Editing Optical telescope (section)

==Characteristics==
[[Image:EightInchTelescope.JPG|thumb|Eight-inch refracting telescope at [[Chabot Space and Science Center]]]]
Design specifications relate to the characteristics of the telescope and how it performs optically. Several properties of the specifications may change with the equipment or accessories used with the telescope; such as [[Barlow lens]]es, [[star diagonal]]s and [[eyepiece]]s. These interchangeable accessories do not alter the specifications of the telescope, however they alter the way the telescope's properties function, typically [[magnification]], apparent [[field of view]] (FOV) and actual field of view.

===Surface resolvability===
The smallest resolvable surface area of an object, as seen through an optical telescope, is the limited physical area that can be resolved. It is analogous to [[angular resolution]], but differs in definition: instead of separation ability between point-light sources it refers to the physical area that can be resolved. A familiar way to express the characteristic is the resolvable ability of features such as [[Moon]] craters or [[Sun]] spots. Expression using the formula is given by twice the resolving power <math>R</math> over aperture diameter <math>D</math> multiplied by the objects diameter <math>D_{ob}</math> multiplied by the constant <math>\Phi</math> all divided by the objects [[apparent diameter]] <math>D_{a}</math>.<ref name="SaharaSkyObservatory">{{cite web|url=http://www.saharasky.com/saharasky/formula.html|title=Telescope Formulae|date=3 July 2012|publisher=SaharaSky Observatory}}</ref><ref name="RyukyuAstronomyClub">{{cite web|url=http://www.nexstarsite.com/_RAC/form.html|title=Optical Formulae|date=2 January 2012|publisher=Ryukyu Astronomy Club}}</ref>

''Resolving power <math>R</math> is derived from the [[wavelength]] <math>{\lambda}</math> using the same unit as aperture; where 550 [[nanometers|nm]] to mm is given by: <math>R = \frac{\lambda}{10^6} = \frac{550}{10^6} = 0.00055</math>.''
<br />''The constant <math>\Phi</math> is derived from [[radians]] to the same unit as the object's [[apparent diameter]]; where the Moon's [[apparent diameter]] of <math>D_{a} = \frac{313\Pi}{10800}</math> [[radians]] to [[arcseconds|arcsecs]] is given by: <math>D_{a} = \frac{313\Pi}{10800} \cdot 206265 = 1878</math>.''

An example using a telescope with an aperture of 130&nbsp;mm observing the Moon in a 550 [[nanometer|nm]] [[wavelength]], is given by: <math>F = \frac{\frac{2R}{D} \cdot D_{ob} \cdot \Phi}{D_{a}} = \frac{\frac{2 \cdot 0.00055}{130} \cdot 3474.2 \cdot 206265}{1878} \approx 3.22</math>

The unit used in the object diameter results in the smallest resolvable features at that unit. In the above example they are approximated in kilometers resulting in the smallest resolvable Moon craters being 3.22&nbsp;km in diameter. The [[Hubble Space Telescope]] has a primary mirror aperture of 2400&nbsp;mm that provides a surface resolvability of Moon craters being 174.9 meters in diameter, or [[sunspots]] of 7365.2&nbsp;km in diameter.

===Angular resolution===
Ignoring blurring of the image by turbulence in the atmosphere ([[astronomical seeing|atmospheric seeing]]) and optical imperfections of the telescope, the [[angular resolution]] of an optical telescope is determined by the diameter of the [[primary mirror]] or lens gathering the light (also termed its "aperture").

The [[Angular resolution#The_Rayleigh_criterion|Rayleigh criterion]] for the resolution limit <math>\alpha_R</math> (in [[radian]]s) is given by
:<math>\sin(\alpha_R) = 1.22 \frac{\lambda}{D}</math>
where <math>\lambda</math> is the [[wavelength]] and <math>D</math> is the aperture. For [[visible light]] (<math>\lambda</math> = 550&nbsp;nm) in the [[small-angle approximation]], this equation can be rewritten:
:<math>\alpha_R = \frac{138}{D}</math>
Here, <math>\alpha_R</math> denotes the resolution limit in [[arcsecond]]s and <math>D</math> is in millimeters.
In the ideal case, the two components of a [[double star]] system can be discerned even if separated by slightly less than <math>\alpha_R</math>. This is taken into account by the [[Dawes limit]]
:<math>\alpha_D = \frac{116}{D}</math>
The equation shows that, all else being equal, the larger the aperture, the better the angular resolution. The resolution is not given by the maximum [[magnification]] (or "power") of a telescope. Telescopes marketed by giving high values of the maximum power often deliver poor images.

