Testing theories in barred spiral galaxies
TTESTING THEORIES IN BARRED SPIRAL GALAXIES
Eric E. Mart´ınez-Garc´ıa
Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, AP 70-264, Distrito Federal 04510,Mexico. [email protected]
ABSTRACT
According to one version of the recently proposed “manifold” theory that explains the originof spirals and rings in relation to chaotic orbits, galaxies with stronger bars should have a higherspiral arms pitch angle when compared to galaxies with weaker bars. A sub-sample of barred-spiral galaxies in the Ohio State University Bright Galaxy Survey, was used to analyze the spiralarms pitch angle. These were compared with bar strengths taken from the literature. It wasfound that the galaxies in which the spiral arms maintain a logarithmic shape for more than 70 ◦ seem to corroborate the predicted trend. Subject headings: galaxies: kinematics and dynamics — galaxies: spiral — galaxies: structure —galaxies: kinematics and dynamics
1. INTRODUCTION
Spiral arms in barred galaxies have been ex-plained in the past as density waves (e.g., Kor-chagin & Marochnik 1975) or spiral waves thatresult from the crowding of gas orbits (Huntley etal. 1978). Kaufmann & Contopoulos (1996) in-voked for the first time the need for chaotic orbitsas building blocks of spirals near the end of thebar. In the Kaufmann & Contopoulos (1996) mod-els, regular orbits dominate the main structure ofthe bar and the outermost portions of spiral arms.The inner portions of spiral arms are supported bychaotic orbits. Recently it has been proposed thatchaotic motion can support the spirals in barred-spiral systems. The new theory proposes that un-stable Lagrangian points ( L or L ) near the endof the bar are the sites where chaotic orbits areguided by invariant “manifolds”, and are the ori-gin of spirals and (inner and outer) rings (Voglis &Stavropoulos 2006a; Patsis 2006; Romero-G´omezet al. 2006; Voglis et al. 2006b,c; Romero-G´omezet al. 2007; Tsoutsis et al. 2008, 2009; Athanas-soula et al. 2009a; Harsoula & Kalapotharakos2009; Athanassoula et al. 2009b, 2010; Contopou-los & Harsoula 2011). In this scenario the spiraldynamics are coupled to the bar, and are driven by the manifolds. This approach has been studiedby two different groups of people.One of those groups (Romero-G´omez et al.2006, 2007; Athanassoula et al. 2009a,b, 2010),considers a continuous flow of orbits along themanifolds emanating from L or L . When spi-rals form, stars move away from the corotationin a radial movement (Athanassoula et al. 2010),and material is needed to replenish the mani-folds. One prediction of this “manifold theory”(or “Lyapunov tube model”), not accounted for inthe density wave scenario, is that stronger barsshould have more open spirals as compared toweaker bars, i.e., the spiral arms pitch angle should increase with bar strength (Athanassoulaet al. 2009b). This kind of correlation was previ-ously predicted by Schwarz (1984), although forgas arms driven by a bar perturbation.Another view of the “invariant manifold the-ory” (Voglis et al. 2006b,c; Tsoutsis et al. 2008,2009) considers the locus of all points with initialconditions at the unstable manifolds that reach alocal apocentric (or pericentric, see Harsoula et The angle between a tangent to the spiral arm at a certainpoint and a circle, whose center coincides with the galaxy’s,crossing the same point. a r X i v : . [ a s t r o - ph . C O ] O c t l. 2011) passage, i.e., the apsidal sections of themanifolds. In this scenario, there is no need forthe replenishment of material to obtain long-livedspirals (see, e.g., Efthymiopoulos, C. 2010). Bothviews of the “invariant manifold theory” predict atrailing spiral pattern for strong perturbations andsimilar pattern speeds for the bar and spiral, i.e.,Ω bar p = Ω spiral p . However, in the view of Voglis etal. (2006b,c) and Tsoutsis et al. (2008, 2009), the“azimuthal tilt” of the spiral response (Tsoutsiset al. 2009), i.e., the difference between the bar’smajor axis and the Lagrangian points L or L atthe moment of the onset of the spiral, determineshow open the spiral arms will be. In this case, thepitch angles are smaller than the ones predictedby Athanassoula et al. (2009b) and become evensmaller for pure bar models when the “azimuthaltilt” is not taken into account (C. Efthymiopoulos,private communication 2011).Patsis et al. (2010) describe one more dynami-cal mechanism that supports spiral arms throughstars in chaotic motion. They propose this mech-anism by describing the spiral arms of the barred-spiral NGC 1300. Together with the bar, thesespiral arms are inside the corotation and are notrelated to the presence of unstable Lagrangianpoints and the associated families of periodic or-bits. This alternative mechanism may be linked tosome range of pitch angles of spiral arms encoun-tered in barred-spiral systems.Do manifolds drive spiral dynamics in barredgalaxies? Or are the dynamics driven by thebar? The bar may drive the dynamics, affectingthe spiral amplitude locally, as reported by Saloet al. (2010) (see also Block et al. 2004) andpreviously discarded (or weakly corroborated) byother authors comparing bar strength to spiralarm strength (Buta et al. 2009; Durbala et al.2009; Seigar & James 1998). Bars driving thedynamics would imply an accordance with (lin-ear) density wave theory. These spirals may be acontinuation of the bar mode, or an independentmode coupled to the bar (e.g., Tagger et al. 1987;Masset & Tagger 1997). In the “Lyapunov tubemodel”, the strength of the bar affects the pitchangle of the spirals, but not its amplitude. Theamplitude of the spirals depends on how much ma-terial is trapped by the manifolds, although, theamplitude of the spirals should in general decreaseoutward (Athanassoula et al. 2010). Grosbøl et al. (2004) investigated the relation between theamplitude of the spirals with the pitch angle innon-barred and weakly barred galaxies.One prediction of the density wave theory (Hozumi2003, see §
