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:
Civil twilight – when the Sun
is below the horizon, but the Sun's centre is less then 6°
below the
true horizon.
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.
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::
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/arcsec2 equals
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.
http://www.eso.org/~fpatat/science/skybright/twilight.pdf
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 ≈ -0,0744x2 - 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.
