Object photons and optimal exposure time

Object photons and optimal exposure time

For astrophotography, it would be advantageous to know before the imaging session how many object photons will arrive in the telescope/on the chip, and whether it is worth imaging an object with the used telescope, and if so, how long the individual and total exposure time should be when using filters. The following theoretical principles and calculations, which will differ somewhat from the real results, serve as a rough guide, as many location and system factors play a role and general assumptions are made.

In astronomy, a reference or comparative measurement for object brightness has been established using the magnitude system. Even in ancient times, the first observers of the sky divided the stars into brightness levels. The dimmer a star appeared, the higher its magnitude. It should be noted that the human eye perceives brightness logarithmically. (https://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law)

Therefore, the individual magnitude levels of the apparent magnitude of a star are also subject to a logarithmic scale, which was defined by Norman Robert Pogson in 1856 in such a way that the perceived brightness value of a star decreases by a factor of 100 every 5 magnitudes (https://en.wikipedia.org/wiki/Apparent_magnitude). The difference in brightness from one magnitude level to the next therefore varies by a factor of ⁵√100 = 10^0.4 ≈ 2.512. Since the perceived brightnesses are directly related to the photon flux, this factor can also be used to draw conclusions about the photon flux of the stars/objects to be compared.

If a reference star with a constant luminosity is assigned the magnitude level 0 (0^m-star), the photon flux can be determined using the magnitude values of the stars.

Source: CactiStaccingCrane, CC0, via Wikimedia Commons (modified)

In our latitudes, the star Vega was chosen as the reference star (0^m-star). By determining the photon flux of Vega and knowing the magnitudes of other stars/objects, these can be set in relation to Vega and thus a statement can be made about exposure times and the signal-to-noise ratio.

To determine how many photons from a star/object hit the upper layer of the Earth's atmosphere, the flux density of photons per wavelength per time and area of the star Vega is required.

Such values can be found, for example, on this page [https://www.stsci.edu/hst/instrumentation/reference-data-for-calibration-and-tools/astronomical-catalogs/calspec] under the archive data [https://archive.stsci.edu/hlsps/reference-atlases/cdbs/current_calspec/] in the file ‘alpha_lyr_stis_011.fits‘.

The specified values are given there in [^erg/_{cm² s Å}] (e.g. for 550 nm: 3,578*10^{-9 erg}/_{cm² s Å}, where 'erg' stands for the energy and 'Å' for angstrom (1Å = 0,1 nm)).

These must now be converted so that they become [^photons/_{m² s nm}].

This formula is listed under: [https://books.google.de/books?id=UC_1_804BXgC&pg=PA29&lpg=PA29&dq=converting+bandwidth+to+wavelength+jansky&source=bl&ots=IW4FpmnEIE&sig= 6PdaCx7NtjPebUYCupYNGIjBVgw&hl=en&sa=X&redir_esc=y#v=onepage&q&f=true]

However, it is somewhat misleading, as the unit Jansky [Jy] is introduced here. However, this is nothing other than: (https://en.wikipedia.org/wiki/Jansky)

The formula can therefore be reformulated as follows:

where c is the speed of light in [^m/_s].

If the photon flux density is therefore multiplied by 10⁷ * c and then divided by the square of the frequency of the respective wavelength (in this example 550 nm, where f = ^c/_λ), this results in a value in ^W/_{m² Hz}.

In order to convert the unit into the number of photons per area and second, 1W = 1^J/_s is used.

In addition, the relationship 1J = 1.509 * 10³³Hz is applied, which results from the direct linear proportionality of the photon energy to the frequency of the corresponding wavelength. The higher the frequency, the higher the energy. If the value of the energy is divided by the corresponding frequency, this results in the factor 1.509 * 10³³Hz.

If the value is multiplied by 1.509 * 10³³Hz the result is a value in [^photons/_{m² s}].

As each flux density is assigned to a corresponding wavelength, the result must also be divided by the respective wavelengths (in this numerical example 550 nm) and a value in [^photons/_{m² s nm}] is calculated.

