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%  "Spin axis of (2953) Vysheslavia and its implications"
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%   by D. Vokrouhlicky, M. Broz, T. Michalowski. F. Colas and L. Sarounova
%
%  manuscript submitted as a Note to the Icarus Journal
%  - ???, 2003
%
%  correspondence to: D. Vokrouhlicky
%                     Institute of Astronomy
%                     Charles University, Prague
%                     V Holesovickach 2
%                     CZ - 18000 Prague 8
%                     Czech Republic
%
%                     E-mail: vokrouhl@mbox.cesnet.cz
%                     Fax #:   (+420) 2 6885095
%                     Phone #: (+420) 2 21912574
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\def\hp{{http://\lb{2}smass.\lb{2}mit.edu}}
\def\hpmira{{http://\lb{2}www.\lb{2}axres.com}}
\def\hppisa{{http://\lb{2}newton.\lb{2}dm.\lb{2}unipi.\lb{2}it}}
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\slugcomment{Submitted to Icarus}
\shorttitle{Spin axis of (2953) Vysheslavia}
\shortauthors{Vokrouhlick\'y et~al.}

\begin{document}
 
\title{Spin axis of (2953) Vysheslavia and its implications}
 
\author{D. Vokrouhlick\'y, M. Bro\v z and J. \v Durech}
\affil{Institute of Astronomy, Charles University,
       V Hole\v sovi\v ck\'ach 2, CZ--18000 Prague 8, Czech Republic\\
       E-mail: vokrouhl@mbox.cesnet.cz}
\author{T. Micha{\l}owski}
\affil{Astronomical Observatory, Adam Mickiewicz University, ul.
       S{\l}oneczna 36, P--60-286 Pozna\'n, Poland}
\author{F. Colas}
\affil{Institut de M\'ecanique C\'eleste, 77 Av. Denfert Rochereau,
       F--75014 Paris, France}
\author{L. \v Sarounov\'a}
\affil{Astronomical Institute, Czech Academy of Sciences, 
       Fri\v cova 298, CZ--25165  Ond\v rejov, Czech Republic}

\newpage

\begin{abstract}
 Photometric observations taken during the last three years are
 used to determine pole orientation of (2953)
 Vysheslavia, about $15$~km size member of the Koronis family.
 We find $\beta_{\rm p} =-62^\circ\pm 8^\circ$ and $\lambda_{\rm p}
 = 182^\circ\pm 3^\circ$ for the rotation pole position (ecliptic
 latitude and longitude), implying obliquity $\gamma = 151^\circ\pm
 9^\circ$; sidereal rotation period is $P_{\rm sid}= 0.262267 \pm
 0.000003$~day. This result is interesting for two reasons: (i)
 obliquity larger than $90^\circ$ is consistent with prediction
 done by Vokrouhlick\'y {\it et~al.} ({\it Icarus} {\bf 150}, 78, 2001)
 that Vysheslavia might have been transported to its metastable
 orbit by the Yarkovsky effect, and (ii) with the obliquity close
 to $180^\circ$, Vysheslavia belongs to one of the two distinct
 groups in the Koronis family found recently by Slivan ({\it Nature}
 {\bf 419}, 49, 2002), further supporting the case of dichotomy
 in the spin axes distribution in this family. We also argue against
 the possibility that Vysheslavia has been placed on the current
 orbit by a recent collisional breakup in the family.
\end{abstract}
\keywords{asteroids, rotation --  Yarkovsky effect}

% For submission only ------------------------------------------------
\mbox{}\newpage
\noindent {\large \bf Editorial Correspondence and Proofs:}\\

\noindent {\bf Running Head:}\\
\noindent Spin axis of (2953) Vysheslavia
\bigskip

\noindent {\bf No. of manuscript pages:} 16\\[-2ex]
\noindent {\bf No. of figures:} 2\\[-2ex]
\noindent {\bf No. of tables:} 1
\bigskip

