2. Numerical method
The calculations were carried out with conventional backwards ray tracing techniques, for details we refer to earlier work (Kayser et al. 1986 (KRS), Schneider and Weiss 1987, Wambsganss et al. 1990). Since we are primarily interested in the standard deviation of the microlensing variation, we found it most convenient to generate a new star field in the lens plane each time a magnitude was calculated (i.e. for each loop). We usually calculated independent magnitudes (1000 loops) for each case with a given set of parameter values (). If we should have calculated a light curve with 1000 "independent" points, we would in some cases have had to introduce a star field of more than stars, a difficult requirement, imposing heavy demands on the computer memory. Also, the correction for the not exact independence of the points in a lightcurve is difficult to estimate.
The size of the star field was in each case kept constant, and for each loop the stars were randomly distributed in the lens plane. The number of stars, however, was not constant for the different loops, but followed a Poisson distribution such that the expected number of stars gave the "correct" value of . Only then will the number of stars projected in front of the source be Poisson distributed in each case (given set of parameter values). With a constant number of stars the distribution would have been binomial, since the number of stars is finite.
From standard statistics we find that with independent values of the magnitude m, it is possible to estimate with an accuracy (standard deviation) of
With our chosen value of we then get .
Another source of statistical error in the determination of is caused by the finite number of traced rays, , which hit the source. With a regular grid of rays traced through the lens plane (), it is found (KRS) that . It is then easy to show that this gives rise to a standard deviation in equal to and that the combined (total) standard deviation is
We usually chose such that . In a few cases for the largest sources approaches .
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998