\section{Introduction}

Spectropolarimetric observations of several Herbig Ae/Be stars have indicated that
magnetic fields are important ingredients of the intermediate-mass star formation process. 
Models of magnetically driven accretion and outflows
successfully reproduce many observational properties of low-mass
pre-main sequence stars (the classical T\,Tauri stars), but the picture is less clear for higher mass stars.
It is important to understand how the magnetic fields in Herbig Ae/Be stars are  generated and how these fields
interact with the circumstellar environment, displaying a combination of disk, wind, accretion, and jets.
To date, these fields have not been studied for a representative number of Herbig Ae/Be stars.
Magnetically controlled behaviour has been found in both the spectroscopic and photometric variability
of the strongly magnetic Herbig Ae star HD\,101412 (Hubrig et al.\ \cite{Hubrig2010,Hubrig2011}).
The small amount of  UVES spectra acquired for this star have been analysed uncovering
 variations in equivalent widths, radial velocities, 
line widths, line asymmetries, and mean magnetic field modulus over the rotation period of 42.1\,d.
Magnetic field configurations are obviously of utmost importance to understanding
 the magnetospheres of Herbig Ae/Be stars.
Since the accreting material has to be lifted from the disk midplane to 
higher stellar latitudes near the stellar photosphere, some coupling
between the accreting plasma and a stellar magnetic field 
is required. Progress in understanding the disk-magnetosphere interaction can, however, only come 
from studying a sufficient number of targets in detail to look for patterns encompassing this type of pre-main sequence stars.

In this work, we present series of mean longitudinal magnetic-field measurements for the Herbig Ae/Be stars HD\,97048, HD\,150193, and 
HD\,176386
obtained at low resolution with the multi-mode instrument FORS\,2 at the VLT, and describe their magnetic field geometries.
Magnetic fields of the order of 120--250\,G were detected in these stars for the first time during our visitor run 
with FORS\,1 in May 2008 (Hubrig et al.\ \cite{Hubrig2009}).
The follow-up program developed to search for magnetic field 
variations over the rotation cycle was allocated for observations with FORS\,2 in 2010. 
The previous spectropolarimetric observations of the star MWC\,480 (=HD\,31648) with FORS\,1 
revealed the presence of a weak longitudinal magnetic field 
$\left<B_{\rm z}\right> = 87\pm 22$\,G and distinct circular polarisation signatures in spectral lines
originating in the circumstellar (CS)
environment (Hubrig et al.\ \cite{Hubrig2006,Hubrig2007}). We report here on three new  high-resolution polarimetric 
spectra obtained with the SOFIN spectrograph installed at the Nordic Optical Telescope, which confirm the presence of a 
magnetic field in this star in both photospheric and CS lines.

\section{Magnetic field measurements and period determination}

\begin{table}
\centering
\caption{
Target stars with multi-epoch spectropolarimetric observations.
}
\label{tab:targetlist}
\centering
\begin{tabular}{llrl}
\hline
\hline
\multicolumn{1}{c}{Object \rule{0pt}{2.6ex}} &
\multicolumn{1}{c}{Other} &
\multicolumn{1}{c}{V} &
\multicolumn{1}{c}{Spectral} \\
\multicolumn{1}{c}{Name} &
\multicolumn{1}{c}{Identifier} &
 &
\multicolumn{1}{c}{Type} \\
\hline
MWC\,480     & HD\,31648     & 7.7 & A3Ve   \\
HD\,97048    & CU\,Cha       & 8.5  & A0pshe    \\
HD\,150193   & V2307\,Oph    & 8.9  & A1Ve     \\
HD\,176386   & CD$-$37 13023 & 7.3  & B9V    \\
\hline
\end{tabular}
\tablefoot{
Spectral types and visual magnitudes are taken from SIMBAD.
}
\end{table}

