The Astrophysical Journal, 580:705–717, 2002 December 1 # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
‘‘ SLOSHING ’’ LIBRATIONS IN LOPSIDED DISK GALAXIES David A. Kornreich Center for Radiophysics and Space Research, Cornell University Space Sciences Building, Ithaca, NY 14853; and Department of Physics and Physical Science, Humboldt State University, Arcata, CA 95521;1 [email protected] R. V. E. Lovelace Department of Astronomy, Cornell University, Cornell University Space Sciences Building, Ithaca, NY 14853; [email protected] and Martha P. Haynes Center for Radiophysics and Space Research and National Astronomy and Ionosphere Center,2 Cornell University Space Sciences Building, Ithaca, NY 14853; [email protected] Received 2002 January 13; accepted 2002 July 29
ABSTRACT A combined particle-mesh N-body/gridded hydrodynamics/stellar evolution code is employed to model the response of a galaxy disk to an impulsive perturbation. The resulting galaxy is examined for asymmetry in morphology and dynamics to answer the question of whether free ‘‘ sloshing ’’ librations in the plane of the disk can explain the relatively large proportion of asymmetric field spirals. The simulation produced a perturbed disk containing streamers of gas and stars, gas shocks, and m ¼ 1 and m ¼ 2 spiral structure. Strong kinematic peculiarities were generated as well. However, although the resulting disk reproduced many of the kinematic features observed in H i synthesis maps, the effects are relatively short-lived, lasting only on the order of a dynamical timescale. The similarities between the model and the tidally deformed galaxy NGC 5474, however, do leave open the possibility of forced sloshing modes playing a role in galaxies undergoing tidal interactions. Subject headings: galaxies: kinematics and dynamics — galaxies: structure — stellar dynamics
1. INTRODUCTION stable to tightly wrapped lopsided m ¼ 1 spiral waves. Jog Although studies of spiral galaxies commonly assume an (1999) examines the response of the disk to lopsided halo potentials and finds that the self-gravity of the disk tends to axisymmetric disk in both morphology and dynamics, there oppose lopsidedness in the halo but that the lopsided poten- is growing evidence that many galaxies lack such symmetry. tial dominates in the outer regions, in general agreement Baldwin, Lynden-Bell, & Sanscisi (1980) were the first to with observations. Lopsidedness may also be the result of examine axisymmetry of galaxies and noted that a signifi- stochastic self-propagating star formation, weak tidal inter- cant fraction are symmetric in neither gas nor optical light. actions (Zaritsky & Rix 1997), disk travel through an inter- Based on more recent studies of optical appearance, as galactic medium, kinematic decoupling (Phookun, Vogel, & shown in Rix & Zaritsky (1995), Zaritsky & Rix (1997), and Mundy 1993; Sellwood & Merritt 1994; Walker, Mihos, & Kornreich, Haynes, & Lovelace (1998, hereafter KHL), Hernquist 1996), or minor mergers (Walker et al. 1996). 30% of disk galaxies exhibit significant ‘‘ lopsidedness.’’ Observations made by KHL such as the Kitt Peak 0.9 m Another significant possibility is that an optical disk may be in a quasi-stationary lopsided state in a symmetric poten- observations of NGC 1073 indicate that even in otherwise tial, as discussed by Syer & Tremaine (1996). In this model, normal galaxies, the optical center of light may be displaced gaseous and stellar matter swirls about the minimum of the from the center of light of the disk isophotes in a significant halo potential in a state not fully relaxed. The result is a lop- fraction of cases. Kornreich et al. (2000, hereafter KHLvZ) sided flow within a symmetric mass distribution. Numerical demonstrated that certain dynamical quantifiers, such as simulations of a similar situation have been done by Levine the normalized difference in approaching and receding rota- & Sparke (1998) using a gravitational N-body tree-code tion curves and changes in observed position angle, also are method for disk galaxies shifted from the center of the main detected in about 30% of otherwise normal disk galaxies. halo potential. In that simulation, a galaxy disk of 20,000 The origin of the lopsidedness in disk galaxies remains unclear. Baldwin et al. (1980) suggest that the lopsidedness gravitating particles distributed in a Kuz’min disk was placed off-center in a static isothermal dark halo potential in outer disks is a kinematic feature that should result in and given the appropriate tangential orbital velocity. They leading spirals. Lovelace et al. (1999) study the gravitational interaction of a series of concentric gravitating rings as a found that when the sense of the disk spin is opposite to that of the orbit about the halo center, the disk remains off-cen- model for galactic structure and predict that disks are un- ter for several galactic rotation periods, so long as the initial offset was within the core radius of the halo. In further work conducted by Noordermeer, Sparke, & Levine (2001), rota- tion curves, velocity maps, and individual orbits were 1 Current address. 2 The National Astronomy and Ionosphere Center is operated by calculated for these models. The results were again that in Cornell University, under a cooperative agreement with the National the case where the disk orbits the halo in a prograde sense, Science Foundation. the asymmetries damped out quickly, but when the orbit is 705 706 KORNREICH, LOVELACE, & HAYNES Vol. 580 retrograde, the resulting rotation curves and velocity maps The GALAXY code also contains several routines that can be highly asymmetric even when the morphological lop- allow the four components of the model galaxy to interact and sidedness is small. Additionally, some of the transients in evolve into one another (Shorey 1990). These are baryonic the prograde orbits produced symmetric rotation curves mass infall from the halo, Jeans collapse of the gas component, and velocity maps but lopsided morphology. star formation triggered by shocks in the gas, star formation Here we study a numerical model of a ‘‘ freely sloshing ’’ due to cloud collision, and stellar supernovae, which return galaxy disk as it oscillates around the center of the halo mass and energy to the gas. The