Limiting magnitude

Formulary for determining the limiting magnitude

1)   Formulas for the visual limiting magnitude (with unaided eye (m_v) and telescope/eyepiece (m_g)):

To be able to see stars, they must have sufficient contrast to the sky background. The respective star must be brighter than the sky background. The pupil opening and the magnification of the telescope also play a role. The pupil opening depends on age and can be determined for a relaxed pupil at night using the following formula: (https://articles.adsabs.harvard.edu/pdf/1990PASP..102..212S (page 213, formula 5))

‘A’ is the age of the observer.

When observing stars, the magnitude scale is used (a detailed description of this scale classification can be found in the section ‘Basics’ – ‘Physical quantities’ – ‘Object photons and optimal exposure time’). The limiting magnitude indicates the largest magnitude number (brightness of a star/object) that is just perceptible.

I)   Visual limiting magnitude (m_v / m_g) by R. Pogson

Norman Robert Pogson established a formula for determining the visual limiting magnitude in the 19th century (https://fr.wikipedia.org/wiki/Magnitude_limite_visuelle), which can be used both for the unaided eye and for telescope use. However, this is a highly generalized formula, as only the pupil resp. telescope aperture is included as a variable here.

A similar formula is also listed in this Wikipedia entry: (https://nl.wikipedia.org/wiki/Grensmagnitude)

Depending on the formula, either the pupil diameter or the telescope aperture in mm resp. cm is entered for ‘D’.

With a 6 mm pupil diameter (approx. 55-year-old observer), this results in a visual limiting magnitude of:

For a telescope with a 250 mm aperture, this results in a limiting magnitude of:

II)   Visual limiting magnitude (m_v) by B. E. Schaefer

For a more precise determination of the visual limiting magnitude, taking into account the brightness of the sky and the atmosphere, this formula by B. E. Schaefer is helpful: (https://articles.adsabs.harvard.edu/pdf/1990PASP..102..212 (page 215, formula 18))

whereby:

F_s – Sensitivity of the observer in detecting point sources (1 – normal, < 1 better, > 1 worse)
k_v – Reduction of brightness by the atmosphere in mag (atmospheric extinction - explanation and determination is dealt with in detail in the section ‘determining the optimum exposure time’)
B_s – Sky brightness in nL (nano Lambert)

To determine the sky brightness ‘B_s’, a sky quality meter (https://en.wikipedia.org/wiki/Sky_quality_meter) or the page https://www.lightpollutionmap.info/ can be used for approximate values. Both methods result in values in ^mag/_arcsec². To convert these into the required unit nL (nano Lambert), the following formula can be used (https://articles.adsabs.harvard.edu/pdf/1990PASP..102..212S, Seite 215, Formel 17):

Example calculation:

F_s = 0.8 (for a slightly trained observer)
k_v = 0.23 mag (reduction in brightness due to the atmosphere at the zenith)
B_s = 215 nL for a sky brightness of 20.5 ^mag/_arcsec²

The aperture (entrance pupil) of a telescope is larger than the pupil of the human eye. The telescope therefore collects more light and concentrates it at the exit pupil (at the end of the eyepiece), where the pupil of the observer is located. The result is increased illuminance - the stars are in fact brightened. At the same time, the selected magnification (depending on the eyepiece used) darkens the background of the sky (reducing luminance). As a result, stars that are normally invisible to the unaided eye become visible through the telescope. (https://en.wikipedia.org/wiki/Limiting_magnitude)

III)   General formula for the visual limiting magnitude (m_g)

In the course of the history of visual sky observation, many formulas have been developed to approximate the limiting magnitude. One that includes both the telescope aperture and the pupil diameter is: (https://de.wikipedia.org/wiki/Scheinbare_Helligkeit)

whereby:

m_v – Own visual limiting magnitude (see formulas above)
D – Telescope aperture
D_e – Pupil diameter

If Schaefer's result is used with m_v = 6.04, D = 250 mm and for D_e = 6 mm, the following limiting magnitude results:

