% iauc197_r21.tex
% IAU Coll. 197 paper on J2/1 resonance.
% Miroslav Broz (miroslav.broz@email.cz), Nov 1st 2004

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\title[The population of asteroids in the 2:1 mean motion resonance ...]%
{The population of asteroids in the 2:1 mean motion resonance
with Jupiter revised}

\author[Bro\v z, Vokrouhlick\'y, Roig, Nesvorn\'y, Bottke and Morbidelli]%
{Miroslav Bro\v z$^1$, D. Vokrouhlick\'y$^1$, F. Roig$^2$, 
 D. Nesvorn\'y$^3$, \break W.F. Bottke$^3$ and A. Morbidelli$^4$}

\affiliation{%
$^1$Institute of Astronomy, Charles University, V~Hole\v sovick\'ach~2,
    18000 Prague, Czech Republic\break
    email: mira@sirrah.troja.mff.cuni.cz, vokrouhl@mbox.cesnet.cz
\\[\affilskip]
$^2$Observat\'orio Nacional, Rua Gal. Jos\'e Cristino 77,
    Rio de Janeiro, 20921-400 RJ, Brasil\break
    email: froig@on.br \\[\affilskip]
$^3$Department of Space Studies, Southwest Research Institute,
    1050 Walnut St., Suite 400, Boulder, CO 80302, USA\break
    email: davidn@boulder.swri.edu, bottke@boulder.swri.edu
\\[\affilskip]
$^4$Observatoire de la C\v ote d'Azur, Dept.~Cassiopee, BP~4224,
    06304 Nice Cedex 4, France\break
    email: alessandro.morbidelli@obs-nice.fr}

\pubyear{2004}
\volume{197} %% insert here IAU Symposium No.
\pagerange{??--??}
\date{?? and in revised form ??}
\setcounter{page}{1}
\jname{Dynamics of Populations of Planetary Systems}
\editors{Z. Kne\v zevi\'c and A. Milani, eds.}

\begin{document}

\maketitle

\begin{abstract}
 We study the population of asteroids inside the 2/1 mean motion
 resonance with Jupiter, in the so called Hecuba gap. Origin of
 these bodies is not well understood: (i) the long-lived (stable)
 population may be primordial, but this contradicts its steep size
 distribution, while (ii) the short-lived (unstable)
 population requires an efficient sustaining mechanism. Our working
 hypothesis is that the unstable asteroids are continuously
 resupplied from outside the resonance by the Yarkovsky effect. As
 a first step toward comparison of such model with observations,
 we report here an update of the observed population of asteroids
 residing in the 2/1 Jovian resonance, mainly because the number of
 cataloged orbits increased substantially during the last few years.
 We found there are $153$~numbered and multi-opposition resonant
 asteroids in total and we classified them into the three
 sub-populations according to their dynamical lifetime. Our work
 also allowed us to derive several important parameters such as
 asteroid locations inside the resonance or size distribution of
 the sub-populations. As a particular novelty, we identified
 $6$~asteroids located inside the high-eccentricity quasi-regular
 stable island, which previously seemed empty.
\keywords{minor planets, asteroids}
\end{abstract}

\firstsection

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                              %%% SEC 1 %%%
\section{Introduction and current state of art}

