Changes between Version 18 and Version 19 of u/BonnorEbert


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Timestamp:
01/15/16 14:54:50 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/BonnorEbert

    v18 v19  
    2121To 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:
    2222
    23 {{{#!latex
    24 \frac{-1}{\rho} \triangledown P - \triangledown\phi_g ~= ~0 \qquad  (1)
     23[[latex($\frac{-1}{\rho} \triangledown P - \triangledown\phi_g ~= ~0 \qquad  (1)$)]]
    2524
    26 }}}
    2725
    28 {{{#!latex
    29 \triangledown^2\phi_g~=~4\pi G \rho \qquad (2)
    30 }}}
    3126
    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($P ~=~ \rho ~C_s^2 \qquad (3)$)]]
    3530
    3631Plugging (3) into (1) for P shows:
    3732
    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 $)]]
    4134
    42 {{{#!latex
    43  \rho(r) ~ =~ \rho_c~e^{(-\phi_g/C_s^2)}\qquad--for~spherical~geometry ~(4),
    44 }}}
     35[[latex($ \rho(r) ~ =~ \rho_c~e^{(-\phi_g/C_s^2)}\qquad--for~spherical~geometry ~(4),$)]]
    4536
    4637which, when inserted into (2), yields:
    4738
    48 {{{#!latex
    49  \frac {1}{r^2} \frac {d}{dr}~(r^2 \frac{d\phi_g}{dr}) ~= ~ 4\pi G \rho_c e^{{(-\phi_g/C_s^2)}}~ \qquad--spherical coordinates
     39[[latex($\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 $)]]
    5040
    51 }}}
    5241
    5342By making the following variable substitutions,
    54 {{{#!latex d
    55 \psi ~= ~ \frac{\phi_g}{C_s^2}
    56 }}}
     43
     44[[latex($d\psi ~= ~ \frac{\phi_g}{C_s^2}$)]]
    5745
    5846
    59 {{{#!latex
    60 \xi ~=~ (\frac{4 \pi G \rho_c}{C_s^2})^\frac{1}{2}~r
    61 }}}
     47[[latex($\xi ~=~ (\frac{4 \pi G \rho_c}{C_s^2})^\frac{1}{2}~r$)]]
    6248
    6349one arrives at the '''''Lane-Emden''''' equation:
    6450
    65 {{{#!latex
    66  \frac{1}{\xi^2}\frac{d}{d\xi}(\xi^2\frac{d\psi}{d\xi}) = e^{-\psi}  \qquad(Lane-Emden Equation)(5)
    67 
    68 }}}
     51[[latex($ \frac{1}{\xi^2}\frac{d}{d\xi}(\xi^2\frac{d\psi}{d\xi}) = e^{-\psi}  \qquad(Lane-Emden ~Equation)~(5)$)]]
    6952
    7053'''Dynamics'''