This can now be carried out for each measured frequency (wavelength), resulting in such a diagram for the star Vega.

It can be seen that the photon flux density varies with the wavelength. In order to make a general statement about how many photons from the spectrum visible to humans arrive at the camera chip, the sensitivity curve would have to be included for each camera chip (e.g. silicon-based or indium gallium arsenide-based chips), and it would not yet be clear which photons may come from a different spectrum. As this is therefore not easy to solve, an average photon flux density over specific sections of the wavelength range can be determined by means of filters instead.

The filter classification according to Johnson/Bessel has become particularly established here. In the 1950s, Johnson developed three filters, U (ultraviolet), B (blue) and V (visual), which divide the electromagnetic spectrum into three ranges (https://en.wikipedia.org/wiki/UBV_photometric_system). M. S. Bessel erweiterte dieses System dann noch mit einem R (rot) und I (infrarot) Filter.

Source: https://sites.astro.caltech.edu/~george/ay122/Bessel2005ARAA43p293.pdf (page 305, table 1)

Template: https://www.baader-planetarium.com/de/blog/coming-soon-neue-baader-sloan-sdss-ugriz-fotometrische-filter/ and https://www.astroshop.de/l-rgb-filter/baader-ubvri-photometric-johnson-bessel-v-filter-4mm-/p,68633

These filters have been adapted a little over time (e.g. by Kron/Cousin in the R and I range https://qd-europe.com/de/en/product/astronomyuvbri-filters/), so that different values can be found in the literature from time to time.

M. S. Bessel published a document in 1979 in which he determined the photon flux density of Vega at the effective wavelengths of the UBVRI filters. (https://iopscience.iop.org/article/10.1086/130542/pdf, page 598, table IV)
Since then, these values have served as a reference for many documents and publications.

Despite the popularity of Bessel's values, the much more recent measurement data from the STScI is used in the following, which differs only slightly from that of Bessel.

The incoming photons can now be determined again from the conversion shown above. In order to be independent of the wavelength at the end (since integration is approximated over a small wavelength range), the corresponding filter bandwidth must be multiplied for the respective UBVRI filter, and the number of photons that hit the earth's atmosphere per m² and s at the respective filter is determined as a good integral approximation.

These values can now be used to draw conclusions about the photon number of other objects.

If the photons from the three bands B, V and R are added together, this results in 31,725,932,744 photons for a 0^m-star that hit 1 m² of the Earth's atmosphere per second. The following formula can then be used to determine the photons that can be expected in the upper layer of the Earth's atmosphere for stars/objects with different magnitude values. (https://en.wikipedia.org/wiki/Magnitude_(astronomy))

Where Flux_ref is the photon flux density of Vega and m_ref is the magnitude of Vega with m_ref = 0 in the desired spectral range.

Number of photons as a function of the apparent magnitude of the star

The next step is to show the extent to which the Earth's atmosphere influences the photon count and how many photons actually arrive in the telescope and on the chip.

When photons pass through our atmospheric layer, they are influenced by three main factors. (https://adsabs.harvard.edu/full/1975ApJ...197..593H&data_type=PDF_HIGH&type=PRINTER&filetype=.pdfInstrumentacin)

      I)      Rayleigh scattering on air molecules

In the case of Rayleigh scattering, the photons hit molecules whose diameters are much smaller in comparison to the wavelength. (https://en.wikipedia.org/wiki/Rayleigh_scattering)

The shorter the wavelength, the more likely a photon is to interact with a molecule. This is why photons in the blue spectrum react more frequently with molecules and are scattered in the atmosphere. As a result, the sky appears blue to us during the day and the sun appears yellowish.