The structure of the Jovian mean motion resonances (MMRs)
has been thoroughly studied for a long time. In the 1980's, there
seemed to be a gap, almost void of any asteroids, in the 2/1 MMR,
and that was in contrast with the 3/2 MMR, where numerous
asteroids (so called Hildas) were observed.
%
The 3-body models of motion in the 2/1 MMR
(Murray 1986; Henrard \& Lema\^{\i}tre 1987; Lema\^{\i}tre \& Henrard 1990)
failed to find any large chaotic regions inside, which could increase
the eccentricity of the resonant objects and drive them to planet-crossing orbits,
a preferred theory for the depletion of resonances since the work of Wisdom (1982).
%
However, the early N-body numerical simulations
(Wisdom 1987) showed the mechanism should work.
%
Improved 3-body models (Morbidelli \& Moons 1993),
with main secular perturbations of Jupiter's orbit,
allowed to calculate borders of secular resonances embedded in the 2/1 MMR,
but the key point was to use at least a 4-body model (Ferraz-Mello 1994,
Henrard \etal\ 1995), because only then low- and high-eccentricity
chaotic regions, caused by overlapping secondary or secular resonances,
are interconnected and allow large scale diffusion.
%
The extensive \hbox{N-body} simulations (Nesvorn\'y \& Ferraz-Mello 1997)
and semi-analytical analysis (Moons \etal\ 1998) allowed to map
the structure of the 2/1 MMR in detail.
Two macroscopic islands of stability, separated by $\nu_{16}$
secular resonance, were denoted A and B (see also Fig.~\ref{fig:arerir}).
In the 1990's, there were already 10 resonant asteroids discovered.

In general, observations support the latest theoretical results,
confirming that the 2/1 Jovian MMR harbors a small population of asteroids.
Recent numerical analysis (Roig \etal\ 2002) allowed to 
classify these objects in three groups:
%
  (i)~{\em Zhongguos\/} residing on stable orbits, whose dynamical
      lifetimes are comparable to the age of the Solar System,
 (ii)~{\em Griquas\/} residing on orbits, whose dynamical
      lifetimes are several hundreds of My, and
(iii)~{\em unstable asteroids\/}, with a typical lifetime of
      $\simeq\!10\,{\rm My}$.
%
Sub-populations (i) and (iii) are distinct not only by their
very different lifetime, but also by their location in the resonance.
Until now, no consistent scenario could make them linked
together or to a common source.

Our goal is to prove that majority of the unstable asteroids
have been pushed to the resonance from the main asteroid belt
by the Yarkovsky effect.
%
Before justifying any such hypothesis, we find useful to revise the 2/1 MMR
asteroid population anew, because the automated search programs significantly
boosted the number of known asteroids and smart identification procedures
allowed to efficiently link past observations with the new discoveries, 
making thus the orbits more accurate. 

In this paper we report results of our new search of the 2/1 
resonant asteroids. The two main achievements are:
%
 (i)~the population has grown to $153$~objects with multi-opposition orbits
     (thus nearly tripled during the last 3~years), and
(ii)~we found the first asteroids in the previously empty island~A.
%
Both results have interesting implications: more numerous populations
allow to better characterize their size distributions --- an information
directly constraining their origin. In the same way, the numbers
of asteroids in the stable islands A and B constrain mechanisms
by which they have been injected in the resonance or later destabilized.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                              %%% SEC 2 %%%
\section{The resonant population revised}

