -0.85 0.09 -0.99 0.10 -0.79 0.11 -0.79 0.08 -0.76 0.07 -0.74 0.10 -1.15 0.11 -0.50 – – 0.14 -0.49 0.14 -0.40 0.10 -1.65 0.08 -1.44 0.08 -1.26 σ?l,GC 0.12 0.10 0.09 0.11 0.14 0.09 0.11 0.09 0.09 0.12 0.11 0.09 0.11 0.10 0.09 0.10 0.08 0.10 0.09 0.11 0.11 0.10 0.11 0.09 0.10 0.08 0.08 0.07 0.06 0.10 0.11 0.10 0.12 0.11 0.10 0.10 0.10 0.12 0.12 0.10 0.08 0.11 0.13 – 0.15 0.16 0.12 0.10 0.09 ?b,GC 0.23 0.22 0.22 0.18 0.14 0.15 0.16 0.41 0.28 0.28 0.23 0.20 0.20 0.12 0.04 0.15 0.13 0.17 0.16 0.06 0.13 0.20 0.16 0.18 0.12 0.19 0.15 0.22 0.24 0.15 0.06 0.15 0.20 0.17 0.17 0.21 0.20 0.08 0.08 0.16 0.15 0.26 0.11 – 0.22 0.24 0.31 0.35 0.35 σ?b,GC 0.04 0.02 0.02 0.04 0.05 0.03 0.03 0.04 0.03 0.03 0.03 0.03 0.03 0.01 0.03 0.04 0.04 0.02 0.02 0.04 0.03 0.02 0.02 0.03 0.02 0.02 0.02 0.04 0.02 0.03 0.03 0.03 0.02 0.03 0.03 0.02 0.03 0.03 0.02 0.02 0.03 0.04 0.03 – 0.03 0.04 0.03 0.03 0.04

o?set. A good correspondence between Tycho-2 and OGLEII proper motions gives us certain con?dence in our measurements. Slightly larger o?sets in ?δ imply that the error in absolute proper motion is at a level of 1 mas yr?1 , which can be improved by using QSOs in the near future. We also compared the measurements in BUL SC1 made by Sumi, Eyer & Wo?niak 2003 (hereafter SC1′ ) with the z proper motions presented in this paper, for which our measurements are above the 10σ level of accuracy. The two sets of proper motions for 1368 cross-identi?ed stars are shown

in Fig. 13 together with the best ?t line. The o?set between the two sets, and the di?erences between individual measurements are within estimated errors. Note: the scale is di?erent than in Fig. 12. Large reduced chi square in ?α are because Sumi, Eyer & Wo?niak 2003 didn’t correct systemz atic distortions (see §3), though their systematic distortions have been reduced at a level of 2 mas yr?1 by dividing images into small chunks. We compared the measurements done by us in one of the overlap regions: between ?elds BUL SC1 and BUL SC45.

10

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Figure 12. Comparison of proper motions ?α ? (thin) and ?δ (thick) for 65 stars cross-identi?ed in our catalogue and in Tycho2 catalogue. The proper motions in our catalogue are transformed to the inertial frame by equations (3) and (4). Dashed lines indicate ?OGLE = ?T ycho?2 , and solid lines represent best ?t with the o?sets: ?α?OGLE ? ?α?Tycho?2 = ?0.23 mas yr?1 and ?δOGLE ? ?δTycho?2 = 0.99 mas yr?1 . ?α ? are shifted for +30 mas yr?1 for clarity.

Figure 13. Same ?gure as Fig. 12 for 1368 stars cross-identi?ed in BUL SC1 by Sumi, Eyer & Wo?niak 2003 (hereafter SC1′ ) and in z SC1 by this work, measured with better than 10σ level accuracy. Dashed lines indicate ?SC1 = ?SC1′ which overlap with the solid line in ?δ , and solid lines represent the best ?t with the o?sets: ?α?SC1 ? ?α?SC1′ = 0.11 mas yr?1 and ?δSC1 ? ?δSC1′ = ?0.03 mas yr?1 . ?α are shifted for +30 mas yr?1 for clarity. Note: the scale is di?erent from Fig. 12.

