Useful Formulas for Amateur Astronomers

Mike Swanson

Professional astronomy is heavily laden in mathematical simulations and complex formulas.  While the amateur astronomer can simply grab some gear or just use their eyes to enjoy the night sky, there are several formulas that become useful as your experience and equipment list grows.  And, you don't need a college mathematics background to make use of them.  Please note that many of these formulas are used in the Scope Calculator spreadsheet available on the Downloads page of our web site.

Magnification of a Telescope
The most commonly used formula in amateur astronomy is used to calculate the magnification of a telescope:

magnification = focal length of telescope / focal length of eyepiece

Example: using a 10mm eyepiece in a telescope with a focal length of 1000mm results in a magnification of 100x (1000 / 10 = 100).

Maximum Magnification of a Telescope
Since we can simply use different eyepieces to reach different magnification, the temptation is to "pump-up" the power as high as possible.  In theory and practice, a telescope with excellent optics is limited to a magnification of about 2 times the aperture (diameter of main object) measured in millimeters.  Example: an 80mm refractor is limited to a maximum magnification of about 160x (80 x 2 = 160).  Multiply inches by 25.4 to convert to millimeters.

Focal Ratio of a Telescope
The focal ratio of a telescope is mostly used when considering exposure time for astrophotography, but it is also a general characteristic of the telescope that can be useful in other discussions.

focal ratio = focal length of telescope / aperture of telescope

The result is written as f/focal ratio.  Example: an 80mm telescope with an 800mm focal length has a focal ratio of f/10 (800 / 80 = 10  - note that both measurements must use the same unit, in this case mm).

Exit Pupil
The exit pupil of an instrument is the cylinder of light leaving the eyepiece.  If the exit pupil is larger than the diameter of the fully opened (dark-adapted) pupil of your eye, some of the light will be wasted.  Younger eyes typically have a maximum pupil of about 7mm; older eyes may be limited to 5 or 6mm.  Various focal lengths and magnifications result in differing exit pupils.

exit pupil for binoculars = aperture of binocular in mm / magnification of binocular

Example: 10x50 binoculars have an aperture of 50 (the second number) and a magnification of 10 (the first number).  The exit pupil of these binoculars would be 5mm (50 / 10 = 5).

exit pupil for telescope = focal length of eyepiece / focal ratio of telescope

Actually just a mathematical rearrangement of the formula given for a pair of binoculars, but this formula turns out to be much easier to work with.  Example: using a 25mm eyepiece in a telescope with a focal ratio of f/10 results in an exit pupil of 2.5mm (25 / 10 = 2.5).

True Field of View
The true field of view (TFOV) of an instrument is a measurement of the actual field of view seen through the eyepiece.  For example, the field of view might show about 1 degree of the sky at a time.  A wider field of view is desirable for extended objects such as large nebula and open clusters.  Calculation of the TFOV requires the apparent field of view (AFOV) of the eyepiece in use.  This is a statistic available from the eyepiece manufacturer, but it is useful to note that most Plossls have an AFOV of about 50 degrees.

TFOV = AFOV of eyepiece / magnification given by eyepiece

Example: a 10mm Plossl with an AFOV of 50 degrees is used in a telescope of 1000mm focal length.  The magnification given by this eyepiece is 100x (1000 / 10) so the TFOV is a half degree (50 / 100 = 0.5).

Resolving Limit
The resolving limit of an instrument is an expression of the smallest detail that can be detected by the instrument.  The unit of measure is arcseconds (1/3600th of a degree) and a common test is detecting separation in the components in a very close double star.  There are two commonly used calculations:

Rayleigh Limit = 5.5 / aperture of telescope in inches

Dawes Limit = 4.56 / aperture of telescope in inches

Example: the Rayleigh Limit for a telescope with a 6 inch aperture is approximately 0.9 arcseconds (5.5 / 6 = 0.92).  To convert an aperture given in millimeters (mm) to inches, simply divide the millimeters by 25.4; for example, an 80mm aperture is 3.15 inches.

Light Gathering Power
This is not really an absolute measurement but rather just a method of comparing two optical instruments.  The larger the light gathering power, the fainter the objects that can be detected (also expressed by the limiting magnitude formula below).

ratio of light gathering power = square of aperture of larger instrument / square of aperture of smaller instrument

Example: an 8 inch telescope gathers 4 times more light than a 4 inch telescope (64 / 16 = 4  - note that both measurements must use the same unit, in this case inches).  Another example: an 80mm scope gathers about 130 times more light than the naked eye (the maximum aperture of the naked eye is about 7mm so 6400 / 49 = 130.6).

Limiting Magnitude of an Optical Instrument
This one is a little complicated.  Limiting magnitude is the magnitude of the faintest object that an average person with fully dark-adapted eyes will be able to detect.

limiting magnitude = 5 x LOG10(aperture of scope in cm)  + 7.5

LOG10 is "log base 10" or the common logarithm.  This formula would require a calculator or spreadsheet program to complete.  Example: considering an 80mm telescope (8cm) - LOG(8) is about 1.9, so limiting magnitude of an 80mm telescope is 12 (5 x 1.9 + 7.5 = 12).  Be certain you multiply 5 times the LOG value before you add 7.5).

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