Signatures of the Galactic bar in high-order moments of proper motions measured by Gaia
Pedro Alonso Palicio, Inma Martinez-Valpuesta, Carlos Allende Prieto, Claudio Dalla Vecchia
aa r X i v : . [ a s t r o - ph . GA ] F e b Astronomy & Astrophysicsmanuscript no. PM_Paper c (cid:13)
ESO 2020February 10, 2020
Signatures of the Galactic bar in high-order moments of propermotions measured by Gaia
Pedro A. Palicio , , , Inma Martinez-Valpuesta , , Carlos Allende Prieto , , and Claudio Dalla Vecchia , Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spaine-mail: [email protected] Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, FranceAccepted XXX / Received YYY
ABSTRACT
Our location in the Milky Way provides an exceptional opportunity to gain insight on the galactic evolution processes, and complementthe information inferred from observations of external galaxies. Since the Milky Way is a barred galaxy, the study of motions ofindividual stars in the bulge and disc is useful to understand the role of the bar. The
Gaia mission enables such study by providing themost precise parallaxes and proper motions to date. In this theoretical work, we explore the e ff ects of the bar on the distribution ofhigher-order moments –the skewness and kurtosis– of the proper motions by confronting two simulated galaxies, one with a bar andone nearly axisymmetric, with observations from the latest Gaia data release (
Gaia
DR2). We introduce the code asgaia to accountfor observational errors of
Gaia in the kinematical structures predicted by the numerical models. As a result, we find clear imprints ofthe bar in the skewness distribution of the longitudinal proper motion µ ℓ in Gaia
DR2, as well as other features predicted for the next
Gaia data releases.
Key words.
Galaxy: structure – Galaxy: evolution — Galaxy: kinematics and dynamics —methods: numerical
1. Introduction
Bars are commonly observed in local disc galaxies, with frac-tions ranging from one- to two-thirds (Eskridge et al. 2000;Marinova & Jogee 2007; Sellwood 2014). The study of motionsof individual stars in the Milky Way is key to understanding thesignature of the bar on the kinematics of these galaxies. Thesee ff ects have been reported on the motion of neutral hydrogen re-gions (de Vaucouleurs 1964), on the orbits of globular clusters(Bobylev & Bajkova 2017; Pérez-Villegas et al. 2018), on thevertex deviation (Zhao et al. 1994), the longitudinal asymmetryin the proper motion dispersion (Rattenbury et al. 2007a), andon the Oort constants (Comeron et al. 1994; Torra et al. 2000;Olling & Dehnen 2003; Minchev et al. 2007; Bovy 2017). Fur-thermore, the presence of the bar has been proposed as an expla-nation for the high-velocity peaks discovered by Nidever et al.(2012) in the bulge (Molloy et al. 2015), and the kinematicsof the Hercules stream (Dehnen 2000; Gardner & Flynn 2010;Pérez-Villegas et al. 2017; Hunt et al. 2018) and other movinggroups (Kalnajs 1991; Minchev et al. 2010).In Palicio et al. (2018), we explored the higher order mo-ments of the line-of-sight velocity distribution ( V los ) to concludethat it is possible to infer the presence of the bar from the skew-ness (Zasowski et al. 2016) and the kurtosis distributions. In thiswork we extend our previous study to analyse the high-order mo-ments of the proper motion, and make predictions for the resultsof the Gaia mission (de Bruijne 2012; Gaia Collaboration 2016),modelling observational errors and constraints. This paper is or-ganised as follows. In Section 2 we introduce the numerical sim-ulations to be used and the modifications adopted to improve themodelling of the Milky Way. In Section 3 we describe the code asgaia , which synthesises the stellar populations of the simula-tion particles including observational errors for
Gaia . The resultsare discussed and summarised in Sections 4 and 5, respectively.
