Warped Galaxies From Misaligned Angular Momenta Victor P. Debattista1 and J. A. Sellwood Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019 [email protected], [email protected] 9 9 9 1 n ABSTRACT a J A galaxy disk embedded in a rotating halo experiences a dynamical friction force 2 which causes it to warp when the angular momentum axes of the disk and halo are 1 misaligned. Our fully self-consistent simulations of this process induce long-lived warps 1 in the disk which mimic Briggs’s rules of warp behavior. They also demonstrate that v random motion within the disk adds significantly to its stiffness. Moreover, warps gener- 3 5 ated in this way have no winding problem and are more pronounced in the extended H I 1 disk. As emphasized by Binney and his co-workers, angular momentum misalignments, 1 which are expected in hierarchical models of galaxy formation, can account for the high 0 9 fraction of warped galaxies. Our simulations exemplify the role of misaligned spins in 9 warp formation even when the halo density is not significantly flattened. / h Subject headings: galaxies: evolution — galaxies: halos — galaxies: kinematics p - and dynamics — galaxies: structure — radio lines: galaxies o r t s a : v i X r a 1Currentaddress: AstronomischesInstitut, Universita¨tBasel,Venusstrasse7,CH-4102Binningen,Switzerland 1 1. Introduction example of a warp formed through the addition of a misaligned torus of matter. An “integral sign” twist has been observed in the If the accreted material is flattened due to its in- extended HI disk ofmany galaxies;insome casesit trinsic spin, a density misalignment is present imme- canalsobe seenin the star light. Briggs(1990)char- diately which exerts a torque to twist the disk, as in acterizedthebehaviorofasampleof12warpedgalax- Jiang&Binney’smodel. Butdissipationlesshaloma- ies as: (1) coplanar inside R25, and warped beyond, terialmaynotbestronglyflatteneddespitestreaming with a straight line of nodes (LON) inside RHo, (2) about a misaligned axis. Nevertheless such material changing near RHo, (3) into a LON on a leading spi- willexertatorqueonthediskthroughdynamicalfric- ral(as seen fromthe inner disk) outside RHo. Bosma tion, causing the disk’s angular momentum vector to (1991) found 12 clearly warped disks in a sample of tip towards alignment with that of the halo, as we 20edge-onsystems;takingintoaccountrandomwarp show in this Letter. orientation,thetruefractionofwarpeddisksmustbe Rotation in the inner halo will cause the disk to larger. This high fraction of warped galaxies implies tilt differentially because the inner disk experiences either that warps are long lived features or that they a strongerdynamical frictiontorque (since the densi- are repeatedly regenerated. ties of both disk and halo are highest) and because If a twisted disk were modeled as a set of uncou- time-scales are shortest in the center. As the in- pled tilted rings in a flattened potential, their chang- ner disk begins to tip, the usual gravitational and ingprecessionrateswithradiuswouldleadtoawind- pressure stresses in a twisted disk become impor- ing problem, similar to that for spirals (e.g. Binney tant. Our fully self-consistent N-body simulations & Tremaine 1987). If warps are long-lived, there- show that this idea is actually quite promising and fore, some means to overcome differential precession leads to warps which are relatively long-lived and of is required (see reviews by Toomre 1983 and Binney the observed form. 1992). Dynamical friction arises through an aspherical Most recent ideas for warp formation rely in some density response in the halo that lags “downstream” way on the influence of the halo. Toomre (1983) and from the disk. In the long run, however, the den- Dekel & Shlosman (1983) suggested that a flattened sity distribution in the halo must become