Light, radar and VHF range

Introduction

When will a lighthouse rise above the horizon, and will it be bright enough to see in tonight's visibility? How far away can the radar see a low coaster, and how far does VHF reach between two antennas? All of these are horizon and attenuation questions. This calculator answers them together: the geographic range of a light from its elevation and the height of eye, its luminous range from its intensity or nominal range and the meteorological visibility, the radar and VHF line-of-sight ranges, and the minimum intensity COLREG requires of a navigation light for a given range.

It is for navigating officers planning landfalls, cadets learning light ranges, and anyone specifying or checking navigation lights.

Formulas

Horizon (Bowditch Table 12):   d = √(2 R h / β),   R = 3440.1 nmi, β = 0.8279   →   1.17 √h(ft) = 2.12 √h(m)
Rounded rule:                  d = 2.08 √h(m);    geometric: d = √(2 R h) = 1.93 √h(m)
Geographic range:              D_geo = d(H) + d(h)
Allard's law:                  E = I · T^d / (1852 d)²,   T = 0.05^(1/V),   E_threshold = 2·10⁻⁷ lx at night
Nominal range:                 luminous range at V = 10 nmi (T = 0.741)
Visible range:                 min(luminous range, geographic range)
Radar and VHF horizon:         d = √(2 · 4/3 · R h) = 2.23 √h(m)
COLREG Annex I minimum:        I = 3.43·10⁶ × 2·10⁻⁷ × D² × 0.8^(−D)  cd

• H — elevation of the light; h — height of eye or antenna; I — luminous intensity in candela; V — meteorological visibility; d, D — distances in nautical miles.

Assumptions and limitations

• Standard refraction; abnormal refraction over cold or warm water changes all horizon ranges.
• Charted light elevations are above a high-water datum; the actual height depends on the tide.
• Allard's law is for a point source at night against a dark background; shore lights raise the threshold.
• Radar and VHF figures are line-of-sight only; detection and reception also depend on power, antenna gain and target size.
• The COLREG intensity uses the regulation's own transmissivity, K = 0.8.

Worked example

A light with a nominal range of 20 nmi stands 40 m above the sea. The observer's eye is 15 m above the water and the visibility is 5 nmi.

1. Horizon from the light: 2.12 √40 = 13.4 nmi; from the eye: 2.12 √15 = 8.2 nmi; geographic range 21.6 nmi.
2. Intensity from the nominal range: I = 2·10⁻⁷ × (1852 × 20)² × 0.741^(−20) ≈ 110 000 cd.
3. In 5 nmi visibility the transmissivity per mile is 0.05^(1/5) = 0.549, and Allard's law gives a luminous range of 11.8 nmi.
4. The light will be seen at about 11.8 nmi, limited by visibility rather than by the horizon. In clear weather it would rise at 21.6 nmi.

FAQ

Related calculators

• CPA and TCPA from radar plots — what to do once the target is detected.
• Air draft and bridge clearance — the other height question on the bridge.
• Tidal height — the state of tide that changes a light's elevation.
• Great-circle distance — the distance to the landfall.