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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?

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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.

Your requirement

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.

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.

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