\noindent{\bf Selection procedure.--}
The primary sample of asteroids we used are the numbered and
multi-opposition bodies from the {\tt AstOrb} database of the Lowell 
observatory ({\tt ftp://ftp.\lb{2}lowell.\lb{2}edu/\lb{2}pub/\lb{2}elgb/}
as of April 2004) whose osculating semimajor axis $a$ and eccentricity
$e$ satisfy $e > {0.45\over 0.14} (3.24 - [a]_{\rm AU}) \mbox{ and }
e > {0.5\over 0.12} ([a]_{\rm AU}-3.34)$. This condition delimits
a broad region near the 2/1 MMR (see Fig.~1 in Roig \etal\ 2002)
containing $\approx\!4000$~asteroids. We numerically integrated their 
orbits for $10\,{\rm ky}$ using {\sc Swift-mvs2-fp} integrator.%
%
\footnote{This is a modification of 
{\sc Swift} package (Levison \& Duncan 1994) with the second-order
symplectic integrator included (see Laskar \& Robutel 2001).
Our program also contains on-line digital filters adapted to
compute both synthetic proper orbital elements for the non-resonant
motion and pseudo-proper elements for the resonant motion.}
%
Our simulation included 4~giant planets. We applied a barycentric 
correction to the initial conditions of the planets' and asteroids' orbits 
(to partially account for the indirect perturbation by terrestrial planets).
The second-order symplectic integrator allows a~longer time-step
than {\sc Swift-mvs}, in our case $91.3125$~days.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
 \includegraphics[height=6.3cm]{follow_mmr_amp5_bw.eps}%
 \hfil
 \includegraphics[height=6.3cm]{follow_mmr_amp6_bw.eps}
\end{center}
\caption{{\it Left:} The behavior of the critical angle $\sigma_{{\rm J2/1}}$
 of the 2/1 MMR with Jupiter for asteroids in the broad region around 
 and in the resonance. The total change $\Delta\sigma$ of $\sigma_{{\rm
 J2/1}}$ (i.e.~the maximum minus
 minimum value) vs. the ratio $r_\sigma$ (defined in the text) is plotted;
 both variables were computed from a $10\,{\rm ky}$ numerical integration
 with an output time-step $10\,{\rm y}$. Bodies for
 which $\sigma_{{\rm J2/1}}$ steadily circulates have either $r_\sigma=1$
 or $r_\sigma=-1$; the four asteroids with $\Delta\sigma > 30,000^\circ$
 and $|r_\sigma|<0.98$ are either highly unstable or exhibit one abrupt
 change of $\sigma_{{\rm J2/1}}(t)$ evolution. Asteroids with $r_\sigma
 \simeq 0$ and $\Delta\sigma \lesssim 360^\circ$ reside inside the resonance.
 {\it Right:} A~zoom in the resonant population. Asteroids highlighted by
 circles were later identified to be located in the stable island~A
 (see the text).}
\label{fig:sigma}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For each of the orbits we computed the evolution of the critical angle
$\sigma_{{\rm J2/1}} = 2\lambda_{\rm J}-\lambda-\varpi$ that characterizes
motion in the 2/1 Jovian MMR ($\lambda_{\rm J}$ and $\lambda$ are the
longitude in
orbit for Jupiter and for the asteroid, and $\varpi$ is the longitude
of pericenter of the asteroid). The time-span of the integration
was $10\,{\rm ky}$ and the output sampling was $10\,{\rm y}$.
We computed two diagnostic parameters (Fig.~\ref{fig:sigma}):
%
 (i)~the ratio $r_\sigma = (n_+-n_-)/(n_++n_-)$, where $n_+$ (or $n_-$)
     is the number of data points for which $\sigma_{\rm J2/1}(t)$
     increased (or decreased) in two successive steps, and
(ii)~the difference $\Delta \sigma$ between the maximum and minimum
     value of $\sigma_{{\rm J2/1}}$.
%
Regularly librating resonant asteroids appear closely clustered near 
the point $(r_\sigma,\Delta\sigma)=(0,180^\circ)$ (in practice we used
a~$|r_\sigma|<0.05$ threshold). Asteroids outside the resonance, for
which the critical angle circulates, have all $r_\sigma\doteq\pm 1$ and
a value of $\Delta\sigma$ depending on the mean rate of their circulation.
%
Since each of the parameters $r_\sigma$, $\Delta\sigma$ should allow
by itself to detect the resonant asteroids, we can expect their values
to be correlated -- Fig.~\ref{fig:sigma} confirms this conclusion.
%
Several intermediate cases are:
%
 (i)~border-line asteroids for which the critical angle
     alternates between periods of libration and circulation
     (full circles), and
(ii)~high-eccentricity asteroids with very unstable orbits
     or with peculiar orbits for which the sense of circulation
     of $\sigma_{{\rm J2/1}}$ changes abruptly at some instant
     (open circles).
%
Bodies with orbits for which $\sigma_{{\rm J2/1}}$ librates,
or temporarily librates, but which are non-resonant were excluded
from Fig.~\ref{fig:sigma}.
%
This is the case of motion near the pericentric and apocentric branches
of periodic orbits in the three body problem. These orbits
would have $|r_\sigma| \in (0.05, 0.98)$ in our plot
and also $N=\sqrt{a}\,(2-\sqrt{1-e^2} \cos I) \le 0.8$
($a$~is the osculating semimajor axis, $e$~the eccentricity and $I$ 
the inclination; see e.g.~Morbidelli \& Moons 1993).
There are many such pericentric alternators with $a\le 3.275\,
{\rm AU}$ and $e\le 0.15$, but we do not study them here.

