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{\centerline{\bf Interpretation of the visibility profiles for double sources}}
\medskip

The visibility function $V(u,v)$ from a circular source is proportional to the
visibility of a unit point source, where the constant of proportionality is 
the flux (F) multiplied by the relative amplitude function $R(u,v)$. The
relative amplitude is an exponential function of the wavenumber 
$k =\sqrt(u^2+v^2)$ and the area $A=\pi a^2$ of the source:

$$
R = e^{-2\pi^2k^2a^2} =  e^{-2\pi k^2 A}
$$

\smallskip

The visibilility of the circular source is therefore,

$$ 
V=F R e^{i{\bf k}\cdot{\bf x}}  \eqno{(1)}
$$

where ${\bf x}$ is the vectorial displacement of the source from map center.

For the two-source case, let the map center be at the geometrical midpoint
of the two sources, which are then located at $-x_1$ and $x_1$.  The
visibility is then

$$
V = R_1\ F_1 e^{-i\bf{k}\cdot \bf{x_1}} + R_2\ F_2 e^{i\bf{k}\cdot \bf{x_1}} \eqno{(2)}
$$

The amplitude $|V|$ may be written as:

$$
|V| = [(R_1F_1)^2+(R_2F_2)^2 + 2(R_1 F_1)(R_2 F_2) cos(2{\bf k}\cdot {\bf x_1})]^{1/2}  \eqno{(3)}
$$
$$
=  [(R_1 F_1-R_2 F_2)^2 + 4(R_1 F_1)(R_2 F_2) cos^2({\bf k}\cdot {\bf x_1})]^{1/2}  \eqno{(3')}
$$

During a 4 s time interval, the RHESSI, $\bf{k}$ vector rotates $360^{\circ}$,
and the visibility is modulated by this rotation.  For a single relatively
coarse subcollimator for which $|k| x_1 < \pi/2$, the amplitude $|V|$ varies
nearly sinusoidally as a function of position angle with a minimum when
$\bf{k}$ is parallel to $\bf{x_1}$ and a maximum when ${\bf k}\cdot {\bf x_1}
=0$. The position angle locations of the minima give the position angle of the
line between the sources. The maxima are $90^{\circ}$ away.

\smallskip

This gives a relation between $F_1/F_2$ and the source areas $A_1$ and $A_2$
(we assume $A_1 > A_2$):

$$ 
(F_1/F_2)e^{-2*\pi k^2[(A_1-A_2)} = (1-|V|_{min}/|V|_{max})/(1+|V|_{min}/|V|_{max}) \eqno{(4)}
$$

For those (finer) grids for which $|k| x_1 \ge \pi/2$, the amplitude $|V|$ can
have more than one extremum when ${\bf k}\cdot {\bf x_1} = \pi/2, 3\pi/2,\dots$.

\smallskip

Taking the ratio of equation (4) for two different values of $|k|$ (i.e two
different subcollimators) makes the flux ratio $F_1/F_2$ cancel out, and the
quantity $(A_1-A_2)$ can be determined.  After that, the flux ratio may be
determined from equation (4) for one of the subcollimators.



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