Changes between Version 18 and Version 19 of u/BonnorEbert
- Timestamp:
- 01/15/16 14:54:50 (9 years ago)
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u/BonnorEbert
v18 v19 21 21 To arrive at the Lane-Emden equation, one begins with the equation of hydrostatic equilibrium, Poisson's equation for gravity, and the isothermal equation of state: 22 22 23 {{{#!latex 24 \frac{-1}{\rho} \triangledown P - \triangledown\phi_g ~= ~0 \qquad (1) 23 [[latex($\frac{-1}{\rho} \triangledown P - \triangledown\phi_g ~= ~0 \qquad (1)$)]] 25 24 26 }}}27 25 28 {{{#!latex29 \triangledown^2\phi_g~=~4\pi G \rho \qquad (2)30 }}}31 26 32 {{{#!latex 33 P ~=~ \rho ~C_s^2 \qquad (3) 34 }}} 27 [[latex($\triangledown^2\phi_g~=~4\pi G \rho \qquad (2)$)]] 28 29 [[latex($P ~=~ \rho ~C_s^2 \qquad (3)$)]] 35 30 36 31 Plugging (3) into (1) for P shows: 37 32 38 {{{#!latex 39 ln ~\rho + \frac{\phi_g}{C_s^2} ~= ~const. \Rightarrow 40 }}} 33 [[latex($ ln ~\rho + \frac{\phi_g}{C_s^2} ~= ~const. \Rightarrow $)]] 41 34 42 {{{#!latex 43 \rho(r) ~ =~ \rho_c~e^{(-\phi_g/C_s^2)}\qquad--for~spherical~geometry ~(4), 44 }}} 35 [[latex($ \rho(r) ~ =~ \rho_c~e^{(-\phi_g/C_s^2)}\qquad--for~spherical~geometry ~(4),$)]] 45 36 46 37 which, when inserted into (2), yields: 47 38 48 {{{#!latex 49 \frac {1}{r^2} \frac {d}{dr}~(r^2 \frac{d\phi_g}{dr}) ~= ~ 4\pi G \rho_c e^{{(-\phi_g/C_s^2)}}~ \qquad--spherical coordinates 39 [[latex($\frac {1}{r^2} \frac {d}{dr}~(r^2 \frac{d\phi_g}{dr}) ~= ~ 4\pi G \rho_c e^{{(-\phi_g/C_s^2)}}~ \qquad--spherical ~coordinates $)]] 50 40 51 }}}52 41 53 42 By making the following variable substitutions, 54 {{{#!latex d 55 \psi ~= ~ \frac{\phi_g}{C_s^2} 56 }}} 43 44 [[latex($d\psi ~= ~ \frac{\phi_g}{C_s^2}$)]] 57 45 58 46 59 {{{#!latex 60 \xi ~=~ (\frac{4 \pi G \rho_c}{C_s^2})^\frac{1}{2}~r 61 }}} 47 [[latex($\xi ~=~ (\frac{4 \pi G \rho_c}{C_s^2})^\frac{1}{2}~r$)]] 62 48 63 49 one arrives at the '''''Lane-Emden''''' equation: 64 50 65 {{{#!latex 66 \frac{1}{\xi^2}\frac{d}{d\xi}(\xi^2\frac{d\psi}{d\xi}) = e^{-\psi} \qquad(Lane-Emden Equation)(5) 67 68 }}} 51 [[latex($ \frac{1}{\xi^2}\frac{d}{d\xi}(\xi^2\frac{d\psi}{d\xi}) = e^{-\psi} \qquad(Lane-Emden ~Equation)~(5)$)]] 69 52 70 53 '''Dynamics'''