%version 13.04.2002 (sended to editor)


\documentclass{aa}
\usepackage{graphicx}
\begin{document}
\title{Spectral features of star spots}

\author{Diana Kjurkchieva \inst{1}, Waldemar Ogloza \inst{2}
    \and Dragomir Marchev \inst{1}}

\offprints{D.Kjurkchieva}

\institute{Department of Physics, Shoumen University, 9700 Shoumen, Bulgaria\\
    and Issac Newton Institute of Chile, Bulgarian Branch\\
    e-mail: d.kyurkchieva@shu-bg.net, d.marchev@shu-bg.net
   \and  Mt. Suhora Observatory, Pedagogical University, Podchorazych 2, 30-084 Cracow, Poland\\
    e-mail: ogloza@ap.krakow.pl}

\date{Received , Accepted }

\abstract{Analytical expressions describing the variations of the
spectral features of the star spots during the star rotational
cycle are obtained. All possible cases corresponding to different
configuration parameters are considered. The obtained expressions
in the inverse problem solutions are quite complex and their
direct using for an analysis of observational data is not
reasonably. But the computer code created on the basis of the
obtained analytical solution provides a fast interactive
calculation of the spot spectral features at each phase,
visualizes the results and displays the visible star disk with the
spot. The comparison of spectral data with families of synthetic
profiles may give information for the combination of the
configuration parameters, i.e. it may provide a preliminary
solution of the inverse problem by trials and errors.
\keywords{Line:profiles -- Methods:analytical -- Methods:numerical
-- Stars:starspots } }

\authorrunning{D.Kjurkchieva, W.Ogloza and D.Marchev}
\titlerunning{Spectral features of star spots}
\maketitle


\section{Introduction}

The most widespread method for an analysis of the observational
data, i.e. for the solution of inverse problems in the astronomy,
is the numerical modelling. It consists in comparison of the
observational curves with families of theoretical curves built on
the basis of numerical solution of integral equations. The goal is
searching for a best fit varying some input parameters of the
model by different procedures (Vogt 1981, Poe $\&$ Eaton 1985,
etc.). The results of the inverse problem solution obtained by
numerical methods are ambiguous (Bopp $\&$ Evans 1973, Dorren
1987, Budding $\&$ Zeilik 1987, Strassmeier $\&$ Bopp 1992, etc.)
due to the impossibility to separate the contributions of the
input parameters. The ambiguity problem can be reduced partially
if an analytical solution of the respective direct problem is
available.

Analytical methods may be used directly for the inverse problem
solutions in the cases of simple geometric configurations
(spherical stars, circle orbits, circle spots, etc.). They are the
basis of so called method of the characteristic points for the
inverse problem solution. The characteristic points of the
observational curves are those at which their course changes
qualitatively. On the other hand they are the points of the
analytical solution at which the considered quantity or its first
derivative has extremum. The analytical equations simplify
considerably at these points and can be used for the determination
of the configuration parameters. Consequently, the bigger is the
number of the characteristic points, the bigger is the number of
the independent equations and correspondingly the bigger is the
number of the obtained parameters in the inverse problem solution.

Earlier the method of the characteristic points was applied mainly
for the determination of the parameters of the eclipsing binaries
where the characteristic points are the phases of contacts. This
approach was developed also for the light curve solution of
spotted stars (Budding 1977; Kjurkchieva 1986; 1987a,b; 1989a,b,c;
1990; Eker 1994). But the number of the derived independent
photometric equations turned out not enough for the determination
of all parameters. Further the method of the characteristic points
was extended for the rotational broadening of the spot spectral
features (Kjurkchieva 1995, hereafter Paper I). This paper is
continuation of Paper I which goal is an investigation of the
profiles of the spectral features of the star spots. This problem
becomes actual because of the increasing power of the
observational tools makes possible the detailed detection of the
spectral profiles.


