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

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3. Results

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

[FIGURE] Fig. 1. The standard deviation [FORMULA] (for [FORMULA] of the mean apparent magnitude from the numerical calculations and [FORMULA] 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 [FORMULA] as a function of [FORMULA], for three different values of [FORMULA] with [FORMULA]. We see that the correlation increases with increasing source size and is larger than 0.9 already for [FORMULA], confirming very well the underlying assumption.

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

b) [FORMULA]
For the case [FORMULA] 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.

[FIGURE] Fig. 3. The standard deviation [FORMULA] for the case [FORMULA] together with [FORMULA] given by Eq. (1), (dashed)
[FIGURE] Fig. 4. The standard deviation [FORMULA] for the case [FORMULA] together with [FORMULA] given by Eq. (1), (dashed)

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

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

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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