2. Simulation data description
We make use of the two simulated galaxies introduced inPalicio et al. (2018) to explore the imprints of the bar on the dis-tribution of proper motions. The initial conditions of these sim-ulations consist on an exponential disc embedded in a ∼ M ⊙ dark matter halo, with a Toomre Q parameter of 1.5. The simu-lations only di ff er in the distribution of baryonic matter: for onesimulation the inner 7 kpc of the disc contains 30% of the totalmass, while for the other this fraction is larger (50%). After 2.52Gyr, the latter simulation develops a 4.5 kpc half-length bar witha pattern speed Ω p ≈
30 km s − kpc − , while the former remainsnearly axisymmetric with weak spiral arms, until the final time( ∼ φ = ◦ with respect to the Sun-Galacticcentre direction (Stanek et al. 1997; Freudenreich 1998;López-Corredoira et al. 2005; Rattenbury et al. 2007b;Shen et al. 2010; Wegg & Gerhard 2013; Cao et al. 2013;Nataf et al. 2015). This rotation is also applied to the unbarredgalaxy to account for the orientation of the spiral arms. Thedensity maps of the two simulations are shown in Figure 1. Wedefine the solar position at R = Z =
25 pc. Since
Gaia providesproper motions in the barycentric rest frame, we correct thevelocities by adding the solar motion component V ⊙ = ( U ⊙ , V ⊙ , Article number, page 1 of 11 & Aproofs: manuscript no. PM_Paper W ⊙ ) = (11.10, 241.92, 7.25) km s − kpc − (Reid & Brunthaler2004; Schönrich et al. 2010) to the velocities given in thegalactocentric rest frame.We adopt the metallicity distribution proposed byPortail et al. (2017) for the central regions of the MilkyWay, which is specified by the relative contribution of four[Fe / H] bins: A (0 < [Fe / H] ≤ < [Fe / H] ≤ < [Fe / H] ≤ -0.5), and D (-1.5 < [Fe / H] ≤ -1). According to thismodel, the bar is dominated by metal-rich stars (the contributionof the A-D bin is 52 ± ± ± ± ± ± ± ± ◦ .The same metallicity distribution is considered for the bulge ofthe axisymmetric model, defined as the region inside the inner1.7 kpc. We adopt Z ⊙ = asgaia code In order to extract stellar population properties from the simula-tion particles, we developed the code asgaia , which allows us totrack the statistics of the underlying stellar population in a moredirect fashion than other codes, such as galaxia (Sharma et al.2011) or snapdragons (Hunt et al. 2015).The main purpose of asgaia is to compare models of theMilky Way galaxy to observations, and in particular statisticsderived from Gaia data, taking into account observational con-straints. It estimates the magnitude in the G band (Jordi et al.2010) and the astrometric errors using the method described inGaia Collaboration et al. (2016) and in the Gaia
Science Perfor-mance web pages. In this context, the magnitude in the G bandand the end-of-mission parallax error ( σ ̟ ) are approximated bythird-order polynomials in V and I .The Gaia selection function is modelled by imposing anadditional brightness restriction of 3 ≤ G <
20 mag. We excludesources brighter than G = Gaia scanning law (see Table 1 in Gaia Collaboration et al.2016). To a first approximation, we assume there are no correla-tions among the errors of the astrometric parameters.To model the stellar populations, asgaia makes use of thesame set of parsec isochrones (Marigo et al. 2008; Bertelli et al.1994) and extinction maps as galaxia and snapdragons , withthe extinction corrections from the Schlegel E( V - B ) map(Schlegel et al. 1998) proposed by Bland-Hawthorn et al. (2010) https://bitbucket.org/pedroap/asgaia.git http://stev.oapd.inaf.it/cgi-bin/cmd and Sharma et al. (2014). The isochrones can be populated ac-cording to the Salpeter (Salpeter 1955) or the Kroupa (Kroupa2001) Initial Mass Functions (IMF), both truncated at the high-est stellar mass of each isochrone.For each simulation particle, asgaia provides the number ofstars potentially observable by Gaia , with their mean parallaxerror at the end of the mission. These quantities are estimateddirectly from isochrone sampling, without creating mock cata-logues, by reducing the computation time and the size of theoutput files. More details on asgaia will be provided in a forth-coming paper.