\begin{table}
\caption[]{
Magnetic field measurements of Herbig Ae/Be stars with FORS\,1/2.
}
\label{tab:log_meas}
\begin{center}
\begin{tabular}{ccr@{$\pm$}lr@{$\pm$}l}
\hline \hline\\[-7pt]
\multicolumn{1}{c}{MJD} &
\multicolumn{1}{c}{Phase} &
\multicolumn{2}{c}{$\left<B_{\rm z}\right>_{\rm all}$ [G]} &
\multicolumn{2}{c}{$\left<B_{\rm z}\right>_{\rm hyd}$ [G]} \\
\hline\\[-7pt]
\multicolumn{6}{c}{HD\,97048} \\
\hline\\[-7pt]
  54609.137 & 0.083 &    164 &  42 &    188 & 47 \\
  55279.131 & 0.349 &  $-$92 &  41 & $-$135 & 58 \\
  55311.109 & 0.467 & $-$163 &  53 & $-$195 & 59 \\
  55312.035 & 0.802 &   $-$6 &  74 &     12 & 91 \\
  55320.094 & 0.426 &  $-$44 &  54 &  $-$62 & 62 \\
  55322.119 & 0.346 & $-$123 &  42 &  $-$62 & 48 \\
  55324.105 & 0.210 &      4 &  59 &      9 & 63 \\
  55326.987 & 0.367 & $-$124 &  34 & $-$156 & 44 \\
  55334.039 & 0.537 & $-$154 &  53 & $-$193 & 61 \\
  55335.119 & 0.094 &    127 & 120 &    135 & 49 \\
  55345.987 & 0.768 &     54 &  59 &     26 & 63 \\
  55347.180 & 0.488 & $-$186 &  61 & $-$231 & 64 \\
  55348.070 & 0.772 &  $-$36 &  38 &  $-$12 & 64 \\
  55349.102 & 0.260 &   $-$7 &  74 &     65 & 78 \\
\hline\\[-7pt]
\multicolumn{6}{c}{HD\,150193}\\
\hline\\[-7pt]
  54609.092 & 0.118 & $-$144 & 32 & $-$252 &  48 \\
  55321.364 & 0.523 & $-$124 & 60 & $-$184 &  62 \\
  55312.319 & 0.648 & $-$97 &  84 & $-$239 & 102 \\
  55320.179 & 0.622 & $-$148 & 69 & $-$147 &  80 \\
  55322.295 & 0.230 & $-$109 & 48 & $-$111 &  64 \\
  55323.311 & 0.003 & $-$39  & 34 &  $-$43 &  46 \\
  55324.285 & 0.744 & $-$107 & 43 & $-$124 &  64 \\
  55325.307 & 0.520 & $-$189 & 45 & $-$228 &  59 \\
  55326.341 & 0.306 & $-$142 & 53 & $-$176 &  70 \\
  55327.340 & 0.066 &  $-$19 & 36 &  $-$15 &  48 \\
  55328.356 & 0.838 &  $-$62 & 74 &  $-$46 &  99 \\
  55334.314 & 0.366 & $-$143 & 32 & $-$132 &  74 \\
  55341.279 & 0.661 & $-$217 & 58 & $-$235 &  74 \\
  55348.125 & 0.864 & $-$120 & 96 & $-$306 & 130 \\
  55354.296 & 0.555 & $-$201 & 55 & $-$229 &  82 \\
\hline\\[-7pt]
\multicolumn{6}{c}{HD\,176386}\\
\hline\\[-7pt]
  54610.272 & 0.174 & $-$119 & 33 & $-$121 & 35 \\
  55312.353 & 0.145 &  $-$10 & 69 &  $-$42 & 88 \\
  55322.313 & 0.224 &  $-$41 & 65 &  $-$44 & 89 \\
  55323.247 & 0.263 &  $-$62 & 62 & $-$126 & 64 \\
  55324.308 & 0.444 & $-$190 & 62 & $-$179 & 65 \\
  55325.331 & 0.582 & $-$201 & 63 & $-$225 & 71 \\
  55326.365 & 0.732 & $-$115 & 54 & $-$119 & 68 \\
  55327.244 & 0.709 &   $-$6 & 64 &  $-$49 & 79 \\
  55328.375 & 0.967 &     99 & 59 &    111 & 79 \\
  55334.339 & 0.602 &  $-$65 & 49 &  $-$50 & 63 \\
  55335.264 & 0.631 & $-$102 & 43 & $-$144 & 54 \\
  55341.317 & 0.364 & $-$101 & 65 & $-$104 & 71 \\
  55349.128 & 0.052 &    115 & 50 &    121 & 68 \\
  55354.226 & 0.724 & $-$130 & 41 & $-$162 & 62 \\
  55355.134 & 0.733 &  $-$25 & 38 &  $-$30 & 61 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{HD97048.pc.ps}
\vskip -0.5mm
\includegraphics[width=0.48\textwidth]{HD97048.pr.ps}
\includegraphics[width=0.48\textwidth]{HD150193.pc.ps}
\vskip -0.5mm
\includegraphics[width=0.48\textwidth]{HD150193.pr.ps}
\includegraphics[width=0.48\textwidth]{HD176386.pc.ps}
\vskip -0.5mm
\includegraphics[width=0.48\textwidth]{HD176386.pr.ps}
\caption{
Phase diagrams with the best sinusoidal fit for the longitudinal magnetic field measurements of
HD\,97048 (top),
HD\,150193 (middle),
and HD\,176386 (bottom).
The residuals (observed -- calculated) are shown in the lower panel. The deviations are mostly of the same 
order as the error bars, and no systematic trends are obvious, which justifies a single sinusoid as a 
fit function. 
}
\label{fig:1}
\end{figure}

