Many spiral galaxies contain a large gaseous hydrogen disk, which overlays the visible stellar disk, the rotating body of stars and gas that gives spiral galaxies their name. Radio observations reveal that the outer regions of the gaseous disk, past the edge of the stellar disk, often deviate from the plane of the stellar disk. This deviation, or warp, commonly takes an antisymmetric form, canted above the plane on one side of the galaxy and below on the opposite, like a horizontal integral sign (see Figure 1). While the number of warped galaxies is uncertain, warps appear in many spiral galaxies, including our own, the Milky Way, and our neighbor, the Andromeda galaxy.
Due to the prevalence of warps, dynamicists conjecture that warps form naturally in the course of galactic evolution. They may result from tidal interactions, tilted sheets of infalling gas, or kinematic misalignments between the galactic disk and halo (e.g. Binney 1992). While the precise cause of warps is unknown, they endure for only a finite time. Calculations of the differential winding time (Binney 1992) and frictional damping time (Nelson & Tremaine 1995) indicate that warps persist for only a few billion years. To explain warp ubiquity, warp excitation must be common, or have been common during the last few billion years. One theory suggests that the gravitational interactions from infalling satellite galaxies provide the warping torque.
Few galaxies exist in isolation. In the same sense that planets frequently possess satellite moons, large galaxies often have orbiting satellite galaxies. Our galaxy has ten known satellites, each containing between 1% and 10% of the mass in the Milky Way's stellar disk (Audouze & Israël 1985). Likewise, the Andromeda Galaxy has seven known satellites. Though too dim to observe at distances much greater than that of Andromeda, low mass satellites greatly outnumber the more massive satellites. Satellites aside, galaxies cluster in gravitationally -bound groups: the Milky Way, the Andromeda Galaxy, the spiral in the constellation Triangulum, and roughly a dozen smaller galaxies form a gravitationally-bound system called the Local Group (see Figure 3).
Due to a process known as dynamical friction, galactic satellite orbits slowly decay (Chandrasekhar 1943). Similar to the drag of atmospheric friction (which brought down Skylab in the 1970s), dynamical friction is a consequence of the satellite losing energy through gravitational interactions. Effectively, the satellite alters the orbits of material in the galaxy, forming a gravitational wake -- an overdensity of matter behind the satellite, akin to waves behind a powerboat (e.g. Mulder 1983). The wake's mass retards the motion of the satellite gravitationally, causing it to lose angular momentum.
Eventually, all satellites fall into their primary galaxy, though it may take many billions of years (Tremaine 1980). The most recently discovered Milky Way satellite contains 0.1% of the disk mass, and is presently passing through the stellar disk (Ibata et al. 1994). For a tiny object like this satellite, passage through the disk is a traumatic event, one that ends in the satellite's destruction and the redistribution of its stars into the disk of the Milky Way (like an egg folded into a cake mix). While the Milky Way barely notices the absorption of a satellite this small, larger satellites cause progressively greater damage. The Large Magellanic Cloud (LMC) is our galaxy's largest satellite, containing roughly 10% of the disk mass and orbital angular momentum equivalent to that of the disk. Its absorption, a few billion years hence, may be sufficient to disrupt the galactic disk.
The Milky Way is warped, the gaseous disk flaring and deviating from the galactic plane by roughly 1.5 kpc at R = 25 kpc (Freudenreich et al. 1994). Early attempts to explain the warp named the LMC as a suspected progenitor (Burke 1957, Hunter & Toomre 1969). Unfortunately, these calculations found the direct tidal force insufficient by an order of magnitude. In theory, a normal mode of oscillation could sustain a primordial warp indefinitely (Binney 1992), but galactic disks appear not to support normal modes due to their floppy edges (Hunter & Toomre 1969). The discovery of massive dark halos suggested a solution: warped matter follows the fundamental axes of the halo far from the core where the disk's self-gravity fails. While Dekel & Shlosman (1983) successfully model this behavior, Dubinski & Kuijken (1995) show disk/halo misalignments transitory. The enormous angular momentum of the disk quickly aligns the halo, extinguishing any warp. This realignment makes suspect attempts to preserve warps with a 'modified tilt-mode' in static halo potentials (Sparke & Casertano 1988).
However, gravitating dark halos actively contribute to orbital dynamics. Mulder (1983) shows that the passage of a satellite through a gravitating background generates a trailing wake which enhances the satellite's apparent mass. In addition to the local trailing wake, the halo responds where there exist low order commensurabilities, resonances between the orbital frequencies of the satellite and halo particles. Weinberg (1995) uses a perturbative expansion to examine the wake's effect in the Milky Way system. Though the high-inclination, retrograde orbit of the LMC induces resonances only weakly, the LMC shifts the center of mass of the combined system, causing a periodic perturbation deep within the galaxy. Weinberg finds this 'remote wake' occurs within 2 disk scale lengths of the galactic core, indirectly influencing the inner galaxy. Combining direct tidal force and this wake, Weinberg calculates that a satellite, similar to the LMC, generates a warp with location and sign similar to that found in the Milky Way, but possessing an amplitude of 0.45 kpc at R = 25 kpc.
In this paper, I investigate the possible relationship between an orbiting satellite and galactic warps. Specifically, I ask if the Large Magellanic Cloud and its gravitational wake can produce a tidal response strong enough to generate a warp like that observed in the Milky Way. While the satellite's direct tidal force is insufficient (Burke 1959; Hunter & Toomre 1969), the wake strongly influences the inner galaxy (Weinberg 1995) and may serve to transmit force indirectly.
To estimate the satellite's influence, I employ an N-body simulation. This computer simulation uses a galactic model in which a collection of point masses, or particles, represent the mass and velocity distribution found in a galaxy. The simulation calculates the gravitational potential of the collection of particles, then finds the force on each particle, updates each particle's velocity, and finally integrates each particle's velocity to find its new position. In the course of a simulation run, the satellite galaxy (also represented by a collection of particles) generates a gravitational wake that causes its orbit to decay. In turn, the satellite and its wake exert a gravitational torque upon the disk of the galaxy.
A set of concentric spinning rings model the galactic disk, where each ring represents an orbit and the spin its orbital velocity. In this model a normal galactic disk is flat, but a warped disk's outer rings progressively tilt away from the plane. These spinning rings are analogous to a gyroscope, tilting and precessing in response to a torque. By measuring the tilt of the disk rings, as they are perturbed by the satellite and wake, I obtain an estimate of the magnitude of the warp generated by the orbiting satellite. I hope to observe a sizable tilt, of order 4°, indicating that the Large Magellanic Cloud is capable of producing the Milky Way's warp, and more generally, that orbiting satellites produce warps.
This paper is structured in six main sections. For readers unfamiliar with astronomy, I discuss the evolution of the universe in §2.1, the properties of spiral galaxies in §2.2, and some discussion of galactic interactions in §§2.3, 2.4, and 2.5. I present the observed properties of warps in §3.1, theories regarding their generation and stability in §3.2, and focus on the Milky Way's warp in §3.3. I describe my experimental setup in §4. My results and analysis appear in §5. Finally, I present a concluding discussion in §6. Potentially unfamiliar vocabulary is defined in Appendix A, various terms and units in Appendix B. Appendix C contains a background of gravitation, and Appendix D an overview of dynamical friction. Appendix E details the static force equations and calculations used to compute §5.5.