The 4/3-Earth radio horizon
The geometric distance to the horizon from a height h above a sphere of radius R is √(2R h + h²). Standard radio atmosphere bends VHF and radar rays downward; the usual engineering model replaces R with k R and takes k = 4/3. In aviation units that simplifies, for h in feet and range in nautical miles, to about
dNM ≈ 1.23 √hft
for the radio horizon, and about 1.06 √h for the geometric horizon (k = 1). Two antennas see each other when the sum of their individual horizons covers the distance between them, so the link range is d1 + d2. This page uses a mean Earth radius of 6,371 km and reports both the exact square-root expression and the 1.23 √h rule.
What the model leaves out
Ducting, inversions, terrain, and bank angle can stretch or cut the path. HF skywave is a different mechanism entirely. A service volume or a radar coverage chart already folds in transmitter power, receiver sensitivity, and obstacles; this calculation is only geometry plus a standard refraction factor. Pair it with the VOR/DME and 1-in-60 tools for station-relative geometry, not as a clearance to leave a frequency.
Sources
- ITU-R P.834 and related radio-meteorological guidance (effective Earth radius factor k ≈ 4/3 in a standard atmosphere).
- FAA Aeronautical Information Manual and Instrument Flying Handbook (VHF navigation and communication service volumes; line-of-sight character of VOR, ILS, and ATC radio).
- Standard geometric horizon √(2R h + h²) on a sphere of mean radius 6,371 km.
Frequently asked questions
Where does the 1.23 √h rule come from?
It is √(2 × (4/3) × R × h) converted so that h is in feet and the range is in nautical miles, dropping the small h² term. For a 1,000-foot antenna it is about 39 NM.
Should I add two aircraft heights or use the higher one?
Add the two individual horizons. A 1,000-foot tower and a 6,000-foot airplane see each other at roughly 1.23√1000 + 1.23√6000 nautical miles, not at the taller height alone.
Does this guarantee radio or radar contact?
No. It is a line-of-sight geometry estimate. Power, interference, terrain, and unusual refraction can make the usable range shorter or, occasionally, longer.