Moonlight and sky brightness · Astr design system

One unit for everything

Light adds as luminance, not as magnitudes. So every contribution is expressed in natural units, where 1.0 is the natural zenith sky brightness (22.0 mag/arcsec², 174 µcd/m²), and the sum is converted afterwards.

Symbol	Meaning	Unit
$r$	artificial brightness over natural brightness (the zone quantity, see Zone scale)	none
$b_{\text{moon}}$	scattered moonlight over natural brightness	none
$1 + r + b_{\text{moon}}$	total brightness, in natural units	none
$\text{SQM}_{\text{eff}}$	effective sky brightness with the moon added	mag/arcsec²
$\text{NELM}_{\text{eff}}$	the naked-eye limiting magnitude that follows from it	magnitude
$\alpha$	moon phase angle, 0 at full moon and 180 at new	degrees
$\rho$	angle between the moon and the part of the sky looked at	degrees
$Z,\ Z_m$	zenith distance of the sky position, and of the moon	degrees
$k_V$	extinction coefficient in the V band: how much the air dims light	mag per airmass
$\tau_{550}$	aerosol optical depth at 550 nm	none
$P / P_0$	station pressure over sea-level pressure	none
$i$	illuminated fraction of the moon, 0 to 1	none

$\text{SQM}_{\text{eff}} = 22.0 - 2.5\log_{10}(1 + r + b_{\text{moon}}) \qquad \text{NELM}_{\text{eff}} = 7.93 - 5\log_{10}\!\left(10^{\,4.316 - \text{SQM}_{\text{eff}}/5} + 1\right)$

SQM is the brightness of the sky in magnitudes per square arcsecond. NELM is the naked-eye limiting magnitude, the faintest star an observer can see, from an empirical fit between sky-meter readings and observers' reports (Unihedron and AAVSO).

Every number below is computed by a TypeScript port that was checked at build time against 162 shared test cases, the same ones the Python and Dart implementations pass.

Moonlight

Scattered moonlight follows Krisciunas and Schaefer (1991), "A model of the brightness of moonlight" (PASP 103, 1033). It is an empirical fit to 33 photometric measurements in the V band, and its authors report agreement with data at the 8% to 23% level. The formulas were checked against the published reference implementation in the specsim package.

Let $\alpha$ be the phase angle in degrees (0 at full moon, 180 at new), $\rho$ the angle between the moon and the sky position, $Z$ and $Z_m$ the zenith distances of the sky position and the moon, and $k$ the V-band extinction coefficient in magnitudes per airmass.

$m = -12.73 + 0.026\,|\alpha| + 4\times10^{-9}\alpha^{4} \qquad I^{*} = 10^{-0.4\,(m + 16.57)}$

$f(\rho) = 10^{5.36}\,(1.06 + \cos^{2}\rho) + 10^{\,6.15 - \rho/40} \qquad X(Z) = \left(1 - 0.96\sin^{2}Z\right)^{-1/2}$

$B_{\text{moon}} = f(\rho)\,I^{*}\,10^{-0.4\,k\,X(Z_m)}\left(1 - 10^{-0.4\,k\,X(Z)}\right)\ \text{nL} \qquad V = \frac{20.7233 - \ln(B_{\text{moon}}/34.08)}{0.92104}$

$b_{\text{moon}} = 10^{\,0.4\,(22.0 - V)} \quad (0\text{ when } Z_m \ge 90^\circ) \qquad \alpha = \arccos(2i - 1)$

Here $i$ is the illuminated fraction of the moon from the ephemeris. The last conversion, from nanolamberts to magnitudes, is Garstang (1986), eq. 19.

An anchor value. The reference implementation documents one worked value: $Z = 20^\circ$ , $Z_m = 70^\circ$ , $\rho = 50^\circ$ , $\alpha = 45^\circ$ and $k = 0.15$ give 19.855 mag/arcsec². The Python, Dart and TypeScript implementations all reproduce it to 0.0005, and this site's build fails if they do not.

Which part of the sky. For the verdict on the sky as a whole, use the zenith: $Z = 0$ and $\rho = Z_m$ . For one object's visibility, use the object's direction, with $\rho$ the angle between the moon and the object computed from their altitudes and azimuths.