4. Results
We consider the ℓ - d b (galactic longitude and plane-projected dis-tance) projection introduced in Palicio et al. (2018) to show themaps of the skewness and kurtosis. Since this face-on projectionkeeps the line of sight fixed, the distance uncertainties shift thesources along one axis, and the increasing size of the bins with d b compensates for the decreasing number of observed stars. Werestrict our study to the disc by imposing the cuto ff in the galacticheight | Z | < x ) = h ( x − h x i ) i σ (1)kurt( x ) = h ( x − h x i ) i σ − σ refers to the standard deviation of the proper motiondistribution and the rightmost term of Eq. 2 subtracts the kurto-sis of a Gaussian distribution. The averages in these equationsare weighted by the mass of the simulation particles or by thenumber of observed stars, depending on the case.In order to illustrate the role of the higher order momentsin the distribution, we select two disc regions in both simula-tions and compute their distributions of proper motions (Fig. 2).These regions correspond to two circular areas of radius 0.6 kpc,centred at ( ℓ , d b ) = (20 ◦ , 7.25 kpc) and (-10 ◦ , 4.5 kpc), with theconstraint | Z | < µ ℓ values when the distribution is skewed pos-itive (negative). In the axisymmetric model, only the distributionof the nearest region is significantly skewed, with skew( µ ℓ ) ≈ -0.330.Since the skewness of the latitudinal proper motion is al-most zero for both galaxies, the histograms of µ b (last columnin Fig. 2) illustrate more clearly the role of the kurtosis in theshape of distributions. As the kurtosis increases, the latitudinalproper motions are more concentrated around the peak, in con-trast to the flatter distribution observed when kurt( µ b ) is negative(lower right panel). This result agrees with the traditional, butcontroversial, interpretation of the kurtosis as an indicator of thepeakedness of distributions, which has been proved to be gener-ally untrue (Kaplansky 1945; Westfall 2014), although but stillvalid for some distribution families. Figure 3 illustrates the maps of the skewness and kurtosis of theproper motions extracted directly from the simulations. In these
Article number, page 2 of 11alicio et al.: Bar signatures in the proper motions
Fig. 1.
Mass-density map for the barred (left panels) and the axisymmetric (right panels) models. The dashed black lines correspond to the edgesof the bar and bulge areas defined in Section 2, while the blue lines represent the line of sight between ℓ = − ◦ and ℓ = ◦ with steps of 15 ◦ .The Sun is denoted by a blue spot at (X, Y, Z) = (0, -8.0, -0.025) kpc. maps, the contribution of each particle is weighted by its mass,and no selection function is applied.The odd columns in Fig. 3 show the higher order moments of µ ℓ (the proper motion along the direction of galactic longitude)for both simulations. As can be seen in the first panel, the largest | skew( µ ℓ ) | values trace the contour of the bar, with opposite signson opposite sides. On the contrary, the unbarred model predictsa wide positive (negative) skewed area at heliocentric distanceslonger (shorter) than R (8 kpc). The major discrepancies arefound between the leading edges and the minor axis of the bar,where the axisymmetric model is skewed to the opposite sidewith respect to the barred model.Regarding the kurtosis, the barred simulation shows a uni-form distribution of kurt( µ ℓ ) < V los (Palicio et al. 2018). In the axisymmetric model, the kurtosis ispositive almost everywhere, with two regions of high kurtosis(kurt( µ ℓ ) >
1) at ( ℓ , d b ) ≈ (0 ◦ , 5 . ◦ , 11 kpc), respec-tively.The even columns in Fig. 3 summarise the results for the lat-itudinal proper motions µ b . Due to the vertical symmetry in thetwo galaxies, the maps of the skewness show a flat distribution ofskew( µ b ) ≈ µ b ) inthe bar region contrast with the uniform distribution of positivekurtosis predicted by the axisymmetric model. As can seen in thesecond row of Figure 3, two areas of opposite sign clearly definethe bar (kurt( µ b ) <