\noindent {\bf Address:}\\
\noindent David Vokrouhlick\'y\\ [-1ex]
\noindent Institute of Astronomy \\ [-1ex]
\noindent Charles University, Prague \\ [-1ex]
\noindent V Hole\v sovi\v ck\'ach 2\\ [-1ex]
\noindent CZ--18000 Prague 8 \\ [-1ex]
\noindent Czech Republic\\
\smallskip

\noindent Phone: (+420) 2 21 91 25 74 \\ [-1ex]
\noindent Fax: (+420) 2 688 50 95\\ [-1ex]
\noindent E-mail: vokrouhl@mbox.cesnet.cz
\clearpage
% For submission only -------------------------------------------------

\newpage

\section{Introduction} 
Vysheslavia, a moderately large asteroid in the zone of the
Koronis family, received considerable attention after Milani 
and Farinella (1995) discovered its very short dynamical lifetime.
According to these authors this asteroid will fall in the 5/2 
mean motion resonance with Jupiter in about $10-20$~Myr
(later confirmed by Vokrouhlick\'y {\it et~al.} 2001); as
a result Vysheslavia must have been placed on its peculiar orbit
a comparable time before the present. This is orders of
magnitude shorter than any relevant timescale, collisional
in particular, of an object of Vysheslavia's size and raises
a question of its origin.

Milani and Farinella (1995) suggested that Vysheslavia may 
outcome of a recent secondary collision (disruptive or non-disruptive), 
though they pointed out that its probability is low (less
than $5$\% in their estimate). Vokrouhlick\'y {\it et~al.} (2001)
proposed a more robust scenario, namely they assumed that
Vysheslavia underwent a slow transport due to the Yarkovsky 
effect from a dynamically more stable area further out from the
5/2 resonance (see also Bro\v z and Vokrouhlick\'y 2001).
With a low value of the surface conductivity (the likely case), 
a specific orientation of the spin axis is needed, namely 
obliquity must be larger than $90^\circ$. Values closer to 
$180^\circ$ are more favorable, since the relevant dynamical
timescale of the Yarkovsky transport is shorter, and thus the
whole hypothesis more robust.

Here we report photometric observations of Vysheslavia that
allowed us to determine pole orientation, and thus obliquity,
confirming the above scenario. Moreover, with a high obliquity 
of $151^\circ\pm 9^\circ$, Vysheslavia further strengthens conclusions
{}from recent observations by Slivan (2002) and Slivan {\it et~al.}
(2003), who found a surprising dichotomy in the obliquity distribution of
large Koronis members. Vysheslavia fits well the retrograde group. In 
the last Section, we collect additional arguments against the recent 
collisional origin of Vysheslavia with help of parameters derived for
the Karin cluster (Nesvorn\'y {\it et~al.} 2002).

\section{Observations}
We have observed Vysheslavia on ten nights during the three last 
apparitions: January 2000, March--April 2001, and September 2002.
The aspect data of the observations and other information are 
summarized in Table I. 
\smallskip

\centerline{EDITOR: PLACE TABLE 1 HERE}
\smallskip

The January 2000 Vysheslavia observation was performed at Saint Veran 
Observatory (France) using a $0.62$-m telescope equipped with a HiSys22 
CCD camera (KAF400 chip) and a non-standard red filter. As a result, no
transformation to the standard photometric system was possible. The data were
processed with the software APHOT developed at Ond\v{r}ejov Observatory. The
aperture photometry was used as described by Pravec {\em et~al.} (1996).

During the next two apparitions the asteroid was observed at 
Pic du Midi Observatory (France) using a $1.05$-m reflector equipped
with Thomson 7863 CCD camera and $R$ filter. A standard reduction was
performed with the ASTROL software developed at the Institut de 
M\'ecanique C\'eleste in Paris. The aperture photometry was obtained 
with the CCLRS STARLINK package (see also Micha{\l}owski {\em et~al.}
2000). Due to non-photometric weather conditions the obtained 
data were not transformed to the standard system and only relative
instrumental magnitudes are used.

