AN ASYMMETRIC STREAMING MOTION IN THE GALACTIC BULGE X-SHAPED STRUCTURE REVEALED BY OGLE-III PROPER MOTIONS

The Milky Way bulge is the closest such structure and the only one in which stars can be well resolved and studied individually. This fact in turn allows a detailed investigation of the properties of the stellar populations within the bulge. The Galactic bulge was not formed via major mergers but was developed via bar buckling (Shen et al. 2010). It was recently found (Nataf et al. 2010; McWilliam & Zoccali 2010) that the Galactic bulge contains an X-shaped structure. This structure is aligned with a bar, which makes a ≈30° angle with Sun—Galactic center direction, and its arms are significantly extended along the line of sight direction. The color–magnitude diagrams (CMDs) constructed in some bulge areas show a double-peaked red clump (RC) because the line of sight crosses two arms of the X-shaped structure. In fields close to the Galactic plane, the two RCs merge and appear single-peaked. These include the well-studied, low-extinction region called Baade's Window at Galactic coordinates (l, b) = (1°, −39). The RC is also single-peaked in the areas where only one arm is seen. X-shaped structures are known in other galaxies (Whitmore & Bell 1988) and may be centered (the four arms cross at the center of the galaxy) or off-centered (the arms cross in two areas within the galactic plane that do not coincide with the galactic center; Bureau et al. 2006).

The double RC in the Galactic bulge was confirmed using the Two Micron All Sky Survey and VISTA Variables in the Via Lactea (VVV) near-infrared data by Saito et al. (2011, 2012). More recently, Cao et al. (2013) analyzed the spatial density of the RC stars and found a clear signature of the X-shaped structure. Li & Shen (2012) found properties of the X-shaped structure in the bulge model of Shen et al. (2010) to be consistent with the observations.

Until now, there have been relatively few studies devoted to the kinematics of both RCs. In some cases, studies were conducted in fields where the X-shaped structure was present but was hard to distinguish, i.e., −5° ≲ b ≲ −3°. Two such examples were reported by McWilliam & Zoccali (2010). The catalog of proper motions by Sumi et al. (2004) based on the second phase of the Optical Gravitational Lensing Experiment (OGLE-II) data was used to calculate a proper motion difference of 1.0 ± 0.06 milliarcseconds per year (mas yr⁻¹) between the two arms of the X-shaped structure in Baade's Window (henceforth, all the differences between the two arms will be given as bright minus faint). The catalog of proper motions in the Plaut field ((l, b) = (0°, −8°)), compiled by Vieira et al. (2007), resulted in a proper motion difference in longitude of 0.19 ± 0.19 mas yr⁻¹ and 0.51 ± 0.18 mas yr⁻¹ in latitude. McWilliam & Zoccali (2010) also claimed a 3.4σ difference in the latitudinal proper motion dispersion without providing the figures themselves. Babusiaux et al. (2010) searched for proper motion differences in Baade's Window using the Sumi et al. (2004) catalog but did not find any statistically significant differences. Recently, Vásquez et al. (2013) reported 3.71 ± 0.33 mas yr⁻¹ and −0.59 ± 0.28 mas yr⁻¹ differences in longitudinal and latitudinal proper motions. In order to select pure bright and faint RC stars, they used arbitrary limits on heliocentric radial velocity (RV) and the claimed differences depend on these limits. We present all the measurements of RV differences in Table 1. In contrast to previous investigations, we study the changes in relative kinematics in steps as small as 01.

There is one problem inseparably involved in the kinematic studies of the Galactic bulge that use RC stars. Not all of the stars in the RC region of the CMD belong to the RC because there are underlying red giants that may be at different distances and thus possess different mean kinematics. As noted by Mao & Paczyński (2002), these stars dilute the measured kinematic differences between the brighter and fainter RC stars. There are two ways one can cope with this fact when comparing the observations with predictions based on Galactic models. The first is to report the raw measured properties and compare them with model predictions, which are diluted in the same manner as the observations (see e.g.,  Ness et al. 2012). The disadvantage of this method is that the above mentioned results cannot be compared in detail with the models as long as one does not specify from which parts of the CMD the stars came from that were used in the calculations. The other approach is presented in this paper. The probability that a given star belongs to the brighter or fainter arm of the X-shaped structure is assigned to each star (based on its extinction-corrected brightness, which for RC stars is a proxy for distance). Using this probability, we find the intrinsic properties of the two arms. The result can be directly compared with the model predictions, which are based on the proper motions of stars selected according to their distances. This technique also allows analytical calculations of some Galactic properties.

