\documentclass[11pt]{aastex}
%\documentclass[11.pt,preprint]{aastex}
%\textwidth 16.cm
%\textheight 22.cm
%\oddsidemargin -1.cm
%\evensidemargin -1.cm
%\topmargin -1.cm
\begin{document}

\include{psfig}

\begin{center}
\large
{\bf  Regular and Chaotic Dynamics in the Mean Motion Resonances: Implications for the Structure 
and Evolution of the Asteroid Belt}
\end{center}
\normalsize
\vspace{1.cm}
\begin{center}
{\bf D. Nesvorn\'y}\\
{\it Southwest Research Institute}\\[0.3cm]
{\bf S. Ferraz-Mello}\\
{\it Univesidade de S\~ao Paulo, Brasil}\\[0.3cm]
{\bf M. Holman}\\
{\it Harvard-Smithsonian Center for Astrophysics}\\[0.3cm]
{\bf and}\\[0.3cm]
{\bf A. Morbidelli}\\
{\it Observatoire de la C\^ote d' Azur, France}\\[1.5cm]
\end{center}
%
% \baselineskip=20pt
%
\begin{center}
{\bf ABSTRACT} \\
\end{center}
%
This chapter summarizes the last decade's achievements in understanding the effect 
of mean motion resonances on asteroid's orbits. The news with respect to 
the beginning of 90s are many. They range from completed theoretical description
of the secular dynamics in the mean motion resonances associated with the Kirkwood gaps
to the discovery of the three--body resonances and slow chaotic phenomena acting 
all over the asteroid belt. Consequences arising from these results required 
re-modeling the meteorite and asteroid delivery to the Earth and a careful consideration 
of dynamical aging of the asteroid families.
%
\section{Introduction}
%
It is evident from the panorama of studies of the mean motion resonances (MMRs) 
in the last decade that a major progress has been done in this subject
with wide consequences for our understanding of asteroids. An attentive observer 
of this branch of the dynamical astronomy could have perhaps noted the following news. 

It was shown that bodies in the main MMRs with Jupiter (3:1, 4:1, and 5:2) can 
reach very high orbital eccentricities (Ferraz-Mello and Klafke 1991, Klafke et al. 1992). 
This result suggested that resonant bodies can be transported not only to the Mars--crossing  
(Wisdom 1982, 1983, 1985), but also the Earth--crossing orbits, and be efficiently extracted from the 
resonances due to the larger mass of the Earth.

The progress in numerical modeling moreover revealed another surprising alternative: the 
resonant asteroids can fall into the Sun (Farinella et al. 1993, 1994), as a consequence
of their eccentricity becoming of order of unity. The current state of our understanding
of the removal of resonant bodies from the 3:1 MMR is that 65--70\% are extracted by 
Earth and Venus, and 25--30\% go directly into the Sun (Gladman et al. 1997).
From this perspective, the formerly suggested Mars--assisted mechanism (Wisdom 1982) 
of clearing the Kirkwood gaps (Kirkwood 1866) seems to play only a complementary rule. 

The innovative semi-numerical treatment of classical perturbation methods introduced by 
Henrard (1990) later allowed the global and realistic description of the secular dynamics 
in the main MMRs with Jupiter, taking into account the precession of the Jupiter's
orbit (Moons and Morbidelli 1995). This supplied convincing evidence that the orbital
evolution of simulated bodies toward the planet--crossing orbits is driven by chaotic secular 
resonances (Morbidelli and Moons 1995). A brief account of this new development is given in 
section 4.

The 2:1 MMR with Jupiter at 3.27 AU is a case of its own, because here the chaotic secular 
resonances are located at high eccentricities (Morbidelli and Moons 1993). Yet, in early 
90s, only a few asteroids were known to exist on resonant orbits, in contrast to the large
non-resonant populations on both sides of the 2:1 MMR.  
The studies devoted to the question of the long--term stability of the resonant orbits were 
marked by the evolution of the numerical methods (Wisdom and Holman 1991, Levison and Duncan 1994)
and the computer speed. 
Important progress was made when the globally chaotic nature of orbits in the 2:1 MMR was demonstrated
(Ferraz-Mello 1994b). Later works (Morbidelli 1996, Nesvorn\'y and Ferraz-Mello 1997, Ferraz-Mello 
et al. 1998a) showed that this resonance is an intermediate case between the unstable MMRs and 
stable 3:2 MMR with Jupiter, the latter hosting the Hilda group. The 2:1 MMR possesses a core with 
dynamical lifetime comparable to the age of the solar system, where currently, nearly thirty  
small asteroids are known (the Zhongguo group). We devote section 5 to this subject. 

A major breakthrough, already signaled by the studies of the slow chaos in the 2:1 MMR,
was obtained about the phenomenon called `stable chaos': as it was previously noted 
a large number of asteroids have strongly chaotic orbits but yet are stable on long 
intervals of time (Milani and Nobili 1992, Milani et al. 1997). The main reason for such 
behavior was revealed by the discovery of the so-called three--body resonances, i.e. MMRs 
where the commensurability of orbital periods occurs between an asteroid and two planets 
(mainly Jupiter and Saturn -- Murray et al. 1998, Nesvorn\'y and Morbidelli 1998). 
The subsequent modeling of these resonances showed that they represent narrow but strongly 
chaotic layers densely intersecting the asteroid belt (section 6.1).   

The slow chaotic evolutions observed in numerical simulations in the narrow MMRs 
in the outer asteroid belt were studied by Murray and Holman (1997).
Although the agreement between the amount of modeled diffusion in this work 
and the one observed in numerical simulations is only approximate, 
the modeling correctly showed that the mechanism driving the chaotic diffusion lies 
in the `multiplet structure' of the narrow MMRs (section 6.2). 

The study of the dynamics in the inner belt (2.1--2.5 AU) revealed surprising instabilities of asteroid
orbits. Numerical simulations showed that many asteroids currently located on non-planet--crossing orbits 
with large eccentricities evolve to Mars--crossing orbits within next 100 My (Migliorini et al. 1998,
Morbidelli and Nesvorn\'y 1999). It turned out that the responsible resonances in this case are the 
MMRs with Mars, previously though unimportant because of the small mass of this planet. The Mars'
MMRs were identified as a major source of the large Near Earth Asteroids (NEA - section 6.3). 
They were also conjectured to disperse the asteroid families in the inner asteroid belt 
(Nesvorn\'y et al. 2001).
 
After introducing basic notation (section 2) and showing the global structure of MMRs in the 
asteroid belt (section 3), we roughly follow the above historical overview. Section 2 is directed
to the reader who wishes to gain some basic insight in the  mathematical methods developed and utilized 
in studies of the MMRs. Perspectives are given in the last section. 
%
\section{Notation and basic terminology}
%
The mean motion resonance occurs when the asteroid has the orbital period commensurable with 
the orbital periods of planets. To fix the notation, we define 
\begin{equation}
\sigma_{{\bf k},k,{\bf l},l,{\bf m},m} = \sum_{j=1}^{N} k_j \lambda_j + k \lambda 
                                       + \sum_{j=1}^{N} l_j \varpi_j  + l \varpi 
                                       + \sum_{j=1}^{N} m_j \Omega_j  + m \Omega \ ,
\label{sigma}
\end{equation}
where $k,l,m$ and ${\bf k}=(k_1,\ldots,k_N),{\bf l}=(l_1,\ldots,l_N),{\bf m}=(m_1,\ldots,m_N)$
are integers with zero sum: $k+l+m+\sum (k_j + l_j + m_j) = 0$ (because of the invariance of 
interaction by rotation, only such combinations exist\footnote{This and other conditions
on integer coefficients required by symmetries of the interaction are known as D'Alembert rules.}%
), and $k \neq 0$ and $\|{\bf k}\| \neq 0$ (i.e., MMRs are related to the fast orbital frequencies, 
unlike the secular resonances). Here, $\lambda,\varpi,\Omega,\lambda_j,\varpi_j,\Omega_j$ are 
orbital angles in usual notation, index $j$ goes over $N$ planets.

The MMR occurs when $\dot{\sigma}_{{\bf k},k,{\bf l},l,{\bf m},m} = 0$, where 
$\dot{\sigma}_{{\bf k},k,{\bf l},l,{\bf m},m}$ is the time derivative of 
$\sigma_{{\bf k},k,{\bf l},l,{\bf m},m}$ given by (\ref{sigma}). 
In the case of asteroid motion, the secular frequencies 
$\dot{\varpi},\dot{\varpi}_j,\dot{\Omega},\dot{\Omega}_j$ are small compared to the orbital 
frequencies $\dot{\lambda},\dot{\lambda_j}$. For this reason, 
$\dot{\sigma}_{{\bf k},k,{\bf l},l,{\bf m},m}  = 0$
is approximately $\sum k_j \dot{\lambda}_j + k \dot{\lambda} = 0$, which can be solved for the
resonant semimajor axis ($a_{\rm res}$) in Keplerian approximation. This means
that resonant conditions $\dot{\sigma}_{{\bf k},k,{\bf l},l,{\bf m},m} = 0$ with unique
${\bf k},k$ but different ${\bf l},l,{\bf m},m$ hold at about the same
semimajor axis. Thus, each $({\bf k},k)$ MMR may be thought to be composed from several resonant 
terms with different ${\bf l},l,{\bf m},m$ (this structure is called the `resonant 
multiplet'). 