The procedure described above led to us to identify $153$~asteroids residing
inside the 2/1 MMR with Jupiter, making the population significantly grown
since the last analysis (Roig \etal\ 2002). The complete list
of the resonant asteroids, which is too long to be published here,
is available at our web site 
{\tt http://\lb{2}sirrah.\lb{2}troja.\lb{2}mff.\lb{2}cuni.\lb{2}cz/\lb{2}yarko-site/}.

\smallskip
\noindent{\bf Resonant asteroids: dynamical lifetime and classification into
sub-po\-pu\-la\-tions.--}
Motion inside MMRs is chaotic enough that any single numerical integration
of a chosen orbit may not necessarily represent the true future motion.
A safer procedure is to consider (for each asteroid) a multitude of orbits
which are close enough to be all compatible with the observations.%
%
\footnote{In practice this means to consider orbits with
slightly modified initial elements, well inside the uncertainty region
of the orbit determination. We call such fictitious orbits,
but otherwise statistically equivalent, ``close clones''.}
%
\ Hence, unlike the previous studies, we used the close-clones technique
to determine dynamical lifetimes of the resonant asteroids:
for each body we generated $12$~clones
by changing the orbital semimajor axis and eccentricity by multiples of 
$\delta a = 10^{-9}\,{\rm AU}$ and $\delta e = 10^{-9}$; orbital inclination
and angular elements of clones were the same as of the original orbit.
We numerically integrated resulting $13\times 153$ orbits using
{\sc Swift-mvs2-fp} integrator upto $1\,{\rm Gy}$ or until the filtered 
mean semimajor axes (i.e.~with fast oscillations removed) showed a
significant departure (larger than $0.05\,{\rm AU}$) from the centre
of the resonance at $3.27\,{\rm AU}$.
This procedure was found reliable enough to characterize
the residence time inside the resonance. Since all
close clones are statistically equivalent realizations of the orbit,
the characteristic dynamical lifetime $t_{\rm J2/1}$ of the corresponding
asteroid was determined as a median value of the individual
lifetimes of the clones.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=12.0cm]{median_clones.eps}
\end{center}
\caption{Median value $t_{\rm J2/1}$ of lifetime vs. difference
 $\Delta t$ between the maximum and minimum lifetime values
 of the close clones for asteroids observed inside the 2/1 MMR with Jupiter.
%
 Asteroids with $t_{\rm J2/1} \le 70\,{\rm My}$ have usually small~$\Delta t$,
 indicating that all close clone orbits are
 unstable with a comparable timescale. Conversely, asteroids with
 $t_{\rm J2/1}\ge 70\,{\rm My}$ have large~$\Delta t$,
 indicating some clones may be long-lived, while others are short-lived.
 This seems to distinguish Griqua-like orbits from the unstable orbits.
 Most Zhongguos (with $t_{\rm J2/1}=1\,{\rm Gy}$) have again small
 or even zero value of $\Delta t$, since nearly all clones have
 the maximum lifetime.}
\label{fig:median}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=8.3cm]{r21_obs_loglftime_bw.eps}
\end{center}
\caption{Distribution of the median lifetime
 values $t_{\rm J2/1}$ for asteroids residing the 2/1 MMR. Note the
 log-binning and scale at the abscissa. Bodies with $t_{\rm J2/1}\le
 70\,{\rm My}$ seem to form a distinct population with unstable orbits;
 Zhongguos on stable orbits ``pile up'' at $t_{\rm J2/1}=1\,{\rm Gy}$,
 limited in fact by the time-span of our integration.
 Griquas, with  $t_{\rm J2/1}\in (70\,{\rm My}, 1\,{\rm Gy})$,
 seem to be adjacent to the Zhongguo group.}
\label{fig:lftime}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Figures~\ref{fig:median} and \ref{fig:lftime} summarize the result.
The first, showing aside the median lifetime $t_{\rm J2/1}$ also the
range $\Delta t$ of the individual close-clone
lifetime values, suggests there is a transition at about $t_{\rm J2/1}
\simeq 70\,{\rm My}$:
%
 (i)~asteroids with smaller values of $t_{\rm J2/1}$ reside
     on unstable orbits so that almost none of the close clones
     happen to have a long dynamical lifetime;
(ii)~asteroids with larger values of $t_{\rm J2/1}$ reside on marginally stable
     or stable orbits, characterized with some (or all) clones having
     very long dynamical lifetimes
     (eventually $\geq 1\,{\rm Gy}$).
%
Additionally, Fig.~\ref{fig:lftime} suggests bodies with $t_{\rm J2/1}<
70\,{\rm My}$ are perhaps composed of two distinct groups: (i) very
short-lived with $t_{\rm J2/1} \le 1\,{\rm My}$, and (ii) short-lived
with $t_{\rm J2/1} \ge 2\,{\rm My}$. The very short-lived
group ($\simeq\!25\,\%$ of the unstable asteroids) might be
populated from a different source than the adjacent main belt,
such as the Jupiter family comets or near-Earth asteroids, but this
issue needs to be studied in care before drawing any conclusions.
The long-lived asteroids ($t_{\rm J2/1} \ge 70\,{\rm My}$) are either
marginally stable Griquas or stable Zhongguos.%
%
\footnote{In fact, our $1\,{\rm Gy}$ simulation
is still inappropriate to quantify lifetimes of Zhongguos, it only indicates
they are longer than $1\,{\rm Gy}$. Griquas are separated from
Zhongguos only conventionally and up to this  time-span they seem to form
a shorter-lived ``tail'' of the Zhongguo population.}
