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. I do not describe the terms here, see page Quick & Dirty Astronomy Glossary for descriptions.
Note: For definitions in a small glossary, see page Quick & Dirty Astronomy Glossary.
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:
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, often using Wikipedia articles. Further definitions can be found on the Internet (I provide a few links...).
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.
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:
The light gathering power of a telescope is expressed in multiples of the light gathering power of the human eye:
(The maximum aperture of the naked eye is about 7 mm)
The magnification of a telescope is calculated from the ratio of the focal length of the telescope and the focal length of the eyepiece:
The maximum practical visual power / useful magnification is more or less determined by the diameter of the primary mirror:
X amounts to:
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:
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 small-scale deep-sky 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 useful magnification and the focal length of the telescope (seems to be my own idea...):
For an exit pupil of 6.5 mm, we get the formula:
For an exit pupil of 7 mm (often used in examples), we get the formula:
The minimum useful visual power (minimum useful magnification, normal magnification) is determined by the size of the 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).
|For Exit Pupil 6.5 mm||For Exit Pupil 7 mm|
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: Sky-Watcher lists 42° as a suitable value for most amateur eyepieces. Obviously, these are Kellner-type eyepieces (Sky-Watcher 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°)
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)
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 2-3 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:
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.
For telescopes, the diameter of the airy disk can be calculated as follows:
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:
(From Oldham Optical UK).
Resolution (or resolving power) is defined as the ability to separate two closely spaced objects (e.g., binary stars in astronomy). There are two empirically found criteria for this ability:
In the practice, rules of thumb are used for both criteria. That is why I leave out the complex mathematics here. The two rules of thumb are:
The resolution depends solely on the aperture of the telescope. Obviously, the Dawes criterion is considered "more practical" (see the article by Stefan Gotthold), and the manufacturers also provide this value (probably also because it looks better).
Example Table: Resolution Values Given by the Manufacturers and the "Rule of Thumb" Formulae
|Telescope||Focal Length *||Aperture||Rayleigh||Dawes||Manufacturer|
|S-W Heritage 76||300||76||1.82"||1.53"||1.51"|
|Meade ETX 90EC||1250||90||1.53"||1.29"||1.3"|
|S-W Heritage 100P||400||100||1.38"||1.16"||1.15"|
|S-W Explorer 150PDS||750||150||0.92"||0.77"||0.77"|
|GSO GSD 680||1200||200||0.69"||0.58"||0.58"|
|Meade Lightbridge 10"||1270||254||0.54"||0.46"||0.45"|
*) Not needed here
The above resolution formulae apply to fine structures such as double stars. In his book Moonhopper, Lambert Spix provides the following empirically determined formulae for the moon, which correspond to much higher resolutions, i.e. enable capturing much smaller structures:
According to Spix, these "theoretic" values have to be doubled in most nights due to atmospheric turbulence. Structures below 0.16" (300 m) are practically invisible even with large telescopes (larger than 10").
Example Table: Resolution Values and Doubled Resolution Values Calculated According to the "Empirical Formulae"
|Telescope||Focal Length *||Aperture||Crater||Groove||Crater||Groove|
|S-W Heritage 76||300||76||0.51"||0.30"||1.03"||0.61"|
|Meade ETX 90EC||1250||90||0.43"||0.26"||0.87"||0.51"|
|S-W Heritage 100P||400||100||0.39"||0.23"||0.78"||0.46"|
|S-W Explorer 150PDS||750||150||0.26"||0.15"||0.52"||0.31"|
|GSO GSD 680||1200||200||0.20"||0.12"||0.39"||0.23"|
|Meade Lightbridge 10"||1270||254||0.15"||0.09"||0.31"||0.18"|
*) Not needed here; 0.52" roughly corresponds to 1 km on the moon.