The power output of the Sun is \(4 \times 10^{26} \mathrm{W}\). (a) If \(90 \%\) of this energy is supplied by the proton-proton chain, how many protons are consumed per second? (b) How many neutrinos per second should there be per square meter at the surface of Earth from this process?

First, we need to find the total energy produced by the proton-proton chain. We know that 90% of the Sun's power output comes from this process, so we can calculate the energy produced as follows: Total energy = Sun's power output × 90% = \(4 \times 10^{26} W \times 0.9\) Now, calculate the total energy: Total energy ≈ \(3.6 \times 10^{26} W\)

In the proton-proton chain, 4 protons are consumed to produce 2 positrons, 2 neutrinos, 6 photons, and 2 helium nuclei. The net effect is the conversion of 4 protons into 1 helium nucleus, releasing energy in the process. The energy released per proton is given by the following formula: Energy released per proton = \( (4m_p - m_{He})c^2\) where \(m_p\) is the mass of a proton, \(m_{He}\) is the mass of a helium nucleus, and \(c\) is the speed of light. However, it is a close enough approximation to use the energy equivalent of the mass defect directly: Energy released per proton ≈ \(26.73 MeV\) We have to convert this energy to joules. Using the conversion: 1 MeV = \(1.602\times 10^{-13} J\) Energy released per proton in Joules ≈ \(26.73 \times 1.602 \times 10^{-13} J\) Energy released per proton in Joules ≈ \(4.29 \times 10^{-12} J\)

Now, we will calculate the number of protons consumed per second using the total energy produced by the proton-proton chain and the energy released per proton: Number of protons consumed per second = \(\frac{Total\:energy}{Energy\:released\:per\:proton}\) Number of protons consumed per second = \(\frac{3.6 \times 10^{26} W}{4.29 \times 10^{-12} J}\) Number of protons consumed per second ≈ \(8.39 \times 10^{37}\)

For every 4 protons consumed, the proton-proton chain reaction produces 2 neutrinos. So, we can find the number of neutrinos produced per second as follows: Number of neutrinos produced per second = \(\frac{1}{2}\) × Number of protons consumed per second Number of neutrinos produced per second = \(\frac{1}{2} \times 8.39 \times 10^{37}\) Number of neutrinos produced per second ≈ \(4.20 \times 10^{37}\)

Assuming that the neutrinos are uniformly distributed in all directions, the number of neutrinos per square meter at the Earth's surface can be calculated by dividing the total number of neutrinos produced per second by the surface area of a sphere with a radius equal to the distance from the Sun to Earth (1 Astronomical Unit or AU, approximately equal to \(1.496 \times 10^{11} m\)): Number of neutrinos per square meter per second = \(\frac{Number\:of\:neutrinos\:produced\:per\:second}{4\pi (distance\:(AU))^2}\) Number of neutrinos per square meter per second = \(\frac{4.20 \times 10^{37}}{4\pi (1.496 \times 10^{11})^2}\) Number of neutrinos per square meter per second ≈ \(6.25 \times 10^{14}\) Thus, the number of neutrinos per second per square meter at the Earth's surface from the proton-proton chain process is approximately \(6.25 \times 10^{14}\).