Radio Horizon & Line-of-Sight (LOS) Calculator

In free space without an atmosphere, the maximum distance at which two antennas can maintain line-of-sight is governed strictly by the Euclidean geometry of a sphere. However, terrestrial radio links operate through the Earth's troposphere—a non-homogeneous dielectric medium where atmospheric pressure, temperature, and water vapor density decrease steadily with altitude.

Because the index of refraction $n$ is slightly greater than unity at sea level ($n \approx 1.000315$) and decreases with height, the upper edge of an electromagnetic wavefront travels through slightly less dense air and moves faster than the lower edge. According to Snell's Law in stratified media, this velocity differential continuously tilts the propagating wavefront downward toward the Earth's surface. Consequently, radio waves follow a curved trajectory with a radius of curvature $r$ that is greater than the Earth's physical radius, extending the radio horizon significantly beyond the visual optical horizon.

Derivation of the 4/3 Effective Earth Radius ($k$-Factor)

To simplify microwave link design without performing complex curved-ray ray-tracing on spherical coordinates, Schelleng, Burrows, and Ferrell (1933) introduced the concept of the effective Earth radius ($R_{\text{eff}} = k \cdot R_e$). By mathematically flattening the Earth's curvature by a factor of $k$, radio waves can be modeled as propagating along straight Euclidean rays:

In standard international telecommunications planning (ITU-R Recommendation P.453), radio refractivity $N = (n - 1) \times 10^6$ exhibits an average vertical lapse rate of $\frac{dN}{dh} \approx -39.2\text{ N-units/km}$ (or $\frac{dn}{dh} \approx -39.2 \times 10^{-6}\text{ km}^{-1}$) in temperate climates. Substituting this standard gradient yields:

Exact Geometric Derivations: Horizon Distance & Mast Heights

Applying the Pythagorean theorem to a right-angled triangle formed by the Earth's center, the antenna top at height $h$, and the tangent point on the horizon:

Substituting metric and imperial units yields the standard industry rules of thumb:

• Metric Formula ($h$ in meters, $d$ in kilometers): $$d_1\text{ (km)} = \sqrt{2 \cdot 1.333 \cdot 6371 \cdot \frac{h_1}{1000}} = \sqrt{16.989 \cdot h_1} \approx 4.12 \cdot \sqrt{h_1\text{ (m)}}$$ For an optical sightline ($k = 1.0$), $d_{\text{opt}}\text{ (km)} \approx 3.57 \cdot \sqrt{h_1\text{ (m)}}$.
• Imperial Formula ($h$ in feet, $d$ in statute miles): $$d_{\text{rad}}\text{ (mi)} \approx 1.414 \cdot \sqrt{h\text{ (ft)}} \quad\text{vs.}\quad d_{\text{opt}}\text{ (mi)} \approx 1.225 \cdot \sqrt{h\text{ (ft)}}$$
• Two-Station Cumulative Line-of-Sight: $$D_{\text{total}} = d_1 + d_2 = 4.12 \left(\sqrt{h_1} + \sqrt{h_2}\right) \quad [\text{km}]$$

Atmospheric Anomalies: Sub-refraction, Super-refraction, and Ducting

Real-world weather phenomena deviate from the idealized $4/3$ standard atmosphere: