I know that one of the Jeans equations can be written in this way [source]:
$$\frac{\partial}{\partial t}\left(\nu \overline{v_{j}}\right)+\frac{\partial\left(\nu \overline{v_{i} v_{j}}\right)}{\partial x_{i}}+\nu \frac{\partial \Phi}{\partial x_{j}}=0 \quad \text { where } \quad \overline{v_{i} v_{j}}=\frac{1}{\nu} \int v_{i} v_{j} f d^{3} \mathbf{v}$$
Assuming the stellar system is:
- in steady state: $\frac{\partial}{\partial t}=0$
- isotropic: $\sigma_{i j}^{2}=\sigma^{2} \delta_{i j}$
- non-rotating: $\overline{v_{i}}=0$
Then the above jeans equation become:
$$-\nu \nabla \Phi=\nabla\left(\nu \sigma^{2}\right)$$
If I know $\nu(r)$ in a spherically symmetric system, I can obtain $\Phi$ using Poisson's equation. My notes claim that I can then solve for $\sigma^2$. I think $\sigma^2$ is. From the above equation:
$$\sigma^2=-\frac{1}{\nu(r)}\int^r\nu(r') \frac{\partial\Phi(r')}{\partial r'} \mathrm{d}r'$$
How do I know the additive constant though, or what is the lower limit of the ingegral for $\sigma^2$?
