Matching Telescope Focal Length and Pixels: Designing an Optimized System

 

Image of the Butterfly Nebula with an optimized setup with high signal-to-noise ratio and high resolution. Credit: Rouzbeh Bidshahri

 

There are several factors to consider when designing a deep sky astrophotography setup, including image scale, resolution, sampling rate, and atmospheric conditions at your site. The telescope aperture and focal length must also be matched to the sensor size and pixel size of the camera. Here, I’ll explain the concepts and compare the results from a well-matched system to one that is not optimized.

 

Sampling Rate 

In modern astrophotography, the optical image produced by a telescope at the focal plane is recorded with a digital sensor. It is a marriage of the analog to the digital and, like in any marriage, these two sides need to be well matched to achieve good results.

When recording the image (signal), we need to have enough sample points (pixels) to ensure it is an accurate reproduction of the original. This is known as the sampling rate, and the closer each pixel is to the next the finer the resulting image scale.

To get the sample points closer to each other, we can increase the focal length of the telescope or use a camera with smaller pixels. This gives us a finer image scale, which is how much of the sky is recorded by each pixel, in arcseconds per pixel, much like dots per inch on a printer (DPI). The smaller (finer) the image scale, the higher the resolution of the image.

 

Oversampling and Undersampling

A common approach is to attempt to image at too fine of an image scale as typically achieved with long focal length scopes. However, there are downsides in doing this such as a reduced  field of view and a dimmer image.

This is referred to as oversampling; using too many pixels for the smallest detail the telescope can resolve. It is inefficient and doesn’t reveal any additional information, merely spreading the incoming information from the sky over too many pixels. This lowers the signal-to-noise ratio (SNR), which can be seen in the grainy image below where stars are fuzzy and soft.

An oversampled image with information spread over too many pixels. Note the fuzzy, soft-looking stars. Credit: Rouzbeh Bidshahri

 

Too few sample points leads to undersampling; decreased resolution and loss of the fine detail the system is capable of resolving.

Below is an example of undersampling with the same target as the oversampled example above. Here the stars appear to be square, clearly not an accurate reproduction of stars. There aren’t enough pixels to record the finer detail that should be available.

An undersampled image with square, blocky-looking stars. Credit: Rouzbeh Bidshahri

 

Optimal Sampling and System Design

So, what is the optimal sampling rate, also referred to as critical sampling, for recording with high efficiency and close to the maximum resolution possible? Let’s first consider the smallest point our telescope can produce.

The signal from the sky, as delivered by the telescope, has a finite resolution, with a limit to the smallest detail that can be discerned and output for the camera to record.

A major factor determining the available resolution is atmospheric seeing, the blurring caused by atmospheric turbulence (assuming good optics and sufficient aperture for our purpose). We can determine this by measuring the size of star images (FWHM). The seeing will vary within a range for any given location and season. This is how I rate seeing in arcseconds:

Very good: 1.0 to 2.0 arcseconds

Typical: 2.0 to 3.0 arcseconds

Mediocre: 3.0 to 4.0 arcseconds

Poor: 4.0+ arcseconds

Classical sampling theory (Nyquist Sampling) calls for sampling by at least twice as much as the smallest available detail. For astrophotography, though, one can go a bit further. My recommendation is to sample at 2.5 to 3.5 times your seeing limit.

You could go over this range but the gain in true resolution over 3.5 times the seeing limit is very small. After about 5 times there is no resolution to be gained, and the SNR and field of view (FOV) are reduced. As an example, the new 8-meter Rubin Observatory also samples at 3.5 times the excellent mountaintop seeing of 0.7 arcseconds, an image scale of 0.2 arcseconds/pixel.

For example, if you have very good seeing of 1.8 arcseconds at your site:

For emphasis on speed and higher SNR, aim for a 2.5x finer sampling rate:
1.8 / 2.5 = 0.72 arcseconds / pixel

For emphasis on resolution, aim for a 3.5x finer sampling rate:
1.8/3.5 = 0.51 arcseconds / pixel

 

Now Design Your Imaging Setup

Pixel Size: Modern imaging cameras are tending towards smaller CMOS pixels, with the majority of amateur sensors currently sized at 3.76 microns. One can easily join several of them together (binning) to synthesize larger pixels but the effective image size is also decreased by 4 or 9 times (for 2x2 and 3x3 binning, respectively).

Focal Length: Use this formula to determine your required focal length based on pixel size and the desired image scale:

Focal Length (mm) = Pixel Size (microns) / Image Scale (arcsec/pixel) x 206

If your pixels are 3.76 microns, with the sampling rates from above:

For a 2.5x finer rate:  Focal Length = 1,077mm

For a 3.5x finer rate:  Focal Length = 1,521mm

 

F-Ratio: For deep sky imaging, my recommendation is a focal ratio between f/4.0 and f/5.0.

Telescopes with longer (larger) f/ratios are smaller and less expensive, but they will be slower at building up the SNR you need.

F- ratios below f/4.0 are often reserved for more exotic designs. They are faster, with more aperture for the same focal length, but will usually cost considerably more. Most are less user friendly in terms of ease of collimation, the availability of back focus, and tilt and defocus tolerance. We’ll use f/4.5 in designing our example system.

