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Astron. Astrophys. 350, L19-L22 (1999)

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4. Optical depth and event rate toward Carina

To put OGLE-1999-CAR-1 into context, in this section we estimate the optical depth, event rate and event duration distribution toward Carina. These can be compared with future observations when more events become available.

One major uncertainty for microlensing toward spiral arms such as Carina is that we do not know the distance to the sources. Several molecular clouds at the same direction have distances between [FORMULA] and [FORMULA] (see Table 1 and Fig. 4. in Grabelsky et al. 1988). We adopt [FORMULA] as the distance to the sources. This assumption is also consistent with the lensed star being a main-sequence star as required by its position in the color-magnitude diagram (Udalski 1999, private communication); we return to this issue in the discussion.

Since the direction toward the lensed star is nearly in the Galactic plane ([FORMULA]) and far away from the Galactic center, we assume that the lenses are entirely contributed by disk stars, and their density profile follows the standard double exponential disk distribution

[EQUATION]

where [FORMULA], r is the lens distance to the Galactic center, [FORMULA], disk scale-length [FORMULA] and scale-height [FORMULA]. The optical depth can then be obtained (e.g., Eq. 9 in Paczynski 1986):

[EQUATION]

independent of the lens mass function and kinematics.

To estimate the event rate and event duration distribution, we have to make assumptions about the lens kinematics and mass function. The motion of lenses can be divided into an overall Galactic rotation of [FORMULA] and a random motion. We assume that the random component follows Gaussian distributions with velocity dispersions of [FORMULA] (cf. Derue et al. 1999). The motion of the Sun relative to the Local Standard of Rest is taken as [FORMULA]. We study three mass functions:

  1. [FORMULA],

  2. [FORMULA],

  3. [FORMULA].

Note that the second is a Salpeter mass function while the third describes the local disk mass function determined using HST star counts (Gould et al. 1997). The predicted event duration distributions for these three mass functions are shown in the left panel of Fig. 2. It is clear that there is a tail toward long durations for all three distributions. The probabilities of having [FORMULA] longer than [FORMULA] are respectively, [FORMULA] 50%, 10% and 20%; so the observed long duration is not statistically rare. The total event rates per million stars per year are found to be

[EQUATION]

for the three mass functions, respectively. The predicted duration distribution and event rate are sensitive to the assumed velocity dispersions. For example, if we adopt [FORMULA] (as in Kiraga & Paczynski 1994), then the events will be on average 30% longer and the event rate decreases by about 30%.

[FIGURE] Fig. 2. The left panel shows the predicted event rate distribution (in units of per million stars per year) as a function of duration toward Carina. The short dashed line is for a [FORMULA]-mass function of [FORMULA], the long dashed line is for a Salpeter mass function, [FORMULA] and the solid line is for a disk mass function, [FORMULA], determined from HST star counts. The shaded region indicates the [FORMULA] range in time-scale for the best parallax fit with blending. The right panel shows the likelihood function and lens mass as a function of the lens distance, respectively.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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