sky brightness and effects of twilight

• see also:
  • astronomic "seeing"
  • light pollution
  • sky transparency and atmospheric extinction

• sky brightness refers to the visual perception of the sky and how it scatters and diffuses light.
• the background brightness of the night sky is an important factor in how well we can see celestial objects such as planets, stars, meteors and comets

• object visibility function = (object's extra-atmospheric luminance) exp(-extinction coeff x extinction function) / (twilight sky luminance + night sky luminance)¹⁾
• extended object's contrast index = -0.4 x (object mpas - sky brightness SQM)²⁾
  • where,
    • object mpas = objMagnitude + (2.5 * (Log10( (PI() / 4) * objMajorAxis_arcsec * objMinorAxis_arcsec) ) )
• the main factors of whether an object will be visible on a clear night are:
  • the brightness of the object as viewed beyond earth's atmosphere
    • this depends also on apparent size and colour of the object
  • the optical extinction due to earth's atmosphere
    • dependent upon:
      • the object's elevation above the horizon (the higher the better) - see sky transparency and atmospheric extinction
      • sky transparency - see sky transparency and atmospheric extinction
  • the limiting night sky brightness (if the sky is brighter than the comet, you will not see it):
    • light pollution
    • airglow (65% of sky brightness at zenith at a dark sky site)
      • occurs all over the Earth
      • at equatorial latitudes, red airglow dominates from oxygen emission 100 to 300 km high.
      • at high latitudes north and south, green airglow dominates from oxygen emission 90 to 100 km high
      • at mid latitudes both red and green airglow is commonly seen
    • zodiacal light (27% of sky brightness at zenith at a dark sky site)
      • intensity of zodiacal light depends on the ecliptic latitude and longitude of the point in the sky being observed relative to that of the Sun
    • starlight (7% of sky brightness at zenith at a dark sky site)
      • eg. Milky Way
      • stars up to V magnitude 16 contribute to the diffuse scattered starlight
      • total brightness of stars reaching earth was calculated to be equivalent to that of 2,000 first-magnitude stars ³⁾
    • moon light
    • (see also light pollution)
  • if near twilight, the contours of background sky brightness during twilight
    • this depends upon:
      • depth of sun below the horizon
      • azimuth angle between sun and the comet
      • elevation of comet above horizon
  • for vision of faint objects in dark skies other factors become important:
    • visual dark adaptation
    • averted vision technique to utilise the rods instead of cones in your retina
    • visual aids such as binoculars or telescopes as these provide:
      • greater light gathering power
        • this vastly increases visibility with the limiting magnitude of a visible object increasing to 10 with a 50mm aperture optic and to 13 with a 200mm diameter telescope
      • greater threshold contrast (ability to see an object whose brightness is near the background brightness)
        • an 8“ aperture telescope has nearly 200x better threshold contrast than the naked eye
    • see astronomic viewing conditions
• a separate issue, is how well the object can be seen through a telescope
  • this largely depends upon how stable the atmosphere is - see astronomic "seeing"

• luminance quantifies the brightness of a surface or light source in a specific direction, measuring luminous intensity per unit projected area

• astronomers use a brightness scale called magnitudes which is a logarithmic scale where each unit change in magnitude is a change in brightness of 2.51189 (the fifth root of 100)
  • the brightest star in the night sky is Sirius at magnitude -1.4. Venus reaches -4.4. The full Moon is about -12.5, and the Sun is -26.7 while the dimmest visual stars at dark sites are around 7.5 magnitude
  • there are about 1.33 photographic “stops” per magnitude (each photographic stop is a twice or half the exposure of the previous stop)
• astronomers describe surface brightness of an extended object in terms of stellar magnitude in an angular area, and commonly use magnitudes per square arc-second
  • if an object is large enough and shows enough contrast with a background, we can perceive surface brightness significantly fainter than 24 magnitudes/square arc-second
  • the surface brightness where the transition between cone dominant (color, photopic vision) and rod dominant (scotopic vision) is between about 19 to 20 magnitudes/square arc-second, dependent on contrast with the background and apparent size of the object. ⁴⁾

• the S10 unit is defined as the surface brightness of a star whose V-magnitude is 10 and whose light is smeared over one square degree, or 27.78 mag / arcsec²
• the total sky brightness in zenith at a dark sky site is therefore ~220 S10 or 21.9 mag/arcsec² in the V-band made up of:
  • air glow is usually around 145 S10
  • zodiacal light is usually around 60 S10
  • starlight is usually around 15 S10

• cd/m² = 10.8×10⁴ × 10^{(-0.4*[value in mag/arcsec²])}
• SQM in mag/arcsec² = Log10([value in cd/m²]/108000)/-0.4

