A neutrino must pass within 1018m of a quark to have any cha

A neutrino must pass within 10^-18m of a quark to have any chance of interacting with it. Thus the maximum target area that a quark presents to a neutrino is about 10^-36m². However, a neutrino moves so quickly and the weak interaction is so weak that experimentally only about 1 in 10¹² typical neutrinos passing a quark within this radius actually interacts significantly with it. Something on the order of 10¹⁴neutrinos per second from the sun fall on 1 m² of the earth\'s surface.

a. What is the (approximate) average time between neutrino interactions with your body when the sun is directly overhead?

b. What is the same at midnight when the entire bulk of the earth is between you and the sun?

Solution

There are roughly 11 * 6.02 * 10^23 protons and neutrons per cubic cm

66.22 * 10^23 =>
6.622 * 10^24

So, let\'s say that there\'s a column of lead that has a cross-section of 10^(-36) m^2, or 10^(-36) * 10^(4) cm^2 => 10^(-32) cm^2 and it is x cm tall

V = 10^(-32) * x cm^3
V = 1 cm^3
x =10^32 cm

So if we had a column that had a cross-section of 10^(-32) cm^2 and a height of 10^(32) cm, there\'d be 6.622 * 10^24 protons or neutrons, or about 6.622 * 10^(24 - 32) protons or neutrons per cm

6.622 * 10^(-8) protons or neutrons per cm

And there\'s a 1 in 10^12 chance of interacting

((10^12 - 1) / 10^12)^n = 0.5
n * ln((10^12 - 1) / 10^12) = ln(0.5)
n = ln(0.5) / ln((10^12 - 1) / 10^12))
n = 6.93147180559598735826952091041202905323... × 10^11

So, after 6.93 * 10^11 possible interactions, there\'s at least a 50-50 chance that an interaction will occur.

6.622 * 10^(-8) protons/neutrons per cm
10^8 / 6.622 cm between each proton and neutron
1.51 * 10^7 cm between each proton and neutron

1.51 * 10^7 * 6.93 * 10^11 =>
10.47 * 10^18 =>
1.05 * 10^19

1.05 * 10^19 cm
1.05 * 10^17 m
1.05 * 10^14 km
10.5 * 10^13 km

The sun is about 150,000,000 km away, or 1.5 * 10^8 km away, so we\'re looking at a column that\'s about 7 * 10^5, or 700,000 times greater than the distance between the earth and the sun.