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\title{\textbf{Practice Work Session P1 (2h)}\\From models to visibilities}
\author{D. S\'egransan, J.-P. Berger, F. Malbet\\
        Observatoire de Gen\`eve\\
        Laboratoire d'Astrophysique de Grenoble (LAOG)}
\date{\textbf{Euro Summer School}\\
  \emph{Observation and Data reduction with the Very Large Telescope
    Interferometer} \\
4-16 June 2006, Goutelas (France)}
\maketitle

\begin{abstract}
  The following exercices are aimed at getting trained to the link
  between the object intensity distribution and the corresponding
  visibility curves. They are intended to be the practical application
  of the ``Visibility modeling'' Tutorial. This practical session is
  meant to be done with the ASPRO software and hand-made uv tables.
\end{abstract}


\section*{Practical considerations.}

To start ASPRO, in the home directory just type "\texttt{\$ aspro}". We
will use for this session the \emph{Full Aspro Interface} in the
\emph{Choose ...} menu. ASPRO is able to do many things but this
session will focus on the \emph{MODEL/FIT} part.

All the exercises should be done assuming a simple yet unrealistic $(u,v)$
coverage. Synthetic $(u,v)$ tables simulating perfect $(u,v)$ coverages
are provided for use:  \texttt{strip-[J,H,K,N]-[000,..,170].uvt},
where the number \texttt{xxx} is the projected baseline angle in degrees.

\section*{Exercise 1: Measuring a diameter.}
Given a star with a 2 mas photospheric radius use the model function
to plot the visibility versus the projected baseline length (baseline
\texttt{radius}).  Use the \textit{MODEL/FIT.UV Plots \& Source
Modeling} menu. At what baseline does the visibility become equal to
zero ? Use this number to evaluate the disk diameter.


\section*{Exercise 2: Binary.}
Display the visibility and phase as a function of projected baseline
(\texttt{strip-K-60}) of a binary with unresolved components with 4
mas separation, flux ratio of 1 and position angle of 30 degrees. Do
the same thing with separation of 12 mas. Comment on the result.

Using the previous model now vary the flux ratio from 1 to 1e-6.
Comment on how the dynamic range requirement to detect the companion
translates into visibility and phase constraints ?  Would the phase
alone be sufficient to constraint binary parameters ?

\section*{Exercise 3: Disk.}
Display the visibility curve of a disk which is assumed to have an
elliptical gaussian shape (\texttt{E\_GAUSS}). Use the minor and major
axes (parameters 4 \& 5) to simulate an inclination. The display
should be done for several PAs (\texttt{strip-*-(0,30,45,60,90)}).
Comment on how the asymetry induced by the inclination changes
the visibility function at a given projected angle.

\section*{Exercice 4: Model confusion and accuracy.}
Use the model function to compute the visibility of a star with a
uniform disk brightness distribution (2 mas radius), circular
gaussian disk (1.2 mas radius) and binary (flux ratio 1, 1 mas separation, 
$45^{\circ}$ PA) with \texttt{strip-K-60} . Compare their
visibilities at 100 m in the K band. How can we distinguish between these various
models? What about measurements at 200 m. What do you conclude?


Construct a 2-component model in which a central unresolved star
(\texttt{POINT}) is surrounded by an inclined extended structure. You
can use an elliptical gaussian distribution (\texttt{E\_GAUSS}) for
that purpose (minor and major axes in the range 0.5 to 15 mas).

Try two scenarii:
\begin{itemize}
\item an extended source easely resolvable but with a flux
contribution much smaller than the star;
\item an extended source smaller but with a larger flux contribution.
\end{itemize}
What are the best baseline lengths to estimate the size and relative
flux contributions with an interferometer?

\subsection*{Exercise 6: Choosing the right baselines.}
In order to determine the parts of the $(u,v)$ plane which constraint
the model most, one can make use of the first derivative of
the visibility with respect to a given parameter.  Choose a uniform
disk model. What is the most constraining part
in the $(u,v)$ plane? For this exercise, in \texttt{UV EXPLORE} panel, use
\texttt{V} vs \texttt{U}, check the \texttt{underplot model image}
option, and choose the appropriate derivative in the \texttt{plot
what...} line.

\subsection*{Exercise 7: An unknown astrophysical object.}

Load the fits table \texttt{fudisk-N.fits} corresponding to the simulation of 
a certain type of astrophysical object (here a disk around an FU Orionis
object) using \textit{OTHER/Display a GDF or
FITS image} menu. If the color scale is not appropriate, check the
\textit{Optional parameters} button and select another color
scale. Notice what is the contrast of the object (NB: the angular units
are in radians).\\ 
Compute the visibility of the model in the N-Band with 
\textit{MODEL/ FIT.UV Plots \& Source Modeling/USE HOMEMADE MODEL}. To
do so, select the appropriate grid \texttt{strip-N-60} in the
\textit{Input Information} menu. Use  \texttt{UV EXPLORE} to plot the
visibility amplitude versus the spatial frequency radius.\\
Repeat these operations in the K band, then the H and J ones.
Compare the visibility profiles. Conclude on the resolution of the
object at these different wavelengths.


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