Zone scale

Definition#

A zone is a band of artificial sky brightness relative to the natural sky, at the zenith, on a clear night with a standard atmosphere and no moon. With $L_{\text{art}}$ the zenith luminance added by artificial light and $L_{\text{nat}}$ the natural zenith luminance:

$r = \frac{L_{\text{art}}}{L_{\text{nat}}}, \qquad L_{\text{nat}} = 174\ \mu\text{cd/m}^2 \;\equiv\; 22.0\ \text{mag/arcsec}^2$

The value 174 µcd/m² for a natural sky is from Falchi et al. (2016). Zero means a natural sky, and one means artificial light equals the natural background.

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.

The ladder#

Zones are a doubling ladder that starts at $r = 0.32$. Each edge is twice the one before it, and a value exactly on an edge belongs to the higher zone.

$\text{zone}(r) = \begin{cases} 1 & r < 0.32 \\ 2 + \lfloor \log_2 (r / 0.32) \rfloor & 0.32 \le r < 40.96 \\ 9 & r \ge 40.96 \end{cases}$

Implementations compare against the list of edges instead of computing a logarithm, so a value on an edge cannot land on the wrong side through rounding.

The nine visibility cards, as the app shows them: the zone, a sky label and the sky brightness at the middle of the zone, with five rating bars.

A doubling is 0.753 magnitudes, so adjacent zones differ by about three quarters of a magnitude of sky brightness.

Why this ladder#

1. It reproduces the production data. Converting the radiance thresholds that built the current zones.db into ratios gives edges within 0.87 to 1.25 times the ladder (table below).
2. It matches a published scale. The Falchi et al. (2016) atlas classes artificial light as a ratio to natural: below 0.01, 0.01 to 0.02, 0.02 to 0.04, 0.04 to 0.08, 0.08 to 0.16, 0.16 to 0.32, with the Milky Way lost at about 1.28 to 2.56 and a top class above about 41 (over 7130 µcd/m²). The doubling between the quoted values is inferred, so confirm it against the paper's legend (unverified).
3. The zones land on things you can see. Zone 4 is the band where the Milky Way fades, and zone 9 is the top class. Zone 1 merges the six darkest published classes, because below $r = 0.32$ the sky is within a third of natural and the differences are not usable.

Zones are not Bortle classes, and the scale does not use Bortle terms.

The earlier specification (version 1.0) used a factor of 2.5 per zone from 0.05 and divided radiance by 0.171. That constant is a luminance in mcd/m², not a satellite radiance in nW/cm²/sr, so its edges did not match the production data.

Conversions#

$\text{SQM} = 22.0 - 2.5\log_{10}(1 + r) \qquad r = 10^{0.4\,(22.0 - \text{SQM})} - 1 \qquad L_{\text{art}} = 174\,r$

$\text{NELM} = 7.93 - 5\log_{10}\!\left(10^{\,4.316 - \text{SQM}/5} + 1\right)$

NELM is the naked-eye limiting magnitude, an empirical fit between sky-meter readings and observers' limiting magnitudes (Unihedron and AAVSO). It is informative: the zone definition does not depend on it. A real sky meter includes starlight above the 22.0 baseline, so dark-site readings sit a little brighter than 22.0 and must be adjusted before comparison.

Try it#

Edge cases#

Current data and the legacy chain#

The current zones.db was built from radiance $R$ in nW/cm²/sr (VIIRS light plus modelled skyglow, see Light pollution data) with a fit and thresholds. This is the legacy chain:

$\text{SQM} = 22.0 - 1.7\log_{10}(1 + 2R) \quad\Rightarrow\quad r = (1 + 2R)^{0.68} - 1$

After calibration the chain becomes $L_{\text{art}} = k\,R_{\text{eff}}$ with $k$ fitted to ground measurements, and the zone is computed from $r = L_{\text{art}} / 174$. Until then, the best available zone is the ladder applied to the legacy ratio.

What changes#

Adopting the ladder on the current data, measured over every stored cell:

• 86.4% of 37,528,537 stored cells keep their zone.
• 13.6% (5,115,892) move by exactly one zone: 2,059,079 up and 3,056,813 down. None moves by more.

Only the 25-city validation set's Anchorage changes zone (8 to 9, at a radiance of 124.66, right on the edge). This measures how the definition changes today's data. It does not measure accuracy, which is the next section.

Names to fix#

The stored record is the H3 index, zone, radiance and sky brightness in 20 bytes, sorted by H3 index behind a 16-byte header, and only cells in zone 2 or above are stored.

Calibration and validation#

The goal is to replace the legacy fit with a physical chain whose error is measured against ground data and published.

