On the morphology of dust lanes in galactic bars
L. Sánchez-Menguiano, I. Pérez, A. Zurita, I. Martínez-Valpuesta, J. A. L. Aguerri, S. F. Sánchez, S. Comerón, S. Díaz-García
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 17 October 2018 (MN L A TEX style file v2.2)
On the morphology of dust lanes in galactic bars
L. S´anchez-Menguiano, , I. P´erez, A. Zurita, I. Mart´ınez-Valpuesta, , J. A. L. Aguerri, S. F. S´anchez, S. Comer´on , and S. D´ıaz-Garc´ıa Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Glorieta de la Astronom´ıa s/n, Aptdo. 3004, E-18080 Granada, Spain Dpto. de F´ısica Te´orica y del Cosmos, Universidad de Granada, Facultad de Ciencias (Edificio Mecenas), E-18071 Granada, Spain Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain Universidad de La Laguna, Dpto. Astrof´ısica, E-38206 La Laguna, Tenerife, Spain Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, A.P. 70-264, 04510, M´exico, D.F. University of Oulu, Astronomy Division, Department of Physics, P.O. Boc 3000, FI-90014, Finland Finnish Centre of Astronomy with ESO (FINCA), University of Turku, V¨ais¨al¨anti 20, FI-21500, Piikki¨o, Finland
Accepted 2015 April 7. Received 2015 April 6; in original form 2014 December 9
ABSTRACT
The aim of our study is to use dynamical simulations to explore the influence oftwo important dynamical bar parameters, bar strength and bar pattern speed, onthe shape of the bar dust lanes. To quantify the shape of the dust lanes we havedeveloped a new systematic method to measure the dust lane curvature. Previousnumerical simulations have compared the curvature of bar dust lanes with the barstrength, predicting a relation between both parameters which has been supported byobservational studies but with a large spread. We take into account the bar patternspeed to explore, simultaneously, the effect of both parameters on the dust lane shape.To that end, we separate our galactic bars in fast bars (1 < R < .
4) and slow bars( R > . G galaxies, obtaining dust lanecurvatures lying within the range covered by the simulations.
Key words: methods: numerical – methods: observational – galaxies: kinematics anddynamics – galaxies: structure – galaxies: spiral.
Bars are common morphological features among spiralgalaxies. Roughly 30-40% of the spiral galaxies have a pro-nounced bar in optical wavelengths and if we take into ac-count weaker bars, the fraction rises up to 60% (de Vau-couleurs 1963; Sellwood & Wilkinson 1993; Marinova & Jo-gee 2007; Barazza, Jogee & Marinova 2008; Sheth et al. 2008;Aguerri, M´endez-Abreu & Corsini 2009; Nair & Abraham2010; Masters et al. 2011). Kinematic data shows the pres-ence of strong non-circular gas motions in bars (e.g. Huntley1978; Zurita et al. 2004), which indicates that the bar con-stitutes a major non-axisymmetric component of the galaxymass distribution (Sellwood & Wilkinson 1993).Stellar bars are thought to be a key mechanism in thedynamical evolution of disc galaxies. For example, they areable to contribute to the redistribution of matter in thegalaxy by exchanging angular momentum with the disc,bulge and halo (e.g. Debattista & Sellwood 1998, 2000;Athanassoula 2003; Cheung et al. 2013 and reviews by Kor- mendy & Kennicutt 2004; Athanassoula 2013 and Sellwood2014).Galactic bars are characterised dynamically by threemain parameters: length, strength and pattern speed. Sev-eral methods have been proposed to measure them. Deter-mining the bar length is not entirely trivial. For instance,early-type galaxies do not usually have obvious spiral struc-ture that demarcates the bar end. In other cases, the pres-ence of a large bulge may also complicate the measurement.For example, Aguerri, Debattista & Corsini (2003) appliedfour different criteria to determine the bar length, whichleads to variations of ∼ . Visualinspection of galaxy images (e.g. Martin 1995), ellipse fittingof the isophotes by locating the maximum in the ellipticity Parameter that quantifies the effectiveness to which a bar po-tential influences the motions of stars. Based on the mean