Vol. Science · Astronomy · Optics
What an aperture, a focal length, and an eyepiece actually decide. The focal ratio is focal length divided by aperture. Magnification is the telescope’s focal length divided by the eyepiece’s, and a Barlow multiplies that effective focal length. The exit pupil is the aperture divided by the magnification. True field is about the apparent field divided by the magnification. Light grasp is the area ratio against a 7 mm comparison pupil. Rayleigh’s criterion is 1.22 λ/D. Dawes’s limit is the empirical 116/D millimetre form, shown beside 4.56/D in inches. How faint a star looks is the magnitude desk. How wide something looks is angular size. What the horizon puts overhead is the local sky. Why the naked-eye limit moves is light pollution.
Simplified optics for a classroom. Not a buying guide. A real view depends on seeing, collimation, optical quality, coatings, a central obstruction this page ignores, the observer, and the sky. Rayleigh’s 1.22 λ/D is a diffraction criterion for two equal point sources and a filled circular aperture. Dawes’s limit is an empirical visual rule for close, similar doubles, not a guarantee. The limiting-magnitude lines are approximations. Hubble and JWST are not eyepiece telescopes.
Two equal Airy disks · separation wider than Rayleigh looks split
Exit pupil beside a 7 mm comparison pupil
Instruments
Naked-eye limit of the sky
Separation in the drawing
Enter an aperture greater than zero.
Enter a focal length greater than zero.
Enter an eyepiece focal length greater than zero.
Enter an apparent field greater than zero.
Enter a Barlow factor greater than zero.
Enter a wavelength greater than zero.
Enter a naked-eye limiting magnitude.
Enter a separation greater than zero.
Read this before the numbers
The arithmetic above is idealised. It assumes a filled circular aperture, perfect optics, no central obstruction, no coating losses, and two equal point sources. A Newtonian or a Cassegrain has a secondary in the beam. Seeing, collimation, and the observer move the real limit by more than the difference between Rayleigh and Dawes. The 50×-per-inch line and the 30×-per-inch line are rules of thumb. Tele Vue prints the 50×-per-inch claim as a myth in the sense that it is not a hard ceiling: a small refractor can look pleasant at higher power and still show no extra detail, and a large aperture often stops sooner because of the air.
The area limiting magnitude only counts photons the aperture collects, then adds that gain to the naked-eye limit you typed. It does not get fainter as you magnify, once the exit pupil fits in the eye. Magnification can still help by darkening the sky background. Bowen’s 1947 formula is the published line that includes magnification. Its constant is 6.5, from his dark-sky observations, not the sky you typed. Schaefer’s 1990 paper is why neither line is a prediction: sky brightness, magnification, throughput, age, and the observer all move the limit, and his own test set scattered by about three quarters of a magnitude. Real limits vary with seeing, observer, and optics.
Words
Classroom picture of an ideal telescope. The 200 mm, 1,200 mm, 25 mm, 50° example is f/6, 48×, exit pupil 4.17 mm, true field 1.042° (62.5′), light grasp 816× against a 7 mm pupil, Rayleigh 0.692″ at 550 nm, Dawes 0.580″ by 116/D and 0.579″ by 4.56/D. The area estimate at naked-eye magnitude 6.0 is 13.28. Bowen’s formula gives 12.94, and the exit pupil is inside the range he stated. Those figures are computed from the formulas on this page. The 7×50, 80 mm, and 200 mm sets are round classroom numbers, not catalog models. Hubble and JWST use the cited apertures and focal lengths and have no eyepiece. Not a buying guide. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The angle
δ = 2·arctan(d / 2D). A telescope’s true field is an angle of that same kind. This desk only estimates the field from the eyepiece.
The scale
Five magnitudes is a factor of 100. The limiting-magnitude lines here are that scale applied to an aperture, with the caveats above.
Viewpoint
What the horizon puts overhead. This desk does not point the telescope.
Night
Why the naked-eye limit you type is brighter in town than the ideal 6.0 to 6.5.
The wavelength
Where a thermal spectrum peaks. Rayleigh’s angle moves with the wavelength you type.
Neighborhood
The sky the aperture is aimed at. Still a schematic.
A classroom calculator. Ideal optics, a filled circular aperture, no obstruction, no atmosphere. Not a buying guide and not a prediction of what you will see.
| Aperture | Clear aperture. Millimetres or inches. An inch is exactly 25.4 mm. The slider runs from 20 mm to 8 m. The number field accepts a wider span, including Hubble and JWST. |
|---|---|
| Focal length | The telescope’s focal length, same unit choice. Focal ratio is this divided by the aperture. A Barlow does not change the number you type. It changes the effective focal length used for magnification. |
| No eyepiece | Turns off magnification, exit pupil, true field, the magnification rules of thumb, and both visual limiting-magnitude lines. Hubble and JWST select this. Resolution and light grasp remain. |
| Eyepiece | Focal length in millimetres. Magnification is telescope focal length times the Barlow factor, divided by this. |
| Apparent field | The eyepiece’s apparent field, in degrees. True field is this divided by magnification. That is the rough approximation. |
| Barlow | A factor of 2 doubles the effective focal length, the magnification, and the effective focal ratio, and it halves the exit pupil and the true field. 1 means no Barlow. A factor below 1 is a reducer, same arithmetic. |
| Wavelength | Used only by Rayleigh’s criterion and the Airy drawing. Default 550 nm, near the CIE photopic peak at 555 nm. Night vision peaks nearer 507 nm. Dawes’s limit does not use this field. |
| Naked-eye limit | The faintest star you can see without the telescope, typed by you. Ideal-sky buttons are NASA’s 6.0 and 6.5. Inner city is Sky & Telescope’s Bortle class 9 figure, 4.0 or less, entered as 4.0. The light-pollution desk is the essay. This number feeds only the area limiting-magnitude line. |
| Separation | Angular separation of the two equal points in the drawing. Albireo sets 34.70″ from the Washington Double Star Catalog row cited on the page. “At Rayleigh” and “At Dawes” set the separation to the limit just computed for the current aperture. |
| 7×50, 80 mm, 200 mm | Round classroom numbers, not products. 50 mm and 175 mm with a 25 mm eyepiece is 7×. 80 mm and 560 mm with a 20 mm eyepiece is f/7 and 28×. 200 mm and 1,200 mm with a 25 mm eyepiece is f/6 and 48×. The 50° apparent field is Hawks’s round Plössl example, typed in, not a measurement. |
|---|---|
| Hubble | 2.4 m and f/24 from the STScI primer, so 57.6 m. Wavelength 500 nm. No eyepiece. NASA’s 189 feet (56.7 m), the 40,000× light comparison, and the 0.043″ FWHM are quoted beside the computed lines. They are not pasted into the formulas. |
| JWST | Fact-sheet 6.5 m and 131.4 m. Wavelength 600 nm, the short edge of 0.6–28.5 μm. An extra line uses the fact-sheet clear aperture, 25 m². The mirror is segmented. Rayleigh here still assumes a filled circle of the fact-sheet diameter. |
| Sliders | Arrow keys nudge the focused slider. The number field is where a cited figure is typed exactly. A fact-sheet value can sit off the slider step. The field keeps it. |
|---|---|
| Presets | Enter or Space selects. Left and Right step through that group. The pressed button is aria-pressed. |
| Touch | Buttons, the question mark, Close, the fields, and the unit menus are at least 44 pixels on a side. |
| This dialog | The question mark opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it. |