Optical Formulae

All parameters derived by the Formulae in this document are purely theoretical and strongly depend
on manufacturing, optical quality, lens count, alignment, coating, air condition, visual acuity, etc.
The values obtained with the formulae can never be reached in practice.
The weakest point of an optical telescope is the air in front of it (Hubble is an exception).

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**A. The Human Eye**

Pupil diameter: 5 to 7.5mm

Focal length: 17mm

Focal ratio: f/2.8 @6mm pupil diameter

Limiting magnitude: 5.5mv

Resolving power: 115.8/pupil diameter

Line resolution: 6 lines/mm

Contrast recognition: >1/4 wavelength

The typical scotopic (dark-adapted) pupil aperture is 6mm, or more for young people.

The photopic (light-adapted) pupil aperture

is about 2.5mm.

The average pupil size based on age can be obtained as: x = 8.1 - (0.04 * age) [mm]

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**B. Conversions**

1 AU = 149,597,871 km

1 feet = 0.3024 meters

1 inch = 25.4 millimeters

1 ounce = 28.3495 grams

1 radians = 57.2958° = 3437.75' = 206265"

**Symbols**

[λ] = wavelength

[π] = 3.14159 (constant)

[α] = right ascension

[δ] = declination

[φ] = angular separation (binary)

["] = seconds of arc

['] = minutes of arc

[°] = angular degrees

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*1) Vixen Sixth-Order Aspherical Catadioptic (VISAC primary mirror plus 3-element field corrector lens)

*2) VMC: Vixen Meniscus Catadioptic

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1. Telescope: Magnification | ||

No | Formulas | Variables |

1.1. | PW = FL_{S}/ FL_{E} | PW: magnification (or power) FL _{S}: focal length of the scope [mm]FL _{E}: focal length of the eyepiece [mm]AP: aperture [mm] EP: diameter of exit pupil [mm] TFOV: true field of view [°] AFOV: apparent field of view [°] |

1.2. | PW = AP / EP | |

1.3. | PW = AFOV / TFOV | |

1.4. | PW = tan(AFOV/2) / tan(TFOV/2) | |

Note: Schmidt-Cassegrain and Maktsutov systems adjust focus by moving the primary mirror changing the focal length of the scope, while the mirrors of Newtonians and Refractors are fixed so is the focal length.Dividing two angles in 1.3. is an approximation since magnification is the tangens of the viewing angles of image and object. The accurate formular is 1.4. especially for wide angle eyepieces over 50° apparent FOV. |

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2. Eyepiece: True Field of View, TFOV | ||

No | Formulas | Variables |

2.1. | TFOV = T_{tr} / 0.9973 x 15 ["] | T_{tr}: transit time [sec]TFOV: true field of view [°] AFOV: apparent field of view [°] PW: magnification (or power) δ = declination angle of a star FSD: eyepiece field stop diameter [mm] FL _{S}: focal length of telescope [mm] |

2.2. | TFOV = AFOV / PW [°] | |

2.3. | TFOV = FSD * 57.3 / FL_{S} [°] | |

Note: Measure the transit time of a star on 0° declination and convert the result to sidereal seconds (1 sidereal sec = 1 time sec / 0.997271) and multiply by 15 to obtain degrees. To convert to arcmin divide by 60, to degrees divide by 3600. If the star is higher or lower 0° declination, set T_{tr} = T_{tr} x cos(δ), change δ to positive if negative: cos(abs(δ)). To obtain the approximate apparent FOV multiply the result in 2.1. with the magnification. |

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3. Eyepiece: Apparent Field of View, AFOV | ||

No | Formulas | Variables |

3.1. | tan(AFOV/2) = tan(TFOV/2) x PW [°] | AFOV: apparent field of view [°] TFOV: true field of view [°] FS _{R}: field stop radius of the eyepiece [mm]FL _{E}: focal length of the eyepiece [mm] |

3.2. | AFOV = 2 x atn(FS_{R} / FL_{E}) [°] | |

Note: The TFOV for 3.1. can be obtained with the transit method in 2.1. |

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4. Telescope: Maximum possible true Field of View, TFOV | ||

No | Formulas | Variables |

4.1. | MFOV = 31.7 x (180 / π) / FL_{S} [°] |
MFOV: maximum field of view [°] FL _{S}: focal length of the scope [mm]π: constant 3.14159 |

