Sphere, geometric
- Radar horizon
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- Target at the entered range
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Radar horizon
How far apart an antenna and a target can be, along the surface, before the curve puts the target below the line of sight. The page gives that distance for a straight ray on the true radius and for the longer standard-atmosphere horizon, and it places a chosen range against both.
- Above both horizons
- Above the radio horizon only
- Below the geometric line of sight
The geometric horizon is a straight ray on the true Earth radius. The longer one is the standard-atmosphere model: a straight ray on a radius 4/3 of the true one, the stand-in for a radio ray that bends slightly downward in ordinary air, and weather can bend the ray more or less.
Sphere, standard atmosphere
- Radar horizon
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- Target at the entered range
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Plane
- Radar horizon
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- Target at the entered range
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The globe and a flat Earth
On a sphere, a straight ray that is not aimed upward meets the surface. The antenna has a horizon, and so does the target. The radar horizon is the sum of those two surface distances. Under a standard atmosphere the radio path is treated as a straight line on a sphere of radius (4/3) R, so the horizon is longer than the geometric one.
A flat-Earth prediction for this page is the plane column. On a plane there is no horizon of this kind: a straight ray at a constant height above a plane does not meet the plane. How far a radar then detects a target is a question of power, target size, clutter, and noise.
Assumptions
- The same sphere and the same horizon function as the other tools. The radius is ….
- Geometric case: k = 1. Standard radio atmosphere: k = 4/3, the linear-gradient value in ITU-R P.834 for paths in roughly the first kilometre of height.
- Both numbers call
combinedHorizonin the shared module. One call uses R. The other uses (4/3) R. The optical coefficient κ is not used here; radio refraction is stronger than the usual optical value, and the 4/3 factor is the radio convention. - This is not a detection-range calculation. A target above the radio horizon can still be missed. Ducting can carry energy past the standard-atmosphere horizon. Diffraction into the shadow is not added on top of the effective radius.
- The bars are to scale. The cross-section uses the true radius and exaggerates heights when the note says so. The straight line is the geometric line of sight. The brown on the mast above the standard-atmosphere horizon is still below the geometric one; that second height is the shared hidden-height function on (4/3) R, drawn on the mast rather than as a second straight ray.
- The practical rule of thumb, distance in nautical miles ≈ 1.22 √(height in feet), is this same standard-atmosphere model with a rounded coefficient. The page reports the arc on (4/3) R, not the rounded rule.
Formulas
Horizon distance from one height h is the shared function: R · atan2(√(h (2R + h)), R).
Radar horizon = horizon(antenna) + horizon(target), evaluated on R and again on (4/3) R.
At a range d, the target is above a horizon when its height is greater than the hidden height at d for that antenna and that radius.
Sources
- Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Géodésique 54(3), 395–405. Mean radius (2a + b) / 3 from a = 6,378,137 m and 1/f = 298.257222101.
- ITU-R Recommendation P.834, Effects of tropospheric refraction on radiowave propagation. For a linear refractivity gradient near the surface, the effective-Earth factor k is 4/3. itu.int/rec/R-REC-P.834
- National Geospatial-Intelligence Agency. The American Practical Navigator (Bowditch). Under standard atmospheric conditions the radar horizon in nautical miles is about 1.22 times the square root of the antenna height in feet. msi.nga.mil/Publications/APN
- Secondary distances use the international nautical mile, 1,852 m.