Telescope Limiting Magnitude Calculator

Background.

This calculator estimates the faintest star an aperture will show a visual observer, and it is deliberately honest about how rough that estimate is. The model is aperture gain: your naked-eye limiting magnitude, plus five times the base-ten logarithm of the effective aperture divided by your own dark-adapted pupil, minus the light your optics lose. The aperture-gain part is exact — it follows from the collecting-area ratio and the definition of the magnitude scale — and everything uncertain is exposed as an input you control rather than baked in as a constant.

What this page does not do matters as much as what it does. Schaefer's 1990 paper in the Publications of the Astronomical Society of the Pacific, which remains the reference work on the subject, describes predicting a visual limiting magnitude as a difficult problem in physiology and shows that a defensible algorithm needs the transmission of the atmosphere and of the telescope, the brightness of the sky, the colour of the star, the age of the observer, the aperture and the magnification — noting that most published formulas do not even consider the magnification. This model includes none of the magnification, colour or extinction physics. It is an aperture-gain calculation with three observer-dependent inputs, and it is offered as an order-of-magnitude guide, not as a prediction of what you personally will see. Two experienced observers at the same eyepiece on the same night routinely differ by a full magnitude.

The page also shows the rule of thumb you will find everywhere else, 2.7 plus five times the log of the aperture in millimetres, and the difference between it and this model. That rule has no primary source, and working backwards from it reveals what it quietly assumes: a naked-eye limit of 6.9 magnitudes and optics that lose nothing. At this page's defaults the two answers differ by 0.93 magnitudes. Neither is chosen for you; both are shown, with the gap named.

One detail worth knowing: a central obstruction removes collecting area, not diameter. A Newtonian with a secondary 25 percent of the primary's diameter loses only about three percent of its light, because the obstruction's area is the square of that fraction. The calculator reports the unobstructed-equivalent aperture so you can see how small that penalty really is.

What is telescope limiting magnitude calculator?

The limiting magnitude of a telescope is the faintest star a visual observer can detect through it. Magnitudes are a logarithmic and inverted scale: a larger number means a fainter star, and a difference of exactly five magnitudes is a factor of exactly one hundred in brightness, a convention set by Pogson in 1856.

A telescope helps by collecting more light than the eye's pupil does. The gain in flux is the ratio of collecting areas, which is the square of the ratio of diameters, and converting that flux ratio to magnitudes gives five times the log of the diameter ratio. A 200 mm aperture against a 7 mm pupil collects 816 times as much light, which is 7.28 magnitudes.

The naked-eye limiting magnitude, or NELM, is the anchor the whole calculation is built on. It encodes your sky brightness and your own eyesight in a single number, and it is the input with the largest influence on the answer.

How to use this calculator.

1. 01Enter the aperture in millimetres.
2. 02Enter the central obstruction as a percentage of the diameter — 0 for a refractor, around 25 % for a Newtonian.
3. 03Enter your own naked-eye limiting magnitude. This is the number that carries the sky, and guessing it badly will dominate the error.
4. 04Adjust the pupil diameter for your age: about 7 mm at 20, 5 mm at 60.
5. 05Adjust the transmission loss if you know your optics, or leave it at the 0.5-magnitude default.
6. 06Read the answer as an order-of-magnitude guide, and read the note for what the model leaves out.

The formula.

The telescope's advantage over the eye is purely one of collecting area. Tele Vue publishes it as aperture gain equals the square of the objective diameter divided by the eye pupil diameter, which is a statement about area with no magnitudes in it at all. Pogson's 1856 definition of the magnitude scale supplies the conversion: a factor of one hundred in flux is exactly five magnitudes, so a flux ratio F corresponds to 2.5 log₁₀ F magnitudes. Composing the two independent statements gives 2.5 log₁₀((D/d)²) = 5 log₁₀(D/d), which is what this page computes. The test suite asserts the composition holds to one part in a billion across apertures from 10 mm to a metre, and checks the clean case: a diameter ratio of ten must give exactly a hundredfold gain and exactly five magnitudes.

