**Figure 3:** Top: gray-level representation of the theoretical twofold PDF of a binary star computed for
**a)**, the twofold PDF
of same binary computed for **b)**,
and the corresponding function **c)**

The use of a reference star is generally needed in speckle interferometry to correct the quantity computed from atmospheric effects. We saw for instance in the previous sections that we derive *Q* computing the ratio of the twofold PDF of the binary star and the twofold PDF of the PSF (even if in this case this does not
exactly correspond to a complete correction of the atmospheric effects). Nevertheless, since seeing conditions can rapidly change
(Coulman 1985), the twofold PDF of the PSF can be badly estimated from the observation of a reference star. In that case, it can be useful to avoid the use of the reference star data. This can be done by using the present technique.

We have considered so far the twofold PDF of a binary star just for the space-lag vector equal to the separation vector , i.e. when the information about the binarity of the object is maximum. Let us now consider the inverse case, i.e. the case for which and are uncorrelated. Within the model assumed here, this occurs whenever and . In practice, and considering the effects due to the real extension of the speckle pattern, we chose to consider the particular vector (with the length ), for which on the one hand and are supposed to be uncorrelated, and on the other the effects due to the low frequencies present in the speckle pattern are supposed to be similar.

Considering again Eq. (5 (click here)), we have , , and that are still statistically independent from one another, all the more so because *d* , and are large in comparison to *s* . Assuming again that the process is stationary in space, we can write:

Taking the same kind of assumption as in Sect. 3 (click here) and following the same process, leads to, if :

and, if :

As in Sect. 3 (click here) one deduces from these quantities that:

where, if :

and, if :

From Eq. (19 (click here)) and Eq. (11 (click here)), we can deduce a relationship between the twofold PDF of a binary star computed for and computed for :

This last relationship is valid as long as is not zero, i.e. for and (if ). Let us now show that has the same kind of behavior and interest as *Q* .
Figure 3 (click here)
illustrates the relationship given in Eq. (22 (click here)), like in
Fig. 1 (click here). The twofold PDF of the binary star speckle pattern computed for appears very similar to the twofold PDF of the PSF, and the above defined clearly shows the same kind of form as *Q* . The result is a little less impressive than in
Fig. 1 (click here), but applications to simulated and real data can lead to an equivalent result. As we shall confirm in what follows, the method suggested in this paper can be used with or without reference star to correct for atmospheric effects (the two versions of the method will be called from now the *standard version* and the *reference-less version*). In the next section we shall among other test the validity of this statement by doing some numerical simulations.