%
\ The respective numbers of asteroids are:
$47$~unstable asteroids,
$31$~Griquas and
$75$~Zhongguos.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=6.7cm]{r21_arer.eps}
\includegraphics[width=6.7cm]{r21_arir.eps}
\end{center}
\caption{Distribution of the resonant asteroids in the space of
 pseudo-proper elements $a_p$, $e_p$, $\sin I_p$. Zhongguos are denoted
 by filled circles, Griquas by open squares and the unstable asteroids
 by crosses. The error bars denote standard deviation of pseudo-proper
 elements computed over $1$~My interval of time. The thin and thick
 solid lines in the $(a_p, e_p)$ projection (left) are the libration
 centre and separatrix of the 2/1 resonance
 (both shown for $I=0^\circ$). Borders of chaotic zones associated
 with embedded secular resonances --- $\nu_5$, $\nu_{16}$ and the
 Kozai resonance --- are the dashed and dashed-dotted lines (all shown
 for $I=10^\circ$; adapted from Moons \etal\ 1998). A~majority of the stable
 asteroids is clustered in the island~B, while a~few of them
 (see Table~\ref{tab:zhonga}) are located in the island~A,
 characterized by a higher mean eccentricity and inclination.
 All unstable asteroids are located in the chaotic zone
 where various secular resonances overlap. Griquas are
 a border-line population mostly at the edge of the B-region.
 In fact, 2-D~projections shown here always lack clarity in showing
 3-D~structures; for that reason we posted a 3-D animation of the
 resonance structure with positions of the embedded asteroids at
 our web-site {\tt
 http://\lb{2}sirrah.\lb{2}troja.\lb{2}mff.\lb{2}cuni.\lb{2}cz/\lb{2}yarko-site/}.}
\label{fig:arerir}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip
\noindent{\bf Resonant asteroids: location inside the resonance.--}
Our numerical integration allowed us to characterize also the location
of asteroids inside the 2/1 MMR. This was done by computing so called
pseudo-proper (resonant) orbital elements (e.g.~\cite{roig_etal2002})
--- semimajor axis~$a_p$, eccentricity~$e_p$ and inclination~$I_p$.
They are defined as osculating orbital elements at the time when the 
orbit intersects a surface of section defined by $\sigma_{\rm J2/1}=
0^\circ \mbox{ and } \dot\sigma_{\rm J2/1} < 0 \mbox{ and } \varpi-\varpi_{\rm J}=0^\circ 
\mbox{ and } \Omega-\Omega_{\rm J}= 0^\circ$, where $(\Omega, \varpi;
\Omega_{\rm J}, \varpi_{\rm J})$ are the longitude of node and pericenter 
of the asteroid and of Jupiter, respectively, and $\dot\sigma_{\rm J2/1}$
is the time derivative of $\sigma_{\rm J2/1}$. In practice, though, we 
used a weakened condition $\sigma_{\rm J2/1}\le 5^\circ\,\mbox{ and }\,\Delta
\sigma < 0\,\mbox{ and }\, |\varpi-\varpi_{\rm J}|\le 5^\circ$ (where $\Delta
\sigma$ is a difference between the two successive values of $\sigma_{\rm 
J2/1}$), in accord with \cite{roig_etal2002}.
The pseudo-proper resonant elements are near cons\-tants
of motion in the averaged three-body problem with Jupiter on a fixed
circular orbit; in reality the motion of both the asteroids and planets
is more complicated and this may cause variations of $(a_p,e_p,I_p)$.
We thus computed mean values of $(a_p,e_p,I_p)$ and standard deviations
$(\sigma_a, \sigma_e,\sigma_I)$ during the first 1\,My of our
integration (only nominal orbits were used for that purpose).
Results are shown in Fig.~\ref{fig:arerir}, where we give positions of the
resonant asteroids projected onto the planes $(a_p, e_p)$ and $(a_p, 
\sin I_p)$. We note there is a strong correlation between
the lifetimes of objects and their positions in the resonance:
%
  (i)~the unstable orbits are located inside the chaotic regions
      where secular resonances overlap,
 (ii)~the stable orbits (Zhongguo-like) are inside the stable islands
      B and A, and
(iii)~the marginally stable (Griqua-like) are right at the outskirts
      of the stable regions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=4.4cm]{r21-size_distribution-unstable.eps}
\includegraphics[width=4.4cm]{r21-size_distribution-griquas.eps}
\includegraphics[width=4.4cm]{r21-size_distribution-zhongguos.eps}
\end{center}
\caption{Cumulative size-frequency distributions of highly unstable
 asteroids (left), Griquas (middle) and Zhongguos (right). The solid
 lines indicate the best fit power-law approximations within the size
 interval delimited by the vertical grey lines. There are statistically
 significant differences in steepness of the size distributions of the
 three sub-populations. We assumed the albedo of all asteroids to be 0.05.
 Arrows in the left-most plot indicate a horizontal displacement
 of the distributions, if we assume different values of the albedo
 --- 0.1 and 0.02.}
\label{fig:sfd}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip
\noindent{\bf Resonant asteroids: size distribution.--}
Having been able to classify resonant asteroids into the three sub-groups 
we are now ready to study some of their parameters indicative of their
origin or evolution. An important information is a size-frequency 
distribution (SFD) shown in Fig.~\ref{fig:sfd}. We assumed the value of
albedo to be $0.05$, appropriate for the prevalent C- and D-type asteroids