We note one more proper motion study in the bulge field that does not show a double RC structure. Based on Hubble Space Telescope data, Clarkson et al. (2008) found proper motions in a field (l, b) = (125, −265). These authors assigned photometric distances to every star and calculated mean proper motions in distance bins. They concluded that the mean circular speed of the bulge stars is well approximated by solid-body rotation to a cutoff radius of 0.3–0.4 kpc with a maximum velocity of 25 km s⁻¹. This velocity corresponds to an angular rotational velocity of 62–83 km s⁻¹ kpc⁻¹. Clarkson et al. (2008) also estimated that biases in the stellar properties decrease the measured rotational velocity by a factor of two. Thus, the angular rotational velocity from that study is between 125 and 167 km s⁻¹ kpc⁻¹.

The aim of this paper is to study the proper motions of the stars belonging to the two southern arms of the X-shaped structure. We use data collected by the OGLE-III survey to calculate the proper motions. The third phase of the OGLE project used a camera with better resolution than the second phase. In addition, the observed sky area was significantly larger and covered some fields with the double RC. The results presented here could not be obtained using the OGLE-II data. We describe the observations used in this study in Section 2. The two following sections present the luminosity function construction and the calculation of proper motions of individual stars. Both of these quantities are used in Section 5 to find the intrinsic proper motion differences of stars in the two arms of the X-shaped structure. We end by discussing the implications of our results.

The OGLE-III project (Udalski 2003) was conducted with the 1.3 m Warsaw Telescope located at the Las Campanas Observatory, Chile. The telescope was equipped with an eight CCD chip mosaic camera. The total camera dimension was 8k × 8k pixels, with a field of view of 35' × 35'. The 15 μm pixels gave a 026 pixel scale, allowing one to benefit from the excellent seeing conditions. The observations were performed with V and I filters with the majority of the observations taken in the I band. The V-band observations are used here only for color information. The exposure time was 120 s.

The main goals of the OGLE-III project were studies of microlensing events that are effectively found only in the Galactic bulge. The observed bulge area was 92 deg² within the range −12° < l < 13° and −7° < b < 6°. During the survey observations, the sky area monitored decreased with time in order to increase the cadence in the fields showing the highest number of microlensing events. The number of epochs per field ranged from a dozen or so to ≈2400, and the time span of observations ranged from about one year to eight years. The photometric maps of the OGLE-III bulge fields were presented by Szymański et al. (2011). We select the fields where the double RC is strongly pronounced and the observing coverage was good enough to obtain proper motion precisions below 1 mas yr⁻¹ in each direction for a single star. We were left with three fields called BLG134, BLG167, and BLG176, where the latter two are neighboring. The sky coordinates of the fields centers, the number of epochs collected, and observing time span are summarized in Table 2. The sky area corresponding to one CCD chip is called a subfield. Different subfields of a given field are distinguished by the numbers 1 to 8, separated by a decimal point.

Table 2. Characteristics of the Observed Fields

Field Name	R.A.	Decl.	l	b	Nepoch	Δt
	J2000.0	J2000.0	(°)	(°)		(yr)
BLG134	17h576	−34°12'	−3.24	−4.88	326	4.6
BLG167	18h035	−31°50'	−0.56	−4.80	360	4.6
BLG176	18h064	−31°15'	0.23	−5.00	355	4.5

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The astrometric measurements analyzed here are independent of the standard OGLE-III reduction, which used Difference Image Analysis. In order to measure the centroids of the stars, we implemented the effective point spread function⁵ (PSF) approach presented by Anderson & King (2000) and Anderson et al. (2006). In our implementation, the grid on which the single effective PSF is defined supersamples the pixels by a factor of four in each direction. The spatial variation of the profile is treated by calculating eight such PSFs (2 × 4 grid) for each CCD chip individually. The effective PSF for a given place of the CCD chip is evaluated by a linear interpolation of the four (out of eight) nearest grid profiles.