0) and the disc (kurt( µ b ) >
0) regions. The kur-tosis is highest beyond the bar at 0 ◦ . ℓ . ◦ (kurt( µ b ) > . µ b ) < − . In contrast to Palicio et al. (2018), we extend the longitudinalrange towards ℓ < ◦ to study the e ff ects of the approaching sideof the bar, which was omitted due to the lack of sources at nega-tive longitudes in APOGEE DR14 (Abolfathi et al. 2018).In order to estimate the number of observed stars N obs andthe mean proper motion error σ µ for each particle, we run as - gaia assuming a Kroupa IMF and a science margin of 20% . Wemultiply the mean proper motion errors by a factor of (60 / n ) . (Mor et al. 2015), where n =
22 is the number of months of datacollection used in Gaia DR2, to correct the values estimated forthe end of the mission. In order to compare the predictions withthe observations, we make use of the optional cuto ff in σ ̟ /̟ implemented in asgaia .Once the proper motion errors for all the particles are known,their contributions to the averages in Eqs. 1 and 2 are weightedby N obs , and smoothed with the Gaussian kernel K ( µ ; ¯ µ, σ µ ) = √ πσ µ exp − ( µ − ¯ µ ) σ µ (3)where µ ∈ { µ ℓ , µ b } and ¯ µ is the proper motion of the particle.This procedure is averaged over 50 realisations of the heliocen-tric distances of the particles, blurred by a Gaussian distributionwith dispersion σ d = σ ̟ d .The predictions for Gaia
DR2 assuming di ff erent cuto ff s in f = σ ̟ /̟ are shown in Fig. 4. In general, the inclusion ofthe proper motion errors blurs the features predicted in the idealcase, and reduces the absolute values of the higher order mo-ments, which implies a higher Gaussianity in the distributionof proper motions. For the barred simulation, the high-skew( µ ℓ )area expected from Fig. 3 is not observed when f = .
15, whilefor higher values of f we can discern a region with skew( µ ℓ ) & The science margin is a contingency factor introduced inGaia Collaboration (2016) to account for the calibration errors and othernon-ideal scenarios, such as crowded regions or background peculiari-ties. We assume the default value (20%) reported in literature.Article number, page 3 of 11 & Aproofs: manuscript no. PM_Paper
Fig. 2.
Probability distribution functions of proper motions for two circular regions centred at ( ℓ , d b ) = (20 ◦ , 7.25 kpc) and ( ℓ , d b ) = (-10 ◦ , 4.5 kpc),with a radius of 0.6 kpc and a galactic height | Z | < ∆ µ = / yr and the total area is normalised to unity. The first (second)column corresponds to the longitudinal (latitudinal) proper motion, while the first (second) values in the insets refer to the barred (axisymmetric)model. The position of the mean values and the ± σ interval are denoted by triangles and error bars, respectively. beyond the near bar arm ( d b & µ ℓ ) < − .
3) atnegative longitudes even for f = .
15, although for f ≥ .
25 itsshape is more consistent with the predictions.For the axisymmetric model, we observe a low-skew( µ ℓ ) re-gion extended along the ℓ ≈ ◦ direction for f = .
15. For highervalues of f , this structure becomes more circular and restrictedto distances d b . . µ ℓ ) ≥ .
30) at d b & . f > .
20, we cannot conclude they are part of the high-skew( µ ℓ )region observed in Fig. 3 since they are not centred at ℓ = ◦ .Regarding the latitudinal proper motions, both models are inagreement with the expected non-skewed distributions of µ b , al-though the barred simulation shows a distortion in the flat patternat d b & f = .
15 due to the low number of particles inthis region (Fig. 5).The most noticeable discrepancy between the ideal and therealistic cases is found in the distribution of kurtosis. For f = .
15, the high-kurtosis regions of both models are completelywashed out by the inclusion of the astrometric errors and theblur in distances. In the case of the barred simulation, this fea-ture is not recovered by increasing f , while the almost circularhigh-kurtosis region centred at ( ℓ , d b ) ≈ (0 ◦ , 4 kpc) is reproducedby the axisymmetric model for f ≥ .