The proper motions of 115 cross-identi?ed stars, and the best ?t with a small o?set between zero points, are presented in Fig. 14. Here proper motions have been transformed into the inertial frame by equations (3) and (4). Good correlations between them with a rather small zero point o?sets: ?α?SC1 ? ?α?SC45 = 0.16 mas yr?1 and ?δSC1 ? ?δSC45 = ?0.07 mas yr?1 gives us certain con?dence in our measurements and in equations (3) and (4) in terms of the relative the zero point. The scatter is also consistent with the estimated errors.

7

POSSIBLE PROBLEMS

There can be various problems associated with proper motions of variable stars. Our ?elds are very crowded, hence many stars may be blended. In the case of blending we measure an average position of a blend of several stars within a seeing disk. If a variable star is blended with other stars which have slightly di?erent positions within a seeing disk, the average position of the blended image may change while one component of the blend varies. This change might mimic a proper motion. As an example we present the time variation of the position of a very long timescale microlensing event candidate (Smith 2003) in the top panel of Fig. 15. The I-band light curve of this event is shown in the bottom panel of Fig. 15 (ID=2859 in variable star catalogue of Wo?niak et al. 2002). z In the top panel we can see an apparent proper motion in the ?rst year which is coincident with the apparent brightness fading, as shown in the bottom panel. After the second year

the proper motion seems to be small, and the position data points are sparse, which coincides with the low brightness of the star and may re?ect the presence of a blend – in poor seeing DoPHOT cannot resolve the two stars, hence there are very few data points in the upper panel. The blend was actually found in the higher resolution OGLE-III images in 2002. A plausible interpretation is that the microlensed star was brighter than the nearby faint ‘companion’ star in 1997, but by 2000 it became the fainter of the two. The position of the ‘companion’ is consistent with the direction of the apparent proper motion. The faint stars seem to have low proper motion. High resolution observations are needed to fully understand this object. As another example we present in Fig. 16 the time variation of the position of a star ID=309705 in the OGLE-II ?eld BUL SC39. Filled circles represent measured positions with type=1 (used in this work), which were at a ?xed location for the ?rst three years but moved signi?cantly in the fourth season. The star ID=309705 identi?ed these centroids at the edge of the search radius (y=-1) for the ?rst 3 seasons and at the center (y=0) in the 4th season. On the other hand, the neighboring star ID=309653 which has a similar brightness of I = 14.8 and position of 0.18 pixels East (positive x) and -1.89 pixels South (negative y), identi?ed the same centroids at the edge of the search radius (y=-1). To see the details of this object, in Fig. 16 we also plot the position measurements which are categorized as a star blended with other stars (type=3, which are not used in

The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

11

Figure 14. Same ?gure as Fig. 12 for 115 stars cross-identi?ed in the overlap region between OGLE-II ?elds BUL SC1 and BUL SC45, and measured with better than 5σ level accuracy. Proper motions are transformed into the value in the inertial frame by equations (3) and (4). Dashed lines indicate ?SC1 = ?SC45 , and solid lines represent the best ?t with the o?sets: ?α?SC1 ??α?SC45 = 0.16 mas yr?1 and ?δSC1 ??δSC45 = ?0.07 mas yr?1 . ?α in SC45 are shifted for +30 mas yr?1 for clarity.

this work) by DoPHOT for this star (crosses) and for a neighboring star ID=309653 (dots), which are shifted by +0.18 pixels in x and -1.89 pixels in y, i.e. these dots are as they are on the CCD, relative to ID=309705. We can see that these positions (crosses and dots) are identi?ed around y=0 (ID=309705) and -2 (ID=309653), respectively, with ?lled circles in-between them. We also show the I-band light curves of ID=309705 (middle panel) and ID=309653 (bottom panel) in Fig. 16. We can see ID=309705 is constant during four seasons but ID=309653 suddenly faded during the 4th season. This is likely to be a R CrB type variable. The simplest interpretation is as follows; small number of data points in the seasons 1997-99 indicates that DoPHOT found that object which is composite of these two stars only on the bad seeing frames as a single star with type=1, in other cases they are separated but categorized as blended - type=3. In 2000 when the star ID=309653 faded, the centroid of the composite moved and ?nally the star ID=309705 became a ”single” star with type=1. So the number of data points is large in 2000. Proper motions of variable stars may have their errors increased not only because of variable contribution of blending, but also because variable stars may change colours, and therefore the coe?cient of di?erential refraction may also change. In rare cases of very long period variables this may be noticeable. Note: we do not treat this kind of objects in a special way, so a reader must be careful when using our catalogue in studies of variable stars. The e?ect of blending changes with seeing may contribute to the scatter of data points, but it is not likely to