In March 2001 Vysheslavia was also observed at
the River Oaks Observatory, New Braunfels (Texas) using a $0.4$-m
Newtonian telescope equipped with a SBIG ST8E CCD camera.
The standard dark substraction, flat fielding and differential photometry 
was performed using Mira software (\hpmira).
\smallskip

\centerline{EDITOR: PLACE FIGURE 1 HERE}
\smallskip

The observations are reported as composite lightcurves (Fig.~1) all
consistent with a synodic period of $6.293\pm 0.002$ hours. The magnitude
offset of each lightcurve was obtained by minimizing the dispersion 
of data points relative to their neighbors. 
The lightcurve amplitude varies from $0.20$ magnitude in 2000 to
$0.18$ magnitude in 2002, with a value of $0.19$ magnitude in 2001
(we estimate the error of the lightcurve amplitude to
be $\simeq 0.02$ magnitude, while the error of the lightcurve
maximum/minimum epoch is $\simeq 10$ minutes).  We preliminarily note, 
that such a small variation during the three different oppositions,
and phase angle ranging from $0^\circ$ to nearly $15^\circ$, already
suggests a non-ecliptic pole for Vysheslavia (in March 2001 the 
phase angle was maximum and this is the reason why the 
secondary amplitude --- see $0.5-0.8$ rotation phase
in Fig. 1b --- was slightly greater than that in April).

\section{Shape and spin model of (2953) Vysheslavia}
Orientation of the spin vector, sidereal period, and triaxial 
ellipsoid model of Vysheslavia have been determined by using
the method described by Micha{\l}owski (1993). Since no 
absolute calibration was done, only the epochs of brightness
maxima and amplitudes have been considered. This is basically
the weighted {\it amplitude-aspect method} (see also Drummond
{\it et~al.} 1988) which models the asteroid as a 
triaxial ellipsoid of uniform surface reflectivity; to a
good approximation the brightness variations are directly
given by the changing cross-section of the asteroid as it
rotates about the shortest axis of the inertia tensor. The
orientation of the spin axis is thus obtained, but this method
cannot by itself resolve the sense of rotation about the axis.
To that end, we complemented the amplitude-aspect analysis by
the {\it epoch method}, namely tracking in time a chosen ``prominent
feature'' on the rotation curve assuming it corresponds
to a pattern at fixed longitude on the asteroid. The
amplitude-aspect-epoch analysis leads to a set of
nonlinear equations for the pole position ($\lambda_{\rm p}$,
$\beta_{\rm p}$), in ecliptic coordinates, the sidereal
rotation period $P_{\rm sid}$ and ratio $a/b$ and $b/c$ 
of the best-fit ellipsoid axes; Micha{\l}owski (1993) discusses
the appropriate iterative method of their solution. At
convergence, we obtained the following best-fit solution:
\begin{eqnarray}
 P_{\rm sid} &\!\!\! =\!\!\! & 0.262267 \pm 0.000003 \;\;
  ({\rm days})\; \nonumber \\ [2pt]
 \lambda_{\rm p} &\!\!\! = \!\!\!& 182^\circ \pm 3^\circ\; , \quad
 \; \beta_{\rm p}  =  -62^\circ \pm 8^\circ \nonumber \\ [2pt]
 a/b &\!\!\! =\!\!\! & 1.18 \pm 0.04 \; , \quad
 b/c  =  1.0 \pm 0.1 \; , \nonumber
\end{eqnarray}
implying a retrograde sense of rotation and a pole far from
ecliptic, in consistency with the above suggestion. This
configuration is unfavourable to solve for the ratio $b/c$, and
the above value of this parameter is only formal
(see eqs.~(3) and (5) in Micha{\l}owski 1993).

Because the pole aspect is close to $90^\circ$, further
observations will keep reducing uncertainty in its determination
but they will likely not improve the shape solution. 