symmetric halo misalignedwith the disk cangive riseto a warp, aboutthediskplane,evenwhileitcontinuestorotate and Toomre (1983), Sparke & Casertano (1988) and aboutamisalignedaxis. Simple time-reversibilityar- Kuijken(1991)foundrealisticwarpmodesinsiderigid gumentsdictatethatasteadysystemcannotsupport halosofthisform. However,angularmomentumcon- net torques (cf. the “anti-spiral” theorem, Lynden- servation requires there to be a back reaction on the Bell & Ostriker 1967; Kalnajs 1971). We therefore halo (Toomre 1983; Binney 1992); Dubinski & Kui- findthedynamicalfrictiontorqueonthediskdoesnot jken (1995) and Nelson & Tremaine (1995) showed persistindefinitely. Wedonotregardthisasaserious thatamobilehaloshouldcauseawarpeddisktoflat- objection to our model, since late infalling material ten quickly (but see also Binney et al. 1998). must constantly revise the net angular momentum of As a warp represents a misalignment of the disk’s the halo. inner and outer angular momenta, Ostriker & Bin- ney (1989) proposed a qualitative model in which 2. Numerical Method the warp is generated by the slewing of the galac- tic potential through accretion of material with mis- Since mild force anisotropies in many grid-based aligned spin. The accretion of satellites by larger N-body methods can cause an isolated disk to set- galaxies,suchasintheMilkyWay,providesdirectev- tle to a preferred plane (e.g. May & James 1984),we idenceoflate-infallingmaterialwithmisalignedangu- adoptacodewithnopreferredplane. Anexpansionof lar momentum. Such misalignments are expected to the potential in spherical harmonics has been widely be generic in hierarchicalmodels of galaxy formation used for N-body simulations both with a grid (van (Quinn&Binney1992)inwhichthespinaxisoflate- Albada 1982) and without (Villumsen 1982; White arriving material, both clumpy and diffuse, is poorly 1983; McGlynn 1984). Here we adopt an interme- correlated with that of the material which collapsed diate approach: we tabulate coefficients of a spher- earlier. Jiang & Binney (1998) calculate a concrete ical harmonic expansion of the density distribution 2 Fig. 2.—Thedisktiltangle(indegrees)asafunction Fig. 1.— The mid-plane force gradient for the disk of radius at intervals of 50 time units (t = 50 to t = described in section 4. The solid line is the exact 950 from bottom up). gradient, the dashed lines show values obtained with lmax =10, 20, 35 & 50 in order of decreasing error. galaxy. on a radialgrid, and interpolate for the gravitational The initial massive disk had velocity dispersions forces between the values on these shells. The radial set by adopting Toomre Q = 1.5 and a thickness of grid smooths the gravitational field, thereby avoid- 0.1 Rd which were both independent of radius. The ing the problem of “shell crossings.” Since there is massless particles startedwith exactly circular orbits no gridding in the angular directions, we retain the in the disk mid-plane. The central mass is a single full angular resolution up to the adopted lmax – the particle with a core radius of 0.3Rd. maximumorderofthe sphericalharmonicexpansion. The initial distribution of halo particles was gen- While avoiding a preferred plane, this method is eratedbythe methodfirstusedby Rahaetal.