The targets for which we present new spectropolarimetric measurements are
listed in Table~\ref{tab:targetlist}.
Multi-epoch polarimetric spectra of the Herbig Ae/Be stars HD\,97048, HD\,150193, and 
HD\,176386 were obtained with FORS\,2\footnote{
The spectropolarimetric capabilities of FORS\,1 were moved to
FORS\,2 in 2009.
}
on Antu (UT1) from 2010 March 23 to 2010 June 7 in service mode.
Using a slit width of 0\farcs4, the spectral resolving power of FORS\,2 achieved
with the GRISM 600B is about 2000.
 A detailed description of the assessment of the longitudinal 
magnetic-field measurements using FORS\,2 was presented in our previous papers 
(e.g., Hubrig et al.\ \cite{Hubrig2004a, Hubrig2004b}, and references therein). 
The mean longitudinal 
magnetic field, $\left< B_{\rm z}\right>$, was derived using 

\begin{equation} 
\frac{V}{I} = -\frac{g_{\rm eff} e \lambda^2}{4\pi{}m_ec^2}\ \frac{1}{I}\ 
\frac{{\rm d}I}{{\rm d}\lambda} \left<B_{\rm z}\right>, 
\label{eqn:one} 
\end{equation} 

\noindent 
where $V$ is the Stokes parameter that measures the circular polarisation, $I$ 
is the intensity in the unpolarised spectrum, $g_{\rm eff}$ is the effective 
Land\'e factor, $e$ is the electron charge, $\lambda$ is the wavelength, $m_e$ the 
electron mass, $c$ the speed of light, ${{\rm d}I/{\rm d}\lambda}$ is the 
derivative of Stokes~$I$, and $\left<B_{\rm z}\right>$ is the mean longitudinal magnetic 
field. 

Three spectropolarimetric observations of MWC\,480 with ${\rm S/N} \ge200$ were obtained on 2009 December 30 and 2010 
December 14, and 2010 December 25 
with the low-resolution camera ($R=\lambda/\Delta\lambda\approx30000$)
of the echelle spectrograph SOFIN (Tuominen et al.\ \cite{Tuominen1999})
mounted at the Cassegrain focus of the Nordic Optical Telescope (NOT).
We used a 2K Loral CCD detector to register 40 echelle orders partially covering
the range from 3500 to 10\,000\,\AA{} 
with a length of the spectral orders of about 140\,\AA{} at 5500\,\AA{}.
The polarimeter is located in front of the entrance slit of the spectrograph and consists of a fixed 
calcite beam splitter aligned along the slit and a rotating super-achromatic quarter-wave plate. Two spectra 
polarised in opposite senses are recorded simultaneously for each echelle order providing sufficient separation 
by the cross-dispersion prism.
Usually two sub-exposures with the quarter-wave plate angles separated 
by $90^\circ$ are necessary to derive circularly polarised spectra.
The spectra are reduced with the 4A software package (Ilyin \cite{Ilyin2000}). 
Bias subtraction, 
master flat-field correction, 
scattered light subtraction, and weighted extraction of spectral orders comprise the standard steps of the image 
processing. A ThAr spectral lamp is used for wavelength calibration, taken before and after each target exposure 
to minimise temporal variations in the spectrograph. 
Wavelength shifts between right- and left-hand side circularly polarised spectra were
interpreted in terms of a longitudinal magnetic field $\left<B_{\rm z}\right>$,
using the moment technique described by Mathys (\cite{Mathys1994}).