In order to assign each given star a probability that it belongs to the brighter or fainter arm, we deredden each star individually using the interpolated extinction maps of Nataf et al. (2013b). The E(V − I) reddening varies between 0.53 mag and 0.81 mag for BLG134 and between 0.62 mag and 1.29 mag for the other two fields. Figure 1 presents a sample CMD after the extinction correction was applied (I₀ and (V − I)₀ are the extinction corrected values of the I band brightness and the (V − I) color). We use a two-dimensional extinction map, so the foreground disk stars are artificially shifted toward bluer colors and brighter magnitudes.

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The luminosity function of the bulge red giants is constructed by taking into account the stars that are redder than the gray line in Figure 1. This line changes its slope at I₀ = 14 mag to exclude the disk RC stars, which are abundant in the CMD region (V − I)₀ ≈ 0.8 mag and I₀ < 13.5 mag. The nearby red giants are shifted toward bluer colors by the extinction correction and thus are not included in our selection box. We argue that this selection of stars removes a large majority of the foreground disk red giants. We verified this fact using the Besançon Galactic model (Robin et al. 2003). A separate luminosity function was constructed for each field; all are shown in the three panels of Figure 2. We found that calculating a separate luminosity function for each OGLE field is a compromise between obtaining good statistics and characterizing the spatial changes in the luminosity function.

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The luminosity function for each field is fit with an analytical function that takes into account both arms of the X-shaped structure. In previous studies (e.g.,  Nataf et al. 2011), simpler functions were used. The number of stars in each arm is a sum of three components: the red giant branch (represented by an exponential function), the RC (represented by a Gaussian), and the red giant branch bump (RGBB Gallart 1998; Nataf et al. 2011, represented by a Gaussian). We take into account the RGBB and ignore the asymptotic giant branch bump that was also found in the Galactic bulge (Nataf et al. 2011) because of two major differences between the two structures. First, the number counts of the asymptotic giant branch bump are at least an order of magnitude smaller. Second, the absolute brightness difference between the RGBB and the RC is significantly smaller than between the asymptotic giant branch bump and the RC (0.7 mag compared with 1.1 mag). The function that we fit has sixteen parameters (two for each of the exponentials and three for each Gaussian), but we make several assumptions to reduce the number of fitted parameters. The slopes of the exponential components were kept the same. The number of red giants at the brightness of the RC was also kept the same. The dispersions of the Gaussians representing the RGBB were the same as the dispersions of the Gaussian representing the RC in each of the arms. The brightness difference between the RGBB and the RC, as well as the number ratio of RGBB to RC stars, were fixed at values found by Nataf et al. (2013a) of 0.737 mag and 0.201, respectively. These parameters depend on the metallicity and thus their true values in the analyzed fields may be different than assumed. We made two exceptions to the fitting procedure described above, which significantly improved our fits. First, in the BLG134 field, the contribution of the fainter RGBB was neglected. Second, in the BLG176 field, the number of the RGBB stars relative to RC stars in the fainter arm was a free parameter with a best fit value of 0.075 ± 0.063, compared with 0.201 in the standard fit. The fits had eight or nine free parameters and between 81 and 101 data points in the luminosity function. We adjust the interval of I₀ over which the fit is performed for each field separately (the actual intervals can be found in from Figure 2). The difference in the extinction-corrected brightness of the two RCs are −0.353 ± 0.034 mag, −0.365 ± 0.051 mag, and −0.387 ± 0.046 mag for BLG134, BLG167, and BLG176, respectively.

Each star was assigned a probability that it belongs to the brighter (p_B(I₀)) or the fainter (p_F(I₀)) arm of the X-shaped structure. This probability is equal to the ratio of the number of stars from a given arm to the total number of stars, both of which are values of the fitted function at the extinction-corrected brightness of the star. The plots of p_B(I₀) and p_F(I₀) are shown in Figure 2.

