 Solution

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

A free-fall timescale estimate follows from

R / t^{2} \sim GM / R^{2}

t_{dyn} \sim \frac{R _{⊙}^{3}}{G M _{⊙}} ≃ 1.6 \times 1 0^{3} s .

Falling to a central point mass gives the more precise factor

π / (22)

, or

1.77 \times 1 0^{3} s

; a distributed star changes an order-one coefficient. For any spherical mass distribution with radius

R

∣Ω∣ = \int_{0}^{M} \frac{G m}{r ( m )} d m \geq \frac{G}{R} \int_{0}^{M} m d m = \frac{G M ^{2}}{2 R} .

The inequality is strict for an extended star; equality is approached by a thin shell at

R

2 U + Ω = 0

, so half the gravitational potential energy released by contraction heats the star and half can be radiated. The Kelvin-Helmholtz cooling time is therefore of order

t_{KH} \sim \frac{G M _{⊙}^{2}}{2 R _{⊙} L _{⊙}} ≃ 1.6 \times 1 0^{7} yr .

A uniform-density model gives

3 G M^{2} / (10 R L) ≃ 9.4 \times 1 0^{6} yr

. The given bound fixes a scale, rather than an exact contraction history; both estimates are far shorter than the Sun's main-sequence lifetime.

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