The frequency analysis  was performed on the longitudinal magnetic field 
measurements 
using a non-linear least squares fit to the multiple harmonics utilizing the Levenberg-Marquardt 
method (Press et al.\ \cite{Press92}) with an optional possibility of pre-whitening the trial harmonics.  
To detect the most probable period, we calculated the frequency spectrum and for each trial frequency we performed a 
statistical F-test of the null hypothesis for the absence of periodicity (Seber \cite{Seber77}).
The resulting F-statistics can be thought of as the total sum including covariances of the 
ratio of harmonic amplitudes to their standard deviations, i.e.\ a signal-to-noise ratio. 
The derived ephemeris for the detected periods are 

\noindent
\begin{eqnarray}
{\rm HD\,97048}: \left<B_{\rm z}\right>^{\rm pos~ext} &=& \nonumber \\
{\rm MJD}55318.41923 &\pm& 0.01372 + 0.69334 \pm 0.00039 E  \nonumber \\
{\rm  HD\,150193}: \left<B_{\rm z}\right>^{\rm pos~ext} &=& \nonumber \\
{\rm MJD}55318.05538 &\pm& 0.03072 + 1.31697 \pm 0.00013 E  \nonumber \\
{\rm HD\,176386}: \left<B_{\rm z}\right>^{\rm pos~ext} &=& \nonumber \\
{\rm MJD}55325.70578 &\pm& 0.02185 + 0.89920 \pm 0.00008 E  \nonumber 
\end{eqnarray}

The logbook of the new FORS\,2 and the old FORS\,1 
spectropolarimetric observations is presented
in Table~\ref{tab:log_meas}.
In the first column, we indicate the MJD value at mid exposure. The phases of the measurements of 
the magnetic field
are listed in Column~2. In Columns~3 and 4, we present the longitudinal magnetic 
field $\left<B_{\rm z}\right>_{\rm all}$ measured using the whole spectrum and the 
longitudinal magnetic field 
$\left<B_{\rm z}\right>_{\rm hyd}$ using only the hydrogen lines. 
All quoted errors are 1$\sigma$ uncertainties.
The FORS\,1 measurements  
published by Hubrig et al.\ (\cite{Hubrig2009}) are presented in the first line, respectively.
Phase diagrams of the data folded with the determined periods are presented in Fig.~\ref{fig:1}.
The quality of our fits is described by a reduced $\chi^2$-value 
that is 0.71 for HD\,98048, 0.32 for HD\,150193, and  
0.92 for HD\,176386.


\begin{table*}
\centering
\caption{
Magnetic field models constrained by the FORS\,1/2 spectropolarimetric observations.
}
\label{tab:dipvals}
\begin{tabular}{cc|r@{$\pm$}lr@{$\pm$}lr@{$\pm$}l}
\hline
\hline
\multicolumn{2}{c|}{Object} &
\multicolumn{2}{c}{HD\,97048} & 
\multicolumn{2}{c}{HD\,150193} & 
\multicolumn{2}{c}{HD\,176386} \\
\hline
$\overline{\left< B_{\rm z}\right>}$ {\rule{0pt}{2.6ex}} & [G]           & $-$14.8 & 14.3 & $-$106.9&  7.4 & $-$36.2 &  15.9 \\
$A_{\left< B_{\rm z}\right>}$        & [G]           & 151.6 & 19.6 & 74.8& 9.8 & 127.5 & 24.7 \\
$v$\,sin\,$i$                        & [km s$^{-1}$] & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & 165& 8 \\
$R$                                  & [R$_{\odot}$] & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & 3.0 & 0.5 \\
$v_{\rm eq}$                         & [km s$^{-1}$] & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & 169 & 28 \\
$i$                                  & [$^{\circ}$]  & 42.8 & 2.5 & 38 & 9 & 78 & 46 \\
$\beta$                              & [$^{\circ}$]  & 84.8 & 5.0 & 41.8 & 10.1 & 37 & 106 \\
B$_{\rm d}$                          & [G]           & 720 & 120 & 590 & 40 & 700 & 650 \\
\hline
\end{tabular}
\end{table*}

