warped disks in spiral galaxies

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New Astronomy 11 (2006) 293–305

Warped disks in spiral galaxies

H.B. Ann *, J.-C. Park

Division of Science Education, Pusan National University, Jangjeon-Dong Gumjeong-Gu, Busan 609-735, Republic of Korea

Received 21 June 2005; received in revised form 26 August 2005; accepted 26 August 2005Available online 27 September 2005Communicated by G.F. Gilmore

Abstract

We have analyzed the disk morphologies of 325 edge-on galaxies to derive the warp statistics in spiral galaxies using Digital Sky Sur-vey. Galaxies were included in our study if their isophotal diameter (D25) satisfied logD25 > 1, and if their major-to-minor axis ratio wasin the range a/b > 9.5. We found that 236 out of the 325 sample galaxies (73%) had warps: 165 S-shaped (51%) and 71 U-shaped (22%).We additionally found that the warp properties (warp angles, warp radius, and warp asymmetry) as well as the warp frequency did notdepend on galaxy morphology. A quite tight anticorrelation was observed between warp radius and warp amplitude, and a positive cor-relation was found between warp asymmetry and warp amplitude. A detailed analysis of the relations between warp parameters andgalaxy properties revealed that strong warps are mostly caused by tidal interactions, whereas weak warps are formed by a variety ofmechanisms including gas accretion. The present results indicate that the fractional warp radius coupled with warp angles representingthe warp amplitude and warp curvature provide useful diagnostic indicators of the origin of warps.� 2005 Elsevier B.V. All rights reserved.

PACS: 98.62 HR

Keywords: Galaxies – Morphology; Galaxies – Photometry; Galaxies – Structure

1. Introduction

The outer parts of the disks in many spiral galaxies,including our own, are warped. Warps are most conspicu-ous in HI observations (Bosma, 1981), although stellardisks can also be highly warped. The best example of opti-cal warps is that of PGC 20348 (Burbidge et al., 1967),which resembles an integral sign. The earliest statistics onthe frequency of optical warps indicated that about 50%of stellar disks are warped: 42 out of 86 northern hemi-sphere NGC galaxies (Sanchez-Saavedra et al., 1990).However, Reshetnikov and Combes (1998) found thatabout 70% of spiral galaxies have warped disks in a studyof 540 spiral galaxies selected from the Flat Galaxy Cata-logue (Karachentsev et al., 1993) (FGC) with a size con-straint of blue angular diameter between 1 0 and 3 0. The

1384-1076/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.newast.2005.08.006

* Corresponding author. Tel.: +82 51 510 2705; fax: +82 51 513 7495.E-mail addresses: [email protected] (H.B. Ann), jcpark@cosmos.

es.pusan.ac.kr (J.-C. Park).

large difference between these two estimates of the fre-quency of optical warps may be due to the differentinclination limits imposed on the two samples, since warpsare more easily detected in galaxies with higher inclina-tions. In the FGC, the selection criterion for the axis ratiois a/b > 7, which is about twice the value used by Sanchez-Saavedra et al. (1990).

A recent analysis of optical warps by Sanchez-Saavedraet al. (2003) revealed that 150 out of 276 spiral and lentic-ular galaxies in the southern hemisphere (d(2000) < 0) havewarped disks with warp angles larger than 4�. Theyselected sample galaxies from the LEDA that have a/b >6 and Bt (total B magnitude) <14.5. The fraction of warpeddisks with warp angles greater than 4� found by Sanchez-Saavedra et al. (2003) (54%) is much larger than the frac-tion of 11% with warp angles in this range found byReshetnikov and Combes (1998) in their study of 540 spir-al galaxies. The large discrepancy between these two statis-tics is quite remarkable, given that they were obtainedusing similar observational materials based on digital sky

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294 H.B. Ann, J.-C. Park / New Astronomy 11 (2006) 293–305

surveys and the same definition of the warp angle. The dis-crepancy becomes more plausible, however, when we con-sider the highly subjective nature of the procedures used tomeasure warp properties. In many cases, the disk bendsfirst slightly in one direction and then turns back in theopposite direction one or two times. This feature, whichis also observed in HI warps (Garcia-Ruiz et al., 2002),leads to confusion between warp types. In some cases,the very existence of a warp is a subject of controversydue to the small departures that must be identified. It isworth noting that, as described by Sanchez-Saavedraet al. (2003), objective procedures to determine warps suchas that employed by Jimenez-Vicente et al. (1998) seem tobe unreliable when applied to a large sample of galaxiesdue to the lack of a good method for treating backgroundstars and spiral arms, which greatly affect the warps.

Along with the statistics of the occurrence rate of warps,most previous warp statistics have focused on the ampli-tude and asymmetry of warps. The amplitude of a warpis usually represented by the warp angle, a, defined as theangle between the average major axis of the inner regionsand the line connecting the galaxy center and the tip ofthe outermost isophotes. Reshetnikov and Combes (1998)used the asymmetry index defined as the ratio of distancesmeasured perpendicular to the major axis from maximumintensity to outer isophotes. Garcia-Ruiz et al. (2002) andSanchez-Saavedra et al. (2003) used asymmetry parametersrelated to the asymmetries in the warp angles measured inone side and the other, although their definitions differedsomewhat. Little attention has been given to the warp ra-dius rw, defined as the radius at which the disk bends awayfrom the plane defined by the inner disk. Garcia-Ruiz et al.(2002) used rw to estimate the warp angles, but did notquantitatively analyze it. Warps in HI gas are usuallyfound beyond the optical radius (e.g., R25) (Briggs, 1990),while optical warps are usually visible near the edge ofthe optical disk. Warps are, however, sometimes found in-side R25. A good example of a galaxy with a short warp ra-dius is the integral sign galaxy PGC 20348, whose warpradius is about 0.5R25. Since the warp radius seems to berelated to the warping force, the warp radius statisticsmay help to disentangle the intricate relations betweenthe warping force and the intrinsic properties of galaxies,such as mass and radius.

The relation between the warp amplitude and the warpasymmetry is difficult to understand. Castro-Rodriguezet al. (2002) found an anticorrelation between these twoparameters, whereas Garcia-Ruiz et al. (2002) showed theopposite relation, specifically, an increase in warp asymme-try with increasing amplitude. Although Garcia-Ruiz et al.(2002) considered far fewer galaxies than Castro-Rodriguezet al. (2002), the correlation between the warp amplitudeand the warp asymmetry in the former study (see Fig. 5of Garcia-Ruiz et al., 2002) is quite robust. Moreover,although the two studies used slightly different definitionsof asymmetry – Castro-Rodriguez et al. (2002) used the dif-ference of warp angles normalized by the sum of the warp

angles, whereas Garcia-Ruiz et al. (2002) used just the an-gle difference between one side and the other – this differ-ence should not give rise to opposing relations betweenthe warp amplitude and the warp asymmetry. Thus, anew statistics is needed to resolve the intriguing contradic-tion in the results of these studies.