IV)   Visual limiting magnitude (m_g) by H. Feijth and G. Comello

A very practical formula, which is based not only on theoretical principles but also on over 100,000 observations, was developed by Henk Feijth and Georg Comello in the mid-1990s and has appeared in various forums ever since. In contrast to the other formulas, which include the telescope, this formula takes into account the obstruction, magnification and transmission of the telescope. (https://www.y-auriga.de/astro/formeln.html (section 14), https://forum.astronomie.de/threads/vergroesserung-berechnen.32683/post-252788 and https://forum.astronomie.de/threads/vergroesserung-berechnen.32683/post-252795)

This formula also appears in the French Wikipedia, except that the obstruction is not included here. (https://fr.wikipedia.org/wiki/Magnitude_limite_visuelle)

The formula based on observations by Henk Feijth and Georg Comello is as follows:

whereby:

m_v – Own visual limiting magnitude (see formulas above)
D – Telescope aperture in [cm]
d – Diameter of the secondary mirror located in the optical path in [cm]
M – Magnification of the telescope eyepiece system
t – Transmission (100% = 1) of the entire system (losses on mirror and lens surfaces)

If Schaefer's result is used with m_v = 6.04, D = 25, d = 4, M = 400 and t = 0.54 the following limiting magnitude results:

V)   Visual limiting magnitude (m_g) by B. E. Schaefer (1990)

A very precise formula for determining the visual limiting magnitude with a telescope was published by B. E. Schaefer in 1990 (https://articles.adsabs.harvard.edu/pdf/1990PASP..102..212S).
This section and all formulas contained therein refer to this publication (unless otherwise noted).

In his calculations, Bradley E. Schaefer refers to a formula by S. Hecht from 1947, which he derived from the publication by H. A. Knoll, R. Tousey, and E. O. Hulburt from the year 1946:

‘C’ and ‘K’ are constants which, depending on the sky brightness ‘B’, refer to day vision (photopic) or night vision (scotopic). The cones of the eye are used for day vision and the rods for night vision. Schaefer defines the transition from day to night vision via the brightness of the sky background ‘B’ at the following numerical value:

Day vision: log(B) ≥ 3,17
Night vision: log(B) < 3,17

The following values result for the constants ‘C’ and ‘K’ for day vision:

These values apply to night vision:

‘I’ is the star brightness in [ftc (foot-candle)], and ‘B’ is the background sky brightness in [nL (nano Lambert)]. Both values are influenced in different ways by the atmosphere as well as by the telescope and the observer characteristics, and must therefore be adjusted with correction factors.

The star brightness 'I' is multiplied by a part of the factors, resulting in ‘I^*’. The background sky brightness ‘B’ is determined by dividing the sky brightness in the V-band (B_s) by a part of the factors.

Overall, this results in the following formula for ’I^*‘:

For an approximate determination of the sky brightness in the V-band, the value of https://www.lightpollutionmap.info/ is used and converted into nano Lambert.

For a sky brightness of e.g. 20.5 ^mag/_arcsec², this formula results in a value of 215 nL for ‘B_s’.

If ‘I^*’ is then known, the value of foot-candle (ftc) can be converted to [mag] using the following relationship:

The correction factors have the following meanings and values, taking into account day and night vision:

a)   F_b – Observation with one or both eyes

Hecht's initial equation is based on observations with both eyes. However, most telescope observations are made with one eye. The correction from binocular to monocular sensitivity is discussed by M. H. Pirenne (1943), who concludes from both observations and theory that the difference in sensitivity is a factor of the square root of two (which corresponds to a magnitude of 0.38 mag). Consequently, the magnitudes must be multiplied by:

b)   F_e – Reduction in brightness when passing through the atmosphere

This factor includes the influence of atmospheric extinction (explanation and determination is dealt with in detail in the section ‘Basics’ – ‘Physical Quantities’ – ‘Object photons and optimal exposure time’) as a function of the observation altitude and the object angle above the horizon. On average, this is 0.2 - 0.3 mag per air mass. The air mass at the zenith is normalized to 1. The closer the object is to the horizon, the greater the air mass (this influence is also discussed in detail in the section ‘Basics’ – ‘Physical Quantities’ – ‘Object photons and optimal exposure time’).