4.2. | MFOV = 50.8 x (180 / π) / FL_{S} [°] | |

Note: Formulas 4.1. and 4.2. are for 1.25" and 2" eyepiece barrel sizes, respectively. The term (180/π) can be replaced by 57.3° which corresponds to 1 radian. |

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5. Telescope/Eyepiece: Relative Brightness | ||

No | Formulas | Variables |

5.1. | RBV= (AP / PW)^{2} |
RBV: relative brightness PW: magnification (or power) AP: aperture of the scope [mm] |

Note: Visual brightness solely depends on aperture, not on focal ratio. Low focal ratio allows shorter photographic exposure times for extended objects, like the Moon or nebulae. |

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6. Telescope/Eyepiece: Linear Field of View | ||

No | Formulas | Variables |

6.1. | LFOV= sin(TFOV) x 1000 [m] | LFOV: linear field of view [m] TFOV: true field of view [°] |

Note: Calculated for a distance of 1000 meters. LFOV= sin(TFOV) * 3048 [feet] for feet in 1000 yards |

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7. Telescope/Eyepiece: Exit Pupil Diameter | ||

No | Formulas | Variables |

7.1. | EP = AP / PW [mm] | AP: aperture of the scope [mm] PW: magnification (or power) |

Note: If the exit pupil diameter is larger than the eye pupil of the observer, the full aperture of the scope is not being used. For instance: observer's pupil diameter = 6mm, exit pupil = 8mm, aperture = 125mm. Used aperture = 125 * 6/8 = 94mm. The eyepiece focal length on a given scope should be so selected as the entire field of the primary lens/mirror passes fully through both the eyepiece and the pupil. |

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8. Telescope: Light Gathering Power and Area | ||

No | Formulas | Variables |

8.1. | LGP = (AP / 7)^{2} |
LGP = light gathering power [x human eye] LGA: light gathering area [mm ^{2}]AP: aperture of the scope [mm] π: constant 3.14159 constant 7: eye pupil diameter [mm] |

8.2. | LGA = π x (AP / 2)^{2} [mm^{2}] | |

Note: Also referred to as "aperture gain", LGP is as compared with the dark-adapted (scotopic) eye. A function of aperture. The light gathering power ratio between two telescopes is T1/T2 = (AP1/AP2)^{2}. |

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9. Telescope: Limiting Magnitude | ||

No | Formulas | Variables |

9.1. | LMAG = 7.5 + 5 x log(AP / 10) [vis mag] | LMAG = limiting magnitude [vm] AP: aperture of the scope [mm] |

Note: Logarithm is decadic. A function of aperture (required in centimeters). |

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10. Telescope: Theoretical Resolution Limit (Resolving Power) | ||

No | Formulas | Variables |

10.1. | RLD = (210589 x λ) / AP ["] | RLD = resolution limit (Dawes) ["] RLR = resolution limit (Rayleigh) ["] λ: wavelength of the light [nm] AP: aperture of the scope [mm] |

10.2. | RLR = (254000 x λ) / AP ["] | |

Note: Resolution limits are usually calculated for λ = 550nm, yellow light to which the human eye is most sensitive (thus simplified: RLD = 115.824 / AP, RLR = 139.7 / AP). Even the largest ground based scopes cannot resolve better than 0.5" due to atmospheric turbulences. A function of aperture. |

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11. Telescope: Airy Disk Diameter (angular and linear) | ||

No | Formulas | Variables |

11.1. | ADDA = 2.24 x λ x 206265) / AP ["] | ADDA = angular airy disk diameter ["] ADDL = linear airy disk diameter [mm] AP: aperture of the scope [mm] FR: focal ratio of the scope (FL / AP) FL: focal length of the scope [mm] |

11.2. | ADDL = 2.24 x λ x FR [mm] | |

Note: Airy disk diameters are usually calculated for λ = 550nm, yellow light to which the human eye is most sensitive. A function of aperture. |

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12. Telescope: Smallest Resolved Features | ||