Extinction

The air dims light on its way through. The moonlight model needs the V-band extinction coefficient, built from three parts:

$k_V = 0.1057\,\frac{P}{P_0} + 0.016 + 1.0857\,\tau_{550}$

The first term is Rayleigh scattering scaled by station pressure (from the international standard atmosphere at the station's elevation), the second is ozone, and the third is aerosol, where $\tau_{550}$ is the aerosol optical depth at 550 nm and 1.0857 is $2.5\log_{10}e$ , which converts an optical depth to magnitudes. With no aerosol data, use $\tau = 0.05$ (a proposal).

The measured median V-band extinction at the Roque de los Muchachos observatory (2400 m) on clear nights is 0.130 mag per airmass (IAC), against 0.127 from this formula.
Case	AOD at 550 nm	Elevation m	P / P0	k_V
Sea level, clean	0.02	0	1.000	0.143
Sea level, hazy	0.1	0	1.000	0.230
Sea level, polluted	0.3	0	1.000	0.447
1000 m, typical	0.05	1000	0.887	0.164
2400 m, clear (La Palma)	0.03	2400	0.746	0.127
4000 m, clear	0.02	4000	0.608	0.102

The two constants are not individually verified (unverified). As a check on the whole formula, a site at 2400 m with an optical depth of 0.03 gives 0.127, against a measured median of 0.130 for the Roque de los Muchachos observatory on clear nights.

Darkness

An hour is dark when the Sun is at or below −18°. Between sunset and that point the sky is still lit by the Sun, so there is no verdict and the app shows twilight or day. How bright the sky is during twilight is measured by Patat et al. (2006) at Paranal, and modelling it is future work.

Worked examples

A site of each zone under a range of moons, computed with the functions above.

Cloud 10%, extinction k = 0.15. Each cell is the state, then the effective sky brightness in mag/arcsec² and the limiting magnitude. Moon positions are altitudes above the horizon.
Site	No moon	10% crescent, 40°	Half, 40°	Full, 20°	Full, 40°	Full, 70°
Zone 1 (r = 0.1)	Milky Way (21.9, 6.6)	Milky Way (21.7, 6.5)	Starry (20.7, 6.0)	Planets (19.0, 4.8)	Few stars (18.5, 4.4)	Few stars (17.7, 3.7)
Zone 3 (r = 0.9)	Milky Way (21.3, 6.3)	Milky Way (21.2, 6.2)	Starry (20.5, 5.8)	Planets (19.0, 4.7)	Few stars (18.5, 4.4)	Few stars (17.6, 3.7)
Zone 5 (r = 3)	Starry (20.5, 5.8)	Starry (20.5, 5.8)	Planets (20.0, 5.5)	Planets (18.8, 4.6)	Few stars (18.4, 4.3)	Few stars (17.6, 3.6)
Zone 7 (r = 12)	Planets (19.2, 4.9)	Planets (19.2, 4.9)	Planets (19.0, 4.8)	Few stars (18.4, 4.3)	Few stars (18.1, 4.0)	Few stars (17.4, 3.5)
Zone 8 (r = 30)	Few stars (18.3, 4.2)	Few stars (18.3, 4.2)	Few stars (18.2, 4.1)	Few stars (17.9, 3.8)	Few stars (17.7, 3.7)	Few stars (17.2, 3.3)
Zone 9 (r = 60)	Too much light (17.5, 3.6)	Too much light (17.5, 3.6)	Too much light (17.5, 3.5)	Too much light (17.3, 3.4)	Too much light (17.2, 3.3)	Too much light (16.9, 3.0)

Moon altitude: 45°
Full moon (solid): 18.4 · NELM 4.3
Half moon (dash): 20.6 · NELM 5.9
10% crescent (dot): 21.7 · NELM 6.5

Effective sky brightness at the zenith as the moon climbs, with extinction k = 0.15, for a site in zone 1 that starts at 21.9 mag/arcsec² with no moon. Dashed lines mark where the state changes. The model has the moon at nearly full strength as soon as it clears the horizon, so the curves start low. Move over the chart to read the values.

A high full moon takes a pristine site to about the limiting magnitude of a zone 8 city, which matches the experience that a full moon wipes out the Milky Way. In a bright city the moon barely matters.

Try it

Artificial ratio r0.100 (zone 1)Cloud cover10%Moon altitude40°Moon illuminated50%Aerosol optical depth0.05Elevation0 m

State: Starry sky
Limiting factor: The moon
Extinction k_V: 0.176
Moonlight: 2.39 × natural
Sky brightness: 20.64 mag/arcsec²
Limiting magnitude: 5.91
Light pollution cost: 0.05 mag
Moon cost: 0.67 mag

Zenith sky at astronomical night. The state and the cause come from the same functions the app uses.