rate of each of these processes potential in the disk plane. The sloshing is ‘‘ free ’’ because is set by the user as initial conditions. we allow the galaxy to oscillate independent of any other gravitating mass. This model will consider the fluid dynam- 2.1.1. Physical Conditions ics of the gas, stellar evolution, as well as N-body particle The initial setup of the galaxy simulation was intended to interactions, within the context of a static dark halo poten- mimic, as closely as possible, the makeup and dynamics of tial. In nature, libration of the galaxy disk might occur after the typical L* Sc spiral galaxy. Because of its simple analytic a perturbation caused by a tidal encounter, mass infall from density profile and well-understood dynamic properties, the the halo or from companion galaxies, or minor mergers. Kuz’min (Toomre model 1) disk with initial Q ¼ 1:1 was In this work, we present the results of a numerical simula- selected to be used for data collection in the simulation. tion combining an N-body code, a hydrodynamics code, Numerical heating in the model raises Q to orders 2–4 in the and a stellar evolution code to model a galaxy undergoing simulation rather quickly, and since we allow transients to ‘‘ sloshing ’’ librations parallel to the disk plane. The com- diminish in the disk before including the perturbation as dis- bined code allows particles (stars and molecular clouds) to cussed below, the exact initial value of Q is unimportant to be integrated alongside the gas component and simulates the bulk of the simulation. star formation, Jeans collapse, and supernova processes. Because the Kuz’min disk is analytically infinite in extent, The libration is set in motion by imparting an impulsive with surface density force to the entire disk at t ¼ 0. If the libration were found to be underdamped and long-lived, then such free modes M a ðRÞ¼ ; ð1Þ may explain the ubiquity of lopsidedness even in isolated 2 ðR2 þ a2Þ3=2 galaxy disks. The resulting models can be analyzed for asymmetry in morphology and dynamics following the dis- where a is the disk scale length and M is the mass of the cussion of KHL and KHLvZ, and the features can be disk as R !1, the disk is arbitrarily truncated at compared to the asymmetric features of disk galaxies Rtrunc ¼ 6a. In this paper, we use the timescale R to refer to examined in those papers. the period of disk rotationpffiffiffi at the radius enclosing half of the In x 2 we discuss the application of the code to the prob- disk mass R1=2 ¼ a 3. For a typical disk galaxy, with 10 lem of librationally induced asymmetry, including the initial M 6 10 M and a 3:5 kpc, R 108 Myr and conditions of the model galaxy disk and its kinematic Rtrunc 21 kpc. The selection of this disk also allowed units features. In x 3 we present the data obtained from the simu- used in the simulation to be more accurately converted to lation and discuss the response of the disk to the initial physical units. impulsive perturbation. Finally, in x 4 we compare the After the selection of the form of the disk density profile, features and asymmetry of the simulation with features the next step was to determine the values and the form of observed in nature and draw some general conclusions. the dark halo and mass fractions for each of the components that was representative of observations of actual galaxies. Based on estimates derived from Navarro, Frenk, & White 2. APPLICATION OF THE COMBINED CODE TO (1996), a static isothermal halo of mass fraction 0.85 was THE PROBLEM OF ASYMMETRY selected. The ratio of halo core radius to disk truncation radius was selected to be 0.39, based on observations by 2.1. Setting the Initial Conditions Rhee (1996). This paper reports the outcome of several runs of the Typical mass fractions of the halo ( fh), molecular gas GALAXY program developed by Schroeder & Comins ( fc), atomic gas ( fg), and stars ( fs) were obtained based (1989), Schroeder (1989), Shorey (1990). The code is a two- on observations of H i and optical fluxes of a large disk gal- dimensional, combined ballistic N-body, hydrodynamics, axy sample covering a wide range of surface brightnesses and stellar evolution code developed at the University of and Hubble types by McGaugh & de Blok (1997). In that Maine and kindly provided by N. F. Comins. study, it was found that the mass ratio of molecular gas to The GALAXY program combines a particle-mesh (PM) atomic gas could be parameterized by Hubble type T: fast Fourier transform (FFT) technique for calculating a MðÞH2 gravitational potential with an Eulerian finite-differencing ¼ 3:7 0:8T þ 0:043T2 : ð2Þ hydrodynamic integration into a single simulation in two MðÞH i dimensions. The grid used for the FFT is the same grid on For ¼ 6, this relation gives MðÞH =MðÞ¼H i 0:448. which the gas flows and is of dimensions N N , where T 2 G G This constrains the simulation parameters NG is the number of grid squares to a side. We have taken N to be 512 for this work. During the integration, each grid G fc=fg ¼ 0:448 : ð3Þ square contains values for the gravitational potential, the mass density, the energy density, the x^-momentum density, Elsewhere in the same study, the mass fraction of gas with and the y^-momentum density of the gas in that square. respect to the baryonic disk fG MG=ðÞMG þ M , where Positions, momenta, masses, and ages of star clusters and MG ¼ Mg þ Mc is the total mass of the gas (both atomic molecular clouds are contained in separate arrays. and molecular) and M* is the mass of stars in the disk, is No. 2, 2002 ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 707 examined. The authors found that for 5 T 7, several times and the effects of adjusting the evolutionary fG ¼ 0:414 0:16. This sets parameters was evaluated in the following way. First, the simulation was run without evolution for a total of 2000 fg þ fc fG ¼ ¼ 0:414 : ð4Þ time steps (10.6 galactic revolutions) with the physical fg þ fc þ fs parameters already determined in equation (5) to allow transients due to the initial setup to dissipate. The final state The solution of equations (3) and (4), with the constraints of the simulation after these 2000 steps was then saved to that f ¼ 0:85 and that the sum of all mass fractions must be h serve as a starting point for following