Calculating how many neutrinos passed through a body

[mattdespat]

Supernova Shelton 1987A, located about 170 000 ly from the Earth, is estimated to have emitted a burst of neutrinos carrying energy ~1046 J. Suppose the average neutrino energy was 6 MeV and your body presented cross-sectional area 5 000 cm^2. To an order of magnitude, how many of these neutrinos passed through you?

My first step was to convert both energies to J, once this was done I was able to find out how many neutrinos were present in the original burst. My issue is at this point, how to connect the amount of neutrinos originally to how many would pass through a body with cross sectional area 5 000cm^2.
Thanks for the help.

 

[zhermes]

Good start!

This is basically the same problem as trying to figure out how many darts randomly thrown at a dart board will hit a given region. In the dart-board case: what fraction of darts would hit the top half of the board?

Can you generalize this idea to your current problem?
Hint: the surface-area of a sphere is $$A = 4\pi r^2$$

 

[mattdespat]

Thanks for the help.

So I took the area of the earth, by using the radius and using the equation above. Then like in the dartboard example half the darts would hit the top half of the dart board, so whatever percentage that the body is of the total area of the Earth would be the same percentage of neutrinos that pass through the body.

My one remaining question is whether it would be appropriate to say that all neutrinos from the original burst would hit earth?

 

[zhermes]

You're exactly on the right track!
The question is the fraction of the total area of the burst which is covered by a person. And your end question is also exactly on the right track---the Earth would definitely not receive all of the neutrinos. Assume that the neutrinos are emitted 'isotropically' (equally in all directions). What spherical surface should you consider then?

 

[asteriatic]

Calculating the number of neutrinos that pass through a body is a complex task as it depends on various factors such as the size of the body, the distance from the source of neutrinos, and the energy of the neutrinos. In this scenario, we can make some approximations and calculate the order of magnitude of neutrinos that may have passed through a body with a cross-sectional area of 5,000 cm^2.

Firstly, we need to determine the number of neutrinos emitted by the supernova explosion. According to the given information, the supernova emitted a burst of neutrinos carrying energy ~1046 J. Assuming that the average energy of each neutrino is 6 MeV, we can calculate the total number of neutrinos using the equation E=mc^2, where m is the mass of the neutrino and c is the speed of light. This gives us a value of approximately 1.1 x 10^58 neutrinos.

Now, to calculate the number of neutrinos that pass through a body with a cross-sectional area of 5,000 cm^2, we need to consider the distance between the body and the supernova. Since the supernova is located about 170,000 light-years away from Earth, we can assume that the body is at a considerable distance from the source. Thus, we can consider the body as a point in space, and all the neutrinos emitted by the supernova will pass through it.

Using the equation for the surface area of a sphere, we can calculate the surface area of a sphere with a radius of 170,000 light-years. This gives us a value of approximately 1.8 x 10^38 cm^2. Now, we can use the ratio of the surface area of the body (5,000 cm^2) to the surface area of the sphere to estimate the number of neutrinos that will pass through the body. This gives us a value of approximately 2.8 x 10^-36 neutrinos.

Therefore, to an order of magnitude, we can estimate that about 10^-36 neutrinos may have passed through the body. However, it is important to note that this is a rough approximation and the actual number may vary significantly depending on various factors.