The crucial point in the calculation of high accuracy proper motions is finding an adequate transformation of grids. We note that the OGLE-III observing strategy was not optimized for precise astrometry: there was typically only one observation of a given field taken per night, the seeing FWHM varied up to a factor of 2.5, exposures were taken at different airmasses, and the airmass could change significantly between any two consecutive exposures. These issues prevent us from using a detailed geometric correction common to all of the subfields of a given exposure, as was done by, e.g., Bellini & Bedin (2010). Instead, we derived the transformations for each subfield individually. This method forced the proper motion zero points to be different for each subfield.

The cross-matching of the star catalogs from each image was done using a simplified version of the algorithm presented by Pál & Bakos (2006). The cross-matched catalogs were used to calculate second order polynomial grid transformations in order to transform the measured positions to a common frame. The stellar centroids from all the frames of a given subfield were transformed to this common frame. After that, we fit a model to each star's centroid that took into account the coordinates for a selected epoch, the proper motion, and the differential refraction coefficient (which describes how atmospheric refraction affects the zenith distances of stars with different colors). The value of differential refraction shift is calculated by multiplying the coefficient by the airmass decreased by unity. The zero point of the proper motions is set by calculating the mean of proper motions for red giants brighter than I₀ = 13.8 mag and with proper motions smaller than 12 mas yr⁻¹ (reference stars). The mean proper motion of the reference stars was subtracted from the proper motion of every star. These proper motions and differential refraction coefficients were used to calculate the residuals of the stellar positions. Those residuals are transformed back from the common frame to the individual frames and the measured positions are subsequently corrected for these residuals, which effectively defines a reference frame in which the bulge red giants have zero mean motion. Then, new grid transformations are found and we use them to calculate the mean position of every reference star. If these positions differ more than 0.15 pixels from the initial common frame, we remove the star from the list of reference stars. We found that this limit separates well the unreliable star detections, e.g., blended objects. The mean positions are used as a new common frame, which in principle is corrected for differential refraction. The grid transformations from the individual frames to that frame are found and final fits of positions for a selected epoch, proper motions, and differential refraction coefficients are performed.

The uncertainties on the proper motions are found using a bootstrap method (Press et al. 1992). From a set of exposures of a given field, we draw with replacement a subset whose number of elements is equal to the original set. Multiple such subsets are drawn and for each of them the procedure of calculating the proper motions described above is repeated. The standard deviation of the proper motion of a given star is taken as the measurement uncertainty. The proper motion uncertainties are about 0.3 mas yr⁻¹ for I = 14 mag and about 0.5 mas yr⁻¹ for I = 16 mag. It was found that the proper motion uncertainties increase near the edges of subfields. This fact is caused by the less well defined grid transformations in these parts of the subfields. We compared the proper motions measured for stars present in overlapping parts of the adjacent subfields and found that the bootstrap estimates are consistent with the measured proper motion differences. The raw measurements of proper motions used in this study will be published as a part of a proper motion catalog covering the entire OGLE-III bulge area.

The intrinsic (i.e., undiluted by red giants) proper motion difference between the brighter and fainter RCs can be found using the calculated proper motions and the probabilities that a given star belongs to either the brighter or the fainter arm. Let the index i label the stars. For each star, we have measured the proper motion in Galactic coordinates: μ_{i, l⋆}, μ_{i, b}⁶ and the corresponding uncertainties ξ_{i, l⋆}, ξ_{i, b}. We also know the extinction-corrected brightness I_{i, 0}, which gives the probabilities p_B(I_{i, 0}) and p_F(I_{i, 0}). For the fainter arm, the average (dispersion) of the proper motions in Galactic longitude is denoted μ_{F, l⋆} (σ_{F, l⋆}), while for Galactic latitude, the corresponding value is μ_{F, b} (σ_{F, b}). For the brighter arm, the corresponding symbols have first changed index from F to B. The likelihood function for a single star ($\mathcal {L}_i$) is defined as