2. GALAXY SAMPLE
The initial galaxy sample consists of 104 galax-ies classified as Fourier bars in Laurikainen etal. (2004). The data were acquired from theOhio State University Bright Galaxy Survey (OS-UBGS) (Eskridge et al. 2002). From this initialsample, it was found that only 84 objects presentspiral-like features. Nevertheless, not all the ob-jects are suitable for this kind of study due toasymmetries, e.g., short, faint, or ragged spiralarms, or prominent rings. The following criteriawere established in order to obtain a sample, in-cluding objects with a morphology candidate tobe explained by “chaotic” spirals.1. The spiral arms must remain logarithmic,i.e., with a constant pitch angle ( i ), at leastfor 50 ◦ in the azimuthal range, α . This wasverified with the “slope method” (see Sec-tion 4.1). The lower limit value of α was cho-sen according to Figure 4 in Athanassoula etal. (2009b), where the manifold loci remain Although the spiral arms may extend further in the diskwith a varying pitch angle, i.e., different slopes in a ln r versus θ map. ∼ ◦ . We consider thatthe manifold loci and the density maximumalong the spirals coincide. According to Pat-sis (2006), spirals supported by chaotic par-ticles may extend up to π/ α toward larger angles will be dis-cussed in Section 6.2. The object presents two spiral arms visuallyconnected to the bar.3. No prominent inner rings (near the bar’send) are present. Ring structures are con-nected to the bar on both sides. The pitchangle definition as applied in this investiga-tion only refers to spiral arms. A dependenceof the inner ring shape on bar strength hasbeen investigated by Grouchy et al. (2010).After applying these selection criteria, the finalsample consists of 27 barred spirals (see, e.g., Ta-ble 1).In order to use the bar strength values of Lau-rikainen et al. (2004, see Section 3), we adopt thesame deprojection parameters of those authors,i.e. the same values for position angle ( φ ) andminor-to-major axial ratios ( q = b/a ). To deter-mine these parameters, Laurikainen et al. (2004)fit ellipses to the outer isophotes on the disk. Theywere based on the OSUBGS B -band images thatare deeper than H -band images.To test the Athanassoula et al. (2009b, 2010)predictions regarding spiral arms pitch angles, weuse the NIR H -band since we are interested in“long”-lived structures rather than young stars,HII regions, or gas that would be present in opticaldata.
3. BAR STRENGTH An exception is NGC 5921 where a ring is present, but itdoes not dominate over the spiral features. With the exception of NGC 1300, for which we adopt φ =100 ◦ ±
14, and q = 0 . ± . The predicted trend in Athanassoula et al.’s(2009a) “manifold models” requires the strengthof the bar at the radius of the Lagrangian points L or L . It should be mentioned that for thesemodels the self-gravity of the spirals was not takeninto account. On the other hand, the addition ofthe spiral potential in Tsoutsis et al.’s (2009) mod-els shifts the positions of the Lagrangian points L or L both in the radial and azimuthal directions.The strength of the bar can be obtained fromthe Laurikainen et al. (2004) radial profiles of theperturbation strength. Laurikainen et al. (2004)used the gravitational torque method (Combes &Sanders 1981; Buta & Block 2001; Block et al.2002) taking care of the artificial bulge stretch (seealso Speltincx et al. 2008). The perturbationstrength is calculated as Q t ( r ) = (cid:16) ∂ Φ( r,θ ) ∂θ (cid:17) max r dΦ ( r )d r , (1)which represents the ratio between the maximumamplitude (over azimuth) of the tangential force,and the mean axisymmetric radial force derivedfrom the m = 0 component of the gravitationalpotential. The potential is inferred from the lumi-nous mass, and can be represented as (see Combes& Sanders 1981; Quillen et al. 1994):Φ( r, θ ) ≈ Φ ( r ) + (cid:88) m =2 , , Φ m ( r ) cos [ mθ ] . (2)The angle θ is given in the deprojected image, and θ = 0 along the bar major axis. For this investi-gation we assume that L = L = L . For realgalaxies, L may differ from L due to odd termsin the gravitational potential.We analyzed three cases in which the bar’sstrength is estimated in three different ways.1. In the first case, the bar’s strength is esti-mated at r = r L . The Lagrangian pointor corotation radius (Sellwood & Wilkinson1993), r L , was obtained from Buta & Zhang(2009), who applied the “potential-densityphase shift method” to the OSUBGS sam-ple. There have been significant discussionson the validity of this method. This is partlybecause Zhang & Buta (2007) found some3ases (e.g., NGC 4665) where r L /r bar <
1, i.e., corotation before the end of thebar. According to Contopoulos (1980), self-consistent bars are not possible to be mod-eled in this regime.One important difference between “mani-fold” models (Romero-G´omez et al. 2006;Voglis et al. 2006b,c; Athanassoula et al.2009a) and the “potential-density phaseshift method” is that Zhang & Buta (2007)and Buta & Zhang (2009) considered po-tentials varying considerably with time.The time-independent (rigid) potentials ofthe “manifold” models generate “passive”chaotic orbit responses. Although the Zhang& Buta (2007) models involve chaos in theindividual stars’ trajectories, “collective dis-sipation” makes possible the existence ofcoherent structures (e.g., spiral arms). Wedefine “ r BZ09 ” as the corotation radius ob-tained from Buta & Zhang (2009). Table 1shows the Q t ( r = r BZ09 ) values for the 27OSUBGS barred galaxies.2. In the second case, we estimate the bar’sstrength at a distance r L = 1 . r bar . Accord-ing to various studies (Athanassoula 1992;Elmegreen 1996; Aguerri et al. 2003), the ex-pected range for the bar length lies between r bar = r L / . r bar = r L / .
