Re: Muon paradox

Aether Regained <AetherRegaind@somewhere.in.the.aether> wrote:

Paul.B.Andersen:

Den 31.03.2025 22:40, skrev LaurenceClarkCrossen:

Do muons move at a different velocity in the laboratory than in the
atmosphere?
>
"No, muons generally do not move at a different velocity in a laboratory
setting compared to their velocity in the atmosphere; they both travel
at speeds very close to the speed of light, typically around 99.8% of
the speed of light" - Google search AI.
>
Then why would they "time dilate" in the atmosphere?

 
The speed of muons is v = ~ 0.999668?c through the atmosphere
which also is within the laboratory with open roof.
? = 38.8.
 
The mean proper lifetime of a muon is t? = 2.2 ?s.
But measured in the Earth's rest frame the lifetime of the muon
is  t? = 2.2e-6?? s = 85.36 ?s (time dilation!).
 
Since muons are created at a height ~15 km, and the time for
a muon to reach the earth is t = 15e3/v = 5.005 s,
then the part of the muon flux that will reach the Earth is
 N/N? = exp(-t/t?) = 0.556, so 55.6% of the muons would reach the Earth.
 
If the lifetime of the muons had been 2.2 ?s, then the part of
the muon flux that will reach the Earth would be:
 N/N? = exp(-t/t?) = 1.32e-10.
So only 0.0000000132% of the muons would reach the Earth.
 
Can toy guess which of them is closest to what is observed?
 

 
@PaulBAndersen
 
There is one flaw I find in the SR explanation, can you confirm if it is
true:
 
What is really measured are these (the facts):
 
1. The mean proper lifetime of a muon is t? = 2.2 ?s.
2. muons are created at a height ~15 km
3. The speed of the muons is ~c, so travel time is ~50.05 ?s
4. muon flux measured on the Earth's surface is about 55.6% of what it
is at 15km.
 
From 1, 2 and 3, the expected muon flux on the Earth's surface is:
 
N/N? = exp(-t/t?) = exp(-50.05/2.2) = 1.32e-10 = 0.0000000132%
 
The important point (the flaw) is that the speed of the muon has not
actually been measured to be 0.999668?c, but instead is computed.
 
N/N? = exp(-t/?t?) = .556 => ? = 38.8 => v = 0.999668?c
 
The SR explanation would have been more convincing, if the speed had
actually been measured to that many significant figures.

Actually the muons have an energy spectrum.
(as indicated by Paul Anderson)

The 0.999668?c is just a typical value for ~c,
in order to have a number to calculate with.
You shouldn't assume that all muons come with precisely this speed.

Some of the cosmic ray muons can be detected,
by observing the electrons or positrons they decay into,
or by observing a reaction they cause.

The energies of the secondary particles can be measured,
from which the energy (hence speed) of the original muon
can be estimated.

Jan