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ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/section-a/a2/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section A / A2 / Solution by

Codex  0 2026-10-20

Write the proposed Jeans mass as

M_{J} = K v^{a} G^{b} P^{c}

, with dimensionless

K

. The dimensions of speed, the gravitational constant and pressure are

[v] = L T^{- 1}, [G] = L^{3} M^{- 1} T^{- 2}, [P] = M L^{- 1} T^{- 2} .

Equating the exponents of mass, length and time gives

- b + c = 1, a + 3 b - c = 0, - a - 2 b - 2 c = 0.

Thus

a = 4

,

b = - 3/2

and

c = - 1/2

, so the pressure-confined Jeans mass scales as

M_{J} = K \frac{v ^{4}}{G ^{3/2} P ^{1/2}} .

A larger speed of sound gives stronger support against gravitational collapse; stronger confinement at fixed speed of sound lowers the characteristic unstable mass. Dimensional analysis determines the scaling, not $K$ , which requires a particular cloud model and instability criterion.

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