\section{Analytical description of the profile of the spot spectral feature}


  \begin{figure}
  \centering
  \includegraphics[]{fig1.eps}
  \caption{The star with a spot}
  \label{Fig1}
  \end{figure}


\begin{figure}
 \centering
 \includegraphics[]{fig2.eps}
\caption[]{The star disk on the visible plane} \label{Fig.2}
\end{figure}

Consider a star with radius $R$ whose rotation axis is inclined
under an angle $i$ in respect to the line-of-sight (Fig.1). The
axis $z$ in Fig. 1 coincides with the projection of star rotation
axis on the visible plane $xz$. A circle spot with angular radius
$\alpha$ at a polar distance $\beta$ on the star surface projects
on the visible star disk as an ellipse (Fig. 2) with semi-axes
(Kjurkchieva 1986):

\[ a=R\sin\alpha;\]
\[b(t)=R \sin\alpha \cos\psi (t)\]
where:

\[\cos\psi (t) =\cos\beta \cos i + \sin\beta \sin i \cos(\omega t), \]
$\omega = 2\pi/T_{rot}$ is the angular rotational velocity of the
star and $t=0$ corresponds to the maximum visibility of the spot.
The radial velocity of an arbitrary point of the visible disk is:

\[ V_{r} =V_{0}\sin i \frac{x}{R} ; \]
where $V_{0}$ is the linear equatorial star velocity.

The spot whose temperature differs from the temperature of the
surrounding photosphere produces spectral features (emission bumps
or depressions) that propagate on the star spectral lines during
the rotational cycle. The stars that rotate fast (for instance RS
CVn-type) have broad spectral lines and the spot features could be
detected and analyzed (Ramsey $\&$ Nations 1980).

In the framework of the linear limb-darkening approximation the
spectral feature of the spot at phase $\tau=t/T_{rot}$ is
described by the integral:

\begin{equation}
 I(\lambda, \tau) = \int_{z_{1}}^{z_{2}} I_{0}(1-u+u\cos\theta)\,dz
\end{equation}
where:
\[  \cos\theta=\frac{\sqrt{R^{2}-x^{2}-z^{2}}}{R} ,\]
$z_{1}$ and $z_{2}$ are the extremum $z$-coordinates of the spot
corresponding for a given value of $x$ (Fig. 2); $u$ is the spot
limb-darkening coefficient; $I_{0}$ is the surface brightness
intensity of the spot at wavelength $\lambda_{0}$. The relation
between the $x$-coordinate of the star disk and the
$\lambda$-position of its contribution to the spectral line is:

\[  x=k(1-\lambda/\lambda_{0}) \]
where
\[k=\frac{cR}{V_{0}\sin i} .\]

After calculation of the integral in the equation (1) we obtained:
\begin{eqnarray}
\  I(\lambda, \tau) &=&I_{0}(1-u)(z_{2}-z_{1}) \nonumber \\
        & & +\frac{I_{0}u}{2R} (R^{2}-x^{2})
\arcsin\frac{z_{2}}{\sqrt{R^{2}-x^{2}}} \nonumber \\
                 & &-\frac{I_{0}u}{2R} (R^{2}-x^{2})
\arcsin\frac{z_{1}}{\sqrt{R^{2}-x^{2}}} \nonumber \\
          & & +\frac{I_{0}u}{2R} z_{2}\sqrt{R^{2}-x^{2}-z_{2}^{2}} \nonumber \\
              & & -\frac{ I_{0}u}{2R} z_{1}\sqrt{R^{2}-x^{2}-z_{1}^{2}} \;.
\end{eqnarray}
Consequently, in order to describe the spectral profile of the
star spot we should obtain the integration limits $z_{2}$ and
$z_{1}$ as functions of $\lambda$ and $\tau$.

In order to do this we enter second reference system
$x^{''}z^{''}$ whose axes are aligned along the axes of the spot
projection ellipse (Fig. 2). Its equation has a canonical form
there (Kjurkchieva 1986):
\[  (x^{''}/a)^{2} +(z^{''}/b)^{2} =1 . \]
The coordinate transformations have the form:
\[  x^{''}=x\cos\gamma - z\sin\gamma \]
\[  z^{''}= z_{C}^{''} + x\sin\gamma + z\cos\gamma \]
where:
\[  \tan\gamma(t) = \cos\chi(t)/\cos\varphi(t);\]
\[  \cos\chi(t) = \sin\beta \sin(\omega t) ;\]
\[  \cos \varphi(t) = \cos\beta \sin i - \sin\beta \cos i \cos(\omega t) ;\]
\[  z_{C}^{''}=-R\cos\alpha \sin\psi (t) .\]

