Radar and Radio Horizon with Standard Refraction

This gives the distance to the horizon for radio or radar from the height $h$ of an antenna, or of an aircraft, above the surface. The air is thinner higher up, so a beam that starts out level curves gently down toward the ground and reaches farther than a straight line would. Recommendation ITU-R P.834 handles that by keeping the beam straight and enlarging the Earth to an effective radius $kR$, $k$ times the real one. The horizon is then the tangent from the antenna to that bigger sphere, $d = \sqrt{h\,(2kR + h)}$, which is about $\sqrt{2kRh}$ when $h$ is small beside the radius.

For the standard atmosphere $k = 4/3$, and the result is about $4.12\sqrt{h}$ kilometres with $h$ in metres. The bare geometry of the library's horizon-distance-from-height page has 3.57 in place of 4.12, so the radio horizon is about 15 percent farther than the geometric one. Because the distance grows only as the square root of the height, a taller mast brings diminishing returns. Two antennas that can just see each other are separated by the sum of their two horizon distances, one for each height. Entering $k = 1$ gives the geometric horizon with no bending.

Two cases printed on Wikipedia, rounded as printed there, show the scale. An eye 1.70 m above the sea, with no bending ($k = 1$), sees 4.7 km; the page gives 4.654 km, and that is the example it offers. A 1 500 m mast or aircraft altitude with $k = 4/3$ sees to about 160 km, from $4.12\sqrt{1\,500}$. The page uses the exact tangent, 159.64 km, which is 0.2 percent less. Run backwards, a horizon of 159.64424575 km at $k = 4/3$ returns the height of 1 500 m, and by hand a 25 m mast at $k = 4/3$ sees 20 609.09 m.

The factor $k$ is an average, and it is the weak point of the method. In strong temperature inversions it can be much larger, with ducting, and in very dry, cold or unstable air it is smaller. Recommendation ITU-R P.834-9 (12/2017), section 2, gives $k = 4/3$ for heights below 1 000 m, so the 1 500 m case above is a step beyond it.