 Solution

ID: past-exam-of-the-radiative-processes-and-high-energy-astrophysics-course-of-the-university-of-oxford/2024/2/b/solution

n_{e} = n_{p}

and

ρ = (m_{p} + m_{e}) n_{e} ≃ m_{p} n_{e}

ℓ = (n_{e} σ_{T})^{- 1}

. Hence

ℓ (r) = L \frac{a r}{sin a r}, L = \frac{m _{p}}{σ _{T} ρ _{c}} = \frac{4 m _{p} R ^{3}}{π σ _{T} M}, \frac{L}{R} = \frac{4 m _{p} R ^{2}}{π σ _{T} M} .

Here and below the tiny electron mass correction can be retained by replacing

m_{p}

m_{p} + m_{e}

L

r = R

reflects the vanishing electron number density.

Using

M_{⊙} = 1.9885 \times 1 0^{30} kg

R_{⊙} = 6.957 \times 1 0^{8} m

m_{p} = 1.6726 \times 1 0^{- 27} kg

, and

σ_{T} = 6.6525 \times 1 0^{- 29} m^{2}

gives

ρ_{c} ≃ 4.64 \times 1 0^{3} kg m^{- 3}, L_{⊙} ≃ 5.42 \times 1 0^{- 3} m, L_{⊙} / R_{⊙} ≃ 7.79 \times 1 0^{- 12} .

These are this profile's values, rather than the Sun's actual central mass density or opacity profile.

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