whereby:

q = 1.0 for day vision and q = 1.2 for night vision
k_v – Extinction coefficient
Z – Angle of the object measured from the zenith, where sec (Z) = ¹/_{cos (Z)}

For day vision, an object at the zenith with an angle Z = 0 and an extinction coefficient k_v ≈ 0.23 results in a factor:

For night vision this results in a factor:

c)   F_t – Obstruction and transmission of the telescope

This factor describes the losses caused by the components of the telescope.
Schaefer proposes the following formula for this factor:

whereby:

D_s – Diameter of secondary mirror
D – Telescope aperture
t_lⁿ – Transmission resp. reflection of all optical surfaces ‘n’

For the sake of simplicity, Schaefer gives all optical surfaces the same transmission or reflection value in this formula. Lenses usually have a transmittance of 0.96 (96%) per surface, but can drop to 0.7 (70%) due to dirt. Depending on the coating, mirrors have a reflectivity of 0.9 (90%) to 0.97 (97%), but this can also drop significantly if they become dirty.

In order to be able to view mirror and lens surfaces separately in the tool, the formula is adapted slightly and extended by r_sⁿ for the mirror surfaces.

Newtonian telescopes usually have a ^Ds/_D-ratio of 0.15 to 0.25. Schmidt-Cassegrain telescopes have a ^Ds/_D-ratio of 0.3 to 0.4.

The following components are used for a numerical example:

• Newtonian telescope with an obstruction (^Ds/_D-ratio) of 0.2 and two cleaned mirrors with a reflectivity of 0.94
• 4-lens eyepiece → 8 lens surfaces, of which the two outer ones are slightly dirty and have a transmission value of 0.9. The inner lens surfaces are clean and uncoated and have a value of 0.96.

d)   F_p – Loss of light due to unsuitable ratio of pupil diameter to exit pupil

This factor should take into account whether the pupil diameter is larger or smaller than the exit pupil of the telescope eyepiece system. (A detailed description of the exit pupil can be found under ‘Basics’ – ‘Physical quantities’ – ‘Magnification with eyepiece and camera’).

If the exit pupil (^D/_M) is larger than the pupil diameter ‘D_e’ of the eye:

whereby:

D – Telescope aperture in [mm]
M – Magnification of the telescope eyepiece system
D_e – Pupil diameter in [mm]

To calculate the pupil diameter ‘D_e’, the above formula is again used as a function of age ‘A’.

The exit pupil only becomes larger than the pupil diameter at very low magnification 'M'.

Otherwise F_p = 1 (if D_e > ^D/_M), which is also desirable in principle, since the entire light captured by the telescope reaches the pupil.

e)   F_a – Ratio of the pupil aperture to the telescope aperture

Since a telescope has a much larger light-collecting surface than the human eye, this correction factor must be included using the following formula:

whereby:

D_e – Pupil diameter in [mm]
D – Telescope aperture in [mm]

With a pupil diameter of 6 mm (approx. 55-year-old observer) and a telescope aperture of 250 mm, for example, this results in a correction factor F_a = 0.000576.

f)   F_m – Influence of used magnification

With extended sources such as the background brightness of the sky, the telescope not only collects additional light, but also distributes the light by magnifying the image. The surface brightness of the sky is reduced by the factor F_m = M² (M – magnification value), as the light is displayed to the observer with a magnification.