$$\begin{eqnarray} \mathcal {L}_i &=& \frac{p_{\rm B}(I_{i,0})}{2\pi \sqrt{\left(\sigma _{{\rm B},l\star }^2+\xi _{i,l\star }^2\right)\left(\sigma _{{\rm B},b}^2+\xi _{i,b}^2\right)}} \nonumber\\ && \times \exp \left(-\frac{(\mu _{i,l\star }-\mu _{{\rm B}, l\star })^2}{2 \left(\sigma _{{\rm B},l\star }^2+\xi _{i,l\star }^2 \right)}-\frac{(\mu _{i,b}-\mu _{{\rm B}, b})^2}{2 \left(\sigma _{{\rm B},b}^2+\xi _{i,b}^2 \right)}\right) \nonumber\\ && + \frac{p_{\rm F}(I_{i,0})}{2\pi \sqrt{\left(\sigma _{{\rm F},l\star }^2+\xi _{i,l\star }^2\right)\left(\sigma _{{\rm F},b}^2+\xi _{i,b}^2\right)}} \nonumber\\ && \times \exp \left(-\frac{(\mu _{i,l\star }-\mu _{{\rm F}, l\star })^2}{2 \left(\sigma _{{\rm F},l\star }^2+\xi _{i,l\star }^2 \right)}-\frac{(\mu _{i,b}-\mu _{{\rm F}, b})^2}{2 \left(\sigma _{{\rm F},b}^2+\xi _{i,b}^2 \right)}\right). \end{eqnarray} \tag{ 1 }$$

The product of likelihoods for all the stars in a given subfield is the function we maximize using the Markov Chain Monte Carlo (MCMC) technique. A separate chain was run for each subfield. After trial and error, we choose the same interval of I₀ brightness between 14 mag and 15 mag. In this range, both RCs are prominent, and that is where the most information on the undiluted proper motions comes from. For stars brighter than I₀ = 14 mag, the probabilities are poorly constrained and the number of brighter stars is smaller. For stars fainter than I₀ = 15 mag, the RGBBs significantly contribute to the luminosity function and the analytical fits presented in Section 3 are slightly poorer. For I₀ ≈ 15.7 mag, the main sequence disk stars start to contribute significantly to the luminosity function. The mean value of the effective number of stars within the range 14 mag < I₀ < 15 mag in the brighter (fainter) arm per subfield, i.e., ∑_ip_B(I_{i, 0}) (∑_ip_F(I_{i, 0})), is 1093 (1499) in the BLG134 field. In the BLG167 field, the corresponding value is 1723 (669); in the BLG176 field, it is 1416 (771). The values in individual subfields do not differ by more than 10% except in subfields BLG167.5 (15% larger than the mean) and BLG176.5 (13% larger than the mean). The average relative contributions of red giants, RC stars, and RGBB stars from the brighter arm as well as red giants, RC stars, and RGBB stars from the fainter arm over the 14 mag < I₀ < 15 mag range are, respectively, 0.238/0.169/0.015/0.297/0.281/0.000 in the BLG134 field. In the BLG167 and BLG176 fields, these numbers are 0.380/0.310/0.031/0.131/0.149/0.000 and 0.344/0.269/0.035/0.162/0.190/0.000, respectively.

In order to illustrate the reliability of our MCMCs, we overlay the p_B(I₀) and μ_{i, l⋆} averaged in 0.2 mag wide I₀ bins. The panels in Figure 3 present such plots for the four sample subfields. The agreement between the measured proper motions and the p_B for I₀ between 14 mag and 15 mag proves the consistency of our extinction correction, luminosity function construction, fitted analytical model, and proper motion calculation procedures.

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The results of the MCMC runs are presented in Table 3. We do not state values of μ_{B, l⋆}, μ_{B, b}, μ_{F, l⋆}, and μ_{F, b}, as their zero points may change in different subfields. Instead, we present their differences, i.e.,  Δμ_l⋆ = μ_{B, l⋆} − μ_{F, l⋆} and Δμ_b = μ_{B, b} − μ_{F, b}, which carry all the astrophysical information.

We note that the measured differences of μ_l⋆ at the brightness of the two RCs range up to 1.2 mas yr⁻¹ in the subfields presented in Figure 3. The values of Δμ_l⋆ for these subfields, which are corrected for dilution by red giants, are about two times larger. Such a large ratio is caused by the contribution of red giant branch stars and the fact that even for the brightness of the brighter RC, some contribution of the fainter arm is seen and vice versa. Among the previous comparisons of the kinematics of the two arms of the bluge, only Uttenthaler et al. (2012) accounted for the probabilities that measured stars belong to either RC. In contrast with our results, they found a very small change in the kinematic properties from 4.4 ± 9.5 km s⁻¹ to 5.2 ± 9.5  km s⁻¹.