Work through the defaults. A 200 mm unobstructed aperture against a 7 mm pupil gives a diameter ratio of 28.5714286, so the light gain is 816.3265306 times and the magnitude gain is 5 log₁₀(28.5714286) = 7.2796600. Adding a naked-eye limit of 6.5 and subtracting half a magnitude of optical loss gives 13.2796600. The classic rule gives 2.7 + 5 log₁₀(200) = 14.2051500, which is 0.93 magnitudes more optimistic.

THE OBSTRUCTION REMOVES AREA. A 25 percent central obstruction gives an effective aperture of 200 × √(1 − 0.25²) = 193.6491673 mm, not 150 mm, and costs only 0.07 magnitudes. A test explicitly asserts the result is not close to the naive 150 mm figure, because treating the obstruction as a diameter loss is the most common error on this topic.

WHY THE CLASSIC RULE DISAGREES. Working backwards from 2.7 + 5 log₁₀(D) with a 7 mm pupil: 5 log₁₀(D/7) = 5 log₁₀ D − 4.2255, so the rule implies a naked-eye limiting magnitude of 6.93 and zero optical loss. That is a better-than-average eye at an excellent site with a perfect telescope. The page reports both numbers and their difference rather than choosing.

WHAT IS NOT MODELLED. No magnification term, no atmospheric extinction — so the figure is for the zenith, and an object low in the sky will be fainter — no colour term, no observer-age term beyond the pupil, and nothing at all about extended objects, which follow surface brightness rather than total flux. Schaefer showed all of these matter.

SIGN AND SCALE. Magnitudes run backwards and logarithmically. More aperture gives a larger limiting magnitude; doubling the aperture adds exactly 5 log₁₀2 = 1.5051 magnitudes; a magnitude of transmission loss subtracts exactly one magnitude. If the aperture is smaller than your pupil the gain goes negative, which is arithmetically right and physically means the instrument is throwing light away — the note says so rather than hiding it.

INVALID DOMAIN. Aperture and pupil must be positive; the obstruction must be between 0 and 90 percent; the naked-eye limit between −5 and 10 magnitudes; the loss between 0 and 5 magnitudes. Each raises an error against its own field.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Schaefer, B.E. (1990). 'Telescopic Limiting Magnitudes.' Publications of the Astronomical Society of the Pacific 102:212, DOI 10.1086/132629. Abstract verified by retrieval 2026-07-29: predicting the faintest star visible through a telescope is 'a difficult problem in physiology'; 'many prediction formulas have been advanced over the years, but most do not even consider the magnification used'; a defensible algorithm accounts for the transmission of the atmosphere and the telescope, the brightness of the sky, the colour of the star, the age of the observer, the aperture and the magnification, tested against 314 observed values. This is the authority for what this page does NOT model. Peer-reviewed, open abstract.
2. [2]Tele Vue Optics, Inc., 'Telescope Formulas'. Manufacturer engineering note, verified by retrieval 2026-07-29 to state: Aperture Gain = '(Objective diameter / Eye pupil diameter)²'. Combined with Pogson's magnitude scale this gives the 5·log₁₀(D/d) gain used here — two independent statements composed, which is this page's second-authority check. Open access.
3. [3]Pogson, N.R. (1856). 'Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857.' Monthly Notices of the Royal Astronomical Society 17:12. Definition of the modern magnitude scale in which five magnitudes is exactly a factor of one hundred in flux. Print / bibliographic reference.
4. [4]Winn, B., Whitaker, D., Elliott, D.B. & Phillips, N.J. (1994). 'Factors affecting light-adapted pupil size in normal human subjects.' Investigative Ophthalmology & Visual Science 35(3):1132–1137. Age dependence of pupil diameter, the reason the pupil is an editable input here rather than a fixed 7 mm. Abstract open access; full text paywalled. Retrieved 2026-07-29.