\section{Discussion}
Our solution for the Vysheslavia pole has several interesting 
implications. First, it fits prediction by Vokrouhlick\'y
{\it et~al.} (2001) and supports the Yarkovsky emplacement
of Vysheslavia onto its unstable orbit. This agrees with
a general model of dynamical evolution of asteroid families
in which they continually spread in semimajor axis due to the
Yarkovsky effect; their sharp termination at the prominent mean
motion resonances (such as the 5/2 mean-motion resonance with
Jupiter; Fig.~2) is one of strong
arguments in favour of this view (Bottke {\it et~al.} 2001, 2003).
Vysheslavia is thus the largest member of the asteroid population ``on 
the brink'', continually resupplied by new objects (for other asteroids
in the Koronis zone see Kne\v zevi\'c {\it et~al.} 1997, Bro\v z and
Vokrouhlick\'y 2001, or Guillens {\it et~al.} 2002 for asteroids
in the vicinity of the 3/1 mean-motion resonance with Jupiter). In 
the Koronis
family, another strong argument for the ongoing spreading in the 
semimajor axes is the existence of the Prometheus clan with a
significantly higher eccentricity beyond the $g+2g_5-3g_6$
secular resonance (Bottke {\it et~al.} 2001).
%Morbidelli and
%Vokrouhlick\'y (2003) discuss that such leakage of the family and
%background asteroids in the main belt is likely a dominant process to
%sustain the near-Earth asteroid population. 
 
An alternative possibility, discussed by Milani and Farinella
(1995), is that Vysheslavia has a collisional origin, either
as a result of a recent disruptive secondary break-up in the
family or a non-disruptive impact on Vysheslavia that would shift
its semimajor axis into the dynamically unstable region.
A possible result of the collision could be a non-principal-axis
rotation of Vysheslavia, provided the collision occurred
recently so that the internal dissipation would not damp the
wobble energy. For a $15$~km size asteroid and $6.29$~hr rotation
period the charateristic damping timescale is $0.5-5$~My,
depending on the physical parameters of the body (e.g. 
Vokrouhlick\'y and \v Capek 2002, Paolicchi {\it et~al.} 2003).
The higher value, if correct, is a non-negligible fraction
of the dynamical lifetime of Vysheslavia's orbit. In this respect
we note that our observations do not support non-principal-axis
rotation and they are well fitted with the shortest axis rotation
mode with a single rotation period over all three oppositions. 
This obviously does not rule out the recent collisional 
origin of Vysheslavia, but makes it less likely. We also note that
collisional lifetime of a hypothetical $30-50$~km size parent 
body of Vysheslavia is likely $\gtrsim 10$~Gy (W.F.~Bottke,
personal communication), so that the likelihood of the recent
secondary breakup in the Koronis family with Vysheslavia as
one of the largest fragments is $\leq (0.5-1)$\% (there might
be less then ten objects of that size in the family close enough
to the 5/2 resonance). Additionally, we note that reflectance spectrum
of Vysheslavia is quite typical in the family (S-type; e.g.
Chapman {\it et~al.} 1989, Bus and Binzel 2002) and does not show 
any anomalous feature that would
be related to its hypothetical younger age. To check this conclusion
we used spectra of the $22$ Koronis members, including Vysheslavia, 
available at the SMASSII web site \hp.
\smallskip