20. We observe an areaof negative kurtosis instead of the high-kurtosis region expectedat ( ℓ , d b ) ≈ (0 ◦ , 11 kpc). In the barred model, it is possible to in- fer the elliptical structure of the ideal kurt( µ ℓ ) distribution for f ≥ .
20. The latitudinal proper motions, however, show irreg-ular kurt( µ b ) patterns at low f and uniform maps at f ≥ . ∼ n =
60 months of observationsto evaluate which features can be resolved at the end of the mis-sion (Figures 6 and 7). The temporal dependence of σ µ as ∼ t − . reduces the parallax errors to approximately 20% of their DR2values. In general, this increases the absolute value of skew( µ ) inboth simulations with respect to the case of n =
22 months (Fig.4). At the end of the mission, the positive skew( µ ℓ ) region ex-pected in the barred galaxy is discernible at lower f ( f & . µ ℓ ) > µ ℓ )region, however, is well defined in both simulations, with de-creasing values of skew( µ ℓ ) as f increases.The additional observing time has no major e ff ects on theskewness of the latitudinal proper motions. The distributions ofskew( µ b ) show the homogeneous pattern similar to that of the n =
22 case, with weaker distortions at d b & f ≤ .
20 com-pared to Fig. 4.As can be seen in Fig. 6, the reduction of the
Gaia astromet-ric errors does not have a significant e ff ect on the maps of thekurtosis, although it is possible to discern di ff erences betweenthe models in the distribution of kurt( µ ℓ ). In the axisymmetric Article number, page 4 of 11alicio et al.: Bar signatures in the proper motions
Fig. 3.
Maps of the higher order moments of the longitudinal (odd columns) and latitudinal (even columns) proper motions. The two left (right)columns correspond to the barred (axisymmetric) model. We apply the restriction | Z | < model, we can identify an almost circular region of high kurtosis( >
1) in the area of skew( µ ℓ ) < d b < µ ℓ ) < -0.5. On thecontrary, the barred simulation shows an irregular area of lowerkurtosis in the near bar arm, which is strongly a ff ected by theimposed cuto ff in f .In general, the increment in the number of sources per bindue to the additional observing time predicts no qualitative ef-fects on the maps for f ≤ .
20, but an enhancement of the fea-tures revealed with n =
22 months. This is not the case of thepositive kurt( µ ℓ ) area found at ℓ > f = .
30, in whichthe additional observing time increases the number of sourcesper bin from ∼ to ∼ . (see Figs. 5 and 7). Similarly,for f = .
15 the increment from ∼ . to ∼ . stars perbin makes it possible to infer part of the high-skew( µ ℓ ) regionexpected beyond the near bar arm.We estimate the significance of the observed features bypropagating the errors in proper motions to the skewness andkurtosis. We can discern the areas of opposite sign in the skew-ness distribution of µ ℓ for n = f ≥ .
