Image Scale Considerations

Let us have a look at this image scale business in more detail because it is important to understand the issues involved. Most of the popular webcams have CCD chips with an array of 640 x 480 pixels. These pixels are, typically, 5.6 microns in size. So the imaging chips themselves are roughly 3.6 x 2.7 mm across. The standard formula for calculating the resolution of a telescope, in arc-seconds, is given either by 138/D (the Rayleigh limit) or 116/D (the Dawes limit), where D is the

Figure 7.2. A TeleVue 5x Powermate attached to a ToUcam Pro webcam ready for insertion into the telescope drawtube. Image: Martin Mobberley.

aperture in millimeters. The first formula is based on the theoretical effects of diffraction at the wavelength of green light, whereas the second is based on practical measurements (by a chap called the Rev. Dawes) on the ability of refractors to visually split close double stars. I will adopt the 116/D formula here. Essentially, the Dawes limit tells us that a telescope of 116-mm aperture will split two double stars of equal brightness that are 1 arc-second apart. A telescope of double the aperture, i.e., 232 mm, will split two stars 0.5 arc-seconds apart. The question is, if we want to capture all the detail that the telescope can theoretically resolve, what focal length do we need to increase the telescope to? This is not as easy a question as it might sound. While the Dawes limit is established in astronomical literature, amateurs have always been able to see much smaller, high-contrast features on the Moon (like lunar rilles) and even on the planets (like the Encke division on Saturn). Resolving two equally bright components of a double star, as their diffraction patterns merge, is a special case and it tells us little about how many pixels should straddle the area in order to capture every last bit of detail. One guideline, often quoted, is the Nyquist sampling theorem, which tells us that to accurately sample a periodic signal of highest frequency f, we need to sample it at twice the frequency. From the terminology the reader will gather that this theorem is generally intended for sampling signals in the world of electronic communications. However, taken at face value it says that if a telescope can resolve 1 arc-second, we should sample the image at a scale of two pixels per arc-second. But pixels are two dimensional and across their diagonal they are 1.414 times bigger, so the diagonal sampling could be interpreted as 1.414 times too coarse. Maybe we need to sample at nearer three pixels per arc-second for a 116mm aperture telescope? With such a telescope and a webcam having 5.6 micron pixels, sampling at two and three arc-seconds per pixel corresponds to f-ratios of 20 and 30, pretty close to what many leading amateurs actually use. Beginners tend to start at f/20: more experienced users tend to drift toward f/30 or more to capture the finest details. Even f/40 is justified under good seeing conditions and with a sensitive webcam.

It should be remembered in this context that not only do planets consist of a myriad of overlapping diffraction patterns from each point on the planet's surface but the webcam user is going to end up stacking hundreds or thousands of images on top of one another. The first point is the reason why we can glimpse such fine high-contrast detail on tiny features like the Moon's lunar rilles, and the second gives us an additional statistical advantage that even the most eagle-eyed visual observer simply does not have. When hundreds of webcam frames are stacked and image processed, quite astonishing details emerge; features like Saturn's Encke division can become almost routine targets given good enough seeing. So, bearing all this in mind does anyone know just how high an f-ratio the planetary imager should really use, in perfect conditions? Is even f/30 enough? Well, in 2004, Damian Peach carried out some interesting experiments with an 80-mm aperture Vixen apochromat of exceptional quality. He imaged high-contrast features like the Moon and sunspots to try to answer this very question. Using only an 80-mm aperture, often under superb seeing conditions (even by large aperture standards), Damian could be sure that the Earth's atmosphere was having no effect on the images he was taking. The lunar pictures Damian obtained under these conditions, when he was temporarily without a decent aperture instrument, were quite an eye opener. To digress for a moment, many years ago I knew the legendary lunar and planetary photographer Horace Dall, inventor of the Dall-Kirkham Cassegrain telescope. He was the best amateur lunar and planetary photographer of his generation and he used a 39-cm aperture Dall-Kirkham telescope. I have compared Dall's best lunar pictures, taken in the 1960s and 70s, with Damian's best webcam images using his 80-mm apochromat. Damian's images resolve fractionally more than Dall's, despite the fact that Dall's telescope had nearly 5 times the theoretical resolution and over 20 times the light grasp! Features like the Hadley rille on the Moon are clearly detectable on Damian's pictures, despite being no more than an arc-second across and the theoretical resolution of the telescope being only 1.45 arc-seconds. Damian concluded that to squeeze the last drop out of the telescope's resolution he had to increase the apochromat's f-ratio to around 45. With the AtiK webcam he was using (also with 5.6 micron pixels) this corresponded to a sampling resolution of 4.5 times finer than the theoretical resolution of the telescope.

With larger apertures, Damian and others tend to use f-ratios of 30-something when the seeing conditions more determine the final outcome with a decent aperture. There are other factors, too, when imaging the planets. Saturn, in particular, is a faint world at high f-ratios. This is hardly surprising as it orbits the Sun at a mean distance of 1400 million kilometers where sunlight is only 1/90th as powerful as here on the Earth. Using a commercial color webcam at f/40 may cause the color balance to fail as the light levels are low. Also, don't forget how small the webcam field of view will become at long f-ratios. Jupiter will fill a webcam screen completely at focal lengths above about 10 meters.

A useful formula that tells you the image scale of your system in arc-seconds per pixel is as follows: image scale = 206 x pixel size in microns/focal length in millimeters. So, as an example, for a 250-mm aperture telescope working at f/30 with a webcam having 5.6 micron pixels: image Scale = 206 x 5.6/(30 x 250) = 0.15 arc-seconds per pixel. Most of the world's top planetary imagers are now working at images scales of between 0.1 and 0.2 arc-seconds per pixel.

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