\centerline{EDITOR: PLACE FIGURE 2 HERE}
\smallskip

Moreover, we can consider the well identified recent break-up
in the Koronis family of a comparable size --- the Karin cluster 
(Nesvorn\'y {\it et~al.} 2002) --- as a ``benchmark'' and deduce
what fraction of observable ejecta from a hypothetical 
secondary break-up event, for which Vysheslavia would be the
largest fragment, would be still surviving. Assuming the
same velocity distribution as observed in the Karin cluster and
the central position of Vysheslavia,
we predict that from initially $20-40$ fragments larger than 
$\simeq 3$~km (depending on steepness of the size distribution),
some $8-16$ should survive $15$~My. None of such objects is
observed in the Vysheslavia vicinity (Fig.~2). Together with
a very low probability of the secondary break-up of a large
member in the family mentioned above, the absence of accompanying
ejecta in the Vysheslavia neighborhood even lessens the
probability of this case.
On the other hand, we recall that a couple of multikilometer 
asteroids, unrelated to Vysheslavia, are also on unstable orbits 
(Bro\v z and Vokrouhlick\'y 2001); this makes more sense if 
we consider them to be members of the population fed by the 
Yarkovsky drift, rather then to be an outcome of several recent 
break-ups at the border of the 5/2 resonance.

Secondly, Slivan (2002) and Slivan {\it et~al.} (2003) analyzed 
spin vectors of the largest members in the Koronis family. They 
found a markedly non-random distribution with two significant 
``clusters'' having obliquity near $50^\circ$ (prograde rotators)
and between $150-180^\circ$ (retrograde rotators). Though 
Vysheslavia is little smaller than 
the asteroids observed by Slivan {\it et~al.} (sizes of $20-40$~km),
it still seems to support the bimodal distribution of the pole
directions in the family, namely it matches characteristics of the
retrograde rotators. Since it is very likely, that the
pole bimodality is due to a so far unidentified dynamical 
process (Vokrouhlick\'y {\em et~al.} 2003), its extension to 
smaller asteroidal sizes may be an
important key. Photometric observations of further $10-20$~km sized
Koronis family members, with the goal to determine their pole, 
is thus an important project for the future.
                                      
\acknowledgments
 We are grateful to Anton Paschke and Lud\v{e}k Va\v{s}ta for their
 help during the observations taken at Saint Veran Observatory.
% and to Bill Holliday for his observations at the River Oaks Observatory.
 This work has been supported by the Grant Agency of the Czech
 Republic, under the grant No.~205/02/0703, and by the Polish KBN
 Grant 2 P03D 007 18.

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\newpage

% TABLE 1 -------------------------------------------------------------
\begin{deluxetable}{lrrrrrrrl}
%\tabletypesize{\scriptsize}
\tablecaption{Aspect data of the (2953) Vysheslavia observations: date is the
 mean epoch of the observation, $r$ and $\Delta$ are the heliocentric
 and geocentric distances, $\phi$ is the phase angle, ($\lambda$, $\beta$)
 are the ecliptic longitude and latitude, in the J2000.0 reference system
 and ($\alpha$, $\delta$) are the J2000.0 right ascension and
 declination. \label{AspDat}}
\tablewidth{0pt}
\tablehead{
\colhead{Date (UT)} & \colhead{$r$} & \colhead{$\Delta$} &
\colhead{$\phi$} & \colhead{$\lambda$} & \colhead{$\beta$} &
\colhead{$\alpha$} & \colhead{$\delta$} & \colhead{Observatory} \\
\colhead{} & \colhead{(AU)} & \colhead{(AU)} & \colhead{($^\circ$)} &
\colhead{($^\circ$)} & \colhead{($^\circ$)} & \colhead{($^\circ$)} &
\colhead{($^\circ$)} & \colhead{}} 
\startdata
\multicolumn{9}{c}{\bf 2000}\\
Jan 16.1 & 2.8822 & 1.9026 &  2.19 & 121.47 & $-1.20$ &
 123.41 & 18.66 & Saint Veran \\
Jan 17.1 & 2.8821 & 1.9010 &  1.77 & 121.26 & $-1.21$ &
 123.19 & 18.70 & Saint Veran \\ [3pt]
\multicolumn{9}{c}{\bf 2001}\\
Mar 16.4 & 2.7858 & 2.0021 & 14.92 & 221.93 & $-1.04$ &
 219.14 & $-16.40$ & River Oaks \\
Mar 20.3 & 2.7850 & 1.9632 & 13.83 & 221.69 & $-1.05$ &
 218.90 & $-16.33$ & River Oaks \\
Mar 21.3 & 2.7848 & 1.9538 & 13.54 & 221.61 & $-1.05$ &
 218.82 & $-16.31$ & River Oaks \\
Apr 26.1 & 2.7779 & 1.7717 &  0.37 & 215.64 & $-0.98$ & 
 212.97 & $-14.32$ & Pic du Midi\\
Apr 27.0 & 2.7778 & 1.7715 &  0.60 & 215.44 & $-0.97$ &
 212.79 & $-14.25$ & Pic du Midi\\
Apr 28.0 & 2.7776 & 1.7715 &  0.98 & 215.22 & $-0.97$ &
 212.58 & $-14.17$ & Pic du Midi\\ [3pt]
\multicolumn{9}{c}{\bf 2002}\\
Sep 02.0 & 2.7951 & 1.8527 &  9.08 & 313.46 &  1.53 &
 315.44 & $-15.33$ & Pic du Midi\\
Sep 03.0 & 2.7953 & 1.8588 &  9.43 & 313.32 &  1.52 &
 315.29 & $-15.37$ & Pic du Midi\\
\enddata
\end{deluxetable} 
% --------------------------------------------------------------------- 

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FIGURE 1 ------------------------------------------------------------ 
\begin{figure}[ht]
\epsfxsize=10truecm
\begin{center}
 \mbox{\epsfbox{vysh.1}}
\end{center}
%\plotone{vysh.1}
\caption{Composite rotation lightcurves of 2953 Vysheslavia in three
 consecutive oppositions: January 2000, March--April 2001 and September 2002.
 The abscissa is the rotational phase with zero points corrected for 
 light-time and corresponding to: 2000, Jan 15.9179 UT, 2001, Apr
 25.9550 UT, and 2002, Sep 2.8146 UT, respectively; 
 on ordinate, relative instrumental magnitudes
 adjusted so that dispersion with respect to the neighbors is minimized.}
\label{f1}
\end{figure}
% ---------------------------------------------------------------------

%\noindent {\bf Figure 1. Vokrouhlick\'y et al., Spin axis of (2953)
% Vysheslavia}

\newpage

% FIGURE 2 ------------------------------------------------------------ 
\begin{figure}[ht]
\epsfxsize=10truecm
\begin{center}
 \mbox{\epsfbox{freqmap.1}}
\end{center}
%\plotone{freqmap.1}
\caption{Koronis family members (dots) projected on the plane of 
 proper semimajor axis and proper eccentricity (Kne\v zevi\'c {\it et~al.}
 2003); Karin cluster is indicated
 in the lower right corner. For an isotropic velocity field, and different
 values orbital parameters, ejecta from a hypothetical secondary collision
 near Vysheslavia (full circle) would fall in the region indicated by dots.
 None of the observed Koronis asteroids falls in this zone, notably in its
 stable part above the 5/2 mean motion resonance with Jupiter. Orbital
 stability is indicated by grey shading obtained with a parameter
 $\sigma =1-n^{(2)}/n^{(1)}$, where $n^{(1)}$ and $n^{(2)}$
 are proper mean motions of test particles calculated in the two
 consequtive time intervals, each spanning $1.2$~Myr (dark zones of small
 $\sigma$ mean stability, white areas of high $\sigma$ value, such as
 the 5/2 resonance, are unstable); this is based of frequency map
 analysis by Robutel and Laskar (2001).
 The family members were identified using the hierarchical clustering
 method and the velocity cutoff of $60$~m/s, e.g. Zappal\`a
 {\it et~al.} (1995), Bottke {\it et~al.} (2001). We first used
 a catalogue of proper elements at \hppisa~computed analytically for
 identification; at a second step, we selected (and
 plotted here) only those asteroids for which synthetic proper elements
 have been computed.}
\label{f2}
\end{figure}
% ---------------------------------------------------------------------

%\noindent {\bf Figure 2. Vokrouhlick\'y et al., Spin axis of (2953)
% Vysheslavia}

\end{document}