4. Elmegreen(1996) and Aguerri et al. (2003) also discussobjects where r bar = r L / .
7. A mean valueof r bar = r L / . r L = 1 . r bar or r L = 1 . r bar , in-stead of r L = 1 . r bar , could be much largerthan deprojecting a galaxy within 10% er-ror in the projection angles. For this studythe bar length, r bar , was taken from Lau-rikainen et al. (2004). In Table 1 we showthe Q t ( r = 1 . r bar ) values adopted for thisinvestigation.3. The third case involves the maximum of theradial Q t ( r ) profiles or Q g . These were tab-ulated in Laurikainen et al. (2004).The adopted Q t ( r ) values from Laurikainen etal. (2004) were computed assuming a constant M/L ratio throughout the disk and an empiricalcorrelation for the vertical scale-height ( h z ). Also, it is assumed that dark matter has little impacton the bar strength. For the Q t ( r ) error calcula-tion shown in Table 1, the Q g error of Laurikainenet al. (2004) was summed in quadrature with theerror inherent to digitization of the Q t ( r ) plotsand the r L (Buta & Zhang 2009) errors for the Q t ( r = r BZ09 ) values.A technique for separating the gravitationaltorques of bars and spirals was developed by Butaet al. (2003, 2005). This technique separates thebar+disk image to obtain the bar strength Q b (atthe respective maximum of Q t ( r )) unaffected bythe spiral gravitational influence. Nevertheless,for the majority of barred galaxies in the OSUBGSsample, the bar strength, Q b , dominates over thespiral arm strength Q s (Durbala et al. 2009). Also,the correction of the spiral arms does not affect thetendencies for Q g in the Hubble sequence (Lau-rikainen et al. 2007). In either case, for this inves-tigation it is assumed that Q g ∼ Q b , and that the Q t ( r ) values are affected by the spirals within theerrors.
4. PITCH ANGLES
Spiral arms pitch angles have been measuredin the literature with different methods. Dan-ver (1942) measured the spiral arms on photo-graphic plates. Kennicutt (1981) measured the spi-ral shapes using the intensity and HII region dis-tributions. Ma et al. (1999) fit the shapes of spiralarms directly on the images. Fourier decomposi-tion methods had also been used (e.g., Considere& Athanassoula 1988; Puerari & Dottori 1992;Saraiva Schroeder et al. 1994; Seigar et al. 2006),yielding similar results as other methods (Con-sidere & Athanassoula 1988; Puerari & Dottori1992).
This method is similar to the one used in Seigar& James (1998). It is assumed that the arms canbe represented by logarithmic spirals, which im-plies a constant pitch angle. Although, variablepitch angles may be a better and more adequate The ADS’s data extraction applet,
DEXTER (Demleitneret al. 2001), was used for this purpose. GC 150 arm II arm I
NGC 210 arm IIarm I
NGC 289 arm IIarm I
NGC 578 arm IIarm I
NGC 864 arm IIarm I
NGC 1073 arm Iarm II
NGC 1187 arm II arm I
NGC 1300 arm II arm I
Fig. 1.— Deprojected H -band images of NGC 150, NGC 210, NGC 289, NGC 578, NGC 864, NGC 1073,NGC 1187, and NGC 1300. The display is in a logarithmic scale. The analyzed arm segments for the “slopemethod” are shown in the figures (solid lines), together with the annulus adopted for the “Fourier method”(dashed lines). 5 GC 1703 arm IIarm I
NGC 1832 arm IIarm I
NGC 3261 arm IIarm I
NGC 3513 arm IIarm I
NGC 3583 arm II arm I
NGC 3686 arm II arm I
NGC 4145 arm II arm I
NGC 3059 arm IIarm I
Fig. 2.— Deprojected H -band images of NGC 1703, NGC 1832, NGC 3059, NGC 3261, NGC 3513, NGC3583, NGC 3686, and NGC 4145. The display is in a logarithmic scale.6 GC 4303 arm IIarm I
NGC 4902 arm IIarm I
NGC 4930 arm IIarm I
NGC 4995 arm IIarm I
NGC 5483 arm IIarm I
NGC 5921 arm IIarm I
NGC 6221 arm II arm I
NGC 6300 arm II arm I
Fig. 3.— Deprojected H -band images of NGC 4303, NGC 4902, NGC 4930, NGC 4995, NGC 5483, NGC5921, NGC 6221, and NGC 6300. The display is in a logarithmic scale.7 GC 6384 arm IIarm I
NGC 7479 arm II arm I
IC 5325 arm IIarm I
Fig. 4.— Deprojected H -band images of NGC 6384, NGC 7479, and IC 5325. The display is in a logarithmicscale. Table 1Perturbation Strengths
Galaxy Q t ( r = r BZ09 ) Q t ( r = 1 . r bar ) Galaxy Q t ( r = r BZ09 ) Q t ( r = 1 . r bar )NGC 150 0.302 ± .
1% 0.352 ± .
1% NGC 3686 0.075 ± .
0% 0.152 ± . ± .
0% 0.058 ± .
4% NGC 4145 0.119 ± .
5% 0.122 ± . ± .
8% 0.099 ± .
1% NGC 4303 0.149 ± .
5% 0.251 ± . ± .
3% 0.040 ± .
5% NGC 4902 0.068 ± .
2% 0.128 ± . ± .
9% 0.236 ± .
7% NGC 4930 0.138 ± .
9% 0.099 ± . ± .
4% 0.386 ± .
4% NGC 4995 0.280 ± .
8% 0.263 ± . ± .
5% 0.206 ± .
9% NGC 5483 0.098 ± .
1% 0.111 ± . ± .
4% 0.277 ± .
0% NGC 5921 0.384 ± .
0% 0.329 ± . ± .
0% 0.058 ± .
6% NGC 6221 0.309 ± .
2% 0.237 ± . ± .
8% 0.149 ± .
1% NGC 6300 0.158 ± .
3% 0.080 ± . ± .
3% 0.268 ± .
9% NGC 6384 0.136 ± .
7% 0.048 ± . ± .
8% 0.117 ± .
7% NGC 7479 0.516 ± .
6% 0.240 ± . ± .
0% 0.210 ± .
9% IC 5325 0.211 ± .
5% 0.102 ± . ± .