In practice, there are two important cases: (i) the two--body MMR, when index 
${\bf k}$ has only one non-zero integer $k_{j_1}$ where $j_1$ denotes, in the asteroid belt, 
either Jupiter or Mars (a few two--body resonances with Saturn and Earth occur but they are of 
minor importance), and (ii) the three--body MMR, when index ${\bf k}$ has two non-zero
integers, $k_{j_1}$ and $k_{j_2}$, corresponding to two planets (mainly to pairs: 
Jupiter--Saturn and Mars--Jupiter). Without lost of generality we assume that $k_{j_1}>0$.

The notation of MMRs that we adopt in the following text is that a $({\bf k},k)$ two--body MMR  
with to Jupiter ($j_1=5$) is $k_5{\rm J}$:--$k$, and with Mars
(${j_1}=4$) is $k_4{\rm M}$:--$k$, where $k_4,k_5$, and $k$ are integers defining the resonant
angle in (\ref{sigma}). In this notation, 2J:1 is the MMR with Jupiter,
where an asteroid has the orbital period of one half of the one of Jupiter. Moreover,
if it is clear from the text which planet is considered, we drop the letter indicating the planet 
(2J:1 becomes 2:1 as frequently used in the literature). Concerning the three--body resonances 
we denote $k_5{\rm J}$:$k_6{\rm S}$:$k$ the MMRs with Jupiter and Saturn, and   
$k_4{\rm M}$:$k_5{\rm J}$:$k$ the MMRs with Mars and Jupiter.

The equations of asteroid's motion in the presence of a MMR can be conveniently written in the
Hamiltonian formalism. In such formalism, the equations of motion derive from a function of 
the canonical variables -- the Hamiltonian. This formulation is useful because it allows us to use
rigorous methods for treating problems, where an integrable system is coupled with small perturbations 
(in current context: asteroid's motion about the Sun is perturbed by the effect of planets).

The classical expression for the Hamiltonian of a small body evolving under the gravitational force 
of the Sun and $N$ planets is\footnote{We adopt units where the product of the gravitational constant 
and the mass of the Sun is 1.}:
\begin{equation}
{\cal H} = {1 \over 2 a } + \sum_{j=1}^{N} \mu_j {\cal R}_j \qquad ,\ 
{\cal R}_j = {1 \over \Delta_j} - {{\bf r}.{\bf r}_j \over r_j^3}
\label{basic_hamiltonian}     
\end{equation}
where $\Delta_j = |{\bf r}-{\bf r}_j|$ and ${\bf r},{\bf r}_j$ are heliocentric
positions of the small body and planet $j$, respectively, and $a$ is the 
semimajor axis of the small body. Here, $1/2a$ is the heliocentric Keplerian part and 
$\mu_j {\cal R}_j$ is the perturbation exerted by the planet $j$ having mass $\mu_j$.

Appropriate canonical variables of the Hamiltonian (\ref{basic_hamiltonian}) are the so-called 
modified Delaunay variables:
\begin{equation}
\begin{array}{lclclcl}
 & & & & & & \nonumber \\
\displaystyle \lambda  \, , & & & \quad & \displaystyle \Lambda & = &\displaystyle  L\, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle p & = &\displaystyle  - \varpi \, , & \quad &\displaystyle P & = &\displaystyle L - G \, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle q & = &\displaystyle - \Omega \, , & \quad &\displaystyle Q & = &\displaystyle G - H \, , \nonumber \\
 & & & & & & \nonumber \\
\end{array}
\label{modified_delaunay}
\end{equation}
where $L = \sqrt{a}$, $G = L \sqrt{1 - e^2}$ and $H = G \cos i$ are the usual Delaunay variables
($e$ and $i$ are the eccentricity and inclination of the small body, respectively). 
The perturbation ${\cal R}_j$ can be written as a function of the orbital elements of
the small body and planet $j$ (see Moons 1993 for such expression with general validity). 
The Hamiltonian (\ref{basic_hamiltonian}) is then expressed in terms of variables 
(\ref{modified_delaunay}). 

To realistically account for motion of planets, the time evolution of the planetary 
orbital elements, provided by the planetary theory (Laskar 1988, Bretagnon 1990), must be 
substituted in ${\cal R}_j$. If we denote the planetary orbital elements by index $o$ (as $o$sculating) 
to distinguish them from the {\it mean} orbital elements (the latter being denoted by: 
$a_j,e_j,i_j,\lambda_j,\varpi_j,\Omega_j$), the general form of the quasi--periodic evolution 
of the planetary elements is:  
\begin{equation}
\begin{array}{lclclcl}
  & & & & & & \nonumber \\ 
\displaystyle a_j^o & = & \displaystyle a_j + \sum_v A_v \cos \Psi_v  \, , & \quad & 
\displaystyle e_j^o \exp {\iota \varpi_j^o} & = &\displaystyle e_j \exp {\iota \varpi_j} + 
\sum_v C_v \cos \Psi_v  \, ,\nonumber \\
  & & & & & & \nonumber \\ 
\displaystyle \lambda_j^o & = &\displaystyle  \lambda_j + \sum_v B_v \cos \Psi_v \, , & \quad & 
\displaystyle i_j^o \exp {\iota \Omega_j^o} & = &\displaystyle i_j \exp {\iota \Omega_j} + 
\sum_v D_v \cos \Psi_v  \nonumber \, , \\
  & & & & & & \nonumber \\                          
\label{planetary_elements} 
\end{array}
\end{equation}
where $\Psi_v = \sum_j (r_j \lambda_j + s_j \varpi_j + t_j \Omega_j)$, the multiindex $v$
denoting different values of integers $r_j,s_j,t_j$, and $\iota = \sqrt{-1}$.
 
By definition, the mean angles evolve linearly with time, with fixed frequencies provided by the 
planetary theory. We will denote by $n_j$, $g_j$, $s_j$ the orbital, perihelion and node frequencies,
respectively. In fact, because $A_v,B_v,C_v,D_v$ in (\ref{planetary_elements}) are small (of order 
of the planetary masses and smaller), the new ${\cal R}_j$ --functions of the  mean elements-- are at the first order 
in planetary masses identical with original functions. There will, however, appear new terms at second 
and higher orders in the planetary masses generated by the substitution of (\ref{planetary_elements})
in (\ref{basic_hamiltonian}). A number of approximations of (\ref{planetary_elements}) 
are used in the literature, varying from the planar model with one planet on the circular orbit 
($a_j^o=a_j,\lambda_j^o=\lambda_j=n_jt,e_j=i_j=0$, where $n_j=\sqrt{(1+\mu_j)/a_j^3}$ 
and $t$ is time) to more realistic ones. 

To summarize, it is understood at this point that although we do not explicitly show such expression,
the Hamiltonian (\ref{basic_hamiltonian}) is a function of (\ref{modified_delaunay}) and planets' 
mean elements (through the substitution in \ref{planetary_elements}). This is an autonomous 
Hamiltonian of $3(N+1)$ degrees of freedom with general form:
\begin{equation}
{\cal H} = {\cal H}_0 + \mu {\cal H}_1 + \mu^2 {\cal H}_2 + {\cal O}(\mu^3)\ ,
\label{2nd_ham}
\end{equation}
where
\begin{equation}
{\cal H}_0 = -{1 \over 2 \Lambda^2} + \sum_{j=1}^N ( n_j \Lambda_j + g_j \Lambda_{g_j} + 
s_j \Lambda_{s_j} ) \ .
\end{equation}
Here, $\mu$ denotes the largest among masses of the involved planets, and $\Lambda_j,\Lambda_{g_j},\Lambda_{s_j}$ 
are the canonical momenta conjugated with the mean angles $\lambda_j,\varpi_j,\Omega_j$.