The reduction in brightness (increase in magnitude value) of an object as a function of the wavelength λ [µm] and the observation height h [km] due to Rayleigh scattering can be calculated as follows: (https://adsabs.harvard.edu/full/1975ApJ...197..593H&data_type=PDF_HIGH&type=PRINTER&filetype=.pdfInstrumentacin)

      II)      Molecular absorption in the lower stratosphere within the ozone layer

Compared with the other factors, molecular absorption in the ozone layer has a minor influence and is therefore determined as follows for all wavelengths and altitudes according to this specification (http://www.icq.eps.harvard.edu/ICQExtinct.html):

      III)      Aerosol scattering due to suspended particles

Atmospheric aerosols come from many sources, such as sea spray, wind-borne desert dust or tree pollen. They vary in complex ways that are difficult to predict accurately. Nevertheless, there are a number of trends that can be used to give a reasonable estimate of the aerosol impact. (https://www.uai.it/pianeti/wp-content/uploads/2021/03/ppr_Sch93-1.pdf)

The calculation is carried out as follows: (http://www.icq.eps.harvard.edu/ICQExtinct.html)

whereby these average values are assumed for the following variables (http://www.icq.eps.harvard.edu/ICQExtinct.html):

A₀ = 0.05 (particle factor, rough average value)
α₀ = 1.3 (wavelength coefficient, value selected according to publications by A. Ångström (1961) https://a.tellusjournals.se/articles/3401/files/656f1b5418dab.pdf (page 215, left column below))
H = 1.5 km (scale height, is usually used in the calculation of aerosol scattering with this value, e.g. B. E. Schaefer (1993) https://www.uai.it/pianeti/wp-content/uploads/2021/03/ppr_Sch93-1.pdf (page 316, table 3))

The unit [^mag/_{air mass}] appears in all three formulas. The air mass (AM) corresponds to the path length of the light through the atmosphere, whereby the shortest path at the zenith (perpendicular to the Earth's surface) is normalized to the value '1'. (https://de.wikipedia.org/wiki/Luftmasse_(Astronomie))

The following approximate formula is used to determine the air mass, assuming that the atmosphere has a constant density and thus a substitute atmosphere ‘y_atm‘ with a layer thickness of 8.5 km (https://en.wikipedia.org/wiki/Air_mass_(astronomy)). The formula also takes into account the dependence on the height of the observer and the object angle Z from the zenith.

whereby:

R_E = 6,375 km (Earth radius)
y_obs – altitude of the observer above sea level
y_atm = 8.5 km (thickness of a substitute atmosphere that “simulates” a constant density that corresponds to the density of our atmosphere)
Z – angle of the observed object measured from the zenith

This results in such a correlation for some example heights:

For an observer at sea level, the air mass at the zenith has the value 1. At an object angle of approx. 42° above the horizon, the air mass increases 1.5 times, and then doubles at 30° above the horizon.

This means that our atmospheric extinctions increase by a factor of two when observing at sea level at an angle of 30° above the horizon.

If the observation of the air mass is now combined with the three formulas for determining the atmospheric extinction, for example, the following values result for the three bands B, V and R for an observer at sea level:

Atmospheric extinction in [mag] as a function of the object angle in the B-band (at 436 nm)

Atmospheric extinction in [mag] as a function of the object angle in the V-band (at 545 nm)

Atmospheric extinction in [mag] as a function of the object angle in the R-band (at 641 nm)

The drop in brightness in the atmosphere is therefore equivalent to the “loss” of photons that no longer reach the chip and must therefore be taken into account in a formula. For the sake of simplicity, it is assumed that the losses within a band are constant.

A formula that determines the number of photons of any star in the focal plane of a telescope was published, for example, by D. J. Schroeder in 2000.

Source: Schroeder, D. J. (2000). Astronomical Optics (2. edition) (page 435, formula 17.3.1.). Academic Press

whereby:

S – photon flux in the focal plane in [^Photonen/_s]
N₀ – illuminance of a 0^m-star (Vega) in the searched frequency band in [^Photonen/_{cm² s nm}]
T – transmission factor of the atmosphere and the optics (lens and mirror surfaces)
D – telescope aperture in [cm]
ε – obstruction of the telescope in [^%/₁₀₀]
Δλ – bandwidth of the filter used for capturing in [nm]
m – magnitude of the star in the corresponding frequency band

This formula applies to point sources, such as stars, and it is assumed approximately that the photons determined for ’N₀‘ are equally distributed within the respective frequency band.