For large ground-based telescopes, the resolution is limited by [[astronomical seeing|atmospheric seeing]]. This limit can be overcome by placing the telescopes above the atmosphere, e.g., on the summits of high mountains, on balloons and high-flying airplanes, or [[space telescope|in space]]. Resolution limits can also be overcome by [[adaptive optics]], [[speckle imaging]] or [[lucky imaging]] for ground-based telescopes.

Recently, it has become practical to perform [[aperture synthesis]] with arrays of optical telescopes. Very high resolution images can be obtained with groups of widely spaced smaller telescopes, linked together by carefully controlled optical paths, but [[List of astronomical interferometers at visible and infrared wavelengths|these interferometers]] can only be used for imaging bright objects such as stars or measuring the bright cores of [[active galaxies]].

===Focal length and focal ratio===
The [[focal length]] of an [[optics|optical]] system is a measure of how strongly the system converges or diverges [[light]]. For an optical system in air, it is the distance over which initially [[collimated]] rays are brought to a focus. A system with a shorter focal length has greater [[optical power]] than one with a long focal length; that is, it bends the [[ray (optics)|ray]]s more strongly, bringing them to a focus in a shorter distance. In astronomy, the f-number is commonly referred to as the ''focal ratio'' notated as <math>N</math>. The [[f-number|focal ratio]] of a telescope is defined as the focal length <math>f</math> of an [[Objective (optics)|objective]] divided by its diameter <math>D</math> or by the diameter of an aperture stop in the system. The focal length controls the field of view of the instrument and the scale of the image that is presented at the focal plane to an [[eyepiece]], film plate, or [[Charge-coupled device|CCD]].

An example of a telescope with a focal length of 1200&nbsp;mm and aperture diameter of 254&nbsp;mm is given by:
<math>N = \frac {f}{D} = \frac {1200}{254} \approx 4.7</math>

Numerically large [[f-number|Focal ratios]] are said to be ''long'' or ''slow''. Small numbers are ''short'' or ''fast''. There are no sharp lines for determining when to use these terms, and an individual may consider their own standards of determination. Among contemporary astronomical telescopes, any telescope with a [[f-number|focal ratio]] slower (bigger number) than f/12 is generally considered slow, and any telescope with a focal ratio faster (smaller number) than f/6, is considered fast. Faster systems often have more [[optical aberrations]] away from the center of the field of view and are generally more demanding of eyepiece designs than slower ones. A fast system is often desired for practical purposes in [[astrophotography]] with the purpose of gathering more [[photons]] in a given time period than a slower system, allowing time lapsed [[photography]] to process the result faster.

Wide-field telescopes (such as [[astrograph]]s), are used to track [[satellite]]s and [[asteroid]]s, for [[cosmic ray|cosmic-ray]] research, and for [[astronomical survey]]s of the sky. It is more difficult to reduce [[optical aberrations]] in telescopes with low f-ratio than in telescopes with larger f-ratio.

===Light-gathering power===
{{further|Etendue}}
[[File:KeckObservatory20071013.jpg|thumb|The [[W. M. Keck Observatory|Keck II telescope]] gathers light by using 36 segmented hexagonal mirrors to create a 10 m (33 ft) aperture primary mirror]]
The light-gathering power of an optical telescope, also referred to as light grasp or aperture gain, is the ability of a telescope to collect a lot more light than the human eye. Its light-gathering power is probably its most important feature. The telescope acts as a ''light bucket'', collecting all of the photons that come down on it from a far away object, where a larger bucket catches more [[photons]] resulting in more received light in a given time period, effectively brightening the image. This is why the pupils of your eyes enlarge at night so that more light reaches the retinas. The gathering power <math>P</math> compared against a human eye is the squared result of the division of the aperture <math>D</math> over the observer's pupil diameter <math>D_{p}</math>,<ref name="SaharaSkyObservatory"/><ref name="RyukyuAstronomyClub"/> with an average adult having a [[pupil]] diameter of 7&nbsp;mm. Younger persons host larger diameters, typically said to be 9&nbsp;mm, as the diameter of the pupil decreases with age.