Substituting $x^{''}$ and $z^{''}$ in the canonical equation of
the projection spot ellipse we obtain:
\begin{eqnarray}
z_{1}&=&\frac{\cos\varphi (R\cos\alpha - x\cos\chi)}{\sin^{2}\chi}
-S\frac{\cos\psi }{\sin^{2}\chi}
\end{eqnarray}
\begin{eqnarray}
\ z_{2}&=&\frac{\cos\varphi (R\cos\alpha -
x\cos\chi)}{\sin^{2}\chi} +S\frac{\cos\psi}{\sin^{2}\chi}
\end{eqnarray}
where
\[S = \sqrt{R^{2}\sin^{2}\alpha \sin^{2}\chi-(R \cos\alpha\cos\chi - x)^{2}} .\]

The expressions (3-4) correspond to those $x$ where the whole
$z$-line of the spot is on the visible star hemisphere. For those
$x$ (and $\tau$) where part of the z-line of the spot is visible
the one or the two integration limits are:
\begin{equation}
z_{1,2} =\pm \sqrt{R^{2}-x^{2}}.
\end{equation}

The visibility of the spot during the star rotational cycle
depends on the configuration parameters. The all possible cases
are shown in Table 1 in which the following abbreviations are used
for the spot visibility: w - wholly; mhe - more than half an
ellipse; lhe - less than half an ellipse; inv - invisible. The
expressions for the characteristic phases $\tau_{i}$ determining
the spot visibility and the characteristic $\lambda_{i}$ at which
the shape of the spot spectral feature changes are respectively:

\begin{table*}[htb]
\caption[]{Spot visibility in dependence on the configuration
parameters}

\begin{flushleft}
\begin{tabular}{|l|l|l|l|l|}
      \hline

Case& Conditions& Visibility & Characteristic & Characteristic \\
  &     &  of the spot   &        phases &     $\lambda$ \\
 \hline