0% 0.207 ± . Note.—
Columns (1) and (4): object name. Columns (2) and (5): perturbation strength (see Section 3)from Laurikainen et al. (2004), at corotation radius from Buta & Zhang (2009). Columns (3) and (6):perturbation strength from Laurikainen et al. (2004), at radius r = 1 . r bar (see Section 3). H -band data), the centers of the objects were deter-mined by fitting ellipses to the central isophotesclose to the bar region. Afterward, the spiralarms were “unwrapped” by plotting them in a ln r versus θ map (e.g., Iye et al. 1982; Elmegreen etal. 1992; Grosbøl et al. 2004). Under this geomet-ric transformation, logarithmic spirals appear asstraight lines. The pitch angle, i , is related to theslope of the line, s , ascot i = k | s | , (3)wherein k is a constant due to “pixelation” andunit conversion.Two arm segments were selected closest to thebar’s end with the condition that the slope, s , wasmaintained nearly constant along them (see Fig-ures 1-4). Due to this “slope restriction”, in manycases the critical segment including the part of thearms attached to the bar, was not able to be con-sidered. The slope ( s ) is determined by first se-lecting for each column in the arm segment thepixels with a maximum in intensity (see as an ex-ample Figure 5, for the case of NGC 1832). Aleast-squares fit is then obtained for the result-ing pixels. As already mentioned, these fits weredone in the H -band aiming to trace PopulationII stars. Young stars and clusters can contributelocally up to 20%-30% of the observed radiationin the NIR (e.g., Rix & Rieke 1993; Rhoads 1998;Patsis et al. 2001; Grosbøl & Dottori 2008). Howthese young objects affect the pitch angles’ mea-surements depends on the star formation condi-tions and young stars kinematics. For this investi-gation, it is assumed that young stars and clusters This was done with the ELLIPSE task in IRAF. IRAF isdistributed by the National Optical Astronomy Observa-tory, which is operated by the Association of Universitiesfor Research in Astronomy, Inc., under cooperative agree-ment with the National Science Foundation. Basically converts θ -axis pixels to radians, and determinesthe equivalence between pixels in the ln r axis and physicalunits of an image. affect the spiral arms pitch angles within the errorsinvolved in the methods applied.As previously mentioned, all the objects withinner rings, asymmetries, unclear, or “logarithmi-cally short” arms were discarded from the analysis.For the remaining 27 objects, the arm segments (Ior II) best determined and with the clearest spiralstructure were also identified. These are markedwith an asterisk (*) in Table 2, together with theadopted radial ranges, ∆ r , tabulated from inner-most ( r ) to outermost radius ( r ), and azimuthalranges, α . Azimuthal ranges are obtained by theequation α = cot i H ln (cid:18) rr (cid:19) (4)and are displayed graphically in Figures 1-4 withregions delimited by solid lines. These values donot indicate the end of the spirals, since spiralarms may extend further with a variable pitchangle (Ringermacher & Mead 2009). Although ifthe extensions of the spirals have reduced ampli-tudes with respect to the logarithmic part, theirphases will be difficult to determine, as will theirpitch angles. The estimations are done indepen-dently of the amplitude (strength) of the spiralitself. Grosbøl et al. (2004, their Figure 8) founda tendency between the amplitude of the m = 2spiral and pitch angles in SA and SAB galaxies.Figure 6 shows a histogram of the maximumazimuthal range distribution (either arm segmentI or II) for each object presented in Table 2. Errors introduced by deprojection parameters( φ and q ) translate into different slopes or devia-tions of a straight line in the ln r versus θ plots.For each object, five deprojected frames were ob-tained to better account for these errors. The im-ages were deprojected with the parameters φ, q ; φ + sd, q ; φ − sd, q ; φ, q + sd ; and φ, q − sd , where sd is the respective standard deviation. Pitch an-gle values were measured and compared to the casewhen φ and q were used as the deprojection pa-rameters (i.e., when sd = 0). The cases with thehighest (positive) or lowest (negative) discrepan-cies were adopted to account for the + σ and − σ errors, respectively (see Table 2).9 able 2“Slope Method” Derived Parameters Galaxy and Segment i H (deg) ∆ r (arcsec) α (deg) Segment i H (deg) ∆ r (arcsec) α (deg)NGC 150 Arm I* 24 . +5 . − . (34.7-55.4) 58 ± . +7 . − . (41.8-51.4) 18 ± . +0 . − . (64.8-107.6) 88 ± . +1 . − . (63.6-105.6) 51 ± . +2 . − . (22.8-28.2) 41 ± . +4 . − . (22.8-30.8) 50 ± . +0 . − . (20.8-50.3) 115 ± . +1 . − . (20.8-50.3) 110 ± . +0 . − . (33.0-68.5) 93 ± . +4 . − . (34.8-53.1) 44 ± . +7 . − . (37.8-55.3) 39 ±
10 Arm II 12 . +4 . − . (48.7-70.0) 93 ± . +3 . − . (35.7-48.8) 25 ± . +3 . − . (25.2-55.6) 82 ± . +7 . − . (84.8-116.9) 47 ±
14 Arm II 18 . +6 . − . (74.3-100.5) 51 ± . +0 . − . (15.6-28.1) 112 ± . +0 . − . (16.8-28.1) 64 ± . +1 . − . (20.0-45.8) 126 ± . +6 . − . (20.0-27.8) 37 ± . +1 . − . (26.8-54.0) 66 ± . +7 . − . (23.6-33.9) 31 ± . +15 . − . (27.4-34.2) 41 ±
14 Arm II 11 . +3 . − . (30.0-36.1) 50 ± . +1 . − . (26.1-72.6) 91 ± . +1 . − . (16.3-72.6) 100 ± . +0 . − . (34.7-59.3) 83 ± . +2 . − . (34.7-68.8) 87 ± . +0 . − . (20.2-46.2) 104 ± . +6 . − . (20.2-43.7) 19 ± . +2 . − . (50.5-87.9) 104 ± . +3 . − . (50.5-87.9) 100 ± . +1 . − . (29.2-60.5) 53 ± . +1 . − . (29.2-57.3) 40 ± . +6 . − . (21.4-30.0) 98 ±
