This question is relevant to considering how close two observers need to be to each other in order to observe the same muon cascades, and hence obtain similar muon detections/have comparable results, for typical muon events.
For a single very high energy primary (proton or alpha), the muon footprint on the ground is reasonably large, but not huge.
A typical muon lateral spread at ground is on the order of a few hundred meters in radius.
Most of the muon density is within roughly 100–300 m of the shower core, with a tail extending farther out at much lower density.
For ultra‑high energy primaries, the footprint grows, and you can get effective muon radii of order 0.5–1 km, again with a long, sparse tail beyond that. The “useful” region where muon density is still significant is typically hundreds of meters, but isolated muons can show up well past a kilometer from the core.
Key dependencies:
– Primary energy: higher energy → more generations in the hadronic cascade → more transverse momentum kicks → larger lateral spread.
– Zenith angle: inclined showers stretch the footprint—projected on the ground, the muon pool can become elongated and extend to kilometer scales.
– Primary type (proton vs alpha): composition changes the detailed muon numbers and distribution, but the radius scale is still in the “hundreds of meters to ~1 km” band for the same total energy.
So if you want a single “typical” number for a vertical, very high energy shower, then this os of order of magnitude: ~200–500 m radius for the main muon pool, with rare muons out to ~1 km.