,
(5-1)
To determine the influence of a satellite on the galactic disk, I performed three simulation runs for ~170 timesteps each, just over two satellite orbital periods, or approximately 3 Gyr. Run0 is the control run, containing no satellite to determine the response of the ring system in an isolated simulation. Run1 and Run2 employ a 0.11 M and 0.32 M satellite respectively. I discuss the calculation of the initial position in §4.2.5.
The control run exemplifies the sensitivity of the ring system to slight perturbations. By the end of Run0, the rings acquire a substantial amount of noise. Unfortunately, the noise appears to amplify monotonically, increasing without bound. The largest inclinations of ~3.2 degrees, appear at the border of the massless rings, 600-650. The entire disk tilts by ~1.5 degrees. The small anticipated tidal response (on order of the noise itself) makes the level of noise objectionable. In retrospect, this particle model would be better suited for higher energy interactions.
To distinguish the disk's tidal response from the background noise, I removed the control inclinations from the satellite runs. Specifically, I calculated inclinations for all rings and averaged them in twenty bins of 50 rings each. I then subtracted the bin average inclinations recorded in the control run from the bin average inclinations of the satellite runs for each timestep. As a comparison, I also performed a linear fit to each bin's inclination over time and subtracted the control run's fit from the satellite run's fits. The maximum RMS error of Run0's fits increased from 0.3 to 0.5 at the edge of the disk, dropping to 0.15 outside the massive disk. In Run2's fits the RMS error increased from 0.4 to 0.6, increasing from 0.2 to 0.4 outside the disk. Higher order polynomial fits failed to reduce the error. The linear fits produced results similar to the unfitted data, generally weaker by ~0.5° due to smoothing of the second perigalacticon response. Further simulations should endeavor to reduce the background noise by increasing the number of halo particles.
Both satellite runs start at an apgalacticon of ~72 kpc, with no radial velocity and a tangential velocity placing the satellites into an elliptical orbit. While the orbit of the LMC is thought to possess an eccentricity between 0.4 and 0.6 (Weinberg 1995), the orbits in this model possess eccentricities of ~0.25. The orbits circularize as they evolve. Table 7 presents a summary of each orbit's extrema.
Table 7. Satellite Orbit Properties
| Apgalacticon | Perigalacticon | Apgalacticon 2 | Perigalacticon 2 | Apgalacticon 3 | |
| Run0 | N/A | N/A | N/A | N/A | N/A |
| Run1 | 72 kpc; t = 0 | 51.9 kpc; t = 45 | 70.0 kpc; t = 96 | 52.5 kpc; t = 151 | N/A |
| Run2 | 72 kpc; t = 0 | 49.1 kpc; t = 45 | 67.1 kpc; t = 90 | 46.7 kpc; t = 135 | 61.4 kpc; t = 176 |
Run1. Falling from 17.9 Rd, the 0.11 M satellite, Figure 12, reaches a perigalacticon of R = 13 Rd at t = 46. Subsequent apgalacticon occurs at t = 97 (R = 17.5 Rd) and perigalacticon at t = 152 (R = 13.1 Rd). The orbit minimally decays during the course of the run.
Run2. The 0.32 M satellite, Figure 13, evolves more rapidly, reaching a perigalacticon of R = 12.3 Rd at t = 45, apgalacticon at t = 90 (R = 16.8 Rd), subsequent perigalacticon at t = 135 (R = 11.7 Rd), and apgalacticon at t = 176 (R = 15.3 Rd). Dynamical friction quickly decays the satellite orbit, dropping apgalacticon from ~18 Rd to ~15 Rd but perigalacticon from ~12.2 Rd to only ~11.7 Rd. The satellite also loses internal energy, producing a dense core. Very little matter is stripped from the satellite (the scatter of points in Figure 19). Several (~5) more orbital times will bring the satellite into contact with the edge of the disk.
The present orbital position of the Magellanic Clouds is just past perigalacticon, 32° below the disk at longitude 280. In Run1, this corresponds to about t = 157, and for Run2, about t = 150. While the estimated perigalacticon position is below the disk (see Figure 10), the actual perigalacticon, influenced by a reactive disk and halo, lies above the disk (Figure 19). So while the galactocentric distance of perigalacticon is equivalent to that of the Magellanic Clouds, the orbital geometry of the model differs from that of the Milky Way.
The satellites in Run1 and Run2 produce similar effects on the disk. As expected, the massive satellite in Run2 tilts the disk to a larger inclination. Figure 15 summarizes the disk inclination for each of the runs. While the line of nodes quickly winds up inside the massive disk, it remains mostly constant within the massless edge rings.
Run1. The largest normalized inclinations appear in the inner disk, tilting to a mean of 1°. The outer disk responds weakly, inclining to only ~0.5°. This light satellite generates a response much too weak to account for the galactic warp. The remainder of the analysis concentrates on the more dramatic behavior of Run2.
Run2. The largest normalized inclination 2.45° appears in the outer disk. The inner disk tilts to a similar inclination of 2.32°. The strongest tidal effects appear shortly after perigalacticon. Figure 16 highlights the response of the outer disk in Run2. The outer rings rapidly tilt following the satellite's passage at t = 40. The second perigalacticon passage at t = 135 similarly perturbs the rings to a higher inclination. Once inclined, the ring model responds with a natural frequency of oscillation as the tilted rings nutate. If an induced tilt fails to damp in less than the satellite's orbital time, successive passages may produce ever larger inclinations.
