Editing Optical telescope (section)

==== Minimum ====
There are two issues constraining the lowest useful [[magnification]] on a telescope:
* The light beam exiting the eyepiece needs to be small enough to enter the pupil of the observer's eye. If the cylinder of light emerging from they eyepiece is too wide to enter the observer's eye, some of the light gathered by the telescope will be wasted, and the image seen will be dimmer and less clear than it would be at a higher magnification.
* For telescope designs with obstructions in the light path (e.g. most [[catadioptric system|catadioptric telescopes]], but ''not'' spyglass-style [[refracting telescope]]s) the magnification must be high enough to keep the central obstruction out of focus, to prevent it from coming into view as a central "black spot". Both of these issues depend on the size of the pupil of the observer's eye, which will be narrower in daylight and wider in the dark.

Both constraints boil down to approximately the same rule: The magnification of the viewed image, <math>\ M\ ,</math> must be high enough to make the eyepiece exit pupil, <math>\ d_\mathsf{ep}\ ,</math> no larger than the pupil of the observer's own eye.<ref name=RASC-OH-2023/> The formula for the eypiece exit pupil is
:<math>\ d_\mathsf{ep} = \frac{\ D\ }{\ M\ } \ </math>
where <math>\ D\ </math> is the light-collecting diameter of the telescope's aperture.<ref name=RASC-OH-2023/>

Dark-adapted pupil sizes range from 8–9&nbsp;mm for young children, to a "normal" or standard value of 7&nbsp;mm for most adults aged&nbsp;30–40, to 5–6&nbsp;mm for retirees in their 60s and 70s. A lifetime spent exposed to chronically bright ambient light, such as sunlight reflected off of open fields of snow, or white-sand beaches, or cement, will tend to make individuals' pupils permanently smaller. Sunglasses greatly help, but once shrunk by long-time over-exposure to bright light, even the use of opthamalogic drugs cannot restore lost pupil size.<ref name=RASC-OH-2023>{{cite book |editor1-first = James S. |editor1-last = Edgar |display-editors = etal |year = 2023 |title = Observers' Handbook |publisher = Royal Canadian Astronomical Society |type = annual |edition = USA |isbn = 978-1-92-787930-6 |publication-date = October 2021 |url = https://secure.rasc.ca/store/product/observer-s-handbook-2023 |access-date = 2024-05-10 }}</ref> Most observers' eyes instantly respond to darkness by widening the pupil to almost its maximum, although complete adaption to [[night vision]] generally takes at least a half-hour. (There is usually a slight extra widening of the pupil the longer the pupil remains dilated / relaxed.)
 
The improvement in brightness with reduced magnification has a limit related to something called the [[exit pupil]]. The [[exit pupil]] is the cylinder of light exiting the eyepiece and entering the pupil of the eye; hence the lower the [[magnification]], the larger the [[exit pupil]]. It is the image of the shrunken sky-viewing aperture of the telescope, reduced by the magnification factor, <math>\ M\ ,</math> of the eyepiece-telescope combination:
:<math>\ M = \frac{\ L\ }{ \ell }\ ,</math>
where <math>\ L\ </math> is the [[focal length]] of the telescope and <math>\ \ell\ </math> is the focal length of the eyepiece.

Ideally, the exit pupil of the eyepiece, <math>\ d_\mathsf{ep}\ ,</math> matches the pupil of the observer's eye: If the exit pupil from the eyepiece is larger than the pupil of individual observer's eye, some of the light delivered from the telescope will be cut off. If the eyepiece exit pupil is the same or smaller than the pupil of the observer's eye, then all of the light collected by the telescope aperture will enter the eye, with lower magnification producing a brighter image, as long as all of the captured light gets into the eye.

The minimum <math>\ M_\mathsf{min}\ </math> can be calculated by dividing the telescope aperture <math>\ D\ </math> over the largest tolerated exit pupil diameter <math>\ d_\mathsf{ep} ~.</math><ref name=RocketMime>{{cite web |title=Telescope equations |date=17 November 2012 |department = Astronomy |website=Rocket Mime |url=http://www.rocketmime.com/astronomy/Telescope/telescope_eqn.html}}</ref><ref name=RASC-OH-2023/>
:<math>\ M_\mathsf{min} = \frac{\ D\ }{\ d_\mathsf{ep} } \ </math>
Decreasing the magnification past this limit will not increase brightness nor improve clarity: Beyond this limit there is no benefit from lower magnification. Likewise calculating the [[exit pupil]] <math>\ d_\mathsf{ep}\ </math> is a division of the aperture diameter <math>\ D\ </math> and the visual magnification <math>\ M\ </math> used. The minimum often may not be reachable with some telescopes, a telescope with a very long focal length may require a longer focal length eyepiece than is available.

An example of the lowest usable magnification using a fairly common 10″ (254&nbsp;mm) aperture and the standard adult 7&nbsp;mm maximum [[exit pupil]] is given by: <math>\ M_\mathsf{min} = \frac{ D }{\ d_\mathsf{ep} } = \frac{\ 254\ }{ 7 } \approx 36\!\times ~.</math> If the telescope happened to have a {{gaps|1|200|mm}} focal length (<math>\ L\ </math>), the longest recommended eyepiece focal length (<math>\ \ell\ </math>) would be <math>\ \ell = \frac{\ L\ }{ M } \approx \frac{\ 1\ 200\mathsf{\ mm\ } }{ 36 } \approx 33\mathsf{\ mm} ~.</math> An eyepiece of the same apparent field-of-view but longer focal-length will deliver a wider true field of view, but dimmer image. If the telescope has a central obstruction (e.g. a [[Newtonian telescope|Newtonian]], [[Maksutov telescope|Maksutov]], or [[Schmidt–Cassegrain telescope]]) it is also likely that the low magnification will make the obstruction come into focus enough to make a black spot in the middle of the image.

Calculating in the other direction, the [[exit pupil]] diameter of a 254&nbsp;mm telescope aperture at 60× [[magnification]] is given by: <math>\ d_\mathsf{ep} = \frac{\ D\ }{ M } = \frac{\ 254\ }{ 60 } \approx 4.2\mathsf{\ mm\ } ,</math> well within pupil size of dark-adapted eyes of observers of almost all ages. Assuming the same telescope focal length as above, the eyepiece focal length that would produce a 60× magnification is  <math>\ \ell = \frac{\ L\ }{ M } = \frac{\ 1\ 200\mathsf{\ mm\ } }{ 60 } \approx 20\mathsf{\ mm} ~.</math>