Microlensing

An extended LMC stellar distribution, both bound and unbound, can enhance the microlensing optical depth caused by self-lensing. We can calculate the optical depth due to microlensing by using the estimated density distribution from the n-body simulation (see Appendix for details). We use the same Galactic halo model adopted by the MACHO collaboration for consistency (e.g. Alcock et al. 1997):

$$\rho_H = 0.0079 \: {R_0^2 + a^2 \over r^2 + a^2} \; M_\odot/pc^3,$$ (2)

where r is the Galactocentric radius, $R_0 = 8.5 {\rm\,kpc}$ is the Galactocentric distance of the Sun and $a \approx 5{\rm\,kpc}$ is the core radius.

The optical depth averaged along the line-of-sight is given by

$$\tau = \int _0 ^\infty \tau (D_s) p (D_s) dD_s \left[ \int _0 ^\infty p (D_s) dD_s \right]^{-1},$$ (3)

where

$$\tau (D_s) = {4 \pi G \over c^2} \int _0 ^{D_s} \rho_d (D_d) {D_d (D_s - D_d) \over D_s} dD_d$$ (4)

is the optical depth due to sources at a distance D_s, $\rho_d$ and $\rho_s$ are the lens and source densities respectively, and

$$p (D_s) dD_s = C \rho_s (D_s) D_s^{2+2\beta} dD_s$$ (5)

(Kiraga & Paczynski 1994) is the probability of finding a source in the interval $\left[D_s, D_s+dD_s\right]$. We take $\beta = -1$, consistent with a fit to the Bahcall-Soneira model (1980).

Figure 12: Left: microlensing optical depth as a function of MACHO fraction in the Galactic halo including MACHOs in LMC halo. The middle (upper, lower) horizontal dotted line show the observed microlensing ($\pm 1\sigma$ confidence limits) from Alcock et al. (1997). Depth computed using the Kiraga & Pacsynzki $\beta$ parameterization with $\beta = -1$. The four curves shows the predicted microlensing of the initial state (solid) and three successive pericenters (short-dash, long-dash, and dash-dot, respectively). Right: shows variation of microlensing optical depth as a function of disk inclination for the final pericenter shown at the left. The five curves show an inclination of 11.25, 22.5, 45, 67.5, and 78.25 degrees from bottom to top (short dash dot, short dash, solid, long dash, long dash dot, respectively).

For these simulation-based estimates, the LMC location is chosen at a point in its orbit that matches its present position. Unfortunately, this does not guarantee that the orientation of the disk in the simulation corresponds to the one observed. Rather than perform expensive iterations, the coordinates are transformed to the observed true orientation. The line-of-sight density distribution is computed using the kernel smoothing procedure described in §3.2.3.

First we assume no Galactic halo MACHOs; both source density $\rho_s$ and deflector density $\rho_d$ include only the stellar LMC distribution. This gives a total optical depth due to LMC self-lensing of 1.4 x 10^-7 at the end of three LMC orbits (5.5 Gyr) in the simulation. This falls shy of the observed value, 2.9^+1.4 _-0.9 x 10^-7, by nearly two standard deviations although precise comparison is impossible since the simulation does not follow the entire LMC history. Nonetheless, self-lensing including the tidally evolved distribution is a significant contribution to the optical depth. The best fit value is $F_{halo}\approx0.21$ for the final orbit. If the Milky Way halo contains MACHOs, it is likely that the LMC halo also contains the same fraction. The LMC halo has one half of the total mass initially. In this case, the best fit is obtained for $F_{halo}\approx 0.18$. Figure 12 (left) shows the run of $\tau$ with F_halo for this latter case.

The increase in the contribution to microlensing optical depth is dominated by the thickened disk rather than the lost stars in this simulation. Although mass is being lost continuously, the density profile near the disk is slowly changing after the first few orbits (as in Fig. 1 and reflected in Fig. 12 (left) for F_halo=0). However, this makes the self-lensing a strong function of disk inclination as shown in Figure 12 (right). For example, an inclination of 67.5, 78.25 degrees would imply F_halo=0.11, 0.0, respectively. This is sensitivity is one-sided; decreasing the inclination below 45 degrees make little change in the $\tau$ estimates.

In summary, the tidal disk heating makes a significant contribution to self-lensing. For no MACHOs, F_halo=0, the optical depth of the tidally evolved disk is three times larger than the initial $\mathop{\rm sech}\nolimits ^2$ disk. This translates to a factor of two difference in the best estimate of F_halo (cf. Fig 1, left) and decreases the significance of rejecting the F_halo=0 hypothesis.