2070 Magh · 5 marks

MUF range numerical: 400 km reflection height

Short answer

Using n=0.8 at 15 MHz, maximum electron density Nmax=10^12 m^-3 gives critical frequency fc=9 MHz. For MUF=20 MHz, sec(theta_i)=20/9, tan(theta_i)=1.9845. With flat earth at height 400 km, range d=2*h*tan(theta_i)=1588 km.

Assume that reflection takes place at a height of 400 km and that the maximum density in the ionosphere corresponds to a 0.8 refractive index at 15 MHz. What will be the range (assume flat earth) for which the MUF is 20 MHz?

The MUF over a flat-earth path is the critical frequency multiplied by the secant of the angle of incidence (the secant law). So we first find the maximum electron density from the given refractive index, then the critical frequency, then the angle and the range.

Given

Virtual height of reflection, h = 400 km
At the layer maximum, n = 0.8 at f = 15 MHz
Required MUF = 20 MHz

Step 1: Maximum electron density

The refractive index of the ionosphere is n² = 1 − 81N/f² (N in electrons/m³, f in Hz).

81·Nmax = f²(1 − n²)
        = (15×10⁶)² × (1 − 0.8²)
        = 2.25×10¹⁴ × 0.36
        = 8.1×10¹³
Nmax    = 1.0×10¹² electrons/m³

Step 2: Critical frequency

fc = 9√Nmax = 9 × √(10¹²) = 9×10⁶ Hz = 9 MHz

Step 3: Angle of incidence for MUF = 20 MHz

MUF = fc·sec θi
sec θi = 20/9 = 2.222
cos θi = 0.45  →  θi = 63.26°
tan θi = √(1 − 0.45²)/0.45 = 1.9845

Step 4: Range (flat earth)

     ionosphere ~~~~~~~~~~~ P ~~~~~~~~~~~ (reflection point)
                           /|\
                          / | \
                      θi /  |  \ θi
                        /   | h = 400 km
                       /    |    \
                     Tx ----M---- Rx      (flat earth)
                     <----- d ----->

For a flat earth, half the range is d/2 = h·tan θi, so:

d = 2h·tan θi = 2 × 400 × 1.9845 = 1587.6 km

Check with the combined formula MUF = fc·√(1 + (d/2h)²):

d = 2h·√((MUF/fc)² − 1) = 800 × √(4.938 − 1)
  = 800 × 1.9845 = 1587.6 km

Answer: Nmax = 10¹² /m³, fc = 9 MHz, θi = 63.26°, range d ≈ 1588 km (about 1590 km).

✓ Answer checked by Claude Opus 5.5 (high effort)

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