26 Arm II 23 . +5 . − . (19.5-34.7) 77 ± . +0 . − . (38.9-58.2) 68 ± . +5 . − . (36.2-53.1) 6 ± . +3 . − . (22.6-34.3) 22 ± . +6 . − . (17.3-37.7) 56 ± . +0 . − . (14.8-39.4) 100 ± . +1 . − . (14.8-34.7) 79 ± . +0 . − . (61.0-95.3) 89 ± . +0 . − . (69.5-91.9) 54 ± . +1 . − . (39.0-57.6) 63 ± . +2 . − . (41.2-56.5) 50 ± . +0 . − . (42.2-78.9) 116 ± . +0 . − . (42.2-78.9) 108 ± . +0 . − . (31.4-84.9) 125 ± . +0 . − . (40.1-57.2) 63 ± . +2 . − . (56.1-95.8) 41 ± . +2 . − . (60.4-95.8) 62 ± . +2 . − . (11.4-23.0) 35 ± . +3 . − . (14.3-27.4) 100 ± Note.—
Column (1): object and spiral arm segment, see Figures 1-4. Columns (2) and (6): H -band pitch angles, i H , in degrees. Columns (3) and (7): radial ranges, r to r , in arcsec. Columns (4) and (8): azimuthal ranges , α = cot i H ln (cid:16) rr (cid:17) , in degrees. Column (5): spiral arm segment (see Figures 1-4) for the same object as Column (1). .2. “Fourier Method” Figure 7(a) plots the pitch angles in arm seg-ment I versus arm segment II for each object asobtained with the “slope method”. Figure 7(b)shows a histogram of the absolute value differ-ence between arm segments I and II. As shown inthe figures, some scatter is present when analyz-ing spiral arm segments within the same galaxies.Since we are interested in comparing single valuesof pitch angles for each object, we need a methodthat provides the “dominant mode” for the pitchangle measurement. The “Fourier method” is per-fectly adequate for this purpose.In this method, it is again assumed that thearms can be represented by logarithmic spirals. The Fourier amplitudes for each component aregiven by A ( m, p ) = (cid:80) Ii =1 (cid:80) Jj =1 I ij (ln r, θ )exp[ − i ( mθ + p ln r )] (cid:80) Ii =1 (cid:80) Jj =1 I ij (ln r, θ ) , (5)where r and θ are the polar coordinates, I ij is theintensity at coordinates ln r , θ , m is the numberof spiral arms (or modes), and p is related to thespiral arms pitch angle ( i H ) bytan i H = − m/p max , (6)where p max corresponds to the maximum of A ( m, p ) and m = 0 , , , , . . . , i.e., the maximumof the Fourier spectrum (see, e.g., Puerari &Dottori 1992; Saraiva Schroeder et al. 1994) formode m . Most of the analyzed objects present m = 2 as the dominant mode for the spiral armsin the H -band (see Table 3), so it was adopted forthis investigation. The exceptions are NGC 3261and NGC 4930 in which m = 1 dominates and wasused instead. For NGC 1300 and NGC 7479, otherFourier modes ( m ) compete with the m = 2 modebecause of the spiral arm segments with variablepitch angles. The pitch angles corresponding tothe m = 2 Fourier mode were adopted for theseobjects in the subsequent analysis.For the galaxies of the sample, it has been real-ized that the presence of foreground stars does not However, a Fourier analysis can be done without the as-sumption of a constant pitch angle. affect the value of the pitch angle in general. Nev-ertheless, caution must be taken when foregroundstars (or objects) compete in extension with spiralarms (see, e.g., annulus for NGC 864 in Figure 1).In these cases the need for masks is required.Objects were deprojected as explained in Sec-tion 4.1. Radial ranges were selected to coverthe spiral segments previously analyzed with the“slope method”. The azimuthal coverage is 2 π ra-dians. The analyzed annuli are shown graphicallyin Figures 1-4 (dashed lines). These are the re-gions where the Fourier analysis was performed.Table 3 shows the results for the Fourier pitchangle values, which agree with the “slope method”within a ∼ ◦ difference (this corresponds to 1 σ in Figure 7) in the majority of the objects. NGC5921 and NGC 6221 present the largest differences( ∼ ◦ ). For two objects, NGC 4995 and IC 5325,the computed pitch angles are close to ∼ ◦ . Thisis due to the fact that the spiral arms have a lowsurface brightness (as compared to the disk) andthe bar component is difficult to isolate in the ana-lyzed annulus. The “slope method”, for the “best-defined arm”, was used instead for these two ob-jects in the subsequent analysis. Errors were determined in the same way as inthe “slope method”. These were added in quadra-ture with the error intrinsic to the method. A pro-gram was built that computes the two-dimensionalfast Fourier transform in Equation 5. The outputof this program is a 128 × m, p ) matrix. Thetwo closest values near p max were used to approx-imate the error of the method.
5. COMMENTS FOR SOME OBJECTS
NGC 210. “Skinny” spiral arms compared withthe bar.
NGC 289.
The outer spiral arms have a greaterpitch angle ( i H ∼ ◦ ) as compared to the innerones ( i H ∼ ◦ ; “slope method”). NGC 578.
Two symmetric spiral arms near thebar.11 zimuthal range (degrees) F r equen cy
40 60 80 100 120 140
Fig. 6.— Histogram of the maximum azimuthal ranges (either arm segment I or II, see Table 2).
Difference between arm I & arm II (degrees) F r equen cy (a) (b) Fig. 7.— (a) Pitch angles (in deg) for arm segments I ( x -axis) vs. arm segments II ( y -axis) for each object.Dotted line: one-to-one relation. (b) Histogram of the absolute value difference between arm segments I andII, obtained for each object with the “slope method”. The standard deviation around the zero differencevalue is 16 . ◦
3. 12ig. 5.— Plot of ln r vs. θ for arm segment I inNGC 1832 ( H -band). Crosses indicate the pointswhere a maximum intensity was found for eachcolumn for the corresponding section in the “un-wrapped” image. The continuous line indicatesthe least-squares fit. NGC 1073.