With a magnification of 400, this results in a correction factor of F_m = 160,000.

g)   F_r – Correction factor for the case of high magnification depending on the seeing

Normally, the factor ‘F_m’ only refers to the background brightness ‘B’, as a point source still appears as a point source under magnification. However, if too much magnification is used, the star will appear as a diffraction spot, giving it an apparent diameter so that the eye perceives the star as an extended source and therefore the brightness differently. The critical magnitude depends somewhat on the background brightness, but for practical purposes the critical magnitude for a “bloated” star is 900”. Schaefer refers to the writings of H. R. Blackwell (1946); J. L. Brown and others (1953) and T. N. Cornsweet (1970).

whereby:

θ – Seeing-value in [‘‘] (more information on the subject of seeing can be found in the chapter ‘Basics’ – ‘Telescope-camera-combination’ – ‘Seeing and FWHM’)
M – Magnification of the telescope eyepiece system

In our latitudes, the seeing is usually 2” – 4”. If we assume a seeing of 3” and a magnification of 400, for example, this results in a value greater than 900”. This would result in a correction value:

h)   F_SC – Stiles-Crawford effect (Brightness perception dependent on light entering the pupil)

The light entering the eye at the outer edge of the pupil is used less efficiently than the light entering the center of the pupil, which is known as the Stiles-Crawford effect.

For day vision (photopic), the drop in efficiency is described by J. A. Van Loo and J. M. Enoch (1975) with the following formula:

For night vision (scotopic), the drop in efficiency is less pronounced and, according to J. A. Van Loo and J. M. Enoch (1975), this formula applies:

This means, for example, that with a 6 mm pupil with a radius r = 3, the efficiency at the edge of the pupil at night is only approx. 84%.

As already explained for the correction factor ‘F_p’, the pupil is fully illuminated at lower magnifications, as the exit pupil of the telescope eyepiece system has a large diameter, which can sometimes be larger than the pupil diameter. Since, depending on the eyepiece selected, different illuminations of the eye pupil can be achieved with one and the same telescope and therefore fainter objects can also be recognized, this must be taken into account using a correction factor.

Schaefer specifies that if the pupil diameter is larger than the exit pupil of the telescope-eyepiece system, the following formula should be used for day vision:

The following applies to night vision:

He therefore defines the edge of the pupil as 1 and assumes an increase in efficiency towards the center of the eye as a function of magnification, making the factor less than 1.

In Garstang's writings from the year 2000, it is pointed out that there is an error in these formulas by Schaefer from the year 1990. (https://articles.adsabs.harvard.edu/pdf/2000MmSAI..71...83G, page 87, top). If the pupil diameter is larger than the exit pupil of the telescope-eyepiece system, the formulas must be applied inversely. In an excerpt from 1993, Schaefer lists the corrected formulas (https://www.uai.it/pianeti/wp-content/uploads/2021/03/ppr_Sch93-1.pdf, page 327, formulas 31).

The corrected inverse formulas are:

For day vision:

For night vision:

whereby:

D_e25 – Average pupil diameter in [cm], determined for a 25-year-old person
D – Telescope aperture in [cm]
M – Magnification of the telescope eyepiece system

With a pupil diameter of 6.8 mm (25-year-old person), a telescope aperture of 250 mm and a magnification of 400 as an example, the result for day vision is:

For night vision, the correction factor is very low:

i)   F_c – Color index (sensitivity depending on the color of a star)

The brightness of stars is usually specified in V-magnitude (around 550 nm), which roughly corresponds to the spectral sensitivity curve of the human eye for day vision (more details on the different frequency bands are explained in the section ‘Basics’ – ‘Physical quantities’ – ‘Object photons and optimal exposure time’). At night, the light sensitivity of the eye shifts towards the blue spectrum, as the rods become active for vision instead of the cones, and these are more sensitive in the blue spectrum.

Source: Frank Murmann, CC0, via Wikimedia Commons (complemented)

This means that two stars with the same V-magnitude but different colors (which corresponds to their temperature) are perceived differently. During the day, both stars would appear equally bright. At night, the blue (hotter) star appears brighter, even though it is not.

Schaefer recommends using the average of the B- and V-magnitude of the star and gives the following formula:

For day vision:

For night vision:

An average value for B - V is 0.7. This results in a correction factor as an example:

j)   F_s – Sensitivity of the observer when detecting point sources

This correction factor takes into account how trained a person is in perceiving point sources. This is difficult to quantify in general, as this ability is very individual. Schaefer sets the numerical value 1 for the average observer. A trained observer has values of less than 1, which can sometimes drop to 0.1, while an untrained observer has values above 1.