values and standard deviations from theirTable 4.c (cid:13) a r X i v : . [ a s t r o - ph . GA ] A p r L. S´anchez-Menguiano et al. profile (e.g. Wozniak & Pierce 1991) or by looking for varia-tions of the position angle (e.g. Wozniak et al. 1995; Aguerriet al. 2009) and Fourier analysis of the surface brightness(e.g. Ohta, Hamabe & Wakamatsu 1990; Aguerri, Beckman& Prieto 1998; Aguerri et al. 2000) are among the severalmethods proposed for measuring the bar length. For detailson different techniques and definitions used to determinethe bar length see, for instance, Michel-Dansac & Wozniak(2006) and Gadotti et al. (2007). Studies have obtained atypical value for the bar length of 6-8 kpc, being bars longerby a factor of ∼ Q b which represents the maxi-mum bar torque applied to a gaseous material in orbital mo-tion relative to its specific kinetic energy. This is one of themost frequently used parameters to describe bar strength.Fourier techniques are also commonly used to measure thebar torque (e.g. Aguerri et al. 2000; Buta, Block & Knapen2003; Mart´ınez-Valpuesta, Shlosman & Heller 2006).The bar pattern speed is one of the most defining dy-namical parameters. It can be parametrised by a distance-independent parameter R = R CR /R bar , where R CR is theLagrangian/corotation radius and R bar is the bar semima-jor axis. Bars that end near corotation (1 < R < .
4) areconsidered as fast bars and shorter bars ( R > .
4) as slowbars (Debattista & Sellwood 2000). For R <
1, orbits areelongated perpendicular to the bar and self-consistent barscannot exist (Contopoulos 1980). The bar pattern speed isalso difficult to measure. Several methods are usually used todetermine it, but the uncertainties in the analysis give incon-clusive results (Knapen 1999). The most direct determina-tion is by applying the Tremaine-Weinberg method, basedon the continuity equation (Tremaine & Weinberg 1984),but it requires imaging data with high signal-to-noise and,therefore, long integration times, which limits its applica-tion to a restricted number of candidates (e.g. Debattista,Corsini & Aguerri 2002; Gerssen, Kuijken & Merrifield 2003;Aguerri et al. 2003; Treuthardt et al. 2007). Aguerri et al.(2015) has applied this method to a larger sample of galax-ies taking advantage of integral field spectroscopy data fromthe CALIFA survey (S´anchez et al. 2012), not finding dif-ferences of pattern speed with morphological types. Recentwork (Font et al. 2014) has presented a new direct methodto locate resonances in a disc using 2-D gas information. Itrequires high velocity and spatial resolution of ionised gasdata. They find that most of the analysed bars are consis-tent with being fast and present a hint of pattern speedsegregation with morphological types.These properties of bars determine their influence in thegalactic dynamics and produce or modify some morphologi-cal features. Due to the difficulties in determining these barparameters, it is highly desirable to find indirect methodsto derive them, like the use of morphological informationof the bar. Previous studies have derived the bar patternspeed matching observational features with resonances, forinstance using rings as indicators (e.g. Buta 1986), even forgalaxies at high redshift (P´erez, Aguerri & M´endez-Abreu2012). One of the most remarkable features of bars is thepresence of dust lanes along them that extend from the nuclear region into the spiral arms. Bar dust lanes havebeen studied from numerical simulations (e.g. van Albada& Sanders 1982; Athanassoula 1992; Patsis & Athanassoula2000; Patsis, Kalapotharakos & Grosbøl 2010; Kim, Seo &Kim 2012), as well as observationally (e.g. Knapen, P´erez-Ram´ırez & Laine 2002; Marshall et al. 2008; Comer´on et al.2009). These show different morphologies; from completelystraight, although they can sometimes curl around the cen-tre, to curved, with the concave sides towards the majoraxis.Studies of gas flows in bars (mainly theoretically fromN-body simulations, e.g. Fux 1999; P´erez, Fux & Freeman2004; P´erez 2008) have