in the outer part of the main asteroid belt; higher or smaller
mean albedo values would displace the distributions toward smaller or
larger sizes as shown in the left panel. The largest members of each
of the sub-populations are of about the same size ($\simeq\!20\,{\rm km}$).
We fitted SFDs with a power-law $N(>\!D)\sim D^\alpha$, usually down to 
the diameter $\simeq\!9\,{\rm km}$, that does not seem to be significantly
affected by the observational incompleteness. We found the following 
values of the SFD exponent $\alpha$:
%
  (i)~$-4.4\pm 0.3$ for Zhongguos,
 (ii)~$-2.6\pm 0.5$ for Griquas, and
(iii)~$-3.7\pm 0.2$ for the unstable asteroids.%
%
\footnote{We warn the results for the long-lived populations --
Zhongguos and Griquas -- are partly ambiguous, because these populations
are not well separated. The current division at $t_{\rm J2/1} = 1\,{\rm Gy}$
was chosen a little arbitrarily.
If the whole long-lived population (i.e. with $t_{\rm J2/1}\ge 70$~My)
is considered together its SFD has an exponent $-3.5\pm 0.2$.}
%
\ In spite of relatively large standard deviations of the exponents,
the SFDs differ from each other with a high statistical significance.
The SFDs of Zhongguos and the unstable asteroids are also significantly steeper
than $-2.5$, what one would expect for a collisionally evolved isolated system
(e.g.~Dohnanyi 1969). Even if generalizations of the original Dohnanyi's 
model are taken into account, such as a size-dependence of the disruption 
$Q$-factor (e.g. O'Brien \& Greenberg 2003),
the SFD of the unstable asteroids is hard to reconcile with a system
evolved solely by collisions.
It is also interesting to compare the SFDs of the resonant asteroids,
the unstable asteroids in particular, with that of a large population
of asteroids adjacent to the 2/1 MMR. Within a comparable interval
of sizes ($9- 50$\,km, say) we found that the cumulative SFD
of the main belt population is well approximated by a power-law
with an exponent $-2.60\pm 0.05$. This is interesting because
its difference from the SFD exponent of the unstable asteroids
is approximately $-1$.%
%
\footnote[1]{In fact, when we discard the extremely unstable asteroids
with $t_{\rm J1/2}\le 1$\,My (see Fig.~\ref{fig:lftime}),
which may have anomalous origin, the size distribution of the remaining
unstable asteroids becomes slightly shallower and the difference
from the main belt population is $\simeq -0.8$.}
%
\ This supports models in which the unstable asteroids
are resupplied with a size-selective process which strength
is inversely proportional to size. The Yarkovsky effect
(e.g. Vokrouhlick\'y \etal, this volume) meets this constraint
(see also Morbidelli \& Vokrouhlick\'y 2003).
%
These models, however, cannot be extended to Zhongguos,
because of their anomalously steep SFD.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{Parameters of Zhongguo- and Griqua-like asteroids found to reside
 in the stable island A of the 2/1 MMR with Jupiter. $a_p$, $e_p$ and
 $I_p$ are pseudo-proper semimajor axis, eccentricity and inclination,
 $\sigma_a$, $\sigma_e$ and $\sigma_I$ their standard deviations during
 a $1\,{\rm My}$ numerical integration, $t_{\rm J2/1}$ median lifetime,
 $H$ absolute magnitude and $D$ diameter,
 calculated from $H$ using the value of albedo 0.05.
 The last column is the classification: G -- Griquas, Z -- Zhongguos.
 Asteroid (4177) Kohman is a border-line case (see the text for discussion).}
\label{tab:zhonga}
\begin{center}
\begin{tabular}{rlrrrrrrrrrc}
\hline
No. &
Name &
\multicolumn{1}{c}{$a_p$} &
\multicolumn{1}{c}{$e_p$} &
\multicolumn{1}{c}{$I_p$} &
\multicolumn{1}{c}{$\sigma_a$} &
\multicolumn{1}{c}{$\sigma_e$} &
\multicolumn{1}{c}{$\sigma_I$} &
\multicolumn{1}{c}{$t_{\rm J2/1}$} &
\multicolumn{1}{c}{$H$} &
\multicolumn{1}{c}{$D$} &
\multicolumn{1}{c}{Class.} \\
%
&
&
\multicolumn{1}{c}{[AU]} &
&
\multicolumn{1}{c}{[deg]} &
\multicolumn{1}{c}{[AU]} &
&
\multicolumn{1}{c}{[deg]} &
\multicolumn{1}{c}{[My]} &
\multicolumn{1}{c}{[mag]} &
\multicolumn{1}{c}{[km]} &
\\
%
\hline
\input zhongguos_a
\hline
\end{tabular}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\smallskip
\noindent{\bf Resonant asteroids: objects inside the stable island
A.--} Among the newly identified asteroids inside the 2/1 MMR with Jupiter
we found $6$~cases that lie above the separatrix of the $\nu_{16}$ 
secular resonance (see Table~\ref{tab:zhonga} and Fig.~\ref{fig:arerir}).
These are the first known objects in the previously void stable island~A.
%
In comparison to B-Zhongguos, A-Zhongguos (and also A-Griquas) have
typically smaller ($<\!90^\circ$) amplitude of $\sigma_{\rm J2/1}(t)$
(see Fig.~\ref{fig:sigma}). There is also a clear secular term
with an approximate amplitude of $20^\circ$ and period $\simeq\!9\,{\rm ky}$.
%
The sizes of objects, excluding Kohman, range from $5.5\,{\rm km}$ 
to $7.5\,{\rm km}$, if we adopt the nominal geometric albedo $0.05$.
%
The ratio of the number of A-Zhongguos (excluding A-Griquas)
and the number of B-Zhongguos is equal to $3/71 \doteq 0.04$.
However, this ratio has only a low statistical significance,
because the number of A-Zhongguos, discovered upto now, is too small
and new discoveries will change the ratio a~lot.