• NELM = 7.93-5*log(10^{(4.316-(SQM/5))+1)5)}

• this is a hand held tool which is pointed at the zenith and uses a silicon photodiode with a filter sensitive to 390–600 nm wavelengths to detect luminance from the night sky. They are much too sensitive for twilight readings unless ND filters are used.
  • eg. Unihedron SQM L
• it converts light to a frequency signal, outputting values in magnitudes per square arcsecond (mag/arcsec²) - the higher the value, the darker the sky
• maximum SQM without light pollution is assumed to be 22 mag/arcsec² at the zenith (highest point in the sky) and this is limited by natural air glow, etc.
  • the max SQM also depends upon the wavelengths of light being detected
    • for Johnson U (blue) it is higher and R (red, and I (infrared) it is lower.

• digital cameras allow simultaneous measurements of large areas of the sky however they have their own issues such as different color spaces to what astronomers use
• makes qualitative assessments easy but accurate quantitative measurements require calibration and adjustment for lens vignetting - each camera will differ
  • calibrating only against known stars seems to get only about 10% to 20% repeatability
  • calibrating against a SQM is far more reliably accurate
• see also:
  • https://www.skyandtelescope.com/wp-content/uploads/SkyglowDigitalCameras.pdf
  • https://clarkvision.com/articles/color-vision-at-night/

• the sun obviously also causes major effects on sky brightness before sunrise and after sunset
• during this time, yellow emissions from the sodium layer and red emissions from the 630 nm oxygen lines are dominant, and contribute to the purplish color sometimes seen during civil and nautical twilight
• if there is any sun above the horizon, only objects with magnitude brighter than about minus 2 (eg. Venus, Jupiter, moon, occasionally Mars at opposition) can be seen - albeit with difficulty in daylight
• remember the sun moves 15deg per hour due to earth's rotation but rate that it disappears depends upon latitude from the equator and the sun's declination (ie. season)
• the sun through the year follows the ecliptic which is tipped 23½° with respect to the celestial equator (the same plane as the equator on Earth).
• depending on the season and the observer’s latitude, the Sun may appear to set at a steep or a shallow angle relative to the horizon.
• at latitude more than 68° north or south, depending on the season, the Sun may not rise or set at all.
• Twilight Stages:
  • Civil Twilight (Sun 0° to 6° down):
    • only the brightest stars (magnitude 0 or brighter, like Sirius, Vega, Arcturus) and planets (Venus, Jupiter) are visible.
  • Nautical Twilight (Sun 6° to 12° down):
    • 1st magnitude stars appear as the sky gets darker.
  • Astronomical Twilight (Sun 12° to 18° down):
    • by the end of this period, 6th magnitude stars (naked-eye limit) become visible, particularly at the zenith.

• SQM rises in an almost linear fashion as the sun drops below the horizon up until 12deg below horizon then increase in darkness tapers off :⁶⁾

for Sun elevation (x) 0° up to -12°, the zenith SQM value ≈ -1.057x + 6.7489

for Sun elevation (x) 12° up to -18° the zenith SQM value ≈ -0.0744x² - 2.5768x - 0.5845

• object visibility function = (object's extra-atmospheric luminance) exp(-extinction coeff x extinction function) / (twilight sky luminance + night sky luminance)⁷⁾
• at 60min before sunrise a star near the sun would need to be brighter than mag 4.5 at 4deg above the horizon to be visible in standard photos, hence comets less bright would not be visible closer to the horizon and their tails which are usually less bright again would be washed out by twilight even higher above the horizon
• eg. a comet mag 4.7 at ~14deg vertical elevation from the sun and 25deg elongation from the sun will be about 3deg above horizon 1hr before sunrise and thus will not be visible due to the sky glow as it is too close to the sun for that brightness
• at sunset, if the object if high enough above the horizon, and bright enough, it may become visible for some time as the twilight darkens just before the comet sets and atmospheric extinction obliterates it
• before sunrise, if the object is bright enough and rises high enough to get past atmospheric extinction effects before twilight impacts, it may be visible for a short period until twilight obliterates it
• latitude may make all the difference
  • for a comet north of the ecliptic and close to the sun, a northern latitude means the comet will have much higher elevation and hence more visibility in twilight
  • for example comet C/2025 R3 Panstarrs on just before sunrise on 16th April 2026:
    • declination +18deg with 28.6deg elongation from the sun which was at declination +10deg
    • at sunrise: in Melbourne, latitude 37.8degS, in was only 13deg above the horizon and invisible prior to sunrise due to twilight sky glow, while in Cairns, Australia, latitude 17degSth it was 21deg above the horizon making it an easier target, although even better further north (Rome at latitude 42degNth it was 25deg above the horizon, at Singapore on the equator, it was ~26deg above the horizon)