shown that dust lanes are sites ofhigh gas and dust density, and therefore, of high absorptionof the starlight. Numerical simulations (Athanassoula 1992;Patsis et al. 2010) have revealed the existence of shocks atthe positions of these narrow lanes. The existence of shocksalong the dust lanes has been observationally shown fromthe analysis of gas kinematics of strongly barred galaxies(e.g. Downes et al. 1996; Mundell & Shone 1999; Zurita et al.2004). These shocks are tracing the channels by which thegas flows towards the galaxy centres where it can contributeto the central mass concentration, for example producingstar formation, pseudo-bulges formation or AGN activity(e.g. Shlosman, Frank & Begelman 1989; Oh, Oh & Yi2012; Wang et al. 2012; Lee et al. 2012).Numerical simulations have pointed out a relation be-tween the shape of the dust lanes and the bar strength,where stronger bars show straighter dust lanes, and betweenthe position of the dust lanes and the bar pattern speed,where dust lanes that are offset from the bar major axis ap-pear for a limited range of bar pattern speeds, below whichthe dust lanes are ‘centred’ (Athanassoula 1992). There havebeen a few attempts to observationally find this relation be-tween dust lane curvature and bar strength (Knapen et al.2002; Comer´on et al. 2009), but without taking into ac-count the bar pattern speed. Even though Knapen et al.(2002) confirmed a correlation between bar strength anddust lane curvature, Comer´on et al. (2009), using a largersample and improving the number statistics from Knapenet al. (2002), showed a large spread in the data and con-cluded that, though bar strength set an upper limit to thedust lane curvature allowed, this parameter by itself did notdetermine the dust lane curvature.In this paper we use dynamical simulations to explorethe effect of bar strength and bar pattern speed on the shapeof the bar dust lanes. To carry out this study we have devel-oped a new methodology to characterise the dust lane shapesbased on the mathematical definition of curvature on a setof simulated galaxies. The final aim of the work would be toderive information about the underlying dynamics of barsby a simple analysis of morphological features, with easiermeasurements.The structure of the paper is organised as follows. InSection 2, we provide a description of the simulations usedin this study. In Section 3, we explain the methodology de-veloped to measure the bar dust lane curvature. In Section4, we present the results found in this work and analyse thedependency of the curvature on bar parameters. Finally, inSection 5 we discuss the results and compare them with pre-vious studies in this topic. c (cid:13) , 000–000 n the morphology of bar dust lanes To study the dynamical response of the gas in the pres-ence of a galactic bar we used an updated version of theFTM 4.4 code from Heller & Shlosman (1994), a three-dimensional hybrid SPH/N-body code. We ran a set of 238simulations, which were performed with 10 isothermal, nonself-gravitating, collisional particles. These are the same sim-ulations used in Comer´on et al. (2009).The potential used in the simulations consists of threeaxisymmetric components associated to the galaxy disc,bulge and halo, each one modelled by a Miyamoto-Nagaiprofile, and a non-axisymmetric component identified withthe bar, modelled by a Ferrers potential (Ferrers 1877) with n = 1, being n the degree of the central density concentra-tion. The halo and disc masses are constant for all the simu-lations, but we vary the bulge to total mass ratio, B/ ( B + D ),by increasing the bulge mass. We have used in the simula-tions two different values of the bar mass, being one bar a50% more massive than the other. For more details about theparameter values used in the potential models, see Comer´onet al. (2009).Regarding the bar pattern speed, we use Ω b equal to 10,20, 30 and 40 km s − kpc − , placing the corotation radiusoutside the bar and obtaining values for R between 1.0 and3.34. Bars with R CR /R bar > . R = 3 . ± .