Many attempts have been made to understand the mech-anisms that drive andmaintain warps (see review by Binney,1992). Among the proposed mechanisms, tidal disturbancesby nearby galaxies and gravitational torques arising from amisaligned halo (Debattista and Sellwood, 2003; Idetaet al., 2000) are thought to be the dominant mechanismsfor interacting galaxies and isolated galaxies, respectively.Accretion of the intergalactic medium (Ostriker and Binney,1989; Jiang andBinney, 1999; Lopez-Corredoira et al., 2002)and a bending instability of self-gravitating disks (Sparke,1995; Revaz and Pfenniger, 2004) have also been proposedas mechanisms for generating and maintaining warps. Thehigh frequency of warped disks in the high redshift popula-tion (Reshetnikov et al., 2002) supports the infall scenarioas well as the tidal mechanism. To date, however, no con-vincing mechanism has been proposed that can explain theorigin and persistence of warps with diverse morphologiesand amplitudes.

Since most previous studies with large sample sizes havebeen confined to galaxies in the southern hemisphere(Reshetnikov and Combes, 1998; Sanchez-Saavedra et al.,2003), it seems plausible that better warp statistics wouldbe obtained by maintaining a large enough sample size forgood statistics, while constraining the selection criteria toavoid the projection effect that hampers the detection ofweak warps if galaxies in both hemispheres are used. Theprimary goal of the present study was to determine the warpstatistics using a well-defined sample of galaxies. To do this,we restricted our sample galaxies to almost perfectly edge-on galaxies (a/b > 9.5) using the Third Reference Catalogof the Bright Galaxies (de Vaucouleurs, 1991)(RC3). Thiswas done in the light of a previous report that a sample ofgalaxies with axial ratios larger than 7 (a/b > 7) still sufferedfrom projection effects in the measurement of warp param-eters (Reshetnikov and Combes, 1998). We also selectedgalaxies larger than 1 0 since the measurement of warpparameters is less accurate in small sized galaxies. The pres-ent paper focuses on the statistics of the warp properties ofdisk galaxies using a large homogeneous sample of galaxies.This paper is organized as follows. Section 2 describes thesample and measurement; Section 3 presents the statisticsof warped disks; and the discussion and conclusions aregiven in Section 4.

2. The sample

2.1. Selection

We selected our sample of galaxies based on the major-axis length of the isophotes at lB = 25 mag/arcsec2 (D25)and the ratio of the major to the minor axis lengths at

H.B. Ann, J.-C. Park / New Astronomy 11 (2006) 293–305 295

the same isophotes (a/b), with the selection criteria ofD25 > 1 0 and a/b > 9.5. These criteria ensure that the sam-ple galaxies are highly inclined galaxies in relatively closeproximity, which are necessary attributes for the easydetection of warps. Since the minimum inclination anglesatisfying a/b > 9.5 is 84�, our sample of galaxies was com-prised of almost perfectly edge-on galaxies in which warpsare easy to detect. Our minimum inclination angle is some-what larger than those used previously (e.g., 81.�7 in Res-hetnikov and Combes (1998)). The resulting sampleconsisted of 349 galaxies. Of these galaxies, 24 were omit-ted from the final sample: 2 galaxies (PGC 2210 andPGC 4435) were found to be insufficiently inclined to fulfillour selection criteria and 22 galaxies were unsuitable forwarp detection due to heavy disturbances of the isophotalmaps by background stars and nearby galaxies. Thus, thepresent statistics are based on the warp properties mea-sured from 325 edge-on galaxies. Fig. 1 shows the fre-quency distribution of the sample galaxies along theHubble type, represented by T. The majority of galaxiesare late type spirals (T > 5), with a peak at T = 6. The smallpeak at T = 0 is due to the classification scheme of RC3,which assigns a value of T = 0 to galaxies of uncertainmorphology. Among the 306 galaxies of known morphol-ogy, only 2 are earlier than T = 3. Galaxies with late typemorphology (T > 8) are also rare in the present sample.The reason for the paucity of galaxies with early type mor-phology is due to the selection criterion for the axis ratio(a/b > 9.5), which excludes galaxies with a large bulge.

2.2. Measurements of warp parameters

The properties of warps can be examined in a variety ofways. If we are interested only in the frequency distribution

Fig. 1. Frequency distribution of sample galaxies along the Hubble type.Galaxies of uncertain morphology are assigned a value of T = 0.

of warped disks as a function of Hubble type, the morphol-ogies of galaxies can be examined in detail, and warpsidentified, by scrutinizing grey scale images using the zoom-ing capability of an image display tool, such as XIMTOOL.If, however, we wish to both identify warps and estimatewarp parameters such as the warp angles and warp radius,it seems better to use isophotal maps, provided the isophotalmaps are sufficiently deep to recognize the weak warps thatare visible in well-contrasted gray scale images.

We obtained isophotal maps of the selected galaxiesusing the digital sky survey. We used the POSSII-blue sur-vey and SERC-J survey, which have resolutions of 1

00and

1.007, respectively. The isophotal maps were obtained using

the relative intensity distribution Irel(x,y) defined as

I relðx; yÞ ¼ðIGþSðx; yÞ � ISðx; yÞÞ

ISðx; yÞ; ð1Þ

where IG+S(x,y) and IS(x,y) are the intensity distributionof the galaxy including the light from the sky backgroundand the intensity distribution of the background local sky,respectively. Since warps are usually visible in the faintouter disks, it is essential to subtract the sky backgroundas accurately as possible. To do this, we applied two-dimensional polynomial fits to the regions surroundingthe target galaxies. In most cases, we used a simple poly-nomial of form IS(x,y) = a0 + a1x + a2y, although insome cases we included higher order terms in x and y.Since the raw relative intensity distribution is very noisy,especially in the fainter outer parts, we applied Gaussiansmoothing after removing bright background stars fromthe images.

The warp parameters were determined by measuringthree lengths: rw, the warp radius, defined as the lengthfrom the center of a galaxy to the point where the diskstarts to bend away from the middle plane of the disk; rlast,the radius at which a vertical line from the tip of the outer-most isophotes meets the mean disk major axis; and h, thedistance of the tip of the outermost isophotes from themean disk major axis. To measure these lengths, we deter-mined the position of the galaxy center and disk middleplane by applying ellipse fittings to the isophotes of the in-ner disk (r < 0.5R25). However, for galaxies in which heavydust obscuration made the isophotes too irregular to be fit-ted by concentric ellipses, we determined the position of thegalaxy center and disk middle plane by visual inspection ofthe isophotal map. In most cases, the outermost isophotesof our sample galaxies reached �4 magnitude below thesurface brightness of the local sky. Thus, the surface bright-ness of the outermost isophotes is about 1 magnitude fain-ter than the surface brightness level at which D25 is defined.