• Data. Globe at Night (CC BY 4.0) and other openly licensed sky-meter networks. The Falchi atlas (CC BY-NC) and the Lorenz atlas are comparison references only, never used to fit a parameter. Use only readings taken in astronomical darkness, moonless and clear, and subtract a natural baseline so a pristine site does not read as artificially lit.
• Parameters to fit. The scale factor $k$ and the kernel's fraction, scale length, power and reference distance. Test whether the 80 km kernel radius truncates real skyglow, since Falchi integrates to 195 km.
• Split by region, not by station, because nearby stations see the same radiance field and would leak.
• Metrics per stratum and overall: bias, mean error and root-mean-square error in mag/arcsec², the zone confusion matrix, the share within one zone, and the station count.
• Adoption rule. The calibrated chain replaces the legacy chain only if its held-out error beats the legacy baseline by a stated margin. Proposed targets, to be set after the baseline is measured: bias within 0.10 mag/arcsec² and error within 0.30. For reference, Falchi reports a residual of 0.15 for the atlas.
• Failure tags. Every failing station gets a cause: aurora or polar noise, equatorial cloud, snow, altitude, local lighting or no data. Reykjavik, Singapore and Ushuaia in the 25-city check already fail this way.

Model issues to test, found by reading the code: the kernel uses a fixed 5.55 km pixel, so its east-west footprint is distorted away from the equator; scatter inside the coarse pixel is excluded; the radius stops at 80 km; no elevation, mountain screening or aerosol variation is modelled; VIIRS is blind below 500 nm, so white LEDs are under-counted; and clouds can amplify skyglow near cities up to ten times (Falchi 2016).

Implementation#

These are the real files, copied into the site and checked against the repository on every build.

1"""Astr Zone scale v2.0 reference implementation.23Spec: docs/astr_zone_scale.md. Test vectors: test/fixtures/astr_zone_scale.vectors.json.4The Dart implementation (lib/core/utils/astr_zone_scale.dart) must give the same results.56r is the ratio of artificial to natural zenith sky brightness, L_artificial / L_natural.7Zone 1 is r < 0.32. Zone n (2 to 8) is [0.32 * 2^(n-2), 0.32 * 2^(n-1)). Zone 9 is r >= 40.96.8Boundaries are lower-inclusive.9"""1011import math1213NATURAL_UCD_PER_M2 = 174.0   # natural zenith brightness, microcandela per m^2 (Falchi et al. 2016)14REFERENCE_SQM = 22.0         # mag/arcsec^2 that corresponds to the natural brightness above15FIRST_EDGE = 0.32            # r at which zone 2 begins16STEP = 2.0                   # each zone edge is twice the previous one17ZONE_COUNT = 918EDGES = tuple(FIRST_EDGE * STEP ** k for k in range(ZONE_COUNT - 1))  # start of zones 2..91920# Legacy v2 chain (interim, used to build the current zones.db). Radiance in nW/cm^2/sr.21LEGACY_THRESHOLDS = ((125.0, 9), (50.0, 8), (20.0, 7), (9.0, 6), (3.0, 5), (1.0, 4), (0.50, 3), (0.25, 2))22LEGACY_SQM_SLOPE = 1.723LEGACY_SQM_GAIN = 2.0242526def _check_ratio(r):27    if isinstance(r, bool) or not isinstance(r, (int, float)) or math.isnan(r):28        raise ValueError(f"r must be a number, got {r!r}")293031def zone_from_ratio(r):32    """Zone 1-9 for an artificial/natural ratio. Negative r counts as 0. NaN raises ValueError."""33    _check_ratio(r)34    zone = 135    for edge in EDGES:36        if r >= edge:37            zone += 138        else:39            break40    return zone414243def sqm_from_ratio(r):44    """Zenith sky brightness in mag/arcsec^2 for a ratio r (r < 0 counts as 0)."""45    _check_ratio(r)46    return REFERENCE_SQM - 2.5 * math.log10(1.0 + max(r, 0.0))474849def ratio_from_sqm(sqm):50    """Inverse of sqm_from_ratio. An SQM above the reference gives 0."""51    if math.isnan(sqm):52        raise ValueError("sqm must be a number")53    return max(10.0 ** (0.4 * (REFERENCE_SQM - sqm)) - 1.0, 0.0)545556def artificial_ucd_from_ratio(r):57    """Artificial zenith luminance in microcandela per m^2."""58    _check_ratio(r)59    return max(r, 0.0) * NATURAL_UCD_PER_M2606162def nelm_from_sqm(sqm):63    """Naked-eye limiting magnitude from an SQM value (empirical, Unihedron/AAVSO relation)."""64    return 7.93 - 5.0 * math.log10(10.0 ** (4.316 - sqm / 5.0) + 1.0)656667def legacy_zone_from_radiance(radiance):68    """Zone assigned by the production pipeline's radiance thresholds."""69    if radiance <= 0:70        return 171    for threshold, zone in LEGACY_THRESHOLDS:72        if radiance >= threshold:73            return zone74    return 1757677def legacy_ratio_from_radiance(radiance):78    """r implied by the legacy SQM fit 22 - 1.7 log10(1 + 2R): r = (1 + 2R)^0.68 - 1."""79    if radiance <= 0:80        return 0.081    return (1.0 + LEGACY_SQM_GAIN * radiance) ** (0.4 * LEGACY_SQM_SLOPE) - 1.0