95, and even in that casethe error may still place the value below the limit of 3.0. Forthis reason, we have discarded those simulations in whichcorotation radius is more than three times the bar radius.After this, the number of simulations drops to 175.We derive analytically the bar strength of the simu-lated galaxies by using the parameter proposed by Combes& Sanders (1981), Q b , which is given by the maximum over r of Q b = F maxT ( r ) < F R ( r ) > = r (cid:12)(cid:12) ∂ Φ( r,φ ) ∂φ (cid:12)(cid:12) max < dΦ (r)dr > , (1)where r is the galactocentric distance, F maxT is the azimuthalmaximum of the absolute tangential force in the bar region, < F R ( r ) > the mean axisymmetric radial force, Φ( r, φ ) thein-plane gravitational potential and Φ ( r ) its axisymmetriccomponent. A way to characterise and quantify the shape of bar dustlanes is by measuring its curvature. As dust lanes are relatedto gas shocks, we measure their curvature in our set of sim-ulated galaxies by deriving the curvature of the gas densityenhancements, using a systematic and easily reproduciblemethod, explained below. The range we cover to measurethe dust lane curvature starts outside the bar inner region,defined by the nuclear ring associated to the ILR in somecases, or a ring without closure in others, and finishes at thebeginning of spiral arms, corresponding to a kink in the dustlane direction. This measurement was performed after twobar rotations in the simulations. Well-defined measurabledust lanes appeared in 126 simulations. This final sample does not exactly match the sample in Comer´on et al. (2009),which comprises 88 simulations. Regrettably, it is not possi-ble to access the sample presented in Comer´on et al. (2009),and therefore, we cannot check the source of the discrepancyin the number of galaxies.To measure the curvature we visually trace one of thedust lanes in each simulation (in our simulations both dustlanes are symmetrical) making use of SAOImage ds9 . Fig-ure 1 shows the outline of a dust lane in three simulatedgalaxies. Then we fit the points tracing the bar dust lanewith a third degree polynomial (Figure 1). To check thegoodness of the fit we calculate the medium residuals of thefit for each galaxy, which is a measure of the discrepancybetween the data and the model. Performing this test, themean value of the residuals for all galaxies is of 0.2 pixels ,with a standard deviation of 0.1 pixels, which reflects thata third degree polynomial fits quite well the data. In theparticular case of the simulated galaxies shown in Figure 1,from left to right the maximum residual for each fitting is0.43, 0.16 and 0.69 pixels, with a mean residual along thefitting of 0.18, 0.08 and 0.38 pixels, respectively.From the fitting of the bar dust lanes we now measuretheir curvature. The curvature of a plane curve gives infor-mation on how fast its tangent vector changes direction asyou travel along the curve, and this is exactly what we wantto know about the bar dust lane shape. If the dust lanekeeps close to the same direction (that is, the dust lane isstraight), the unit tangent vector changes very little and thecurvature is small (the opposite if the dust lane undergoesa tight turn). To measure bar dust lane curvature we usethe mathematical definition of curvature for a plane curvewritten in cartesian coordinates in the form y = f ( x ) κ ≡ d φ d s = | y (cid:48)(cid:48) ( x ) | (cid:0) y (cid:48) ( x )) (cid:1) / (2)where φ is the tangential angle and s is the arc length. Ithas units of inverse distance. For more details about themathematical deduction of the curvature of a function see,for example, Weisstein (2003).We should not forget that this mathematical curvatureis not a constant function. Therefore, we have adopted as‘curvature of a bar dust lane’ the absolute mean value ofthe curvature along the bar dust lane. This parameter iszero for straight bar dust lanes and progressively increasesas these deviate from straight lines.Besides, we should also notice that the curvature, as de-fined before, is a measurement that depends on the absolutebar size. As in the simulations this parameter is constant, wedo not have to deal with this issue. However, to apply thismethod to observational data, we need to ‘normalise’ thecurvature to obtain a dimensionless parameter which takesinto account the size of the bars, multiplying the mean cur-vature by the bar semimajor axis.To show that the method is viable and easy to im-plement on observational studies, we have applied it to An astronomical imaging and data visualisation application de-veloped by Smithsonian Astrophysical Observatory (Joye & Man-del 2003). Taking the unit length of simulations to 10 kpc, each pixelcorresponds to 60 pc.c (cid:13)000
