Accurately compute ionospheric critical frequency (foF2, foE) and maximum usable frequency (MUF) from electron density and elevation angle. Essential for HF radio propagation prediction, space weather analysis, and amateur radio.
In radio propagation, the critical frequency (fc) is the highest frequency that will be reflected back to Earth when transmitted vertically upward into the ionosphere. It depends directly on the electron density of the ionospheric layer. The formula derives from the plasma frequency of a magnetoionic medium:
f_c = 9 √(N / 10¹²)
where f_c is in MHz, N is electron density in electrons per cubic meter (e⁻/m³).
For example, if N = 1.0×10¹² e⁻/m³ (typical daytime F2 layer), then f_c = 9·√(1) = 9.0 MHz.
The concept of critical frequency emerged from early 20th‑century experiments by Appleton and Barnett (1924) who demonstrated the existence of the ionosphere. Appleton received a Nobel Prize for his work. The secant law (MUF = f_c · sec φ) was formulated to predict oblique propagation. During the 1930s–40s, it became essential for long‑distance shortwave communication. Today, real‑time ionosondes measure critical frequencies to issue HF propagation forecasts (e.g., NOAA/SWPC).
The critical frequency equals the plasma frequency of the ionized layer: ω_p = √(N e² / (ε₀ m)). In practical units: f_p (Hz) ≈ 8.98 √N (with N in m⁻³). To express in MHz: f_c (MHz) = 8.98×10⁻⁶ √N. The commonly used engineering approximation is f_c (MHz) = 9 √(N / 10¹²), which is accurate within 0.2% for N near 10¹².
The Maximum Usable Frequency (MUF) for an oblique path at elevation angle Δ (above horizon) is given by the secant law:
MUF = f_c · sec φ = f_c / sin Δ
where φ = 90° – Δ is the incidence angle relative to vertical. This assumes a flat ionosphere and no earth curvature; for longer paths, more sophisticated models (e.g., Breit and Tuve’s theorem) are used, but the secant law remains a robust first approximation.
All angles in degrees. The calculator also shows the plasma frequency, which equals f_c.
| Layer | Height (km) | Typical N (e⁻/m³) | Critical frequency (MHz) | Notes |
|---|---|---|---|---|
| D layer | 60–90 | 10⁸ – 10¹⁰ | 0.1 – 0.9 | Absorbs HF, no reflection |
| E layer | 90–150 | 10¹⁰ – 2×10¹¹ | 0.9 – 4.0 | Sporadic E can reach higher |
| F1 layer | 150–250 | 2×10¹¹ – 5×10¹¹ | 4.0 – 6.4 | Daytime only |
| F2 layer | 250–400 | 5×10¹¹ – 3×10¹² | 6.4 – 15.6 | Main reflective layer for HF |
A broadcaster wants to select a frequency for a 9000 km path. Assuming the F2 layer critical frequency foF2 = 12 MHz (typical high sunspot number) and an elevation angle of 12° at the path midpoint. Using the calculator: sin(12°) ≈ 0.208, so MUF = 12 / 0.208 ≈ 57.7 MHz. However, the working frequency is usually taken as 85% of MUF (Optimal Working Frequency, OWF) ≈ 49 MHz, which falls into the VHF band and may not reflect efficiently. This indicates the need for a lower angle or a different layer. In practice, multi‑hop propagation uses lower elevations. Our calculator helps quickly test scenarios.
For a real curved Earth and ionosphere, the secant law is modified by the ionospheric transmission factor (ITF) or MUF factor. The effective MUF = f_c · k · sec(φ₀), where k accounts for the Earth's curvature. Typically, for a 3000 km hop, k ≈ 1.0 to 1.2. Our calculator provides the basic secant law value; for professional planning, refer to ITU‑R P.533 or VOACAP.
function criticalFrequency(N) {
// N in electrons/m³
return 9.0 * Math.sqrt(N / 1e12); // MHz
}
function MUF(fc, elevationDeg) {
let sinEl = Math.sin(elevationDeg * Math.PI / 180);
if (sinEl <= 0) return NaN;
return fc / sinEl;
}