All analyzed fields are close to b = −5°. The range of Galactic longitudes is significantly larger. The BLG167 and BLG176 fields span from l = −09 to l = 06. The BLG134 field is located around 2° from these two fields and is therefore discussed separately.

Figure 4 presents the proper motion differences Δμ_l⋆ and Δμ_b versus Galactic longitude. We are analyzing the fields close to l = 0°, so the purely cylindrical rotation identified from RV surveys (Kunder et al. 2012; Ness et al. 2013) should result in constant values of Δμ_l⋆ and Δμ_b close to zero. However, this expectation actually only holds for negative l. For positive l, both Δμ_l⋆ and Δμ_b are linear functions of l. We fit a five parameter phenomenological model to both Δμ_l⋆(l) and Δμ_b(l). The BLG176.2 subfield is not included in this fit since Δμ_l⋆ and Δμ_b derived in this subfield are the most outlying points in both panels of Figure 4. For l < l_break, we assume

$$\begin{equation*} \Delta \mu _{l\star }(l) = \Delta \mu _{l\star }^{{\rm const}} \end{equation*}$$

$$\begin{equation*} \Delta \mu _{b}(l) = \Delta \mu _{b}^{{\rm const}} \end{equation*}$$

and for l ⩾ l_break, we assume

$$\begin{equation*} \Delta \mu _{l\star }(l) = \Delta \mu _{l\star }^{{\rm const}} + \frac{d\,\Delta \mu _{l\star }}{d\,l}\cdot \left(l-l_{\rm break}\right) \end{equation*}$$

$$\begin{equation*} \Delta \mu _{b}(l) = \Delta \mu _{b}^{{\rm const}} + \frac{d\,\Delta \mu _{b}}{d\,l}\cdot \left(l-l_{\rm break}\right). \end{equation*}$$

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The fit results in an l_break value of −0114 ± 0060, i.e., 2σ away from 0°. The parameters for the longitudinal proper motion are $\Delta \mu _{l\star }^{{\rm const}} = 2.04\pm 0.07 \,{\rm mas \,yr^{-1}}$ and (d Δμ_l⋆/d l) = 1.23 ± 0.32 mas yr⁻¹ deg⁻¹. For the latitudinal proper motion, we obtain $\Delta \mu _{b}^{{\rm const}} = -0.14\pm 0.08 \,{\rm mas \,yr^{-1}}$ and (d Δμ_b/d l) = −3.27 ± 0.43 mas yr⁻¹ deg⁻¹. The fit results in χ²/dof = 42.1/25.

The most striking result of Figure 4 is the presence of significant changes in the proper motion differences with Galactic longitude for l > −01. We interpret this result as a signature of the asymmetric streaming motions of stars in the X-shaped structure. The fact that the mean motion of the stars belonging to a certain structure is different from the mean motion of this structure is known, e.g.,  in the case of Galactic spiral arms. The measurements presented in this paper are insufficient to determine in which of the arms the streaming motion occurs. Vásquez et al. (2013) very recently presented evidence of streaming motions within the X-shaped structure using RV measurements. None of the published models of the X-shaped structure (Li & Shen 2012; Ness et al. 2012; Robin et al. 2012) give predictions of the streaming motions that we report.

