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\begin{document}
\title{Evidence for a binary lens in the MACHO LMC \#1 microlensing event}
\author{M. Dominik \and A. C. Hirshfeld}
\date{Received 9 November 1995 / Accepted 18 February 1996}

\maketitle

\begin{appendix}
\renewcommand{\theequation}{\thesection\arabic{equation}}
\addtocounter{figure}{4}

\section{Light curve of an extended source}
\setcounter{equation}{0}
Let $\mu_p(\vec y\,)$ denote the total magnification of a point source
at the position $\vec y$ in the source plane.
The magnification for an extended source with intensity $I(\vec y\,)$ is
then given by
\begin{equation}
\mu_e = {\int I(\vec y\,) \,\mu_p(\vec y\,) \, d^2 y \over 
	\int I(\vec y\,) \, d^2 y}\,,
\end{equation}
where the integration is performed over the source plane.

For a source with uniform brightness extending over a region $F$,
the magnification follows as
\begin{equation}
\mu_e = {\int\limits_F \mu_p(\vec y\,) \,d^2 y \over \int\limits_F d^2 y}\,,
\end{equation}
where the integration is performed over the region $F$. 

For an axisymmetric lens, $\mu_p(\vec y\,)$ depends only on $|\vec y\,| 
\equiv r$, and by using polar coordinates $(r,\vartheta)$, $\mu_e$ can
be rewritten as
\begin{equation}
\mu_e = {\int\limits_F d\vartheta\, \mu_p(r)\, r\,dr \over 
	\int\limits_F d\vartheta \,r\,dr}\,.
\end{equation}
Since the integrand does not depend on $\vartheta$, the integration
over $\vartheta$ yields the range $\Delta\vartheta(r,F)$ 
of $\vartheta$--values, and only an integral over $r$ remains.
The integral in the denominator is simply the area of $F$.

Let us now consider a circular source of radius $R$ at a distance $d$.
For this case the magnification is
\begin{equation}
\mu_e(d,R) = {1 \over \pi R^2}\,\int r\, \mu_p(r)\,\Delta\vartheta(r,d,R)\,dr
\,.
\end{equation}

For the various ranges of $r$ and $\vartheta$, three cases 
have to be distinguished.
For the case $d > R$, $r \in [d-R,d+R]$ and $\Delta\vartheta(r,d,R)
= 2 \arccos {d^2+r^2-R^2 \over 2dr}$. 
For the case $d < R$, $\Delta\vartheta = 2\pi$ for $r \in [0,R-d]$ and
$\Delta\vartheta(r,d,R) = 2 \arccos {d^2+r^2-R^2 \over 2dr}$ for
$r \in [R-d,R+d]$.
For $d = R$, $r \in [0,2R]$ and $\Delta\vartheta(r,d,R) = 2
\arccos {r \over 2R}$.

Using these results, $\mu_e(d,R)$ can be written as
\begin{equation}
\mu_e(d,R) = \left\{\begin{array}{lcl}
{2 \over \pi R^2} \int\limits_{d-R}^{d+R} r\, \mu_p(r)\, \arccos {
d^2+r^2-R^2 \over 2dr}\,dr & {\rm for} & d > R \\
{2 \over \pi R^2} \left[\int\limits_{R-d}^{R+d} r\, \mu_p(r)\, \arccos {
d^2+r^2-R^2 \over 2dr}\,dr + \pi \int\limits_0^{R-d} r\, \mu_p(r)\,dr
\right] & {\rm for} & d < R \\
{2 \over \pi R^2} \int\limits_0^{2R} r \,\mu_p(r)\,\arccos{r \over 2 R}\,dr
& {\rm for} & d = R 
\end{array}
\right.\,.
\end{equation}