20, with ∆ skew( µ ℓ ) > σ (18 σ ) for the barred (axisymmetric) model. At the end of themission, the estimated significance of ∆ skew( µ ℓ ) is greater than140 σ for the barred simulation and 88 σ for the axisymmetricmodel. We compare our predictions with the latest
Gaia
Data Release(Gaia Collaboration et al. 2018), which provides parallaxes, po-sitions, and proper motions for more than 1.33 million sourcesobserved during 22 months of mission. We make use of thedistances estimated by Bailer-Jones et al. (2018) because theinverse of the parallax leads to biased distances (Kovalevsky1998; Bailer-Jones 2015; Astraatmadja & Bailer-Jones 2016;Luri et al. 2018) for large relative parallax errors. These dis-tances correspond to the mode (median) of the unimodal (bi-modal) posterior probability distribution function (PDF) calcu-lated using only the astrometric data, without invoking the pho-tometric data of the sources or the extinction. The likelihood ofthis Bayesian model is assumed to be Gaussian, while the prior is the exponential distribution introduced in Bailer-Jones (2015), inwhich its length scale depends on the star position on the sky. Weadopt one-half of the separation between the r lo and r hi param-eters as an estimation of the uncertainty ( δ d ). These quantitiescorrespond to the bounds of the highest density interval (HDI)or the equal-tailed interval (ETI), depending on the shape of theposterior probability distribution (p = − ◦ ≤ ℓ < ◦ , 3 ≤ d b <
12 kpc and | Z | < ≤ G <
20 mag. We exclude the starswith large fractional errors in proper motions by imposing δµ ≤ . µ , with µ = µ ∗ α + µ δ ( µ ∗ α ≡ µ α cos δ ) and δµ = µ − q(cid:0) µ ∗ α δµ ∗ α (cid:1) + ( µ δ δµ δ ) + ρ (cid:0) µ ∗ α δµ ∗ α (cid:1) ( µ δ δµ δ ) , (4)where ρ is the correlation between the equatorial proper motions µ ∗ α and µ δ . Once these constraints are applied, we create foursubsets based on the relative distance errors f = Gaia
DR2 x with x = f . These distancecuto ff s are larger than the widely used 10% relative error becausewith such a restrictive constraint we get less than 100 stars inmost of the bins beyond ∼ f . The contribution of each starto the averages involved in the computation of the higher ordermoments is weighted by the inverse of the distance uncertainty(1 / δ d ).It is worth noting that our aim is to demonstrate whether it ispossible to detect the predicted structures in the data available,not to reproduce them quantitatively. The comparison betweenthe simulations and the Gaia proper motions must be understoodin a qualitative way, since their particular values depend on barparameters such as the shape or the pattern speed, whose studyis beyond the scope of this paper.The results for all the
Gaia data subsets are summarised inFigure 8. For the
Gaia
DR2 data set, we can discern the asym-metric low-skew( µ ℓ ) area expected from the barred simulation at Article number, page 5 of 11 & Aproofs: manuscript no. PM_Paper
Fig. 4.
Maps of the higher order moments of µ ℓ (odd columns) and µ b (even columns) predicted for the Gaia
DR2 (22 months of observations).The same convention as in Fig. 3 is used. White areas, if present, correspond to empty bins. From top to bottom: σ ̟ /̟ = .
15, 0 .
20, 0 .
25, and0 . Fig. 5.
Maps of the number of stars per bin estimated for the barred(left panel) and axisymmetric (right panel) models after 22 months ofobservations with
Gaia . White areas, if present, correspond to emptybins. From top to bottom: σ ̟ /̟ = .
15, 0 .
20, 0 .
25, and 0 . ℓ < ◦ , while the high-skewness region is not observed due to thelack of sources beyond ∼ µ ℓ ) becomes more evident as f increases. Onthe contrary, the high-skewness region is not recovered, althoughminor areas of positive skew( µ ℓ ) can be inferred at ℓ & ◦ and d b ≈ µ b ) map expected from the vertical sym-metry of the Galaxy, with minor distortions in the distant re-gions. These distortions can be explained by the low number ofsources contained in the farther bins. Figure 9 shows the mapsof skew( µ b ) for eight di ff erent subsampling realisations of themodels with the same spatial distribution as Gaia
DR2 . As canbe seen, the larger discrepancies from the flat skew( µ b ) ≈ f ) to a more complexpattern (high f ). For f = .
30, the observations reveal an ellipti-cal area of kurt( µ ) . µ ) > ff set of ∼ -2 in the rest ofthe disc. We attribute the discrepancy in kurtosis to three mainfactors:1. Since the kurtosis involves the fourth-order moments, it re-quires a larger number of sources compared to the otherlower order estimators, such as the dispersion or the skew-ness, to get reliable results. 2. The kurtosis constitutes a measurement of contribution of theoutliers (Kaplansky 1945; Ali 1974; Decarlo 1997; Westfall2014), which in the case of the proper motions are notbounded, and are di ffi cult to include in the models. The anal-ysis in detail of the observed proper motions in the highlypopulated areas of Fig. 8 reveals a single-peaked distributionof µ ℓ with a flattened maximum, while the distribution of µ b shows a much sharper mode.3. We verified that the kurtosis is more sensitive to the distanceand proper motions errors, and to the size of the bins, thanthe other estimators. This can be seen in the evolution of themaps as f increases. We must emphasise that in order to ex-plore the regions beyond ∼ ff larger than the usual 10%, which implicitly leadsto larger errors and blurred features.