For extended objects, the detector area projected onto the sky in square arc seconds must also be taken into account in the formula (Source: Schroeder, D. J. (2000). Astronomical Optics (2. edition) (page 441, formula 17.5.4.). Academic Press). Since the aim is to determine how many electrons are to be triggered in a pixel, the detector area is the pixel area. To express this projection, this is therefore the image scale in [^"/_Pixel] (see section ‘Basics’ – ‘Telescope-camera combination’ – ‘Image scale’) which is squared and multiplied by the above formula. The focal length is also included in the image scale.

Another factor that has an influence on the conversion of the photons hitting a pixel into electrons is the quantum efficiency QE of the chip and the set amplification factor (Gain). The gain factor is not the number of the Gain, but the number of electrons required to “trigger” an ADU (analog-digital unit) as stated in the data sheet. This also changes the full well capacity and the readout noise.

Source: https://www.zwoastro.com/product/asi294/ (edited)

It should also be noted that the transmission values of today's RGB filters and also of a Bayer matrix are usually higher than those of the Johnson-Bessel filters used for the reference determination. This leads to more photons, especially in the B-band, than with the Johnson-Bessel B-band filter.

Template: https://www.baader-planetarium.com/en/filters/photometric-filters/baader-ubvri-bessel-r-filter-photometric.html and https://www.astroshop.de/l-rgb-filter/optolong-lrgb-filter-set-2-/p,67534

The areas under the respective curves are approximately the same for the V and R bands. In order to better compare the photon fluxes (from reference determination and from our own system) in the B-band, a factor of around 1.2 (97% / 80%) should be applied to the photon flux density N₀ (illuminance).

The formula changes as follows due to the influencing factors mentioned:

whereby:

S – generated electrons in [^e-/_{pixel s}]
N₀ – illuminance of a 0^m-star (Vega) in the searched frequency band in [^photons/_{cm² s nm}]
T – transmission factors of the atmosphere, filters and optics (lens and mirror surfaces)
QE – quantum efficiency of the camera in the corresponding frequency band (from data sheet)
g – gain factor
D – telescope aperture in [cm]
ε – obstruction of the telescope in [^%/₁₀₀]
P – pixel size of the chip in [µm]
f – focal length of the telescope in [mm]
Δλ – bandwidth of the filter used for capturing in [nm]
m – surface brightness of the star/object/sky background in the corresponding frequency band

The surface brightness of the object (actually [^mag/_arcsec²]) is used without a unit of measurement, as the photon flux density of a patch of sky with a size of one square arcsecond (arcsec²) is considered here. (Source: technical discussion with Frank (freestar8n) from the forum Cloudy Nights)

In the formula, within a certain frequency band (B, V or R), the surface brightness 'm' of an object (and thus the photon flux density) is compared with the photon flux density ’N₀‘ of a 0^m-star over a certain bandwidth 'Δλ' and estimated. It is assumed that the reference star emits the photons per wavelength approximately uniformly on average within the bandwidth 'Δλ'.

However, this procedure means that the formula can only be applied to broadband sources such as galaxies, globular clusters or sky glow. For emission line sources that only emit one or a few waves of a certain wavelength (e.g. Hα and / or OIII), this procedure would lead to a contradictory result. If the bandwidth of a narrowband filter for an emission line source is used in the formula, this would pretend that object photons that enter the equation via the surface brightness are limited, although the narrowband filter allows precisely these photons of the emission nebula to pass through.
However, this effect is just right for broadband sky glow. The photons of the sky background are limited by the narrowband filter, and the object emitting at this filter bandwidth is thus visualized with greater contrast.