runs. unity, yields the mass fraction parameters used in the The total mass of gas lost off the edges of the simulation simulation: 2 3 2 3 was tracked for each time step and equalled 0.0422% of the fh 0:85 mass of the halo after transients had dissipated. Adding to 6 7 6 7 this value the 5 M mass of star formation per year desired, 6 f 7 6 0:0429 7 6 g 7 6 7 the total mass expected to be lost from the gas phase per 4 5 ¼ 4 5 : ð5Þ R fc 0:0192 was 1.55% the mass of the halo, and thus Rif was set to 1.55, fs 0:0879 to balance the gas mass lost. The simulation was then con- tinued from the previously saved state several times with The remaining gas parameter, representing the average evolution activated, varying the remaining evolutionary internal energy of the gas in a cell, was set via a trial-and- parameters. It was found that the values error process. A few trial runs indicated that values for the 2 3 2 3 initial temperature between 10 and 100 K all yielded qualita- Rcf 0:9 6 7 6 7 tively similar results. Therefore, an average gas temperature 4 Rsfc 5 ¼ 4 0:9 5 ð6Þ of 20 K was adopted for this simulation. Rsfs 0:95 2.1.2. Numerical Conditions produced the desired rate of star formation, a constant The number of particles used in the simulation depends number of molecular clouds, and a constant mass of the gas on two considerations: speed of execution and size of the phase. Once these parameters were determined, the simula- data dump. The more particles in the simulation, the slower tion was then continued for 1000 time steps beyond the orig- its execution and larger the data output. It is also important inally saved 2000 time steps, to allow transients due to the to recognize that, barring the few particles that are ejected activation of evolution to dissipate. The final state of the from the simulation, the absolute number of stars will simulation after the total of 3000 time steps (15.9 R) was always increase monotonically because of evolution. Many then saved as the starting point for the experiment proper. of these ‘‘ stars ’’ will be inactive remnants, but they are still treated as particles in the simulation. In this application, the 2.2. The Experiment initial number of star clusters was set at 105 and the number of clouds was set at 5000. These values resulted in approxi- The GALAXY code was used to examine the physics and mate program execution times of 1.2 hr and data dump sizes observable effects of a disk galaxy sloshing within the halo of 300 MBytes per galactic rotation when run on a Sun potential parallel to the disk plane. For the purposes of this Microsystems Ultra 5 400 MHz computer. paper, the details of the initial impulse are not considered and may be considered to arise from tidal forces, minor mergers, or interactions with other galaxies. The initial 2.1.3. Evolutionary Conditions impulse imparted to the disk was simulated here by instan- Rates of evolution in the simulation are controlled taneously imparting a velocity equal to one-third of the disk through the evolutionary efficiency parameters Rcf, Rsfc, rotational velocity in the +x^-direction for all gas cells, star and Rsfs, representing the evolutionary rates of cloud forma- clusters, and cloud particles. The simulation was then tion, star formation due to collapse, and star formation due allowed to run for 2000 time steps (10.6 R). The state of the to shocks, respectively, and the percentage infall rate from system was dumped to an output file every 10 time steps the halo Rif. Except for Rif, which represents a percentage (0.056 R). In addition, a second simulation was run for the rate per R of smooth, continuous infall onto the disk from same period for purposes of comparison, using the same ini- the halo, these parameters all fall between zero and unity tial conditions but with no impulse imparted. and represent the probability of cloud and star formation Evolutionary quantities such as the total amount of mass when the local conditions are favorable. These must be set converted into stars, clouds, and gas, as well as total masses such that the rate of star formation is physically reasonable of these components, were calculated for each time step. and the masses of the gas and cloud components remain as Figure 1 illustrates the masses of each component as a func- constant as possible. The evolutionary parameters, how- tion of time, normalized to t ¼ 0. The gas and cloud compo- ever, depend on the mechanics of the simulation, not only nents drop precipitously following the initial impulse, as a on physical constraints. Therefore, it was necessary to run result of significant losses off the edge of the simulation. The the simulation a number of times, examining the rates of gas gas mass recovers by t ¼ 5 R because to gas re-infall from mass escape off the grid boundaries, total halo infall, and beyond the simulation, while the cloud mass does not mass collapsing into and out of the cloud state, and iterating recover beyond McðtÞ=Mcð0Þ 0:95 within the simulated the values of the evolutionary parameters to make the proc- time period for several reasons: clouds, once lost off the grid, esses balance. escape the simulation; the depressed gas mass inhibits cloud In an average L* spiral, we expect on the order of 5 M of formation, and rapid star formation depletes the molecular star formation per year due to cloud collisions (Kennicutt clouds. This rapid star formation, caused by shocks due to 1998). Given these constraints, the simulation was run the impulse, keeps the mass of the stellar component from 708 KORNREICH, LOVELACE, & HAYNES Vol. 580
angle ¼ 270 . Because the initial perturbation was made in the +x^-direction, , then, represents the angle (in the plane of the galaxy) between the observed major axis and the direction of the perturbing impulse. Several intermediate data files were generated for given values of and i in formats readable by the GIPSY software package. Two of these represent the zeroth- and first- moment maps of the gas component, and the remainder represent maps of the star and cloud particle components. For both the perturbed and unperturbed simulations, moment 0, star, and cloud maps were made at i ¼ 0 . Moment 0 and moment 1 maps for the gas were also calcu- lated at i ¼ 45 ; ¼ fg0 ; 45 ; 90 ; 135 .