A&$\alpha + \beta + i < \pi/2$                    &  w (Fig.3)                 &  0, 0.5                   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$ \\
B&$\beta+i<\pi/2$; $\alpha+\beta<\pi/2$;          &  w, mhe (Fig.4)            &  0, $\tau_{1}$, 0.5       & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$   \\
 &$\beta+i+\alpha>\pi/2$; $\beta-i+\alpha<\pi/2$  &                            &                           & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$   \\
C&$\beta+i>\pi/2$; $\alpha+\beta<\pi /2$;         &  w, mhe, lhe (Fig.5)       &0,$\tau_{1}, \tau_{3}$,0.5 & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$    \\
 &$\alpha + \beta -i <  \pi /2$                   &                            &                           & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$   \\
D&$\beta-i-\alpha<\pi /2<\beta+i-\alpha$;         &  w, mhe, lhe, inv (Fig.6)  &0,$\tau_{1}, \tau_{2}$,    & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$     \\
 &$ \alpha + \beta  <  \pi /2$                    &                            &$\tau_{3}$, 0.5            & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$ \\
E&$\beta-\alpha-i >\pi/2$                         &  inv         &   & \\
F&$\alpha+\beta>\pi/2$; $\beta+i<\pi/2$;          &  w, mhe (Fig.7)            &0, $\tau_{1}, \tau_{4}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$  \\
 &$i>\beta$; $\alpha+i-\beta<\pi/2$               &                            & 0.5                       & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
G&$\alpha+\beta>\pi/2$; $\beta+i>\pi/2$;          &  w, mhe, lhe (Fig.8)       &0, $\tau_{1}, \tau_{3}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$ \\
 &$i>\beta$; $\alpha+i-\beta<\pi/2$               &                            &$\tau_{4}$, 0.5            & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
H&$\alpha+\beta>\pi /2$; $\beta+i<\pi/2$;         &  w, mhe             &0, $\tau_{1}, \tau_{4}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$   \\
 &$i<\beta$; $\alpha+\beta-i<\pi /2$              &                            &$\tau_{5}$, 0.5            & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
K&$\alpha+\beta>\pi/2$; $\beta+i>\pi/2$;          &  w, mhe, lhe       &0, $\tau_{1}, \tau_{3}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$  \\
 &$i<\beta$; $\alpha-\beta+i<\pi/2$               &                            &$\tau_{4}$, $\tau_{5}$, 0.5& $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
L&$\alpha-\beta+i>\pi/2$; $i+\beta>\pi/2$;        &  mhe
&0, $\tau_{1}$, 0.5          & $\lambda_{1}^{min}$;
$\lambda_{1}^{max}$  \\
 &$\alpha+\beta<\pi/2$; $i<\beta$                 &                            &                           & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$   \\
M&$\alpha+\beta-i>\pi/2$; $i+\beta>\pi /2$;       &  mhe, lhe
&0, $\tau_{3}$, 0.5          & $\lambda_{1}^{min}$;
$\lambda_{1}^{max}$    \\
 &$\alpha+\beta<\pi/2$; $i>\beta$                 &                            &                           & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$     \\
N&$\alpha+\beta>\pi/2$; $\alpha+i-\beta>\pi/2$;   &  mhe                       &0, $\tau_{1}, \tau_{4}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$    \\
 &$i+\beta<\pi/2$; $i >  \beta$                   &                            & 0.5                       & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
P&$\alpha+\beta>\pi/2$; $\alpha+i-\beta>\pi/2$;   &  mhe, lhe                  &0, $\tau_{1}, \tau_{3}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$ \\
 &$i+\beta<\pi/2$ ; $i >  \beta$                  &                            &$\tau_{4}$, 0.5            & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$;$\lambda_{3}$ \\
Q&$\alpha+\beta>\pi/2$; $\alpha+\beta-i>\pi/2$;   &  mhe                       & 0, $\tau_{1}, \tau_{4}$,  & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$  \\
 &$i +  \beta  <  \pi /2$; $i <  \beta$           &                            & $\tau_{5}$, 0.5           & $\lambda_{2}^{min}$; $\lambda_{2}^{max}$; $\lambda_{3}$ \\
R&$\alpha+\beta>\pi/2$; $\alpha+\beta-i>\pi /2$;  &  mhe                       &0, $\tau_{1}, \tau_{3}$,   & $\lambda_{1}^{min}$; $\lambda_{1}^{max}$    \\
 &$i +  \beta  >  \pi /2$;  $i <  \beta$          &                          &$\tau_{4}$, $\tau_{5}$, 0.5& $\lambda_{2}^{min}$; $\lambda_{2}^{max}$; $\lambda_{3}$  \\

\hline
\end{tabular}
\end{flushleft}
\end{table*}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig3a.eps}
\includegraphics[width=5cm]{fig3b.eps}
\caption[]{Case A} \label{Fig.3}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig4a.eps}
\includegraphics[width=5cm]{fig4b.eps}
\caption[]{Case B} \label{Fig.4}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig5a.eps}
\includegraphics[width=5cm]{fig5b.eps}
\caption[]{Case C} \label{Fig.5}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig6a.eps}
\includegraphics[width=5cm]{fig6b.eps}
\caption[]{Case D} \label{Fig.6}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig7a.eps}
\includegraphics[width=5cm]{fig7b.eps}
\caption[]{Case F} \label{Fig.7}
\end{figure}


\begin{equation}
\tau_{1}=\frac{1}{2\pi}\arccos\frac{\sin\alpha-\cos\beta\cos i}{\sin\beta\sin i};
\end{equation}

\begin{equation}
\tau_{2}=\frac{1}{2\pi}\arccos\frac{-\sin\alpha-\cos\beta\cos i}{\sin\beta\sin i};
\end{equation}

\begin{equation}
\tau_{3} =\frac{1}{2\pi}\arccos(-\cot \beta \cot i) ;
\end{equation}

\begin{equation}
\tau_{4} =\frac{1}{2\pi}\arcsin\frac{\cos\alpha}{\sin\beta};
\end{equation}