At the first order in $\mu$, the elimination of the fast orbital angles by the Lie--series canonical 
transformation (see, for instance, Hori 1966) is equivalent to averaging of (\ref{2nd_ham}) over 
$\lambda_j$, $j=1,\ldots,N$, maintaining only the terms in which $\lambda,\lambda_{j_1},\lambda_{j_2}$ appear in 
resonant combinations $\sigma_{{\bf k},k,{\bf l},l,{\bf m},m}$ (corresponding to the $({\bf k},k)$ MMR in question). 
This procedure is straightforward if one utilizes the {\it resonant} variables in (\ref{2nd_ham}):
\begin{equation} 
\begin{array}{lclclcl}
 & & & & & & \nonumber \\
\displaystyle \sigma & = & \displaystyle {k_{j_1} \lambda_{j_1} + k_{j_2} \lambda_{j_2} + k\lambda + 
(k_{j_1} + k_{j_2} + k) p \over k_{j_1} + k_{j_2} + k } 
\, , & \quad &\displaystyle  S & = &\displaystyle  P\, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle \nu & = & \displaystyle - {k_{j_1} \lambda_{j_1} + k_{j_2} \lambda_{j_2} + k\lambda 
\over  k_{j_1} + k_{j_2} + k}  
\, , & \quad & \displaystyle N & = & \displaystyle {k_{j_1} + k_{j_2} + k \over k}\Lambda + P + Q\, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle \sigma_z & = & \displaystyle { k_{j_1} \lambda_{j_1} + k_{j_2} \lambda_{j_2} + k\lambda 
+ (k_{j_1} + k_{j_2} + k) q \over k_{j_1} + k_{j_2} + k}   
\, , & \quad &\displaystyle S_z & = &\displaystyle Q \, , \nonumber \\
 & & & & & & \nonumber \\
\displaystyle \lambda_{j_1}  
\, , & & & \quad &\displaystyle \widetilde{\Lambda}_{j_1} & = &\displaystyle \Lambda_{j_1} - 
{k_{j_1} \over k} \Lambda 
\, , \nonumber \\
 & & & & & & \nonumber \\
\displaystyle \lambda_{j_2}    
\, , & & & \quad &\displaystyle \widetilde{\Lambda}_{j_2} & = &\displaystyle \Lambda_{j_2} - 
{k_{j_2} \over k} \Lambda 
\, , \nonumber \\
 & & & & & & \nonumber \\
\displaystyle \lambda_{j}\, , & & j \neq j_1,j_2 \, , & \quad &\displaystyle \Lambda_{j}\, , & &
j \neq j_1,j_2 \, , \nonumber \\
 & & & & & & \nonumber \\
\end{array}
\label{threebody_var}
\end{equation}
where, in the case of a two--body MMR with $j_1$-th planet, $k_{j_2}=0$. The Hamiltonian
(\ref{basic_hamiltonian}) written in canonical variables (\ref{threebody_var}) is then simply 
averaged over $\lambda_j$, $j=1,\ldots,N$. The resulting Hamiltonian of the two--body MMR is: 
\begin{equation}
{\cal H}^{\rm 2BR} = - {1 \over 2 \Lambda^2} + {k_{j_1} \over k} n_{j_1} \Lambda + 
\sum_{j=1}^N (g_j \Lambda_{g_j} + s_j \Lambda_{s_j}) + \mu \bar{\cal H}_1(S,N,S_z,\sigma,\nu,\sigma_z) + 
{\cal O}(\mu^2)\ , 
\label{2b_normalf}
\end{equation}
where 
\begin{equation}
\mu \bar{\cal H}_1 = {1 \over 2\pi} \sum_{j=1}^N \mu_j \int_0^{2\pi} {\cal R}_j\, {\rm d}\lambda_j\, .   
\label{barH}
\end{equation}

In the case of a three--body MMR, $\bar{\cal H}_1$ in (\ref{barH}) contains only terms independent of 
$\sigma$, because ${\cal R}_j$ depends solely on variables of the small body and $j$-th planet.
The resonant terms of three--body MMRs appear at {\it second} and higher order terms in $\mu$. These terms 
are generated by the substitution of the planetary orbital elements (\ref{planetary_elements}) (so-called 
`indirect' contribution) and by iterating the Lie--series transformation elimination of fast 
orbital angles to higher orders in $\mu$ (so-called `direct' contribution). This procedure was described in 
Nesvorn\'y and Morbidelli (1999). The resonant Hamiltonian of the three--body MMR is
\begin{equation}
\begin{array}{lcl}
& &   \nonumber \\
\displaystyle {\cal H}^{\rm 3BR} & = & \displaystyle - {1 \over 2 \Lambda^2} +  
{k_{j_1} n_{j_1} + k_{j_2} n_{j_2} \over k}  \Lambda +  \sum_{j=1}^N (g_j \Lambda_{g_j} + s_j \Lambda_{s_j}) + \nonumber \\
 & &   \nonumber \\
& + & \displaystyle \mu \bar{\cal H}_1 (S,N,S_z,\nu,\sigma_z) + 
\mu^2 \bar{\cal H}_2(S,N,S_z,\sigma,\nu,\sigma_z) + {\cal O}(\mu^3) \ .  \nonumber \\
 & &   \nonumber \\ 
\label{3b_normalf}
\end{array}
\end{equation}
Note that (\ref{2b_normalf}) and (\ref{3b_normalf}) have $2N+3$ degrees of freedom (i.e., $N$ less than 
(\ref{2nd_ham})).

Thus, we learn an important difference between the two-- and three--body MMRs, which is that 
magnitudes of the resonant terms are proportional to the planetary mass and to the planetary mass 
squared, respectively. As the planetary masses in our solar system are $<10^{-3}$ (in solar units), this
shows that the two--body MMR is expected to have a larger effect on the asteroid orbit than
the three--body MMR. To document this fact, let us consider an Hamiltonian with only one $\sigma_{{\bf k},k,{\bf l},l,{\bf m},m}$ 
(i.e. fixing ${\bf k},k,{\bf l},l,{\bf m},m$, and ignoring terms in (\ref{2b_normalf}) and (\ref{3b_normalf})
with other than this multiindex). 
Such Hamiltonian accounts for a single isolated multiplet term of the $({\bf k},k)$ MMR. The phase portrait 
of trajectories of a single multiplet term is basically equivalent to the phase portrait of 
trajectories of a pendulum. The width in semimajor axis, represented by the maximal extent of 
the pendulum separatrices, is:
\begin{equation}
\Delta a = 8 a_{\rm res}^{3/2} \sqrt{\beta \over 3},
\label{rwidth}
\end{equation}
where $\beta$ is the coefficient in front of the cosine of the specific multiplet term in the Fourier 
expansion of the resonant parts in (\ref{2b_normalf}) and (\ref{3b_normalf}) 
(Nesvorn\'y and Morbidelli 1999). In the case of the two--body MMR, this coefficient is 
$\propto \mu_{j_1} P(e,e_{j_1},i,i_{j_1})$, where $P$ is a polynomial in $e,e_{j_1},i,i_{j_1}$ with the 
total power of each term being at least $\rho = |k_{j_1} + k|$ (called the resonant order). 
From (\ref{rwidth}), the width of the two--body resonance is then 
$\propto \sqrt{\mu_{j_1}} a_{\rm res}^{3/2} P^{1/2}$. For the three--body MMR, 
$\rho = |k_{j_1} + k_{j_2}  + k|$ and its width is $\propto \mu a_{\rm res}^{3/2} P^{1/2}$. 
Consequently, the two--body MMR is generally larger than the three--body MMR due to the mass 
factor. On the other hand, there are typically more three--body MMRs than two--body MMRs 
within a semimajor axis interval, because three integers ($k_{j_1},k_{j_2},k$) allow for a larger 
number of combinations than two integers ($k_{j_1},k$). In some sense, the larger density of three--body
MMRs in the orbital space compensates for their smaller widths, so that both two-- and three--body 
MMRs are important for the asteroid dynamics. 
%
\section{Overlapping MMRs and the global structure of the asteroid belt} 
%
Let us consider the two--body resonances only. Figure 1 shows the structure of resonant trajectories for 
the 3J:2 (a,b), 3J:1 (c), 5J:2 (d) and 5M:9 (e,f). The model used for the Jupiter's MMRs (a-d) is 
an approximation of (\ref{2b_normalf}) assuming coplanar orbits and the circular orbit of Jupiter. 
In such case the first--order resonant Hamiltonian is:
\begin{equation}
{\cal H}^{\rm PC} = - {k_5^2 \over 2 k^2} (N-S)^{-2} + n_5 (N-S)
+ \mu_{5} \bar{\cal H}_1 (S,N,\sigma). 
\label{2b_circular_planar}
\end{equation}
It does not depend on $S_z,\sigma_z$ because of the coplanarity, and does not depend on $\nu$ because
${\cal R}_{5}$ in (\ref{basic_hamiltonian}) is invariant by rotation around an axis perpendicular to 
the orbital plane. Consequently, $N=\sqrt{a}(1-\sqrt{1-e^2}+(k_{5}+k)/k)=const$ and trajectories can 
be obtained as level curves of the Hamiltonian (\ref{2b_circular_planar}). 

\begin{center}
EDITOR: PLACE FIGURE 1 HERE
\end{center}

In Fig. 1, two $N=const$ manifolds are shown for the 3J:2 (first--order MMR, $a_{\rm res} = 3.97$ AU) 
corresponding to eccentricities under (a) and above (b) the Jupiter--crossing limit. The 
Jupiter--crossing limit is approximately $e>0.31$, where the resonant orbit may intersect the orbit 
of Jupiter. The bold line in (b) is where collisions take place. There are two equilibria 
in Fig. 1a: the stable one at $\sigma=0$ (libration center) and the unstable one at $\sigma=\pi$. 
The trajectories connected to the unstable equilibrium 
are called separatrices. The trajectories enclosing the stable equilibrium are the resonant ones. 
They are characterized by oscillations of $\sigma$ about 0 (so-called `libration').
Above the Jupiter--crossing limit (Fig. 1b), the trajectories near the libration center are 
protected from collisions. This happens due to the so-called `resonant phase--protection mechanism' which 
guarantees that conjunctions with Jupiter occur when the resonant asteroid is near perihelion of its orbit, 
i.e. far from Jupiter.

The 3J:1 MMR (second order) has two libration centers at $\pi/2$ and $3\pi/2$ (Fig. 1c) and occurs 
closer to the Sun ($a_{\rm res} = 2.5$ AU), where collisions with Jupiter cannot happen on elliptic 
heliocentric orbits. The 5J:2 resonance (third order) has, in addition to the libration center at 0, 
two other centers at $2\pi/3$ and $4\pi/3$ (Fig. 1d). In general, the two--body MMRs at the 
interior of planet's orbit ($|k_{j_1}|>|k|$) have libration centers at $\sigma_{\rm c}= 2 j_1 \pi/\rho$ if 
$\rho$ is odd and at $\sigma_{\rm c}= (2j_1+1) \pi/\rho$ if $\rho$ is even. The exterior resonances have
the libration centers at $\sigma_{\rm c}= (2j_1+1) \pi/\rho$ if $k_{j_1} \ne 1$. Conversely, if $k_{j_1} = 1$,
the structure of the exterior resonance is characterized by so-called asymmetric librations where
none of the $2\rho$ libration centers is at $2 j_1 \pi/\rho$ or $(2j_1+1) \pi/\rho$ (Message 1958, Beaug\'e 1994). 
			 
