 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Physics course of the University of Oxford

Year 4 of the physics course of the University of Oxford

Astrophysics course of the University of Oxford

Past exam of the astrophysics course of the University of Oxford

2018

Interpret $\sigma_{\mathrm{DM}}$ as a constant one-dimensional, isotropic <velocity dispersion>. The effective <pressure> is $p=\rho\sigma_{\mathrm{DM}}^2$, corresponding to the stated gas-temperature analogy. For $\rho=A/r^2$, the spherical enclosed mass is
$$
M_{\mathrm{sph}}(r)=4\pi\int_0^r s^2\rho(s)\,ds=4\pi Ar.
$$
The two sides of the <hydrostatic equilibrium> equation are then
$$
\frac1\rho\frac{dp}{dr}=-\frac{2\sigma_{\mathrm{DM}}^2}{r},\qquad
-\frac{GM_{\mathrm{sph}}(r)}{r^2}=-\frac{4\pi GA}{r}.
$$
Equating them fixes the <singular isothermal sphere>:
$$
\boxed{A=\frac{\sigma_{\mathrm{DM}}^2}{2\pi G},\qquad
\rho(r)=\frac{\sigma_{\mathrm{DM}}^2}{2\pi Gr^2}.}
$$
For collisionless <dark matter> this is an isotropic Jeans-equation construction, not a claim that the particles thermalize by gas collisions. The central singularity and infinite total mass show that the untruncated profile is an idealization.

Solution

Past exam of the astrophysics course of the University of Oxford / 2018 / 5 / a / Solution

Past exam of the astrophysics course of the University of Oxford / 2018 / 5 / a / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the astrophysics course of the University of Oxford / 2018 / 5 / a / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)