\begin{equation}
\tau_{5}  =\frac{1}{2\pi} \arccos(\tan i\cot \beta )    ;
\end{equation}

\begin{equation}
\lambda_{1}^{min}=
\lambda_{0}\left\{1-\frac{R\cos(\alpha-\chi)}{k}\right\};
\end{equation}

\begin{equation}
\lambda_{1}^{max}=
\lambda_{0}\left\{1-\frac{R\cos(\alpha+\chi)}{k}\right\}\;
\end{equation}

\begin{equation}
\lambda_{2}^{min} = \lambda_{0}\left\{1-\frac{R}{k}\frac{\cos\alpha \cos\chi
-\delta}{\sin^{2}\psi}\right\};
\end{equation}

\begin{equation}
\lambda_{2}^{max} = \lambda_{0}\left\{1-\frac{R}{k}\frac{\cos\alpha \cos\chi
+\delta}{\sin^{2}\psi}\right\};
\end{equation}

\begin{equation}
\lambda_{3}=\lambda_{0}\left\{1-\frac{c}{V_{0}\sin i}\right\}.
\end{equation}
where

\[\delta=\cos\varphi \sqrt{\sin^{2}\alpha - \cos^{2}\psi} .\]


\section{Inverse problem solution}

In the framework of the method of the characteristic points we
obtained very simple equations for the rotational broadening of
the spot spectral features at some phases (Paper I):

\begin{equation}
\frac{\Delta\lambda^{sp}(0)}{\Delta\lambda^{st}} = \sin \alpha
\end{equation}

\begin{equation}
\frac{\Delta\lambda^{sp}(0.25)}{\Delta\lambda^{st}} = \sin \alpha \cos \beta
\end{equation}

\begin{equation}
\frac{\Delta\lambda^{sp}(\tau_{1})}{\Delta\lambda^{st} } =
\frac{\sin \alpha }{\sin i} \sqrt{\cos^{2}\beta-2\sin\alpha
\cos\beta \cos i+\sin^{2}\alpha}
\end{equation}
where $\Delta\lambda^{sp}(\tau)$ was the rotation broadening of
the spot spectral feature at phase $\tau$ and $\Delta\lambda^{st}$
was the rotation broadening of the corresponding star line.

In order to use the obtained analytical solution from the previous
section for an analysis of spectral data of spotted star, i.e. for
the inverse problem solution, we have to find those phases $\tau$
and/or wavelengths $\lambda$ at which the expression (2)
simplifies. For an analysis of the observational data it would be
easiest to compare the maximum intensities of the spot spectral
features at different phases. However it turned out impossible to
obtain analytical expression for the maximum value of
$I(\lambda,\tau)$ because the spot spectral features are
asymmetric due to the limb-darkening effect. But we found that the
expression (2) simplifies at wavelength

\begin{equation}
\lambda_{C}(\tau)=[\lambda^{max} ( \tau)+\lambda^{min}(\tau)]/2
\end{equation}
where $ \lambda^{max}(\tau)$ and $\lambda^{min}(\tau)$ are the
maximum and minimum wavelength of the profile of the spot spectral
feature at phase $\tau$. In fact, $\lambda_{C}(\tau)$ is related
with the configuration parameters by the equation (Paper I):

\begin{equation}
\lambda_{C}(\tau)=\lambda_{0} [1+(V_{0}/c)\cos\alpha \sin i\sin\beta
\cos(2 \pi \tau )].
\end{equation}
Then the intensities I($\lambda_{C}, \tau$) can be compared in the
inverse problem solution. For the sake of brevity further we will
use the symbol I$(\tau)$ instead of I$(\lambda_{C},\tau)$ and will
write some of the equations obtained on the basis of the
analytical solution.