Limits

Twilight brightness is not modelled; the model is gated at −18°.
Moonlight is modelled in the V band, while a sky meter is broader.
Moonlight reflected from clouds, airglow variation, the zodiacal light and the observer's altitude are ignored.
The artificial ratio $r$ comes from satellite data, so its error carries through.
The Rayleigh and ozone constants are not individually verified.

Validation plan

Moonlight. Compare the effective sky brightness with Globe at Night and sky-meter network readings taken with the moon up (the data record date and time, so the moon can be computed), in dark and suburban strata. Report bias and error in mag/arcsec².
Limiting magnitude. Compare the model's limiting magnitude with observers' reports for the same nights.
Cloud. Compare forecast cover with satellite cloud masks for the sampled nights.

Implementation

Excerpts of the real files, cut out by name, copied into the site and checked against the repository on every build.

astr_sky.py · pressure_ratio

1def pressure_ratio(elevation_m):2    """Station pressure over sea-level pressure, international standard atmosphere (valid below 11 km)."""3    return (1.0 - 2.25577e-5 * elevation_m) ** 5.25588

astr_sky.py · extinction_k_v

1def extinction_k_v(aod550=0.0, elevation_m=0.0):2    """V-band extinction coefficient: Rayleigh scaled by pressure, ozone, and aerosol optical depth."""3    return RAYLEIGH_K_SEA_LEVEL * pressure_ratio(elevation_m) + OZONE_K + TAU_TO_K * max(aod550, 0.0)

astr_sky.py · scattering_airmass

1def scattering_airmass(zenith_deg):2    """Krisciunas & Schaefer (1991) eq. 3."""3    s = math.sin(math.radians(zenith_deg))4    return (1.0 - 0.96 * s * s) ** -0.5

astr_sky.py · phase_angle_from_illumination

1def phase_angle_from_illumination(illumination):2    """Moon phase angle in degrees (0 = full, 180 = new) from the illuminated fraction 0..1."""3    return math.degrees(math.acos(max(-1.0, min(1.0, 2.0 * illumination - 1.0))))

astr_sky.py · moon_brightness_v

1def moon_brightness_v(obs_zenith_deg, moon_zenith_deg, separation_deg, phase_angle_deg, k_v):2    """Scattered moonlight in V mag/arcsec^2 (Krisciunas & Schaefer 1991). math.inf when the moon is down."""3    if moon_zenith_deg >= 90.0:4        return math.inf5    alpha = abs(phase_angle_deg)6    m = -12.73 + 0.026 * alpha + 4e-9 * alpha ** 4                                     # eq. 97    i_star = 10.0 ** (-0.4 * (m + 16.57))                                              # eq. 88    rho = separation_deg9    f = 10.0 ** 5.36 * (1.06 + math.cos(math.radians(rho)) ** 2) + 10.0 ** (6.15 - rho / 40.0)  # eq. 2110    x_obs = scattering_airmass(obs_zenith_deg)11    x_moon = scattering_airmass(moon_zenith_deg)12    b_nl = f * i_star * 10.0 ** (-0.4 * k_v * x_moon) * (1.0 - 10.0 ** (-0.4 * k_v * x_obs))  # nanoLamberts13    return (20.7233 - math.log(b_nl / 34.08)) / 0.92104                                # Garstang 1986 eq. 19

astr_sky.py · moon_ratio

1def moon_ratio(obs_zenith_deg, moon_zenith_deg, separation_deg, phase_angle_deg, k_v):2    """Scattered moonlight in natural units (0 when the moon is below the horizon)."""3    v = moon_brightness_v(obs_zenith_deg, moon_zenith_deg, separation_deg, phase_angle_deg, k_v)4    if math.isinf(v):5        return 0.06    return 10.0 ** (0.4 * (REFERENCE_SQM - v))

astr_sky.py · angular_separation_deg

1def angular_separation_deg(alt1_deg, az1_deg, alt2_deg, az2_deg):2    """Angle between two directions given as altitude and azimuth in degrees."""3    a1, a2 = math.radians(alt1_deg), math.radians(alt2_deg)4    d_az = math.radians(az1_deg - az2_deg)5    c = math.sin(a1) * math.sin(a2) + math.cos(a1) * math.cos(a2) * math.cos(d_az)6    return math.degrees(math.acos(max(-1.0, min(1.0, c))))