There are two kinds of warp angle, a and b, both ofwhich seem to be related to the warping force and thedynamical properties of disks. The warp angle, a, whichis most widely used to represent the warp amplitude (San-chez-Saavedra et al., 1990; Reshetnikov and Combes, 1998;Garcia-Ruiz et al., 2002; Sanchez-Saavedra et al., 2003), iscalculated using the relation


tan a ¼ hrlast

. ð2Þ

The angle b, by contrast, seems to be better for represent-ing the curvature of the warped disk, and is determined bythe relation

tan b ¼ hrlast � rw

. ð3Þ

As is apparent from (2) and (3), a is the angle between thedisk major axis and the line connecting the galaxy centerand the tip of the outermost isophotes, whereas b is the an-gle between the disk major axis and the line from rw to thetip of the outermost isophotes. The errors associated withthe measurement of a are usually smaller than those for bdue to the dependence of b on rw, which is most uncertainowing to the frequent variations of the position angles ofthe outer disks. In Table 1, we list the warp parametersmeasured in the present study for 104 galaxies that havea > 3�a, where �a is the error associated with the measure-ment of a, along with some basic parameters of the samplegalaxies.

For comparison, in Table 2, we list the warp angles ofgalaxies whose warp properties were investigated in previ-ous studies. Among 10 common galaxies, 9 galaxies haveidentical warp types: 6 S-shaped warps and 3 no warp.PGC 71948 is classified as a U-shaped warp, although San-chez-Saavedra et al. (2003) classified it as a no-warp galaxy.Thus, the present classification of warp type is consistentwith previous ones. Warp angles agree with those of Res-hetnikov and Combes (1999), but show some discrepancieswith those of Sanchez-Saavedra et al. (2003), whose esti-mates are, on average, 30% larger than ours.

3. Results

3.1. Type examples

Warps are usually characterized by an integral-signmorphology, although some warps are cup-shaped. Res-hetnikov and Combes (1998) classified warps into twotypes, S-shaped and U-shaped. Subsequently, Sanchez-Saavedra et al. (2003) added an additional type, L-shaped,to describe disks warped in one side only. Yet another type,the N-shaped warp, introduced by Sanchez-Saavedra et al.(1990, 2003), seems to be the same as the S-shaped warp,and hence we do not consider it as an independent type.In the present analysis, we followed the classification ofReshetnikov and Combes (1998) because there seems tobe little or no L-shaped warp if we consider any deviationfrom the middle plane defined by the inner parts of the diskas a real feature. It is worth noting that a significant frac-tion of warps show a wavy pattern due to the variationof the position angles of the major axis across the middleplane. This wavy pattern seems to be caused by a morphol-ogy in which the disk is first bent slightly in one direction,then turns back and bends in the opposite direction. Awavy pattern could also arise, however, from bending of

the spiral arms rather than bending of the entire disk.The presence of a wavy pattern in a warped disk makesit difficult to identify the warp type when the warp angleis small. Fig. 2 displays isophotal maps of representativeS- and U-shaped warps along with two extreme warps.

3.2. Frequency distribution of warps

We found 236 warped disks among the 325 sample gal-axies analyzed. Thus about one third of the galaxies consid-ered had no warps. S- and U-shaped warps were found in165 (51%) and 71 (22%) galaxies, respectively. The fractionof warped disks in the present sample, 73% is in goodagreement with that found by Reshetnikov and Combes(1998). However, whereas Reshetnikov and Combes(1998) found similar fractions of S- and U-shaped warps,we found more than twice as many S-shaped warps asU-shaped warps. This discrepancy may be due to differ-ences in the selection criterion for the axis ratio: we useda/b > 9.5, whereas Reshetnikov and Combes (1998) useda/b > 7. This difference means that our analysis is less af-fected by the projection effect, which is more severe inU-shaped warps than in S-shaped warps (Reshetnikovand Combes, 1998). The fraction of the present sample ofgalaxies that was found to be strong S-shaped warps withwarp angle (a) greater than 4� (�15%) was similar to thatfound by Reshetnikov and Combes (1998) (11%) but about28% of that found by Sanchez-Saavedra et al. (2003).Given that strongly warped galaxies are easy to identify,the discrepancies in these statistics can be attributed to sys-tematic errors associated with the warp angle measure-ments. The larger fraction of strong S-shaped warpsidentified by Sanchez-Saavedra et al. (2003) is likely dueto their tendency to estimate higher values of the warp an-gles, as shown in Table 2. This demonstrates an inherentweakness in warp statistics, including the present one,which arises from the necessarily subjective nature of theanalysis.

The frequency distribution of warp types is plotted inFig. 3, where solid lines represent the total sample of galax-ies that are suitable for warp detection, and the dotted anddashed lines represent the number of galaxies that havewarp angle, a, greater than 4� and 5.�5, respectively. Asshown in Fig. 3, the fractions of U-shaped warps decreasesmore rapidly than S-shaped warps for large a. There seemsto be no significant dependence of warp frequency on gal-axy morphology, except for a slightly higher frequency atlater types (Fig. 4).

3.3. Warp angles

Warp angles are the most important quantities that areassumed to be related to the strength of the warping force.In the present work, we estimated two warp angles, a andb. Various groups have estimated a, which is thought torepresent warp amplitude, whereas only Sanchez-Saavedraet al. (2003) have reported on b. Fig. 5 shows plots of a and

