# Star equations of hydrostatic equilibrium for a mix of 2 fleebly interacting gases/fluids and interaction term

For a single matter species, the equations of hydrostatic equilibirum for a star are $$\begin{eqnarray} \nabla^2 \phi &=& 4\pi G \rho\\ \vec{\nabla} P + \rho \vec{\nabla}\phi &=&0 \end{eqnarray}$$ These can be combined with an Equation of State $$P=f(\rho)$$ (degenerate case), or with both the equation of state $$P=f(\rho,T)$$ and the heat transport equation (non-degenerate case), to find a solution. These equations have also a generelisation in GR, that are called TOV equations.

When generalising to 2 components, that have 2 different EoS, however, I was only able to find the ones that assume the 2 components do not interact, that are an easy generalisation of the above, with $$\rho\rightarrow \rho_A +\rho_B$$, and 2 copies of the second equation, one with $$P\rightarrow P_A$$, $$\rho\rightarrow \rho_A$$, and another one for the $$B$$ component. In the general case where they interact very weakly, however, the equations should take the form

$$\begin{eqnarray} \nabla^2 \phi &=& 4\pi G (\rho_A+\rho_B)\\ \vec{\nabla} P_A + \rho_A\vec{\nabla}\phi &=&\vec{F}_{AB}\\ \vec{\nabla} P_B + \rho_B\vec{\nabla}\phi &=&- \vec{F}_{AB} \end{eqnarray}$$

I would expect that the term $$\vec{F}_{AB}$$ should depend on the interaction rate/cross section of interaction of the 2 components. However, I could not find/derive any specific form that makes sense. My ideas were:

• It should be antisymmetric for exchange $$A\leftrightarrow B$$,
• Should be proportional to the interaction rate of the 2 components, and therefore to the number densities of both species, the interaction cross section and the relative speed,
• To give a force, it should also be proportional to the average momentum exchanged, that means another power of relative speed and one of reduced mass of the 2 particles,
• Should have dimensions of force per unit volume,
• Should be a local operator assuming interactions are only contact interactions and not long-range,
• It should include some derivative/gradient, to allow boundary conditions in $$r=0$$ to be regular, and to make it such that it is zero in an isotropic medium.

However, all these assumptions do not seem to be compatible with each other... I suspect at least one is wrong... Does anybody know the solution?

For $$\vec F_{AB}$$ to have any effect, you need to have nonzero momentum. Then you can parameterize this term either as in Benitez-Llambay+2019, and the momentum equation for species i becomes
$$\rm \frac{Dv_i}{Dt} = -\frac{\nabla P_i}{\rho_i} - \sum_{i\neq j}\alpha_{ij}(v_i-v_j)$$ where it has to be fulfilled that $$\alpha_{ij} \rho_i = \alpha_{ji} \rho_j$$ and the alphas can be computed from $$\alpha_{ij} = k_B T \rho_j /(m_{i} b)$$, and the $$b$$'s are the binary diffusion coefficients, coming from lab measurements or kinetic theory.
can be usually satisfied via the momentum diffusion operator $$\nabla(\mu \nabla v)$$, with $$\mu$$ being the kinematic viscosity, which would appear if internal friction is taken into account.