Solution of Boltzmann equation

Figure 1: LMC disk heating by the Milky Way. Contours and wire frame show the cumulative distribution of stars at a height Z or larger. The curves show mass fractions 1, 10^-1, 10^-2, 10^-3, 10^-4 from bottom to top.

Figure 2: The projected surface density distribution for the edge on view of disk at four times shown in Fig. 1. The smooth color variation from red to green to blue reflects logarithmically change in projected surface density over six orders of magnitude from the peak.

Figure: As in Fig. 2 but showing the phase space density in the E and $\cos\beta\equiv J_z/J$ plane. NB: the evolution is both axisymmetric and symmetric around the midplane ($\cos\beta=0$).

To estimate the evolution, I present a solution of the time-dependent collisional Boltzmann equation for orbits in a fixed potential. The angular momenta of individual stars change during passage through resonances as the disk slowly evolves. The change in the conserved angular momenta depends on the direction that an orbit crosses a particular resonance (see Henrard 1982 for discussion). A galaxy will have different phase-space densities on either side of the resonance resulting in a net gain or loss for the passage. The net change in the phase-space distribution function, then, due to the resonant heating takes the form of a collisional Boltzmann equation where the right-hand-side collision term depends on the gradient of the phase-space distribution function (see Appendix for additional detail). For simplicity, we assume that the background gravitational potential is constant in time, dominated by the halo. The now linear partial differential equation may be solved by finite-difference on a three-dimensional grid (e.g. E, J, J_z). The z-axis is perpendicular to the disk plane. The ratio of the z-axis angular momentum to the total angular momentum is the cosine of the orbital-plane inclination angle, $\beta$: $\cos\beta = J_z/J$. At every time step, the potential is recomputed and any phase space whose stars have apocenters larger than the tidal radius are deleted from the grid. Although, these weakly bound stars may linger near the tidal boundary for some time in reality (Lee & Ostriker 1987), this one way tidal boundary is easy to implement. A W₀=1.5 King model was chosen to represent the LMC gravitational potential and approximately fits the rotation curve.

Figure 1 shows the cumulative distribution of mass above the disk plane as the system evolves, M(Z). After approximately 1 Gyr, 1% of the disk mass has a height larger than 6 kpc and 10% above 3 kpc. The thickening occurs from the outside in, appearing as a flared population that fills in at smaller radii with time. This leads to a very thick disk or flattened spheroid population.

Figure 2 shows the edge-on projected surface mass density. One sees that the tidal envelope is filled in a gigayear, and over longer time scales the disk scale height is increasing (cf. the 10^-1 contour in Fig. 1). This trend is more apparent in phase space: the orbits at low binding energy are heated first and those at successively higher binding energy as time goes on. This is clearly seen in the energy-orbital inclination (E-$\cos\beta$) projection of the phase space distribution (Fig. 3): the high binding energy inclined orbits--the upper and lower left corners-are successively filled in with time.

Also worthy of note is that $\log M(Z)$ is roughly linear with Z at times larger than 1 Gyr. This suggests an exponential profile which has been recently reported for the RR Lyrae distribution in the LMC halo (Kinman et al. 1991). The sharp roll over at the tidal radius is suggestive of the observed star-count profile but may be an artifact of the one-way tidal boundary.