1 Introduction

The Keck telescopes (Gillingham 1997) have demonstrated operationally the long held dream (see for instance Horn D'Arturo 1955) of providing very large mirror surfaces using a huge number of small mirrors co-aligned and co-phased actively in order to mimic a single telescope aperture. In principle there is no limit (other than the cost) to the size of these telescopes and several studies on giant telescopes ranging from 25 to 100 m (Angel 1990; Ardeberg et al. 1992; Ardeberg et al. 1998; Diericx & Ardeberg 1998; Gilmozzi et al. 1998; Owner-Petersen 1996; Sebring et al. 1998) have appeared in the literature. While photon collection gain is more than enough to make these giants an attractive idea, it is only through adaptive optics that one would obtain $\lambda/D$ diffraction limited capabilities on such instruments. With the exception of some specific (but very appealing, like imaging of extra-solar planets, see for instance Angel 1994) cases, sky coverage is severely limited if one needs to use on-axis Natural Guide Stars (NGSs) as in the conventional adaptive optics schemes.

Laser Guide Stars (LGSs) are at a first glance an obvious choice to overcome such a limitation (Foy & Labeyrie 1985). Assuming that laser firing, Rayleigh pollution and absolute tip-tilt indetermination (Pilkington 1987) problems are somewhat solved through one of the several approaches that have appeared in the literature (Baruffolo et al. 1998; Belen'kii 1994, 1995, 1996; Esposito et al. 1998; Foy et al. 1992, 1995; Lukin 1996; Ragazzoni 1995, 1996ab, 1997; Ragazzoni et al. 1995; Ragazzoni & Rigaut 1998; Riccardi et al. 1997; Rigaut & Gendron 1992; Whiteley et al. 1998), one is still faced with conical anisoplanatism problem (Gardner et al. 1990; Esposito et al. 1996; Neymann 1996; Ragazzoni et al. 1998) due to the finite height of the LGS itself (Happer et al. 1994).

Fried & Belsher (1994) defined a diameter d₀ such that conical anisoplanatism can be neglected. For any reasonable ground-based site this d₀ is expected to be of the order of a few meters. Stitching or butting (Fried 1995) to solve the conical problem requires approximately (D/d₀)² LGSs. Even assuming that these LGSs could be generated by Rayleigh scattering (a somewhat less expensive and technically easier choice), at least few hundreds of LGSs should be fired and sensed making this approach a formidable one. The tomographic approach (Tallon & Foy 1990; Ragazzoni et al. 1999) to solve conical anisoplanatism is much easier. The number of LGSs is of the same order of the number of significant layers. With something like $\approx$4 LGSs fired at significantly different directions in the sky one could realize a LGSs adaptive optics system for such giant telescopes. This approach is not free from technical problems. The angles at which the LGSs are fired scales with the telescope diameter D and in order to recover these one would need to look several arcminutes away from the optical axis of the instrument. That translates into meters on the focal plane scale and would possibly require dedicated off-axis optical relays to compensate for the (presumably huge) aberrations at such large distance from the optical axis.

I pointed out (see for instance Ref. 19 in Gilmozzi et al. 1998) that the angular sky area where the LGSs are fired can be so large, when the diameter D increases, that one could have some reasonable chance of finding a corresponding number of NGSs (see Fig. 1). This raises the hope that under certain conditions and for a telescope aperture with a diameter larger than a given size one could realize nearly all-sky coverage adaptive optics without the need of any LGS!

  

Figure 1: In this schematic representation the LGS-based tomographic technique (left) is illustrated in a sectional view. In this case LGSs can be fired at some precise locations on the sky and some conical beams explore the various perturbing layers. Tilt is to be retrieved in some other way. On the right side the NGS-based case is briefly sketched: the stars have different brightness and are unevenly located on the sky. On the other hand the beams are cylindrical and tip-tilt is retained

The absolute tip-tilt problem is automatically solved in this case. Tomographyc retrieval would not be more difficult than using LGSs (maybe even simpler because of the cylindrical shape of the NGS's beams instead of the LGS's conical ones).

This paper tries to give an answer to the following questions: i) "At which telescope diameter $D_{\rm c}$ this technique is more effective than traditional adaptive optics in terms of sky coverage?" and ii) "At which telescope diameter D₅₀ (or D₉₀) are NGSs solely are able to provide some reasonable sky coverage of, say 50% (or 90%)?"