Telescope Calculations
On this page, I present some simple formulas for telescopes and the calculation
results for telescopes that I own, owned, or find interesting. In addition,
I provide a few useful links.
Overview of Formulae
Note: For definitions in a small glossary, see page Quick & Dirty
Astronomy Glossary.
Introductory Notes
As a telescope owner, you may have some requirements, but these can be satisfied
by the different types of telescopes only to a certain degree:
 A high magnification (visual power), to resolve details on the moon,
on planets, and in galaxies and nebulae
 A large field of view for a easier orientation around the sky
 A sufficient brightness to be able to recognize distant stars, galaxies,
and nebulae, but also details on the moon and on planets.
The following calculations enable telescope users to determine some characteristics
of their telescopes and eyepieces and thus, to better judge what these can
do and what not. Regrettably, some "astronomy jargon" is needed here. Therefore,
I try to explain some of the used terms in a small glossary below, often using
Wikipedia articles. Further definitions can be found on the Internet (I provide
a few links...).
Calculations
Aperture
The term aperture refers to the diameter of the opening of a telescope.
For mirror telescopes this is either the diameter of the primary mirror or
a value that takes care of obstructions that limit the light receiving area.
Examples (Aperture = Diameter of Primary Mirror)
 Heritage 76: 76 mm
 Heritage 100P: 100 mm
 Heritage 114P: 114 mm
 Heritage P130: 130 mm
 Newton 6": 150 mm
 Dobson 8": 200 mm
 Dobson 10": 254 mm
 ETX 90/EC: 90 mm
 Skymax102: 102 mm
Focal Ratio
The focal ratio of a telescope is given by the ration of the focal
length of the telescope and the diameter of the primary mirror:
 Focal Ratio (f) = (Focal length of the telescope) / (Diameter of primary mirror)
Examples (Aperture = Diameter of Primary Mirror)
 Heritage 76: 300 mm / 76 mm = 3.95 (about 4)
 Heritage 100P: 400 mm / 100 mm = 4
 Heritage 114P: 500 mm / 114 mm = 4.4
 Heritage P130: 650 mm / 130mm = 5
 Newton 6": 750 mm / 150 mm = 5
 Dobson 8": 1200 mm / 200 mm = 6
 Dobson 10": 1270 mm / 254mm = 5
 ETX 90/EC: 1250 mm / 90mm = 13.89
 Skymax102: 1300 mm / 102 mm = 12.75
Light Gathering Power
The light gathering power of a telescope is expressed in multiples
of the light gathering power of the human eye:
 light gathering power = (aperture in mm)² / 49
(The maximum aperture of the naked eye is about 7 mm)
Examples (Aperture = Diameter of Primary Mirror)
 Heritage 76: 76 mm² / 49 = 118
 Heritage 100P: 100 mm² / 49 = 204 (200 officially...)
 Heritage 114P: 114 mm² / 49 = 265
 Heritage P130: 130 mm² / 49 = 345
 Newton 6": 150 mm² / 49 = 459
 Dobson 8": 200 mm² / 49 = 816
 Dobson 10": 254 mm² / 49 = 1317
 ETX 90/EC: 90 mm² / 49 = 165
 Skymax102: 100 mm² / 49 = 212
Magnification (Visual Power)
The magnification of a telescope is calculated from the ratio of the
focal length of the telescope and the focal length of the eyepiece:
 Magnification = (Focal length of the telescope) / (Focal length of the eyepiece)
Examples (10mm Eyepiece)
 Heritage 76: 300 mm / 10 mm = 30 x
 Heritage 100P: 400 mm / 10 mm = 40 x
 Heritage 114P: 500 mm / 10 mm = 50 x
 Heritage P130: 650 mm/10 mm = 65 x
 Newton 6": 750 mm / 10 mm = 75 x
 Dobson 8": 1200 mm / 10 mm = 120 x
 Dobson 10": 1270 mm / 10 mm = 127 x
 ETX 90/EC: 1250 mm / 10 mm = 125 x
 Skymax102: 1300 mm / 10 = 130 x
Maximum Practical Visual Power (Maximum Usable Magnification)/Minimum
Usable Focal Length of Eyepieces
The maximum practical visual power / usable magnification is
more or less determined by the diameter of the primary mirror:
 Maximum Practical Visual Power = (Primary Mirror Diameter) * X = Aperture * X
X amounts to:
 1.5 for Newtonian telescopes (including Dobsons), rich field refractors,
Schmidt Cassegrain telescopes
 2 for refractors from f/8, Maksutov telescopes
Note: Stoyan (Deep Sky Reiseführer) speaks of the "beneficial" visual
power,
at which the airy disk is still not resolved and at which the
magnitude limit of the telescope is reached. It is calculated as:
 Beneficial Visual Power = Aperture / 0.7 = (Primary Mirror Diameter)
/ 0.7
This corresponds more or less to a factor X of 1.5 (exact: 1.43) in the first
formula. Since this leads to very similar data, I do not list the results
obtained by this formula.