An example gathering power of an aperture with 254&nbsp;mm compared to an adult pupil diameter being 7&nbsp;mm is given by: <math>P = \left(\frac {D}{D_{p}}\right)^2 = \left(\frac {254}{7}\right)^2 \approx 1316.7</math>

Light-gathering power can be compared between telescopes by comparing the [[area]]s <math>A</math> of the two different apertures.

As an example, the light-gathering power of a 10-meter telescope is 25x that of a 2-meter telescope: <math>p = \frac {A_{1}}{A_{2}} = \frac {\pi5^2}{\pi1^2} = 25</math>

For a survey of a given area, the field of view is just as important as raw light gathering power. Survey telescopes such as the [[Large Synoptic Survey Telescope]] try to maximize the product of mirror area and field of view (or [[etendue]]) rather than raw light gathering ability alone.

===Magnification===
The magnification through a telescope makes an object appear larger while limiting the FOV. Magnification is often misleading as the optical power of the telescope, its characteristic is the most misunderstood term used to describe the observable world.{{clarify|date=April 2019}} At higher magnifications the image quality significantly reduces, usage of a [[Barlow lens]] increases the effective focal length of an optical system—multiplies image quality reduction.

Similar minor effects may be present when using [[star diagonal]]s, as light travels through a multitude of lenses that increase or decrease effective focal length. The quality of the image generally depends on the quality of the optics (lenses) and viewing conditions—not on magnification.

Magnification itself is limited by optical characteristics. With any telescope or microscope, beyond a practical maximum magnification, the image looks bigger but shows no more detail. It occurs when the finest detail the instrument can resolve is magnified to match the finest detail the eye can see. Magnification beyond this maximum is sometimes called ''empty magnification''.

To get the most detail out of a telescope, it is critical to choose the right magnification for the object being observed. Some objects appear best at low power, some at high power, and many at a moderate magnification. There are two values for magnification, a minimum and maximum. A wider field of view [[eyepiece]] may be used to keep the same eyepiece focal length whilst providing the same magnification through the telescope. For a good quality telescope operating in good atmospheric conditions, the maximum usable magnification is limited by diffraction.

====Visual====
The visual magnification <math>M</math> of the field of view through a telescope can be determined by the telescope's focal length <math>f</math> divided by the [[eyepiece]] focal length <math>f_{e}</math> (or diameter).<ref name="SaharaSkyObservatory"/><ref name="RyukyuAstronomyClub"/> The maximum is limited by the focal length of the [[eyepiece]].

An example of visual [[magnification]] using a telescope with a 1200&nbsp;mm focal length and 3&nbsp;mm [[eyepiece]] is given by: <math>M = \frac {f}{f_{e}} = \frac {1200}{3} = 400</math>

==== 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>

====Optimum====
The following are [[rule of thumb|rules-of-thumb]] for useful magnifications appropriate to different type objects:
* For small objects with low surface brightness (such as [[galaxies]]), use a moderate magnification.
* For small objects with moderate surface brightness (such as [[planetary nebulae]]), use a high magnification.
* For small objects with high surface brightness (such as [[planet]]s), use the highest magnification that the current night's "seeing" will allow, and consider adding in [[astronomical filter]]s to sharpen the image.
* For large objects (such as the [[Andromeda Galaxy]] or wide-field [[diffuse nebulae]]), regardless of surface brightness use low magnification, often in the range of minimum magnification.
* For very to extremely bright, large objects (the [[Moon]] and the [[Sun]]) narrow-down the aperture of the telescope by covering it with a piece of cardboard with a small hole in it, and insert filters as-needed to both cut down excess brightness and to enhance the contrast of surface features.

Only personal experience determines the best optimum magnifications for objects, relying on observational skills and seeing conditions, and the status of the pupil of observer's eye at the moment (e.g. a lower magnification may be required if there is enough moonlight to prevent complete dark adaption).