An interesting case is the one of the two--body resonances with Mars due to the large modulation of their
widths with the evolution of perihelion longitudes and eccentricities. Figures 1e and 1f show the trajectories 
near the 5M:9 MMR ($a_{\rm res}=2.253$ AU) 
in the coplanar model with Mars on the fixed {\it elliptic} orbit ($e_4 = 0.065$), assuming two 
values of perihelion longitudes: $\varpi-\varpi_4=\pi$ in Fig. 1e and $\varpi-\varpi_4=0$ in Fig. 1f. In general, 
when  $\varpi-\varpi_{j_1}=\pi$, the MMRs have maximum widths. For other phases of secular angles, the sizes 
of resonant islands are smaller. 
            
The resonant width (i.e. the width of the libration island) can be conveniently estimated measuring the 
distance between separatrices for $\sigma=0$ (for interior resonances of odd order) or $\sigma=\pi/\rho$ 
(for interior resonances of even order and exterior resonances with $k_j \neq 1$). For the exterior 
resonances with $k_j = 1$, the separatrix distance is measured at $\sigma_c$, which must be 
determined beforehand as an extreme of (\ref{2b_circular_planar}). Iterating the algorithm over levels 
of $N=const$, the resonant width $\Delta a$ is obtained in dependence on $e$.  

\begin{center}
EDITOR: PLACE FIGURE 2 HERE
\end{center}

Figure 2 shows the MMRs in the asteroid belt. It is an extension of a similar result obtained by Dermott
and Murray (1983), but accounting for all MMRs with $\Delta a > 10^{-4}$ AU. We also plot in this figure
the asteroids with magnitudes up to the level of completeness (Jedicke and Metcalfe, 1998). 
The orbital distribution of these bodies is thus unaffected by observational biases. 
In the region of the 2J:1, the resonant orbital elements of all known resonant objects regardless 
of size are shown. 
     
To correctly interpret this figure, one should be aware about the now--classical result of
Chirikov (1979) known as the `resonance overlap criterion'. This criterion affirms that whenever two 
resonances with similar sizes overlap, the corresponding resonant orbits become chaotic.  
A simple version of this criterion was used by Wisdom (1980) to show the onset of chaos in the vicinity 
of a planet due to the overlap of the first--order MMRs with 
Jupiter. This criterion can be used in the current context to explain why there are so few asteroids located 
above the threshold in $e$ for the resonance overlap.  Moreover, we know from the experience obtained in the 
last few years with the secular dynamics inside the MMRs that MMRs themselves are usually 
characterized by chaotic dynamics (next section). In resonances like the 3J:1 and 5J:2 MMRs (at 2.5 and 2.82 AU),
and many others, the chaos causes large--scale instabilities: the resonant bodies evolve to high--$e$ orbits
and are removed from resonant orbits by encounters with Mars or Jupiter (Gladman et al. 1997). 
Such resonances are associated to the Kirkwood gaps in the asteroid distribution. 
Smaller MMRs albeit being strongly chaotic, do not generate strong instabilities whenever eccentricities 
are not near planet--crossing limit. Large first--order MMRs are characterized by either marginal 
instabilities (like 2J:1 at 3.27 AU) or possess cores with stability exceeding the solar system age 
(like the 3J:2 MMR at 3.97 AU and the low--$e$ region in the 4J:3 MMR at 4.29 AU, where (279) Thule is located). 

The basic impression one has from Fig. 2 is that the MMRs delimit the orbital space inhabited by the
observed asteroids in $a$ and $e$. Due to chaos and instabilities generated by MMRs, this is quite intuitive. 
Most of the job in removing the material from the belt was apparently done by several large Jupiter's MMRs 
(as 3J:1, 5J:2, 7J:3, 9J:4, 7J:4, 5J:3, etc.) and many narrow Mars' MMRs. It is not more than a curiosity that 
a few MMRs with Saturn and Earth also exist in the inner and central asteroid belts (6S:1 at 2.89 AU, 7S:1 
at 2.61 AU, 2E:7 at 2.305 AU, 2E:9 at 2.73 AU, etc.).   

We have accounted solely for two--body MMRs in Fig. 2. This happens to be a good approximation when
considering the gross structures of the asteroid belt in $a$ and $e$, but becomes less acceptable when 
one's objective is to understand the dynamics of real bodies. The three--body resonances are usually as 
narrow as two--body resonances with Mars ($<10^{-2}$ AU), but generally being of a low order, they have 
non-negligible sizes down to small eccentricities. Moreover, the three--body MMRs are numerous.
%
\section{Chaotic secular dynamics in the MMRs with Jupiter: the Kirkwood gaps}
%
The bodies inside the wide MMRs with Jupiter undergo an important secular dynamics, 
in particular for what concerns the evolution of $e$ (Wisdom 1982, Ferraz-Mello and Klafke 1991,
Morbidelli and Moons 1995, Gladman et al. 1997). We briefly review how this can be 
theoretically justified, restricting for simplicity to the case where the asteroid and planets 
have coplanar orbits, although elliptic and precessing (see Morbidelli and Moons, 1993, for a discussion 
of the inclined case). We start from the resonant Hamiltonian (\ref{2b_normalf}), that we rewrite as
${\cal H}^{\rm 2BR}={\cal H}^{\rm PC} + {\cal F}$ where ${\cal H}^{\rm PC}$ is given in (12), 
and ${\cal F}={\cal H}^{\rm 2BR} - {\cal H}^{\rm PC}$. The function ${\cal F}$ can be considered 
as a perturbation of ${\cal  H}^{\rm PC}$, because -- for D'Alembert rules --
the former is proportional to the planetary eccentricities.

Because ${\cal H}^{\rm PC}$ is integrable, we introduce the Arnold action--angle variables in the MMR region 
where $\sigma$ librates. Following Henrard (1990), the transformation has the form:
\begin{equation}
\begin{array}{lclclcl}
&&&&&& \nonumber \\
\displaystyle \psi_\sigma &=& \displaystyle \frac{2\pi}{T_\sigma}t \ , & \quad & 
\displaystyle J_\sigma &=& \displaystyle \frac{1}{2\pi} \oint S \, d\sigma\ , \nonumber \\
&&&&&& \nonumber \\
%\psi_z &=& \sigma_z - \rho_z(\psi_\sigma, J_\sigma, J_z, J_\nu)
%         & \quad , \quad
%         & J_z &=& S_z
%           \nonumber \\
\displaystyle \psi_\nu &=& \displaystyle \nu - f(\psi_\sigma, J_\sigma, 
%J_z, 
J_\nu)\ , & \quad & \displaystyle J_{\nu} &=& N\ , \nonumber \\
&&&&&& 
\label{Aaesp}
\end{array}
\end{equation}
where the integral is done over a trajectory of $S,\sigma$ for $N=const$ (Fig. 1), $T_\sigma$ is a period
of the trajectory, and $f$ is a periodic function of $\psi_\sigma$, with null average. 

We then write ${\cal H}^{\rm PC}$ and ${\cal F}$ as functions of these variables. 
By construction, ${\cal H}^{\rm PC}$ now depends only on the new action variables $J_\sigma, J_\nu$. 
By averaging ${\cal F}$ over $\psi_\sigma$, we obtain an Hamiltonian of the form:
\begin{equation}
{\cal H}^{\rm SEC}=\sum_j g_j\Lambda_{g_j} + {\cal H}^{\rm PC}(J_\sigma,J_\nu)
+\bar{\cal F}(J_\sigma,J_\nu,\psi_\nu,\mbox{\boldmath $\varpi$})\ ,
\label{H-sec-mmres}
\end{equation}
where $\mbox{\boldmath $\varpi$} = (\varpi_1,...,\varpi_N)$.
Because this Hamiltonian is independent of $\psi_\sigma$, the action $J_\sigma$ is now a constant 
of motion. Thus, (\ref{H-sec-mmres}) describes the {\it secular} dynamics inside the MMR, 
namely the evolution of the eccentricity (through $J_\nu$) due to the motion of the 
perihelia of the asteroid and the planets ($-\nu$ and $\varpi_j$ respectively).

The mean frequency of the longitude of perihelion can be computed as:
\begin{equation}
-\dot\psi_\nu={\partial {\cal H}^{\rm PC} \over \partial {J_\nu}}\ .
\end{equation}
A first--order perihelion secular resonance occurs when $\dot\psi_\nu+g_j=0$, where $g_j$ is the frequency 
of $\varpi_j$. 

