Changes between Version 20 and Version 21 of SelfGravityDevel


Ignore:
Timestamp:
03/05/12 22:11:54 (13 years ago)
Author:
Baowei Liu
Comment:

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  • SelfGravityDevel

    v20 v21  
    6565
    6666The free-fall time is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse.
    67 Using the Gauss's theorem, the gravitional acceleration $g_{r}$ is given by
     67Using the Gauss's theorem, the gravitional acceleration [[latex($g_{r}$)]] is given by
    6868{{{
    6969#!latex
    … …  
    9393\end{equation}
    9494}}}
    95 Suppose the gas molecule is initially located at $r=R$. So the total energy
     95Suppose the gas molecule is initially located at [[latex($r=R$)]]. So the total energy
    9696{{{
    9797#!latex
    … …  
    100100\end{equation}
    101101}}}
    102 Now assume [[latex($M_{r}\approx M_{r}(R)$)]]. Equation (8) gives
     102Now consider gas collapse. So [[latex($M_{r}= M_{r}(R)$)]]. Equation (8) gives
    103103{{{
    104104#!latex
    … …  
    107107\end{equation}
    108108}}}
    109 Let [[latex($r=R\cos^{2}\eta$)]]. From the initial state we have [[latex($\eta(t=0)=0$)]]. So Equation (10) reduces to
     109Let
    110110{{{
    111111#!latex
    112 \begin{equation}\label{eq:reduced}\tag{11}
     112\begin{equation}\tag{11}
     113r=R\cos^{2}\eta
     114\end{equation}
     115}}}. From the initial state we have [[latex($\eta(t=0)=0$)]]. So Equation (10) reduces to
     116{{{
     117#!latex
     118\begin{equation}\label{eq:reduced}\tag{12}
    113119\cos^{2}\eta\frac{d\eta}{dt}=\left(\frac{GM_{r}(R)}{2R^{3}}\right)^{\frac{1}{2}}
    114120\end{equation}
    115121}}}
    116 Solve Equation (11) we have
     122Solve Equation (12) we have
    117123{{{
    118124#!latex
    119 \begin{equation}\label{eq:sol}\tag{12}
     125\begin{equation}\label{eq:sol}\tag{13}
    120126t=\left(\frac{R^{3}}{2GM_{r}(R)} \right)^{\frac{1}{2}}\left(\eta+\frac{\sin2\eta}{2} \right)
    121127\end{equation}
    … …  
    124130{{{
    125131#!latex
    126 \begin{equation}\ref{eq:freeFall}\tag{13}
     132\begin{equation}\ref{eq:freeFall}\tag{14}
    127133\begin{aligned}
    128134 t_{ff}&=\left(\frac{R^{3}}{2GM_{r}(R)}\right)^{\frac{1}{2}}\frac{\pi}{2}\\
    … …  
    134140{{{
    135141#!latex
    136 \begin{equation}\label{rhoBar}\tag{14}
     142\begin{equation}\label{rhoBar}\tag{15}
    137143\rho_{0}=\frac{M_{r}(R)}{\frac{4\pi R^{3}}{3}}
    138144\end{equation}
    … …  
    142148
    143149== Implementation ==
    144 
    145 
     150=== Uniform Density Cloud ===
     151The definition of [[latex($\eta$)]] in Equation (11) can also be written as
     152{{{
     153#!latex
     154\begin{equation}\tag{16}
     155\cos\eta=\left(\frac{\rho}{\rho_{0}} \right)^{-\frac{1}{6}}
     156\end{equation}
     157}}}
    146158=== Boundary Conditions ===
    147159