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Opus in profectus

# Quantum Flavordynamics

## Practice

### practice problem 1

Two related problems in solar physics…
1. How many solar neutrinos pass through your fingernail every second?
2. How many solar photons land on your finger every second when you are in direct sunlight?

#### solution

1. Use the total power of the Sun (P1), which includes the electromagnetic (3.828 × 1026 W) and neutrino (0.088 × 1026 W) luminosities. Divide by the energy released per reaction in the proton-proton chain (E2 = 26.7 MeV). Spread the radiation over the surface of a sphere (A1 = 4πr12) the size of the Earth's orbit (r1 = 1 au). Multiply by the area of a fingernail (A2 = 1 cm2) for a duration of one second (t1 = 1 s). Double this number since the proton-proton chain releases two neutrinos per reaction.

n =
 E1 A2 = 2P1t1 A2 E2 A1 E2 4πr12
n =
 2(3.916 × 1026 W)(1 s) (1 cm2) 26.7 MeV 4π(1 au)2
n =  6.5 × 1010 neutrinos

That's 65 billion neutrinos through your fingernail every second. In order to capture the full number, you need to hold your fingernail perpendicular to the direction of the Sun, however. Tilt your fingernail and the number decreases.

This number is not significantly affected by the material between your fingernail and the Sun. Even at midnight, when the whole mass of the Earth is between you and the Sun, 65 billion neutrinos pass through your fingernail every second. Neutrinos interact with matter via the weak force, which means they only interact on "direct hits" with certain subatomic particles, which means they interact with almost nothing.

2. Use the electromagnetic luminosity of the Sun (P1 = 3.828 × 1026 W). Divide by the energy (E2 = hc) of a photon near the peak wavelength (λ2 = 500 nm). Spread the radiation over the surface of a sphere (A1 = 4πr12) the size of the Earth's orbit (r1 = 1 au). Multiply by the area of a fingernail (A2 = 1 cm2) for a duration of one second (t1 = 1 s). Do not double this since these photons were emitted from the surface of the Sun, not the core where the reactions take place.

n =
 E1 A2 = 2P1t1 A2 E2 A1 E2 4πr12
n =
 (3.828 × 1026 W)(1 s) (1 cm2) (hc/500 nm) 4π(1 au)2
n =  3.4 × 1017 photons

That's 340 quadrillion photons landing on your fingernail every second. As with the neutrino number calculated above, this number assumes the fingernail is aligned perpendicular to the direction of the Sun.

Unlike the neutrino number, however, this number is greatly affected by the material between your fingernail and the Sun. The atmosphere absorbs some of the Sun's photons even when the sky is crystal clear. Aerosols like dust and water droplets absorb some more. And of course at midnight, when the Earth is between you and the Sun, absolutely no photons make the direct trip from the Sun to your fingernail. Photons interact with matter via the electromagnetic force, which means they interact with anything with charge, which means they basically interact with almost everything.

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