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 (the Hubble Space Telescope is an exception).

Table of Contents

  1. Magnification (power)
  2. True field of view
  3. Apparent field of view
  4. Maximum possible true FOV
  5. Relative brightness
  6. Linear field of view
  7. Exit pupil diameter
  8. Light gathering power/area
  9. Limiting magnitude
  10. Resolution limit
  11. Airy disk diameter
  12. Smallest resolved features
  13. Sensor image scale
  14. Sensor field of view
  15. Optimal Focal Ratio
  16. Minimum aperture to split a binary
  17. Minimum magnification to split a binary
  18. Longest useful eyepiece focal length
  19. Effective focal length
  20. Light recording power
  21. Linear resolution
  22. Star transit time
  23. Field stop diameter
  24. Image scale at prime focus
  25. Unguided exposure time
  26. Sensor magnification
  27. Telescope exposure time

Supplements

  1. The human eye
  2. Conversions and symbols
  3. Telescope types
  4. Telescope variations


 

 

1. Telescope: Magnification
NoFormulaeVariables
1.1.PW = FLS/ FLEPW: magnification (or power)
FLS: focal length of the scope [mm]
FLE: 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.

 

 

2. Eyepiece: True Field of View, TFOV
NoFormulaeVariables
2.1.TFOV = Ttr / 0.9973 x 15 ["]Ttr: 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]
FLS: focal length of telescope [mm]
2.2.TFOV = AFOV / PW [°]
2.3.TFOV = FSD * 57.3 / FLS [°]
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 Ttr = Ttr 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.

 

 

3. Eyepiece: Apparent Field of View, AFOV
NoFormulaeVariables
3.1.tan(AFOV/2) = tan(TFOV/2) x PW [°]AFOV: apparent field of view [°]
TFOV: true field of view [°]
FSR: field stop radius of the eyepiece [mm]
FLE: focal length of the eyepiece [mm]
3.2.AFOV = 2 x atn(FSR / FLE) [°]
Note: The TFOV for 3.1. can be obtained with the transit method in 2.1.

 

 

4. Telescope: Maximum possible true Field of View, TFOV
NoFormulaeVariables
4.1.MFOV = 31.7 x (180 / π) / FLS [°] MFOV: maximum field of view [°]
FLS: focal length of the scope [mm]
π: constant 3.14159
4.2.MFOV = 50.8 x (180 / π) / FLS [°]
Note: Formulae 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.

 

 

5. Telescope/Eyepiece: Relative Brightness
NoFormulaeVariables
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.

 

 

6. Telescope/Eyepiece: Linear Field of View
NoFormulaeVariables
6.1.LFOV = 2 * (Tan(TFOV / 2 / RAD) * 1000) LFOV: linear field of view [m]
TFOV: true field of view [°]
RAD = 57.2958 (radians)
Note: Calculated for a distance of 1000 meters.

 

 

7. Telescope/Eyepiece: Exit Pupil Diameter
NoFormulaeVariables
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.

 

 

8. Telescope: Light Gathering Power and Area
NoFormulaeVariables
8.1.LGP = (AP / 7)2 LGP = light gathering power [x human eye]
LGA: light gathering area [mm2]
AP: aperture of the scope [mm]
π: constant 3.14159
constant 7: eye pupil diameter [mm]
8.2.LGA = π x (AP / 2)2 [mm2]
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.

 

 

9. Telescope: Limiting Magnitude
NoFormulaeVariables
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).

 

 

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

 

 

11. Telescope: Airy Disk Diameter (angular and linear)
NoFormulaeVariables
11.1.ADDA = 2.43932 x λ / 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)
λ: wavelength [nm]
11.2.ADDL = 2.43932 x λ x FR [µm]
Note: Airy disk diameters are usually calculated for λ = 0.000550mm, yellow light. If airy disk size is larger sensor pixel size * 2.5 then the image is diffraction-limited.

 

 

12. Telescope: Smallest Resolved Features
NoFormulaeVariables
12.1.SRFM = ((231.65 / AP) x 3476) / ø [km]
alternatively:

SRFM = 384400 * Sin(231.65 / AP) [km]		 SRFM = for Lunar craters [km]
SRFS for Sun spots [km]
AP: aperture of the scope [mm]
ø: apparent diameter of Moon or Sun ["]
12.2.SRFS = ((231.65 / AP) x 1391000) / ø [km]
alternatively:

SRFS = 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.

