The brightness (SQM) value of the twilight sky

To develop an empirical formula I used an Unihedron "Sky Quality Meter - L" to measure the brightness of the night sky in  SQM units mag/arcsec2 as function of the Sun elevation at one night.

As background information, there  are three stages of twilight,  civil, nautical, and astronomical on the basis of the Sun's elevation:

Twilight definitions:
  1. Civil twilight – when the Sun is below the horizon,  but the Sun's centre is less then 6° below the true horizon.
  2. Nautical twilight – when the Sun's centre is between 6° and 12° below the horizon. For nautical purposes sailors can distinguish a visible horizon for reference.
  3. Astronomical twilight – when the Sun's centre is between 12° and 18° below the horizon. Below that the sky is truly dark and no remnant of the Sun's afterglow can be seen. 
For twilight the elevation is defined the angle of the geometric center of the Sun with the true horizon. Note that the apparent horizon is about 34 arc minutes lower. Note also that Sunset is defined if the upper limb of the sun is below the apparent horizon.  The Sun elevation is for these reason sometimes expressed in distance from the zenith.

The maximum darkness the sky can reach is a SQM value of around 21.75 to 22 mag/arcsec2due to the natural sky background brightness. The natural sky brightness is caused by the zodiacal light, stars, Milky Way and Gegenschein.. In Johnson U (blue) it is higher and  R (red, and I (infrared) it is lower.

Currently only one serie of twillight measurements where taken on 2017-6-26 at a site with best SQM value of 20.7 in the past. The measurements where corrected for the artifical background  with assistance of the following formualas::

[value in cd/m2] = 10.8×104 × 10(-0.4*[value in mag/arcsec2])

[value in mag/arcsec2] = Log10([value in cd/m2]/108000)/-0.4

The best site SQM value of the site in the past where about 20.7 mag/arcsec2 equal a sky brightness of  0,00057 cd/m2. The natural brightness without light pollution (literature)  is assumed to be 21.75 mag/arcsecequals  0,00022 cd/m2. So the measured values where first converted to a linear range in  cd/m2 , then corrected with  - (0,00057 - 0,00022)  cd/m2 and then converted back to the SQM logaritmic range in mag/arcsec2..

Since I don't have access to a true dark site, I have added value (pink) from the following ESO graph:

Comparison between the zenith twilight-sky brightness measured at Paranal (small symbols) and the Crimean Astrophysical Observatory (large symbols) for the V passband.

This result in the following graph combining my (blue) measurments with pink ESO values:

The both lines match well. From the curves it possible to find an approximation formula:

For Sun elevation 0° up to -12°, SQM value ≈ -1.057x + 6.7489
For range - 12° up to -18° the SQM value ≈  -0,0744x
2 - 2,5768x - 0,5845

The measurements are depending on many factors such as air transparancy so are an indication only. More measurments are required to get a higher accuracy.

Date: 2017-6-28
email:i Han