Note: For smallscale deepsky objects Stoyan (Deep Sky Guide) proposes
to go far beyond the beneficial visual power up to the maximum visual
power, which is twice as high as the beneficial visual power
(and corresponds to a factor X of 3). Depending on the telescope type and the
seeing (air turbulence), this is, however, not always possible. Smaller telescopes
reach their maximum visual power easier because it is lower than that of large
telescopes and thus, the seeing has less influence.
The minimum (practically) usable focal length of eyepieces is
calculated from the maximum
usable magnification and the focal length of the telescope (seems to be
my own idea...):
 Minimum usable focal length of eyepiece = (Focal length of the telescope)
/ (Maximum usable magnification)
Examples
 Heritage 76: 76 mm * 1.5 = 114 x > 300 mm / 114 x = 2.63 mm
 Heritage 100P: 100 mm * 1.5 = 150 x > 400 mm / 150 x = 2.67 mm
 Heritage 114P: 114 mm * 1.5 = 171 x > 500 mm / 171 x = 2.92 mm
 Heritage P130: 130 mm * 1.5 = 195 x > 650 mm / 195 x = 3.3 mm
 Newton 6": 150 mm * 1.5 = 225 x > 750 mm / 225 x = 3.3 mm
 Dobson 8": 200 mm * 1.5 = 300 x > 1200
mm / 300 x = 4.0 mm
 Dobson 10": 254 mm * 1.5 = 381 x (according to Meade: 600 x)
> 1270 mm / 381 x = 3.3 mm
 ETX 90/EC: 90 mm * 2 = 180 x (according to Meade: 325 x) > 1250 mm
/ 180 x = 7 mm
 Skymax102: 102 mm * 2 = 204 x > 1300 mm / 204 x = 6.37 mm
Maximum Usable Focal Length of Eyepiece/Minimum Usable Visual Power (Minimum
Usable Magnification)
The maximum usable focal length of eyepiece is determined by
the exit pupil and the focal
ratio of the telescope:
 Maximum Usable Focal Length of Eyepiece = (Focal Ratio)
* (Exit Pupil)
For an exit pupil of 6.5 mm, we get the formula:
 Focal Length of Eyepiece = (Focal Ratio) * 6.5 mm
For an exit pupil of 7 mm (often used in examples), we get the formula:
 Focal Length of Eyepiece = (Focal Ratio) * 7 mm
The minimum usable visual power (magnification) is determined
by the size of the exit pupil.
 Minimum Usable Visual Power (Magnification) = (Focal Length of Telescope)
/ (Maximum Usable Focal Length of Eyepiece) = (Focal Length of Telescope)
/ ((Focal Ratio)
* (Exit Pupil)) = (Diameter of primary mirror) / (Exit Pupil)
If magnification
is too low, parts of the light that
leaves the eyepiece cannot be utilized by the human eye (the exit
pupil is too large).