===Field of view===
Field of view is the extent of the observable world seen at any given moment, through an instrument (e.g., telescope or [[binoculars]]), or by naked eye. There are various expressions of field of view, being a specification of an [[eyepiece]] or a characteristic determined from an [[eyepiece]] and telescope combination. A physical limit derives from the combination where the FOV cannot be viewed larger than a defined maximum, due to [[diffraction]] of the optics.

====Apparent====
Apparent field of view (commonly referred to as AFOV) is the perceived angular size of the field stop of the [[eyepiece]], typically measured in [[Degree (angle)|degrees]]. It is a fixed property of the eyepiece's optical design, with common commercially available eyepieces offering a range of apparent fields from 40° to 120°. The apparent field of view of an eyepiece is limited by a combination of the eyepiece's field stop diameter, and focal length, and is independent of magnification used.

In an eyepiece with a very wide apparent field of view, the observer may perceive that the view through the telescope stretches out to their [[peripheral vision]], giving a sensation that they are no longer looking through an eyepiece, or that they are closer to the subject of interest than they really are. In contrast, an eyepiece with a narrow apparent field of view may give the sensation of looking through a tunnel or small porthole window, with the black field stop of the eyepiece occupying most of the observer's vision.

A wider apparent field of view permits the observer to see more of the subject of interest (that is, a wider true field of view) without reducing magnification to do so. However, the relationship between true field of view, apparent field of view, and magnification is not direct, due to increasing distortion characteristics that correlate with wider apparent fields of view. Instead, both true field of view and apparent field of view are consequences of the eyepiece's field stop diameter.

Apparent field of view differs from true field of view in so far as true field of view varies with magnification, whereas apparent field of view does not. The wider field stop of a wide angle eyepiece permits the viewing of a wider section of the [[real image]] formed at the telescope's focal plane, thus impacting the calculated true field of view.

An eyepiece's apparent field of view can influence total view brightness as perceived by the eye, since the apparent angular size of the field stop will determine how much of the observer's retina is illuminated by the [[exit pupil]] formed by the eyepiece. However, apparent field of view has no impact on the apparent [[surface brightness]] (that is, brightness per unit area) of objects contained within the field of view.

====True====
True FOV is the width of what is actually seen through any given eyepiece / telescope combination.

There are two formulae for calculating true field of view:

# Apparent field of view method given by <math>v_{t} = \frac {v_{a}}{M}</math>,  where <math>v_{t}</math> is the true FOV, <math>v_{a}</math> is the apparent field of view of the eyepiece, and <math>M</math> is the magnification being used.<ref name=":0">{{Cite web|date=2017-11-20|title=Simple Formulas for the Telescope Owner|url=https://skyandtelescope.org/observing/stargazers-corner/simple-formulas-for-the-telescope-owner/|access-date=2022-01-28|website=Sky & Telescope|language=en-US}}</ref><ref name=":1">{{Cite web|title=Determine Your True Field of View - Astronomy Hacks [Book]|url=https://www.oreilly.com/library/view/astronomy-hacks/0596100604/ch04s15.html|access-date=2022-01-28|website=www.oreilly.com|language=en}}</ref>
# Eyepiece field stop method given by <math>v_{t} = \frac {d_f}{f_t} \times 57.3</math>, where <math>v_{t}</math> is the true FOV, <math>d_{f}</math> is the eyepiece field stop diameter in millimeters and <math>f_{t}</math> is the focal length of the telescope in millimeters.<ref name=":0"/><ref name=":1"/>

The eyepiece field stop method is more accurate than the apparent field of view method,<ref name=":1"/> however not all eyepieces have an easily knowable field stop diameter.

====Maximum====
Max FOV is the maximum useful true field of view limited by the optics of the telescope. It is a physical limitation where increases beyond the maximum remain at maximum. Max FOV <math>v_{m}</math> is the barrel size <math>B</math> over the telescope's focal length <math>f</math> converted from [[radian]] to degrees.<ref name="SaharaSkyObservatory"/><ref name="RyukyuAstronomyClub"/>

An example of max FOV using a telescope with a barrel size of 31.75&nbsp;mm (1.25 [[inches]]) and focal length of 1200&nbsp;mm is given by: <math>v_{m} = B \cdot \frac {\frac {180}{\pi}}{f} \approx 31.75 \cdot \frac {57.2958}{1200} \approx 1.52^\circ</math>