For a single emission line, a combination of 'N₀' and 'Δλ' is required, which results in the photon flux density actually received. If 'N₀' is large or is far away from the wavelength of the emission line, 'Δλ' must be adjusted accordingly. (Source: Schroeder, D. J. (2000). Astronomical Optics (2. edition) (page 442). Academic Press) Here, 'Δλ' is only used as a correction factor to bring the total flow to the correct value. (Source: technical discussion with Frank (freestar8n) from the forum Cloudy Nights)

As this cannot be realized without a reasonable amount of effort, other procedures must be chosen.

Procedure for broadband objects:

In order to express the atmospheric extinction in [mag] as a transmission factor in the formula, the logarithmic magnitude scale must be taken into account. As already explained at the beginning, the photon flux is reduced by a factor of 10^0.4 with an increase in magnitude of 1. The following formula is therefore used to convert a magnitude into a transmission factor, taking into account the ratio of the photon fluxes: (https://en.wikipedia.org/wiki/Apparent_magnitude)

The value of the atmospheric extinction in [mag] can be used for Δm.

Please note that in the formula for a flat object, the value of the apparent magnitude ‘m’ must be specified as the surface of [^mag/_arcsec²]. If a star is to be used as a point source in this formula, it can be considered as a surface via the seeing, or the initial formula can be used.

To get from the apparent magnitude to a surface brightness, this formula is used: (https://en.wikipedia.org/wiki/Surface_brightness)

whereby:

m – apparent magnitude of the star/object
A – apparent size (area in the sky) of the star/object in [arcsec²]

D. J. Schroeder's formula can be used to determine both the electrons emitted by an object and the electrons generated by the sky background. Determining the electrons generated by the sky background is important in order to be able to estimate how long the exposure can be in order to obtain background limited images. The exposure should only be long enough so that the photons of the sky background (which are received as noise) do not dominate over the object photons. The background thus limits the exposure time, which can be determined using the following formula: (Source: https://www.youtube.com/watch?v=3RH93UvP358, from minute 49:00)

whereby:

t_H – background limited exposure time
R_A – readout noise of the camera (from data sheet)
S_H – electron rate due to light pollution (electrons generated by the sky background per pixel per second) in [^e-/_{Pixel s}]

C is a factor that takes into account how much percentage deviation E from an unavoidable minimum possible noise should still be permitted.

E should be between 1% and 10%, according to Dr. Robin Glover. (Source: https://www.youtube.com/watch?v=3RH93UvP358, from minute 49:30)

The photons generated by the sky glow are included in the calculation for the signal-to-noise ratio as noise component S_H (see section ‘Basics’ – ‘Physical quantities’ – ‘Noise’)

whereby:

SNR – signal-to-noise ratio
N_P – number of single exposures
S_P – signal electrons of the object
t – single exposure time
S_P * t is the photon noise from the object
S_H * t is the photon noise from the sky background
S_D * t is the dark current noise
R_A – readout noise of the camera (from data sheet)

If a signal-to-noise ratio is specified in this formula and the background-limited exposure time t_H is used for t, the formula can be converted to N_P and the required number of background limited exposures is calculated for a specific exposure time with a specified signal-to-noise ratio. However, a shorter time than t_H can also be selected.

Specifying a suitable signal-to-noise ratio for astrophotography is not easy. Bright objects, such as the centers of galaxies, quickly reach a high SNR, whereas the dark sky background requires many exposures for a low-noise appearance. In general, an SNR of at least 3 should be achieved in astrophotography. Anything from an SNR of 10 - 20 results in satisfactory images.
Sometimes it happens that very bright object centers burn out, while the remaining areas could still be exposed for a long time. If brightness values are known for individual areas, it is possible to check whether the full well capacity has been exceeded with the gain and background limited exposure time used. To do this, all photons that hit a pixel and are responsible for filling the pixel memory must be added together over the exposure time. The readout noise is not added here, as it only occurs during readout and not when the pixel memory is filled. (see section ‘Basics‘ – ‘Physical quantities‘ – ‘Noise‘)

whereby:

S_FWC – total number of electrons generated in the pixel
S_P – signal electrons of the object
S_H – signal electrons of the sky background
S_H – signal electrons of the dark current
t – Single exposure time (usually background limited exposure time t_H)

Summary of the procedure for broadband objects:

1)   Calculate the electrons of the sky background using the following formula:

where m_H is the value of the surface brightness of the light pollution.
With the values for the sky glow from https://www.lightpollutionmap.info, the light no longer passes through the atmosphere (https://arxiv.org/pdf/astro-ph/0108052). Therefore, only the transmission of the optics plays a role in the formula for 'T'.

2)   Calculate the object electrons using the following formula:

where m_P is the value of the surface brightness of the object.

3)   Insert the electron number of the sky background of 1) into the following formula to determine the exposure time for background limited exposures:

4)   Calculation of the required number of images for background limited exposures and specified SNR when S_H, S_P and t_H are used in the following formula:

5)   Check whether the full well capacity has been exceeded with the used gain and background limited exposure time:

Example 1: (Messier 33 with a Newtonian telescope and a mono camera with LRGB filter set)
Example 2: (Messier 33 with a Newtonian telescope and a colour camera (OSC))

Procedure for objects that emit in the narrowband range:

As already described, Schroeder's formula cannot be used for emission line objects without problems. Therefore, a different procedure must be used.

It is helpful if information is given directly for the photon fluxes of the individual wavelengths. Such information exists for planetary nebulae, for example. There are no such databases for emission nebulae, and so far there are usually only magnitude data in the visual range (V-band), although they often emit at several wavelengths (e.g. Hα and OIII). It is therefore only possible to make a very rough estimate of how many photons come from which wavelength in order to determine how long and with which filter an exposure must be made.

In the range of the Hα line (656 nm), the transmission of the V-band filter is still approx. 5%. This means that only 5% of the Hα photons reach the chip. In the OIII range (501 nm) it is still 76%.

The following rough overview applies to the emission line filters:

The following image shows an example of how the percentages for OIII and Hα are determined:

Template: https://www.baader-planetarium.com/en/filters/photometric-filters/baader-ubvri-bessel-r-filter-photometric.html and https://www.astroshop.eu/l-rgb-filters/baader-filters-ubvri-photometric-johnson-bessel-v-filter-4mm-/p,68633

As mentioned at the beginning, a 0^m reference star in the V-band has a photon flux density of 8,443,532,998 [^photons/_{s m²}].

The formula 10^-0,4*m = ^Flux / Flux_ref shown at the beginning can be used to determine relative to the reference star (0^m-star) via the magnitude specification (which was determined with the reference filter system) how many photons of the corresponding emission lines that are effective in the V-band reach the chip together after they have passed through the filter.

However, since the photons are not limited to the same extent when using narrowband filters as is the case with V-band filters, it is important to know how many photons the emission nebula originally emitted.

If it is assumed that only the photons of the emission lines that are effective in the V-band have determined the V-magnitude value, the photon flux arriving at the chip can be calculated back and it is thus possible to determine how many photons the fog emits. (Source: technical discussion with Frank (freestar8n) from the forum Cloudy Nights)

For this purpose, it is only necessary to estimate the ratio in which the nebula emits the photons of the individual emission lines from empirical values or from images of the nebula. To make this easier to understand, the following diagram is intended to help using the example of an emission nebula that emits OIII and Hα photons:

Using the same procedure, the other two bands (B and R) can also be considered, and formulas can now be derived from this context to determine the photon flux density of the individual emission lines:

In general:

whereby:

N₀ – illuminance of a 0^m-star (Vega) in the searched frequency band in [^photons/_{s cm²}]
m – apparent magnitude of the object
T_λB – transmission values of the considered emission lines in the corresponding frequency band
r_λB – defined ratios of the effective emission lines in the considered frequency band
r_λ – defined ratio of the emission line sought
Σr_λB – Sum of the ratios of all effective emission lines in the considered frequency band

Case 1 (indication of the magnitude in the B-band):

IThe emission lines HeII, Hβ and OIII are effective in the B-band and must be set in relation to each other. The sum of the ratios must be '1' (corresponds to 100%).

