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4 Discussion

4.1 Other possible sites in Antarctica

The South Pole does not lie on one of the highest points of the antarctic plateau, and is therefore not expected to represent the best seeing conditions available in Antarctica. The results obtained here lead us to consider the possibility that the $0.3-0.4\hbox{$^{\prime\prime}$}$ seeing measured above the 100-200 m high boundary layer might extend to ground level at other sites on the plateau.

\epsfbox{fig_8.eps}\end{figure} Figure 8: Contour map of surface wind speeds over Antarctica, from Dopita 1993, based on results of Schwerdtfeger 1984

The surface winds at the South Pole are of katabatic origin, due to the descent of cold air from the higher regions of the plateau. The strength and direction of these winds over the continent are largely dependent on the local topography, as illustrated in Fig. 6 (Dopita 1993; Schwerdtfeger 1984). The strong wind shears observed occur at the boundary between this katabatic flow and the upper atmosphere winds, which are generally geostrophic (Schwerdtfeger 1984). At Dome A (4200 m, 82$^\circ$S 80$^\circ$E), Dome C (3300 m, 74$^\circ$S 123$^\circ$E), and Dome F (3810 m, 77$^\circ$S 40$^\circ$E), the three most significant local "peaks'' on the antarctic plateau, these winds do not exist. Dome A, in particular, being the highest point on the plateau, is at the origin of the katabatic flow. It is likely that, at all of these sites, very little optical turbulence is produced in the boundary layer since, despite the possible presence of strong positive temperature gradients such as those measured at the South Pole, there is little mechanical mixing of the different temperature layers. If the free atmosphere turbulence at these sites is similar to that at the South Pole, we would expect to observe very good seeing from surface level: quite possibly the best seeing conditions available anywhere on the earth's surface.

Table 3: Summary of the calculated values of the astrophysical parameters defined in Eqs. (7-11), based on the 15 balloon flights. The limit of the boundary layer is taken to be 220 m. Values for Cerro Paranal were obtained using the average $C_{\rm N}^{2}$ profiles measured at this site (Fuchs 1995), although no wind data were available. Results for La Palma are from Vernin & Tuñon-Muñóz (1994)

...{2}{c}{--} & \multicolumn{2}{c}{--}\\ \noalign{\smallskip}

In order to measure the important site characteristics at these locations, an automated astrophysical observatory (AASTO) has been built, which contains several instruments for measurement of the seeing and the atmospheric transmission over a wide range of wavelengths. The AASTO is designed to function independently for a full winter season, and is scheduled to be deployed at Dome C in 1999, and at Dome A in 2000.

The construction of a permanent station at Dome C has begun (the France/Italy Concordia Project), and is expected to be completed by 2000. This will enable the operation of a similar range of longer term site-testing experiments to those currently being conducted at the South Pole.

In addition, while there are currently no plans for astronomical site-testing at Dome F (also known as Dome Fuji), a temporary winter station is currently being operated there by Japanese scientists, whose experiments include meteorological balloon sondes, to be launched during 1997. Access to these data would provide some very enlightening comparisons to the measurements made at the South Pole.

4.2 Other astrophysical parameters

The measured $C_{\rm N}^{2}$ profiles allow calculation of several parameters necessary to determine the feasibility of image correction techniques (adaptive optics and speckle interferometry) at the site, as well as other quantities such as the scintillation index. Using the notation:

{\cal C}(x, \eta)=\int x^{\eta}(h)C_{\rm N}^{2}(h) {\rm d}h,~~{\cal C}(0)=\int 
C_{\rm N}^{2}(h) {\rm d}h\nonumber\end{displaymath}   
these quantities are given by Eqs. (7-11) (Roddier 1981; Roddier et al. 1982). The subscripts AO and SI refer to adaptive optics and speckle interferometry respectively.

Isoplanatic patch:

\theta_{\rm AO}=0.31r_{0}\left(\frac{{\cal C}(h, 5/3)}{{\cal C}(0)}\right)^{-
3/5}\end{displaymath} (6)

\theta_{\rm SI}=0.36r_{0}\left(\frac{{\cal C}(h, 2)}{{\cal C...
 ...eft(\frac{{\cal C}(h, 1)}{{\cal C}(0)}\right)^{2}\right)^{-1/2}\end{displaymath} (7)

Coherence time:

\tau_{\rm AO}=0.31r_{0}\left(\frac{{\cal C}(U, 5/3)}{{\cal C}(0)}\right)^{-3/5}\end{displaymath} (8)

\tau_{\rm SI}=0.36r_{0}\left(\frac{{\cal C}(U, 2)}{{\cal C}(...
 ...eft(\frac{{\cal C}(U, 1)}{{\cal C}(0)}\right)^{2}\right)^{-1/2}\end{displaymath} (9)