3.2. Data Reduction and Analysis The analysis of the morphological symmetry properties Fig. 1.—Masses of each component as a function of time in the of the models was performed before reading into GIPSY. perturbed model normalized by their values at t ¼ 0. Shown are the total simulation mass (solid line), the halo mass (dotted line), the hydrodynamic The measurement is conducted as described in KHL and mass (short-dashed line), the stellar mass (long-dashed line), and the KHLvZ by dividing the simulation grid into eight sectors molecular cloud mass (dash-dotted line). centered symmetrically on the center of mass of the disk and summing the total number of stars and the total mass of gas in each sector. The greatest difference between star number dropping, and in fact the total mass of stars initially or gas mass between two sectors is then taken and normal- increases slightly. Most of the star formation prior to ized by the total in all sectors, yielding models of the optical t ¼ 2 R is balanced by losses off of the grid, but it dominates and H i asymmetry measurements, Af ;R and Af ;H i, respec- the rising mass of the stellar component after that time. The tively. To suppress asymmetries due to the square shape of t rate of mass increase in the stellar component after ¼ 2 R the grid, and to parallel the analysis of actual galaxies, only is equal to that observed throughout the unperturbed points of radius 20 < r < NG=2 grid units are considered. It model. The decrease in the mass of the halo component rep- should be noted here that in performing similar analyses of resents the infall into the disk from the halo. observed galaxies, the wedge pattern is generally centered The format of the data dumps includes the density, inter- on the center of light of the object, whereas here we use the nal energy, and average momentum vector for each gas cell, disk center of mass, as it is easier to localize in the simula- age of each star cluster particle, and position, mass, and tion. Similar centering issues are discussed in KHL and momentum for each particle. Thus, because the number of KHLvZ, where it is found that in most cases these centers star cluster particles necessarily increases during the run, it are indeed well colocated, although exceptions may be was found that individual data dumps at each stage became found among highly distorted galaxies. The results of the progressively very large and that the entire output file analysis of the perturbed model are presented in Figure 2 as approached or exceeded the largefile limit of the operating 32 a plot of the morphological asymmetry parameter Af versus system, which in this case is 2 bytes (4.2 GBytes). Because simulation time. files larger than this limit require special treatment, it was Figure 2 demonstrates that the morphology of the disk decided to break the simulation up into two 1000 time-step responds to the impulse by reaching an asymmetry peak at chunks, so that the entire history of the model could be time t ¼ 0:36 R for the stars and t ¼ 0:58 R for the gas. recorded easily. g s These peaks, at Af ¼ 0:105 and Af ¼ 0:426, lie a factor of 5 above the baseline asymmetry values exhibited by the 3. DATA PRESENTATION AND REDUCTION unperturbed model and calculated as part of Table 1. The delayed response time and smaller response magnitude in 3.1. Initial Data Acquisition and Transportation to GIPSY the gas is due to fluid viscosity and turbulence of the gas Following the simulation proper, the data output file was component. The gas behaves as a viscous fluid because of interpreted for reading into the GIPSY data reduction envi- the Fourier grid, which provides a numerical viscosity. The ronment (van der Hulst et al. 1992; GIPSY 20003). During Reynolds number is of the order of number of grid elements this process, the raw output file was read, a morphological symmetry analysis was performed following the discussion TABLE 1 of KHL (see x 3.2), the simulation plane was reprojected for Unperturbed Asymmetry Parameters a desired viewing angle, and the various components were separated from one another. The reprojection was com- Parameter x x x pleted by rotating the simulation about the ^z-axis by an Ag ...... 0.018 0.006 angle , and then about the new x^0-axis by the inclination f As ...... 0.038 0.004 angle i. Thus the x^0-axis becomes the observed major axis of f the galaxy, which is always conventionally set to a position S2 ...... 3.74 1.81 hiD ðrÞ (deg)...... 0.57 0.50 hiD ð Þ (deg) ...... 8.87 3.61 jjD ðrÞ (deg)...... 19.0 6.75 3 > See the Groningen Image Processing System at jjD ð Þ (deg) ...... 19.0 6.71 http://thales.astro.rug.nl/~gipsy. > No. 2, 2002 ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 709
counterclockwise. Figure 5 presents the H i velocity field at i ¼ 45 , ¼ 0. Finally, Figure 6 presents the analogous gas component data for the unperturbed model for all times simulated, t < 10:6 R.