Figure 15. Top panel: Same plots as Fig. 3 for the star ID=244353 in the OGLE-II ?eld BUL SC5. Filled circles represent actual positions and open circles are correspond to positions corrected for di?erential refraction, with the o?set of ?1 pixels. Bottom panel: I-band light curve of same star (ID=2859 in catalogue of Wo?niak et al. 2001), a possible very long microlensing z event.

have a seasonal e?ect. Hence, we think that seeing variations are not a major problem. The probability of blending is much larger for fainter stars, in particular those close to I = 18 mag. This may produce a bias in their proper motions, most likely reducing their formal proper motion, as the blended stars may have a di?erent proper motion vector, so the average value is likely to be reduced. We also do not provide a special treatment for this kind of objects, so a reader must be careful when using our catalogue for faint stars.

8

DISCUSSION AND CONCLUSION

We have measured proper motions for 5,080,236 stars in all 49 OGLE-II GB ?elds, covering a range of ?11? < l < 11? and ?6? < b < 3? . Our catalogue contains objects with proper motions up to ? = 500 mas yr?1 and I-band magnitudes in the range 11 ≤ I ≤ 18. The accuracy of proper motions in our catalogue is better than 1 mas yr?1 for 12 < I < 14.

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One should keep in mind that all measurements of ? presented here are not absolute, but relative to the astrometric reference frame which is roughly that of the Galactic Center (GC) with a small o?set seen in the lower panel of Fig. 5. However, as demonstrated in §6, by using the crude estimation for the proper motion of the GC in our reference frame and formula equations (3) and (4), we can obtain crude proper motions in an inertial frame. From the comparison with these inertial values and the Tycho2 catalogue, this transformation seems to work well with errors at a level of 1 mas yr?1 . From comparison of ? measured in the overlap region of ?elds BUL SC1 and BUL SC45 (Fig. 14), this transformation works very well in the relative o?set from ?eld to ?eld. These zero points for proper motions can be improved by using background quasars which may be detected in the near future using the OGLE-II variability catalogue (Wo?niak et al. 2002; Eyer 2002; Dobrzycki et al. z 2003). As demonstrated by Sumi, Eyer & Wo?niak (2003), the z proper motions based on OGLE-II data can be used to clearly detect the presence of a strong streaming motion (rotation) of stars in the Galactic bar. While the reference frame established from all stars is not well de?ned with respect to the inertial frame, the relative motions of groups of stars within a given ?eld are well determined. Though our primary goal is to constrain the Galactic bar model with the future analysis of our catalogue, we provide proper motions for all stars with I < 18 mag in all 49 OGLE-II GB ?elds because this catalogue might be useful for a variety of projects. An analysis of the catalogue is beyond the scope of the present study.

ACKNOWLEDGMENTS We are grateful to B. Paczy?ski for helpful comments and n discussions. We acknowledge M. Smith for carefully reading the manuscript and helpful comments. We are also thankful to the referee, F. van Leeuwen for suggestive comments. T.S. acknowledge the ?nancial support from the Nishina Memorial Foundation and JSPS. The paper was partly supported by the Polish KBN grant 2P03D02124 to A. Udalski. This work was partly supported with the following grants to B. Paczy?ski: NSF grants AST-9820314 and AST-0204908, and n NASA grants NAG5-12212, and grant HST-AR-09518.01A provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.

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The Catalog of stellar proper motions in the OGLE-II Galactic bulge ?elds

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Figure 16. Top: Same plots as Fig. 3 for star ID=309705 in OGLE-II ?eld BUL SC39. Filled circles represent actual positions with type=1 (used in this work), crosses correspond to position measurements with type=3 (not used) and dots indicate positions measurements of neighboring star ID=309653 with type=3 which are shifted by +0.18 pixels in x and -1.89 pixels in y, i.e. these dots are as they are on CCD, relative to ID=309705. This is a probable case of a blend with one component being a variable star.


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