95, and even in that casethe error may still place the value below the limit of 3.0. Forthis reason, we have discarded those simulations in whichcorotation radius is more than three times the bar radius.After this, the number of simulations drops to 175.We derive analytically the bar strength of the simu-lated galaxies by using the parameter proposed by Combes& Sanders (1981), Q b , which is given by the maximum over r of Q b = F maxT ( r ) < F R ( r ) > = r (cid:12)(cid:12) ∂ Φ( r,φ ) ∂φ (cid:12)(cid:12) max < dΦ (r)dr > , (1)where r is the galactocentric distance, F maxT is the azimuthalmaximum of the absolute tangential force in the bar region, < F R ( r ) > the mean axisymmetric radial force, Φ( r, φ ) thein-plane gravitational potential and Φ ( r ) its axisymmetriccomponent. A way to characterise and quantify the shape of bar dustlanes is by measuring its curvature. As dust lanes are relatedto gas shocks, we measure their curvature in our set of sim-ulated galaxies by deriving the curvature of the gas densityenhancements, using a systematic and easily reproduciblemethod, explained below. The range we cover to measurethe dust lane curvature starts outside the bar inner region,defined by the nuclear ring associated to the ILR in somecases, or a ring without closure in others, and finishes at thebeginning of spiral arms, corresponding to a kink in the dustlane direction. This measurement was performed after twobar rotations in the simulations. Well-defined measurabledust lanes appeared in 126 simulations. This final sample does not exactly match the sample in Comer´on et al. (2009),which comprises 88 simulations. Regrettably, it is not possi-ble to access the sample presented in Comer´on et al. (2009),and therefore, we cannot check the source of the discrepancyin the number of galaxies.To measure the curvature we visually trace one of thedust lanes in each simulation (in our simulations both dustlanes are symmetrical) making use of SAOImage ds9 . Fig-ure 1 shows the outline of a dust lane in three simulatedgalaxies. Then we fit the points tracing the bar dust lanewith a third degree polynomial (Figure 1). To check thegoodness of the fit we calculate the medium residuals of thefit for each galaxy, which is a measure of the discrepancybetween the data and the model. Performing this test, themean value of the residuals for all galaxies is of 0.2 pixels ,with a standard deviation of 0.1 pixels, which reflects thata third degree polynomial fits quite well the data. In theparticular case of the simulated galaxies shown in Figure 1,from left to right the maximum residual for each fitting is0.43, 0.16 and 0.69 pixels, with a mean residual along thefitting of 0.18, 0.08 and 0.38 pixels, respectively.From the fitting of the bar dust lanes we now measuretheir curvature. The curvature of a plane curve gives infor-mation on how fast its tangent vector changes direction asyou travel along the curve, and this is exactly what we wantto know about the bar dust lane shape. If the dust lanekeeps close to the same direction (that is, the dust lane isstraight), the unit tangent vector changes very little and thecurvature is small (the opposite if the dust lane undergoesa tight turn). To measure bar dust lane curvature we usethe mathematical definition of curvature for a plane curvewritten in cartesian coordinates in the form y = f ( x ) κ ≡ d φ d s = | y (cid:48)(cid:48) ( x ) | (cid:0) y (cid:48) ( x )) (cid:1) / (2)where φ is the tangential angle and s is the arc length. Ithas units of inverse distance. For more details about themathematical deduction of the curvature of a function see,for example, Weisstein (2003).We should not forget that this mathematical curvatureis not a constant function. Therefore, we have adopted as‘curvature of a bar dust lane’ the absolute mean value ofthe curvature along the bar dust lane. This parameter iszero for straight bar dust lanes and progressively increasesas these deviate from straight lines.Besides, we should also notice that the curvature, as de-fined before, is a measurement that depends on the absolutebar size. As in the simulations this parameter is constant, wedo not have to deal with this issue. However, to apply thismethod to observational data, we need to ‘normalise’ thecurvature to obtain a dimensionless parameter which takesinto account the size of the bars, multiplying the mean cur-vature by the bar semimajor axis.To show that the method is viable and easy to im-plement on observational studies, we have applied it to An astronomical imaging and data visualisation application de-veloped by Smithsonian Astrophysical Observatory (Joye & Man-del 2003). Taking the unit length of simulations to 10 kpc, each pixelcorresponds to 60 pc.c (cid:13)000 , 000–000
L. S´anchez-Menguiano et al.
Figure 1.