Table 1Warp parameters and basic properties of 104 selected galaxies

Galaxy a2000 d2000 T logD25 Warp ae aw be bwrwRopt

Ca

PGC 70 0.00025 20.3333 6.0 1.26 S 4.5 4.0 14 11 0.7 –PGC 1797 0.48975 31.3931 6.0 1.04 U 3.0 4.5 8 11 0.6 –PGC 2190 0.61053 �56.9067 �0.7 1.28 S 2.0 5.5 11 14 0.7 –PGC 2261 0.63181 32.6875 6.0 1.11 S 6.0 5.0 24 17 0.7 –PGC 2747 0.78400 30.3386 6.0 1.34 S 5.5 3.5 17 9 0.6 –PGC 4148 1.16556 20.7714 6.0 1.19 S 4.0 6.0 9 17 0.6 –PGC 5000 1.37592 �29.9822 0.0 1.17 S 5.5 4.0 31 14 0.8 –PGC 7806 2.05061 �9.6569 6.8 1.45 S 4.5 1.5 14 4 0.7 sPGC 7944 2.08336 24.6667 10.0 1.04 U 4.5 1.5 11 5 0.6 –PGC 8353 2.18442 6.6664 8.0 1.04 U 1.0 6.5 3 17 0.6 cPGC 8499 2.22008 �70.9136 5.0 1.19 S 4.0 5.5 9 16 0.6 –PGC 8618 2.25006 18.6667 6.0 1.00 S 4.0 5.5 14 17 0.7 –PGC 8621 2.25006 49.8500 7.0 1.22 S 1.0 6.5 3 18 0.6 –PGC 9917 2.60025 5.4500 6.0 1.00 S 5.5 6.0 18 13 0.6 –PGC 11289 2.98461 32.6322 4.0 1.04 U 4.5 5.0 15 14 0.7 –PGC 13178 3.55017 72.1833 6.0 1.04 S 4.0 1.5 12 11 0.8 –PGC 14007 3.86675 2.3667 7.0 1.00 S 2.5 7.5 14 15 0.7 –PGC 14504 4.12850 25.7761 10.0 1.14 U 4.0 1.5 13 14 0.8 –PGC 15181 4.46669 1.0333 8.0 1.16 S 4.0 2.0 9 12 0.7 –PGC 15693 4.63333 72.3167 7.0 1.16 S 7.5 3.5 23 17 0.7 –PGC 19285 6.58339 85.3500 5.0 1.00 U 2.0 8.0 9 30 0.8 –PGC 19652 6.80025 66.2333 4.0 1.39 S 3.5 2.5 9 7 0.6 –PGC 19674 6.82942 29.5253 7.0 1.06 U 5.0 4.5 12 10 0.6 –PGC 20348 7.18925 71.8350 7.0 1.52 S 11.5 19.5 19 32 0.4 cPGC 20486 7.23358 48.7167 6.0 1.11 U 2.5 6.0 11 23 0.8 –PGC 21451 7.63333 70.8667 6.0 1.04 S 8.0 3.5 18 9 0.6 cPGC 21680 7.73342 47.7333 6.0 1.00 S 2.5 7.0 9 19 0.7 sPGC 22032 7.86533 27.3008 3.0 1.07 S 5.0 3.5 14 9 0.6 sPGC 22482 8.00025 59.1167 6.0 1.03 S 6.0 3.5 16 10 0.6 –PGC 22921 8.16947 24.8925 6.0 1.33 S 1.5 5.5 5 21 0.7 sPGC 24982 8.88358 18.6667 6.0 1.14 U 3.0 5.0 10 16 0.7 cPGC 25232 8.98347 39.2094 8.0 1.61 S 3.0 2.5 9 10 0.7 sPGC 25673 9.13333 20.4833 4.0 1.00 U 1.0 5.0 4 10 0.6 –PGC 26086 9.23358 40.0500 3.0 1.29 U 2.0 4.5 8 16 0.7 sPGC 27371 9.61667 13.8833 7.0 1.14 S 4.0 3.0 13 10 0.7 –PGC 28248 9.82392 14.6572 0.0 1.16 S 3.5 4.5 13 24 0.8 –PGC 28700 9.94350 20.6481 6.0 1.47 S 4.5 3.5 14 11 0.7 cPGC 29156 10.05011 13.0833 6.0 1.00 S 3.5 5.0 9 12 0.6 sPGC 29261 10.08339 44.5000 7.0 1.05 S 7.0 2.0 17 7 0.6 sPGC 29956 10.25019 7.3167 7.0 1.40 U 3.0 2.0 12 9 0.8 sPGC 30714 10.45033 28.6414 3.0 1.52 S 2.0 2.5 8 7 0.7 –PGC 30832 10.47653 3.5608 8.0 1.16 U 4.5 1.5 13 5 0.7 sPGC 32644 10.86681 10.0167 6.0 1.19 S 4.0 4.5 9 11 0.6 –PGC 34861 11.36667 35.7000 6.0 1.00 S 6.0 5.0 19 20 0.7 –PGC 34869 11.36667 69.6333 7.0 1.44 S 4.0 4.0 17 21 0.8 –PGC 34870 11.36669 34.9333 6.0 1.11 U 1.5 5.0 8 23 0.8 –PGC 35017 11.40003 24.6167 4.0 1.22 U 4.0 2.0 14 14 0.8 –PGC 35235 11.45525 38.6642 6.0 1.23 S 2.5 3.5 8 9 0.6 –PGC 35320 11.46678 9.1000 6.0 1.27 S 2.0 5.5 7 13 0.7 sPGC 35803 11.58336 15.9500 6.0 1.23 S 4.0 4.0 14 13 0.7 sPGC 36988 11.83336 6.9833 7.0 1.22 S 2.5 2.0 11 10 0.8 sPGC 37088 11.85011 32.5500 6.0 1.07 S 5.0 3.0 18 22 0.8 –PGC 38748 12.17694 18.8233 6.0 1.48 S 6.0 5.0 14 14 0.6 cPGC 38933 12.21092 34.7033 6.0 1.07 U 3.0 5.0 10 17 0.7 –PGC 39432 12.29281 22.5392 7.0 1.74 U 1.0 2.5 3 9 0.7 cPGC 40839 12.45600 10.8667 6.0 1.24 U 4.0 2.0 10 13 0.7 sPGC 41051 12.46683 31.4833 6.0 1.04 S 4.5 5.5 14 24 0.7 –PGC 41119 12.48506 44.6553 7.0 1.39 S 2.0 8.5 4 18 0.5 sPGC 45006 13.04050 �17.6797 5.0 1.41 S 3.0 3.5 7 10 0.6 –PGC 45451 13.11675 32.8500 6.0 1.20 S 4.5 3.0 24 27 0.9 –PGC 47812 13.56672 32.2000 3.0 1.04 S 6.5 1.5 19 11 0.8 –PGC 47996 13.60003 37.0333 7.0 1.14 U 6.0 2.5 19 13 0.8 –PGC 48251 13.63356 14.7333 4.0 1.04 U 3.5 5.0 13 17 0.7 –

(continued on next page)


Table 1 (continued)

