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Angular Separation

The apparent visual distance between astronomical objects is often measured by angular separation.  The angle between the objects with the vertex at the observer's location is the angular separation, for example, 6.5 arc seconds of separation.

When a telescope projects an astronomical image onto a sensor such as a CCD camera, the separation can be measured as the linear distance between where the two objects strike the sensor, for example, 45 micrometers of separation.

Lastly, the separation on the surface of the CCD sensor can be measured as the number of fractional pixels between them, for example, 4.7 pixels of separation.

There are mathematical relationships between all three of these measurements - arc seconds, pixels, and linear distance - as shown below.  Therefore, for a given telescope (focal length) and camera setup (pixel size), if any one of the measurements is known, then they are all known.  These mathematical relationships allow us to use any measurement of separation which is convenient for the purpose.

Any theory or analysis which is true and accurate for one measure of angular separation is likewise true and accurate for all of them.  Arc seconds, pixels, and linear distance can be considered interchangeable for the purpose of analysis.

angular height

angular pixels

F = telescope effective focal length, typically millimeters
θ = angular separation, typically arc seconds or arc minutes
h = linear distance, typically micrometers (microns)
hp = pixel size, typically micrometers (microns)
d = pixel distance, as a fractional number of pixels

Some care must be taken to keep the decimal point in the right place, for example, when calculating linear distance h in micrometers while using focal length F in millimeters.

Analysis of CCD images usually begins with fractional pixel distances because pixel distances are inherent in the image itself.  For example, analysis of focus quality can be conveniently performed using fractional pixel distances and are usually most meaningful during image processing.

On the other hand, image resolving power and astronomical seeing are often measured as arc seconds because arc seconds are a universal measure and do not depend upon individual equipment for focal length and pixel sizes.

Note, CCD cameras may have differing vertical and horizontal pixel sizes, for example, non-square pixels.  The equations above can be easily adjusted for these pixel size differences if precise theoretical accuracy is required in that case.

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