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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-72/3/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 3 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For a singular isothermal sphere, spherical integration of the mass density gives

M (< R) = \int_{0}^{R} 4 π r^{2} \frac{σ ^{2}}{2 π G r ^{2}} d r = \frac{2 σ ^{2} R}{G} .

(1)

The spherical shell theorem makes the radial attraction

GM (< R) / R^{2}

. Balancing it with circular centripetal acceleration yields

M (< R) = \frac{2 σ ^{2} R}{G}, V_{c} (R) = \frac{GM ( < R )}{R} = 2 σ .

(2)

Thus the model has a flat galaxy rotation curve, resembling the outer galaxy rotation curves of many spiral galaxies. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported elliptical galaxy does not itself rotate at this test-particle circular speed.

The model has divergent central mass density, although finite enclosed mass at each finite radius, and an infinite total mass because

M (< R) \propto R

. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic velocity dispersion, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the dark matter halo; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.

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