26 | | |
27 | | |
28 | | |
29 | | |
| 26 | Substituting the Poisson equation into the momentum equation we have |
| 27 | |
| 28 | [[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right ) \right ]$)]] |
| 29 | |
| 30 | |
| 31 | |
| 32 | |
| 33 | |
| 34 | For the energy equation, we can only find a conservative approach - if we consider the evolution of the total combined energy (hydro + gravitational). |
| 35 | |
| 36 | [[latex($\frac{d \left(E + \frac{1}{2} \left (\rho - \rho_0 \right) \phi \right)}{dt} = -\nabla \cdot \mathbf{F_g} $)]] |
| 37 | |
| 38 | |
| 39 | Combining this with the our previous energy equation we arrive at |
| 40 | |
| 41 | [[latex($\nabla \cdot \mathbf{F_g} = -\frac{1}{2} \frac{d \left (\rho \phi \right)}{dt} + \frac{1}{2} \rho_0 \frac{d \phi}{dt}+ \rho \mathbf{v} \cdot \nabla \phi$)]] |
| 42 | |
| 43 | Expanding the time derivative and using the continuity equation, we arrive at |
| 44 | |
| 45 | |
| 46 | [[latex($\nabla \cdot \mathbf{F_g} = \frac{1}{2} \phi \nabla \cdot \left ( \rho \mathbf{v} \right) - \frac{1}{2} \left (\rho - \rho_0 \right) \dot{\phi} + \rho \mathbf{v} \cdot \nabla \phi$)]] |
| 47 | |
| 48 | |
| 49 | Now from time differencing the poisson equation, we arrive at |
| 50 | |
| 51 | [[latex($\nabla^2 \dot{\phi} = 4 \pi G \dot{\rho} = - 4 \pi G \nabla \cdot (\rho \mathbf{v})$)]] |
| 52 | |
| 53 | and multiplying both sides by [[latex($\frac{\phi}{8 \pi G}$)]] gives |
| 54 | |
| 55 | [[latex($\frac{1}{2} \nabla \cdot (\rho \mathbf{v}) \phi = -\frac{1}{8 \pi G} \phi \nabla^2 \dot{\phi}$)]] |
| 56 | |
| 57 | |
| 58 | Adding the LHS and subtracting the RHS and substituting |
| 59 | |
| 60 | [[latex($\rho - \rho_0 = \frac{1}{4 \pi G} \nabla^2 \phi$)]] |
| 61 | |
| 62 | into the divergence equation gives us |
| 63 | |
| 64 | [[latex($\nabla \cdot \mathbf{F_g} = \nabla \cdot \left (\phi \rho \mathbf{v} \right) + \frac{1}{8 \pi G} \left (\phi \nabla^2 \dot{\phi} - \dot{\phi}\nabla^2\phi \right) $)]] |
| 65 | |
| 66 | Now we can use the relationship |
| 67 | |
| 68 | [[latex($\nabla \cdot \left [ \phi \nabla \dot{\phi} - \dot{\phi} \nabla \phi \right ]= \phi \nabla^2 \dot{\phi} - \dot{\phi} \nabla^2 \phi$)]] |
| 69 | |
| 70 | to arrive at |
| 71 | |
| 72 | [[latex($\mathbf{F_g} = \phi \rho \mathbf{v} + \frac{1}{8 \pi G} \left (\phi \nabla \dot{\phi} - \dot{\phi}\nabla \phi \right) $)]] |
| 73 | |
| 74 | |
| 75 | [[CollapsibleStart(Notes)]] |