There are a few astronomy formulas that all of us as backyard stargazers need to use from time to time. To save you having to remember them, we’ve compiled this alphabetical list of astronomy equations for you to consult at any time.
Exit Pupil Diameter
Exit Pupil Diameter (mm) = Diameter of Objective (mm) / Magnification
Example: If your telescope is 4″ (100mm) in diameter and you have a magnification on 80x, the size of the exit pupil is 100 / 80 = 1.25mm.
The exit pupil of a telescope’s eyepiece is the diameter of the cylinder of light which hits your eye. This needs to be no larger than your pupil to see it all (around 7mm) and not so small that it appears as a tiny dot of a picture in the centre of your vision.
Increasing magnification results in a decrease in both surface brightness and exit pupil.
Field of View
True Field of View = Apparent Field of View / Magnification
Example: If your eyepiece has an apparent field of view of 50° and gives a magnification of 40x (see formula for magnification, below) then your true field of view is 50 / 40 = 1.25°.
Changing the eyepiece affects how much of the sky can been seen. Field of view is the portion of sky you can see with your telescope. In the example, your view through the eyepiece is a circle of sky 1.25° across.
More specifically, telescope eyepieces are manufactured with a set apparent field of view (an angular measurement describing how many angular degrees of sky can been seen with the eyepiece when it’s not attached to a telescope)
Focal Ratio = Focal Length of Telescope (mm) / Diameter of Objective (mm)
Example: If your telescope has a focal length of 700mm and the diameter of your main mirror or lens is 4″ (100mm), then your focal ratio is 700mm/100mm = 7.
Optical Tube Assemblies (OTA) are manufactured with a fixed focal ratio. This ratio is expressed as an “f/number”. In the example above, this would be written as f/7.
Telescopes with focal ratios from f/4 to f/5 are considered “fast” and are best for low power wide field observing and astrophotography. Those with focal ratios from f/11 to f/15 are considered “slow” and better for higher power lunar, planetary, and binary star observing and high power photography. Medium f/6 to f/10 focal ratios work well with either.
Magnification = Telescope Focal Length / Eyepiece Focal Length
Example: If your telescope has a focal length of 500mm and your eyepiece has a focal length of 10mm, the magnification that combination will give you is 500mm / 10mm = 50x
This is the most commonly referred to formula for astronomers because every time we change an eyepiece, we change our magnification.
Optical tube assembly (OTA) and eyepieces are manufactured with fixed focal lengths. The focal lengths of your telescope and eyepiece determine the magnification you will obtain by using the telescope and eyepiece together.
The simple rule is to increase your magnification, decrease the focal length of your eyepiece. You can read more about how eyepieces work here.
Theoretical Maximum Magnification
Maximum Magnification (theoretical) = 50 * Diameter of Objective (inches)
Example: If your telescope has a diameter of 4 inches, then its theoretical maximum magnification is 50 x 4 = 200x
This is only theoretical though because Earth’s atmosphere will limit the useful maximum magnification of any size scope to around 200x, maybe up to 300x with a high quality large scope on a perfect night.
Above that, you’ll be magnifying atmospheric disturbance so much that your view will be ruined.
Minimum Magnification = Diameter of Objective (mm) / 7
Example: If your telescope has a diameter of 4 inches (100mm), then its minimum (useful) magnification is 100 / 7 = 14.3x
While decreasing the magnification increases both field of view and image brightness, there is a limit at which doing so is no longer helpful.
When the telescope’s exit pupil diameter exceeds that of the observer’s pupils, increasing magnification is no longer helpful. This is because your eyes can’t physically see the whole image your scope is presenting.
This formula assume the average human observer will have a dark-adapted pupil diameter of 7mm. If you know yours is smaller, adjust the formula accordingly.
Resolving Power = 120 / Diameter of Objective (mm)
Example: If your telescope’s main mirror or lens has a diameter of 4″ (100mm), then your resolving power is 120mm/100mm = 1.2.
This is a measure of the smallest separation between two stars that a telescope can resolve. The answer you get for this formula is in arcseconds. A 4″ telescope can resolve, as per the example, can theoretically show two stars 1.2″ apart as individual stars. Closer together than that and they will appear as a single body.
Stellar Magnitude Limit
Stellar magnitude Limit = 2 + (5 * Log (Diameter of Objective in mm))
Example: This is the trickiest formula in the list and will require a calculator. Start with your objective diameter in mm. Assume a 4″ scope has a 100mm objective. Enter 100 into a calculator and press the [log] button. Next, multiply the answer by 5 and, finally, add 2. A 100mm lens has a stellar magnitude limit of 12.
The stellar magnitude limit is the faintest magnitude that your telescope can possibly see, regardless of how much magnification is used. It is a measure based on light-gathering power.
Your eyes, maxing out at a pupil diameter of 7mm and can see stars only to around magnitude 6.2. A telescope’s objective has a greater surface area than our pupils and as result a greater light grasp. The increase in light grasp corresponds to an increase in stellar limiting magnitude.
If the object you are trying to see (eg a comet, asteroid) is less bright than your telescope’s star magnitude limit, you won’t be able to see it regardless of the magnification.
If you are a little daunted by the formula, this link will do the math work for you.
If you have any other formulas that you’d either like explaining, or you think will be helpful for your fellow astronomers, share them at Adam@LovetheNightSky.com and they could be on this page soon.