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\section{Binary stars}  Binary stars \textit{Binary stars}  are systems which are very useful when determining stellar masses and sizes. Binaries are two stars in mutual gravitational interaction orbiting their common center of mass (see Fig. 5). Higher order binaries like triples and quadruples exist as well. Most stars are born as multiples. We distinguish:  \begin{itemize}  \item{Visual binaries: \item{\textit{Visual binaries}:  Two stars are seen separately} \item{Spectroscopic binaries: \item{\textit{Spectroscopic binaries}:  Stars are not seen separately, but the spectrum shows two set of lines moving in opposite directions due to the Doppler shift.} \item{Eclipsing binaries: \item{\textit{Eclipsing binaries}:  Stars are not seen separately, but one star eclipses the other in regular intervals.} \end{itemize}  Binaries are useful for two reasons: a) Orbits determined by gravity, i.e. they can be used to determine masses. b) Eclipses determined by sizes, i.e. can be used to determine radii.  \subsection{Doppler effect}  The wavelength that an observer measures depends on the relative motion between the observer and the light source. If the source is moving towards the observer, the light is blueshifted ($\lambda < \lambda_0$). If the source is moving away from the observer, the light is redshifted ($\lambda >\lambda_0$). This is called the Doppler effect \textit{Doppler effect}  (analogous to the Doppler effect with sound waves). The shift in wavelength due to the Doppler effect in the light of a star is this: 
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a^3 = (m_1 + m_2) P^2  \end{equation}  This is Kepler's \textit{Kepler's  3rd law. law}.  If $a$ and $P$ are measured,either from observing the orbits in a visual system or from observing the Doppler shifts in a spectroscopic system,  the sum of the stellar masses can be calculated. Factors that are neglected here: elliptic orbits, inclination of orbits against the line of sight (for spectroscopic binaries) or the sky (for visual binaries). \subsection{Eclipsing binaries}  Fig. XXX shows the lightcurve of an eclipsing binary system. From the lightcurve, $t_1$, $t_2$, $t_3$, $t_4$ can be measured. The geometry of the system then gives expressions for the radii $R_1$ and $R_2$ of the stars:  \begin{equation}  t_4 - t_1 / P = (R_1 + R_2) / (2 \pi R)  \end{equation}  \begin{equation}  t_3 - t_2 / P = (R_1 - R_2) / (2 \pi R)  \end{equation}  With these two equations the radii can be derived in units of the orbital separation $R$. The radial velocities $v$ and the period $P$ yield $R$, and thus the radii of the stars. This only works in an eclipsing binary which is also a spectroscopic binary. This rare type of system is therefore extremely important in astronomy  \section{Synopsis}