Your own value can be determined, for example, using Schaefer's formula shown above for determining the visual limiting magnitude ‘m_v’ by converting to ‘F_s’, if a star can be defined when looking into the night sky that is just visible to the observer.

For the following example calculation, the value for a slightly trained observer is set at F_s = 0.8.

All example values for night vision are now summarized once again in order to use them as examples in the above-mentioned initial formulas:

Initial formulas:

Given:

C = 1.58489*10^-10, K = 0.012589, B_s = 215 nL

F_b = 1.4142, F_e = 1.29, F_t = 1.86, F_p = 1, F_a = 0.000576, F_m = 160,000, F_r = 1.633, F_SC = 0.999991153, F_c = 0.55, F_s = 0.8

What initially seems somewhat illogical is the fact that the limiting magnitude increases when a higher age and thus a smaller pupil opening is entered while the equipment remains the same. This results from the fact that only a very small part of the pupil is used due to the usually high magnifications of the telescope eyepiece systems. A person with a smaller pupil diameter is not restricted by the magnification compared to an observer with a large pupil opening. As the observer with the smaller pupil size can see faint objects just as well, this is included in the final result as a kind of “bonus weighting”. According to this formula, the observer with the smaller pupil diameter is therefore theoretically able to see fainter objects, although this is not the case in reality. If, on the other hand, no telescope is used, the smaller pupil diameter also results in a reduced limiting magnitude.

VI)   Visual limiting magnitude (m_g) by Garstang (1999)

R. H. Garstang published a document in 1999 (https://articles.adsabs.harvard.edu/pdf/1999JRASC..93...80G) in which he redefined the starting formula

shown by Schaefer (1990) using differently selected constants ‘C’ and ‘K’. ‘C’ is now an illuminance in [lx] to which the variable ‘i₀’ is assigned. ‘K’ is removed from the brackets and is therefore changed to a different value.

whereby:

i₀ = 2.908*10^-9 lx
K = 0.115

He then extended the equation so that the seeing was also taken into account (page 81, formula 2).

By subsequently generalizing some of the correction factors from Schaefer's 1990 publication and introducing simplifications, he then developed the following formula (page 82, formula 7). For better understanding, the variables/factors from Schaefer (1990) are used here; Garstang used other terms:

whereby:

i₀ = 2.908*10^-9 lx
K = 0.115
α = 0.000154
y = 0.000062
z = 0.276
D_e – Pupil diameter in [cm]
D – Telescope aperture in [cm]
t – Obstruction and transmission of the telescope
B_s – Sky brightness at the observation location in [nL]
M – Magnification of the telescope eyepiece system
θ – Seeing in [‘]

Since ‘i₀’ is given in [lx] in Garstang's work, he also uses a different formula to convert it into the limiting magnitude (page 81, formula 3):

After calculating the limiting magnitude, this should be offset against the loss of atmospheric extinction. On average, the extinction coefficient ‘k_v’ reduces the brightness by an additional 0.2 - 0.3 mag/air mass.

Example calculation:

If the example figures from the section by Schaefer (1990) are again selected for the variables, the following relationship results:

D_e = 0.6 cm, D = 25 cm
t = 0.96⁶ * 0.90² * 0.94² * [1 - (0.2)²] = 0.54
B_s = 215 nL
M = 400, θ = 3‘‘ = 0.05‘
k_v ≈ 0.23 mag

Including the extinction of 0.23 mag, the value drops to:

VII)   Visual limiting magnitude (m_g) by Garstang (2000)

In 2000, R. H. Garstang published an expanded version of his writings from 1999 (https://articles.adsabs.harvard.edu/pdf/2000MmSAI..71...83G). His original formula

referred exclusively to night vision (scotopic - index ‘s’). His aim was to integrate day vision (photopic - index ‘p’) so that there would be a standardized formula for ‘I’. In addition, this time most of the correction factors from Schaefer's 1990 publication were to be included without simplifications and conversions. ‘i₀‘ also became a variable ‘C‘ again.