5. Summary
The analysis of the higher order moments of the proper motioncan reveal signatures of the bar on the galactic kinematics. Inparticular, it is possible to discern the bar shape and orientationfrom the areas of opposite sign in skew( µ ℓ ), in contrast to the al-most circular regions predicted by the axisymmetric model. Partof this structure, the low-skew( µ ℓ ) region, is observed in the mostrecent Gaia data release, although it requires a distance error cut-o ff that is less restrictive than the usual 10%.On the contrary, the maps of the observed kurt( µ ℓ ) andkurt( µ b ) show an area of positive kurtosis which di ff ers signif-icantly from the almost zero values predicted by the simulations.Increasing the relative distance error cuto ff , we find a tentativeelliptical structure compatible in location and orientation withthe nearest bar arm. Based on our tests, we attribute the discrep-ancy in the kurtosis maps to a greater sensitivity of the fourth-order moment on the errors and outliers. Our simulations withthe code asgaia suggest that the expected features in kurt( µ ) willnot be much easily resolved in the Gaia end-of-mission data, butthe contrast between the high- and low-skewness region in µ ℓ will allow us to di ff erentiate between the models.For the latitudinal proper motion, no features are expectedin the distribution of skew( µ b ) since both galaxies are verticallysymmetric. We also find significant e ff ects induced by the selec-tion function, which distorts the flat pattern of skew( µ b ), reducesthe kurtosis, and makes the distributions of proper motions moreGaussian. Acknowledgements.
C.A.P. is thankful to the Spanish Ministry of Economyand Competitiveness (MINECO) for support through grant AYA2017-86389-P. CDV and PAP acknowledge financial support from MINECO throughgrant AYA2014-58308-P. CDV also acknowledges financial support fromMINECO through grant RYC-2015-18078. We acknowledge the contribu-tion of Teide High-Performance Computing facilities to the results of thisresearch. TeideHPC facilities are provided by the Instituto Tecnológico y deEnergías Renovables (ITER, SA). URL: http://teidehpc.iter.es .This work has made use of data from the European Space Agency(ESA) mission
Gaia ( ), pro-cessed by the Gaia
Data Processing and Analysis Consortium (DPAC, ). Fundingfor the DPAC has been provided by national institutions, in particular theinstitutions participating in the
Gaia
Multilateral Agreement.
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Article number, page 7 of 11 & Aproofs: manuscript no. PM_Paper
Fig. 6.
Same as Fig. 4, but at the end of the
Gaia mission (60 months of observations).Article number, page 8 of 11alicio et al.: Bar signatures in the proper motions
Fig. 7.
Same as Fig. 5, but at the end of the
Gaia mission (60 months ofobservations).
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Article number, page 9 of 11 & Aproofs: manuscript no. PM_Paper
Fig. 8.
Maps of the skewness (odd rows) and kurtosis (even rows) of the observed longitudinal (first column) and latitudinal (second column)proper motions of
Gaia
DR2. The number of sources per bin is shown in the third column. Each block corresponds, from top to bottom, to the f = σ ̟ /̟ cuto ff of 0.15, 0.20, 0.25, and 0.30. The Galactic centre and the bar are denoted by the black spot at d b = R and the solid line,respectively. The half-length and orientation angle of the bar are set to 4.5 kpc and 25 ◦ , respectively.Article number, page 10 of 11alicio et al.: Bar signatures in the proper motions Fig. 9.
Maps of skew( µ b ) for eight di ff erent subsamplings of the models with the same spatial distribution as the Gaia
DR2 subset. The first andsecond columns correspond to the barred simulation while the two last columns show the maps expected for the axisymmetric model. The numberof observing months is set to n =
22 and the maximum relative parallax error is f = ..