For one component with density $\rho_k=A_kr^{-2\beta_k}\Psi^{p_k}$, direct velocity integration, or the <Spherical Jeans equation>, gives
$$
P_{r,k}=\rho_k\sigma_{r,k}^2
=\frac{\rho_k\Psi}{p_k+1}.
$$
Each component has velocity-anisotropy parameter $\beta_k$. Therefore the combined coefficient is the radial-pressure-weighted mean
$$
\widehat\beta
=\frac{\beta_1P_{r,1}+\beta_2P_{r,2}}
{P_{r,1}+P_{r,2}}.
$$
Here
$$
q(r)=\frac{P_{r,2}}{P_{r,1}}
=\frac34\frac{\rho_2}{\rho_1}
=\frac{\sqrt{br}}{D(r)},
$$
so
$$
\boxed{
\widehat\beta(r)=
\frac{3+2q(r)}{4[1+q(r)]}}.
$$
Since $q\sim\sqrt{r/b}/2$ at the origin and $q\sim\sqrt{b/r}$ at infinity,
$$
\boxed{
\widehat\beta\longrightarrow\frac34
\quad\text{as }r\to0
\quad\text{and as }r\to\infty}.
$$
The second, less radial component reduces the anisotropy only at intermediate radii.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-320/2/c/solution