deletions | additions       diff --git a/sectionResults__subs.tex b/sectionResults__subs.tex  index c939e89..3d6bc58 100644  --- a/sectionResults__subs.tex  +++ b/sectionResults__subs.tex 
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\begin{equation}  \label{eqn:kepler}  P \(P  = \frac{2\pi a^{3/2}}{\sqrt{G(M+m)}} \)  \end{equation}  G=6.67 $\times$10$^{-11}m^{3}kg^{-1}s^{-2}$  \subsection{Stars Without Close Companions}  \subsection{Probability of Chance Alignments}  This section addresses the likelihood of visual binaries resulting from chance alignments with background stars. The primary objective stars included in this study were very bright, so short exposure times were used to obtain unsaturated images with a useful S/N ratio. As a result, observations of dim background stars are unlikely. Using a method introduced by Cirudillo, the observer can calculate the probability that a star of equal or greater brightness than the observed companion has randomly aligned with the primary objective \cite{Romero2007}. Calculation of probability of chance overlap shown in Table~\ref{table:Probability} Table\ref{tab:Probability}  are negligible. The bright stars in the sample are all nearby and many have appreciable proper motions. Common proper-motion of the primary star and the putative companion, or an orbit, are the only secure method to be sure of a binary system. For thoroughness, follow-up observations of the binary companions have been made to verify common proper motion and binarity.  On a case-by-case basis, the equation used by Ciradullo et al. , Equation~\ref{eqn:ciradullo}, gives the probability that an observed companion is a random interloper. The probability \(P\) on the left hand side is the probability that the observed companion is not physically associated with the primary star. The term \rho is the angular separation of the primary and companion stars. The right-hand side contains the term \(A\) the “area of the sky where we have searched for stars brighter than the secondary (expressed in the same unit as \rho )“. The term \(N\) is the number of stars brighter than the secondary found in \(A\). For \(A\), a circle with a radius of one degree was searched around each of the primary stars. Because the binary companions are so bright ( K mag \lt \textless  6.24) as well as their companions (K mag \lt \textless  8.96), the number of field stars brighter than the companions found in \(A\) was relatively small, resulting in a near 0 chance of coincidental overlap. \begin{equation}  \label{eqn:ciradullo} \label{eq:ciradullo}  \(P = 1 - \left(1 - \frac{\pi\rho^2}{A}\right)^N \)  \end{equation}