No | Formulas | Variables |

12.1. | SRF_{M} = ((231.65 / AP) x 3476) / ø [km]SRF _{M} = 384400 * Sin(231.65 / AP) [km] |
SRF_{M} = for Lunar craters [km]SRF _{S} for Sun spots [km]AP: aperture of the scope [mm] ø: apparent diameter of Moon or Sun ["] |

12.2. | SRF_{S} = ((231.65 / AP) x 1391000) / ø [km]SRF _{S} = 149597871 * Sin(231.65 / AP) [km] | |

Note: The average value for the diameters of Moon and Sun is 1800". The constant value 231.65" is twice the Dawes limit based on 550nm wavelength (refer to 10.). A more practicable value, however, would be 4 x Dawes limit. The constant values 3476 and 1391000 are the diameters in kilometers of the Moon and the Sun, respectively. The constant values 384400 and 149597871 (1AU) are the mean distances in kilometers of the Moon and the Sun, respectively. The main mirror of the Hubble Space Telescope is 2400mm across resolving Moon features less than 100 meters and sun spots smaller than 35 km across. A function of aperture. |

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13. CCD: Resolution per Pixel | ||

No | Formulas | Variables |

13.1. | RPP = 205 x PS / FL ["] | RPP = resolution/pixel ["] FL: focal length of the scope [mm] PS: pixel size [µm] |

Note: CCD pixels can be of square or rectangular shape. Often, the diagonal size of the CCD chip is used to calculate its field of view: D = √x^{2} + y^{2}. |

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14. CCD: True Field of View | ||

No | Formulas | Variables |

14.1. | TFOV_{CCD} = Atn(X / FL) [°] |
TFOV_{CCD} = CCD's true field of view [°]FL: focal length of the scope [mm] X: width of pixel array [mm] Y: height of pixel array [mm] |

14.2. | TFOV_{CCD} = Atn(Y / FL) [°] | |

Note: CCD width and height refer to the pixel area, not to the CCD chip dimension. Often, the diagonal size of the CCD chip is used to calculate its field of view: D = √x^{2} + y^{2}. Quarter inch CCD webcams have a fairly narrow field comparable to a ca 5mm eyepiece. |

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15. CCD: Optimum Pixel Size | ||

No | Formulas | Variables |

15.1. | PS = ADS x FL / 205 [µm] | ADS = airy disk size ["] FL: focal length of the scope [mm] PS: pixel size [µm] |

Note: The size of a star depends on seeing conditions. Below 4" is considered good seeing. The optimum pixel size is slightly under the size of the star, however, it varies with the object to be imaged. By adjusting the focal length of the scope (barlow or reducer) the CCD camera can be optimized, while some models allow binning which combines four or nine physical pixels into one virtual pixel. |

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16. Telescope: Minimum Aperture to Split a Binary | ||

No | Formulas | Variables |

16.1. | AP = 115.824 / φ [mm] | AP: aperture of the scope [mm] φ: angular separation of the binary pair ["] |

Note: The constant value 115.824" is the Dawes limit at 550nm wavelength (refer to 10.). A more practicable value would be twice or more the Dawes limit.Ability to resolve a binary also depends on the magnitude difference of the pair, seeing conditions and visual acuity. |

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17. Telescope: Minimum Magnification to Split a Binary | ||

No | Formulas | Variables |

17.1. | PW = 480 / φ | PW: magnification (or power) φ: angular separation of the binary pair ["] |

Note: The constant value 480 marks the minimum angle in arcsec of two close stars the human unaided eye can distinguish. The ability to resolve a binary also depends on the magnitude difference of the pair, seeing conditions and visual acuity. |

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18. Telescope/Eyepiece: Longest Useful Eyepiece Focal Length | ||

No | Formulas | Variables |

18.1. | FL_{E} = FR x EP [mm] |
FR: telescope focal ratio EP: maximum exit pupil diameter [mm] |

Note: A too large eyepiece focal length can entail an exit pupil which is larger than the size of the observer's pupil, resulting in loss of telescope aperture (refer to 12.). |

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19. Telescope/Eyepiece: Effective Focal Length | ||

No | Formulas | Variables |

19.1. | EFL = FL_{S} x (DF - FL_{E}) / FL_{E} [mm] |
EFL = effective focal length [mm] DF = distance between eyepiece (field stop plane) and plane of film/CCD [mm] FL _{S}: focal length of the scope [mm]FL _{E}: focal length of the eyepiece [mm]DL = focal length of camera lens [mm] PW = magnification (or power) AP = scope aperture [mm] |