(1991) notwell-suitedtorepresentationofdisks. Thevertical whichgivesanexactequilibriumisotropichalointhe restoringforcetothediskmid-planeconvergesslowly potential of the disk and central mass. The halo ex- with increasing lmax, as shown in Figure 1. Most of tends to a radius rtrunc = 30 Rd. Because the disk our simulations included terms to lmax = 10 only; hasmass,thehaloisnotpreciselyspherical;itsinitial tests with higher lmax (and fewer particles) suggest axis ratio varies from closely spherical at r ≥ 4Rd to these models overestimate the magnitude and dura- c/a ≃ 0.7 near r = 1.5 Rd. The initial halo angular tion of the warp in the massive part of the disk, al- momentum was created by selectively reversing halo though milder and shorter-lived warps still develop. particlevelocities aboutachosenaxis,whichfor Run Moreover, the warp in the test-particle layer beyond 1 is tipped awayfromthe disk spin(z-)axisby 45◦ in the edge of the massive disk is unaffected by force the x-direction. We chose a value of the dimension- resolution. less λ ≡ GLqM|E5| ≃ 0.07 for our halo models; here L, E and M are respectively the total angular momen- 3. Initial model tum, energy and mass of the halo. A value of 0.07 is typical in hierarchicalclustering models (Barnes & While we have found that a massive halo is able Efstathiou 1987; Steinmetz & Bartelmann 1995). to produce spectacular warping, we here prefer more realistic minimal halo models (cf. Debattista & Sell- We work in units where G = M (= Mdisk + wood 1998). Our galaxy model has three massive Mhalo) = Rd = 1; the unit of time is therefore (R3/GM)1/2. A rotation period in the disk plane at components: an exponential disk of length-scale R , d d truncated at 8 R , a polytropic halo and a central R=1is32. Wesetlmax =10,usedaradialgridwith d 200 shells and a time step of 0.05. The disk and halo softenedpointmass. Theratiodisk:halo:centralmass components are represented by a total of 106 equal is 1:9:0.2,chosento give a roughlyflat rotationcurve mass particles. Our simulations conserve energy to out to ∼ 15 R . We also include a disk of test par- d better than 0.04%. ticles, extending well outside the massive disk, that is intended to mimic the neutral hydrogen layer of a 3 Fig. 3.— Contours of the disk density in Run 1 at t=400projectedontothe(x,z)plane. Thedot-dash lineindicatesthecross-sectionofthelayerofmassless particles. Bythistime,theinnerdiskhastilted∼10◦ away from its original plane, which was horizontal. 4. Results 4.1. Warp in the massive disk The outer disk lagsas the inner partofthe disk in Run1beginstotilt,causingawarptodevelopalmost at once. Figure 2 shows that the approximately rigid tiltoftheinnerdiskincreasesrapidlyatfirstandthen moreslowly,whiletheradiusatwhichthewarpstarts also moves outwardsovertime. The disk in Run 1 at t=400 is shown in Figure 3. Fig. 4.— Polar angle plots showing the shape of the Figure4 showsthe warpofthe massivediskin the warped disk. The points indicate the angles (θ,φ) form of a Tip-LON diagram (Briggs 1990) at equally of the symmetry axis of an annular piece of the disk spaced times. Each point represents the direction of (width 0.8 R ) relative to the inner disk axis at the d thenormaltothebest-fitplaneofanannularpieceof giventime. Theradialcoordinateisthepolarangleθ the disk, with the center of the disk defining the ori- and the azimuthal coordinate shows the azimuth, φ, gin. The normalto the inner disktilts initially in the of the LON of each annulus. The boundary circle is (x,z)planewhiletheouterdiskisleftbehind,thereby at θ =10◦. Disk rotation is counter-clockwise. shifting outer-disk points along the φ = 180◦ direc- tion (e.g. t = 100) while the almost flat inner part 4.2. Secular evolution of the disk gives rise to the concentration of points in the center. The warp reaches a maximum angle of Several changes occur as the model evolves: the ∼ 7◦ at t ≃ 350 and it takes roughly 700 time units dynamicalfrictionforce driving the warpdecaysover (∼ 20 disk rotations at R =R ) for most of the disk d time, as expected from §1, causing the inner disk to to settle back to a common plane. tilt more slowly (Figure 2). The radius at which the The leading spiral, reminiscent of Briggs’ (1990) warpstartsmovesoutwardsandthe amplitudeofthe thirdrule,developsthroughclockwisedifferentialpre- warp (the difference in tilt angle between the inner cessionintheouterparts. Precessionisaconsequence