Case 2 (specification of the magnitude in the V-band):

The emission lines Hβ, OIII, HeI, Hα, NII and SII act in the V-band. These must be set in relation to each other. The sum of the ratios must be '1' (corresponds to 100%).

Case 3 (indication of the magnitude in the R-band):

The emission lines HeI, Hα, NII and SII act in the R-band. These must be set in relation to each other. The sum of the ratios must be '1' (corresponds to 100%).

Example:

Given an emission nebula with mag_V = 9, it emits weakly at OIII and strongly in the Hα range with a ratio of 0.25 : 0.75.

These values can now be used further. It should be noted that this is the extrapolated number of photons emitted directly by the nebula. A modified form of Schroeder's equation can be used for the further calculation, but the transmission of the narrowband filter must now also be taken into account.

However, if the photon fluxes of the individual wavelengths are already available in databases, as in the case of planetary nebulae, the following procedure can be used to obtain a rough estimate of the exposure time and the number of images required for a desired SNR.

Databases for the photon fluxes for planetary nebulae can be found under the following data:

• In the document https://arxiv.org/pdf/1211.2505 many Hα flux values of planetary nebulae are given, which can also be searched online in the Vizier database:
  https://vizier.cds.unistra.fr/viz-bin/VizieR-3?-source=J/MNRAS/431/2/fluxes&-out.max=50&-out.form=HTML%20Table&-out.add=_r&-out.add=_RAJ,_DEJ&-sort=_r&-oc.form=sexa

• Another source for photon fluxes of different wavelengths can be found here:
  http://www.eso.org/sci/libraries/historicaldocuments/Strasbourg-ESO_catalogue/Strasbourg-ESO_Catalogue_of_Galactical_Planetary_Nebulae_Part_I.pdf
  http://www.eso.org/sci/libraries/historicaldocuments/Strasbourg-ESO_catalogue/Strasbourg-ESO_Catalogue_of_Galactical_Planetary_Nebulae_Part_II.pdf
  These are also listed online in the Vizier database:
  https://vizier.cds.unistra.fr/viz-bin/VizieR-3?-source=V/84/hbeta
  https://vizier.cds.unistra.fr/viz-bin/VizieR-3?-source=V/84/intens
  The photon flux density of an emission line (usually Hβ (486 nm), rarely also Hα (656 nm)) forms the basis of measurement and is normalized by 100%. All other emission lines are then specified in the ratio of whole numbers to this emission line.

• The spectral lines of many planetary nebulae are shown graphically at https://web.williams.edu/Astronomy/research/PN/nebulae/search/index.php#galactic_milky_way and the photon fluxes can be read off in [^erg/_{cm² s Å}].

In the first two literature sources, the photon fluxes are given in [^mW/_m²] which is equivalent to [^erg/_{cm² s}]. (http://www.eso.org/sci/libraries/historicaldocuments/Strasbourg-ESO_catalogue/Strasbourg-ESO_Catalogue_of_Galactical_Planetary_Nebulae_Part_I.pdf (page 8, section b))
The emission line value therefore indicates the energy with which the photons hit one cm² of the Earth's atmosphere per second.

The photon fluxes are given in logarithm and can be converted using the following relationship:

However, the values are required in [^photons/_{cm² s}]. For this purpose, the conversion method shown at the beginning can be selected, or the following formula can be used: (https://www.pveducat...ght/photon-flux)

whereby:

h = 6.62607015 * 10^-34 Ws² – Planck constant
c = 299,792,458 ^m/_s – speed of light
λ – wavelength in [m] of the corresponding emission line

With the third source, the photon fluxes are available in [^erg/_{cm² s Å}]and can be converted to [^Photonen/_{cm² s}] using the conversion shown at the beginning or the formula just applied.

