# Limit size

The limit size or limit brightness is a term from observational astronomy and describes the momentary transparency of the atmosphere. ${\ displaystyle m_ {gr}}$

It is given in magnitudes ("star size", mag) and is the apparent brightness of the weakest stars that an individual observer in the night sky can just about perceive. It can differ between different observers depending on their visual acuity, experience and age up to about 1 mag.

## Limit size for the naked eye

If one observes with open eyes - i.e. without optical or technical aids - the limit size of the natural, star-clear sky is indicated. It mainly depends on the weather and climatic conditions of the location, on its light pollution and on the eye of the observer. Diminishing influences are also possible moonlight , residual twilight light and insufficient adaptation of the eye to the darkness . The latter is largely achieved after about 10 minutes and completely after 20–30 minutes.

All of the following values ​​apply to indirect vision (a bit of "looking the other way"); In a direct view, the limit size is about 0.5 mag less favorable. In contrast, particularly sharp eyes increase the limit size by 0.5 to 0.8 mag.

In the open country far outside of cities, the limit size is

• in particularly clear desert or mountain regions 6 to 7  mag (for visibility comparisons of meteors ( ZHR ) 6.5 mag is therefore assumed)
• in Europe an average of 5 to 6 mag - d. H. Stars 5th to 6th mag are just visible, which means about 500 to 2,000 stars.

In cities, commercial and street lighting can reduce the limit size by up to 3 mag, and air pollution ( haze ) even further:

• On the outskirts and in the outskirts, the limit size is around 4 mag (3.5 to 4.7 mag depending on visual acuity)
• in very brightly lit cities , the limit size can drop to 1-2 mag, i. H. in extreme cases, only the brightest five to ten stars of the first magnitude are visible.

## Limit magnitude in the telescope

When using binoculars or an astronomical telescope , the limit size shifts to significantly weaker stars, theoretically in the area ratio of the aperture (free opening) of the telescope to the pupil size of the eye - provided that the exit pupil (the light beam emerging from the eyepiece ) is not larger than the pupil of the eye:

${\ displaystyle {\ frac {Q _ {\ mathrm {v, with}}} {Q _ {\ mathrm {v, without}}}} = {\ frac {A_ {Obj}} {A_ {Aug}}} = \ left ({\ frac {D_ {Obj}} {D_ {Aug}}} \ right) ^ {2}}$

With

• the amount of light with or without a telescope${\ displaystyle Q _ {\ mathrm {v}}}$
• the aperture area or objective area ${\ displaystyle A_ {Obj}}$
• the pupillary surface of the eye${\ displaystyle A_ {Aug}}$
• the aperture or the objective diameter ${\ displaystyle D_ {Obj}}$
• the pupil diameter of the eye.${\ displaystyle D_ {Aug}}$

In addition, the loss of light within the optics must be taken into account, which is usually estimated at around 20 percent; if the remuneration is excellent , it is lower.

Example: The pupil size prevailing in the dark in small children (who probably rarely use a telescope) is 8 mm, later 7 mm, and decreases to 5–6 mm with age. A field-glasses 7 x 50 (.. D h 50 mm lens -diameter or aperture) has the advantage over the 7 mm of the eye 50 times the surface: . ${\ displaystyle \ left ({\ tfrac {50} {7}} \ right) ^ {2} \ approx 7 ^ {2} \ approx 50}$

With its exit pupil of also 7 mm, it brings young people with a dark-adapted pupil a factor of 50 or 4.2  size classes , with 25% light loss still 3.9. At a good observation site in Central Europe, they can still see stars of 9th to 10th magnitude. Compared to adolescents, only half of the light reaches the pupil of senior citizens, both without and with binoculars, i.e. H. the relative amplification factor through the binoculars does not change; absolutely, however, old people see fewer stars than younger people.

The magnitude gain by using a telescope can be calculated directly from the diameters involved as follows:

${\ displaystyle m_ {gr, with} -m_ {gr, without} = 5 \ cdot \ log _ {10} \ left ({\ frac {D_ {Obj}} {D_ {Aug}}} \ right)}$

With an eight-inch model , the standard instrument used by amateur astronomers , the amount of light that falls on the eye decreases due to the larger aperture of around 200 mm (assuming a magnification of at least 30 ×) compared to the above. Binoculars up to 16 times, i.e. the range by another 3. With a limit size of now about 13 mag (the loss of light is lower in modern Cassegrain systems ) you can z. B. in the globular cluster  M13 ( constellation Hercules ) already recognize numerous stars at the edge. For the central star in the Ring Nebula (14.7 mag, i.e. 1.7 mag weaker), the images of which are often seen on giant telescopes , a telescope mirror with at least twice as large a diameter would be required.