and outer disk) also decreases. The massive disk is ofgravitationalcouplingbetweenthe innerandouter almost coplanar again by time 1000 (Figure 4). Spi- disk (Hunter & Toomre 1969). The extremely slow ral arms and a weak bar also drive up the velocity precessionofthe outermostedge ofthe diskindicates dispersion of the particles to Q≃2.5. that it is subject to a very mild torque, arising al- The inner disk tilts remarkably rigidly indicating most exclusively from the distant, tipped inner disk. strong cohesion which arises from two distinct mech- Theouterdisk wouldprecessmorerapidlyifthe halo anisms. Most studies have focused on gravitational density distribution were flattened. forces,whichHunter&Toomre(1969)foundwerein- 4 4.3. Test particle layer The sheet of test particles is intended to approx- imate a gaseous disk. Being massless, it does not induce a response from the halo, but is perturbed by forces from the tilted massive disk and its associated halo response. Within 8 R the test particles simply d tilt or warp with the disk. Outside the massive disk, however,thedisturbanceforcesfromthehaloandthe massive disk drop off rapidly and the plane of this dynamically-cool sheet hardly moves at large radii. Fig. 5.— The warp angle as a function of radius for It therefore appears warped relative to the plane of three different models. Solid line: our standard (run the tilted disk, andremains soevenby the endofthe 1), dashed line: hotter disk (run 2) and dot-dashed simulation when the stellar disk has mostly settled. line: greater misalignment (run 3). All warp angles are at t=450. 4.4. Further Simulation The spin axis of the halo in Run 1 was initially adequate to persuade the outer disk to precess along inclined at 45◦ to that of the disk. In Run 3, we set with the inner disk in a steady mode. However, the this angle to 135◦ thereby reversing the sign of J , z diskisalsostiffenedbytheradialepicyclicexcursions thecomponentofthehalo’sangularmomentumalong of the stars which communicate stresses across the the z-axis. The larger misalignment angle causes the disk. innerdisktotiltfasterandfurtherandgivesrisetoa Both self-gravity is strongest and random motion largerwarp,asshowninFigure5. Dynamicalfriction is greatest in the inner disk, where the coupling is also lasts longer. Other properties of the warp are evidently strong enough to preserve its flatness. The similar to those in Run 1; in particular, the warp settling of the disk to ever larger radii should be de- begins at a similar radius at equal times. scribableintermsofthegroupvelocityand/ordamp- ingofbendingwavesinawarmandfinitelythickdisk, 5. Conclusions but the absence of a dispersion relation valid in this regimeprecludesacomparisonwiththeory. Itisinter- Our simulations have confirmed that dynamical estingthateachannulussettlesasitsprecessionangle friction from a halo having angular momentum mis- reaches ∼ 180◦ (Figure 4), thereby preventing exces- alignedwith thatof the disk causesa transientwarp. sive winding of the warp. The settling of each ring Thewarphastwopropertiescommonlyobserved: the after half its precession period could be coincidental LON traces out a leading spiral relative to the inner butwehaveseenitinmanymodels. Onepossiblerea- disk and lasts longest in the HI layer. sonisthatawarmdiskcannotsupportbendingwaves By driving the warp, we side-step the most trou- withwavelengthshorterthanthe averageepicycledi- blesome difficulties faced by other warp mechanisms. ameter;apredictionbasedonthisideaisonlyroughly Thebaneofglobalmodewarpmodels,thatforcesare in accord with the radially dependent settling time, simply too weak to overcome differential precession however. It should be noted that whether settling is near the edge, has become a strength in our mecha- describedby groupvelocity,wavedamping