whereby:

C_p = 4.276*10^-8, C_s = 3.451*10^-9
K_p = 0.00151, K_s = 0.109
α_p = 0.00581, α_s = 0.000235
y_p = 0.00129, y_s = 0.00002
z_p = 0.0587, z_s = 0.174
B – Sky brightness at the observation location in [nL]
θ – Seeing in [‘]

In order to determine a corrected ‘I^*’ again, the formula for ‘I_p’ must be offset with correction factors for day vision and ‘I_s’ with the correction factors for night vision from Schaefer's 1990 document. To introduce these correction factors, they are calculated in the same way as in Schaefer's (1990) formula. The constants ‘C_p’ and ‘C_s’ are multiplied by the respective 'F₁' (see end of section Schaefer (1990)) for day and night vision. The value ‘B_s’ is divided by the respective correction factors of ‘F₂’ (see end of section Schaefer (1990)) to replace ‘B’ in the formula.

The following correction factors apply:

The magnification ‘M’ is multiplied by the imaged object diameter ‘θ’ (the seeing for stars). This means that the factor ‘F_r’ is no longer required in Garstang's formulas.

Since ‘I^*’ is output in [lx], Garstang uses the following formula for the conversion to magnitudes, as he did in his 1999 publication:

Note:

Garstang also gives his own formula for the pupil diameter, which includes the brightness of the sky (page 86, formula 6):

where ‘A’ is the age and ‘B’ is the sky brightness, which still has to be offset against the respective correction factors for day (photopic) and night (scotopic) vision. (page 86, paragraph 2)

However, since the pupil diameter is required via the factors ‘F_a’ and ‘F_p’ to determine the correction factors for the sky brightness, this creates circular references that are somewhat more difficult to calculate with.

If the circular references are to be avoided, Schaefer's formula for determining the pupil diameter can be used for the sake of simplicity. This simplification results in the following relationship:

In order to maintain consistency in this chapter, Garstang's formula is used for the calculations and 10 calculation loops are run through.

If the example figures from the section of Schaefer (1990) are used for better comparability with Schaefer (1990), the following values result for the formulas

mentioned:

F_b = 1.4142, F_ep = 1.24, F_es = 1.29, F_t = 1.86, F_pp = 1, F_ps = 1,
F_ap = 0.000532 (after 10 loops with F_2p, approx. 55-year-old observer),
F_as = 0.000533 (after 10 loops with F_2s, approx. 55-year-old observer),
F_m = 160,000, F_SCp = 0.585895196, F_SCs = 0.999991153, F_c = 0.55, F_s = 0.8

These can now be used to calculate the limiting magnitude.

Given:

B_s = 215 nL
θ = 3‘‘ = 0.05‘
M = 400

Using the formulas, it is also possible to determine the visual limiting magnitude ‘m_v’ (page 87, chapter 3). Some of the correction factors that relate to the telescope properties are omitted, as only observation with the unaided eye is referred to here. For the same reason, the correction factor ‘F_a’ must also be determined differently. Instead of determining the ratio of the pupil aperture to the telescope aperture, the ratio of a pupil diameter ‘D_e23’ of a 23-year-old average observer (calculated from the formula of Schaefer (1990) in [cm]) to the pupil diameter ‘D_e’ of the observer is formed (Garstang's formula is used here).

If the example figures from the section of Schaefer (1990) are used for better comparability with the formula for the visual limiting magnitude given by Schaefer (1990), the following values result (for visual observation, the magnification ‘M’ is 1 and the telescope diameter corresponds to the pupil diameter, which is included in ‘F_sc’):

F_ep = 1.24, F_es = 1.29,
F_ap = 1.2402 (after 10 loops with F_2p, approx. 55-year-old observer),
F_as = 1.2289 (after 10 loops with F_2s, approx. 55-year-old observer),
F_SCp = 0.58251071, F_SCs = 0.99999098, F_c = 0.55, F_s = 0.8

Given:

B_s = 215 nL
θ = 3‘‘ = 0.05‘
M = 1

VIII)   Visual limiting magnitude (m_g) by Garstang (2000) with generalized formula by Bowen (1947)

In the same document from the year 2000 (https://articles.adsabs.harvard.edu/pdf/2000MmSAI..71...83G) R. H. Garstang describes in chapter 6 an extended formula that can be derived from the publication by I. S. Bowen from 1947 (https://iopscience.iop.org/article/10.1086/125960/pdf), and which is included here for the sake of completeness.