It often occurs in MMRs with Jupiter that the secular resonances associated with the $g_5$ and $g_6$ 
frequencies are located close to each other. To study the effects of the interaction between these two 
resonances, we construct a two--resonance model by retaining from (\ref{H-sec-mmres}) the harmonic of 
$\bar{\cal F}$ with arguments $\sigma_5=\psi_\nu+\varpi_5$ and/or $\sigma_6=\psi_\nu+\varpi_6$. 
Thus, we consider the Hamiltonian
\begin{equation}
{\cal H}^{5,6}= g_5\Lambda_{g_5}+g_6\Lambda_{g_6}+{\cal H}^{\rm PC}(J_\sigma,J_\nu)+
\bar{\cal F}^{5,6}(J_\sigma,J_\nu,\sigma_5,\sigma_6)\ .
\label{nu5+nu6-mmres}
\end{equation}
This is a non-integrable Hamiltonian that must be studied numerically. The dynamics can be represented on 
the $\sigma_5,J_\nu$ plane, for instance through a section at $\sigma_6=0$. The sections depend 
parametrically on $J_\sigma$, roughly corresponding to the libration amplitude of $a$ ($A_a$).

\begin{center}
EDITOR: PLACE FIGURE 3 HERE
\end{center}

In the case of the 3:1 MMR with Jupiter (Fig. 3), most of the phase space is covered by
a chaotic region, which extends up to $e=1$ (top borders of the panels). 
Only the orbits with small $A_a$ and $e$ (the smooth curves at the bottom of the two top 
panels of Fig. 3) still have a regular dynamics. These orbits however periodically reach a 
Mars--crossing eccentricity ($e>0.3$); the encounters with Mars give impulse changes to 
$a$ and $e$, which --~although generally very small~-- effectively modify $A_a$. At large $A_a$, 
the chaotic region generated by the  
overlapping of the $\nu_5$ and $\nu_6$ resonances extends to all eccentricities (bottom panels 
of Fig. 3), and the asteroid can therefore rapidly and chaotically evolve to very large $e$.  
This combination of large--scale chaos of the secular dynamics and weak Martian encounters  
explains the behavior of 3:1 resonant asteroids observed in numerical integrations of the full 
equations of motion (see Morbidelli et al. chapter), and explains the formation of a gap in the 
asteroid distribution. 

The same happens in many other major MMRs with Jupiter that are associated with 
a Kirkwood gap. Moons and Morbidelli (1995) have shown that also the 4J:1, 5J:2 and 7J:3 MMRs are 
dominated by the chaotic region generated by the overlapping of the $\nu_5$ and $\nu_6$ resonances, 
so that also the asteroids in these MMRs can reach very large $e$ on a few My timescale.

Large evolutions of $e$ in many MMRs are typically caused by the secular resonance $\nu_5$ itself.
The structure of the $\nu_5$ resonance can be computed in models neglecting $g_6$ (Ferraz-Mello and Klafke 1991, 
Klafke et al. 1992, Moons and Morbidelli 1995). These models show that by the effect of $\nu_5$, the 
eccentricity suffers large variations, while $\varpi-\varpi_5$ oscillates around 0 or $\pi$. 
In such models, however, only bodies with certain initial orbits can evolve from low to 
very high eccentricities (Klafke et al. 1992). Other parts of the orbital space are characterized by 
regular motion bounded at moderate $e$. 
This regular motion almost completely vanishes when accounted for $g_6$ (as shown above, Fig. 3).  
Consequently, in a globally chaotic environment, orbits evolve according to the underlying structure 
of $\nu_5$, and despite their initial location in the orbital space, reach very large $e$. 
This is usually accomplished by a series of small transitions due to the interaction 
$\nu_5$ and $\nu_6$, and a few large events, when $e$ increases due to the effect of $\nu_5$ 
(Morbidelli and Moons 1995). 
%
\section{Transient and stable populations of the 2J:1 and 3J:2 MMRs}
%
In the 2J:1 and 3J:2 MMRs, simple models with fixed Jupiter's orbit also show a high--$e$ 
regime of motion associated with the $\nu_5$ resonance. In this case, however, the high--$e$ regime is 
separated from low $e$ by regular orbits, which are robust and persist in the models 
accounting for $g_6$ (Henrard and Lema\^{\i}tre 1987, Morbidelli and Moons 1993, Ferraz-Mello 1994b, 
Michtchenko and Ferraz-Mello 1995, Moons et al. 1998). 
Consequently, unlike in the 3J:1, 4J:1, 5J:2 and 7J:3 MMRs, the secular dynamics does not 
explain why the observed orbital distribution of asteroids should 
display a gap at the place of the 2J:1 MMR (the Hecuba gap -- see Fig. 2), where only
a few tens of small resonant asteroids reside. Conversely, the 3J:2 MMR hosts some 260 resonant 
asteroids known at the time of writing this text (the Hilda group), with about thirty bodies 
exceeding 50 km in diameter. 

This puzzling difference, known as the `2:1 versus 3:2 paradox', led some authors to investigate
the possibility of opening the Hecuba gap during the primordial stages of the solar system
formation, invoking effects by mutually displacing resonances and asteroids in $a$
(Henrard and Lema\^{\i}tre 1983). It become clear later that although plausible, such effects are 
not strictly required to explain the lack of asteroids in the 2J:1 MMR.  

An important difference between the 2J:1 and 3J:2 MMRs was noted by Ferraz-Mello (1994a,b).
He computed the maximum Lyapounov Characteristic Exponents (LCE\footnote{%
The maximum LCE measures the ratio of divergence of nearby orbits and is an indicator
of chaos (Benettin et al. 1976).}) of a number of resonant orbits in both MMRs. 
It turned out that in the 2J:1 MMR, LCE $\sim10^{-5}$--$10^{-3.5}\ {\rm yr}^{-1}$, while in the 3J:2 MMR, 
LCE $<10^{-5.5}\ {\rm yr}^{-1}$. This indicated that orbits the 2J:1 MMR are chaotic on short
timespans (Morbidelli 1996), at variance with 3J:2 MMR where most trajectories are 
only weakly chaotic. 
 
\begin{center}
EDITOR: PLACE FIGURE 4 HERE
\end{center}

The comparative study (Nesvorn\'y and Ferraz-Mello 1997) of the MMRs employing the Frequency 
Map Analysis (Laskar 1988, 1999) provided additional clues. Figure 4 shows how the magnitude of
the chaotic evolution varies in the orbital space of the 2J:1 and 3J:2 MMRs. The color coding
represents the value of $\log_{10}|\delta \dot{\varpi}|$, where $\delta \dot{\varpi}$ 
is a relative change of the perihelion frequency per 0.2 My. This quantity is a powerful indicator 
of the rate of chaotic evolution (chaotic diffusion) suffered by orbits in the integration
timespan. The results were extrapolated to larger time intervals assuming a chaotic random walk 
of orbits (see Ferraz-Mello et al. 1998a).

In regions corresponding to smallest magnitudes of the chaotic diffusion (deep blue), the orbits
relatively evolve by less than 10\% on 1 Gy. Such orbits should be stable over the age of the solar
system. Conversely, in regions where $ \log_{10}|\delta \dot{\varpi}| > -2.5$ (red, yellow), 
the perihelion frequency of a resonant body should typically change by more than 100\% in less 
than 1 Gy. Such evolutions should be enough to destabilize the orbit. Thus, the orbits with 
$ \log_{10}|\delta \dot{\varpi}| > -2.5$ are expected to be unstable. Despite the large 
uncertainties involved in the extrapolation from My to Gy timescales, Figure 4 provides 
a convincing argument suggesting at least 1--2 orders in magnitude shorter
lifetimes in the 2J:1 than in the 3J:2 MMR. The lifetimes in the most stable regions of the 
2J:1 MMR were estimated to be of order of $10^9$ years (Morbidelli 1996, Nesvorn\'y and Ferraz-Mello 1997).

Recently, these results has been put on a firmer ground by direct simulation of 50 test bodies 
in the 2J:1 MMR over 4 Gy (Roig et al. 2002). This simulation showed that the most stable region
of the 2J:1 MMR is characterized by marginal instabilities: there is about 50\% probability that a 
body started at $3.2<a<3.3$ AU, $0.2<e<0.4$, small $i$, and $\sigma=\varpi-\varpi_J=0$, 
escapes from the resonance in 4 Gy due to the diffusive chaos. Thus, the lack
of a larger asteroid population in the 2J:1 MMR is, at least partially, due to the slow removal of
the primordial population in last 4 By. The group of resonant asteroids, now accounting for some
30 small bodies known as the Zhongguo group, is localized in a very small region in the resonance 
where lifetimes generally exceed 4 Gy. 

The origin of the Zhongguo group is unknown. This group has a steep size distribution, which
would be expected from the collision injection at the breakup of the parent body of the Themis family
(Morbidelli 1996). The large ejection velocities needed for such injection, an offset in 
$e$ between the Themis family and the Zhongguo group, and possibly incompatible spectral 
type of (3789) Zhongguo however suggest this to be an unlikely origin (Roig et al. 2002).  
The Zhongguo group seems to be an order of magnitude more depleted than what would be expected from 
a dynamically eroded population similar to the Hilda group. Assuming Zhongguos to be dynamically
primordial bodies in the 2J:1 MMR, their additional depletion could have been achieved by collision 
processes or during the primordial stages of the solar system formation. 

