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Fast or slow? Your scope's focal ratio affects your images, and here's how.

One of the hotly debated topics among deep-sky astro-photographers today is the importance of telescope f/ratio for deep-sky imaging. It's a rather complex issue, but it can be boiled down to this: you can gain signal-to-noise ratio (SNR) but lose resolution by decreasing the f/ratio. Higher SNR is always good. Reduced resolution can be good or bad depending on the circumstances. But decreasing the f/ratio can also have other side effects, as we shall see.

Fast versus Slow

Imagine that you have two telescopes with the same aperture, but one is f/8 and the other f/4. These scopes will collect the same amount of light; the number of photons collected doesn't change with the f/ratio, only the way these photons are distributed on your camera's detector. Let's say you chose a galaxy that will fit on the chip at both f/ratios and you use the same exposure time. With the f/8 telescope, your galaxy will cover four times as much area on the chip as it covers at f/4. With the f/4 telescope, however, the image is brighter because the light is projected onto a smaller area.

But what does this mean for the SNR of one pixel when you take identical exposures through these two telescopes? Dark current and read noise will be the same, but since the signal (light from the galaxy) per pixel is more concentrated at f/4, the SNR will improve. Photon noise (shot noise) will increase at f/4, but not as much as the signal, so the SNR is still better. Skyglow and light pollution will rise the same way the signal does. As a result, you will achieve a better SNR by decreasing the f/ratio. The SNR will not be four times better by halving the f/ratio, as one might think, but the improvement is significant.

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You also boost the dynamic range of your image: the tonal range of your galaxy covers more gray levels as seen by the camera. That means smoother tones in your image, particularly when you stretch the image in processing. And of course the field of view increases as the f/ratio decreases.

So what's the price of all this? You can't expect an improved SNR without somehow paying for it. The price is paid in resolution. Your object only covers one-fourth as many pixels at f/4 as it does on the f/8 instrument. Whether that's good or bad depends on your ultimate goal for the image. And there are other problems related to telescopes with fast f/ratios. They are harder to make and are therefore usually more expensive. They need a higher degree of mechanical precision in the optical system, and collimation and focusing are much more critical. It's often difficult to achieve a well-corrected and well-illuminated field of view at a fast f/ratio, particularly when using large detectors. Also, it can be difficult to align large-chip cameras square to the optical axis. For these reasons, there are limits to how fast a telescope can be for astrophotography, but f/4 to f/5 is not unusual today.

Resolution and Sampling

Let's return to the problem of resolution. Compared to f/8, at f/4 your galaxy (or other deep-sky object) has only half the linear resolution, and one-fourth the measured area on the chip. If you're trying to resolve the finest details that your local seeing conditions will allow, a fast f/ratio may cause problems. To find out, you need to know what pixel resolution your scope and camera deliver.

Pixel resolution depends on your telescope's focal length and your camera's pixel size. The smallest resolvable detail, according to the Nyquist-Shannon sampling theorem, should cover two pixels. Less than that means undersampling and lost detail, while more becomes oversampling and longer-than-necessary exposure lengths. Many of us live in places where the seeing is often poor or average. Where I live in Denmark, the smallest resolvable detail in exposures longer than a few seconds is usually about 3 to 5 arcseconds. This means that around 1.5- to 2-arcsecond-per-pixel resolution will be close to optimal. Of course, seeing conditions can vary on different nights and at different locations, but learning what your average conditions are will be good enough. Pixel resolution is calculated using this formula:

Arcseconds per pixel = (pixel size/FL) x 206.3

where pixel size is measured in microns and focal length in millimeters.

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If your criterion is optimal sampling at a fast f/ratio, you can use the above formula to determine the best combination of telescope and camera. Let's say that I want to use a camera with 9-micron pixels and an f/5 telescope. An instrument with 8 to 10 inches (200 to 250 mm) of aperture would be about right for my rather typical seeing conditions. If you have excellent seeing on a regular basis you can use a bigger scope--or a camera with smaller pixels. So the best telescope for a specific camera is the one that has a fast f/ratio and matches your seeing to your pixel resolution.

The pixel resolution for a given camera is solely a function of the telescope's focal length, but the aperture comes into play for the desired f/ratio. So optimal sampling with any given camera corresponds to a specific focal length. The most effective way to image at that focal length is with a fast f/ratio. One way to look at the f/ratio question is that it's just a way to increase the aperture, and thus the amount of light collected, at a preferred focal length, or at a preferred pixel resolution. Since all focal lengths are good for something, in principle a fast f/ratio is always preferred.