Examples (Maximum Usable Focal Length of Eyepieces/Minimum Usable Visual
Power (Minimum Usable Magnification)
For Exit Pupil 6.5 mm 

For Exit Pupil 7 mm 
 Heritage 76: 3.95 * 6.5 mm = 26.0 mm
> 300 mm / 26.0 mm = 11.68
x
 Heritage 100P: 4 * 6.5 mm = 26.0 mm
> 400 mm / 26.0 mm = 15.38
x
 Heritage 114P: 4.4 * 6.5 mm = 28.5 mm
> 500 mm / 28.5 mm = 17.54 x
 Heritage P130: 5 * 6.5 mm = 32.5 mm
> 650 mm / 32.5 mm = 20 x
 Newton 6": 5 * 6.5 mm = 32.5 mm
> 750 mm / 32.5 mm = 23 x
 Dobson 8": 6 * 6.5 mm = 39.0 mm
> 1200 mm / 39.0 mm = 30.77
x
 Dobson 10": 5 * 6.5 mm = 32.5 mm
> 1270 mm / 32.5 mm = 39.08
x
 ETX 90/EC: 13.89 * 6.5 mm = 90.3 mm
> 1250 mm / 90.3 mm = 13.84
x
 Skymax102: 12.75 * 6.5 mm = 82.8 mm
> 1300 mm / 82.8 mm = 15.7 x


 Heritage 76: 3.95 * 7 mm = 27.65 mm
> 300 mm / 27.65 mm = 10.85
x
 Heritage 100P: 4 * 7 mm = 28.0 mm
> 400 mm / 28.0 mm = 14.29
x
 Heritage 114P: 4.4 * 7 mm = 30.7 mm
> 500 mm / 30.7 mm = 16.29 x
 Heritage P130: 5 * 7 mm = 35.0 mm
> 650 mm / 35.0 mm = 18.57
x
 Newton 6": 5 * 7 mm = 35.0 mm
> 750 mm / 35.0 mm = 21.43 x
 Dobson 8": 6 * 7 mm = 42.0 mm
> 1200 mm / 42.0 mm = 28.57
x
 Dobson 10": 5 * 7 mm = 35.0 mm
> 1270 mm / 35.0 mm = 36.29
x
 ETX 90/EC: 13.89 * 7 mm = 97.23 mm
> 1250 mm / 97.23 mm = 12.86
x
 Skymax102: 12.75 * 7 mm = 89.22 mm
> 1300 mm / 89.22 mm = 14.57 x

Field of View
The apparent field of view determines the angle that is shown
by an eyepiece as a section of the sky. It depends on the type of the eyepiece
an is usually given by the manufacturer of the eyepiece. See the glossary for
more information.
Note: SkyWatcher lists 42° as a suitable value for most amateur
eyepieces. Obviously, these are Kellnertype eyepieces (SkyWatcher delivers
these together with its budget telescopes).
The true field of view determines the size of objects that
can be observed in a telescope
(Example: The moon corresponds to a field of view of about 0.5°)
 True field of view = (Apparent field of view) / Magnification = (Apparent field of view) * (Focal length of the eyepiece) / (Focal length of the telescope)
Examples (10 mm Eyepiece, 42°)
 Heritage 76: 42° / 30 = (42° * 10 mm) / 300 mm = 1.4°
 Heritage 100P: 42° / 40 = (42° * 10 mm) / 400 mm = 1.05°
 Heritage 114P: 42° / 50 = (42° * 10 mm) / 500 mm = 0.84°
 Heritage P130: 42° / 65 = (42° * 10 mm) / 650 mm = 0.65°
 Newton 6 ": 42° / 120 = (42° * 10 mm) / 750 mm = 0.56°
 Dobson 8": 42° / 120 = (42° * 10 mm) / 1200 mm = 0.35°
 Dobson 10": 42° / 127 = (42° * 10 mm) / 1270 mm = 0.33°
 ETX 90/EC: 42° / 125 = (42° * 10 mm) / 1250 mm = 0.34°
 Skymax102: 42° / 130 = (42° * 10 mm) / 1300 mm = 0.32°
Measuring the Field of View
The true field of view F of an eyepiece may not always be known and can be determined using a stopwatch (thanks to Jörg Meyer!):
Locate a star of known declination d close to the celestial equator and place it at the eastern edge of the field of view in the eyepiece (motor off!). Measure the time t that the star needs to move through the field of view and enter it into the following equation:
F = (t * 15 * cos d) / 60 (arc minutes)
Exit Pupil
The exit pupil determines, how bright the
image of a certain object, for example, the moon, will appear in the eye
piece. For the same exit pupil it will appear with the same brightness, irrespective
of the telescope, its aperture, and its magnification.
If the exit pupil of an eyepiece is too small, objects appear too dim (below
1 mm for deep sky objects, below 0.7 mm for planets, below 0.5 mm for the moon
and bright double stars), if it is larger than that of the human eye (>7
mm), only part of the light hits the human eye. For galaxies, choose an exit
pupil of 23 mm, not at all the maximum magnification (from the Internet).
The exit pupil can be determined in two ways, both of which lead to the same
formula:
 Exit pupil = (Diameter of primary mirror) / Magnification = (Diameter
of primary mirror) * (Focal length of the eyepiece) / (Focal length of
the telescope)
 Exit pupil = (Focal length of the eyepiece) / (Focal ratio) = (Diameter
of primary mirror) * (Focal length of the eyepiece) / (Focal length of
the telescope)
Thus, depending on your point of view, the exit pupil of an eyepiece can
be calculated either from the magnification or the focal ratio of a telescope
or a binocular.