The slow diffusive chaos in the 2J:1 MMR is where the concept of a resonance between an asteroid
and {\it two} perturbing planets appeared. As originally pointed out (Ferraz-Mello 1997, 
Michtchenko and Ferraz-Mello 1997, Ferraz-Mello et al. 1998b) the
slow chaos in the 2J:1 MMR is probably due to the commensurability between the periods of 
the Great Inequality ($\sim 880$ yr) and the one of $\sigma=2\lambda_5-\lambda-\varpi$ 
(300--500 yr). The effect of this resonance was put into evidence numerically by artificially 
changing the Great Inequality period to 440 yr. In such situation, which could have occurred when Jupiter
and Saturn were slightly closer to each other during the primordial migration phase (Fernandez and Ip 1984), 
the `beat' between the periods of the Great Inequality and of $\sigma$ was approximately 1:1, and the 
instabilities in the 2J:1 MMR were significantly accelerated (inset in Fig. 4). 
%
\section{Numerous narrow MMRs driving the chaotic diffusion}
%
The narrow MMRs (three--body, high--order two--body, and Mars' resonances) in the asteroid belt 
are usually not powerful enough to destabilize bodies continuously re-supplied into them by 
the Yarkovsky force (chapter by Bottke et al.) and collisions (Morbidelli et al. 1995, Zappal\`a et al. 2000). 
Consequently, the narrow MMRs do not open gaps in the asteroid distribution and are populated 
by many asteroids at present.    
%
\subsection{Three--body MMRs}
%
The growing evidence that many asteroids move on chaotic orbits (Milani and Nobili 1992, 
Milani and Kne\v{z}evi\'c 1994, Holman and Murray 1996, \v{S}idlichovsk\'y and Nesvorn\'y 1997) 
has challenged the view of the asteroid belt as an unchanging, fossil remnant of its 
primordial state. For example, the asteroid (490) Veritas at $a=3.174$ AU has the Lyapunov 
time\footnote{The Lyapunov time is defined as an inverse of the LCE} 
$\sim 10,000\ {\rm yr}$, showing unpredictability of the orbit over $>10^5$ yr intervals.
It was moreover noted that $e$ and $i$ of this object chaotically evolve outside the 
borders of the Veritas family (of which (490) Veritas is the largest fragment) in 50 My. Based on this fact, 
the age of the Veritas family was hypothesized to be of this order (Milani and Nobili 1992).

In current understanding, (490) Veritas is one among many objects in the asteroid belt, residing in 
the three--body MMRs. Convincing demonstration of this fact was provided by estimates
of the LCE for large samples of real and fictitious asteroids (Nesvorn\'y and Morbidelli 1998, 
Murray et al. 1998, Morbidelli and Nesvorn\'y 1999). Figure 5 shows the profile of the LCE computed 
for test bodies initially at $e=0.1$ and $i=0$. The peaks correspond to chaotic regions. Some of the main peaks 
may be identified with the two--body MMRs (3J:1 at 2.5 AU, 5J:2 at 2.82 AU, 7J:3 at 2.96 AU, etc.), 
while most of the narrow peaks are related with the three--body MMRs. Moreover, many two--body 
MMRs with Mars occur in the inner asteroid belt (2.1--2.5 AU). It has been shown that a 
large fraction of real asteroids reside in narrow resonances (Nesvorn\'y and Morbidelli 1998). 
(490) Veritas evolves in the 5J:-2S:-2, the angle $\sigma_{5,-2,-2,0,0,-1,0,0,0}$ having irregular 
oscillations about 0 correlated with oscillations of the semimajor axis about $a_{\rm res} = 3.174$ AU.
 
\begin{center}
EDITOR: PLACE FIGURE 5 HERE
\end{center}

This finding raised questions concerning implications of such a complex chaotic structure of the
asteroid belt. Asteroids were shown to escape from the main belt to the Mars-- and Jupiter--crossing
orbits due to a slow chaotic evolution of eccentricities in narrow resonances.
Realistic modeling including also Yarkovsky effect demonstrated that
the narrow resonances are an important source of the Near--Earth object population 
(Migliorini et al. 1998, Morbidelli and Nesvorn\'y 1999, Bottke et al. 2002). 
Recently, narrow MMRs were also hypothesized to disperse the asteroid families
in $e$ and $i$, possibly creating their often asymmetric shapes (Nesvorn\'y et al. 2002). 
If this process is as general as conjectured, many asteroid families could have been created as small 
groupings and later become dynamically dispersed by the chaotic diffusion in narrow MMRs and by the Yarkovsky 
effect. This mechanism may eventually solve the paradox between hydrocode simulations of breakup events 
suggesting creation of tightly grouped collision swarms, and the observed, largely dispersed asteroid 
families. 

Theoretical modeling of the three--body MMRs revealed several clues for understanding the
behavior of asteroids in the numeric simulations. The resonant Hamiltonian of several three--body 
MMRs with Jupiter and Saturn in the planar model has been computed by Nesvorn\'y and Morbidelli (1999). 
In that work, the Hamiltonian of the $k_5J:k_6S:k$ MMRs (\ref{3b_normalf}) was written as the Fourier 
expansion:
\begin{equation}
\begin{array}{rcl}
& & \nonumber \\
\displaystyle \bar{\cal H}_1 & = & \displaystyle 
{1 \over a_5} \sum_{v}
{\cal P}_{1,v} (\alpha_{\rm res})\ e_5^{s_5} e_6^{s_6} e^s 
\cos(l_5 \varpi_5 + l_6 \varpi_6 + l \varpi) \ ,\nonumber \\
& & \nonumber \\
\displaystyle \bar{\cal H}_2 & = & \displaystyle 
{1 \over a_5} \sum_{v}
{\cal P}_{2,v} (\alpha_{\rm res})\ e_5^{s_5} e_6^{s_6} e^s 
\cos( \sigma_{{\bf k},k,(l_{5},l_{6}),l,{\bf 0},0} ) \ ,\nonumber \\
& & \nonumber \\
\end{array}
\label{h12}
\end{equation}
where $\alpha_{\rm res} = a_{\rm res}/a_5$, and the multiindex $v$ denoting different values
of integers $s_5,s_6,s,l_5,l_6,l$. Denoting:
\begin{equation}
\beta = {\mu_5^2 \over a_5} \sum_{v,\ l_5=l_6=0} {\cal P}_{2,v}\ e^s  
\label{beta}
\end{equation}
these coefficients, truncated at two lowest orders in $e$, are given in Table 1. 
 
\begin{center}
EDITOR: PLACE TABLE 1 HERE
\end{center}

Ignoring all but one resonant multiplet term in (\ref{h12}), the complete Hamiltonian (\ref{3b_normalf})
can be reduced to the Hamiltonian of a simple pendulum: ${\cal H} = \alpha (S-N)^2 + \beta \cos \sigma$, 
$N$ representing a constant parameter (Nesvorn\'y and Morbidelli 1999). The resonant full width 
can then be computed from (\ref{rwidth}). Similarly, the period of small amplitude librations is:
\begin{equation}
T = {a_{\rm res} \over |k|} \sqrt{1 \over 3 \beta}\ , \qquad {\rm (in\ years)}\ .
\label{rperiod}
\end{equation}
The position of the libration center is determined by the signs of $\alpha$ and $\beta$.
As $\alpha=-3 \dot{\lambda}^2/2a_{res}^2$ is always negative, the resonant center is at 0
when $\beta>0$, and at $\pi$ when $\beta<0$. Moreover, according to a simple argument 
by Benettin and Gallavotti (1986), the Lyapunov time is of the order of the libration
period, so that $T$ provides a rough estimate of the LCE.

Table 1 shows $\Delta a$, $T$ and $\sigma_c$ for the most important three--body MMRs with $e=0.1$. 
The values computed from (\ref{rwidth}) and (\ref{rperiod}) are in good agreement 
with numerical results (Fig. 5). In particular, (490) Veritas was
numerically found to evolve in the 5J:-2S:-2 MMR with $0.004$ AU oscillations of $a$ and the libration 
period of several $10^3$ yr, both values being in agreement with the ones in Table 1.

Understanding the chaos and the long--term evolution of asteroids in the three--body MMRs 
requires to account for more than one multiplet term in (\ref{h12}). Nesvorn\'y and Morbidelli
(1999) showed that different multiplets of the three--body MMR generally overlap, thus creating 
a chaotic domain at the position of the MMR. Quantitative estimates of the LCE and diffusion timescales 
in this domain showed that significant chaotic evolutions of $e$ and $i$ of resonant bodies occur on long timespans 
(Murray and Holman 1997; Murray et al. 1998). Conversely, the semimajor axis of resonant bodies remains locked 
in the interval spanned by the resonant multiplet and although behaving stochastically on short timespans, 
never directly evolves outside the resonant borders, unless two or more different MMRs overlap (the case 
of high $e$ in Fig. 2).   
%
\subsection{Narrow two--body MMRs with Jupiter}
%
As discussed above, the details of the multiplet structure of each two--body and three--body resonance 
affect the extent of the chaotic zone associated with the resonance, the typical Lyapunov time for a
small body in that resonance, and the time scale on which such a body might chaotically diffuse in the 
space of orbital elements.  Useful analytic models of these effects can be obtained using some simplifying
assumptions (Holman and Murray 1996; Murray and Holman 1997; Murray et al. 1998). We show an example of such
modeling for a two--body resonance (see also Morbidelli 2001).

To most simply account for the multiplet structure of a two--body (${\bf k},k$) MMR, we restrict ourselves 
to the planar elliptic three--body model. The resonant Hamiltonian of such model can be written:
\begin{equation}
{\cal H} = - {1 \over 2 \Lambda^2} + {k_{j_1} \over k} n_{j_1} \Lambda + c_0(\Lambda,P) + 
\sum_{s=0}^{|k_{j_1} + k|} c_s (\Lambda,P)
\cos(k_{j_1}\lambda_{j_1} + k\lambda + s p - (k_{j_1} + k - s) \varpi_{j_1})\ ,
\label{multiplet_hamiltonian}
\end{equation}
where $c_0$ is a part of $\mu \bar{\cal H}_1$ in (\ref{2b_normalf}) independent of angles.   
In (\ref{multiplet_hamiltonian}), we retained only those terms of the Fourier expansion of (\ref{2b_normalf}) 
that are lowest order in the eccentricities of the planet and the small body. 
For simplicity, we arbitrarily set $\varpi_{j_1} = 0$, and drop it from (\ref{multiplet_hamiltonian}). 

