| 395 | | Surprisingly this cubic has no real positive roots! To see why consider the radius ratio of the isolated parent and child clump. From the Jeans relation, the child clump radius should be .7071 and it would occupy 35% of the volume. This would then give a mean parent density of 2*.35+.65 = 1.35 that will require the parent radius be shrunk. This however will only increase the mean density so the solution is stuck between shrinking the parent cloud to smaller and smaller radii is too high. To lower the mean parent density we can increase the radius of the parentTo increase the mean parent density we need to If we include this additional contribution to the parent clump, the parent clump radius would shrink and the child clump would occupy even more then 35% of the volume. On the other hand since the ratio of |
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| 398 | | If we have multiple levels |
| | 395 | Surprisingly this cubic has no real positive roots! To see why consider the radius ratio of the isolated parent and child clump. From the Jeans relation, the child clump radius should be .7071 and it would occupy 35% of the volume. This would then give a mean parent over density of 2*.35+.65 = 1.35 that will require the parent radius be shrunk. This however will also increase the mean density at a rate faster then making the clump smaller can compensate for. At higher density contrasts and lower filling fractions, this obstacle can be overcome and a solution exists. |
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| | 398 | If we have multiple levels we can start on the highest level 'L' where [[latex($\xi_L=1$)]] and solve for the filling fraction for level 'L-1'. This can then be used to calculate [[latex($\overline{\rho_{L-1}}$)]] and [[latex($\xi_{L-1}$)]] which then allows us to solve for the filling fraction for the next higher level and so on. Finally once we have [[latex($\xi_0$)]] and [[latex($r_0$)]] we can calculate [[latex($\rho_0$)]] which can then be used to calculate [[latex($\rho_1,\rho_2,...,\rho_L$)]] |
