deletions | additions       diff --git a/sectionResults__subs.tex b/sectionResults__subs.tex  index d11b224..bbd0703 100644  --- a/sectionResults__subs.tex  +++ b/sectionResults__subs.tex 
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I subsequently found MWC 237 in the CCDM catalog \cite{Dommanget2002} as the visual binary CCDM J15329+3122AB. The source data for the CCDM entry were observed in 1971, at which time the binary separation was $\rho$=0.5" with position angle PA=203$\deg$. The 2009 Gemini data show $\rho$=0.85" and PA=198.5$\deg$. CCDM gives proper motion for the primary component only. If AB is a true binary system, then it shares a common proper motion and the relative position change is due to orbital motion. With two widely spaced measurements of $\rho$ and PA 38 years apart, an estimate of the orbital period and inclination can be made. By fitting an ellipse, centered on the primary, to the two positions along the orbit, the semi-major axis can be measured. Then by assuming the orbit to be circular, the ratio of major to minor axes of the fitted ellipse provides the inclination of the orbital plane to the line of sight. The period of this orbit is given by Kepler's Law (Equation~\ref{eq:kepler}), using masses for A and B derived from their spectral types and absolute magnitudes. Finally we can test whether the distance traveled by B in the 38 years between measurements is consistent with the orbital solution.  \begin{equation}  \label{eq:kepler} \label{eqn:kepler}  \(P = \frac{2\pi a^{3/2}}{\sqrt{G(M+m)}} \)  \end{equation}  G=6.67 $\times$10$^{-11}m^{3}kg^{-1}s^{-2}$ 
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\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{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{eq:ciradullo}, 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 \textless 6.24) as well as their companions (K mag \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{eq:ciradullo} \label{eqn:ciradullo}  \(P = 1 - \left(1 - \frac{\pi\rho^2}{A}\right)^N \)  \end{equation}