Disk warping by the Magellanic Clouds

by Martin D. Weinberg & Leo Blitz

Submitted to ApJ Letters

Abstract

We show that a Magellanic Cloud origin for the warp of the Milky Way can explain most quantitative features of the outer HI layer recently identified by Levine, Blitz & Heiles (2005). We construct a model similar to that of Weinberg (1998) that produces distortions in the dark matter halo, and we calculate the combined effect of these dark-halo distortions and the direct tidal forcing by the Magellanic Clouds on the disk warp in the linear regime. The interaction of the dark matter halo with the disk and resonances between the orbit of the Clouds and the disk account for the large amplitudes observed for the vertical m=0,1,2 harmonics. The observations lead to six constraints on warp forcing mechanisms and our model reasonably approximates all six. The disk is shown to be very dynamic, constantly changing its shape as the Clouds proceed along their orbit. We discuss the challenges to MOND placed by the observations.

Keywords

Galaxy: structure, Galaxy: Disk, Galaxy: kinematics and dynamics

Introduction

The warp of the outer Milky Way, known since 1957 (Kerr et al. 1957), has been quantitatively determined for the first time by Levine et al. (2005). It can be described as a superposition of three and only three of the lowest order vertical harmonics of a disk: a dish-shaped m=0, an integral-sign-shaped m=1, and a saddle-shaped harmonic} m=2. The lines of nodes for each are close to coincident and nearly radial. The amplitudes of each reaches 7--10% of the radius of the disk. A number of possible warp producing mechanisms have been suggested including long-lived eigenmodes, forcing by halo triaxiality, persistent cold-gas accretion, and tidal excitation. We show here that the origin of this warp can be well-described as the tidal interaction of the Magellanic Clouds with the disk and dark matter halo of the Milky Way. The interaction of the dark matter halo with the disk and resonances between the orbit of the Clouds and the disk account for the large amplitudes of the three harmonics and their approximate shape and orientation.

Observations

Levine et al. (2005) found that a dynamical model for the warp must satisfy six observational constraints:

1. The three lowest-order harmonics: m = 0, 1 and 2 are necessary and sufficient to describe the global shape of the warp; higher-order global harmonics are typically an order of magnitude or more weaker.
2. The m = 1 warp has the largest amplitude everywhere in the outer disk.
3. The m = 0 and m = 2 warps are comparable in amplitude to the m = 1 warp, but are smaller at all radii.
4. The m = 1 warp has a measurable amplitude at the galactocentric radius of the Sun, R_°$, but the m = 0 and m = 2 warps begin near the edge of the stellar disk, at R = 2R__°.
5. All three harmonics grow approximately linearly with radius, reaching amplitudes of 1--2 kpc at about $R$ = 30 kpc.
6. The amplitudes of each of the harmonics reach 5--10% of the radius of the disk. The lines of maximum descent of the m = 1 and m = 2 warps are coincident within about 12 degrees in galactocentric azimuth φ and show little evidence of precession. The lines are located near φ = 90_^°.

Methodology

We use the procedure described by Weinberg (1998, W98) to couple the halo response to the tidal excitation theory presented by Hunter & Toomre (1969, HT). This assumes that the disk remains thin and that gas dissipation is unimportant for the dynamics. HT derived an expression for the bending of a thin disk by the Magellanic Clouds and concluded that the direct excitation of the disk by the Clouds produces a warp of only a few hundred parsecs, an order of magnitude less than the amplitude of the observed warp. W98 wedded the halo excitation presented in Weinberg (1989) to the HT approach, allowing the disk to feel both the tidal field from the Clouds directly as well as the force from the dark-matter halo wake excited by the Clouds. The assumption of linearity limits the predictions to modest amplitudes.

We use perturbation theory rather than N-body simulation because of the intrinsic difficultly and subtlety in obtaining accurate multiple time scale results from a simulation. The excitation hierarchy of satellite orbitj-->halo wake-->disk bending modes results in multiple interleaved time scales: the orbital periods of the Clouds, the pattern speeds of the halo wakes, and the pattern speeds of the bending modes. We also have multiple spatial scales. For example, a simulated N-body disk must be capable of supporting the bending modes. In addition, particle simulations can degrade the resonant dynamics as described by Weinberg & Katz (2005). With such difficulties, one should verify the dynamics of each component of the mechanism before putting everything together in an N-body simulation, although this is rarely done.

Model

The perturbation theory includes the force from an extended satellite. However, the halo and disk excitation depend only on the lowest-order hamonics while the spatial extent changes the higher-order harmonics, and thus the extent plays little role. Therefore, as long as the Clouds remain bound, their masses may be added for our estimates. We use the disk profile from W98. Our halo is an NFW profile with c=15 (Navarro et al. 1997) and virial mass 20 times the disk mass. The orbital plane was computed as described as in W98. We used the radial velocity and proper motion from Kallivayalil et al. (2005) and the distance modulus from Freeman (2001) to derive a space velocity and computed the resulting orbit in the spherical dark matter halo. We adopted the LMC mass from Westerlund (1997) of 2 × 10¹⁰ M_°, although there is considerable variance in the literature. The current position of the LMC is shown as is the current state of the warp. This orbit will carry it toward the NGP.

Although our calculation assumes a collisionless and vertically thin medium, it is more generally applicable for the following reasons. Firstly, although the dispersion relations for a multicomponent and collisionless media differ at small scales (e.g. Jog 1996, Rafikov 2001), the large-scale warp will be governed by inertia and the gravitational restoring force, not by local pressure. In addition, the disk self gravity is dominated by the inner stellar disk; the outer gas layer plays only a minor role in establishing the modes. Therefore, our collisionless results are likely to be similar to a multicomponent calculation. Secondly, the vertical restoring force depends very weakly on the thickening as long as the vertical degree of freedom does not couple to the bending (this may be demonstrated by straightforward but tedious algebra). Therefore, the modes will be largely unchanged by thickening over short time scales. However, the challenging problem of vertical coupling is important and remains to be investigated thoroughly.

The warp is a very dynamic structure based on the temporal evolution of the model. This can be seen in the AVI file of the simulations which can be found at in the following pages. Also included at this site are comparisons of the m = 0,1,2 evolution. Rather than a static structure that might be expected for a warp in response to a triaxial halo, a warp that results from the Magellanic Clouds is continuously changing shape because of the varying amplitudes and phases of the various modes. The image looks rather like a flag flapping in the breeze as the Clouds completes an orbit of the Milky Way.