astr_sky.py · sqm_effective

1def sqm_effective(r_art, b_moon=0.0, b_twilight=0.0):2    """Effective sky brightness in mag/arcsec^2: natural + artificial + moon + twilight, in natural units."""3    return REFERENCE_SQM - 2.5 * math.log10(1.0 + max(r_art, 0.0) + max(b_moon, 0.0) + max(b_twilight, 0.0))

astr_sky_model.dart · pressureRatio

1/// Station pressure over sea-level pressure, international standard atmosphere (below 11 km).2static double pressureRatio(double elevationM) =>3    math.pow(1 - 2.25577e-5 * elevationM, 5.25588).toDouble();

astr_sky_model.dart · extinctionKV

1/// V-band extinction coefficient in mag per airmass.2static double extinctionKV({double aod550 = 0, double elevationM = 0}) =>3    _rayleighKSeaLevel * pressureRatio(elevationM) + _ozoneK + _tauToK * math.max(aod550, 0);

astr_sky_model.dart · scatteringAirmass

1/// Krisciunas & Schaefer (1991) eq. 3.2static double scatteringAirmass(double zenithDeg) {3  final double s = math.sin(_rad(zenithDeg));4  return math.pow(1 - 0.96 * s * s, -0.5).toDouble();5}

astr_sky_model.dart · phaseAngleFromIllumination

1/// Moon phase angle in degrees (0 full, 180 new) from the illuminated fraction 0 to 1.2static double phaseAngleFromIllumination(double illumination) =>3    _deg(math.acos((2 * illumination - 1).clamp(-1.0, 1.0)));

astr_sky_model.dart · moonBrightnessV

1/// Scattered moonlight in V mag/arcsec² (Krisciunas & Schaefer 1991). Infinity when the moon is down.2static double moonBrightnessV({3  required double obsZenithDeg,4  required double moonZenithDeg,5  required double separationDeg,6  required double phaseAngleDeg,7  required double kV,8}) {9  if (moonZenithDeg >= 90) return double.infinity;10  final double alpha = phaseAngleDeg.abs();11  final double m = -12.73 + 0.026 * alpha + 4e-9 * math.pow(alpha, 4); // eq. 912  final double iStar = math.pow(10, -0.4 * (m + 16.57)).toDouble(); // eq. 813  final double f = math.pow(10, 5.36) * (1.06 + math.pow(math.cos(_rad(separationDeg)), 2)) +14      math.pow(10, 6.15 - separationDeg / 40); // eq. 2115  final double xObs = scatteringAirmass(obsZenithDeg);16  final double xMoon = scatteringAirmass(moonZenithDeg);17  final double bNl = f *18      iStar *19      math.pow(10, -0.4 * kV * xMoon) *20      (1 - math.pow(10, -0.4 * kV * xObs)); // nanoLamberts21  return (20.7233 - math.log(bNl / 34.08)) / 0.92104; // Garstang 1986 eq. 1922}

astr_sky_model.dart · moonRatio

1/// Scattered moonlight in natural units (0 when the moon is below the horizon).2static double moonRatio({3  required double obsZenithDeg,4  required double moonZenithDeg,5  required double separationDeg,6  required double phaseAngleDeg,7  required double kV,8}) {9  final double v = moonBrightnessV(10    obsZenithDeg: obsZenithDeg,11    moonZenithDeg: moonZenithDeg,12    separationDeg: separationDeg,13    phaseAngleDeg: phaseAngleDeg,14    kV: kV,15  );16  if (v.isInfinite) return 0;17  return math.pow(10, 0.4 * (AstrZoneScale.referenceSqm - v)).toDouble();18}

astr_sky_model.dart · angularSeparationDeg

1/// Angle between two directions given as altitude and azimuth in degrees.2static double angularSeparationDeg(double alt1, double az1, double alt2, double az2) {3  final double a1 = _rad(alt1);4  final double a2 = _rad(alt2);5  final double c = math.sin(a1) * math.sin(a2) + math.cos(a1) * math.cos(a2) * math.cos(_rad(az1 - az2));6  return _deg(math.acos(c.clamp(-1.0, 1.0)));7}

astr_sky_model.dart · sqmEffective

1/// Effective sky brightness in mag/arcsec²: natural + artificial + moon + twilight, in natural units.2static double sqmEffective(double rArt, {double bMoon = 0, double bTwilight = 0}) =>3    AstrZoneScale.referenceSqm -4    2.5 * _log10(1 + math.max(rArt, 0) + math.max(bMoon, 0) + math.max(bTwilight, 0));