The value of $\Delta \mu ^{{\rm const}}_{l\star }$, obtained from the BLG167 field (see Figure 4), can be used to constrain the angular velocity of the Galactic bar. We denote d_B and d_F as the distances from the observer to the points where the line of sight crosses the brighter and fainter arm, respectively. For small values of distance modulus differences ΔI_RC of the two RCs, the distance difference divided by the mean of the two distances, (d_B − d_F)/((d_B + d_F)/2), can be approximated by ln10ΔI_RC/5. We are limited to l ≈ 0°, where the bar velocity equals the tangential velocity as seen by an observer in an inertial frame. The bar angular velocity Ω multiplied by d_B − d_F equals $d_{\rm B}\mu ^{{\rm inert}}_{{\rm B}, l\star }-d_{\rm F}\mu ^{{\rm inert}}_{{\rm F}, l\star }$ (the superscript inert corresponds to the proper motions measured in an inertial frame). If the relative differences in distances are smaller than the relative uncertainties in the measured proper motions, one can approximate $d_{\rm B}\mu ^{{\rm inert}}_{{\rm B}, l\star }-d_{\rm F}\mu ^{{\rm inert}}_{{\rm F}, l\star } \approx (\mu ^{{\rm inert}}_{{\rm B}, l\star }-\mu ^{{\rm inert}}_{{\rm F}, l\star })(d_{\rm B}+d_{\rm F})/2$. This result leads to the following equation in the rotating frame (μ_{Sgr A*, l⋆} is the proper motion of Sgr A*):

$$\begin{equation*} \Omega = \mu _{{\rm Sgr\,A*}, l\star }+\frac{5\Delta \mu _{l\star }^{\rm const}}{{\rm ln}10\Delta I_{{\rm RC}}}. \end{equation*}$$

Substituting the $\Delta \mu _{l\star }^{\rm const}$, ΔI_RC with BLG167 of −0.365 ± 0.051 mag and μ_{SgrA*, l⋆} = −6.38 mas yr⁻¹, as measured by Reid & Brunthaler (2004), we obtain Ω = −18.5 ± 1.8 mas yr⁻¹ = −87.9 ± 8.2 km s⁻¹ kpc⁻¹.⁷ This result is marginally consistent with the results of Clarkson et al. (2008), without correcting for biases in their analysis. If we assume that the Galactic angular velocity is constant up to some Galactic radius R' and for larger radii the circular speed is the same as the local circular velocity of the Sun (V_c = 238 ± 9 km s⁻¹; Schönrich 2012), then we can estimate R' = V_c/|Ω| = 2.71 ± 0.28 kpc. The bar angle relative to Sun–Galactic center line is ≈30° (e.g.,  Cao et al. 2013). Thus, we can estimate that the bar points that are 2.71 ± 0.28 kpc away from the Galactic center are observed at l = 1287 ± 24 and l = −727 ± 044 (these uncertainties do not take into account the uncertainty of the bar angle).

The proper motion differences in the BLG134 field are similar to those found in BLG167. The two most outlying subfields are BLG134.8 and BLG134.5. The former has the smallest longitude among the subfields analyzed here, while the latter is the closest to the Galactic plane among the BLG134 subfields. Thus, these discrepancies may be caused by a significantly smaller contribution of the brighter arm in these subfields.

The ratio of proper motion dispersions in both arms along the latitudinal direction can be used as an independent estimate of the distance ratio of the two arms if the same tangential velocity distribution is assumed in each arm. We note that this fact may not be the case since in a given sight line, the two arms are at the different distances from the Galactic plane.

There are several studies devoted to proper motion dispersions in the Galactic bulge (Spaenhauer et al. 1992; Kuijken & Rich 2002; Vieira et al. 2007 and references therein). These dispersions were found to be anisotropic with σ_l⋆/σ_b ≈ 1.2. We do not confirm this finding when the two arms are treated separately. The values of σ_{B, l⋆}/σ_{B, b} and σ_{F, l⋆}/σ_{F, b}, deduced from the posterior probability distribution of the MCMC, are presented in last two columns of Table 3. In only five cases are the ratios significantly different from unity: BLG167.8 and BLG176.8 for the brighter arm, as well as BLG167.7, BLG167.8, and BLG176.8 for the fainter arm. In each of these cases, the longitudinal proper motion dispersion is smaller than the latitudinal dispersion.

We thank A. Robin and S. Kozłowski for discussions. The anonymous referee is acknowledged for comments that helped to clarify the text. This work was supported by the Polish Ministry of Science and Higher Education through the program "Iuventus Plus" award No. IP2011 043571 to R.P. A.G. acknowledges supported from NSF grant AST 1103471. The OGLE project has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. 246678 to A.U.