Gas density maps of the bar region for three simulated galaxies with one of the dust lanes marked (white crosses). Bar majoraxis is parallel to the horizontal in all frames. We have also plotted the dust lane fitting using a third degree polynomial (red line), seetext for details. Ellipticity, bar strength and the ratio of corotation radius to bar semimajor axis ( R ) are given in this order in the upperleft-hand corner of each frame. From left to right, the dust lane curvatures (as explained in Sect. 3) are 1.0, 1.3 and 2.3. a pilot sample. This sample comprises the galaxies fromComer´on et al. (2009) for which there are measurementsof their bar pattern speed in Rautiainen et al. (2008). Q b values are taken from D´ıaz-Garc´ıa et al. (in prep), who in-ferred the gravitational potential from the 3.6 µ m imagingof the Spitzer Survey of Stellar Structure in Galaxies sample(S G) following the recipe from Laurikainen & Salo (2002)and Salo et al. (2010). To obtain the curvature values, wehave followed the same procedure as for the simulations,using g or colour-index g − r images, proxies for dust distri-bution, from SDSS when available and otherwise, other op-tical images from the NASA / IPAC Extragalactic Database(NED). Table 1 shows the names of the galaxies included inthe study, as well as some of their properties and the de-rived normalised curvatures, for both bar dust lanes whenpossible, and just for one when the other is not well-defined. We want to assess the influence of bar strength and barpattern speed on the curvature of bar dust lanes. For thatpurpose, we have developed in this work a new methodol-ogy to quantitatively measure bar dust lane curvatures (seeSection 3).In Figure 2 we show the derived dust lane curvaturesas a function of the bar ellipticity distinguishing betweenfast (1 < R < .
4) and slow ( R > .
4) bars. The red starsare the mean curvature values for each ellipticity. The bluecircular markers are the values for galaxies with a bar massof M b = 1 . × M (cid:12) and the yellow squares for M b =1 . × M (cid:12) . Finally, the purple triangles are the valuesfor the real galaxies included in the study. In the cases inwhich it has been possible to measure both dust lanes, theyare distinguished using both filled and unfilled markers.Regarding the mean curvature values, it is clear that forfast bars the curvature is smaller for higher ellipticity values, ∼ ksheth/S4G/ Figure 2.
Dust lane curvatures as a function of the bar ellipticity (cid:15) for fast bars, i.e., 1 < R < . R > . M b =1 . × M (cid:12) and the yellow squares for M b = 1 . × M (cid:12) .The purple triangles correspond to the values for the real galaxies.c (cid:13) , 000–000 n the morphology of bar dust lanes Table 1.
Properties of the real galaxies used in the study.Name i d PA d R a b ε b Q b κ (NGC) (deg) (deg) (kpc) R fromRautiainen et al. (2008) (Column 4), the deprojected bar semi-major axis and ellipticity (Columns 5 and 6) from Mu˜noz-Mateoset al. (2013), the bar strength (Column 7) from D´ıaz-Garc´ıa et al.(in preparation) and the dust lane curvature measured as definedin Section 3 (Column 8). and slow bars show similar values of the mean curvature forall ellipticity values. There seems to be a lack of points forfast bars at low ellipticities and low curvatures, this effect isnot so evident for slow bars. The most striking result fromthis figure is the absence of high curvature points for slowbars. In fact, the average curvature value for slow bars atthe lowest ellipticity is 0.95 while the value for fast bars is1.90. We show later in this section that the distribution ofpoints for both fast and slow bars is clearly different. As itcan be seen from Figure 2, the values for the real galaxies arewithin the range of parameters covered by the simulations.Figure 3 presents the curvature against the barstrength, again distinguishing between fast (top panel) andslow bars (bottom panel). The marker symbols and coloursused are the same as in the previous figure, being the redstars the mean curvature values for bar strength bins of 0.1.We observe similar trends to those found in Figure 2,indicating that, in this case, the ellipticity is a good proxyfor bar strength. Previous studies comparing bar strengthand ellipticity for a large sample of galaxies have reacheda similar conclusion (e.g. Laurikainen, Salo & Rautiainen2002). For slow bars, we have almost constant curvaturevalue for the whole range of bar strengths while for fast barswe found higher curvature values at low bar strength. Asshown in Figure 2, there is a lack of low curvature points forfast bars at low bar strengths and a deficit of high curvaturepoints for slow bars. Also in this case, the values for the realgalaxies are within the range of parameters covered by thesimulations.We perform a two-sample Kolmogorov-Smirnov test tocheck if the differences found in the results between fast andslow bars are statistically significant, obtaining a P-value of12%. The significance level of the K-S test is 5%, meaningthat values below this limit come from different distribu-tions. Therefore, the two complete samples seem to havesimilar distributions. However, if we restrict the comparisonrange to weak bars, with Q b (cid:54) .
