Comparison of limiting magnitudes

The limiting magnitude of reference source which can be used for adaptive optics is set by the need to measure the reference source image position in each of the $\sim r_{0}$ diameter subapertures in about one tenth of the atmospheric coherence time for the subapertures. This is a similar problem to the correction of image position for Lucky Exposures, and I will now compare the two limiting magnitudes directly.

In the simplest approximation, the limiting magnitude for measurement of image position is set by the requirement for a minimum number of photons in the image core. The number of photons in the image core is proportional to the photon flux density $I$ from the star at the observing wavelength, the collecting area of the aperture $A$, the exposure time $T$ and the Strehl ratio of the image $\mathcal{S}$. If the number of photons required in the image core is the same in both cases, and the losses in the optics and the detector are the same, then from Equation 1.10 the limiting magnitude for adaptive optics will be poorer by:

$$\Delta m=2.5 \log \left (\frac{A_{AO}T_{AO} \mathcal{S}_{AO}}{A_{LE}T_{LE} \mathcal{S}_{LE}} \right )$$ (1.11)

where the subscripts $AO$ and $LE$ refer to the adaptive optics and Lucky Exposures cases respectively. For the case described in Figure 1.4b the telescope diameter for the Lucky Exposures case is seven times the adaptive optics subaperture diameter. Passive Lucky Exposures observations can have ten times longer exposure times than adaptive optics wavefront sensors, but the Strehl ratio in the subapertures of an adaptive optics wavefront sensor is typically twice that in a Lucky Exposure. This means that Lucky Exposures should be able to use stars which are about seven magnitudes fainter than would be required for near diffraction-limited imaging with adaptive optics. The faintest reference stars which provide good adaptive optics correction at I-band are $I\sim 10$ (Graves et al. , 1998), in broad agreement with the arguments here.

Recent studies (e.g. Ragazzoni & Farinato (1999)) have shown that novel wavefront sensors such as Pyramid sensors can improve the reference star limiting magnitude for adaptive optics by several magnitudes at extremely large telescopes, but the gains for moderate sized telescopes such as those described in this thesis are relatively small. The limiting reference star magnitude is still not competitive with Lucky Exposures.

One approach which may overcome the problems with the reference source limiting magnitude for adaptive optics is the use of artificial reference stars, typically provided by light scattered from a high power laser pointing along the line of sight of the telescope. A number of observatories are currently developing such laser systems.