All three Herbig Ae/Be stars exhibit a single-wave variation in 
the longitudinal magnetic field during the stellar rotation cycle.
These observations are usually considered as evidence 
of a dominant dipolar contribution to the magnetic field topology. 
Assuming that the studied Herbig Ae/Be stars are  oblique dipole rotators,
the magnetic dipole axis tilt $\beta$ is constrained by 
$\left< B_{\rm z}\right>^{\rm max}/\left< B_{\rm z}\right>^{\rm min} = \cos(i+\beta)/\cos(i-\beta)$,
where the inclination angle $i$ is derived from resolved observations of their disks.
In Table~\ref{tab:dipvals}, we show for each star in rows~1 and 2 
the mean value $\overline{\left< B_{\rm z}\right>}$ and the amplitude of the field variation $A_{\left< B_{\rm z}\right>}$.
For the star HD\,97048, Lagage et al.\ (\cite{Lagage2006}) used VISIR observations of the
mid-infrared emission of polycyclic hydrocarbons at the surface of the disk to determine
$i=42.8\pm2.5^{\circ}$.
Using this value, we obtain a magnetic obliquity of $\beta=84.8\pm5.0^{\circ}$ and the dipole strength 
$B_{\rm d}=720\pm120$\,G.
The disk inclination of HD\,150193, $i=38\pm9^{\circ}$, was determined by Fukagawa et al.\ (\cite{Fukagawa2003}) using 
Subaru near-infrared imaging.
With this value, we find a magnetic obliquity $\beta=42\pm10^{\circ}$ and 
the dipole strength $B_{\rm d}=590\pm40$\,G.

\begin{figure}
\centering
\includegraphics[angle=270,totalheight=0.30\textwidth]{new176386.ps}
\caption{
Determination of the $v$\,sin\,$i$ value for HD\,176386.
The line profile of \ion{Mg}{ii}~4481 in the NTT/EMMI spectrum is fitted by the corresponding
synthetic spectrum assuming $T_{\rm eff}=9900$\,K and log\,$g=3.7$.
}
\label{fig:4}
\end{figure}

\begin{figure}
\centering
\includegraphics[angle=270,totalheight=0.30\textwidth]{82g4vt4.ps}
\caption{
Determination of the $v$\,sin\,$i$ value for MWC\,480.
The line profile of \ion{Mg}{ii}~4481 in the SOFIN spectrum is fitted by the corresponding
synthetic spectrum assuming $T_{\rm eff}=8500$\,K and log\,$g=3.5$.
}
\label{fig:chucka}
\end{figure}

For HD\,176386, the inclination angle $i$ is not known because both IR (Siebenmorgen et al.\ \cite{Siebenmorgen2000}) and far-UV 
spectroscopy (Martin-Za\"idi et al.\ \cite{MartinZaidi2008}) suggest that the circumstellar 
matter has the form of an envelope rather than the form of a disk.
To constrain the magnetic field geometry of this star, we determined that
$v$\,sin\,$i=165\pm8$\,km\,s$^{-1}$ from our NTT/EMMI spectrum obtained for this star in 1995 July 8. In Fig.~\ref{fig:4},
we present the line profile
of \ion{Mg}{ii}~4481 overplotted by the corresponding synthetic spectrum assuming
$T_{\rm eff}=9900$\,K and log\,$g=3.7$, and 
solar abundances. We note that the literature values for $T_{\rm eff}$ widely differ from each other, from $T_{\rm eff}=10715$\,K 
as given by van den Ancker et al.\ (\cite{Ancker1997}) to $T_{\rm eff}=9527$\,K derived by Meyer \& Wilking (\cite{Meyer2006}).
The equatorial rotation velocity can be calculated using the relation
$v_{\rm eq}=50.6$\,$R/P$, where $R$
is the stellar radius in solar units and P is the rotation period in days. 
Taking into account the uncertainties in atmospheric parameter 
determinations for HD\,176386, we adopt the radius $R=3\pm0.5$\,$R_\odot$.
With this radius and the period $P=0.899$\,d, we
obtain $v_{\rm eq}=169\pm28$\,km\,s$^{-1}$, which leads to the poorly constrained
$i=78\pm46^{\circ}$,
magnetic obliquity $\beta=37\pm106^{\circ}$, and the dipole strength $B_{\rm d}=700\pm650$\,G.
The parameters of the magnetic field dipole models for all three stars are listed in the last two rows of Table~\ref{tab:dipvals}.