The approaches to the magnitude gain of long-range optical instruments discussed so far remain incomplete in that they take into account the amount of light caused by the aperture, but not the contrast of the image . On a starry night, the sky background has a typical luminance of up to  cd / m . Decreases with increasing magnification of a given optical instrument (i. E. With decreasing diameter the exit pupil ), the area brightness of the displayed sky background from, the contrast rises to the star image, the circle of confusion (ideally) has a diameter unchanged. For this reason, the limit size is not only a function of the objective diameter, but also the magnification of the instrument. One tries to take this fact into account with the help of the astro indices - empirically determined scale laws for telescope performance in relation to the limit sizes of stars. The Canadian astronomer Roy Bishop introduced the visibility factor${\ displaystyle 3 \ cdot 10 ^ {- 4}}$${\ displaystyle 3 \ cdot 10 ^ {- 3}}$${\ displaystyle ^ {2}}$${\ displaystyle V}$

${\ displaystyle I _ {\ mathrm {v}} = V \ cdot D_ {Obj}}$

one in which the magnification and the lens diameter appear in the same weighting. Such an astro index does not provide an absolute value for the limit size of an instrument, but it does provide differences in limit sizes of different instruments in direct comparison. For example, if binoculars with the code number 10x50 have a performance index of  mm, and a second one with the code number 20x100 has an index of  mm, the result is a magnitude gain of ${\ displaystyle I _ {\ mathrm {v}} (\ mathrm {10x50}) = 500}$${\ displaystyle I _ {\ mathrm {v}} (\ mathrm {20x100}) = 2000}$

${\ displaystyle m _ {\ mathrm {20x100}} -m _ {\ mathrm {10x50}} = {\ frac {\ log _ {10} [I _ {\ mathrm {v}} (\ mathrm {20x100}) / I_ { \ mathrm {v}} (\ mathrm {10x50})]} {\ log _ {10} 2.512}} = 1.51}$ like

when using the larger 20x100 instrument over the 10x50 binoculars. Empirical observations with different telescopes moving the amateur astronomer Alan Adler to this as Adler Index or Astro-index became known law

${\ displaystyle I _ {\ mathrm {A}} = V \ cdot D_ {Obj} ^ {1/2}}$

to propose, in which the magnification plays a more dominant role for the achievable limit size than the objective diameter. However, Beat Fankhauser raised the objection that the eagle index violates the conservation law of radiation flux and is therefore physically inconsistent. Fankhauser also showed that every law of scale is the general condition

${\ displaystyle I = \ left (V ^ {a} \ cdot D_ {Obj} ^ {b} \ right) ^ {2 / (a ​​+ b)}}$

with any exponent , must be sufficient to ensure that the radiation flow is maintained. The index he proposed, ${\ displaystyle a}$${\ displaystyle b}$

${\ displaystyle I _ {\ mathrm {F}} = \ left (V ^ {2} \ cdot D_ {Obj} \ right) ^ {2/3}}$

fulfills this condition and, like the eagle index, gives the enlargement a high weighting.

In a more systematic way, scale laws for the limit sizes of stars can be derived from the models of telescope performance and the contrast thresholds derived from them, as suggested by Max Berek , for example . Consistent application of this approach to the calculation of limit values ​​in the starry sky leads to the index

${\ displaystyle I _ {\ mathrm {B}} = V ^ {3/4} \ cdot D_ {Obj} ^ {5/4}}$

in which, in turn, the objective diameter plays a more important role than the magnification for the limit size class.

The variety of astro indices traded in the specialist literature suggests that a review of the various approaches for calculating limit values ​​in visually used instruments on the basis of precise observation data is still pending. Using the example of published data on limit values, collected with binoculars of various key figures and quality levels, it was shown that the scattering of the data thwarted a reliable evaluation of the different astro indices. This is probably due not least to the different quality standards of the instruments in terms of transmission , imaging performance (i.e. size of the circle of confusion of the star image) or contamination with stray light .

## Individual evidence

1. ^ R. Brandt, B. Müller and E. Splittgerber, Sky observations with binoculars , Johann Ambrosius Barth Leipzig, p. 20 (1983)
2. ^ Roy Bishop, Observer's Handbook , Royal Astronomical Society of Canada, p. 63 (2009)
3. a b Lambert Spix, Fern-Seher , Oculum-Verlag Erlangen, p. 24 (2013)
4. ^ Alan Adler, Some Thoughts on Choosing and Using Binoculars for Astronomy , Sky & Telescope, Sept. 2002
5. Beat Fankhauser, A New Performance Size for Binoculars , ORION, No. 387, 2/2015
6. Max Berek: On the basic physiological law of the perception of light stimuli. In: Zeitschrift für Instrumentenkunde. Volume 63, 1943, pp. 297-309.
7. Max Berek: The useful power of binocular earth telescopes. In: Journal of Physics. Volume 125, No. 7-10, 1949, pp. 657-678.
8. a b H. Merlitz: Hand binoculars: function, performance, selection , 2nd edition, Verlag Europa-Lehrmittel, ISBN 978-3-8085-5775-4 , pp. 149–150 (2019)
9. Ed Zarenski, CN Report: Limiting Magnitude in Binoculars https://www.cloudynights.com/documents/limiting.pdf (2003)