The energy radiated by the Sun is generated primarily by the fusion of hydrogen into helium-4. In stars the size of the Sun, the primary mechanism by which fusion takes place is the proton-proton chain. The chain begins with the following reactions:
2 p → X₁+e⁺+X₂ (0.42 MeV) (A4-1)
p+X₁ → X₃+γ (5.49 MeV) (A4-2)
The amounts listed in parentheses are the total kinetic energy carried by the products, including gamma rays. p is a proton, e⁺ is a positron, γ is a gamma ray, and X₁, X₂, and X₃ are particles for you to identify.
The density of electrons in the Sun's core is sufficient that the positron is annihilated almost immediately, releasing an energy x:
e⁺+e⁻ → 2 γ (x) (A4-3)
Subsequently, two major processes occur simultaneously. The "pp I branch" is the single reaction
2 X₃ → ⁴He+2 X₄ (y), (A4-4)
which releases an energy y. The "pp II branch" consists of three reactions:
X₃+⁴He → X₅+γ (A4-5)
X₅+e⁻ → X₆+X₇ (z) (A4-6)
X₆+X₄ → 2 ⁴He (A4-7)
where z is the energy released in step A4-6.
a. Identify X₁ through X₇. X₂ and X₇ are neutral particles of negligible mass. It is useful to know that the first few elements, in order of atomic number, are H, He, Li, Be, B, C, N, O.
b. The mass of the electron is 0.51 MeV/c², the mass of the proton is 938.27 MeV/c², and the mass of the helium-4 nucleus is 3727.38 MeV/c². Find the energy released during the production of one helium-4 nucleus, including the kinetic energy of all products and all energy carried by gamma rays.
c. Find the unknown energies x and y above.
d. Step (A4-6) does not proceed as follows because there is insufficient energy.
X₅ → X₆+e⁺+X₇
What constraint does this fact place on z?
e. In which of the reaction steps is the energy carried by any given product the same every time the step occurs? Assume that the kinetic energy carried in by the reactants in each step is negligible, and that the products are in the ground state.