Fig. 2 Coordinate system. The subscripts i, p, and d refer to the intruder, particle and disk respectively. (a), (b) Side and top view, respectively. (c) Initial configuration of the simulation (not to scale). The arrow indicates the direction of the intruder’s initial motion relative to the plane of the disk galaxy. M i, M d, m are the masses of the intruder, disk, and particle respectively. R p is the vector from the origin to the particle, which makes an angle φ p with the +z axis. The vector r p is the projection of R p onto the xy plane, which makes an angle θ with the +x axis. R i is the vector from the origin to the intruder, which makes an angle φ i with the +z axis. The vector r i is the projection of R i onto the xy plane. R d is the vector from the origin to the disk, which makes an angle φ d with the +z axis. The vector r d is the projection of R d onto the xy plane. The particle orbits about the tilted disk with angular velocity ω, while tracing out an angle ψ. The disk is tilted relative to the +z axis by an amount α. The vector r pd points from the center of the disk to the particle and the vector r ip points from the intruder to the particle, respectively. (a)(b)(c) The Theory We propose that the overall shapes of the warps in the disk galaxies we are studying are driven, in part, by ‘pseudoforces’. These forces come into play as the stars and other matter in the galaxy are pulled out of the initial plane of its orbit by the intruder’s gravitational attraction. This disturbance causes all or part of the disk to tilt slightly towards the intruder. However, this deflection does not immediately generate the ‘integral’ shape distribution. Rather, it gives rise to the three pseudoforces of classical mechanics (see below), which have components perpendicular to the plane of the tilted disk. (Prior to the tilt, all these pseudoforces are in the plane of the disk.) The pseudoforces on the tilted disk, we believe, cause the deflections of particles into the integral sign distribution. For reference, the pseudoforces are given by: The coordinates are as shown in Figure 2. From the equations above we will estimate the magnitude and direction of the deflection on each side of the disk due to each of the pseudoforces. We plan to compare the analytical magnitudes of the deflections with warp amplitudes obtained from the simulations. Inertial Effects from Companion Galaxies Driving Warps in Disk Galaxies N. F. Comins 1, N. Palestini 1, E. F. Borra 2, R. Hohlfeld 3,4,5, L. Wholly 1 1 Department of Physics and Astronomy, University of Maine 2 Département de Physique, Université Laval 3 Center for Computational Science, Boston University 4 Wavelet Technologies, Inc. 5 PhaseCapital LP Abstract Most disk galaxies, including the Milky Way, are observed to be warped 1 (Figure 1). The mechanisms for this warping are not well understood. We explore warping caused by the passage of a distant galaxy (the ‘intruder’). For this presentation, we have carried out N-Body simulations with plausible interaction parameters that give significant warps having amplitudes comparable to those observed in some disk galaxies. We propose that the distortion on the disk due to the intruder is caused in part by the effects of pseudoforces that occur while the two galaxies interact. These pseudoforces are highly non-linear dynamics whose theoretical equations we are now studying. Further observations of galaxies with these signature integral-shaped warps, located in the vicinity of massive, relatively distant neighbors, would support the existence of this mechanism. The Simulations We set up each simulation using GENICS 2, which generates the initial positions and velocities for 20,000 particles in our test galaxy. We insert the intruder as a single particle. This data set is then uploaded to Boston University’s computer Twister 3 and evolved using GADGET-2 4. We have created simulations covering all classes 5 of warps, but are focusing on the canonical ‘integral’-shaped ones. In each simulation the mass of the disk is 1 in natural units, thus each particle has a mass of 1/20,000. For the intruder, we set an initial position, velocity and its total mass. Figure 2 shows the coordinate system we use for our calculations and the initial relative orientation of the galaxy and intruder. Results In the figures below, the natural coordinate system of GENICS and GADGET has the origin fixed at the initial position of the center of the disk. (The coordinate system of Figure 2 is set up by the authors to clarify the equations of motion of the three bodies of interest.) Figure 3 shows a side view of the density contours of an unwarped galaxy simulation (no intruder). The red line in (b) through the contour maxima indicates that the disk is flat over its entire extent. Figure 4 shows the same view for a perturbed galaxy whose intruder has passed perpendicularly to the initial plane of the disk. At the time of the snapshot, the intruder is up and to the left. With the same red reference line drawn, secondary red lines follow the warped contours in the simulation’s outer regions. In a second distant encounter run (Figure 5), we show alternate views of the same data. As a further comparison with our distant encounter results, we present in Figure 6 the results of a close encounter started at the same initial angle as the run in Figure 5. References 1 K. Kuijken & I. García-Ruiz: Galactic Disk Warps. In: ASP Conf. Ser. 230, Galaxy Disks and Disk Galaxies, ed. by J. G. Funes & E. M. Corsini (Sheridan Books, Ann Arbor 2001) pp 401–408 2 Schwarzmeier J., “On the Simulations of Galaxy Dynamics and Their Application to Physics Education.” Ph.d. diss., U. West Bohemia, IBM pSeries 655 shared memory cluster consisting of 72 processors on the Boston University Campus ( 4 Springel V., Mon. Not. R. Astron. Soc. 364, 1105 (2005). 5 Kahn, F. D., and Woltjer, L., ApJ. 130, 705 (1959). 6 Binney, J., Jiang, I., & Dutta, S., MNRAS, 297, 1237 (1998.) 7 Tubbs, A. D. & Sanders, R. H. Astrophys. J. 230, 736 (1979). 8 Shen, J., Sellwood, J. A., MNRAS, 370, 2 (2006). Acknowledgements Funding for this project has been provided by the Maine Space Grant Consortium. A special thanks to the Center for Computational Science at Boston University on whose supercomputer cluster, Twister, the simulations were run. Figure 1. Warped spiral galaxy ESO (Hubble) Discussion The simulations show that gravitational interactions between galaxies lead to warping of galactic disks. Presently we are investigating the persistence and stability of warps. Kahn and Woltjer 5, Binney et al. 6, Tubbs and Sanders 7, and Shen and Sellwood 8, have detailed arguments regarding the persistence of warps. If pseudoforces drive the warps, then the effect should persist for more than a few rotations. Our latest simulations are able to explore this issue. So far, we see that in long runs in which the intruder passes once and then recedes, the disk maintains its warp for at least three rotations. In many cases, observations suggest that the intruder galaxies are often bound to the warped galaxy. We are presently setting up and running simulations of bound systems. Fig. 3 Unperturbed Disk Galaxy Containing 20,000 particles, it is shown at time t=1 billion years. (a) particle plot. (b) Density contour plot of same data. (a) (b) Fig. 4 Warp Resulting from Distant Intruder (a) Particle plot (b) density contour plot of a warped disk galaxy containing 20,000 particles. The intruder, with mass M=95, is moving perpendicular to the plane of the disk. This is at time t= 375 million years. (a) (b) Fig. 6 A Close Encounter Simulation This intruder has mass M=.75 and is initially moving at 30° relative to the initial plane of the disk. (a) particle data. (b) Corresponding contour plot tilted to the same planar angle as distribution in (a) (a) (b) Fig. 5 Second distant encounter simulation Intruder has mass M=45 and is moving at an angle of 30° relative to the initial plane of the disk. (a) Particle plot of the warp (b) Top view of simulation in (a). (c) density plot of the simulation. (c) (b) (a)