3.2.1. Morphological Response For simplicity in the following discussions of the model galaxy response to the perturbation, the cardinal directions ‘‘ north,’’ ‘‘ south,’’ ‘‘ east,’’ and ‘‘ west,’’ will be used to refer to the +y^, y^,+x^, and x^ axes of the model respectively, as it is presented in Figures 3–6. As illustrated in Figure 3, in comparison to Figure 6, the fluid gas disk in the perturbed model responds to the impulse in a radially dependent fashion. Motion of the nucleus of the galaxy, that region within 3 kpc of the dynam- ical center, is efficiently damped by the high density of vis- Fig. 2.—Morphological asymmetry parameter Af observed in the cous gas and the depth of the potential well. The outer perturbed simulation vs. time in units of rotation periods at the radius regions of gas, however, are sufficiently underdamped to enclosing half the disk mass, R, since the perturbation. Af is calculated by comparing total masses in eight symmetric sectors surrounding the disk exhibit a significant response to the perturbation. The outer center of mass. The asymmetries in both the stellar (solid line) and gaseous envelope of gas travels in the direction of the perturbation (dotted line) components are presented. The two components are shown and collects at a shock front, which propagates outward. together to demonstrate the comparative behaviors of Af with time; how- ever, the absolute quantitative measures of the two components are not This shock front manifests first as a spiral arm of density 19 2 directly comparable, since the gas component data was convolved with a 1:7 10 cm connected with the northern nuclear region. 19 2 beam approximating the VLA C Array before the analysis. By t ¼ 0:32 R, a dense lump of 2:5 10 cm forms in the arm, which has reconnected with the southern nuclear region, forming a ring. The front continues to propagate (K ¼ 512) in one direction. After reaching its peak, the outward, increasing in surface density, carrying 20% of the asymmetry in both components decays steeply, reaching the gas mass of the galaxy with it. The front exits the simulation baseline values after t ¼ 6 R. grid by t ¼ 0:6 R, leaving behind two streamers. Gas lost off These baseline values agree with asymmetry values the grid is recovered in part through the northern streamer, obtained from the unperturbed model and represent asym- which falls back onto the main disk from t ¼ 0:9 R to metry associated with the m ¼ 1 spiral structure seen in the t ¼ 1:5 . Material in the southern streamer continues to unperturbed model and the perturbed model at late times. R flow outward and disconnects from the main disk at t ¼ R. This spiral structure is produced mainly as a consequence of What remains is a trailing outer spiral arm left over from our choice for the halo core radius. Runs with significantly the northern streamer, and inner spiral structure at the larger halo core radii did not exhibit this effect. However, 1:5 1019 cm 2 level which is also observed in the unper- we chose to stay with smaller core radii as being more turbed model. This inner spiral structure persists in both physical. models for the remainder of the simulation and is caused by The data were then translated into GIPSY data cubes, evolution, as models without evolution do not show this simulating observed angular sizes by placing the simulation structure. The inner arm is a result of a combination of i grid at a distance of 10 Mpc. The H moment maps were supernova shock-wave propagation through low-density convolved with a 1900 1900 beam to simulate the data sets 4 regions and Jeans collapse of medium-density regions to obtained by KHLvZ with the Very Large Array in its C clouds. These processes tend to segregate the gas into high- configuration. The moment maps are presented in Figures density (>1:5 1019 cm 2) regions where the percentage 0 3–6. At the simulated distance of 10 Mpc, 1 ¼ 2:9 kpc. mass converted into clouds per time step is low and low- i The effective radius Reff, which encloses 50% of the H density (<1:5 1019 cm 2) regions, which are further mass, is depicted in each of these figures for visual reference. depleted by passing supernova shocks. Trailing spiral struc- This radius was chosen because it could be precisely deter- ture such as that seen in Figure 6 is the result. mined from the simulation; for disk galaxies it is equal to The stars, too, are affected by the perturbation in a radi- the optical radius R25 to within 20% (Rhee & van Albada ally dependent way, as demonstrated by Figure 4. There is 1996). little change in position of the stellar nucleus within a radius Figure 3 presents the gas component at inclination i ¼ 0 of 10, while stars inhabiting the outer regions dominate the as a function of time for times t < 2:18 R, by which time the dynamic response. Like the gas component, northern and model has qualitatively recovered from the impulse. After southern streamers form, creating a ring at times t ¼ 2:18 R, the morphology of the perturbed model is quali- 0:4 R < t < R. In the stellar component, the northern tatively similar to that of the control model. Figure 4 streamer dominates and is fed by star formation in the gas presents the stellar component of the perturbed model on shock front. The stellar features are more affected by the the same scale. As seen in these figures, the galaxy rotates galaxy’s rotation than the gas and can be seen to revolve about the galactic center to a much greater extent. The northern streamer takes a shape suggestive of a tidal tail by 4 The VLA is a facility of the National Radio Astronomy Observatory, t ¼ R, while the southern streamer dissipates beyond a which is operated by Associated Universities, Inc. under a cooperative radius of 1<5. The remainder of the southern streamer and agreement with the National Science Foundation. the inner 1<5 of the northern streamer become dominant 710 KORNREICH, LOVELACE, & HAYNES Vol. 580
Fig. 3.—Unprojected H i surface density mosaic for the perturbed model. Each frame is labeled by the time since the perturbation in units of R. Contours are atfg 0:5; 1; 1:5; 2; ... 1020 cm 2. The sense of disk rotation is counterclockwise, and the perturbation is in the x-direction. The circle in the first frame is of radius Reff and is included to indicate the edge of the visible disk and for cross-reference with other figures.
trailing m ¼ 2 spiral arms for the remainder of the simula- radial velocity in the northwest of the plot for times t < 2 R tion. Also present for the remainder of the simulation are and acting also to significantly distort the shape of the stellar arcs formed by stars falling back onto the disk. minor axis. The dynamical center, although morphologically rela- 3.2.2. Dynamic Response tively undisturbed by the impulse, nevertheless suffers sig- The velocity fields of the perturbed model are illustrated nificant dynamical distortion. At times 0:25 R < t < 0:5 R, in Figure 5. These figures were constructed by viewing the the apparent position angle in the dynamical center varies gas at an inclination of i ¼ 45 with no rotation in the plane by as much as 45 from the known true position angle of of the galaxy, such that the dynamical major axis of the 270 . galaxy is also the direction of the perturbation. It should be The velocity fields of the gas were also viewed from other noted that although velocities in the simulation are known a perspectives, all at inclinations of 45 but with the galaxy priori at all points on the grid, actual observations would having been rotated counterclockwise (in the direction only be able to trace the velocity field to some minimum of disk rotation) in the plane of the disk by value of NH i.Byt ¼ 0:16 R, streaming motions of some ¼f45 ; 90 ; 135 g. Similar features, as well as features 30 km s 1 are apparent along the minor axis of the per- not seen at ¼ 0 , are observed at the other azimuthal turbed model, particularly in the south. At times angles, as illustrated in Figure 7. 0:2 R < t < 0:6 R, strong deformations of the minor axis Strong distortions of the minor axis are observed at all , are apparent, creating S-shaped minor-axis contours, with as are S-shaped contours. Additionally, outer velocity con- apparent position angle changes with radius of more than tours parallel to the minor axis and intersecting the major 90 . During this same period, in which the gas shock domi- axis are observed on the receding sides of the ¼ 90 and nates the morphology in the west, contours of constant ¼ 135 mosaics during the reinfall phase at times radial velocity are disconnected, especially between the 1:6 R t 2:55 R. A shift in the dynamical center in the shock and the dynamical center, but also in regions on the direction of the impulse (north in this figure) is observed as eastern (approaching) side. In fact, a region of anomalously well, especially in the ¼ 90 mosaic during the initial slow rotation persists in the northeast of the model until outflow, 0:32 R t 0:96 R. Deviations from the known t ¼ 1:8 R. The northern streamer also affects the dynamics position angle of 270 are also observed during this time for for significant times, manifesting as a region of constant all . No. 2, 2002 ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 711
Fig. 4.—Stellar component mosaic for the perturbed model. Each frame is labeled by the time since the perturbation in units of R. Shading is linear and proportional to the number of stars at each point. The circle is of radius Reff.