The largest inclinations occur at the far edge of the disk. This suggests that the disk's self-gravity helps resist deformation, or it may simply be that the massive rings respond only sluggishly. As the massive rings set into motion, their slewing drags the adjacent rings in response. In Figure 14, plots of inclination versus radius reveal wave-like deformations propagating outwards.
In Run2, the disk primarily tilts along the satellite's orbital path, with the disk edge tilted towards perigalacticon (Figures 17, 18, and 19). This generally agrees with the orientation of the Milky Way's warp (see §3.3 ), though the difference in perigalacticon position makes a direct comparison problematic.
Nelson and Tremaine (1995) approximate the increased potential due to the wake as
,
(5-1)
where Us is the direct tidal potential, Mtot is the total mass within the satellite's orbit, M(r) is the mass within the radius of interest, and kl is a dimensionless positive number that parametizes the halo's response. They suggest 0 < kl 1, with kl ~ 1. For example, if the mass within the LMC's present distance of 50 kpc is 3 times larger than the mass within 15 kpc, the LMC's potential at 15 kpc would be enhanced by a factor of 3.
I use Equation (5-1) to estimate the satellite mass enhancements within the halo model. The results, summarized in Table 8, suggest a 2-3 times mass enhancement, similar to Weinberg's (1995) conclusions.
Table 8. Mass Enhancement due to the Wake
| Edge of Disk | ||
| Satellite Distance | 20 kpc | 16 kpc |
| 50 kpc | 2.44 | 3.29 |
| 45 kpc | 2.29 | 3.09 |
Warps are not a high energy phenomenon. The energy contained by a warp is (Nelson & Tremaine 1996)
. (5-2)
In the case of the Milky Way, the warp starts at approximately the solar radius, increasing as h(R) = (R - R0), with Å 0.1. The surface density of HI is approximately 0 exp[-(R - R0) / Rd], where 0 = 10 Mu pc-2. Nelson and Tremaine (1996) find
E Å ~106 Mu Vc2. (5-3)
The energy contained by the warp is no more than the kinetic energy of a large globular cluster, or very small satellite.
Some insight into the dynamics of satellite warping may be obtained by considering the influence of a stationary mass Ms on a ring with fixed origin, initially in the x-y plane (See force diagrams in Appendix E). Taking Ms to lie along the x-axis, we note two facts from symmetry: a) the z-axis rotation of the ring is neither advanced nor delayed; and b) the gravitational force will cause the ring to rotate on its y-axis, the side of the ring near Ms rotating towards positive z and the opposite opposing that rotation. The net rotation is towards positive z on the near side of the ring for all positions off the z-axis. We may also note that the force of rotation is zero for all locations on the z-axis, and similarly zero for all positions in the x-y plane.
When center fixed, the sole motion of a ring under the influence of such a gravitational force is a rotation about the y-axis due to a torque N of the form
Ny = Integral[(F(q) · r sin(q))] dq (5-4)
evaluated about the ring's circumference. As the rotation occurs along only one axis, we may rewrite Equation 5-4 as
Ny = Fz · r , (5-5)
where Fz is a summation of the forces about the ring, canceling in all but the z direction. The moment of inertia of a such a rotating ring is
Iy = M · r2 . (5-6)
Newton's 2nd Law for rigid bodies states
,
(5-7)
where and are the angular acceleration and velocity respectively. Combining Equations 5-5, 5-6. and 5-7, we find the angular acceleration
= 2Ny/(M · r2) , or, (5-8)
= 2Fz/(M · r) , (5-9)
in units of radians/s2. The Milky Way's warp is ~1 kpc at r = 16 kpc, an inclination of = 3.6 or 0.0624 radians. Using model units (see §4.2 ), the radius of the ring, r = 4 Rd and the mass, Mr = 3.2 x 10-4 M. The LMC is at R = 12.5 Rd, an inclination of -32, and a mass of Ms = 0.3 M. The net force in the z-direction, Fz, is difficult to evaluate directly because the equation, presented in Appendix E, involves an elliptic integral. By dividing the ring into finite segments, we can obtain a numerical approximation for the force at each ring segment and by summation the total torque Ny.
Calculated numerically, Ny 6.1 x 10--7, therefore from Equation 5-8, the angular acceleration = 2.4 x 10-4. The acceleration is the second derivative of the inclination , or = t2/2. Solving for t finds that, given a static torque and discounting any restoring potential, a 0.3 M satellite will tilt an isolated ring by ~4 in t = 24, ~427 million years.
In the Run2 model, the satellite's orbital period is roughly t = 90, traversing each orbital quadrant in t ~ 23. For a static approximation, a tilting time of t = 24 is too long. A unrealistic close perigalacticon of 20 kpc is required to reduce the tilting time to a swift ~117 million years. Therefore, even at the most generous - a satellite allowed to exert force from a fixed position with no restoring force - the LMC's direct tidal force is insufficient.
In a more realistic calculation, the halves of the near orbit apply torque in opposing directions. This calculation also neglects self-gravity, a restoring force on each ring k from all rings j != k. Calculations involving self-gravity were performed by Hunter & Toomre (1969) who found that self-gravity suffices to prevent the warping of the Milky Way under the influence of the LMC.