Spiral arms difficult to trace (lowsignal-to-noise ratio).
NGC 1187.
The two arm features analyzed arevisually attached to the bar. A third arm feature,not visually attached to the bar, is present. Theradial ranges for the “Fourier method” were mod-ified with respect to the “slope method” to allowa better signal-to-noise ratio in the ln r versus θ map. NGC 1300.
Two well-defined logarithmic spi-ral arms, although short in azimuthal range. Theadopted deprojection parameters were changed ascompared to the ones of Laurikainen et al. (2004).This was done because the values provided in Lau-rikainen et al. (2004) do not agree with the outerisophotes of the OSUBGS images. An average be-tween Hyperleda (Paturel et al. 2003), RC3 (deVaucouleurs et al. 1991), and a visual determina-tion of the outer isophotes was used.The deprojection parameters from Lindblad etal. (1997) were also tried for the pitch angle mea-surements. These parameters, φ = 87 ◦ ± ◦ and q = 0 . ± .
05, are based on H I data and are inde-pendent of kinematical or dynamical criteria (seealso Kalapotharakos et al. 2010). Using these pa-rameters, spiral arms are difficult to follow in aln r versus θ map (assuming a logarithmic geome-try). For arm region I, a pitch angle of 21 . ◦ ± ◦ was obtained. Arm region II was not possible tomeasure via the “slope” method. The pitch anglesobtained by applying the “Fourier” method led tovalues with a contrary sign to the one expected,i.e., an inverse sense of winding for the spiral arms. NGC 1703.
Difficult to analyze the spiral armsin the inner regions due to few pixels in a ln r versus θ map. NGC 1832.
The bar region is distorted (notstraight).
NGC 3059. “Hard to follow” logarithmic shapefor the spiral arms.
NGC 3583.
Two symmetric spiral arms can beappreciated in the outer disk. The region close tothe bar presents a structure similar to a ring or atight spiral arm.
NGC 4145.
Double bar system?
NGC 4303.
This object presents three mainspiral arms.
NGC 4902.
Three spiral regions are present in13his object.
NGC 5921.
This object presents an inner ringand spiral features.
NGC 6300.
This object presents spiral featuresand apparently a ring feature.
NGC 6384.
Spiral arms with bifurcations.
NGC 7479.
In general the spiral arms for thisobject do not present a clear logarithmic geometry.
IC 5325.
This object presents four well definedsegments of spiral arms. Only the ones near thebar’s end were analyzed.
6. RESULTS AND DISCUSSION
Figure 8 shows the results for the pitch an-gle, i H (Fourier method, except for NGC 4995,and IC 5325, see Section 4.2), versus perturba-tion strengths, Q t ( r = r BZ09 ). A first inspec-tion of the data, where the “azimuthal range” is α > ◦ , shows considerable scatter around thepredicted correlation for models A (Ferrers 1877,bar potential) and D (Dehnen 2000, bar potential)in Athanassoula et al. (2009b). However, if the α criterion is changed to logarithmic spiral seg-ments that extend up to α > ◦ , α > ◦ , and α > ◦ , the scatter is reduced. The reducedPearson’s chi-square, χ /n , obtained as χ = n (cid:88) k =1 ( i k − i p ) i p , (7)where i k is the k th Fourier-measured pitch angleand i p is the predicted pitch angle value for modelsA and D in Athanassoula et al. (2009b), gives theresults 3.10, 1.55, 1.83, and 2.00 for α > ◦ ( n =27), α > ◦ ( n = 17), α > ◦ ( n = 13), and α > ◦ ( n = 5), respectively.Figures 9 and 10 show the results for the cases Q t ( r = 1 . r bar ) and Q g , respectively. For α > ◦ , This is obtained with the “slope method” via Equation 4.It is the “maximum” azimuthal range that is taken intoaccount, i.e., the greatest value of α for either arm segmentI or II. A third model with a Barbanis & Woltjer (1967) bar po-tential (BW model) was considered in Athanassoula etal. (2009a,b). This model agrees with model D up to Q t ( r = r L ) ∼ .
2, and deviates toward higher pitch an-gles afterward, up to ∼ ◦ at Q t ( r = r L ) ∼ . reduced Pearson’s chi-square values obtained as χ = n (cid:88) k =1 ( Q k − Q p ) Q p , (8)where Q k is the k th bar strength value correspond-ing to the k th Fourier-measured pitch angle and Q p is the predicted bar strength value for models Aand D in Athanassoula et al. (2009b), yield the re-sults 0.049, 0.075, and 0.084 for the Q t ( r = r BZ09 ), Q t ( r = 1 . r bar ), and Q g plots, respectively.According to this result, the best concordancewith the Athanassoula et al. (2009b) model is ob-tained by comparing the pitch angles with Q t ( r )given at Buta & Zhang (2009) bar corotation radii( r = r BZ09 ). This last point is not discussedin Athanassoula et al. (2009b).One important aspect in the Athanassoula etal. (2009b) prediction is that the self-gravity ofthe spirals was not taken into account. The poten-tial created by the “confined” chaotic orbits is ne-glected. Contrarily, Tsoutsis et al. (2009) empha-size the contribution of the spiral part for studyingthe dynamics of the “chaotic” spirals. Also, realis-tic bar potentials are hard to model. If many dif-ferent realistic potentials are used, the predictedcorrelations may become broader (Athanassoulaet al. 2009b, 2010). This may explain in Fig-ure 8 the tendency of the points (squares andcircles) to be above the predicted correlation for Q t ( r = r BZ09 ) < . The modal approach explains the density wavephenomena as generated by intrinsic mechanismsin the disk (Bertin et al. 1989a,b; Bertin & Lin1996). Normal modes of oscillation generate spon-taneously and evolve according to the physical anddynamic properties of the system. Three physi-cal properties determine the morphology in diskgalaxies: the disk mass, the gas content, and thestellar velocity dispersion. When the disk mass is“high”, bar structures are generated as oscillatingmodes of the system. The modal theory considersbars and spirals equally, i.e., as normal modes ofoscillation in the disk.Based on the dispersion relation, linear densitywave theory predicts (Hozumi 2003) that the pitch14 able 3“Fourier Method” Derived Parameters