If the spot is visible wholly during the whole rotation cycle then
the following equations could be used:

\begin{equation}
\frac{I(0)}{I(0.5)}=\frac{\alpha + f +c}{\alpha+ b +d}
\end{equation}

\begin{equation}
\frac{I(0)}{I(0.25)}=\frac{\alpha+ f +c}
                   {i(1-\cos^{2}\alpha\sin^{2}\beta) + e}
\end{equation}
where:
\[f=2(1/u-1)\sin\alpha \cos(\beta-i);\]
\[b=2(1/u-1)\sin\alpha \cos(\beta+i);\]
\[c=0.5\sin(2\alpha )\cos(2i-2\beta);\]
\[d=0.5\sin(2\alpha )\cos(2i+2\beta);\]
\[e=2(1/u-1)\sin\alpha\cos i +
    0.5(\sin^{2}\alpha -\cos^{2}\alpha\cos^{2}\beta)\sin 2i.\]

The left sides of the equations (21-22) contain the quantities
that can be measured from the observational data. Consequently,
these equations as well as the equations (16-17) can be used for
determination of the parameters $i$, $\alpha$, $\beta$, $u$.

If the spot is partially and wholly visible during the cycle then
one can use the equations (6), (16), (18) and the equation:

\begin{equation}
\frac{I(0)}{I(\tau_{1})}=
\frac{\alpha+a+c}{(1/u-1)\sin^{2}\alpha/\sin\chi(\tau_{1})+0.5XP+0.5Q}
\end{equation}
where
\[ X = \sqrt{1-\cos^{2} \alpha \cos \chi ( \tau_{1})};\]
\[P = \arcsin{\frac{z_{2}(\tau_{1})}{RX}}-\arcsin{\frac{z_{1}(\tau_{1})}{RX}}
; \]
\[Q =\frac{z_{2}(\tau_{1})}{R^{2}} \sqrt{(RX)^{2}-z_{2}^{2}(\tau_{1})}
   - \frac{z_{1}(\tau_{1})}{R^{2}} \sqrt{(RX)^{2}-z_{1}^{2}(\tau_{1})}. \]

If the spot is invisible during some part of the rotational cycle
then one can use the equations (6),(7), (16), (23) and the
equation:

\begin{equation}
\frac{I(0)}{I(\tau_{3})}=\frac{\alpha+a+c}{(1/u-1)\sin i
(1-\cos\alpha)/\cos\beta + E +F}
\end{equation}
where:

\[E=\frac{\sin i}{2\cos\beta}\left\{1-\cos\alpha
-\frac{\cos^{2}\beta}{\sin^{2} i}\right\} \sqrt{2(1-\cos\alpha)-h}
;\]

\[ F=\frac{\cos^{2}\beta}{2\sin^{2} i}[\pi /2
  -\arcsin(1-h)] ;\]

\[ h=\sin^{2} i \frac{(1-\cos\alpha)^{2}}{\cos^{2}\beta}.\]


Of principle, the new-obtained equations (21-24) together with the
equations (16-18) as well as the equations derived from the light
curve solution (Kjurkcheva 1987b) present a full system of
independent equations for determination of the configuration
parameters. But the most of the new-obtained analytical
expressions (8-15) and (21-24) are quite long to be applied
directly for an analysis of observational spectral data and for
calculation of the unknown parameters. Moreover, the determination
of the characteristic phases and wavelengths will not be precise
enough due to the low resolution of the present spectral data. The
higher time resolution of the photometric observations makes them
more appropriate for determination of the characteristic phases.
Consequently, for the inverse problem solution by the method of
the characteristic points simultaneous photometry and spectroscopy
of the spotted star covering the whole rotational cycle is
necessary.

\section{Computer code}

In order to visualize the results of the obtained analytical
solution we wrote the computer program SPSPF. Its input parameters
are $i$, $\beta$, $\alpha$, $V_{0}$, $\lambda_{0}$ and $u$. At the
very beginning the program checks the combination of the input
parameters $i$, $\alpha$, $\beta$ to what case (from A to R)
corresponds. An output table contains the values of:
characteristic phases $\tau_{i}$, characteristic wavelengths
$\lambda_{i}$, the rotational broadening of the spot spectral
feature $\Delta \lambda^{sp}(\tau)$, $\lambda_{C}(\tau)$ and
I$_{C}(\tau)$. The program calculates the profile of the spot
spectral feature for each (or for wanted) phase according to the
equation (2) and draws the profile together with the spot contour
on the visible star disk.

