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4. Avoiding the use of a reference star

 

  figure569
Figure 3: Top: gray-level representation of the theoretical twofold PDF of a binary star computed for tex2html_wrap_inline2085 a), the twofold PDF of same binary computed for tex2html_wrap_inline2199 b), and the corresponding tex2html_wrap_inline2201 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 tex2html_wrap_inline1981 equal to the separation vector tex2html_wrap_inline1927, 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 tex2html_wrap_inline1969 and tex2html_wrap_inline1973 are uncorrelated. Within the model assumed here, this occurs whenever tex2html_wrap_inline2213 and tex2html_wrap_inline2215. In practice, and considering the effects due to the real extension of the speckle pattern, we chose to consider the particular vector tex2html_wrap_inline2217 (with the length tex2html_wrap_inline2219), for which on the one hand tex2html_wrap_inline1969 and tex2html_wrap_inline1973 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 tex2html_wrap_inline1929, tex2html_wrap_inline2075, tex2html_wrap_inline2229 and tex2html_wrap_inline2231 that are still statistically independent from one another, all the more so because d , tex2html_wrap_inline2235 and tex2html_wrap_inline2237 are large in comparison to s . Assuming again that the process is stationary in space, we can write:


 eqnarray604

Taking the same kind of assumption as in Sect. 3 (click here) and following the same process, leads to, if tex2html_wrap_inline2241:


 eqnarray628

and, if tex2html_wrap_inline2243:


 equation643

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


 equation651

where, if tex2html_wrap_inline2241:


 eqnarray659

and, if tex2html_wrap_inline2243:


 equation670

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 tex2html_wrap_inline2029 and computed for tex2html_wrap_inline2217:


 equation680

This last relationship is valid as long as tex2html_wrap_inline2253 is not zero, i.e. for tex2html_wrap_inline2255 and tex2html_wrap_inline2257 (if tex2html_wrap_inline2241). Let us now show that tex2html_wrap_inline2201 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 tex2html_wrap_inline2217 appears very similar to the twofold PDF of the PSF, and the above defined tex2html_wrap_inline2201 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.


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