1 Introduction

When evaluating justification for new facilities, one may question whether and by how much the new technique is superior to already existing ones. We describe a general approach which leads to a quantitative answer, expressed as a relative figure of merit of two observational techniques. The approach is based on the comparison of the corresponding errors in the model parameters determination. The derived figure of merit can also be used in the closely related problem of optimization of complex observations, thus contributing to maximizing their scientific output.

We illustrate the developed method by comparing the technique of optical long baseline interferometry to the classical spectroscopy, both used to find the parameters of the gaseous envelope around the P Cyg star. For this first trial application, we use simulated data, obtained by solving the radiative transfer problem for grids of physical models of the envelope and then adding the photon noise.

The chosen example of the interferometry vs. spectroscopy comparison appears to be timely. Indeed, in recent years, interferometry has been undergoing steady progress witnessed by publication of fringe visibilities for circumstellar envelopes of various types sometimes combined with a good spectral resolution (Mourard et al. [1989]; Vakili et al. [1994], [1997]; Quirrenbach et al. [1993], [1994], [1997]; Harmanec et al. [1996]; Stee et al. [1995]; see also expected performances of interferometric arrays under construction, VLTI, von der Lühe et al. [1997]; Petrov et al. [1998], and CHARA, McAlister et al. [1997]). While these studies are fully justified, for the knowledge of the flux distribution on small angular scales can unveil important new features of astrophysical objects, it is also important to question whether this effort is justified for all classes of objects and for all kinds of physical problems. As a matter of fact, interferometric observations as compared to those at a single telescope are more complex, more expensive, more time consuming, and, when possible to compare, have a larger accumulated measurement error due to their complexity. Furthermore, in the foreseeable future the instruments of this type will remain orders of magnitude less numerous and by all given reasons much less available for the community than single telescopes. Thus, it makes sense to evaluate in quantitative terms what are the targets for which the interferometric instrumentation has the highest potential in terms of astrophysical information as compared to single telescopes.

Also, the optimization of interferometric observations may be a crucial aspect. Indeed, the existing and forthcoming long baseline interferometers comprise only few individual apertures, covering a small set of spatial frequencies during one observing cycle. On the other hand, the information provided by observations can depend critically on spatial frequencies. Certainly, in such a situation the scientific yield of the interferometric array could be greatly improved if thorough model calculations can indicate the configurations expected to be the most useful for determination of intended physical parameters of the object.

Not surprisingly, the relative figure of merit derived in the present article depends not only on the experimental techniques, but also on the chosen set of model parameters, reflecting the fact that the physical model makes a necessary part of the merit problem. However, in modern astronomy, designing and building instrumentation on the one hand, and modeling objects on the other hand, were pushed to such a level of sophistication that they are as a rule two distinct activities. The optimum use of interferometry needs a close collaboration of these two communities.

After these preliminary remarks, let us specify that in the present first trial application, our discussion will be limited to the comparison of observations with a two-aperture optical long baseline interferometer (hereafter OLBI) and the classical single telescope spectroscopy (hereafter spectroscopy), both applied to study a circumstellar gaseous envelope formed by the wind of a massive star. The OLBI observations are assumed to have the same spectral resolution as spectroscopy. The discussion will be further limited to spectral profiles and visibilities in the H$\alpha$ line of the P Cyg envelope, allowing us to use published high quality observations both in OLBI and spectroscopy, and thus to provide a clear numericnalial illustration.

In our earlier attempts to solve the OLBI vs. spectroscopy merits problem (Burgin & Chalabaev [1992]; Bourguine & Chalabaev [1994]), we used qualitative comparison of observables, visibility and spectral profile, computed for a small number of envelope models. The results were ambiguous. Analysing them, we arrived at the firm conviction that meaningful conclusions can be obtained only if (1) intercomparisons of different types of observations are performed using a clearly defined figure of merit, indicating how much information on the studied object is provided by various techniques, and (2) sufficiently large ranges in the space of envelope model parameters are analyzed.

In the case of P Cyg, the required number of computed models (see Sect. 5) turned out to be 10⁴-10⁵. An efficient simplified envelope model code, computing the emergent spectral profile and visibilities for the H$\alpha$ line, was developed and is described in Sect. 4. We considered only spherically symmetric outflow models, the choice of which is justified in Sect. 4.