To calculate pixel scale, one needs telescope focal length and pixel width or height in microns.
pixel scale or sampling in arcseconds = (206.265) * (pixel size in microns) / (focal length in mm)
focal length = (206.265) * (pixel size in microns) / pixel scale
The constant 206.265 converts microns to mm and radians to arc seconds.
A micrometer or micron is one millionth of a meter, or equivalently one thousandth of a millimeter (mm)
[360 deg / (2 pi radians)] x (3600 arc sec / deg) x ( 1 mm / 1000 microns) = 206.26480624709635515647335733078 ~= 206.265
or just about 206 as many use as rule of thumb.
This is a straightforward calculation if you know the two input parameters.
Focal ratio is the ratio of the focal length to telescope aperture
Focal ratio = focal length / aperture
For example, if focal length = 3048 mm and aperture = 30.48 mm, Focal ratio = 10 or telescope is at f /10.
If you have 9 micron square CCD pixels, then your pixel scale is 206.265 x 9 / 3048 or 0.6 arc sec per pixel.
This has meaning when compared to your "seeing" measured in arc secs of FHWM (Full Width Half Maximum) of the stars in your images and is referred to as "sampling". You have near optimal sampling when your pixel scale is about half of the value of your seeing. If your pixel scale is less than optimal, the sampling is termed "under sampled". If your pixel scale is larger than optimal, the sampling is termed "over sampled".
Focal reducers reduce the effective focal length of your imaging setup and increase your pixel scale.
Binning your CCD camera imaging chip 2x2 (versus high resolution 1x1) doubles the size of each pixel and increases your pixel scale.
Barlow lenses increase the effective focal length of your imaging setup and decrease your pixel scale.
Astrometrica calculates these parameters for you if you can interpret the values provided after doing a plate solution. The default values of its configuration settings must be initially modified by a new user for their imaging setup. Thus, one must be able to calculate or estimate a focal length as an input parameter for this program.
This is a copy of several lines in a recent Astrometrica.log file from my own astrometry:
The column labeled FHWM reports the seeing of the stars in the image and shows that sthe seeing is about 5 to 6 arc seconds. The pixel size is calculated at 2.58 arc sec or about half the seeing. I was using 2x2 binning on a SBIG ST-8XME CCD camera. There is a f/6.3 focal reducer a filter wheel and an SBIG AO-7 unit in the imaging setup, but due to the spacing of the components resulting in an effective focal ratio of near 4.7. This also gives a nice size field of view to capture more stars which is better for accurate astrometry. The SNR of the stars measured for astrometry is also given. It is important to have sufficient SNR for an accurate centroid calculation of stars and the object to be measured. The Fit RMS is the most important columns because it shows the bottom line of the astrometric calculation and the goal is to have fractional arc second accuracy is all measurements..
Pixel scale is used to determine how long of an exposure one can take of a moving object. That will be the subject of another topic.
The larger your pixel scale the longer you can expose your target per image.
The larger your pixel scale, you provide allow more photons from your target to fall on a pixel which improves the SNR of your target, improves the calculation of the target centroid in the image, improves astrometric precision, and results in smaller residuals.(only up to optimal)
Too high a pixel scale beyond optimal however, is not good for astrometry because stars will be undersampled and are not spread out over one or more pixels. It would be like trying to accurately measure the position of a fly with a ruler with only inch or centimeter tick marks. and is of no astrometric value no matter how reference stars you have imaged..
IAU Minor Planet Center Guide to Minor Body Astrometry: