PhysicsCalcs
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Radar Horizon Calculator

Calculate radar line-of-sight distance with Earth curvature and atmospheric refraction.

Interactive calculator

Radar Horizon Calculator

Calculate radar line-of-sight distance accounting for Earth curvature and atmospheric refraction (k-factor).

Try an example

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Enter antenna and target heights.
  • Default k = 4/3 for standard atmosphere.
  • Results show radar and geometric horizons.

Table of Contents

Key Takeaways

  • Radar horizon: d ≈ 4.12(√h₁ + √h₂) km for standard atmosphere (k = 4/3), heights in metres.
  • The k-factor models atmospheric refraction: k = 4/3 is the standard, k = 1 is geometric (no refraction).
  • Refraction extends the radar horizon about 15% beyond the geometric horizon.
  • Both antenna height and target height contribute to the total detection range.
  • Super-refraction (k > 4/3) occurs near warm water and can dramatically extend range.

What Is the Radar Horizon?

The radar horizon is the maximum distance at which a radar system can detect a target, limited by Earth’s curvature and atmospheric refraction. Unlike the visual horizon, radar waves bend slightly downward due to atmospheric density gradients, effectively extending the detection range by about 15% under standard conditions.

Formulas

d=2kRehr+hr2+2kReht+ht2d = \sqrt{2kR_e h_r + h_r^2} + \sqrt{2kR_e h_t + h_t^2}

Simplified (h « R):

d≈4.12(hr+ht) kmd \approx 4.12\left(\sqrt{h_r} + \sqrt{h_t}\right) \text{ km}

Where k = 4/3, R_e = 6371 km, heights in metres.

The k-Factor

k-FactorConditionEffect
1.0No atmosphere (geometric)Baseline horizon
4/3 (1.333)Standard atmosphere~15% extension
2.0Super-refraction~22% extension
< 1.0Sub-refractionReduced range

Applications

  • Marine radar — ship-to-ship and ship-to-shore detection range.
  • Air traffic control — radar coverage for aircraft detection.
  • Military radar — low-flying target detection limits.
  • Radio communications — VHF/UHF link planning.

How to Use

  1. Enter radar antenna height and target height.
  2. Set k-factor (default 4/3 for standard atmosphere).
  3. Click Calculate to see detection range with geometric comparison.

Examples

Ship radar (25m antenna, surface target)

d ≈ 4.12 × √25 = 4.12 × 5 = 20.6 km

ATC radar (30m, aircraft at 10,000m)

d ≈ 4.12 × (√30 + √10000) = 4.12 × (5.48 + 100) = 434 km

FAQ

What is the k-factor in radar horizon calculations?›

The k-factor (or effective Earth radius factor) models how atmospheric refraction bends radio waves. k = 4/3 (1.333) is standard for temperate climates. The effective Earth radius is R_eff = k × R_Earth, making calculations equivalent to straight-line propagation on a larger Earth.

How does the radar horizon differ from the visual horizon?›

The radar horizon is typically farther because radio waves bend more than visible light in the atmosphere. Standard visual horizon uses k ≈ 1.07-1.17, while radar uses k ≈ 4/3 (1.333). This means radar can 'see' about 15% farther than the eye.

What is ducting and how does it affect radar?›

Atmospheric ducting occurs when extreme temperature inversions trap radio waves in a layer near the surface, effectively making k → ∞. This can extend radar range hundreds of kilometers beyond normal, but is unpredictable and can also create blind spots.

Why is the simplified formula d ≈ 4.12√h used?›

This approximation drops the h² term (valid when h << R_Earth) and uses R_Earth = 6371 km with k = 4/3: d = √(2 × 4/3 × 6371000 × h) ≈ 4120√h metres = 4.12√h km. It's accurate to within 1% for heights below 1000 m.

Sources

  1. Skolnik, M. – Introduction to Radar Systems (3rd ed.)
  2. ITU-R P.834 – Effects of Tropospheric Refraction
  3. Wikipedia – Radar Horizon
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

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