Galaxy a2000 d2000 T logD25 Warp ae aw be bwrwRopt

Ca

PGC 49292 13.86683 68.4500 4.0 1.00 S 5.5 6.0 17 13 0.6 –PGC 50021 14.04508 9.1617 7.0 1.04 S 3.0 13.5 8 26 0.6 cPGC 50126 14.06667 11.9833 6.0 1.00 S 4.5 3.5 27 16 0.8 –PGC 51046 14.28347 82.6333 6.0 1.07 U 2.5 5.5 9 14 0.7 –PGC 51599 14.45003 50.5667 6.0 1.07 U 4.5 4.5 18 27 0.8 –PGC 52921 14.82014 29.7469 4.0 1.11 S 4.0 3.0 18 10 0.8 –PGC 53350 14.91678 37.4167 6.0 1.04 U 4.5 1.5 12 11 0.8 sPGC 53563 14.98350 27.3167 6.0 1.00 S 7.5 5.5 23 17 0.7 –PGC 54262 15.20069 1.6981 7.0 1.39 S 3.5 4.0 11 16 0.7 sPGC 55881 15.71683 33.2833 6.0 1.00 S 6.0 4.0 18 12 0.7 sPGC 55902 15.73336 7.8333 0.0 1.00 S 23.0 5.5 33 9 0.4 cPGC 55919 15.73345 11.5333 3.0 1.19 S 3.5 4.5 12 13 0.7 cPGC 57854 16.33333 37.5667 7.0 1.04 U 3.0 11.5 11 21 0.6 sPGC 58545 16.58336 40.9833 6.0 1.16 S 4.0 1.5 17 5 0.7 sPGC 59310 16.95011 38.6667 7.0 1.11 S 4.0 6.0 8 20 0.6 sPGC 60049 17.30022 29.8500 6.0 1.04 S 3.5 6.5 13 18 0.7 sPGC 60286 17.43339 11.3000 8.0 1.00 S 5.0 7.5 15 16 0.6 cPGC 60370 17.46692 29.3000 6.0 1.07 U 6.0 3.5 17 10 0.7 –PGC 61777 18.35025 21.1500 8.0 1.04 U 2.5 4.0 8 9 0.6 –PGC 63317 19.51672 42.2000 6.0 1.14 U 3.5 4.5 15 12 0.7 –PGC 63395 19.59508 �57.5194 6.7 1.54 S 1.5 3.0 7 12 0.8 –PGC 63592 19.73922 �27.4058 4.5 1.47 S 4.0 2.0 12 4 0.6 –PGC 64429 20.28897 �38.6747 7.0 1.35 U 1.5 4.5 4 12 0.6 –PGC 65683 20.90005 17.7667 6.0 1.14 S 4.0 2.5 18 13 0.8 –PGC 67158 21.67478 �26.5258 4.0 1.41 S 3.0 3.5 13 11 0.7 –PGC 67550 21.87647 28.3061 7.0 1.31 S 1.0 3.5 6 12 0.8 –PGC 68305 22.21686 �62.0672 7.0 1.45 S 5.0 6.0 16 18 0.7 –PGC 68611 22.33353 35.2167 6.0 1.04 S 5.0 6.0 21 15 0.7 –PGC 69571 22.71667 �3.7833 0.0 1.12 U 6.0 6.5 20 16 0.7 –PGC 70026 22.92847 31.7706 7.0 1.16 U 4.5 1.5 14 9 0.8 –PGC 70040 22.93333 12.7833 3.0 1.24 S 5.0 3.5 13 8 0.6 sPGC 70175 22.98678 13.6058 8.0 1.54 S 2.5 1.5 11 8 0.8 sPGC 70321 23.03353 26.0333 0.0 1.14 S 3.5 4.5 8 23 0.7 –PGC 70541 23.13194 5.1611 6.0 1.04 S 5.5 2.0 22 7 0.8 –PGC 70708 23.21670 6.4000 5.0 1.54 S 3.0 2.0 13 9 0.8 –PGC 70734 23.22883 29.0092 7.0 1.35 U 4.0 2.5 11 5 0.6 cPGC 71652 23.51672 9.2167 8.0 1.04 S 5.5 4.0 15 14 0.7 –PGC 71839 23.59542 32.3858 6.0 1.34 S 4.0 1.0 12 3 0.7 –PGC 71876 23.60567 �57.6300 7.0 1.34 S 2.5 4.5 9 18 0.7 –PGC 71948 23.63039 �47.7264 4.8 1.72 U 4.0 3.5 18 10 0.7 –PGC 72261 23.73786 �80.1756 5.0 1.30 U 1.5 4.0 8 23 0.8 –

a Companionship of host galaxies. �s� and �c� represent companionship from SDSS and NED, respectively, and �–� indicates unknown companionship.

Table 2Comparisons with previous works

Galaxy T Warp ae aw WA1 Warp Ref.

PGC 4010 7 S 4.6 4.1 3.5 S b

PGC 8499 6 S 4.2 5.7 3.5 S b

PGC 28909 5 – 0 0 0–0 – c

PGC 39886 6 – 0 0 0–0 – a

PGC 44358 7 – 0 0 0–0 – c

PGC 45006 5 S 3.1 3.4 5–7 N c

PGC 63395 7 – 0 0 2–3 S c

PGC 64597 5 S 3.0 2.9 3–3 S c

PGC 67158 4 S 3.0 3.4 ?–4 S c

PGC 71948 5 U 0.0 2.9 0–0 – c

a Sanchez-Saavedra et al. (1990).b Reshetnikov and Combes (1999).c Sanchez-Saavedra et al. (2003).1 WA is warp angle (a). When given a single value, it is a mean warp angle, while it gives two values connected by � the left one is ae and the right one

is aw.


Fig. 2. Isophotal maps of four galaxies. PGC 55919 and PGC 26086 show representative S- and U-shaped warps, respectively, and PGC 20348 and PGC55902 display extremely strong warps. The outermost isophotes are 4 magnitudes below the local sky brightness. The box size of each diagram is 200

00in

one dimension.

Fig. 3. Frequency distribution of warp types. Solid lines represent thewhole sample, and dotted and dashed lines indicate galaxies with a > 4�and a > 5.�5, respectively.

Fig. 4. Histograms of warps along the Hubble type. Solid lines representgalaxies with a > 0 and dotted lines indicate those with a > 4�.


b as a function of galaxy morphology. Both warp angles re-main almost constant with varying the galaxy morphology.Warp angle, b, shows a somewhat larger scatter than a,which can be attributed, at least in part, to the large errorsassociated with measurement of the warp radius. However,

Fig. 5. Distribution of warp angles along the Hubble type.

0

20

40

S

0 2 4 6 8 100

5

10

15

20 U

S

10 20 30

U

Fig. 6. Histograms of warp angles. Solid lines represent warps in the eastside of disks and dotted lines indicate those in the west side.


the larger scatter in b may reflect the intrinsic scatter in bdue to its dependence on variables other than the warpingforce, since similar intrinsic scatters in a and b would be ex-pected if they were similarly correlated with the warpingforce. Two galaxies, PGC 20348 at T = 7 and PGC 55902at T = 0, have large values of a = 15.�4 and 14.�6, respec-tively. Such a large values of a would be consistent withwarps formed by extremely strong torques, most likelycaused by the presence of nearby companion galaxies ofcomparable mass. In support of this hypothesis, wesearched NASA Extragalactic Database (NED) and foundthat PGC 20348 and PGC 55902 each have two compan-ions of comparable mass within 0.5 Mpc: PGC 20362 andPGC 20398 for the integral-sign galaxy PGC 20348 andPGC 55870 and PGC 55889 for PGC 55902 which showsa peculiar morphology suggesting tidal features.

Fig. 6 presents the frequency distributions of warp an-gles a and b for each warp type. In this figure, the solidand dotted lines represent the distributions in the eastand west sides of the disk, respectively. The distributionsfor the whole sample resemble those of the S-shaped warps,which are characterized by peaks at a � 3� and b � 10�,respectively, with tails at large warp angles. The distribu-tion of a for the S-shaped warps is very similar to that re-

ported by Reshetnikov and Combes (1999). Thedistributions of warp angles in the east and west sides ofthe disk are similar for the S-shaped warps, whereas thosefor the U-shaped warps show some discrepancies, probablydue to small number statistics. The mean values of a and bare 3.4� and 12.2� for the S-shaped warps, and 3.1� and12.0� for the U-shaped warps, respectively. The high-endtails in the S-shaped warps are thought to be due to thepresence of large warps rather than to statistical fluctua-tions, because the number of S-shaped warps is sufficientlylarge to ensure good statistics. None of the U-shaped warpshave a value of a greater than 8�. This may indicate that thedriving force of the U-shaped warps is weaker than that ofthe highly warped S-shaped warps. In the histograms of afor the S-shaped warps, we have omitted the two galaxiesPGC 20348 and PGC 55902, which have extremely largewarp angles.

Fig. 7 shows the relation between a and b for the 104galaxies that have a > 3�a. In this plot, circles and trianglesdenote the galaxies with S-shaped warps and U-shapedwarps, respectively, and filled symbols are used to indicategalaxies that have companions based on redshift data fromthe Sloan Digital Sky Survey (SDSS) and NED. We usedNED and the Data Release 3 (DR3) version of the SDSSto find companions that are close to host galaxies, i.e.,DD < 0.5 Mpc and Dz < 0.002. Most of the companion gal-axies found from the SDSS are satellite galaxies that aremuch smaller than the host galaxies while those fromNED are comparable mass companions. There seems tobe a weak correlation between a and b for the whole sam-ple, with no significant difference between S- and U-shapedwarps. The value of b varies over a wide range for a given

0 5 10 150

10

20

Fig. 7. Relation between the warp angles a and b. Galaxies known to havecompanions are denoted by filled symbols and galaxies that have noinformation on their companionship are indicated by open symbols.S-shaped warps are represented by circles and U-shaped warps areindicated by triangles. The two galaxies with error bars represent PGC20348 (upper one) and PGC 55902, respectively, which are known to havecompanions of comparable mass. The solid and dotted lines are least-square fits to the filled symbols with (solid line) and without (dotted line)the two extreme warps.

0

5

10

15

20 40 60 800

5

10

15

Fig. 8. Warp amplitude (a) as a function of galaxy size (D25) for S-shapedwarps (upper panel; circles) and U-shaped warps (lower panel; triangles).The two filled circles with error bars represent PGC 20348 (upper one) andPGC 55902, respectively.


a. Sanchez-Saavedra et al. (2003) found a similar correla-tion, but claimed that a does not increase with b for strongwarps. However, as demonstrated in our data by the pres-ence of two extreme warps (denoted by filled circles with er-ror bars), a does indeed increase with b for strong warps. InFig. 7, the filled circle with small error bars represents theintegral-sign galaxy PGC 20348 and that with larger errorbars indicates PGC 55902, a galaxy with a very peculiarmorphology. The errors cited for PGC 20348 and PGC55902 are typical of large and small galaxies, respectively,since errors in the warp angles primarily depend on themeasurement errors, which are relatively smaller in largergalaxies. As can be seen in Fig. 7, the galaxies with com-panions show a much tighter correlation than the wholesample, and are located near the lower envelope. The dot-ted and solid lines in Fig. 7 represent linear least-square fit-tings to the galaxies with companions with (solid line) andwithout (dotted line) the two extreme warps. The goodnessof the fits are 0.80 and 0.91, respectively for dotted line andsolid line. Since the locations of the two extreme warps arewithin �2r from the dotted line, their warps are thought tobe caused by the same mechanism but with differentstrength if a and b are mainly determined by the warpingforce. Thus, it seems likely that the lower envelope repre-sents the correlation between a and b expected for the casein which external torque due to nearby sources is the dom-inant force determining a and b. The present results there-fore suggest that warp curvature represented by b isstrongly correlated with warp amplitude when these two

parameters depend directly on the warping force. However,the wide range of b for a given a seems to indicate thatwarp curvature is related to the dynamical structure of gal-axies as well as to the warping force in a more complicatedway.

Reshetnikov and Combes (1998) and Castro-Rodriguezet al. (2002) claimed that an anticorrelation exists betweenthe warp amplitude and galaxy radius. However, as shownby Castro-Rodriguez et al. (2002), this relation is notstrong. Fig. 8 shows the distribution of a as a function ofD25. In this plot, we can discern a very weak anticorrelationfor the S-shaped warps but no correlation for U-shapedwarps. Reshetnikov and Combes (1998) interpreted theanticorrelation between warp amplitude and galaxy radiusin terms of the rareness of the massive companions that arerequired to produce large warp amplitudes in large disks.The interpretation that large warps have a tidal originwas supported by the observation by Garcia-Ruiz et al.(2002) of an anticorrelation between warp amplitude andgalaxy radius for HI warps in interacting galaxies. In theirsample of HI warps, they found this type of anticorrelationonly for interacting galaxies and found no relation forother galaxies, regardless of their environment. The ab-sence of a significant anticorrelation between warp ampli-tude and galaxy mass or radius for non-interactinggalaxies in Garcia-Ruiz et al. (2002) is probably due tothe weakness of the relationship between a and D25 shownin the present study. The weakness of the anticorrelationbetween warp amplitude and galaxy mass or radius is alsoapparent in a larger sample of optical and radio warps(Castro-Rodriguez et al., 2002).