1import 'dart:math' as math;23/// Astr Zone scale v2.0.4///5/// Spec: `docs/astr_zone_scale.md`. Test vectors: `test/fixtures/astr_zone_scale.vectors.json`.6/// The Python reference (`scripts/astr_zone.py`) must give the same results.7///8/// `r` is the ratio of artificial to natural zenith sky brightness. Zone 1 is9/// `r < 0.32`. Zone n (2 to 8) is `[0.32 * 2^(n-2), 0.32 * 2^(n-1))`. Zone 9 is10/// `r >= 40.96`. Boundaries are lower-inclusive.11abstract final class AstrZoneScale {12  /// Natural zenith brightness in microcandela per square metre (Falchi et al. 2016).13  static const double naturalMicroCdPerM2 = 174;1415  /// Sky brightness in mag/arcsec² that corresponds to the natural brightness.16  static const double referenceSqm = 22;1718  /// The ratio at which zone 2 begins.19  static const double firstEdge = 0.32;2021  /// Each zone edge is this multiple of the previous one.22  static const double step = 2;2324  static const int zoneCount = 9;2526  /// The ratio at which zones 2 to 9 begin.27  static const List<double> edges = <double>[28    0.32, 0.64, 1.28, 2.56, 5.12, 10.24, 20.48, 40.96, //29  ];3031  /// Legacy radiance thresholds (nW/cm²/sr) of the production pipeline, highest first.32  static const List<(double, int)> legacyThresholds = <(double, int)>[33    (125, 9), (50, 8), (20, 7), (9, 6), (3, 5), (1, 4), (0.5, 3), (0.25, 2), //34  ];3536  /// Zone 1-9 for a ratio. A negative [r] counts as 0. Throws if [r] is NaN.37  static int zoneFromRatio(double r) {38    _checkNumber(r, 'r');39    int zone = 1;40    for (final double edge in edges) {41      if (r >= edge) {42        zone++;43      } else {44        break;45      }46    }47    return zone;48  }4950  /// Zenith sky brightness in mag/arcsec² for a ratio (a negative [r] counts as 0).51  static double sqmFromRatio(double r) {52    _checkNumber(r, 'r');53    return referenceSqm - 2.5 * _log10(1 + math.max(r, 0));54  }5556  /// Inverse of [sqmFromRatio]. An SQM above the reference gives 0.57  static double ratioFromSqm(double sqm) {58    _checkNumber(sqm, 'sqm');59    return math.max(math.pow(10, 0.4 * (referenceSqm - sqm)).toDouble() - 1, 0);60  }6162  /// Artificial zenith luminance in microcandela per square metre.63  static double artificialMicroCdFromRatio(double r) {64    _checkNumber(r, 'r');65    return math.max(r, 0) * naturalMicroCdPerM2;66  }6768  /// Naked-eye limiting magnitude from an SQM value (empirical, Unihedron/AAVSO relation).69  static double nelmFromSqm(double sqm) {70    return 7.93 - 5 * _log10(math.pow(10, 4.316 - sqm / 5).toDouble() + 1);71  }7273  /// Zone assigned by the production pipeline's radiance thresholds.74  static int legacyZoneFromRadiance(double radiance) {75    if (radiance <= 0) return 1;76    for (final (double, int) entry in legacyThresholds) {77      if (radiance >= entry.$1) return entry.$2;78    }79    return 1;80  }8182  /// The ratio implied by the legacy SQM fit `22 - 1.7 log10(1 + 2R)`: `(1 + 2R)^0.68 - 1`.83  static double legacyRatioFromRadiance(double radiance) {84    if (radiance <= 0) return 0;85    return math.pow(1 + 2 * radiance, 0.4 * 1.7).toDouble() - 1;86  }8788  static double _log10(double x) => math.log(x) / math.ln10;8990  static void _checkNumber(double value, String name) {91    if (value.isNaN) {92      throw ArgumentError.value(value, name, 'must be a number');93    }94  }95}