3, where we observe dif-ferences between both distributions, the P-value is now 2%.
Figure 3.
Dust lane curvatures as a function of the bar strength Q b for fast bars, i.e., 1 < R < . R > . M b =1 . × M (cid:12) and the yellow squares for M b = 1 . × M (cid:12) .The purple triangles correspond to the values for the real galaxies. This result indicates that slow and fast bars have a differ-ent distribution of curvature for this range of Q b values. Atlow bar strengths, the statistics of our sample is consider-ably reduced ( ≈
30 elements). Because of this we have alsoperformed an Anderson-Darling test, more appropriate forsmall samples, obtaining a P-value of 5%, reinforcing theobserved trends.
In this work we have used numerical simulations to explorethe effect of two bar parameters, bar strength and bar pat-tern speed, on the bar dust lane shape. To characterise theshape of dust lanes, we have developed a new method basedon the mathematical curvature of a plane curve, providing ameasurement which quantifies this shape, and we have thenused these measurements to study the influence of the barparameters. To do that, we have analysed 126 barred simu-lations, covering bar strengths from 0 to 1 and R between1.0 and 3.0 for two bar masses. We have also applied thismethod to a set of observational data.Athanassoula (1992) studied the shape of bar dust lanesand its dependency on a set of model parameters, like the c (cid:13)000
In this work we have used numerical simulations to explorethe effect of two bar parameters, bar strength and bar pat-tern speed, on the bar dust lane shape. To characterise theshape of dust lanes, we have developed a new method basedon the mathematical curvature of a plane curve, providing ameasurement which quantifies this shape, and we have thenused these measurements to study the influence of the barparameters. To do that, we have analysed 126 barred simu-lations, covering bar strengths from 0 to 1 and R between1.0 and 3.0 for two bar masses. We have also applied thismethod to a set of observational data.Athanassoula (1992) studied the shape of bar dust lanesand its dependency on a set of model parameters, like the c (cid:13)000 , 000–000 L. S´anchez-Menguiano et al.
Lagrangian radius, the axial ratio of the bar and the barquadrupole momentum Q m as a measurement of the barmass. However, the characterisation she made of the shape ofthe dust lanes was purely visual, not providing a method toquantify its curvature and it is therefore difficult to directlyapply on observations.Athanassoula (1992), analysing simulations for fastbars, deduced that curved dust lanes result from weak barsand straight ones originate from strong bars. Our simula-tion results confirm this prediction. We extend the study toslow bars and find that the curvature does not vary with barstrength. Athanassoula (1992) also included in the study theinfluence of the bar pattern speed, but more related to theposition of the dust lanes than to the shape, which is thepurpose of this work. She found a tight range of Lagrangianradii, which corresponds to fast bars in the distinction wehave made, that leads to dust lanes that are ‘offset’ from thebar major axis. For smaller values, corresponding to slowbars, the dust lanes are ‘centred’ and for larger values theyturn their convex sides towards the bar major axis, corre-sponding to unrealistic dust lanes.A methodology similar to the one presented here, isused by Knapen et al. (2002) and Comer´on et al. (2009) onreal galaxies. These works also developed a way to quan-tify the curvature of dust lanes using real galaxies. Theirmethod is based on measurements of the change in the an-gle of the tangent to the curved dust lanes in the rangewhere the curvature is constant, Knapen et al. (2002) ex-pressing the result in units of degrees per kpc and Comer´onet al. (2009) multiplying that angle by the angular radiusat which the torque is maximal, to take into account thesize of the host bars. This method has the advantage of be-ing simple and rather straightforward, but it also presents anumber of drawbacks. First, it depends a lot on which partof the dust lane is selected to measure the curvature; andtherefore, it is not easy to replicate their results. And sec-ond, they also assume a constant curvature for the bar dustlanes, which is not always the case (52% of our simulationsshow