In order to study the dynamical asymmetry produced by using the observed rotation curves: the perturbation in a quantitative way, rotation curves were S ðV sin iÞ produced for both models by applying the GIPSY routine 2 rot R rotcur (Begeman 1989) to the first-moment map at each time jjjj VrotðrÞ sin i jjVrotð rÞ sin i dr step for each of the four extracted values of . Because of 102 R : ð7Þ 1=2½ jjV ðrÞ sin i þ jjV ð rÞ sin i dr the large number of rotation curves involved, it was impos- rot rot sible to fit the curves iteratively, but the a priori knowledge To quantify the deviations from constant position angle, of the inclination and position angle allowed the efficient two measurements were taken. First, the maximum differ- searching of parameter space. For each time step, a rotation ence in position angle as a function of radius curve was constructed using both sides of the velocity field, with the inclination fixed at the known value of 45 leaving jjD ðrÞ >¼ max j ðrnÞ ðrmÞj ; ð8Þ position angle variable, but with a first ‘‘ guess ’’ the known position angle of 270 . Independent receding and approach- where n and m are indices of rings in the tilted-ring fit on the ing rotation curves were then constructed iteratively from same side (approaching or receding) of the fit, and second, the solution found using both sides. These curves were used the average of position angle differences between sides: to quantify dynamical asymmetry following the discussion X of KHLvZ. D ð Þ¼1N jj ð rnÞ ðrnÞ : ð9Þ Quantifying dynamical asymmetry was performed in n KHLvZ by examining the rotation curves of target galaxies. Note that the latter two parameters are introduced simply In a perfectly axisymmetric disk, approaching and receding as D ðrÞ and D ð Þ in KHLvZ. rotation curves should be identical, and the kinematic posi- In this case, the dynamical parameters of rotation curve tion angle should be constant as a function of radius and asymmetry between the approaching and receding sides. The measure R of rotation curve symmetry is performed by superposing the 2 R jjjjVrotðrÞ jjVrotð rÞ dr approaching and receding sides of the rotation curve and S2ðVrotÞ 10 ; ð10Þ 1=2½ jjVrotðrÞ þ jjVrotð rÞ dr integrating the differences between them, normalized by half of the velocity width. In KHLvZ, measurements were taken the maximum change in position angle with radius jjD ðrÞ >, 712 KORNREICH, LOVELACE, & HAYNES Vol. 580
Fig. 5.—H i velocity field mosaic for the perturbed model at i ¼ 45 , ¼ 0 . Each frame is labeled by the time since the perturbation in units of R. 1 Contours and gray scales are at intervals of 20 km s , with negative contours shown as dashed lines. The ellipse is of semimajor axis Reff. and the average difference in position angle between and are anticorrelated. The greater the magnitude of the approaching and receding sides at constant radius hiD ð Þ peak, the later the peak is reached. For instance, the greatest were calculated. Here we expand this analysis to also calcu- response was seen at ¼ 90 , in which the rotation curve late the average difference in position angle with radius on asymmetry reaches a value of S2 ¼ 33:7 at time t ¼ 0:80 R, the same side, hiD ðrÞ and the maximum change in position while the least response was seen at ¼ 135 , where the angle at constant radius between sides, jjD ð Þ >. rotation curve asymmetry only reaches a value of S2 ¼ 12:5 Once calculated for each time step and value of , the at time t ¼ 0:63 R.Byt ¼ 1:5 R, the rotation curve asym- parameters were Hanning-smoothed with time to reduce metry at all azimuthal angles is indistinguishable from that noise associated with the inability to manipulate and iterate in the unperturbed model. the fits manually. Average values and standard deviations The greatest response in both measures of D ð Þ are of the dynamical variables were then calculated for the observed at the high azimuthal angles ¼ 90 and unperturbed model by taking the mean and rms of the varia- ¼ 135 , where the respective peak values are bles over all values of t and . The results are given in Table jjD ð Þ >¼ fg74 ; 66 and hiD ð Þ ¼ fg51 ; 47 . Again, 1 along with the average values of Af in the unperturbed the parameters decay to the values observed in the unper- s model. All parameters except for Af are indistinguishable turbed model at time t 1:5 R. Peak values of jjD ð Þ > from zero at the 3 level. observed in the low azimuthal angles ¼ 0 and ¼ 45 The dynamical parameters are presented as functions of only just reach the 5 noise values seen in the unperturbed time in Figure 8 for the perturbed model for times t 4 R. model, while the peak value of hiD ð Þ does not even reach At times t > 4 R, the parameters for the perturbed model the 5 value for ¼ 0 . are qualitatively similar to and have the same average values Values of jjD ðrÞ > reach a narrow peak at t ¼ 0:61 R, and standard deviations as those for the unperturbed except for ¼ 0, in which the response is wider in time and model. reaches a peak at t ¼ 0:4 R. However, these differences are These figures show that the quantitative dynamical within 3 noise levels of the narrow peak in the ¼ 90 response of the disk to an instantaneous perturbation is sig- component. The greatest response is seen at ¼ 135 , nificant for times t < 1:5 R and that the response depends where the parameter reaches values of jjD ðrÞ >¼ 99 . None on the azimuthal angle . The parameter S2 is affected by of the values of hiD ðrÞ observed in the perturbed disk the perturbation by increasing to a maximum value within exceed the 5 value observed in the unperturbed disk, t < R followed by a rapid decay to the baseline value. The except for the ¼ 135 component, which reaches an 11 magnitude of the peak and the time to reach it depend on , peak at t ¼ 0:67 R. No. 2, 2002 ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 713
20 Fig. 6.—H i surface density mosaic for the unperturbed model. Each frame is labeled by the time in units of R. Contours are atfg 0:5; 1; 1:5; 2; ... 10 2 cm . The circle is of radius Reff.