\begin{table}
\centering
\caption{
Magnetic field measurements of MWC\,480 with SOFIN.
}
\label{tab:log_sofin}
\begin{tabular}{cr@{$\pm$}l}
\hline
\multicolumn{1}{c}{MJD} &
\multicolumn{2}{c}{$\left<B_{\rm z}\right>$ [G]} \\
\hline
\hline
53296.350 & 87  & 22 \\
55195.086 & $-$952 & 177 \\
55544.090 &  458   & 149 \\
55555.982 &  189   & 204 \\
\hline
\end{tabular}
\end{table}

The logbook of our new SOFIN spectropolarimetric observations of MWC\,480 is presented in Table~\ref{tab:log_sofin}.
In the first column, we list the MJD
values for the middle of the spectropolarimetric observations and in Col.~2 we present the mean longitudinal magnetic 
field $\left<B_{\rm z}\right>$.
In the first line, we list the earlier FORS\,1 measurement 
published by Hubrig et al.\ (\cite{Hubrig2006}).
All quoted errors are 1$\sigma$ uncertainties.
Since only four magnetic field measurements are available, no search for a periodicity
can be undertaken.
On the other hand, the rotation period can be estimated using the known inclination angle 
$i=36\pm1^{\circ}$ (Pi\'etu et al.\ \cite{Pietu2006}), the radius $R=1.67$\,$R_\odot$ (Blondel \& Tijn A Djie \cite{Blondel2006}),
and $v$\,sin\,$i=90\pm2$\,km\,s$^{-1}$ estimated on our SOFIN spectra.
In Fig.~\ref{fig:chucka}, we present the line profile
of \ion{Mg}{ii}~4481 overplotted by the synthetic spectrum 
assuming $T_{\rm eff}=8500$\,K and log\,$g=3.5$.
With this value for $v$\,sin\,$i$, we obtain
$v_{\rm eq}=153\pm8$\,km\,s$^{-1}$. 
By substituting these values into the relation $P=50.6$\,$R/v_{\rm eq}$,
it follows that $P=0.531\pm0.029$\,d.
This period is probably very close to
the true rotation period, since Grady et al.\ (\cite{Grady2010})
report that their FUSE observations integrated over the 
individual FUSE orbits indicate a marginal detection of variability
on the timescale of 0.5\,d.
  

\section{Magnetic field and CS environment of MWC\,480}

In our earlier studies based on the low-resolution polarimetric FORS\,1 spectra, 
we reported the detection of distinct Zeeman features in several Herbig Ae/Be stars at the positions of 
the \ion{Ca}{ii} H\&K lines 
(Hubrig et al.\ \cite{Hubrig2004c,Hubrig2006,Hubrig2007,Hubrig2009}).
The \ion{Ca}{ii} H\&K lines
frequently display variable multi-component complex structures in both the Stokes~$V$
and Stokes~$I$ spectra and it has been suggested that they originate
 at the base of the stellar wind, as well as in 
the gaseous accretion flow. 
The star MWC\,480 exhibits a significant spectral variability, which implies that there is a 
strong interaction between the star and its CS environment.
The temporal behaviour of the complex variable structure of H$\alpha$, \ion{He}{i}, and the \ion{Na}{i}~D line
was monitored in several previous  studies (e.g., Beskrovnaya \& Pogodin \cite{Beskr2004}; Kozlova et al.\ \cite{Kozlova2007}),
which showed that the stellar wind is non-homogeneous and consists of several components, 
differing from each other in terms of their velocities.
Furthermore, structural variations in the stellar wind have been 
found to correlate with changes in the accretion process in the envelope.  

\begin{figure}
\centering
\includegraphics[width=0.40\textwidth]{herrbigd.eps}
\caption{
The behaviour of the \ion{Fe}{ii} line at $\lambda$5018.4 at the three observational epochs
2009 December 30 (bottom),
2010 December 14 (middle),
and 2010 December 25 (top).
}
\label{fig:fe_velocity}
\end{figure}

