7 Final remarks

In this section we discuss to what extent our results are relevant for real observations, and how the methods developed here could be generalized to more complex cases.

7.1 Additional sources of measurement errors

Although general formalism presented in Sect. 2 remains valid for any source of errors provided that the linearized least-square method is applicable, Eqs. (11) and (18) need to be modified if the accuracy of measurements is not shot noise limited.

In the simplest case of uncorrelated errors in spectral channels, this modification reduces to multiplication of the integrands in Eqs. (11) and (18) by corresponding ratios (shot noise error)/(total error). If these ratios are nearly constant across the observed spectral range, then after appropriate scaling our results can be directly applied to real observations.

The assumption of uncorrelated errors may be invalid for interferometric observations, the observables $\hat W_i$ being obtained as a result of much more complicated processing of raw observational data (cf. Mourard et al. [1994]) then that considered in Appendix A. This point must be analysed further, and in future studies, it may be necessary to use directly Eq. (1) instead of Eq. (18).

In any case, a generalization on arbitrary measurement errors could be easily incorporated into the used code with no considerable increase in the required computer resources.

7.2 Observation in several spectral lines

The theory developed in Sect. 2 can be applied to the case of multiple observed lines without any modifications. Formally, it reduces to changing integration domains in Eqs. (11) and (18).

Rapid evaluation of source function, on the contrary, would require a detailed study and development of efficient methods specific to each spectral line involved, because different lines can be emitted in different regions of the envelope and originate from different physical processes.

7.3 Multi-baseline OLBI of spherically symmetric objects

The method used here can be generalized to visibility measurements done simultaneously at more than one baseline: the only modification required is the replacement of RHSes of Eq. (18) by the sum of corresponding expressions over a given set of projected baseline lengths, the directions of the baselines being unimportant because of circular symmetry of intensity distribution. This can be done by modifying only the code performing the second stage of calculations (see Sect. 4.4) and using the data prepared for the single-baseline case during the most time consuming first stage of computation.

7.4 Asymmetric brightness distribution and information on fringe phase

Although time-averaged structure of P Cyg wind can be considered spherically symmetric, polarimetric (Taylor et al. [1991]) and interferometric (Vakili et al. [1997]) observations revealed the existence of time-dependent "clumpy'' structures in the envelope, giving rise to deviations from spherical symmetry.

In this case, measurements of the fringe phase can yield important information. This contrasts with the spherically symmetric case, where the fringe phase is irrelevant to physical modeling of the object.

On the other hand, a unique determination of model parameters using only spatially unresolved spectroscopy appears impossible for the complex physical models resulting in two-dimensional brightness distributions, so that even the formulation of the comparison problem for spectroscopy and interferometry can hardly be done. The choice of the optimal set of baseline lengths for interferometry still remains the important problem, and the method developed here can be used for such an optimization after certain modifications.

First, the fringe phases (or observable linear combinations thereof) should be included in the set of observables defined in Eq. (17), and Eq. (18) should be modified accordingly. This would require comprehensive error analysis depending on the particular method used for extracting phase information from measurements.

Second, errors in parameter determination for observations performed with interferometer configurations B₁ and B₂ (here B₁ and B₂ denote corresponding sets of simultaneously used projected baseline lengths) can be compared using the ratio $C^{\rm I}(\vec\Theta,\mathcal T,\mathcal N,B_1) /C^{\rm I}(\vec\Theta,\mathcal T,\mathcal N,B_2)$instead of $R^{{\rm IS}}(\vec\Theta,\mathcal T,\mathcal N,B)$.

Acknowledgements

The authors are grateful to A. Labeyrie, D. Mourard and F. Vakili for encouraging discussion at the beginning of this project. We thank C.Bertout who provided the code for computation of the emergent intensities. MSB is grateful to the staff of Observatoire de Grenoble for their hospitality during his stay in Grenoble. Suggestions by the referee P. Stee and correctons of English by S. McKenzie helped to improve the article, we are thankful to them. This research made use of SIMBAD astronomical database, created and maintained by the CDS, Strasbourg, and NASA's Astrophysics Data System Abstract Service. Part of this work was supported by CNRS through the grants PICS No. 194 "France - Russie, Astronomie à haute résolution angulaire", GdR "Milieux circumstellaires", and GdR and PN "Haute résolution angulaire en Astronomie".