4. DISCUSSION inner arms. The extension in UGC 6429 even exhibits two column density peaks, just as the simulation does for times 4.1. Comparison of Generated Features with Observation 0:4 R < t < 0:6 R. The similarity ends here, however; for, In this paper, we have used a combined N-body/hydro- while in the case of UGC 6429, the eastern extension does dynamics/stellar evolution code to simulate the dynamics not appear to affect the kinematics of the galaxy, the appear- and morphology of a galaxy given an impulsive ‘‘ kick.’’ ance of the feature in the simulation is associated with a The simulated galaxy exhibits a number of qualitative strong kinematic asymmetry. and quantitative features observed in actual instances of There were also many asymmetric features observed in galaxy asymmetry. In stellar morphology, the asymmetric KHLvZ which were not reproduced by this simulation. spiral structure observed in the model, where one arm domi- Incomplete rings such as those seen in NGC 991 (KHLvZ, nates the other in surface brightness, is observed in five of Fig. 2), the appearance of compression such as that seen in the 11 asymmetric sample galaxies discussed in KHL. The NGC 1042 (KHLvZ, Fig. 3), and the appearance of off-cen- remainder of the asymmetric galaxies discussed there are ter high column density regions such as in UGC 12742 asymmetric in shape, where one spiral arm is more curved (KGLvZ, Fig. 11) were not reproduced. These features may than the other, or where only one spiral arm is visible. These be the result of travel through an intergalactic medium or morphological features are also reproduced by the model at some other mechanism not modeled in this simulation. various times. Just after the impulse, structure is observed Quantitatively, the simulation produced short-lived on only one side of the nucleus, while at later times maximum morphological asymmetry parameters of order (0:8 R < t < 2 R), the main spiral arms generated in the 0.1 for both the gas and the stellar components, values stars are asymmetric. It should be noted, however, that none which are an order of magnitude higher than those of the galaxies in KHL shows the presence of a streamer, obtained for real galaxies in Table 4 of KHLvZ. While such as that present in the simulation for t < 4 R. the gas component only reaches just this order of magni- In gas morphology, the expanding shock front seen in the tude, maximum Af values for stars are higher than 0.4. simulated galaxy is similar to the eastern extension observed This high degree of asymmetry may be due to a number in UGC 6429 (KHLvZ, Fig. 6). Quantitatively, too, the col- of factors, but the predominant factor, at least in the stel- umn densities of both the model extension and the extension lar component, is likely to be due to the neglect of stellar in UGC 6429 are observed to be of the same order as the age. When a color and luminosity function is convolved Fig. 7.—H i velocity field mosaics for the perturbed model at i ¼ 45 and (top to bottom) ¼ 45 , ¼ 90 , and ¼ 135 . Each frame is labeled by the time in units of R. Contours and gray scales are at intervals of 20 km s 1, with negative contours dashed. ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 715
Fig. 8.—Dynamic asymmetry parameters for the perturbed galaxy as a function of time in terms of R. Top to bottom, left to right: Rotation curve asymmetry S2, maximum change in position angle with side jjD ð Þ >, average change in position angle with side hiD ð Þ , maximum change in position angle with radius jjD ðrÞ >, and average change in position angle with radius hiD ð Þ . Results are shown for ¼ 0 (solid line), ¼ 45 (dotted line), ¼ 90 (dashed line), and ¼ 135 (dot-dashed line). with the ages of stars in the simulation, and this in turn tent, and in fact a great deal of the detected asymmetry is passed through a simulated optical R-band filter, the lies in the low-density regions such as the streamers. measured stellar asymmetry may decrease. Also of inter- Dynamically, certain qualitative features in the simula- est here is the relative importance of the low surface tion were again similar to those observed in nature. The brightness and low surface density regions in the galaxy. S-shaped minor axis contours produced in the velocity Such regions require long exposure or integration times field are nearly identical to those observed in NGC 991, to be detected by telescopes, and even then may be NGC 1042, and NGC 5474 by KHLvZ. Moreover, detected at low S/N and are easily overlooked. In the occurrences of outer contours paralleling the minor axis simulation, however, these issues are lessened or nonexis- and intersecting the major axis was also observed in 716 KORNREICH, LOVELACE, & HAYNES Vol. 580
NGC 4688, NGC 5474, and NGC 5701. Finally, saddle baseline simulation value range of jjD ðrÞ >¼ 19 6=75. points in the velocity field are observed both in these gal- Larger values of 43.4, 95.4, and 43.0, however, are measured axies and in the simulation for times t < 0:74 R. These for NGC 1042, NGC 5474, and NGC 5701, respectively. features are illustrated in Figure 9, which compares fea- Although admittedly a very small sample, this represents a tures seen in the simulated velocity fields and the range of values supported by the sloshing model. observed velocity field in the peculiar disk galaxy NGC Of all the galaxies considered in KHLvZ, NGC 5474, a 5474 described in KHLvZ. This figure indicates regions galaxy known to be tidally perturbed by its dominant neigh- of analogous kinematic features in NGC 5474 and in two bor M101, is most consistent with the free sloshing mode frames of the simulation. Outer contours parallel with studied in this simulation. It exhibits, in fact, every measura- the minor axis, saddle points, and S-shaped contours are ble effect discussed here, except for a significantly