Let us assume that the muon production happens at an altitude of about 15 kilometers above the surface of the Earth, and that the produced muons have a velocity of 0.99 c. Thus, in the Earth system, it takes the muon approx. 50500 nanoseconds to reach the surface of the Earth. In a non-relativistic calculation, what is the probability for a muon to reach the surface of the Earth?
Follow up question asks: What is the probability for a muon to reach the surface of the Earth, taking into account time dilation of its mean-life in the Earth system?
mean life:2193 nanoseconds
Here is the relativistic (correct) answer:
In the muon's own coordinate system, the time that it takes to reach the earth is
t' = 50500*10^-9 s * sqrt[1 - (0.99)^2]
= 5*10^-5 s * 0.141
= 7.05*10^-6 s
The fraction that arrive without decaying is
P = exp(-t'/2.193*10^-6) = 0.040
For a nonrelativistic calculation, you would get
P = exp(-50500/2193),
a much lower number.
Note that we are using mean life and not half life numbers for the lifetime. They are not quite the same.
To calculate the probability for a muon to reach the surface of the Earth in a non-relativistic calculation, we can consider the mean life of the muon in the Earth system.
In a non-relativistic context, the probability for a muon to survive without decaying for a certain time period can be described by an exponential decay function. The probability of survival, P, can be determined using the equation:
P = e^(-t/T)
where t is the time period and T is the mean life of the muon.
Using the given mean life of the muon in the Earth system as 2193 nanoseconds, and considering it takes approximately 50500 nanoseconds for the muon to reach the Earth's surface, we can substitute these values into the equation:
P = e^(-50500/2193)
Calculating this expression will give us the probability for the muon to survive without decaying while reaching the Earth's surface using non-relativistic calculations.
Now, let's discuss the second part of your question, which takes into account time dilation due to the relativistic effects.
When a muon moves at a high velocity close to the speed of light, it experiences time dilation, which means that time appears to pass more slowly for the moving muon compared to an observer at rest relative to the muon.
The time dilation factor, γ, can be calculated using the equation:
γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the muon and c is the speed of light.
In this case, the muon's velocity is given as 0.99c, so we can substitute this value into the equation to find the time dilation factor, γ.
Next, we need to account for the time dilation when considering the mean life of the muon. The time dilation factor, γ, can be applied to the mean life (T) of the muon in the non-relativistic case:
T' = T / γ
where T' is the time dilation corrected mean life.
Finally, using this time dilation corrected mean life, T', and the time it takes for the muon to reach the Earth's surface, we can calculate the probability for the muon to survive using the exponential decay equation as mentioned earlier:
P' = e^(-t/T')
Substituting the known values of T' and t, you can compute the probability for the muon to reach the Earth's surface, taking into account time dilation.