Hues of the Heavens
Astrophotography by Ivan Bok
Sub-Pixel Autoguiding
How a 30 mm guidescope can guide a 2000 mm FL telescope
Note: For those who already know that autoguiding is, please skip section 1
1. Introduction
Astrophotography of deep sky objects necessitates long exposures, because the light originating from distant stars and nebulae need to traverse across the vast distances of interstellar and intergalactic space. While long exposures are technologically easy to achieve by just about any image sensor in the modern day, the photography of celestial bodies presents a particular challenge that no terrestrial feature has: the fact that the Earth spins.
Creating sharp images of the stars when exposures can take several minutes or even hours therefore requires a tracking mount to compensate for our planetary spin. For the most part, this is achieved with equatorial mounts.
1.1 Autoguiding vs Tracking
Tracking: A motorised mount tracks the motion of the stars, compensating for the effect of Earth rotation
Autoguiding: A setup (consisting of a second telescope and camera) that increases the accuracy of a tracking mount by monitoring and tracking errors and sending correction signals to the mount
The term ‘autoguiding’ is a misnomer of sorts. Many people conflate the terms ‘tracking’ and ‘autoguiding’. Where tracking refers to the automatic motion of motorised mounts in following the motion of the stars, which is an almost necessary prerequisite to any deep sky astrophotography, autoguiding is a more complex process used to increase the accuracy of a mount that already has tracking capability.
So what is the purpose of autoguiding if mounts can already track on their own? The answer lies, for the most part, in mechanical imperfections in every single equatorial mount that exists. Manufacturing tolerances mean that some mounts, more so than others, have gears that are less precisely fabricated. This means that for high precision tracking of the night sky, the mechanics of many stock tracking mounts may not be sufficient for the demands of the very long exposures required for imaging.
Enter autoguiding. Autoguiding is a closed-loop process. In essence, a second smaller telescope (ignoring off-axis guiding for now) and camera are mounted in parallel to the main imaging telescope and camera. The smaller telescope and camera will henceforth be simply referred to as the autoguider.
Where the main imaging system shoots the actual images, the autoguider sequentially shoots short exposures (typically on the order of 1 second). In short, the process happens in the following sequence:
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The autoguider continuously takes short exposures, sending the images to a computer
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The computer, which is loaded with autoguiding software (such as phd2), monitors the position of a selected star
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If the position of the star ever so slightly deviates from its original position, the computer will send a correction signal to the mount to move the star back into its original position
Autoguiding therefore provides a closed feedback loop, since any deviation in the position of the star will be picked up and corrected by the software and mount. When the process is working correctly, tracking errors will not accumulate over very long exposures, which is ideal for imaging faint targets.
2. Sub-Pixel Guiding
With the basics of autoguiding out of the way, I will now bring in another important principle known as sub-pixel guiding. Oftentimes, it may seem unintuitive that a very small, short focal length (i.e. low magnification) telescope is capable of guiding a much larger one. At first glance, it is reasonable to assume that a guidescope should ideally have a focal length comparable to the main imaging system. However, this is not true because of sub-pixel guiding.
Personally, I have used my little 30 mm guidescope (at 120 mm focal length) to successfully autoguide up to 2500 mm of focal length. I have also heard of guidelines stating that a guidescope needs to be at least 1/3rd the focal length of the main imaging telescope for guiding to work successfully. However, as experience suggests, this is simply not true. Theoretically, there is a sound (and intuitive) reasoning why this works. To understand this, let us build a simple model of how autoguiding works.
3. A Simple Model to Explain Sub-Pixel Guiding
3.1 Model Assumptions
Before we create the model, there are a few simplifying assumptions that we have to make. While these are not entirely realistic, they are necessary for us to perform a quantitative analysis. After the model is explained, we will consider what happens when these assumptions are relaxed. If the assumptions confuse you, feel free to skip them as they are only there for the more technically minded to scrutinise.
Assumptions:
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Infinite signal-to-noise ratio (or at least noise is negligible relative to CCD bit depth)
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Diffraction-limited optics with perfect seeing (i.e. no atmospheric distortion)
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8-bit Monochrome sensor (no Bayer matrix), autoguiding only in a single axis
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Star profile only extends across 2 pixels, and has a uniformly-distributed point-spread function (PSF)
With these assumptions in mind, let’s consider what happens when a star is being registered on the image sensor. When light is collected by a single photosite (i.e. single pixel) and converted to digital signals, it is registered as a value between 0 to 255. The more intense the light received, the greater this value becomes. If no light is received, the photosite reads a value of zero. If too much light is received and the photosite becomes fully saturated, we read a value of 255. Ideally, we do not want our guide star to be too bright, because it will fully saturate the photosite.
3.2 Creating the Model
Consider a star with a uniformly distributed PSF that is positioned exactly between 2 photosites, each of which having a width of 1 μm. Let’s call this position A. At this position, both photosites register the same star intensity, because both sides receive an equal amount of light from the star. Additionally, we denote d1 and d2 as the length of the star profile that is registered by pixels 1 and 2, respectively. In position A, the star profile is such that d is equal for both photosites, which is the reason both pixels detect the same light intensity.
However, let’s now consider what happens when there is a slight shift in the position of the star due to (for e.g.) tracking error. This shift causes it to move towards position B. The amount of movement here is 0.004 μm, or 1/256th the width of the pixel. This shift results in a change in the distribution of the star’s light between the 2 sensors, which causes pixel 1 to read a slightly higher value of the light intensity (64/255) compared to its original value. Similarly, pixel 2 records a lower value of the light intensity (62/255) since a smaller proportion of the star’s profile now lies above it.
This model therefore gives us the critical conclusion: in perfect conditions, the maximum theoretical guiding accuracy is at a scale that is 1/bit depth of the width of the pixel. For an 8-bit sensor, this means that the autoguider can theoretically detect any movement as small as 1/256th the width of the pixel!
3.3 Relaxing the Assumptions
In reality, sub-pixel guiding accuracies approaching this theoretical limit can rarely ever be achieved. This is due to the fact that we have made several assumptions that are unrealistic in a practical setting. In particular, this is the assumption of infinite SNR and the lack of seeing limitations. While the other assumptions also do not hold in practice, they are not the reason behind why we cannot guide at the theoretical limit.
When looking back at the assumptions, one striking point of note is that the factors limiting guiding accuracy (i.e. low SNR) have no relation to guidescope focal length or pixel pitch. For a given star, the SNR is dependent on the clear aperture (not aperture ratio) of the guidescope and the sensitivity of the guide camera. To guide with fainter stars, a larger aperture is required, which may not be necessarily accompanied with a longer focal length or faster optics (which only affect the photon flux of extended objects and not point sources).
However, most guide cameras are sensitive enough today to get sufficient SNR from stars at just about anywhere you point the telescope in the night sky for decent guiding capability. While SNR can be increased with larger guidescope apertures, it is usually not necessary because even a small setup with a 30mm guidescope is already limited by atmospheric seeing. As such, there is often no benefit to using large guidescopes even when imaging at long focal lengths.