Models that would attempt to explain the origin and evolution
of the stable population in the 2/1 MMR should consider the differences
between island~A and island~B population as important constraints.
For instance, \cite{ferrazmello_etal1998}
discussed a possibility, that Zhongguos are relics of a primordial
population. They proved the island~A would become significantly more
(but not completely) depleted if the mutual position of Jupiter and
Saturn were such that the period of Great Inequality in Jupiter's
motion would be shorter. (This is a plausible scenario since the giant
planets likely migrated during the early evolution of the Solar system.)
On the other hand, the steep SFD of Zhongguos (Fig.~\ref{fig:sfd})
would indicate a recent collisional origin of a majority of this
population. However, the mean values of both eccentricity and inclination
for A-Zhongguos and B-Zhongguos differ so much (the ejection 
velocities would be of order 1\,km/s), that a possibility of
a collisional break-up taking part in the island B and populating with
fragments the island A is very unlikely.

We finally note that the asteroid (4177)~Kohman, previously classified 
as Griqua-like by \cite{roig_etal2002}, is a border-line case
residing very close to the separatrix of the $\nu_{16}$ resonance.
Indeed, we found the critical angle
of this resonance ($\sigma_{16}=\Omega - \Omega_{\rm S}$)
exhibits periods of librations and circulations in Kohman's case.%
%
\footnote{The close proximity of $\nu_{16}$ resonance leads to a relatively
larger value of $\sigma_I$ for this body;
similarly, the asteroid 1999~VU$_{218}$ is near the resonance,
though in this case the critical angle circulates.}
%
With its large size (17\,km in diameter) this body would far dominate
the island A population.