Defining the canonical variables:
\begin{equation}
\begin{array}{lclclcl}
 & & & & & & \nonumber \\
\displaystyle \Psi & = &\displaystyle  {1 \over k} (\Lambda - \Lambda_0) \, ,
& \quad & \displaystyle \psi & = &\displaystyle  k_{j_1}\lambda_{j_1} + k\lambda\, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle I & = &\displaystyle  P \, , & \quad &\displaystyle \phi & = &\displaystyle p \, ,
\nonumber \\
 & & & & & & \nonumber \\
\displaystyle \widetilde\Lambda_{j_1} & = &\displaystyle \Lambda_{j_1} - {k_{j_1} \over k} \Lambda \, , 
& \quad &\displaystyle \lambda_{j_1}\, , &  & \nonumber \\
 & & & & & & \nonumber \\
\end{array}
\end{equation}
where $\Lambda_0=\sqrt{a_{\rm res}}$ is the unperturbed MMR location. Keeping the 
lowest order terms in $\Psi$ and $I$ in the Taylor expansion of first three terms of
Hamiltonian (\ref{multiplet_hamiltonian}), we get:
\begin{equation}
{\cal H} = {{1 \over 2} \beta \Psi^2} + 2\epsilon A I + \sum_{s=0}^{|k_{j_1}+k|} c_s(I)
\cos(\psi + s \phi),
\label{simple_mult_hamiltonian}
\end{equation}
where $\beta$ and $2 \epsilon A = \partial c_0/\partial I (\Lambda_0,0)$ are the constants associated 
with the Taylor expansion, and $c_s(I)=c_s(\Lambda_0,I)$. 

To apply the resonance overlap criterion to the Hamiltonian (\ref{simple_mult_hamiltonian}) we must 
compute the separation and widths of the individual multiplet terms in the resonance. The
separation in $\Psi$ between two such terms is given by $\beta \delta\Psi = 2\epsilon A $, while the 
width of each multiplet is $\Delta \Psi = 2\sqrt{c_s(I,P_{j_1})/\beta}$. The stochasticity
parameter (Chirikov 1979, Lichtenberg and Lieberman 1983) is then
\begin{equation}
K_{\rm eff} = \left(\pi{\Delta\Psi\over\delta\Psi}\right)^2 =
\left(c_{s_0}\over\beta\right)\left({\pi\beta}\over{\epsilon A}\right)^2,
\label{keff}
\end{equation}
where $c_{s_0}$ is the coefficient of a representative multiplet term (Murray and Holman 1997). 
The orbits in the MMR will be chaotic if $K_{\rm eff} > K_{\rm crit}$, where $K_{\rm crit}\sim 1$.  
The extent of the chaotic zone in $\Psi$ (and
therefore in semimajor axis) is roughly given by the $\Psi$ range of those multiplet terms
that overlap. The Lyapunov time in the chaotic zone given by
\begin{equation}
T_L \approx {\pi\over{\epsilon A}}/\ln\left[1 + {K_{\rm eff}\over 4} +
\sqrt{{K_{\rm eff}\over2} + \left({K_{\rm eff}\over 4}\right)^2}\right] \sim {\pi\over{\epsilon A}},
\label{lt}
\end{equation}
which indicates that the Lyapunov time is comparable to the precession period.

For $K_{\rm eff} > K_{\rm crit}$, the diffusion in $\Psi$ and $I$ can be described by the Fokker--Planck 
equation (Lichtenberg and Lieberman 1983, Murray and Holman 1997). The associated diffusion coefficients
are
\begin{equation}
D_\Psi = {K_{\rm eff}^2\over2}\left({\epsilon A}\over{\pi\beta}\right)^2 =
{1\over2}\left({c_{s_0}\over\beta}\right)^2\left({\pi\beta}\over{\epsilon
A}\right)^2 = {1\over2}c_{s_0}^2{\pi\over{\epsilon A}},
\end{equation}
and
\begin{equation}
D_I = s_0^2 D_\Psi = {s_0^2\over2}c_{s_0}^2{\pi\over{\epsilon A}}.
\end{equation}
In the model of Murray and Holman (1997) the resonant bodies execute random walks in $\Psi$ 
(or semimajor axis) and $I$ (or eccentricity). These random walk are subject to boundary conditions.
The permissible values of $\Psi$ lie within the range of values for which the resonant multiplet 
terms overlap (the chaotic zone). The values of $I$ range from $I=0$ (circular orbit) to $I=I_{\rm max}$, 
where $I_{\rm max}$ corresponds to planet crossing. The bounding values of $\Psi$ and $I=0$ can be modeled 
as reflecting barriers, and the value of $I=I_{\rm max}$ as an absorbing barrier. For a resonant asteroid
at $I_0$, the solution of the Fokker--Planck equation yields typical values of the {\it removal time}, $T_{\rm R}$, 
of
\begin{equation}
T_{\rm R} \sim {{I_{\rm max}I_0}\over\sqrt{D_I(I_{\rm max})D_I(I_0)}}\ .
\label{removal_time}
\end{equation}

Figure~6 shows removal times for outer belt asteroids in a number of two--body MMRs. The solid symbols are the 
results of numerical integrations; the open symbols are analytic predictions from (\ref{removal_time}). 
The values of $I_0$ were determined from short numerical integrations of resonant asteroids, hence the scatter 
in the predicted removal times. In most cases the agreement between the numerical experiments and 
the analytic predictions agrees to an order of magnitude. The results of these calculations indicate that 
for high--order MMRs in the outer belt, removal times are in some MMRs substantially shorter than 
the age of the solar system (the case of the 7J:4, 5J:3, and 8J:5 MMRs) and in other resonances substantially 
longer than the age of the solar system (the 13J:7 and 11J:6 MMRs).

\begin{center}
EDITOR: PLACE FIGURE 6 HERE
\end{center}

Figure~6 shows that resonances like the 7J:4, 5J:3, and 8J:5 MMR are expected to correspond to gaps in 
the asteroid distribution. Indeed, no large outer belt asteroids are observed in them (Fig. 2).
The outer asteroid belt however seems more depleted than what would be expected from the effect of chaotic 
resonances. This led some authors to suggest that depletion occurred during the primordial radial migration    
of the planets (Liou and Malhotra 1997). The same scenario can also nicely explain the presence of many asteroids in
the 3J:2 MMR, which could have been captured by the resonance accompanying the inward migration of the
Jupiter. Alternatively, the outer asteroid belt could have been depleted by the effect of large
Jupiter--scattered planetesimals (Petit et al. 1999).  
%
\subsection{MMRs with Mars}
%
It may seem surprising at a first glance that the MMRs with Mars are important in the asteroid
belt because Mars is a factor of $\sim 3,000$ less massive than Jupiter. Thus, according to the
discussion in sect. 2, the two--body MMRs with Mars should be in general a factor $\sqrt{3,000}\sim 55$ smaller
than the two--body MMRs with Jupiter. In the inner asteroid belt, however, this factor may be compensated by
the small distance between an asteroid and Mars. Indeed, the width of any MMRs near planet--crossing
limit becomes large. According to Fig. 2 (see also Fig. 5), the MMRs with Mars provide a dominant
source of chaos in the inner asteroid belt, overweighting in size and number the MMRs with Jupiter.

\begin{center}
EDITOR: PLACE FIGURE 7 HERE
\end{center}

The dynamical interaction of asteroids with the MMRs with Mars is complex. As an example, Fig. 6 shows 
the evolution of $a$, $\sigma=9\lambda_4 - 22\lambda+13\varpi$, and $e^{13/2}$ of (41) Daphne. 
This asteroid is temporally captured in the 3M:11 MMR when $e$ becomes large during the 
secular oscillations. The irregular behavior of $a$ and $\sigma$ witnesses the chaoticity of the orbit,
which has the Lyapunov time $<10^4$ yr (\v{S}idlichovsk\'y 1999). 

Very little analytical work has been done on the subject of the Mars' MMRs. Figures 1e-f are apparently
highly idealized approximations of the real dynamics (Fig. 7). To understand the real dynamics, it 
is important to account for the rich multiplet structure of the Mars' MMRs and significant 
time--modulation of each multiplet's width with largely varying $e_4$ (interesting MMRs with Mars
are usually of very high orders). 

The chaotic diffusion in the MMRs with Mars provide transfer routes from the asteroid belt
to Mars--crossing orbits. This source was hypothesized to be one of the major sources of large 
NEAs (Migliorini et al. 1998, Morbidelli and Nesvorn\'y 1999, Bottke et al. 2002). The detailed
mechanism of this transfer involves complex interaction of the Yarkovsky effect and MMRs.
In particular, the Yarkovsky effect is thought to be responsible for refilling MMRs 
by new objects; otherwise the space occupied by the active MMRs would be cleared of the
asteroids, and gaps in the asteroid distribution at resonant semimajor axes would be created.
Such gaps are not currently observed. 
%
\section{Perspectives}
%
In the last decade, we witnessed a fascinating progress from studies of separate MMRs in simplest
physical models to the global understanding of structure and effects of MMRs in the whole asteroid
belt. Nevertheless, we feel that many open problems await solutions. The least developed is 
the modeling of narrow resonances, especially the three--body MMRs and the MMRs with Mars. 
The intricate interaction of MMRs and the Yarkovsky effect is an interesting subject for future studies.
Basically, one would like to know how the probability of capture of a drifting body into the narrow MMRs 
depends on the magnitude and direction of the semimajor axis drift, and whether the phase of secular 
angles plays any role in the capture process. Ideally, when capture occurs, one would also like to 
statistically estimate the change of $e$ and $i$ of an asteroid during the time spent on a resonant 
orbit. Although some hints about these processes may be derived from recent numerical simulations,
the general trends in the parameter space are unknown. Yet, the interaction of narrow MMRs with 
the Yarkovsky effect is important for such issues as the origin of NEAs and dynamical dispersion
of the asteroid families. 

