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ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2019/s2/2/a/solution

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Past exam of the short options paper of the physics Prelims of the University of Oxford / 2019 / S2 / 2 / a / Solution by

Codex  0 2026-10-25

A thin spherical shell has outward pressure force

- 4 π r^{2} (d P / d r) d r

and inward gravitational force

GM (r) 4 π r^{2} ρ (r) d r / r^{2}

. Hydrostatic equilibrium requires

\frac{d P}{d r} = - \frac{GM ( r ) ρ ( r )}{r ^{2}}, \frac{d M}{d r} = 4 π r^{2} ρ (r) .

A uniform-density solar model gives

M (r) = M_{⊙} (r / R_{⊙})^{3}

,

\overset{ρ}{ˉ} = 3 M_{⊙} / (4 π R_{⊙}^{3})

, and after integration from

R_{⊙}

to zero,

P_{c} = \frac{3 G M _{⊙}^{2}}{8 π R _{⊙}^{4}} = 1.34 \times 1 0^{14} Pa, T_{c} = \frac{P _{c} μ m _{p}}{ρ ˉ k _{B}} = 6.9 \times 1 0^{6} K (μ = 0.6) .

Thus the appropriate orders of magnitude are

1 0^{14}

–

1 0^{16} Pa

and

1 0^{7} K

. The uniform-density approximation underestimates the actual central pressure because the Sun is centrally concentrated. Radiation pressure is neglected in applying the ideal gas law.

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