Bowen's 1947 formula for the limiting magnitude is as follows:

whereby:

D – Telescope aperture
M – Magnification of the telescope eyepiece system

Garstang derived two formulas from this, taking into account the ratio of the exit pupil of the telescope eyepiece system to the pupil diameter.

whereby:

D – Telescope aperture in [cm]
M – Magnification of the telescope eyepiece system
D_e – Pupil diameter in [cm] calculated according to the formula from the publication by Schaefer (1990)
t – Obstruction and transmission of the telescope
B_s – Sky brightness at the observation location in [nL]
F_s – Sensitivity of the observer in detecting point sources (1 - normal, < 1 better, > 1 worse)
k_v – Reduction of brightness by the atmosphere in [mag] (Atmospheric extinction - explanation and determination is dealt with in detail in the section ‘Basics’ – ‘Physical Quantities’ – ‘Object photons and optimal exposure time’)

The following is an example calculation if the exit pupil of the telescope is smaller than the pupil diameter. The numerical values from the examples above are used again.

D_e = 0,6 cm (for an approx. 55-year-old observer), D = 25 cm
t = 0.96⁶ * 0.90² * 0.94² * [1 - (0.2)²] = 0.54
B_s = 215 nL
M = 400,
k_v ≈ 0.23 mag
F_s = 0.8

The following is an overview of the example results with the following boundary conditions:

Pupil diameter D_e = 6 mm = 0.6 cm, Telescope aperture D = 250 mm = 25 cm
Sky brightness B_s = 215 nL = 20.5 mag/arcsec²
Observer sensitivity F_s = 0.8
Magnification M = 400, Seeing θ = 3‘‘ = 0.05‘
Atmospheric extinction k_v = 0.23 mag
Transmission and obstruction from the telescope t = 0.54

Each of the formulas established over the years has its justification. The true value probably lies in the average of the purely theoretical formulas and those that also contain practically determined values.

2)   Formulas for the imaging limiting magnitude:

As with the visual limiting magnitude, there are several formulas that can be used to approximately determine where the limiting magnitude for a system lies for a selected total exposure time.
However, there are many factors that influence this value, so that the results of all formulas are only approximate values. The true limiting magnitude can only be determined by taking a long exposure and evaluating the stars using software.

A formula for point source, for example, is:

Source: Martin A. & Koch B. (2009). Digitale Astrofotografie, Grundlagen und Praxis der CCD- und Digitalkameratechnik (1. Edition) (page 233, formula 47). Oculum-Verlag GmbH, Erlangen

whereby:

t – Total exposure time in [s]
D – Lens diameter
z – Diameter of the star disk with z = ^{Seeing["] * Telescope focal length}/_206264,8"
S – ISO value of the camera
K – Camera-specific correction value (the filter transmission and the deviation of the true ISO value from the set ISO sensitivity are included here)
m_H – Sky brightness in [^mag/_degree²]

A formula for extended objects is as follows:

Source: Martin A. & Koch B. (2009). Digitale Astrofotografie, Grundlagen und Praxis der CCD- und Digitalkameratechnik (1. Edition) (page 234, formula 49). Oculum-Verlag GmbH, Erlangen

whereby:

t – Total exposure time in [s]
N – F-number
S – ISO value of the camera
K – Camera-specific correction value (the filter transmission and the deviation of the true ISO value from the set ISO sensitivity are included here)
A – Area of the captured object in [degree²]
m_H – Sky brightness in [^mag/_degree²]

When using these formulas, it is difficult to determine the camera-specific value. And for cameras that have a gain setting instead of an ISO setting, the use of these formulas is not suitable anyway. Unfortunately, ISO values cannot be converted into gain values (or vice versa), because ISO values date back to the days of film and refer to the sensitivity of the film material. Gain values describe the amplification of an electronic signal.

