A published range is not a promise that every observer will see a lighthouse at that distance. Before using the number, ask which limit it describes: light passing through the atmosphere, or a line of sight passing over the curved Earth. The two questions require different inputs.
Three ranges and the inputs they need
NOAA's Coast Pilot 9, chapter 1, distinguishes nominal, luminous and geographic range. Nominal range is a light's visibility rating at a reference meteorological visibility of 10 nautical miles. Luminous range adjusts the light's visibility to the atmospheric conditions being considered. Neither includes the height of the light, the observer's eye height or the Earth's curvature.
Geographic range addresses that geometry using both the light's elevation and the observer's eye height. It is therefore possible for two people viewing the same light through similar air to have different horizon limits. A tower's height above its foundations is not automatically the elevation of its light above the relevant sea-level datum.
- Nominal: How is this light rated under the reference visibility?
- Useful for comparing a stated lighting performance; it is not a weather forecast.
- Luminous: How far could this light be detected through the air under the assumed visibility?
- Useful for the brightness-and-atmosphere question; it does not remove the horizon.
- Geographic: How far apart could this light and this observer have a line of sight over the horizon?
- Useful for geometry; it does not guarantee that the light is bright enough to detect.
A worked geometry example, with every assumption exposed
The following is our own teaching calculation, not a surveyed station or a navigation table. Model Earth as a sphere of radius 6,371,000 metres, put the light 30 metres above the model sea surface and the observer's eyes 5 metres above it. Ignore atmospheric refraction, waves and intervening land. One nautical mile is 1,852 metres.
The straight-line distance to the tangent point from height h is sqrt((R + h)² − R²), equivalently sqrt(2Rh + h²). Calculate one contribution for the light and another for the observer, then add them.
- Light contribution:
sqrt(2 × 6371000 × 30 + 30²) / 1852 ≈ 10.56 NM. - Observer contribution:
sqrt(2 × 6371000 × 5 + 5²) / 1852 ≈ 4.31 NM. - Combined geometric line-of-sight distance: approximately 14.87 NM.
This uses tangent distances; the corresponding surface arc gives the same rounded result at these heights. Raising the hypothetical eye height to 10 metres changes its contribution to about 6.10 NM, giving 16.65 NM in total. The lamp has not become more powerful: only the observer's geometry has changed. Published geographic-range tables may use atmospheric refraction assumptions that this deliberately simple model excludes.
Combine limits without confusing the units
Now suppose a separate luminous-range assessment for the first scenario gives 12 NM. That is a hypothetical input, not a number derived from the height calculation. The air-and-light limit would be shorter than 14.87 NM. If instead that assessment gave 20 NM, the simple geometric limit would be shorter. Raising the observer would help in the second situation; it would not turn the first situation's 12 NM luminous range into 16.65 NM.
Taking the smaller value is a useful first comparison under matching assumptions. It is not a guarantee of detection. An intervening headland, an obscured sector or distracting shore lights introduces questions the spherical model never asked. Nor should a distance observed when a light rises above or dips below the horizon be assigned wholly to the light's own height: the observer contributes too.
A reading order that avoids false precision
- Identify the range label and the publication's convention before copying a number.
- Keep metres, feet and nautical miles explicit; identify the elevation datum.
- Separate the visibility assumption from the two height inputs.
- Check the light's character and sectors as well as its range. Reading a light characteristic explains that separate identification task.
The value of the example is its audit trail: another reader can substitute a height and reproduce the arithmetic. For passage planning, use current official charts, light lists and notices with their stated conventions rather than this simplified educational model.
Spot the input that an answer cannot supply
Suppose a visitor knows only that a tower is thirty metres tall. The calculation still cannot proceed responsibly. Is the lantern at the top of those thirty metres? Does the tower stand on a cliff? Which water-level datum was used? The worked example avoids those ambiguities by specifying the light's elevation above its model surface, rather than copying a building dimension.
There is another missing input if someone asks for luminous range from the same information. Neither the tower height nor our geometric result supplies atmospheric visibility or the light's nominal rating. Substituting 14.87 NM into a luminous-range calculation as though it were a lamp specification would mix two independent limits.
A reproducible answer should therefore include a small input record: the quantity requested, the elevations, the visibility assumptions, the units and the treatment of refraction. An unexplained extra decimal place cannot compensate for an absent input.
