Charles Beck edited PH1 stable?.tex  almost 9 years ago

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\subsection{Inclination}  Finally, for one last test of the systems stability, we will compare its inclination with Halliday \& Resnick's model. The planet's inclination is 87.4$^{\circ}$ and the mass ratio is $m_\star /M_\star /M_\odot  = .000507/1.936 < 0.05$. As Halliday \& Resnick showed, the smaller the $m_\star /M_\star /M_\odot  $ ratio is, the less impact the inclination has on stability. At a mass ratio of 0.05, every inclination was found to be stable in their integration so anything less than 0.05 mass ratio can be assumed stable. Our ratio is much less than 0.05; therefore, the planet is stable! Unfortunately, Schwamb et al. were unable to report an inclination for the outer binary so we can not make an approximation of its stability based on Halliday \& Resnick's model. We can, however, find the range of inclinations in which the binary \textit{would} be stable. The mass ratio for the outer binary to the inner binary is $m_\star /M_\star /M_\odot  = 1.500/1.936 \approx .77$. For large mass ratios, inclination must be very small for the system to be stable. This ratio of 0.77 is very large so an inclination of $i_0 \approx 0^{\circ}$ would be required for the outer binary to be considered stable.