3.4. Warp radius

No detailed statistics have been reported on the warp ra-dius, although it is commonly thought that warps begin atthe edges of disks. The mean warp radius of the presentsample, in units of R25, and the observed optical radius,Ropt, were found to be rw

R25¼ 0:9� 0:3 and rw

Ropt¼ 0:7� 0:1,

respectively. Thus, the disks begin to warp near R25, whichis located at �0.75Ropt. The present estimate of Ropt seemsto well represent the radius where the disk ends, because itwas measured at the isophotes �4 mag fainter than the skybrightness. We determined the warp radius rw (kpc) for thegalaxies within our sample whose distances are known fromradial velocities. Assuming H = 75 km/s/Mpc, the meanwarp radius of the whole sample is 14.4 ± 10.6 kpc. Thelarge standard deviation of 10.6 kpc implies that there is alarge intrinsic scatter in the warp radius. Fig. 9 shows thedistribution of fractional warp radius (rw/Ropt), warp radiusand D25 as a function of Hubble type. The distribution ofrw/Ropt is almost flat between T = 4 and 7, where the num-bers of galaxies are large enough for good statistics. By con-trast, the distribution of rw shows a tendency of decreasingwarp radius with increasing T, although the error bars arelarger than the difference between the warp radii. This trendmight be due to the dependence of galaxy radius on themorphology shown in the lower panel of Fig. 9. For galax-ies later than T = 7, rw/Ropt increases with increasing T,

Fig. 9. Distribution of size parameters along the Hubble type. Error barsindicate standard deviations.

whereas rw remains almost constant and D25 decreases con-tinuously. This indicates that the disks likely bend awayfrom the middle plane at a larger distance from the centerin the later Hubble types. The large magnitudes of the errorbars are due to the intrinsic scatter in the warp radius.

Fig. 10 shows a plot of the fractional warp radius (rw/Ropt) as a function of a for the galaxies that have a > 3�a.A fairly tight anticorrelation is clearly observed betweenrw/Ropt and a, especially for galaxies with companions(filled circles). This anticorrelation indicates that disks withlarger warp amplitudes begin to bend away at smaller radii.This trend is expected based on the fact that the resistanceof a disk to warping increases with decreasing radius, andhence stronger warping forces are required for disks thatbend at smaller radii. PGC 20348 and PGC 55092 as shownby the two filled circles with error bars have the smallestvalues of rw/Ropt, indicating that they are exposed to thestrongest warping forces. As is the case for the relation be-tween a and b, the extreme warps do not much affect theanticorrelation observed for galaxies with companionssince the relations between rw/Ropt and a for galaxies with(solid line) and without (dotted line) the two extreme warpsare not much different. The value of rw/Ropt varies over awide range for a given a in the whole samples, but it is con-fined to a narrow range for the galaxies with companions.This observation is consistent with a model in which strongwarps are caused by tidal interactions with nearby compan-ions, which determine the warp radius as well as the warpamplitude, whereas weak warps are caused by a variety ofmechanisms including weak tidal interactions, misalignedhalos, and gas accretion. It is worth noting that there seemsto be a limiting rw/Ropt defined by the upper envelope ofthe anticorrelation for warps caused by tidal interactions.

0 5 10 15

0.4

0.6

0.8

Fig. 10. Fractional warp radius versus warp amplitude. Galaxies withknown companions are indicated by filled symbols. The remaining opensymbols represent galaxies that have no information on their compan-ionship. The filled circle with smaller error bars represents PGC 20348 andthe larger one PGC 55902.

0 10 20

0.4

0.6

0.8

Fig. 11. Fractional warp radius versus warp curvature. Galaxies withknown companions are indicated by filled symbols. The remaining opensymbols represent galaxies that have no information on their compan-ionship. The solid and dashed lines represent the least-square fits to thefilled symbols and open symbols, respectively.

0 5 10 150

5

10

15

20

Fig. 12. Warp asymmetry versus warp amplitude. Galaxies with knowncompanions are indicated by filled symbols. The remaining open symbolsrepresent galaxies that have no information on their companionship.


Fig. 11 shows the distribution of rw/Ropt as a function ofb. There is no correlation between rw/Ropt and b for thewhole sample. But if we divide the sample into two sub-groups according to their companionship, the warped diskswith companion galaxies display an anticorrelation betweenrw/Ropt and b (solid line), whereas the disks with unknowncompanionship show a positive correlation (dashed line).Both of the correlations have a goodness of �0.60 whichis smaller than those of other correlations in Figs. 7 and10. The positive correlation between rw/Ropt and b is easilyunderstood since b should be large for a given a when disksstart to bend close to the edges of the disks. By contrast, theanticorrelation between rw/Ropt and b can be understood inthe same way as the fairly tight anticorrelation observed be-tween rw/Ropt and a for galaxies with companions; that is,the stronger the warping force, the shorter the warp radius.However, this anticorrelation is much weaker than that be-tween rw/Ropt and a and strongly depends on the two ex-treme warps since the goodness of the anticorrelationbecomes one third of the anticorrelation represented bythe solid line in Fig. 10 if we exclude the two extreme warps.Thus, it is of interest to fill the gaps between the two extremewarps and the others to see whether the anticorrelation be-tween rw/Ropt and b is real. If it is real, the present resultsindicate that the diagram of rw/Ropt versus b may providea diagnostic tool for segregating warps according to theirdriving mechanisms.

3.5. Warp asymmetry

A majority of the sample galaxies have non-negligibleasymmetries in both the warp angles and warp radius.The asymmetry in the warp radius, defined as the difference

between the warp radii in the east and west sides of thedisk, is closely related to the intrinsic lopsidedness of galax-ies observed in at least 30% of spiral galaxies (Rix andZaritsky, 1995; Zaritsky and Rix, 1997; Kornreich et al.,1998). However, the asymmetry in the warp angles seemsto be related to the intrinsic properties of warps (Garcia-Ruiz et al., 2002). The asymmetry in the warp angle hasbeen defined in two ways: Garcia-Ruiz et al. (2002) definedthe warp asymmetry as the simple difference between thewarp angles in the east and west sides of the disk, i.e.,

Aa ¼ jae � awj; ð4Þwhere ae and aw are the warp angles in the east and westsides of the disk, respectively; whereas Castro-Rodriguezet al. (2002) and Sanchez-Saavedra et al. (2003) definedthe warp asymmetry as |ae � aw|/(ae + aw). We used thedefinition of Garcia-Ruiz et al. (2002).

