Image Scale - Angular Field of View

To calculate the angular field of view of your scope or camera lens, you need to know the focal length of the observing instrument and the size of the sensor at the focal plane.

With this information, you can calculate the angular field of view of your telescope in arcseconds, or arcminutes, or degrees. You can also calculate your image scale on a pixel basis in arcseconds per pixel, an important calculation for high-resolution planetary work when trying to match the sample size to the Airy Disk size and the seeing.

Formula 8 Image Scale in arcminutes per millimeter

S = (3437.8 x D) / FL

Where:

S = Angular field of view in arcminutes
3437.8 = arcminutes in one radian
D = Dimension of sensor (mm)
FL = the Focal Length of the scope in millimeters

Example: What is the angular field of view in arcminutes of a 22.5 x 15mm sensor on a 5 inch f/8 scope with 1016mm of focal length?

For the 22.5mm side:

S = (3437.8 * 22.5) / 1016
S= 77,350.5 / 1016
S = 76.13 arcminutes

For the 15mm side:

S = (3437.8 * 15) / 1016
S= 51,576 / 1016
S = 50.75 arcminutes

The angular field of view in arcminutes of a 22.5 x 15mm sensor on a 5 inch f/8 scope is 76.13 x 50.75 arcminutes.

To change arcminutes to degrees, simply divide by 60:

75.9 arc min / 60 = 1.265 degrees
50.6 arc min / 60 = 0.843 degrees

Note that for shorter focal lengths (less than 50mm) the formula above is not accurate and the following should be used:

S = 2 x arc tan ( 0.5 x D / FL) , where S is degrees or radians depending on what mode the calculator/program is set to.

Formula 9 Image scale in arcseconds per millimeter

S = (205740 * D) / FL

Where:

S = the Image Scale
206265 = arcseconds in one radian
D = Dimension of sensor (mm)
FL = the Focal Length of the scope in millimeters

Example: What is the angular field of view in arcseconds of a 22.5 x 15mm sensor on a 5 inch f/8 scope with 1016mm of focal length?

For the 22.5mm side:

S = (205740 * D) / FL
S = (206265 * 22.5) / 1016
S = 4,640,962.5 / 1016
S = 4567.8 arcseconds

For the 15mm side:

S = (205740 * D) / FL
S = (206265 * 15) / 1016
S = 3,093,975 / 1016
S = 3045.2 arcseconds

The angular field of view in arcseconds of a 22.5 x 15mm sensor on a 5 inch f/8 scope is 4567.8 x 3045.2 arcseconds.

To convert this to arcminutes, simply divide by 60: 4567.8 arcseconds divided by 60 = 76.13 arcminutes, and 3045.2 / 60 = 50.75 arcminutes.

So the angular field of view in arcminutes of a 22.5 x 15mm sensor on a 5 inch f/8 scope is 76.13 arc min x 50.75 arc min.

Formula 10 Image scale in arcseconds per micron

S = (206265/FL) / 1000

Where:

S = the Image Scale
206265 = arcseconds in one radian
FL = the Focal Length of the scope in millimeters
1000 = number of microns in a millimeter

Example: What is the angular field of view per micron for a 5 inch f/8 scope with 1016mm of focal length:

S = (206265/FL)/1000
S = (206265/1016)/1000
S = 203/1000
S = 0.203 arcseconds per micron

So, the angular field of view for a 5 inch f/8 scope is 0.2 arcseconds per micron.

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