In our high-resolution SOFIN spectra, the contamination of the spectra by the CS material is clearly visible.
In Fig.~\ref{fig:fe_velocity}, we show the presence of a sharp blue component in the Mult.~42 \ion{Fe}{ii} line
at $\lambda$5018.4 during the first and the second observational epochs in 2009 and 2010.
During the last observation, the wings of the \ion{Fe}{ii} line appear in emission.
In addition, two other lines, $\lambda\lambda$5169.0 and 4923.9 
of the same multiplet, exhibit a similar behaviour. The shift of this sharp blue component,
reaches $\sim$153\,km\,s$^{-1}$ during the first SOFIN observation in 2009,
and is of the same order as the shifts measured for the blue 
components of the \ion{Na}{i}~D,
the \ion{Ca}{ii} H\&K lines, and the cores of the hydrogen lines. 



\begin{figure*}
\centering
\includegraphics[width=0.30\textwidth]{herrbigia.eps}
\includegraphics[width=0.30\textwidth]{herrbigib.eps}
\includegraphics[width=0.30\textwidth]{herrbigic.eps}
\caption{
The  variability and complex structure of \ion{Na}{i}~D (left) and \ion{Ca}{ii} K\&H
(middle and right) in the spectra
of MWC\,480 at the same observational epochs as in Fig.~\ref{fig:fe_velocity}.
}
\label{fig:na_i}
\end{figure*}






In agreement with previous studies, the \ion{Na}{i}~D lines are highly variable, but 
the \ion{Ca}{ii} H\&K lines also show a complex structure consisting of several components and pronounced variability. 
The behaviour of the \ion{Na}{i} and \ion{Ca}{ii} doublets
at three different observation epochs is presented in Fig.~\ref{fig:na_i}.
Owing to strong blending, it is impossible to measure the shift between the line profiles in right- and left-hand side
circularly polarised spectra for all line components. The compilation of 
measurements of the component positions in the \ion{Ca}{ii} and \ion{Na}{i} doublet lines and our attempts to determine
the longitudinal magnetic field are presented in Table~\ref{tab:fields}.

Weak magnetic fields of the order of several tens of Gauss were detected in the CS components of 
the \ion{Na}{i} and \ion{Ca}{ii} doublet lines and
can probably be considered as CS magnetic
fields. If we assume a dipole configuration for the magnetic field geometry,
then we expect to measure a magnetic field strength of $-$76\,G detected in the CS component of the \ion{Ca}{ii} K line 
at a distance of $\sim$2.3 stellar radii.
This consideration is fully in line with the generally assumed scenario of magnetospheric
accretion in Herbig Ae and T\,Tauri stars. The stellar magnetic field truncates the accretion disk at a few stellar radii and 
gas accretes along magnetic channels from the protoplanetary disk to the star. The disk is truncated at a radius R$_{\rm T}$, which 
depends upon the strength of the surface magnetic field, the mass accretion radius, the stellar mass,
 and the stellar radius 
(e.g.\ Bouvier et al.\ \cite{Bouvier2007}).
For a dipole configuration of the magnetic field geometry, a magnetic field strength of $-$120\,G detected in the CS component 
of the \ion{Na}{i}~D is expected to be measured at a distance of $\sim$2.0  stellar radii. 





\section{Discussion}

Using FORS\,2 low-resolution multi-epoch polarimetric spectra, we have determined
rotation/magnetic periods for the three Herbig Ae/Be stars HD\,97048, HD\,150193, and HD\,176386.
In the framework of the rigid rotator model usually assumed for magnetic stars
on the upper main-sequence,
the period of the magnetic field variation corresponds to the stellar rotation period. Our study indicates that dipole 
models provide a satisfactory fit to the acquired magnetic data. 

\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{histMWC480.eps}
\caption{
Available {\em Hipparcos} measurements for MWC\,480 with outliers observed during HJD 8674.75-8674.94. 
} 
\label{fig:hist}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{grafMWC480.eps}
\includegraphics[width=0.45\textwidth]{perMWC480.eps}
\caption{
{\em Hipparcos} observations of MWC\,480 and the corresponding frequency periodogram.
The six outliers are indicated by an ellipse.
}
\label{fig:zdena}
\end{figure}

On the other hand, our search for rotation-modulated photometric
variability using the available photometric databases has been unsuccessful: no clear variation was found for the three 
stars in either the ASAS database (Pojma\'nski \cite{pojm02}) or {\em Hipparcos} photometry. 
The star MWC\,480 was not observed by ASAS,
and only {\em Hipparcos} photometry is available for a period search. It displayed moderate light 
changes of up to 0.098\,mag on timescales from tens of hours to several days. The behaviour of MWC\,480 resembles the characteristics 
typical of other photometrically monitored Herbig Ae/Be stars, for which light variations are likely of stochastic nature and 
caused by the 
pre-main-sequence disk accretion phenomena (e.g.\ Rucinski et al.\ \cite{Rucinski2010}). 
Using archive {\em Hipparcos} data, we have found an indication of a variability with a period of 1.6\,d.
However, it seems that six outliers appear
during the period HJD~8674.75--8674.94.
This group of outliers is located on the right side of the histogram presented in Fig.~\ref{fig:hist}. 
Excluding these outliers from the analysis,
we found no indication for periodicity.
In Fig.~\ref{fig:zdena}, we present {\em Hipparcos}
observations of MWC\,480 and the corresponding frequency periodogram. 

\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{perV380.eps}
\includegraphics[width=0.45\textwidth]{phaV380.eps}
\caption{
Frequency periodogram from {\em Hipparcos} observations of V380\,Ori.
The red arrow in this figure denotes the rotation period determined by Alecian et al.\ (\cite{Alecian2009}). 
The lower panel presents the corresponding phase diagram. No periodicity is detected.
}
\label{fig:zdenb}
\end{figure}

Kozlova et al.\ (\cite{Kozlova2007}) studied the spectral 
variability of MWC\,480 in the region of H$\alpha$. 
They found an equivalent width variability on 
a timescale of about 1200\,d and suggest that the most plausible reason for the detected cyclic variability is the 
reconstruction of the inner structure of the CS gas envelope, caused by the presence of a low mass 
companion or planet around the star.
It is not yet clear how and whether this periodicity is related to the appearance of the red-shifted component
in the \ion{Ca}{ii} H\&K lines during the second and third epochs of the SOFIN observations.

Our experience shows that the photometric periodic variability of Herbig Ae stars is generally difficult to detect.
As an example, we show in Fig.~\ref{fig:zdenb} {\em Hipparcos} photometric data for V380\,Ori,
for which the rotation period $P=4.31276\pm0.00042$\,d
was determined from magnetic field measurements (Alecian et al.\ \cite{Alecian2009}). As we show in this figure, no periodicity 
can be detected in the photometric data.

Until recently, rotation periods had been known for
only three Herbig Ae stars, V380 Ori, HD\,101412, and HD\,135544B.
They had been determined from direct measurements of longitudinal
magnetic fields (for V380\,Ori and HD\,101412), photometry (for HD\,101412), or radial velocities (for HD\,135544B)
(Alecian et al.\ \cite{Alecian2009}; Hubrig et al.\ \cite{Hubrig2011}; M\"uller et al.\ \cite{Muller2011}).

The longitudinal magnetic fields of HD\,97048, HD\,150193, and HD\,176386 are not very strong compared to the kG fields detected in 
V380\,Ori, HD\,101412, and Z\,CMa (Alecian et al.\ \cite{Alecian2009}; Hubrig et al.\ \cite{Hubrig2010,Hubrig2011}; 
Szeifert et al.\ \cite{Szeifert2010}).
Grady et al.\ (\cite{Grady2010}) placed limits on the field strength for MWC\,480 assuming that this star
is accreting magnetospherically.
Their result, $B_{\rm aver}=3^{+5}_{-2}$\,kG, appears comparable 
with our measurements presented in this work.



The detection of a weak longitudinal magnetic field in the blue-shifted CS components of the \ion{Na}{i} and 
the \ion{Ca}{ii} doublet lines in the polarimetric spectra of MWC\,480 implies that  for Herbig Ae/Be stars
the Doppler-shifted spectropolarimetric contributions
from photospheric and circumstellar environmental diagnostic lines should be investigated using high-resolution
polarimetric observations over the rotation period. These studies will allow us to apply 
the technique of Doppler Zeeman tomography to determine the correspondence between the magnetic field structure and 
the radial density and temperature profiles.

{
\acknowledgements
This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France.
MAP and RVY acknowledge the support obtained by RFBR grant No\,07-02-00535a and Sci.Schole No\,6110.2008.2.
}

