deletions | additions       diff --git a/section_The_Simulation_Process_Holman__.tex b/section_The_Simulation_Process_Holman__.tex  index 5497f6a..5982b45 100644  --- a/section_The_Simulation_Process_Holman__.tex  +++ b/section_The_Simulation_Process_Holman__.tex 
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Their results showed that the particle was more stable the smaller the initial semimajor axis was. At a starting semimajor axis of $a=0.17$, no planet survived more than 30 binary periods. Then , as the semimajor axis decreased, the planets lasted more binary periods. The planet closest to its parent star lasted longer than any of the other stars with the same semimajor axis for every trial but one. Interestingly, though, the increase in binary periods lasted is very slow until a semimajor axis of $a=0.14$ where it jumps to a median binary periods lasted of 175.5. Then, at the critical semimajor axis, there is a huge jump where every planet lasts 10,000 binary periods. This implies a couple of things; first, the distance from the star is not as influential in the stability as the semimajor axis. Second, the critical semimajor axis is a threshold. The semimajor axis does not gradually become more stable, it is either stable or not.    \subsubsection{Comparison with Earlier Work}  The authors then compare their results to that of Rabl and Dvorak who did an almost identical simulation 11 years prior \cite{1988A&A...191..385R} \cite{1988A&A...191..385R}. One major difference was simply the computing power. Rabl and Dvorak only ran the integration for 300 binary periods because computers were slow and expenxive meaning running the simulation longer would hvae been uneconomical.  Even with the lower precision, however, this simulation found very similar results. Looking at Rabl and Dvorak's comparable table ($e=0.5$), the same patterns arise: as the semimajor axis decreases, the stability increases. The planet closest to the stars is generally more stable compared to further planets with the same semimajor axis. And, most importantly, at $a=0.12$, every planet remained stable for more than 300 binary cycles.   Holman and Wiegert calculated a best fit line for their critical semimajor axes which came to  \begin{equation}  a_c = .274-.338e+.051e^2 ,  \end{equation}  which they compared to Rabl and Dvorak's:  \begin{equation}  a_c = .262-.254e-.06e^2 .  \end{equation}  As depicted in \textbf{HWvRD}, the expressions are clearly not equal, but are extremely close.