For a point mass lens $\mu_p(r) = {r^2 +2 \over r \sqrt{(r^2+4)}}$, and 
the second integral for the case $d < R$ can be solved analytically.
This yields the final result for the magnification of a circular source
of uniform brightness with radius $R$ caused by a point mass lens, where
the center of the source is at a distance $d$:
\begin{equation}
\mu_e(d,R) = \left\{\begin{array}{lcl}
{2 \over \pi R^2} \int\limits_{d-R}^{d+R} {r^2+2 \over \sqrt{r^2+4}} 
\arccos {d^2+r^2-R^2 \over 2dr}\,dr & {\rm for} & d > R \\
{2 \over \pi R^2} \int\limits_{R-d}^{R+d} {r^2+2 \over \sqrt{r^2+4}}
 \arccos {d^2+r^2-R^2 \over 2dr}\,dr 
+ {R-d \over R^2} \sqrt{(R-d)^2+4} & {\rm for} & d < R \\
{2 \over \pi R^2} \int\limits_0^{2R} {r^2+2 \over \sqrt{r^2+4}}
 \arccos{r \over 2 R}\,dr
& {\rm for} & d = R 
\end{array}
\right.\,.
\end{equation}
This expression can also be found in (Schneider et al. 1992, p. 313).
For $d = 0$, the above formula yields
the well known result (e.g. Schneider et al. 1992, p. 38)
\begin{equation}
\mu_e(0,R) = \sqrt{1+{4 \over R^2}}\,,
\end{equation}
while for $d = R$, the integral can be solved to yield the result 
of Maeder (1973)
\begin{equation}
\mu_e(R,R) = \frac{2}{\pi}\,\left[\left(1+{1 \over R^2}\right)
\arcsin {1 \over \sqrt{1+{1 \over R^2}}} + {1 \over R}\right]\,.
\end{equation}
Note that the integrands of the remaining integrals do not contain 
singularities. For their calculation an adaptive integration routine 
is recommended.  
We note that an expression for $\mu_{\rm e}(d,R)$ involving elliptical
integrals has been derived following a different approach by
Witt \& Mao (1994).

\section{Finding solutions for the two-point-mass lens near the positions of
\setcounter{equation}{0}
the point masses and near the projected source position} 
For a two-point-mass lens with a large separation of the masses, one expects
three images if the source is located outside the caustics, where one
image can be found near the projected source position and one image near each 
point mass. 
The iterative scheme described here can be used to obtain an adequate
initial guess for these image positions, which can be used for further
calculation with a 2-dimensional root-finder (e. g. Powell's root-finder
as described by Dominik (1995) ).

For a mass $m_1$ in $(\chi,0)$ and a mass $m_2$ in $(-\chi,0)$, the
lens equation reads
\begin{eqnarray}
y_1(x_1,x_2) & = & x_1 - m_1\,{x_1 - \chi \over \left(x_1-\chi\right)^2
+x_2^2} - m_2 \,{x_1 + \chi \over \left(x_1+\chi\right)^2+x_2^2} \\
y_2(x_1,x_2) & = & x_2 - m_1\,{x_2 \over \left(x_1-\chi\right)^2
+x_2^2} - m_2 \,{x_2 \over \left(x_1+\chi\right)^2+x_2^2} \,.
\end{eqnarray}

To get a soulution near $(\chi,0)$, we substitute $x_1 = \chi + 
\varepsilon_1$ and $x_2 = \varepsilon_2$, where the $\varepsilon_i$ are thought
to be small compared with $\chi$. 
For the lens equation, one obtains
\begin{eqnarray}
y_1(\varepsilon_1,\varepsilon_2) & = & 
\chi+\varepsilon_1 - m_1 \,{\varepsilon_1 
\over \varepsilon_1^2
+\varepsilon_2^2} - m_2 \,{2 \chi + \varepsilon_1 
\over \left(2\chi+\varepsilon_1\right)^2+\varepsilon_2^2} \\
y_2(\varepsilon_1,\varepsilon_2) & = & \varepsilon_2 - m_1 \,{\varepsilon_2 
\over \varepsilon_1^2
+\varepsilon_2^2} - m_2 \,{\varepsilon_2 
\over \left(2\chi+\varepsilon_1\right)^2+\varepsilon_2^2}  \,.
\end{eqnarray}
Defining
\begin{equation}
\zeta = {\varepsilon_1 \over \varepsilon_1^2+\varepsilon_2^2}\,, \quad
\eta =  {\varepsilon_2 \over \varepsilon_1^2+\varepsilon_2^2}
\end{equation}
and solving for $\zeta$ and $\eta$ yields
\begin{eqnarray}
\zeta(\varepsilon_1,\varepsilon_2)
 & = & {\chi+\varepsilon_1-y_1 \over m_1} - {m_2 \over m_1} \,
{2 \chi+\varepsilon_1 \over \left(2\chi+\varepsilon_1\right)^2+
\varepsilon_2^2}\,, \\
\eta(\varepsilon_1,\varepsilon_2)
 & = & {\varepsilon_2-y_2 \over m_1} - {m_2 \over m_1} \,
{\varepsilon_2 \over \left(2\chi+\varepsilon_1\right)^2+
\varepsilon_2^2} \,.
\end{eqnarray}
This equation can be solved iteratively by initially setting
$\varepsilon_1^{(0)} = 0$ and $\varepsilon_2^{(0)} = 0$ and 
\begin{eqnarray}
\zeta^{(i)} & = & \zeta(\varepsilon_1^{(i)},\varepsilon_2^{(i)})\,, \\
\eta^{(i)} & = & \eta(\varepsilon_1^{(i)},\varepsilon_2^{(i)})\,, \\
\varepsilon_1^{(i+1)} & = & 
	{\zeta^{(i)} \over {\zeta^{(i)}}^2 + {\eta^{(i)}}^2}\,, \\
\varepsilon_2^{(i+1)} & = & 
	{\eta^{(i)} \over {\zeta^{(i)}}^2 + {\eta^{(i)}}^2} \,.
\end{eqnarray}