 

Determining Telescope Size:

We now know the major parameters for our example: 

Best case sky conditions = 1.8 arcseconds

High-resolution emphasis sampling Rate = 3.5x finer 

Image scale = 0.51 arcseconds / pixel

Required focal length = 1,521mm

Preferred f-ratio = 4.5

Required Telescope Aperture = (Focal Length / F-ratio) = 1,521mm/f4.5 = 338mm aperture or 13.3 inches

 

Tests and Results

To illustrate the impact of each of these factors, I conducted a series of real-world tests with a well-matched system and one that is not optimized, comparing them for field of view (FOV), image brightness, signal-to-noise ratio (SNR), and integration time. I found the most interesting results to be the comparisons of true resolution and image sharpness.

A 14-inch Planewave CDK was used for testing with a camera using 3.76 micron pixels. This telescope has been modified to switch focal lengths quickly (described for AGT here). The Soul Nebula was selected as the target, and it was imaged for two hours (10 x 600-sec) in both configurations. The seeing at my site is rarely below 2.0 arcseconds.

The two systems for testing have these characteristics:

1. Oversampled Setup: Native focal length of 2565mm, f-ratio of f/7.2, image scale of 0.3 arcseconds/pixel.


2. Optimized Setup: Reduced focal length of 1700mm, f-ratio of f/4.75, image scale of 0.45 arcseconds / pixel. 

 

Field of View

The most obvious effect of reducing the focal length is the gain in the field of view (FOV). A larger FOV is usually preferred as it allows a greater number of objects (or larger objects) to fit in the frame. If the FOV is larger than needed, the frame can always be cropped. A focal length that is too long creates a small and limited FOV that can only be used by upgrading to a camera with a larger sensor (assuming the telescope can cover the larger sensor).

Below is an animation of the native FOV of the test system compared to the lower focal length FOV.

 

FOV gain with a reduced focal length. Credit: Rouzbeh Bidshahri

Image Brightness

Lower f-ratio systems are faster and create brighter images at the focal plane. The image below shows the brighter image resulting from switching from the f/7.2 test configuration to f/4.75. The images are stretched the same amount.

 

Both nebulosity and stars are more than twice as bright using the shorter focal length as measured by pixel ADU values, as expected. Credit: Rouzbeh Bidshahri

 

Signal to Noise (SNR)

Every system will have some noise, but the brighter image from the faster system provides more signal to overcome noise better and faster. With an optimized system, shorter sub frames can be used over a shorter total integration time to achieve the same SNR as an oversampled system.

Below is a comparison of stacked images: one from 10x600-sec frames at the slower f-ratio and the other only 6x600-sec frames at the reduced (faster) f-ratio. The SNR achieved in each image is similar despite the difference in total integration time.

 

Less integration time was required to achieve the same SNR with the faster configuration: Oversampled 10x600-sec vs. Optimized 6x600-sec, both with an equal measured SNR. Credit: Rouzbeh Bidshahri

 

Loss of Resolution 

What about the higher resolution requirements for imaging small objects like galaxies?

This is probably the main argument used to recommend longer focal lengths. It’s generally believed that a longer focal length will record smaller details, but that’s not always the case.

An optimized system does not have less resolution (the finest detail that can be seen in the data) than an oversampled one. The target object will be larger with a longer focal length system but that is image scale, not resolution. Resolution is usually determined by the atmosphere. With perfect seeing, telescope design characteristics such as aperture will determine the maximum resolution possible.

Having captured as much resolution as possible from the object, we can up-sample that information in post processing and enlarge the image to match the image produced by the longer focal length system.

This is demonstrated with the comparison below, with the same target imaged in the two test configurations on the same night. The smaller image of the target recorded with the optimized, lower focal length system was up-sampled 1.5 times to match the image scale of the longer focal length setup, with no loss in resolution.

 

Image resolution compared on the same night (Note: target’s altitude was lower for the optimized setup). Credit: Rouzbeh Bidshahri

 

The previous test was actually a bit unfair as the object’s altitude was lower by the time we switched to the reduced system. The test was repeated the next night with the target at the same altitude as the native longer focal length setup.

Again, all other parameters were kept constant. The optimized setup actually performed better in terms of image sharpness, with smaller, sharper stars.

Image sharpness compared at the same altitude on two successive nights. Credit: Rouzbeh Bidshahri

Final Words

Almost any combination of telescope and camera will produce an image, but a great deal can be gained using a well-matched system.

With the current trend towards smaller CMOS pixels, oversampling is becoming more common. While we can bin pixels, given the choice, it would be preferred to use a sensor in its native format. The same applies to optics; a telescope that is closer to your requirements in its native form is also preferred. An optimized system will generally have an advantage in field of view, image brightness, and signal-to-noise ratio compared to an oversampled setup.

Remember not to confuse a larger displayed image with higher resolution. Once we have captured the most resolution possible with our setup and site conditions, small objects can be enlarged in post processing just as well.

To achieve higher resolution, a larger telescope or longer focal lengths is not always the answer. A more important factor is a location with excellent seeing.

 

A gallery with examples of images captured with these parameters is available at https://rouzastro.com/nebulae/