dust lanes with non-constant curvature).Our methodology does not assume any a priori shapefor the dust lanes and the results are easy to reproduce.We select the beginning and end of the dust lanes in suchway that we avoid morphological structures, such as nuclearrings and spiral arms, that may be masking the actual dustlane curvature. As we discussed in Sect. 3, we define theend of the dust lane as a kink in the dust lane directionwhich corresponds to the beginning of the spiral arms. Formost of the galaxies, this parameter ranges within a similargalactocentric radius, indicating that our method measuresdust lanes close to the bar ends in almost all cases. We havechecked the effect of shortening the dust lanes in the curva-ture determination to test those cases where the dust lanesdo not reach the end of the bar. We have done this test on 40galaxies obtaining that the mean curvature is only changedby 19% on average.Knapen et al. (2002) results agree with the correlationbetween the dust lane curvature and the bar strength foundby Athanassoula (1992). However, Comer´on et al. (2009),using a larger sample, obtained a large spread in the relationthat made them state that bar strength by itself did notdetermine the dust lane curvature.None of these observational studies take into account the bar pattern speed as an important parameter influencingthe bar dust lane shapes. However, numerical simulations(e.g. van Albada & Sanders 1982; Athanassoula 1992; Fux1999; P´erez et al. 2004; P´erez 2008) have shown that thelocation of gas shocks is linked to the bar potential andthe pattern speed. We take a step forward adding to thedistribution of bar strengths a segregation with bar patternspeed, dividing the sample in fast and slow bars, to find arelation between both parameters and dust lane curvature.The result obtained in the performed Kolmogorov-Smirnovtest supports our assumption that the bar pattern speedhas influence in the shape of bar dust lanes. We find thatthis inverse relation between the dust lane curvature and thebar strength only holds for fast bars, obtaining for slow barsapproximately a constant curvature for all the values of thebar strength.The distribution of points shown in Figure 3 indicates alarger probability of finding slow bars among galaxies withweak bars and straight dust lanes. This is an interestingeffect to be checked observationally.We have tested this method on a set of S G galaxiesfor which we have pattern speed and bar strength measure-ments. We have followed the same procedure as describedfor the simulations. Despite the fact that the sample is notlarge enough to confirm the trends found here, the obtainedvalues lie within the range of values covered by the simula-tions.Although all parameters have been measured in a sim-ilar way for both simulations and observations, the halopotential is not included in the measurement of the barstrength in the S G galaxies. In principle, this could mod-ify the results since the dust lane morphology depends onthe whole galaxy potential. However, the analysed sampleis comprised of high surface brightness galaxies for whichis commonly assumed a maximum disc, implying that thecontribution of the dark halo in the optical disc is minimum(e.g. van Albada & Sancisi 1986; Broeils 1992; Weiner, Sell-wood & Williams 2001; P´erez et al. 2004). Because of thiswe do not expect the results to be substantially changed.Furthermore, the fact that the bar strength follows the el-lipticity trend also reinforces this assumption.In summary, we have developed a method to measuredust lane curvatures that is robust and easily applicableto observational data. A future work using a larger sampleof galaxies with measurements of bar strength and patternspeed is necessary to test the trends found with the simula-tions.
ACKNOWLEDGEMENTS
We acknowledge financial support from the Spanish
Minis-terio de Econom´ıa y Competitividad via grants AYA2011-24728 and AYA2012-31935, and from the ‘Junta de An-daluc´ıa’ local government through the FQM-108 project.We also acknowledge support to the DAGAL Network fromthe People Programme (Marie Curie Actions) of the Eu-ropean Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement number PITN-GA-2011-289313. c (cid:13) , 000–000 n the morphology of bar dust lanes REFERENCES
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