| | 39 | To model the ring, we imagine the following situation: |
| | 40 | |
| | 41 | 1. a uniform thermal pressure distribution in the ring (so ignore gradients in pressure) |
| | 42 | 2. steady state (so ignore time derivatives, implying the bulk fluid is at rest so throw out terms with u also) |
| | 43 | 3. the ring is puffed out from a spherical flow of gas expelled from the collision region |
| | 44 | 4. there is no field within the cylinder of the colliding flows themselves (which is a good approximation given how dynamically weak the field is there) |
| | 45 | 5. to track the ring we define 2 radii, the outer radius of the ring [[latex($r_o$)]], and the inner radius of the ring, [[latex($r_i$)]] (both relative to the center of the collision region) |
| | 46 | 6. this ring contains the same amount of flux as it spreads out, by flux freezing. this means that as the ring grows the field strength is geometrically diluted |
| | 47 | 7. as far as the ram pressure pushing this ring out, we assume the velocity of the ejecta is a constant with radius, and only the density is decreasing by geometrical dilution. |
| | 48 | 8. we take the ram pressure of the ejecta to be the incoming ram pressure. this is a reasonable approximation as the shocked sound speed should be of order the incoming velocity. |
| | 49 | |
| | 50 | This gives: |
| | 51 | |
| | 52 | [[latex($0 = -\bold{\nabla}(\frac{B^2}{2 \mu}) + \frac{B^2}{\mu} \frac{\hat{\bold{n}}}{R_c}$)]] |