high value indicated in the figure by A, B, and C, respectively. of hiD ð Þ . Other galaxies studied in that paper exhibit The other qualitative dynamic features, deviation of the some of the effects of free sloshing, but none as consistently. derived position angles and dynamical centers from their Evolutionary effects did not appreciably contribute to the known values, were of course impossible to observe in discussion here in most respects. We completed several runs actual galaxies where the ‘‘ true ’’ values of these quantities with no evolution (e.g., mass infall from the halo and star are not known a priori. Nevertheless, the displacement of formation) and several with partial evolution (one or the the dynamical center at early stages of sloshing, along with other) when running this simulation. The absence of mass the morphological damping of the inner few kiloparsecs of infall from the halo resulted in qualitatively similar results both the stellar and gaseous components implies that the in kinematics and morphology and did not affect the damp- kinematic and morphological centers of a galaxy will be ing timescale by more than 3%. However, the lack of infall decoupled during sloshing. Such an effect is certainly did result in steadily decreasing gas component masses. observed in NGC 5474, and should be most easily observ- Lack of star formation resulted in smaller values of the mor- able in the future in inclined galaxies. phological asymmetry parameters in the stellar component, The large maximum values of the rotation curve asymme- owing to absence of star formation at the shock front, but try parameter S2 obtained in the model were not observed in did not appreciably affect the gas kinematics or damping KHLvZ, but both sets of values lie within a factor of 2 of timescale. each other. Nonetheless, that S2 values given in KHLvZ are It should be noted that in this simulation, the halo is cal- significantly greater than the value of S2 ¼ 3:74 1:81 culated as a static potential and does not respond to the obtained from the unperturbed simulation is a confirmation gravitation of the disk. Because the halo is treated as static, that these galaxies are indeed kinematically asymmetric. momentum is not strictly conserved in the model. This is All values of D ð Þ, on the other hand, measured in likely to be a valid approximation to the behavior of real observed galaxies in KHLvZ are entirely consistent with the disk galaxies because of the high halo to disk mass ratio. In unperturbed baseline value of hiD ð Þ ¼ 8=87 3=61 cases where the halo response to a perturbation is large or obtained by the model. Values of D ðrÞ obtained there rapid compared to the disk, we might expect to see even occur over a wide range, many of which are within the greater asymmetries than are observed in this model,
Fig. 9.—Comparison of features seen in the numerical simulation with features actually observed in peculiar disk galaxy NGC 5474. Shown are the simulated velocity field at ði; ;tÞ¼ð45; 90; 1:92; upper leftÞ, ði; ;tÞ¼ð45; 0; 0:43; lower leftÞ, and the observed velocity field of NGC 5474 from Fig. 9 of KHLvZ, (right). Analogous kinematic features are indicated: (A) outer contours perpendicular to the major axis; (B) saddle points; and (C) S-shaped velocity contours. No. 2, 2002 ‘‘ SLOSHING ’’ LIBRATIONS IN DISK GALAXIES 717 because the disk will then be responding to a perturbed, and Here only a small part of the possible parameter space therefore presumably asymmetric, halo potential. of the perturbation model has been studied. It will be This model is also qualitatively consistent with the analy- desirable in the future to study a variety of strengths of sis of prograde N-body disks conducted by Noordermeer et sloshing perturbations and perturbative impulses, as well al. (2001). In both models, the disk sinks to the center of the as nonimpulsive perturbations. It will also be interesting halo potential quickly, but the transient phenomena in the to study the effects of galaxy disks centered in but rotation curves and velocity fields are very similar. inclined with respect to oblate halos as proposed by Lovelace et al. (1999). Galaxies of differing mass fraction 4.2. Conclusions and evolutionary tracks need to be examined, as do the We have studied here the effects of imparting a librational effects of Lindblad and corotation resonances on the per- energy to the disk of a galaxy embedded within a static iso- turbed system. One important possibility here is the study thermal halo in order to study the effects and evolution of a of a harmonic driving force to the disk of the galaxy, freely sloshing disk. While many of the features reproduced simulating the effects of an orbiting companion perhaps. by the simulation are observed in nature, it is also important It will also be desirable to study the effects of ram pres- to note the short damping timescales involved in the free sure on simulations such as these in an attempt to repro- mode. Most observable effects of the perturbation are duce the types of asymmetry seen in, for instance, NGC damped out within 4 R, and many are damped out within 1042 and NGC 5701. 2 R. That the timescales involved are of the order of a dynamical time indicates that free sloshing is probably not the dominant cause of most asymmetries found in disk gal- This work has been partially supported by NSF grant axies. However, the observation that the tidally disturbed AST 99-00695 to M. P. H. and by National Space Grant galaxy NGC 5474 consistently exhibits nearly all of the College and Fellowship Program grant NGT-40019 to effects observed in the simulation does indicate that driven D. A. K. We also gratefully acknowledge Neil F. Comins sloshing modes may play a part in galaxy asymmetry. for providing the GALAXY code.
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