The image near $(-\chi,0)$ can be found in an analogous way by 
substituting $x_1 = -\chi+\varepsilon_1$ and $x_2 = \varepsilon_2$.
The quantities $\zeta$ and $\eta$ then follow as
\begin{eqnarray}
\zeta(\varepsilon_1,\varepsilon_2)
 & = & {-\chi+\varepsilon_1-y_1 \over m_2} - {m_1 \over m_2} \,
{-2 \chi+\varepsilon_1 \over \left(-2\chi+\varepsilon_1\right)^2+
\varepsilon_2^2}\,, \\
\eta(\varepsilon_1,\varepsilon_2)
 & = & {\varepsilon_2-y_2 \over m_2} - {m_1 \over m_2} \,
{\varepsilon_2 \over \left(-2\chi+\varepsilon_1\right)^2+
\varepsilon_2^2} \,.
\end{eqnarray}

The image near the projected source position can be found in a more general way.
The lens equation reads
\begin{equation}
{\vec y} ({\vec x}) = {\vec x} - {\vec \alpha}({\vec x}) \,,
\end{equation}
or for ${\vec x} = {\vec y} + {\vec \varepsilon}$
\begin{equation}
{\vec y} ({\vec \varepsilon}) = {\vec y} 
+ {\vec \varepsilon} - {\vec \alpha}({\vec y}+{\vec \varepsilon}) \,,
\end{equation}
which immediately yields the fixed point equation
\begin{equation}
{\vec \varepsilon} = {\vec \alpha}({\vec y}+{\vec \varepsilon})\,.
\end{equation}
This implies an iteration scheme
\begin{equation}
{\vec \varepsilon}^{\,(0)} = 0, \quad 
{\vec \varepsilon}^{\,(i)} =  {\vec \alpha}({\vec y}
+{\vec \varepsilon}^{\,(i-1)})\,.
\end{equation}

Note that this scheme cannot be used for the images near the point masses
because the deviation angle $\vec \alpha$ diverges there.

\section{A degeneracy in light curves produced by binary sources behind
point-mass lenses}
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Since the two components of the binary source may be located on the same
side or on opposite sides of the 
MACHO trajectory without changing the distance
functions $u_1(t)$ and $u_2(t)$, two different physical configurations
producing the same light curve exist. We call the case where the components
are on the same side the {\em cis}-configuration, the case where
the components are on opposite sides the {\em trans}-configuration.

Let $2 \rho$ denote the distance between the closest approaches of the MACHO
to the two components and $2 \lambda$ denote the distance between the
source components, both measured in projected Einstein radii $r_E'$.
It follows that
\begin{equation}
\rho = {t_{\rm max,2}-t_{\rm max,1} \over t_{\rm c}}\,.
\end{equation}
The angle $\beta$ between the direction from source component 1 to source component
2 and the MACHO trajectory is given by
\begin{equation}
\beta = \arctan {r_{\rm min,2} \pm r_{\rm min,1} \over 2 \rho}
=  \arctan \left({r_{\rm min,2} \pm r_{\rm min,1} \over 
  t_{\rm max,2}-t_{\rm max,1}} \, {t_{\rm c} \over 2}\right)
 \,,
\end{equation}
where the upper sign refers to the trans-configuration and the lower sign
refers to the cis-configuration.
The distance between the components follows by
\begin{equation}
\lambda = \rho / \cos \beta\,.
\end{equation}

An example for this degeneracy is shown in Fig.~\ref{BSmcps}, which shows
magnification contour plots for the BS-fit to MACHO LMC \#1 and the blue
data.

\begin{figure}[h]
\caption[ ]{Magnification contour plots for the BS-model fitted to
the MACHO LMC \#1 event for the blue data. All distances in multiples
of Einstein radii. 9 contours are shown for $\Delta {\rm mag} = 0.5..2.5$ in steps by
0.25. (a) cis-configuration, (b) trans-configuration}
\label{BSmcps}
\end{figure}


\end{appendix}

\begin{thebibliography}{}
\bibitem{}
Dominik M., 1995, A\&AS 109, 597
\bibitem{}
Maeder A., 1973, A\&A 26, 215
\bibitem{}
Schneider P., Ehlers J., Falco E. E., 1992, Gravitational Lenses,
Springer, Berlin
\bibitem{}
Witt H., Mao S., 1994, ApJ 430, 505


\end{thebibliography} 


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