Astron. Astrophys. 325, 877-880 (1997)

3. Results

a)
For the case we have calculated for three different source sizes and a large range of -values. We have plotted in Fig. 1 the values of together with as a function of . We see that is always smaller than for the same value of and . Except for -values close to one, where the amplification gets very large and approaches zero, compare Deguchi and Watson (1987), is a good approximation for large sources. As a rule of thumb, represents a reasonable approximation to for . The deviation of from is then less than for .

Fig. 1. The standard deviation (for of the mean apparent magnitude from the numerical calculations and according to the analytical formula, Eq. (1), (dashed line)

The underlying assumption for the derivation of Eq.(1) was that the variation in m was due to Poisson fluctuations in the number of stars projected in front of the source. In order to test this, we have investigated the correlation between m and the actual number of stars (N) projected in front of the source. The results are shown in Fig. 2, where we have plotted the correlation coefficient as a function of , for three different values of with . We see that the correlation increases with increasing source size and is larger than 0.9 already for , confirming very well the underlying assumption.

Fig. 2. The correlation between and the number of stars N projected in front of the source for . Results are plotted for (upper curve), and (lower curve)

b)
For the case we have not succeeded in deriving a simple analytical formula similar to Eq. (1).

Some of the results from our numerical calculations are plotted in Figs. 3 and 4.

Fig. 3. The standard deviation for the case together with given by Eq. (1), (dashed)

Fig. 4. The standard deviation for the case together with given by Eq. (1), (dashed)

The standard deviation and the analytical approximation given by Eq. (1), are plotted as a function of for and .

It is generally found that also when shear terms are included, represents an upper limit for . Except for a narrow interval around , we even find that the value of is smaller than for . For a large range of -values we see however that the effect of the shear is rather small. An obvious exception is of course the "forbidden" intervals around where the amplification again gets very large and approaches zero. Eqs.(1) and (2) can therefore still be used to estimate an upper limit of the source size if is available from observations:

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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