Short answer first, then the nuance.
TL;DR (rule-of-thumb)
For 1420 MHz hydrogen-line pyramidal horns:
- Safe / conventional half-flare angles:
~10–25° (E- and H-plane) - Upper practical limit before problems appear:
~30° half-angle - Beyond ~30–35° half-angle:
You will start to see measurable degradation.
That’s why almost all serious 21 cm horns cluster around 15–25°.
What “problems” actually appear — and why
Horn flare angle is fundamentally a phase-error trade-off.
1️⃣ Phase error across the aperture (the big one)
As the flare angle increases, the wavefront at the aperture becomes more spherical instead of planar.
- At ≤20°: phase error is small → clean main lobe
- At 25–30°: phase error becomes noticeable
- >30°: phase error causes:
- Reduced forward gain
- Broader main beam than predicted
- Elevated sidelobes
This is the dominant failure mode for hydrogen-line work.
2️⃣ Sidelobe growth & baseline ripple (radio astronomy killer)
Steep flares excite higher-order field variations at the aperture.
For radio astronomy this causes:
- Standing-wave ripple between horn & feed
- Spectral baseline structure
- Increased sensitivity to ground spillover
Even if gain looks OK on paper, spectral quality suffers.
3️⃣ Mode purity at the throat
Very steep flares can partially excite unwanted waveguide modes:
- TE₂₀ / TM modes appear sooner
- Pattern symmetry degrades
- Polarisation purity worsens
This usually shows up before catastrophic gain loss.
Quantitative engineering guidance (classic horn theory)
From standard horn antenna theory (as in your McMaster notes):
- Acceptable maximum phase error across aperture:
≈ λ/8 to λ/6
This phase-error constraint maps to flare angle roughly as:
| Half-flare angle | Status |
| ≤15° | Excellent |
| 15–22° | Optimal |
| 22–28° | Acceptable |
| 28–32° | Marginal |
| >32° | Poor |
At λ ≈ 21 cm, this lines up exactly with what’s seen in real 1420 MHz horns.
Why hydrogen-line horns are more conservative than radar horns
Radar and comms horns sometimes push 35–40° half-angles because:
- They care about instantaneous gain, not spectral flatness
- Sidelobes can be filtered or ignored
Hydrogen-line work cares about:
- Baseline smoothness
- Spillover noise
- Stability over MHz-wide bandwidths
So radio astronomy horns deliberately stay conservative.
Practical recommendation (the one builders actually follow)
For a hydrogen-line horn you want to measure sky temperature, not just detect a signal:
- Target:
E-plane: 18–25°
H-plane: 15–22° - Absolute max if size-constrained:
~30° half-angle, but expect compromises
This explains:
- The SETI League / astronomy.me.uk values (~22–27°)
- The Harvard horn being slightly gentler (~20–22°)
Both are intentionally below the danger zone.