Another collection of formulas for point sources is listed in this document: (https://www.interstellarum.de/wp-content/uploads/2019/12/is91.pdf, page 49)

For normal or telephoto lenses:

For refracting telescopes:

For reflector telescopes:

whereby:

m_Sky – Measured SQM value [^mag/_arcsec²]
f – Focal length in [mm]
B – F-number (e.g. 5.6 at aperture f/5.6)
t – Total exposure time in [s]
ISO – ISO amplification value
D – Optics diameter in [mm]
O – Obstruction factor (^{secondary mirror diameter}/_{primary mirror diameter}), e.g. 0.25 at 25% obstruction

In principle, the third formula (for reflecting telescopes) can also be used here for the other two types of telescopes. Without obstruction, the expression ^{√D²-(D*O)²}/₇ is reduced to ^D/₇. And if the aperture of a lens is known, the focal length and aperture can be used instead of the f-number (which is nothing other than the quotient of focal length to aperture → B = ^f/_D). The focal length is thus reduced from the formula ^f/_B*7 and ^D/₇ remains again.

But even these formulas are only suitable for cameras that have an ISO setting.

The following values are assumed for an example calculation:

m_Sky = 20.5 ^mag/_arcsec²
D = 250 mm
O = 30%
T = 18,000 s (5 hours total exposure time)
ISO = 1,600

Such a formula can be used for cameras with a Gain setting:

Source: McLean Ian S. (2008). Electronic imaging in astronomy - Detectors and instrumentation (2. Edition) (page 349, formula 9.32). Springer Berlin Heidelberg NewYork

whereby:

The formula can therefore be summarized as follows if ^S/_N = 1 is set:

Limiting magnitude of a point source in [mag]:

Limiting magnitude of a flat object (n_pix = 1) in [^mag/_arcsec²]:

The following is a numerical example for a exiended object taken with a 250/1000 Newtonian telescope and a ZWO ASI294MM:

Given:

τ = 0.67 (2 mirrors, flattener, filter and protective glass in front of chip)
QE ≈ 0.8 (for 80% quantum efficiency of the chip in the V-band) (http://www.astrosurf.com/buil/asi294mm.html)
λ = 0.545 µm
∆λ = 0.1 µm
D = 25 cm with obstruction of 25%
→ A_tel = ^π/₄ * D² * (1 - ε²) = 460.19 cm² (source: Schroeder, D. J. (2000). Astronomical Optics (2. Edition) (page 435). Academic Press)
F_λ(0) = 10,052 [^photons/_{cm² s nm}] = 3.663774 * 10^-12 [^W/_{cm² µm}] (https://www.stsci.edu/~strolger/docs/UNITS.txt)
→ (see section ‘Basics’ – ‘Physical quantities’ – ‘Object photons and optimal exposure time’)
h = 6.62607015*10^-34 Js = 6.62607015*10^-34 Ws²
c = 299,792,458 ^m/_s = 299,792,458*10^6 µm/_s
g = 1^e-/_ADU (at gain 120 (from data sheet)) → 1 ^e-/_s
f = 1,000 mm
B_Sky ≈ 1.42 ^e-/_{pixel s} (at m_Sky = 20.5 ^mag/_arcsec², ∆λ = 100 nm)
T = 18,000 s (5 hours total exposure time)

After an exposure time of 5 hours, the imaging limiting magnitude for a flat object is 26.11 mag/arcsec². This corresponds approximately to a galaxy with a brightness in V_mag = 8 mag and an extension in both directions of 70', in which the brightness is evenly distributed. In galaxies, however, the center is usually much brighter than the edge, so that the center is always well imaged and only the edge structures require longer exposure times.

The following diagrams once again illustrate the influence of exposure time and telescope aperture, which can be easily influenced in astrophotography.