Examples (10 mm Eyepiece)
 Heritage 76: 76 mm / 30 = (76 mm * 10 mm) / 300 mm = 2.53 mm
 Heritage 100P: 100 mm / 40 = (100 mm * 10 mm) / 400 mm = 2.5 mm
 Heritage 114P: 114 mm / 50 = (114 mm * 10 mm) / 500 mm = 2.28 mm
 Heritage P130: 130 mm / 65 = (130 mm * 10 mm) / 650 mm = 2 mm
 Newton 6": 150 mm / 750 = (150 mm * 10 mm) / 750 mm = 2 mm
 Dobson 8": 200 mm / 120 = (200 mm * 10 mm) / 1200 mm = 1.67 mm
 Dobson 10": 254 mm / 127 = (254 mm * 10 mm) / 1270 mm = 2 mm
 ETX 90/EC: 90 mm / 125 = (90 mm * 10 mm) / 1250 mm = 0.72 mm
 Skymax102: 102 mm / 130 = (102 mm * 10 mm) / 1300 mm = 0.78 mm
Airy Disk
For telescopes, the diameter of the airy disk can be calculated as
follows:
 D = 2.43932 * λ * (focal ratio of telescope) (D = Diameter
of airy disk in mm; λ = wavelength in mm, e.g. 546 nm = 0.000546 mm)
Example: For a focal ratio = F/4 and a wavelength of 546 nm,
D = 0.00533 mm
Another formula for the diameter of the airy disc is:
 A = 7200 * arctan (1.21966 * λ / d) (A = angular diameter
of the airy disk in arc seconds, d = diameter of telescope main mirror in
mm)
(From Oldham
Optical UK).
Resolution
For the theoretical resolution of a telescope results (formula according to
Rayleigh):
 Resolution of telescope α = (λ /D) * 360°/2π = (λ /D)
* 57,3° = (λ /D) * 3438' = (λ /D) * 206265"
with λ = wave length of light in mm, D = diameter of lens or mirror
(aperture) in mm
This value is, however, of purely theoretical nature. According to Stefan
Gotthold, a factor of 1.22 is used to obtain the "practical" resolution.
Using a mean wavelength of light (usually green, λ = 555 nm = 0.000555
mm), one arrives at the following rule of thumb (found at Gotthold, Oden):
 Resolution of telescope (") α = 138 / diameter of lens or mirror
(mm) (Rayleigh criterion: diffraction disks do not touch)
(I get a factor of 139,7 for 555 nm  but I will trust the common factor
of 138...)
Examples:
 For a Newtonian telescope with a mirror of 150 mm diameter (e.g. my Explorer
150PDS) this results in a resolution of α = 138/150 = 0.92",
for a Newton telescope with a mirror of 100 mm diameter (e.g. my Heritage
100P) in one of α = 138/100 = 1.38".
 A small refractor with 60 mm front lens diameter has a resolution of α =
138/60 = 2.3", one with 80 mm front lens diameter one of α =
138/80 = 1.725".
 A very large SchmidtCassegrain telescope with a mirror diameter of 350
mm (for example, a C14) has a resolution of 0.4".
(After: Was
ist eigentlich ... die Auflösung? (Peter Oden, Abenteuer Astronomie)
and Mathematik
in der Astronomie (Teil 4): Das Auflösungsvermögen von Teleskopen (Stefan
Gotthold, clearskyblog))
Typically, however, telescope manufacturers provide the resolution according
to the Dawes criterion, which looks a little bit better:
 Resolution of telescope (") α = 116 / diameter of lens or mirror
(mm) (Dawes criterion: diffraction disks form an "8" pattern)
Examples:
 For a Newtonian telescope with a mirror of 150 mm diameter (e.g. my Explorer
150PDS) this results in a resolution of α = 116/150 = 0.77", for
a Newton telescope with a mirror of 100 mm diameter (e.g. my Heritage 100P)
in one of α = 116/100 = 1.16".
 A small refractor with 60 mm front lens diameter has a resolution of α =
116/60 = 1.93", one with 80 mm front lens diameter one of α =
116/80 = 1.45".
 A very large SchmidtCassegrain telescope with a mirror diameter of 350
mm (for example, a C14) has a resolution of 0.33".
References