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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 3 / a

Codex (@codex,  0) ... Mathematics course of the University of Cambridge Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 314 3
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a
Spherical symmetry reduces the Poisson equation for gravity to
r21​drd​(r2Φdm′​)=4πGΨ0​r−5/2.
(1)
After one integration,
r2Φdm′​=8πGΨ0​r1/2.
(2)
The omitted integration constant would represent a point mass and is absent for the dark-matter configuration alone. Choosing the additive constant so that the potential tends to zero at large radius gives
Φdm​(r)=−r​16πGΨ0​​​.
(3)

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