A modified version of Schroeder's formula can now be used for the further procedure with narrowband objects, in which the photon flux density is used directly instead of the illuminance of a 0^m‑star and the surface brightness within a certain bandwidth.

As a small area of the sky section is mapped onto a pixel in square arc seconds via the image scale, the complete photon flux density of the planetary nebula must not be used in the formula. The individual photon fluxes of the respective wavelengths must still be divided by the apparent size (area in the sky) of the nebula, as only a small section of the nebula is observed with one pixel. It should be noted that this is a great simplification, as by simply dividing by the area it is assumed that the nebula emits the photons constantly distributed over the entire area. (Source: technical discussion with Frank (freestar8n) from the forum Cloudy Nights)

whereby:

S – generated electrons in [^e-/_{Pixel s}]
F_λ – photon flux density at the corresponding wavelength in [^photons/_{cm² s}]
A_obj – apparent size (area in the sky) of the object in arcsec²
T – transmission factor of the atmosphere and the optics (lens and mirror surfaces as well as filter transmission)
QE – quantum efficiency of the camera in the corresponding frequency band (from data sheet)
g – gain factor
ε – obstruction of the telescope in [^%/₁₀₀]
D – telescope aperture in [cm]
P – pixel size of the chip in [µm]
f – focal length of the telescope in [mm]

The transmission factor for the atmosphere is always the one at which the respective filter is particularly “effective”.

HeII-filter           ➔ T_Atm from B-band
Hβ-filter             ➔ T_Atm from B-band
OIII-filter            ➔ T_Atm from V-band
HeI-filter            ➔ T_Atm from R-band
Hα-filter             ➔ T_Atm from R-band
NII-filter             ➔ T_Atm from R-band
SII-filter             ➔ T_Atm from R-band

Summary of the procedure for narrowband objects:

1)   Calculate the electrons of the sky background using the following formula:

where m_H is the value of the surface brightness of the light pollution.
With the values for the sky glow from https://www.lightpollutionmap.info, the light no longer passes through the atmosphere (https://arxiv.org/pdf/astro-ph/0108052). Therefore, only the transmission of the optics plays a role in the formula for 'T'.
The values of the frequency band at which the filter is particularly “effective” are used for N₀.

HeII-filter           ➔ T_Atm from B-Band (this time not multiplied by 1.2, because only a very narrow bandwidth is considered)
Hβ-filter             ➔ T_Atm from B-Band (this time not multiplied by 1.2, because only a very narrow bandwidth is considered)
OIII-filter            ➔ T_Atm from V-Band
HeI-filter            ➔ T_Atm from R-Band
Hα-filter             ➔ T_Atm from R-Band
NII-filter             ➔ T_Atm from R-Band
SII-filter             ➔ T_Atm from R-Band

2)   Calculate the object electrons using the following formula:

where F_λ for an emission nebula is estimated using the following formula:

For a planetary nebula, the photon flux densities from the databases are used and converted into [^photons/_{cm² s}] using the following formulas:

The photon flux densities of Hα and NII are so close together that they are usually transmitted simultaneously by a narrowband filter (>4 nm), which must be taken into account in the calculation.

3)   Insert the electron number of the sky background of 1) into the following formula to determine the exposure time for background limited exposures:

4)   Calculation of the required number of images for background limited exposures and specified SNR when S_H, S_P and t_h are used in the following formula:

5)   Check whether the full well capacity has been exceeded with the used gain and background limited exposure time:

Example 3): Messier 8 (Lagoon Nebula) with a refractor and a mono camera with 12 nm filter
Example 4): Messier 8 (Lagoon Nebula) with a refractor and a colour camera (OSC) with Duo Narrowband filter for OIII and Hα (2x7 nm) and increased gain
Example 5): Abell 21 (planetary nebula) with an SC telescope and a mono camera with 12 nm filter

The basics of this chapter have been implemented in a calculation tool, which can be accessed in the 'Tools' section.