Galaxy m m m m m m m m m m m m i H (deg) i B (deg) ∆ r (arcsec)NGC 150 0.416 1.000 0.344 0.213 0.395 0.167 27 . +0 . − . . +1 . − . (34.7-55.4)NGC 210 0.525 1.000 0.189 0.478 0.255 0.220 16 . +0 . − . . +0 . − . (63.6-107.6)NGC 289 0.103 1.000 0.309 0.213 0.091 0.109 19 . +0 . − . . +1 . − . (22.8-30.8)NGC 578 0.656 1.000 0.289 0.255 0.080 0.116 24 . +1 . − . . +0 . − . (20.8-50.3)NGC 864 0.721 1.000 0.462 0.361 0.307 0.325 20 . +1 . − . . +0 . − . (33.0-68.5)NGC 1073 0.778 1.000 0.515 0.506 0.307 0.349 34 . +1 . − . . +1 . − . (37.8-70.0)NGC 1187 0.668 1.000 0.460 0.436 0.655 0.239 21 . +1 . − . . +1 . − . (29.0-59.4)NGC 1300 0.617 1.000 1.001 0.854 0.399 0.330 11 . +15 . − . . +7 . − . (74.3-116.9)NGC 1703 0.245 1.000 0.255 0.371 0.161 0.196 18 . +1 . − . . +1 . − . (15.6-28.1)NGC 1832 0.242 1.000 0.492 0.399 0.100 0.128 25 . +1 . − . . +1 . − . (20.0-45.8)NGC 3059 0.728 1.000 0.371 0.475 0.424 0.694 27 . +3 . − . . +0 . − . (23.6-54.0)NGC 3261 1.043 1.000 0.246 0.226 0.259 0.322 9 . +13 . − . . +20 . − . (27.4-36.1)NGC 3513 0.481 1.000 0.229 0.472 0.141 0.220 25 . +1 . − . . +1 . − . (27.8-72.6)NGC 3583 0.446 1.000 0.470 0.225 0.300 0.146 24 . +2 . − . . +2 . − . (34.7-68.8)NGC 3686 0.450 1.000 0.886 0.271 0.276 0.493 14 . +0 . − . . +0 . − . (20.2-46.2)NGC 4145 0.448 1.000 0.250 0.348 0.240 0.275 23 . +1 . − . . +0 . − . (50.5-87.9)NGC 4303 0.260 1.000 0.174 0.294 0.149 0.186 42 . +2 . − . . +1 . − . (29.2-60.5)NGC 4902 0.467 1.000 0.210 0.665 0.277 0.401 20 . +2 . − . . +3 . − . (19.5-34.7)NGC 4930 1.081 1.000 0.550 0.272 0.181 0.223 30 . +2 . − . . +0 . − . (36.2-58.2)NGC 4995 0.898 1.000 0.356 0.595 0.234 0.204 90 . +5 . − . . +5 . − . (22.5-37.7)NGC 5483 0.163 1.000 0.134 0.439 0.158 0.280 26 . +1 . − . . +1 . − . (14.8-39.4)NGC 5921 0.188 1.000 0.454 0.524 0.152 0.313 30 . +4 . − . . +2 . − . (61.0-95.3)NGC 6221 0.602 1.000 0.315 0.413 0.115 0.235 35 . +4 . − . . +1 . − . (39.0-57.6)NGC 6300 0.199 1.000 0.201 0.330 0.157 0.283 23 . +0 . − . . +0 . − . (42.2-78.9)NGC 6384 0.501 1.000 0.574 0.312 0.385 0.273 26 . +1 . − . . +0 . − . (31.4-84.9)NGC 7479 1.073 1.000 0.656 1.200 0.610 0.473 26 . +3 . − . . +1 . − . (56.1-95.8)IC 5325 0.367 1.000 0.426 0.176 0.200 0.090 69 . +3 . − . . +4 . − . (11.4-27.4) Note.—
Column (1): galaxy name. Column (2): ratio between the maximum amplitudes of Fourier modes m = 1 and m = 2, in the H -band. Columns (3), (4), (5), (6) and (7): ratio between the maximumamplitudes of the respective Fourier modes, in the H -band. Column (8): H -band (see Section 4.2) pitchangles, in degrees. Column (9): B -band (see Section 6.1) pitch angles, in degrees. Column (10): radialranges, r to r , in arcsec. i H in deg vs. perturba-tion strength Q t ( r = r BZ09 ) for the 27 galaxiesselected for analysis (see Section 2). Lagrangianradius, r L = r BZ09 , from Buta & Zhang (2009).The dashed line corresponds to the predicted cor-relation for models A and D in Athanassoula etal. (2009b). Data are separated by α > ◦ (allpoints), α > ◦ (triangles), α > ◦ (squares),and α > ◦ (circles). Fig. 9.— Same as Figure 8 for perturbationstrength Q t ( r = 1 . r bar ).Fig. 10.— Same as Figure 8 for perturbationstrength Q g .16ngle should increase with increasing velocity dis-persion, or that tan i ∝ c r / Σ , (9)where i is the arms pitch angle, c r is the radialvelocity dispersion, and Σ is the surface densityof the disk. Spiral structure shows different mor-phologies when observed in optical versus NIRbands (Block & Wainscoat 1991; Grosbøl & Patsis1998). NIR bands can trace both the old popula-tions of bar and spiral arms, assuming that redyoung stars do not contribute globally to the ob-served radiation (Rix & Rieke 1993). Also, olderpopulations have a higher velocity dispersion com-pared to younger ones (Barbanis & Woltjer 1967;Wielen 1977; Nordstr¨om et al. 2004; Binney &Tremaine 2008, section 8.4). For most galaxies atthe arm location, we have that ∼
98% (by mass)of the stars belong to evolved populations (seeGonzalez & Graham 1996, and references therein).Nevertheless, young stars contribute to most of thelight in optical wavelengths. According to this,NIR images of spiral perturbations should presenthigher pitch angles compared to optical ones. Az-imuthal age (color) gradients (e.g., Gonzalez& Graham 1996; Mart´ınez-Garc´ıa et al. 2009a,b;Mart´ınez-Garc´ıa & Gonz´alez-L´opezlira 2011) mayalso affect the pitch angles observed in the op-tical versus NIR bands, but these are very diffi-cult to trace by just comparing the light distri-butions in two bands (Gonzalez & Graham 1996;Seigar & James 1998). Besides, azimuthal gradi-ents are not located continuously along the spi-ral arms but in specific regions (Gonzalez & Gra-ham 1996; Mart´ınez-Garc´ıa et al. 2009a; Mart´ınez-Garc´ıa & Gonz´alez-L´opezlira 2011).From Equation 9, taking into account thatyoung and old stars are similarly affected by thegravitational potential of the disk (which dependson the surface density), we obtain i B = arctan (cid:40)(cid:18) c r B c r H (cid:19) tan i H (cid:41) , (10)where i B is the B -band pitch angle, i H is the H -band pitch angle, c r B is the radial velocity disper-sion of young stars, and c r H is the radial velocitydispersion of old stars.In the case of the invariant manifold theory, where chaotic orbits are “confined” in the spirallocus, no difference between pitch angles of spi-ral arms traced in different wavelengths is pre-dicted (Athanassoula et al. 2010).Seigar et al. (2006) found a nearly 1:1 correla-tion between pitch angle measurements in the B and H bands, for 57 galaxies in the OSUBGS (Es-kridge et al. 2002) sample. Nevertheless, based onthe sample of five non-barred and weakly barredspirals, Grosbøl & Patsis (1998) notice that themain two-armed spiral is tighter when measuredin bluer colors. For the barred-spirals data pre-sented in this investigation, we measured the pitchangles in the B -band images for the same ob-jects analyzed in the H -band from the OSUBGSsample, applying the “Fourier” method. The B -band images were registered to the H -band im-ages, so the high-resolution data ( B -band) weredegraded to the low-resolution data ( H -band inthis case). Annulus regions were selected in thesame positions as the H -band, and the pitch an-gles were measured identically with the methoddescribed in Section 4.2. The results are shown in Figure 11 (and Table 3) where a tendency of ∼
30% of the points toward higher H -band pitchangles is observed. Although, if we apply the sameazimuthal range ( α ) criteria as in Figures 8, 9,and 10, we can notice that ∼
80% of the α > ◦ data lie very close to the 1:1 relation as expected(independently of α ) from Athanassoula et al.(2010). A comparison of the different treatments of themanifolds viewed as apsidal sections (Voglis et al.2006b,c; Tsoutsis et al. 2008, 2009), or as tubesthat guide chaotic orbits (Romero-G´omez et al.2006, 2007; Athanassoula et al. 2009a,b, 2010), re-quires a different analysis involving separating spi-rals and bars. This will be covered in a subsequentpublication. The same dominant modes, m = 1 or m = 2 (see Sec-tion 4.2), as measured in the H -band were adopted for the B -band pitch angle measurements. NGC 4995 and IC 5325 were excluded from this analysis(see the last paragraph in Section 4.2). H -band pitch angle vs. B -band pitchangle, obtained via the “Fourier” method. Dottedline: one-to-one relation; short-dashed line: den-sity wave theory prediction from Equation 10, as-suming c r H ∼ √ c r B ; long-dashed line: densitywave theory prediction from Equation 10, assum-ing c r H ∼ c r B .
7. CONCLUSIONS
The results of this investigation show the fol-lowing.1. Although the adopted deprojection param-eters may introduce some biases (see, e.g.,Barnes & Sellwood 2003), a trend can beobserved where some strong barred spiralshave more open spiral arms when comparedto galaxies with weaker bars. This kindof trend was also discussed in Block et al.(2004), where a similar behavior was found.The correlation predicted by the manifoldmodels of Romero-G´omez et al. (2006, 2007)and Athanassoula et al. (2009a,b, 2010) isbetter reproduced by observations on twoconditions.(a) The corotation values obtained withthe “potential-density phase shift method” (Buta& Zhang 2009) are adopted.(b) The spirals logarithmic geometry ismaintained for large azimuthal ranges, α > ◦ .2. The ∼
60% of the 27 galaxies on the ana-lyzed sample seem to reproduce the investi-gated correlation.3. The pitch angles calculated via the “Fouriermethod” in the B (young stars) and the H (mostly old stars) bands yield similar valuesfor ∼
80% of the objects where the azimuthalrange, α , is greater than 70 ◦ . This kind ofbehavior is expected in the “Lyapunov tubemodel” (Athanassoula et al. 2010), althoughno restriction on the azimuthal range wasgiven by the authors.4. Other possible mechanisms to generate spi-ral features in barred galaxies, such as bar-driven spirals (e.g. Salo et al. 2010), modelswhere the Lagrangian points of the systemare specified by both bar and spirals (e.g.,Tsoutsis et al. 2009), or chaotic spirals in-side corotation (thus not related with thepresence of unstable Lagrangian points; Pat-sis et al. 2010), cannot be excluded by thepresent investigation.18 cknowledgments I am grateful to the anonymous referee for manyimportant remarks and helpful comments thathave improved this paper. I acknowledge post-doctoral financial support from UNAM (DGAPA),M´exico. I thank Christos Efthymiopoulos for clar-ifying my inquiries about spiral arms driven by“manifolds”. This work made use of data fromthe Ohio State University Bright Spiral GalaxySurvey, which was funded by grants AST-9217716and AST-9617006 from the United States NationalScience Foundation, with additional support fromthe Ohio State University.
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