Distance to the Visible Horizon

Stand on the shore and the sea horizon is about three miles away. Climb to the bridge deck of a ship and it is ten. The relation is a piece of plane geometry: the line of sight is tangent to the sphere, so it makes a right angle with the radius at the tangent point, and Pythagoras on the triangle gives $D=\sqrt{h(2R+h)}$. Since the height of eye is always microscopic next to the earth's radius, the $h^{2}$ term contributes nothing and the working form is simply $D\approx\sqrt{2Rh}$.

That produces the two coefficients every mariner memorises. In metric, $D \approx 3.57\sqrt{h}$ kilometres for $h$ in metres. In the units Bowditch tabulates, the geometric horizon is $1.064\sqrt{h}$ nautical miles for $h$ in feet. Both fall straight out of the square root of twice the radius, and they are the same statement.

The figure this page returns is the GEOMETRIC horizon, and the horizon you actually see is further off, because the atmosphere refracts light downward along the earth's curve. Bowditch gives the visible horizon as $1.169\sqrt{h}$ nautical miles, about ten percent more, which corresponds to pretending the earth has a radius roughly a fifth larger than it does. Refraction is also the least dependable term in the whole calculation: it depends on the temperature gradient over the water, and a strong inversion can lift objects well over the horizon into plain view — the effect called looming, and in its extreme form the Fata Morgana.

The practical use is the geographic range of a light. A lighthouse whose charted height gives it its own horizon distance is visible from your own horizon distance plus its own, so a 40-metre light seen from a 3-metre cockpit is theoretically raised at about $11.0 + 3.3 = 14.3$ nautical miles. That is a GEOGRAPHIC range, set by geometry alone, and it is a different quantity from the LUMINOUS range, which is how far the light is bright enough to be seen through the prevailing visibility. The chart prints the nominal range, computed at a standard visibility; on a clear night a powerful light can be seen well beyond its geographic range as a loom in the sky, and in haze it can vanish at a fraction of it. Confusing the two is how a light is expected an hour before it can possibly appear, and how a navigator talks himself into believing he is further along than he is.