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ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015/7/b/solution

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Past exam of the astrophysics course of the University of Oxford / 2015 / 7 / b / Solution by

Codex  0 2026-10-25

For spherical gravity,

d Φ/ d r = GM (\leq r) / r^{2}

. Inserting

β = 1 - σ_{t}^{2} / σ_{r}^{2}

into the Spherical Jeans equation and solving for the total enclosed mass gives

M (\leq r) = - \frac{r ^{2}}{G ν} [\frac{d ( ν σ _{r}^{2} )}{d r} + \frac{2 β ν σ _{r}^{2}}{r}] = - \frac{r σ _{r}^{2}}{G} (\frac{d ln ν}{d ln r} + \frac{d ln σ _{r}^{2}}{d ln r} + 2 β) .

(1)

The mass includes all sources of the potential, including dark matter or a central compact object; the stellar tracer density need not follow total mass. The bracket must yield nonnegative enclosed mass in a physically admissible model. Density, radial stress and anisotropy are independent observational/model inputs before this equation becomes a mass estimate.

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