Library / Sensors & detection / SEN-06
Sensors & detection
Radar horizon
Line-of-sight range between two antennas, including the slight bending of radio waves in the atmosphere (4/3 earth radius).
- Ref
- SEN-06
- Rev
- v1
- Field
- Sensors & detection
- Result
- d [km]
h₁ and h₂ are the heights of the two antennas above the surface, in metres, and the answer is in kilometres. The constant 4.12 equals 3.57 × √(4/3): 3.57 is the geometric horizon constant for the Earth’s radius, and the 4/3 factor allows for the downward bend of radio waves in a normal atmosphere. For the visible horizon of the eye, the constant is about 3.57.
Line-of-sight range between two antennas, including the slight bending of radio waves in the atmosphere (4/3 earth radius).
Calculator
- Horizon distance of antenna 1 alone20.6 km
- Horizon distance of antenna 2 alone13.03 km
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 25 m mast sees a 10 m target out to 33.6 km. Raising the antenna to 100 m extends this to 54 km; to double the range of a mast you need to raise it four times.
- The result is a limit set by geometry, not a detection range: a target at the horizon must still return a strong enough echo.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h1 Height of antenna 1 | 0.5 to 10,000 m | 15.94 | 425 | 409.1 | +10% | +2.99 % |
| h2 Height of antenna 2 or target | 0.5 to 10,000 m | 23.51 | 432.6 | 409.1 | +10% | +1.89 % |
Sweep of Height of antenna 1
| Height of antenna 1 (m) | Horizon distance (km) |
|---|---|
| 0.5 | 15.94 |
| 1.724 | 18.44 |
| 5.946 | 23.07 |
| 20.5 | 31.68 |
| 70.71 | 47.67 |
| 243.8 | 77.36 |
| 840.9 | 132.5 |
| 2,900 | 234.9 |
| 10,000 | 425 |
Sweep one variable
Variables and how each one contributes
Horizon distance
What the equation gives you. Everything else feeds it.
Height of antenna 1
The height of antenna 1 above the surface, in m. Square root: four times the height doubles its contribution. A 10 % rise in h₁ raises d by about 3 %.
Height of antenna 2 or target
The height of the second antenna or the target above the surface, in m. Square root, like h₁ but smaller at the default since it is lower: a 10 % rise in h₂ raises d by about 1.9 %.
Worked example with the values above
- Square roots: √25 = 5, √10 = 3.162.
- Sum: 5 + 3.162 = 8.162.
- d = 4.12 × 8.162 = 33.63 km.
Assumptions and limits
- A smooth Earth (or sea) and standard atmospheric refraction (effective Earth radius 4/3 of the real radius).
- Line of sight is the only path; terrain and buildings are not considered.
Common mistakes
- Entering heights in feet or the target’s range in the wrong unit.
- Treating the horizon range as the radar’s detection range.
Statistics and uncertainty
Since d goes as the square root of the heights, errors are halved: a 10 % error in a height gives 5 % in the range. Real conditions matter more: ducting can double the range over the sea and sub-refraction can cut it by a third, so the 4/3 value is an average, not a guarantee.
Where it is used
- How far away a low-flying target or ship can be seen from a mast.
- Siting antennas for line-of-sight radio links.
Maths you need
- Square roots: effect of a height on range
- Earth geometry: a tangent to a sphere