Fig. 12 shows a plot of the warp asymmetry Aa as afunction of a for galaxies with a > 3�a. As shown by dottedand solid lines, which indicate the least-square solutions forgalaxies with (solid line) and without (dotted line) the twoextreme warps, there is a positive correlation between thewarp asymmetry and warp amplitude. Garcia-Ruiz et al.(2002) found a similar relation for HI warps, although theirdata showed a tighter correlation at small warp angles(a [ 7�) but larger scatter for strongly warped galaxies.They additionally showed that the largest asymmetrieswere observed in galaxies that displayed obvious tidal fea-tures. This relation of large asymmetry with strong tidalinteractions is also observed in our data (Fig. 12); specifi-cally, the galaxies in our sample showing large asymmetries(Aa > 7�) are also the galaxies with companions. Fig. 13shows the relation between the warp asymmetry and the

0.4 0.6 0.80

5

10

15

20

Fig. 13. Warp asymmetry versus fractional warp radius. Galaxies withknown companions are indicated by filled symbols. The remaining opensymbols represent galaxies that have no information on theircompanionship.


warp radius. Here again, we can discern a weak anticorre-lation between Aa and rw/Ropt for warps caused by tidalinteractions. One of the extreme warps (PGC 55902) hasa large Aa that deviates more than 4r from the dotted linethat represents a least-square solution for galaxies withcompanions excluding the two extreme warps.

4. Discussion and conclusions

The fraction of warped disks in the present sample, 73%,is in qualitative agreement with previous reports that �50%(Sanchez-Saavedra et al., 1990), �70% (Reshetnikov andCombes, 1998), and �54% (Sanchez-Saavedra et al.,2003) of spiral galaxies have warped disks. The discrepan-cies among the various studies can be attributed to differ-ences in the sample selection criteria, the lower limit ofthe warp angle used to generate the statistics, and the qual-ity and depth of the digitized photographic plates. Moregenerally, the subjective nature of warp analysis may alsohave contributed to the different fractions reported.

Our observation of no significant relation between warpfrequency and galaxy morphology confirms the findings ofReshetnikov and Combes (1998) and Sanchez-Saavedraet al. (2003). In addition, we found no significant depen-dence of warp parameters (warp angles, warp radius, andwarp asymmetry) on galaxy morphology. Moreover, wefound no correlation between warp type and galaxymorphology.

Our analysis revealed a quite strong anticorrelation be-tween the fractional warp radius rw/Ropt and the warpamplitude represented by the warp angle a, with a smallerscatter, for the galaxies with companions. This anticorrela-tion suggests that the warp radius at which a disk bends

away from its middle plane is mainly determined by thewarping force. However, our observation that rw/Ropt var-ies over a wide range for a given a for a [ 5� may indicatethat rw/Ropt depends not only on the warping force but alsoon the dynamical properties of the galaxy such as the mass.In contrast to warps with large amplitudes, which arethought to be induced by strong tidal interactions withnearby companions, weak warps exhibit large variationsin rw/Ropt at a given amplitude, suggesting that the warpingis caused by several mechanisms. These mechanisms mayinclude torque due to a misaligned prolate halo (Idetaet al., 2000), gas accretion (Lopez-Corredoira et al.,2002), and a bending instability (Sparke, 1995; Revaz andPfenniger, 2004) as well as weak tidal interactions.

We found that the relation between the fractional warpradius rw/Ropt and the warp angle b is an effective indicatorof the mechanism of warp formation. Specifically, galaxieswith companions displayed an anticorrelation between rw/Ropt and b, similar to that between rw/Ropt and a, whereasthe other galaxies showed a positive correlation with b. Butwe need more evidence for the former anticorrelation be-cause, as shown in Fig. 11, it is much affected by the pres-ence of two extreme warps (PGC 20348 and PGC 55902).However, if warps in galaxies with companions are formedby tidal interactions, the warp angle b should correlate withthe warping force in the same way as the warp angle a does,and disks should bend away from the middle plane atshorter distances from the center of galaxies when the ap-plied warping force is strong. If warps are formed by othermechanisms such as in-falling gas, it is possible that theouter parts of the disks bend quite sharply with little distur-bance of the inner disk. Thus, it seems likely that the disksbend gently, with a warp radius inversely proportional tothe strength of the applied warping force when the primarycause of warping is tidal interactions.

Our observation of a weak anticorrelation between thewarp amplitude and radius for S-shaped warps is consistentwith previous findings by Reshetnikov and Combes (1998)and Castro-Rodriguez et al. (2002). Although this anticor-relation may suggest a tidal origin for warps (Reshetnikovand Combes, 1998), especially for large warps, the weak-ness of the relation found in the present study as well asin previous works (Garcia-Ruiz et al., 2002; Castro-Rodri-guez et al., 2002) suggests that warps are formed by a vari-ety of mechanisms. For example, small warps can be easilyformed by gas accretion (Lopez-Corredoira et al., 2002)and a bending instability (Revaz and Pfenniger, 2004).

The warp asymmetry, defined as the difference in a be-tween the east and west sides of the disk, shows an anticor-relation with the fractional warp radius but a positivecorrelation with the warp amplitude represented by a.These opposite relations, which likely stem from the anti-correlation observed between a and rw/Ropt, indicate thatthe warp asymmetry is proportional to the warping force.However, the warp asymmetries of warps with extremelylarge amplitudes do not conform to this linear relation.Similar behavior has been observed for HI warps by


Garcia-Ruiz et al. (2002), who showed that large warpasymmetries are associated with strong tidal interactions.Consistent with this, the morphology of the galaxy thathave the largest asymmetry (PGC 55902) in the presentsample shows tidal signatures. Thus, it seems likely that ti-dal interactions are the primary source of large warps withhigh degrees of asymmetry.

The effect of galaxy environment on warp frequency issomewhat controversial. Reshetnikov and Combes (1998)claimed that the frequency of optical warps depends onthe environment, in the sense that the detection probabil-ity of warps in rich environments is higher than that inpoor environments. The observation of a higher probabil-ity of detecting warps in rich environments indicates thattidal interactions cause warps, or at least reinforce them(Reshetnikov and Combes, 1998; Reshetnikov et al.,2002). A similar conclusion was drawn by Schwarzkopfand Dettmar (2001), who, in a study of a large sampleof edge-on galaxies, found warps in all merging systemsbut in only half of their control sample. However,Garcia-Ruiz et al. (2002) found a higher fraction ofwarped disks in poor environments than in rich environ-ments, although the asymmetry and warp amplitude washigher in rich environments. When we checked our sam-ple using redshift data from the SDSS to determinewhether warp frequency is environment-dependent, wefound that of the 75 galaxies that have been observedin DR3, 56 had companions and 19 did not. The fractionof warped disks in galaxies with no companions was 81%,whereas that in galaxies with companions was 64%. Thus,consistent with the findings of Garcia-Ruiz et al. (2002),our results show a tendency toward a lower frequencyof warps in rich environments than in poor environments.However, our results and those of previous studies (e.g.,Reshetnikov et al., 2002) clearly show that tidal interac-tions are the primary mechanism responsible for largewarps.

Acknowledgements

H.B.A. thank Y.Y. Choi for her kind help in the retrie-val of the SDSS data. This work was supported by a KoreaResearch Foundation Grant (KRF-2002-015-CP0151).

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