19.2. | EFL = DL x PW / AP [mm] | |

Note: The effective focal ratio is then obtained by: EFR = EFL / AP. This formular is applied for afocal photography (camera over eyepiece). |

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20. Telescope: Light Recording Power | ||

No | Formulas | Variables |

20.1. | LRP= r^{2}/FR^{2} |
LRP: light recording power r: radius of aperture [mm] FR: focal ratio of the scope |

Note: A convention typically used to compare the ability of optical systems to record light. |

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21. Telescope: Linear Resolution | ||

No | Formulas | Variables |

21.1. | LR = 0.001/(FR x λ) [lines/mm] | LR: linear resolution [lines/mm] FR: focal ratio of the scope λ = wavelength [nm] |

Note: Fast focal ratios provide higher linear resolution. The typical value for λ is 550nm, yellow light. |

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22. Telescope: Star Transit Time | ||

No | Formulas | Variables |

22.1. | t = TFOV * 240 * Cos(Abs(δ)) [sec] | TFOV: true field of view [°] |

Note: δ is the declination of the star, either positive or negative. |

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23. Eyepiece: Field Stop Diameter | ||

No | Formulas | Variables |

23.1. | fs = FL_{E} * AFOV / 57.3 [mm] |
FL_{E}: focal length of eyepiece [mm]AFOV: apparent field of view [°] |

Note: |

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24. Telescope: Image Scale at Prime Focus | ||

No | Formulas | Variables |

24.1. | x = 3438 * 1/ FL_{S} [arcmin/mm] |
FL_{S}: focal length of telescope [mm] |

Note: |

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25. Telescope: Unguided Exposure Time | ||

No | Formulas | Variables |

25.1. | x = (1000 * w / 36) / (FL_{S}*cos(δ)) [sec] |
w: effective width of film or CCD chip FL _{S}: focal length of lens or telescope [mm]δ: either positive or negative declination [°] |

Note: The maximum exposure time up to which stars show no trails. |

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Pupil diameter: 5 to 7.5mm

Focal length: 17mm

Focal ratio: f/2.8 @6mm pupil diameter

Limiting magnitude: 5.5mv

Resolving power: 115.8/pupil diameter

Line resolution: 6 lines/mm

Contrast recognition: >1/4 wavelength

The typical scotopic (dark-adapted) pupil aperture is 6mm, or more for young people.

The photopic (light-adapted) pupil aperture

is about 2.5mm.

The average pupil size based on age can be obtained as: x = 8.1 - (0.04 * age) [mm]

Contents

1 AU = 149,597,871 km

1 feet = 0.3024 meters

1 inch = 25.4 millimeters

1 ounce = 28.3495 grams

1 radians = 57.2958° = 3437.75' = 206265"

[λ] = wavelength

[π] = 3.14159 (constant)

[α] = right ascension

[δ] = declination

[φ] = angular separation (binary)

["] = seconds of arc

['] = minutes of arc

[°] = angular degrees

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C. Telescope Types | ||
---|---|---|

Common | Alternative | Elements |

Refractor | Dioptric | Lenses only |

Reflector | Catoptric | Mirrors only |

Cassegrain | Catadioptric | Compound |

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D. Telescope Variations | |
---|---|

Type | Variations |

Refractors | Keplerian (1 lens) |

Achromat (2 lenses) | |

Fraunhofer Achromat (2 lenses) | |

Apochromat (3+ lenses) | |

Neo-Achromat (Vixen design, 4 lenses) | |

Reflectors | Newtonian |

Schmidt-Newtonian | |

Dobsonian | |

Cassegrains | Schmidt-Cassegrain |

Maktsutov-Cassegrain | |

Ritchey-Cretien (eg Hubble) | |

Dall-Kirkham Cassegrain | |

Dillworth Cassegrain | |

Klevstov Cassegrain | |

VISAC^{*1} and VMC^{*2} (Vixen design) | |

Dilworth Catadioptric |

*1) Vixen Sixth-Order Aspherical Catadioptic (VISAC primary mirror plus 3-element field corrector lens)

*2) VMC: Vixen Meniscus Catadioptic

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