Large Instruments

The previous statement suggests that there's an optimalsize telescope for your camera and your local seeing. But if that's true, why is it that amateurs and professionals desire bigger scopes? Keck, VLT, and other giant scopes feature f/ratios in the same range as our small scopes, and the cameras on these scopes have about the same pixel size as ours. According to the earlier logic, these large scopes should not be able to "go deeper." While it's true that professional observatories often benefit from superb seeing and telescopes that have adaptive optics, that only improves the resolution, not the per-pixel SNR.

As aperture increases, a telescope collects more light. If the f/ratio and the camera are the same, the per-pixel SNR will not change, but the SNR for the entire object will nevertheless improve.

There's a more subtle way in which big scopes work: Most deep-sky objects are not point sources. A star is a point-source--its angular size is smaller than the theoretical resolution of any telescope, and in an ideal world any star would be seen by a single pixel. While that's not the case in the real world, stars are still more or less seen by your camera as point sources. So you could change the f/ratio, and hence the focal length, of your telescope as much as you like without any measurable effect on the brightness of a star. The only way to make the star brighter is to increase the telescope's aperture.

Non-point-source objects behave differently. The surface brightness of such an object, as seen by your camera, is a function of f/ratio. If you double the aperture but retain the same f/ratio, the scope collects four times as much light, but this light is distributed over four times more of the chip (because the focal length also doubles). The surface brightness doesn't change. So any size telescope will see the same surface brightness as long as the f/ratio is the same. If you reduce the f/ratio, however, the surface brightness of an extended object will always increase.

The point of all this discussion is that most deep-sky objects feature knots and edges, and thus fall somewhere in-between point-source and evenly lit objects. Increasing the aperture will help deep-sky objects as well as stars; even if you use the same camera you used for your "optimal" scope and as a result oversample the image. This also helps large scopes, which tend to have rather slow f/ratios, to nevertheless work well for deep-sky imaging. As the old saying goes, aperture rules.

Small Telescopes

The prior discussion makes it sound like small telescopes are no good for imaging! Rest assured they are, just not for the same things larger scopes are good for. Deep-sky objects come in all shapes and sizes, and small scopes are great for wide-field images--you can fit big deepsky objects and large areas of sky into their field of view. Wide-field imaging is one of the most rewarding disciplines in astrophotography, and it can only be done with small scopes or camera lenses. When imaging wide fields, we don't worry about optimal sampling. Undersampling is not only acceptable in wide-field imaging, it's beneficial because more light reaches each pixel, thus making up for the small aperture. Undersampling is comparable to reducing the f/ratio in that regard. As mentioned earlier, a fast f/ratio will help at whatever focal length you need for your intended image, so every aperture has its merit.

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Let's return to the SNR. There's a general belief that reducing the f/ratio only helps reduce noise in the faint parts of a deep-sky object, such as extended nebulae or the dim regions of a galaxy, because the bright regions are recorded well, even at a slower f/ratio. It's certainly true that noise is always most apparent in the faint parts of the image, but it's not true that SNR is unimportant in the bright areas. At least not if you want to show things such as the bright cores of galaxies, or the Trapezium region of the Orion Nebula, with as much contrast and detail as can be squeezed out of the image. Good SNR is important in the "highlights" as well as the "shadows," but it's true, after all, that the faint parts are where it matters the most.

Let's go back to our two telescopes mentioned earlier. If you want to collect the same number of photons per pixel with the f/8 scope as you do with the f/4 scope, you need to quadruple the exposure time. That's bad enough, but it's not the only problem. Deep-sky images are usually made by stacking a series of sub-exposures. While you could take sub-exposures that are four times longer, most astrophotographers would simply take four times more sub-exposures. The read noise from four times as many exposures would mean that the per-pixel SNR still wouldn't be as good as with the f/4 scope.

The f/ratio skeptics sometimes say that the advantages of fast f/ratios only apply to old-fashioned film cameras, because film suffers from reciprocity failure. It's true that the f/ratio was more important for film-based deep-sky imaging, but the discussion above shows that the f/ratio is important in CCD imaging too.

Jesper Sorensen images from his backyard in Copenhagen, Denmark, with a variety of telescopes chosen for specific tasks.

Comparing f/4 and f/8

Using fast f/ratio instruments (top) for most deep-sky imaging has the advantage of better signal-to-noise ratios compared to slower instruments. For example, it would theoretically take four times longer exposure time to record a comparable signal with a similar-sized instrument having twice the f/ratio. But speed comes with a price. The image scale on your CCD detector at f/4 will be half the size of that recorded on the f/8 instrument. This lowers the image resolution.

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Title Annotation:Imaging and the f/ratio
Author:Sorensen, Jesper
Publication:Sky & Telescope
Geographic Code:1USA
Date:Nov 1, 2010
Words:1973
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