 

 

13. Image Sensor: Image Scale
NoFormulaeVariables
13.1.x = (206.265 * px / FLS) [arc sec / pixel] px: Pixel size of the image sensor [µm]
FLS: focal length of lens or telescope [mm]
Note: The angle of sky covered by a single sensor pixel (resolution per pixel).

 

 

14. Image Sensor: Field of View
NoFormulaeVariables
14.1.x = (2 * atan(d / (2 * FLS))) * 57.296) [°] d: Sensor dimension, width, height or diagonal [mm]
FLS: focal length of lens or telescope [mm]
Note: The angle of sky covered by the image sensor also known as aperture angle (x = d / (2 * PI * fl) * 360 [°]).

 

 

15. Image Sensor: Optimal Focal Ratio
NoFormulaeVariables
15.1.x = px * 5 (or 6, depending on seeing) px: Pixel size of the image sensor [µm]
Note: Optimal focal ratio for a given sensor pixel size.

 

 

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

 

 

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

 

 

18. Telescope/Eyepiece: Longest Useful Eyepiece Focal Length
NoFormulaeVariables
18.1.FLE = 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.).

 

 

19. Telescope/Eyepiece: Effective Focal Length
NoFormulaeVariables
19.1.EFL = FLS x (DF - FLE) / FLE [mm] EFL = effective focal length [mm]
DF = distance between eyepiece (field stop plane) and plane of film/CCD [mm]
FLS: focal length of the scope [mm]
FLE: 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).

 

 

20. Telescope: Light Recording Power
NoFormulaeVariables
20.1.LRP = r2 / FR2 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.

 

 

21. Telescope: Linear Resolution
NoFormulaeVariables
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.

 

 

22. Telescope: Star Transit Time
NoFormulaeVariables
22.1.t = TFOV * 240 * Cos(Abs(δ)) [sec] TFOV: true field of view [°]
240 = Earth rotation in sec/degree
Note: δ is the declination of the star, either positive or negative.

 

 

23. Eyepiece: Field Stop Diameter
NoFormulaeVariables
23.1.fs = FLE * AFOV / 57.3 [mm] FLE: focal length of eyepiece [mm]
AFOV: apparent field of view [°]
Note:

 

 

24. Telescope: Image Scale at Prime Focus
NoFormulaeVariables
24.1.x = 3438 * 1/ FLS [arcmin/mm] FLS: focal length of telescope [mm]
Note:

 

 

25. Telescope: Unguided Exposure Time
NoFormulaeVariables
25.1.x = (500 * w / 36) / (FLS*cos(δ)) [sec] w: effective width of film or CCD chip
FLS: focal length of lens or telescope [mm]
δ: either positive or negative declination [°]
Note: The maximum exposure time up to which stars show acceptable trails.

 

 

26. Image Sensor: Magnification
NoFormulaeVariables
26.1.x = FLS / diagonal diagonal: Diagonal sensor size [mm]
FLS: focal length of lens or telescope [mm]
Note: The magnification is approximate (analog to telescope FL / eyepiece FL).

 

 

27. Telescope: Exposure Time
NoFormulaeVariables
27.1.x = t * (FR1/FR2)² [sec] t: exposure time [sec]
FR1, FR2 focal ratio of telescopes being compared.
Note: How much shorter or longer the exposure time with a telescope of FR1 versus FR2.

 

 

I. 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]

 

 

II. 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

 

 

III. Telescope Types
CommonAlternativeElements
RefractorDioptricLenses only
ReflectorCatoptricMirrors only
CassegrainCatadioptricCompound

 

 

IV. Telescope Variations
TypeVariations
RefractorsKeplerian (1 lens)
Achromat (2 air-spaced lenses)
Doublet (ED) Apochromat (2 lenses)
Triplet Apochromat (3 lenses)
Quadruplet Apochromat (3 lenses + flattener lens)
ReflectorsNewtonian
Schmidt-Newtonian
Dobsonian
CassegrainsSchmidt-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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