Scintillation index:  
\sigma_{\rm I}^{2}=19.12\lambda^{-7/6}{\cal C}(h, 5/6).\end{displaymath} (10)

4.2.1 Isoplanatic patch, coherence time

The isoplanatic angle, $\theta$, depends on the altitude distribution of the turbulent cells producing the seeing, according to Eqs. (7-8). This angle represents the maximum angular distance between the source of interest and a reference star for which the wavefront distortions are coherent and may, in principle, be fully corrected. The maximum integration time, $\tau$, for which a given correction remains accurate is limited by temporal isoplanatism, due to the movement of turbulent cells across the field of view at the wind velocity, according to Eqs. (9-10). The values of these quantities, calculated from the measured $C_{\rm N}^{2}$ profiles are shown in Table 3, along with corresponding values from La Palma and Cerro Paranal. The results at the South Pole are somewhat better than at the other sites, especially during the best observing conditions. The improvement is minor, however, especially in the values of $\theta_{\rm 
AO/SI}$, and does not greatly relieve the problem that the probability of finding a suitable reference source within this area of the sky is extremely low ($\sim$10-3 at visual magnitude mv=16). The relatively large difference between $\tau_{\rm SI}$ and $\tau_{\rm AO}$ is due to the different $C_{\rm N}^{2}$ and $\vec{U}$ profiles, compared with other sites, as discussed in Sect. 3.4.

In practice, varying degrees of partial correction may be sought, and the performance of such systems has been analysed in detail (e.g. Cowie & Songaila 1988; Wilson et al. 1996). Evidently, the maximum image "reconstruction angle'' (the term coined by Cowie & Songaila) depends on the degree of correction required. A trade-off must be made between the quality of the corrected image and access to enough suitable reference sources to obtain adequate sky coverage.

As discussed previously, the large majority of the optical turbulence at the South Pole occurs in the lowest $\sim$200 m. Thus, and due to the strong height dependence of $\theta_{\rm 
AO/SI}$ shown in Eqs. (7-8), we may expect that an image correction system that removed the boundary layer component of the seeing, leaving the remainder uncorrected, would be effective over a substantially greater area of the sky than such a system operating at the best mid-latitude sites.

In a simple approximation, angles for partial image correction were calculated by varying the upper limits, $h_{\rm u}$, of the integrals in Eqs. (7-8). The residual seeing, $\varepsilon_{\rm r}$, related to each of these angles, may be found by setting $h_{\rm u}=h_{0}$ in Eq. (3), and using the values calculated in Sect. 3.4 (Fig. 7). As expected, the reconstruction angle increases rapidly for corrections limited to the boundary layer seeing (i.e. $h\simeq200$ m, $\varepsilon_{\rm r}\simeq0.3\hbox{$^{\prime\prime}$}$ on average), as shown in Table 3 ($\theta_{\rm SI}=1\hbox{$^\prime$}58\hbox{$^{\prime\prime}$}$, $\theta_{\rm AO}=1\hbox{$^\prime$}6\hbox{$^{\prime\prime}$}$), and Fig. 9. A system operating at $\varepsilon_{\rm r}=0.3\hbox{$^{\prime\prime}$}$ at Cerro Paranal would require that $h_{\rm u}$ be much higher (see Fig. 7), resulting in substantially smaller reconstruction angles ($\theta_{\rm SI}=13\hbox{$^{\prime\prime}$}$, $\theta_{\rm AO}=10\hbox{$^{\prime\prime}$}$). Thus, the difference between the sites at the same level of partial image correction is very large.

\epsfbox{}\end{figure} Figure 9: $\theta_{\rm 
AO/SI}$, as a function of the residual seeing $\varepsilon_{\rm r}$. Solid lines are derived from the average $C_{\rm N}^{2}$ profiles at the South Pole, and the dashed lines from the Cerro Paranal data. The curves are obtained by varying the upper and lower limits of Eqs. (7)-(8) and Eq. (3), respectively. The value of $\theta$ for $\varepsilon=0$ is the isoplanatic angle

\end{figure} Figure 10a and b: Percentage sky coverage as a function of the residual seeing, for a) adaptive optics and b) speckle interferometry, in the visual magnitude range m v=14-16. Solid lines show the results for the South Pole; dashed lines for Cerro Paranal. Calculations are averaged over all galactic latitudes

The improvement in terms of sky coverage associated with the increased partial correction angles can be found by determining the probability of finding an appropriate guide star of magnitude mv within an angle $\theta$, at Galactic latitude b (Olivier et al. 1993):

P_{\rm sky}=1-\exp\left[-\pi\theta^{2}\Sigma(m_{ v},b)\right]\end{displaymath} (11)
where $\Sigma(m_{ v},b)$ is the number density of stars of a given mv and b. Values of $\Sigma(m_{ v},b)$ are tabulated in various astronomical data books (e.g.  Allen 1973).

