Skip to content

Muons

Space and time · level 7, lesson 3 of 6

A question to start

Muons are made about 15 km up, and they decay in about 2.2 millionths of a second. Even at the speed of light, that is only 660 m. So how do so many reach the ground?

The idea

When cosmic rays hit the top of the atmosphere, they make showers of particles. Many of them are muons, heavy cousins of the electron. At rest, a muon lives on average 2.2 µs before it decays.

Most muons are made at around 15 km up. Even moving at nearly \(c\), a muon covers only about 660 m in 2.2 µs. Without relativity, almost none would get down to sea level. Yet about 1 muon per square centimetre per minute arrives.

The answer is time dilation. Muons move so fast that, from the ground, their clocks run many times slower. Their lifetime, as we measure it, stretches to tens of microseconds: enough to cross the atmosphere.

In 1977, at CERN, muons were kept going round a ring at \(\gamma = 29.33\). Their measured lifetime was 64.4 µs, 29.33 times their lifetime at rest, just as relativity predicts.

× no time dilation:decays after 660 mwith it:reachesthe ground15 km upsea level

Worked examples

1. The trip time. 15 km at nearly \(c\): \(15{,}000 \div 300{,}000{,}000 = 50\) µs. That is about 23 lifetimes at rest. The fraction left would be about \(e^{-23}\): roughly one in ten billion.

2. With \(\gamma = 10\). The lifetime we measure is \(10 \times 2.2 = 22\) µs. Now 50 µs is only about 2.3 lifetimes, and \(e^{-2.3} \approx 0.1\): about one in ten survives.

3. The CERN check. \(29.33 \times 2.2 \approx 64.5\) µs, close to the measured 64.4 µs.

Try it

  1. What is a muon's lifetime at rest?
  2. How far would a muon go in that time at nearly \(c\)?
  3. Why do muons reach the ground?
  4. A muon has \(\gamma = 20\). How long does it live, as measured from the ground?
Answers
  1. About 2.2 µs.
  2. About 660 m.
  3. Time dilation: their clocks run slow as seen from the ground, so they live long enough to cross the air.
  4. \(20 \times 2.2 = 44\) µs.

Common mistake

"The muons are just faster than light." They move at just under \(c\). What changes is how long they live, as seen from the ground.

Where it leads