orpreces- nism: theweakcouplingoftheouteredgecreates the sion angles, it should be more rapid in a disk with warp. Furthermore, the gradual settling of the warm stronger forces towards the mid-plane. disk avoids any winding problem. Todemonstratetheimportanceofrandommotion, The massive disk can warp slightly, but is largely we ran a new simulation (Run 2) identical to Run 1 rigid both because of gravitational restoring forces except with Q = 4.0 initially in the disk. The warp andradialpressure. Thesizeandlifetimeofthewarp wasmuchreduced,asshowninFigure5,eventhough in the massive disk are probably somewhat overesti- the inner disk tilted by an angle comparable to that mated because our numerical method does not yield in Run 1. thefullgravitationalrestoringforce. Thisworrydoes not affect the conclusions about the warp in the ex- 5 tended H I layer, which has very little mass and Dekel, A. & Shlosman, I. 1983, in IAU Symposium rigidity. 100, Internal Kinematics and Dynamics of Galax- We have deliberately adopted an almost spherical ies, ed. E. Athanassoula (Dordrecht: Kluwer), 187 haloinordertoshowthatwarpscanbe formedwith- Dubinski, J., & Kuijken, K. 1995,ApJ, 442, 492 out misaligned density distributions. Rotating halos Hunter, C., & Toomre, A. 1969, ApJ, 155, 747 are likely, of course, to be slightly flattened also, in whichcasethediskwillrespondtobothtypesofforc- Jiang, I.-G. & Binney, J. 1998, MNRAS, submitted ing. This will lead to warps that precess, HI layers (astro-ph/9807161) thatdo experience forces,andsoon. Studies ofthese Kalnajs, A. J. 1971,ApJ, 166, 275 cases will be reported in a future paper. Kuijken, K. 1991,ApJ, 376, 467 Asnotedabove,weexpectthenetangularmomen- Lynden-Bell, D., & Ostriker, J. P., 1967, MNRAS, tumofthehalotoberevisedasmaterialcontinuesto 136, 293 straggleinlongafterthemaingalaxyhasreachedma- turity. Every change to the halo’s spin vector can be May, A., & James, R. A. 1984, MNRAS, 206, 691 expectedtoaffectthediskthroughfriction,evenifthe McGlynn, T. A. 1984,ApJ, 281, 13 arrivingmaterialistornapartatlargedistancebythe Nelson, R. W., & Tremaine, S. 1995, MNRAS, 275, tidalfieldofthehostgalaxy. Ourpictureissimilarto 897 that proposedby Ostriker & Binney (1989),but who envisage warps as being driven from the outside by Ostriker,E.C.,&Binney,J.1989,MNRAS,237,785 a misalignment of the inner disk with the flattened Quinn, T., & Binney, J. 1992, MNRAS, 255, 729 outer halo. In practice, both mechanisms must be Raha, N., Sellwood, J. A., James, R. A., & Kahn, F. inextricably linked. On-going infall makes it hardly D. 1991, Nature, 352, 411 surprising that warps are detected in most disks. Sparke, L. S., & Casertano, S. 1988, MNRAS, 234, ConversationswithScottTremaineandAlarToomre 873 andthereportofthereferee,JamesBinney,weremost Steinmetz, M., & Bartelmann, M. 1995, MNRAS, helpful. The authorswishto thank the IsaacNewton 272, 570 Institute, Cambridge, England for their hospitality Toomre, A. 1983, in IAU Symposium 100, Inter- for part of this project. This work was supported nal Kinematics and Dynamics of Galaxies, ed. E. by NSF grantAST 96/17088and NASA LTSA grant Athanassoula (Dordrecht: Kluwer), 177 NAG 5-6037. van Albada, T. S. 1982, MNRAS, 201, 939 REFERENCES Villumsen, J. V. 1982,MNRAS, 199, 493 Barnes, J., & Efstathiou, G. 1987, ApJ, 319, 575 White, S. D. M. 1983, ApJ, 274, 53 Binney, J. 1992, ARA&A, 30, 51 Binney,J.,&Jiang,I.-G. &Dutta,S.1998,MNRAS, 297, 1237 Binney, J., & Tremaine, S. 1987, Galactic Dynamics, (Princeton University Press: Princeton) Bosma,A.1991,in WarpedDisks andInclinedRings Around Galaxies, ed. S. Casertano, P. D. Sackett &F.H.Briggs(Cambridge: CambridgeUniversity Press), 181 Briggs, F. H. 1990, ApJ, 352, 15 Debattista, V. P., & Sellwood, J. A. 1998, ApJ, 493, L5 This 2-column preprint was prepared with the AAS LATEX macrosv4.0. 6