From Fig. 9 and Eq. (12), we have a relationship between $P_{\rm sky}$ and the uncorrected component of the seeing. Figures 10a,b shows this relationship for various values of m v averaged across all galactic latitudes, for both speckle interferometry and adaptive optics. Corresponding calculations based on the available data from Cerro Paranal are included for comparison.

Figure 10 shows that the sky coverage is up to around 75% for image quality $\varepsilon_{\rm r}=0.3\hbox{$^{\prime\prime}$}$, using reference sources in the range m v=16 to m v=14. This value decreases somewhat at the bright end, in the adaptive optics case (Fig. 10a). The percentage of sky covered at Cerro Paranal over this range of $\varepsilon_{\rm r}$ and m v is around $1-2\%$. 75% sky coverage would be possible at this site only with images in the range $\varepsilon_{\rm r}=0.4-0.5\hbox{$^{\prime\prime}$}$.

Note that these values are calculated for an "average'' number density of stars at each magnitude; the higher density near the galactic equator improves the resolution by $\sim$$0.05\hbox{$^{\prime\prime}$}$ for a given value of $P_{\rm sky}$. A further improvement of approximately the same magnitude should also be obtained in the infrared, bringing the residual seeing down to the range $\varepsilon_{\rm r}\sim0.1-0.2\hbox{$^{\prime\prime}$}$.

As noted above, these results assume that partial corrections may be performed based directly on the altitude distribution of the turbulent layers. This is not possible in practice, but, rather, any method of partial correction must take into account the power spectrum of the turbulence. However, the comparison between the two sites certainly remains valid, and the results are a good qualitative indication of the attainable image resolution at the site.

Some of the practical methods that have been developed to increase the workable angle of adaptive optics systems indicate that the achievable image quality may be significantly better than the values stated here. Cowie & Songaila (1988) were able to obtain $0.1-0.2\hbox{$^{\prime\prime}$}$ images at Mauna Kea over an angle of up to $\sim$30$\hbox{$^{\prime\prime}$}$ (a factor of five greater than the isoplanatic angle at this site) by relaxing the requirement of full isoplanicity at high frequencies.

Another method that has been investigated is the "multiconjugate" approach (e.g. Tallon et al. 1992), in which a number of deformable mirrors are used, each placed conjugate with a turbulent layer. Theoretical calculations have been performed by Wilson & Jenkins (1996), to determine the relative performance of image correction systems using pupil and turbulence conjugation. It was found that the sky coverage was greater using turbulence conjugation, compared with pupil conjugation, by a factor of 2-3 (at the same level of image correction).

Turbulence conjugation is a particularly powerful technique in situations where the bulk of the image degradation is concentrated in a small number of turbulent layers, which is clearly seen to be the case at the South Pole (Figs. 4a-f). Since most of the boundary layer turbulence is confined to $\sim 1-3$ intense turbulent layers, with perhaps 2-3 weaker layers throughout the free atmosphere, it can be envisaged that such a system operating at this site would be capable of very high resolution imagery over a large proportion of the sky.

4.2.2 Scintillation index

The scintillation index, $\sigma_{\rm I}^2$ (Eq. (11)), is a measure of the amplitude fluctuations in the received signal from an astronomical source, caused by atmospheric turbulence. It is an important limiting factor in observations that measure small variations in the flux of a source, including observations of variable stars, and asteroseismology. The h5/6 dependence of $\sigma_{\rm I}^2$ indicates that this quantity, too, should have a lower value at the South Pole than at any other site.

The results for $\lambda=0.5~\mu$m, summarised in Table 3 and Fig. 11, show that the average value of $\sigma_{\rm I}^2$ is about 40% of the corresponding value calculated from the Cerro Paranal data (which, again, represents close to the best conditions measured at a mid-latitude site). These results, together with the long continuous observations possible during the polar night, indicate that the antarctic plateau is an ideal site for the types of observations mentioned above.

\epsfbox{}\end{figure} Figure 11: Distribution of the value of $\sigma_{\rm I}^2$ calculated for each of the fifteen balloon flights. The highest measured value, $\sigma_{\rm 
I}^{2}=0.118$, is statistically more than 3$\sigma$ above the mean, and the $C_{\rm N}^{2}$ data from this flight were omitted when calculating the average value shown in Table 3

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