The work of SPSPF is illustrated in Figures 3-7 for the input
parameters given in Table 2 for the cases A, B, C, D and F. For
comparability all figures correspond to equal values $V_{0}$, $u$
and $\lambda_{0}$.

\begin{table}[htb]
\caption[]{Configuration parameters}
\begin{flushleft}
\begin{tabular}{|l|llllll|}
\hline   & A& B& C& D& F \\ \hline
  i      &20&45&60&70&45\\
$\beta$  &30&30&40&60&40\\
$\alpha$ &20&20&20&20&60\\
                                                      \hline
\end{tabular}
\end{flushleft}
\end{table}

Moreover the code SPSPF has the following subroutines:

(a) For calculation and drawing of the variations of the
rotational broadening of the spot spectral feature during the
cycle (Fig. 8);

(b) For calculation and drawing of the variations of the intensity
I$_{C}(\tau)$ of the spot spectral feature during the cycle (Fig.
9);

(c) For calculation and drawing of the light curve corresponding
to the input parameters (Fig.10).

(d) For determination of the characteristic phases and wavelengths
of observational photometric and spectral data by numerical
calculation of the corresponding derivatives.

The fortran code, data and figures corresponding to all cases are
accessible in: www.as.ap.krakow.pl/publ.html


\begin{figure}
\centering
\includegraphics[width=8cm]{fig8.eps}
\caption[]{Phase variations of the rotation broadening of the
         spot spectral features for the different cases}
\label{Fig.8}
\end{figure}


\begin{figure}
\centering
\includegraphics[width=8cm]{fig9.eps}
\caption[]{Phase variations of $I_{C}(\tau)$ for the different
cases} \label{Fig.9}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8cm]{fig10.eps}
\caption[]{The light curves for some cases} \label{Fig.10}
\end{figure}



\section{Conclusion}

The analytical techniques have been always useful tools for the
solutions of inverse problems in the astronomy. They can be
applied direvtly for simple geometric configurations. The complex
configurations (as the geometry of Roche surfaces) require a large
number of calculations and correspondingly the use of digital
computers. In this aim new procedures of numerical calculations
were developed: Wilson-Devinney code (Wilson 1983), Binary Maker 2
(Bradstreet 1993), LIGHT2 (Hill $\&$ Rucinski 1993), Doppler
imaging (Vogt $\& $ Penrod 1983, Vogt et al. 1987, Rice et al.
1989, Piskunov et al. 1990, Strassmeier et al. 1991, Vincent et
al. 1993, Strassmeier et al. 1993, Strassmeier 1994, Kurster 1993,
Kurster et al., 1994, Hatzes 1995, etc.), Zeeman-Doppler imaging
(Semel 1989, Donati $\&$ Semel 1990, Brown et al. 1991, Donati
1995) and Doppler tomography (Jankov $\&$ Foing 1992, Richards et
al. 1995). But the analytical solutions are useful in these cases
too because they provide tests of the numerical techniques and can
be used for fast numerical calculations.

Of principle, the obtained analytical expressions in this paper
may be applied for an analysis of spectral data of spotted star.
But the precision of the present spectral observations does not
allow to investigate the shape of the spot spectral feature in
details, to determine its characteristic wavelengths and
correspondingly to use the equations (11-15). Moreover, the low
time resolution of the spectral data does not allow the
determination of the characteristic phases and the using of the
equations (6-10) for the inverse problem solution.

But the results of this paper may be used for very fast
computation of the spot spectral feature by our computer code
created on the basis of the analytical solution. Then the
comparison of the spectral data with families of synthetic
profiles may give some initial information for the appropriate
combination of configuration parameters, i.e. it may provide a
preliminary solution of the inverse problem by trials and errors.

The obtained results can be applied also for Ap stars taking into
account that the quantity I$^{sp}$ is a function of the abundance
instead of function of the temperature.

The derived analytical expressions may be used as broadening
functions in the technique of Rucinski (1999).

Some of the obtained results (equation 16) was applied yet in the
analysis of the spectral data of the short-period RS CVn-star SV
Cam (Kjurkchieva et al. 2002).


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\end{document}

