Results

The galaxy is constructed to be in equilibrium in absence of the time-dependent tidal field. During the first $6\times10^8$ years¹), the system achieves a new approximate equilibrium. A slow ramp-up of the tidal terms in equation (1) yield larger initial transients. During the initial virialization phase, the disk responds strongly in its outer parts to the full non-inertial set of forces although the total mass involved is small. The inner disk oscillates as it phase mixes under the fully consistent self-potential and external galactic potential. Both effects have little effect on scale height. A simulation with the same initial conditions but with n_max increased by a factor of two shows the same behavior.

The LMC disk precesses under the torque from the Galaxy. Uncorrected, this would cause the expansion plane to drift away from the true disk plane defined by the instantaneous mean angular momentum. To follow this, the disk bodies are ranked by binding energy and the lowest 2% are used to determine the disk angular momentum vector and expansion center. To damp any numerical feedback, both the expansion plane and center are determined from a 100 time step running average. This precession is shown in Figure 4 which shows the azimuth of the LMC disk's angular momentum axis in the original frame. The disk also nutates, as seen in Figure 5, because of the initial transient. The initial angle between the mean angular momentum vector perpendicular to the disk and the Galactic center is $45^\circ$. Although a non-nutating system might be achieved by iterating the initial angular momentum vector, there is little reason to assume this is closer to the natural state, so no corrections have been made.

Figure 4: Change in azimuth of the precessing disk with time. The vertical dotted lines indicate perigalatica.

Figure 5: Colatitude of angular momentum axis of precessing disk with time. For comparison with Fig. 4, the vertical dotted lines indicate perigalatica.

At the same time, the LMC halo and disk have a mutual self-gravitating response that causes a slow m=1 oscillation of the disk density center. The existence of such weakly damped modes can be demonstrated analytically using the methods described by Weinberg (1994d). Because of these modes and the fact that mass and momentum loss in the Galactic tidal field is asymmetric, the potential expansion is chosen to track the density center. Although the force solver can handle this situation, the secular heating of disk increases. Tests without the tidal field confirm that this effect does not dominate or obscure the tidal heating (cf. Fig. 6).

The disk thickness is estimated from the density distribution in a column through the LMC at a radius of a disk scale length. The disk plane is inferred by the same method used to orient the force expansion (see §3.2). The line of sight inclination is chosen to be $45^\circ$ and azimuthally averaged at one LMC disk scale length. The line of sight quantities are computed by selecting tracers in a pencil along the line of sight and estimating the one-dimensional density distribution using optimal kernel smoothing (Silverman 1986). The quantity $\sigma_d$ denotes the half width corresponding to the mass enclosed within one Gaussian standard deviation. To convert to scale height h of the equivalent isothermal slab, an explicit evaluation determines that $\sigma_d\approx1.8h$. With a $45^\circ$ inclination, this is $\sigma_d\approx2.6h$. The variance of the velocity distribution of stars along the line of sight is denoted $\sigma^2_v$.

Figure 6 shows h and $\sigma _v$. There are two clear trends: 1) the thickness of the disk increases; and 2) the velocity dispersion very slowly decreases. The slope, shown as a straight solid line in Figure 6, is $70{\rm\,pc}/{\rm\,Gyr}$. The evolved n-body disk has an approximately exponential profile, as also predicted by the semi-analytic computation. The linear increase in h with time is predicted by the underlying resonance theory and is a natural consequence of secular evolution.

The decrease in velocity dispersion seems counterintuitive at first glance but is often seen in globular cluster evolution. In a fixed gravitational potential, the heating would go into kinetic energy. However, the work done on the self-gravitating disk decreases the depth of the potential well and, by the virial theorem, also decreases the kinetic energy. The relative velocity dispersion tends to increase owing to the increased orbital eccentricity and a larger projection of the velocity along the line of sight caused by increasing orbital inclination. However this is not enough to offset the overall decrease in kinetic energy and the magnitude of the dispersion decreases.

In summary, the effect of the heating is significant although not as dramatic as in the analytic computation that ignores the disk's self gravity.

Figure 6: The scale height of the density distribution, h (left-hand axis: solid, long dash, dotted) and the root variance of the velocity distribution, $\sigma _v$ (right-hand axis: upper dashed curve) for a line-of-sight inclined $45^\circ$ to the LMC disk. The scale height of the LMC disk increases at a rate of 70 pc/Gyr (solid segment). The dotted and long-dashed curves show the secular evolution in absence of the Galactic tidal field for m=0 terms only and all terms, respectively. The velocity dispersion is nearly constant (note the small range in velocity) but decreases slowly on average after virialization.

Figure 7 describes the mass loss as a function of time. Mass beyond the LMC tidal radius is assumed to be lost. Loss of dark halo material, disk stars and total are shown separately with the orbital pericenters indicated as vertical dotted lines. Roughly 10% of the halo and 3% of the disk is lost by $T=6{\rm\,Gyr}$. The halo material is lost episodically, with the peak loss just past every pericenter. The disk stars are lost at a roughly steady rate. A total mass of $1.4\times10^9{\rm\,M_\odot}$ or 7% of the original mass is lost by $T=6{\rm\,Gyr}$. The spatial distribution of the LMC disk at this point is shown in Figure 8 in both edge-on and face-on projection. The colors here range from blue to red, orange, yellow and finally white as the mass density increases logarithmically. The outlying stellar spheroid that has been heated out of the disk is fairly tenuous and the edge-on disk is thinner than it appears. Figure 9 highlights the distribution of lost mass for both stars and halo material. In this figure, the lowest densities are shown as white. Recall, the relative number of points at different radii in these plots do not trace mass; the lower binding energies are preferentially represented as described in §3.2.2 in order to better resolve the mass loss.

Figure 7: Mass loss as a function of time for total mass (left), LMC disk (center), and LMC halo (right). In each panel, the left axis describes mass in solar masses and the right describes mass fraction. The vertical dotted lines indicate perigalatica.

Figure 8: Edge-on (left) and face-on (right) views of the LMC disk after about 5 Gyr. The points are color coded to indicate mass density on a logarithmic scale from blue to yellow. The distance top to bottom is approximately $20{\rm\,kpc}$.

Figure 9: Large-scale views of evolved LMC, highlighting the low-density ejected material (white). The sharp edge is caustic due to stars lost at a previous perigalacticon. The distance top to bottom is approximately $50{\rm\,kpc}$. The radius of the well-defined disk is approximately $10{\rm\,kpc}$.

Figure: The projected stellar density perpendicular to the orbital plane. The Cloud is near pericenter and moving in the $\hat y$ direction. The five contours are spaced logarithmically and correspond to $1.5\times 10^{-3}{\rm\,M_\odot }/{\rm\,pc}^2$ to $1.5\times 10^1{\rm\,M_\odot }/{\rm\,pc}^2$. Each unit of length in the simulation is $7{\rm\,kpc}$; the points $(\pm 7,\pm 7)$ are labelled. To provide a sense of scale, the `O' and `X' denote the Galactic center and an offset of $8{\rm\,kpc}$, respectively.

Figure: The projected stellar density on the sky in Aitoff projection. The five contours are spaced logarithmically and correspond to $2\times10^1$ to $2\times10^5$ solar masses per square arc minute. The simulation is shown with the LMC close to its current position.