\section{Conclusions}

We revised the currently observed population of asteroids residing
in the 2/1 mean motion resonance with Jupiter. We identified
153~resonant asteroids, and among them six new located
in the stable island~A. The properties of the resonant
population will serve as important constraints for models
of its origin and evolution (e.g. for the model involving
the Yarkovsky effect to push main belt asteroids
onto unstable resonant orbits).
%
To make such models even more constrained, the following further
observations would be particularly useful:

\begin{itemize}
\item accurate astrometry of faint asteroids to complete the known population
 down-to smaller sizes, with the goal to reduce uncertainty in determination
 of the power-law representation of the size-frequency distribution;
\item spectral or multi-color observations to investigate similarity
 or diversity of the resonant sub-populations and of possible sources,
 as far as the surface mineralogy is concerned;
\item lightcurve observations of the unstable asteroids with the goal
 to obtain (or at least constrain) their pole positions.%
%
\footnote[2]{Note the Yarkovsky-driven origin of the unstable population would,
 for instance, predict most of these asteroids should have prograde rotation.}
\end{itemize}

However, spectral and lightcurve observations of a large number
of small main-belt asteroids will probably not be available until we have
data from space-born projects (such as GAIA).

\begin{acknowledgments}
 We thank the referee, Sylvio Ferraz-Mello, for valuable comments and discussion.
 The work of MB and DV was supported by the Grant Agency of the Czech Republic.
\end{acknowledgments}

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\end{document}