Concerning the asteroid population of large MMRs in the asteroid belt, few puzzles remain, 
possibly related to the processes involved in the early stages of the solar system formation. 

We note that a large number of asteroids exist in the 3J:2 MMR, while both the 2J:1 and 4J:3 MMRs 
are substantially depleted. Although some depletion of latter resonances can be explained by the chaotic processes 
acting in last 4 By, additional depopulation likely occurred during the 
formation of the asteroid belt. Is the resonant structure of the outer asteroid belt a natural outcome 
of the accretion itself (Petit et al. 1999) or possibly a sign of the primordial planetary migration? 
We wonder if an unique answer to this question can be given. Apparently, a detailed modeling 
of the primordial phases is required. Such quantitative modeling should also show whether the 
current depletion of the outer belt between the 2J:1 and 3J:2 MMRs (3.4--3.9 AU) requires the
primordial migration (Liou and Malhotra 1997). Moreover: what effect the planetary migration would 
have in the primordial main asteroid belt? First attempts to quantify this effect by direct simulations 
(Gomes 1997, Levison et al. 2001) show a large efficiency of the secular resonance $\nu_6$ in removing the 
asteroid population. Our hope is that the increasingly detail information about the current 
asteroid belt provided by the observations, and future progress in theoretical modeling, will eventually 
disclose some marks which the primordial processes had imprinted in the orbital structure of the belt,
and which have been preserved until now. Understanding the effects of mean motion resonances will
certainly be part of this story.  
%
\begin{thebibliography}{widest-label}
\bibitem[label]{cite_key}
%
\bibitem[Beauge(1994)]{1994CeMDA..60..225B} 
Beaug\'e C.\ (1994) Asymmetric librations in exterior resonances.  
{\it Celest. Mech. and Dynam. Astron., 60}, 225-248.\  
%
\bibitem[Benettin et al.(1976)]{1976PhRvA..14.2338B} 
Benettin G., Galgani L., Strelcyn J.\ (1976) 
Kolmogorov entropy and numerical experiments.\  
{\it \pra, 14,} 2338-2345.\  
%  
\bibitem[Bretagnon and Simon(1990)]{1990A&A...239..387B} 
Bretagnon P., Simon J.-L.\ (1990) 
General theory of the Jupiter-Saturn pair using an iterative method.\  
{\it \aap, 239,} 387-398.\ 
%
\bibitem{42}
Bottke, W.~F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H., Michel, P., Metcalfe, T.~S. (2002)
Debiased orbital and size distributions of the Near--Earth Objects.
submitted to {\it Icarus}.
%
\bibitem[37]{chir79} 
Chirikov, B. V. (1979) 
A universal instability of many--dimensional oscillator systems, 
{\it Phys. Rep., 52}, 263--379.
%
\bibitem[Dermott and Murray(1983)]{1983Natur.301..201D} 
Dermott S.~F., Murray C.~D.\ (1983) 
Nature of the Kirkwood gaps in the asteroid belt.\  
{\it \nat, 301,} 201-205.\  
% 
\bibitem[1]{1}
Farinella, P., Froeschl\'e, C., and Gonczi, R. (1993) 
Meteorites from the asteroid 6 Hebe.
{\it Celest. Mech. Dyn. Astron., 56,} 287-305.
%
\bibitem[Farinella et al.(1994)]{1994Natur.371..315F} Farinella P., 
Froeschle C., Froeschle C., Gonczi R., Hahn G., Morbidelli A., Valsecchi 
G.~B.\ (1994) Asteroids falling onto the Sun.\  {\it Nature, 371,} 315-317. 
%
\bibitem[Fernandez and Ip(1984)]{1984Icar...58..109F} 
Fernandez J.~A., Ip W.-H.\ (1984) 
Some dynamical aspects of the accretion of Uranus and Neptune - The exchange of orbital 
angular momentum with planetesimals.\  
{\it Icarus, 58,} 109-120.\  
% 
\bibitem[2]{2}
Ferraz-Mello, S. (1994a) 
Kirkwood gaps and resonant groups. 
{\it IAU Symposium, 160,} 175-188.
%
\bibitem{3}
Ferraz-Mello, S. (1994b) 
The dynamics of the asteroidal 2:1 resonance. 
{\it Astron. J., 108,} 2330-2337.
%
\bibitem{4}
Ferraz-Mello, S. (1997)
A symplectic mapping approach to the study of the stochasticity in asteroidal resonances. 
{\it Celest. Mech. Dyn. Astron., 65,} 421-437.
%
\bibitem{5}
Ferraz-Mello, S., and Klafke, J.~C. (1991) 
A model for the study of very-high-eccentricity asteroidal motion. The 3:1 resonance. 
In {\it Predictability, Stability and Chaos in N-Body Dynamical Systems}  
(A.E.Roy, ed.), Plenum Press, New York, ({\it NATO Adv. Stud. Inst. Ser B Phys.} {\it 272}) 177-184.
%
\bibitem{6}
Ferraz-Mello, S., Nesvorn\'y, D., and Michtchenko, T.~A. (1998a)
Chaos, diffusion, escape and permanence of resonant asteroids in gaps and groups.
In {\it Solar System Formation and Evolution} (D.Lazzaro et al. eds.), ASP,
San Francisco, {\it (ASP Conf. Ser.} {\it 149}), 65-82.
%
\bibitem{7}
Ferraz-Mello, S., Michtchenko, T.A., and Roig, F. (1998b), 
The determinat role of Jupiter's Great Inequality in the depletion of the Hecuba gap. 
{\it Astron. J., 116,} 1491-1500.
%
\bibitem{8}
Gladman, B.J., Migliorini, F., Morbidelli, A., Zappal\`a, V., Michel, 
P., Cellino, A., Froeschl\'e, C., Levison, H.F., Mailey, M., and Duncan, M. (1997)
Dynamical Lifetimes of objects injected into asteroid belt resonances.
{\it Science, 277,} 197-201.
%
\bibitem[Gomes(1997)]{1997AJ....114..396G} 
Gomes R.~S.\ (1997) Dynamical Effects of Planetary Migration on the Primordial Asteroid Belt.\  
{\it \aj, 114,} 396-401.\
%  
\bibitem[Henrard(1990)]{1990CeMDA..49...43H} 
Henrard J.\ (1990) 
A semi-numerical perturbation method for separable Hamiltonian systems.\  
{\it Celestial Mechanics and Dynamical Astronomy, 49,} 43-67.\  
%
\bibitem[Henrard and Lemaitre(1983)]{1983Icar...55..482H} 
Henrard J., Lemaitre A.\ (1983) 
A mechanism of formation for the Kirkwood gaps.\  
{\it Icarus, 55,} 482-494.\  
% 
\bibitem{9}
Henrard, J., and Lema\^{\i}tre, A. (1987) 
A perturbative treatment of the 2/1 Jovian resonance. 
{\it Icarus, 69,} 266-279.
%
\bibitem[Holman and Murray(1996)]{1996AJ....112.1278H} 
Holman M.~J., Murray N.~W.\ (1996) 
Chaos in high-order mean resonances in the outer asteroid belt.\  
{\it \aj, 112,} 1278.
% 
\bibitem[Hori(1966)]{1966PASJ...18..287H} 
Hori G.\ (1966) 
Theory of general perturbation with unspecified canonical variable.\  
{\it \pasj, 18,} 287. 
% 
\bibitem{10}
Jedicke, R., and Metcalfe T.~S. (1998)
The orbital and absolute magnitude distributions of main belt asteroids.
{\it Icarus, 113,} 245-260.
%
\bibitem{31}
Kirkwood, D. (1866)
In {\it Proceedings of the American Association for the Advancment of Science for 1866}.
%
\bibitem{11}
Klafke, J.C., Ferraz-Mello, S., and Michtchenko, T. (1992)
Very-high-eccentricity librations at some higher-order resonances. 
{\it IAU Symposium, 152,} 153-158.
%
\bibitem[Levison and Duncan(1994)]{1994Icar..108...18L} 
Levison H.~F., Duncan M.~J.\ (1994) 
The long-term dynamical behavior of short-period comets.\  
{\it Icarus, 108,} 18-36.\
%  
\bibitem[Levison et al.(2001)]{2001Icar..151..286L} 
Levison H.~F., Dones L., Chapman C.~R., Stern S.~A., Duncan M.~J., Zahnle K.\ (2001) 
Could the Lunar ``Late Heavy Bombardment'' have been triggered by the formation of 
Uranus and Neptune?.\  
{\it Icarus, 151,} 286-306.\  
% 
\bibitem[Lichtenberg and Lieberman(1983)]{1983rsm..book.....L} 
Lichtenberg A.~J., Lieberman M.~A.\ (1983) 
Regular and stochastic motion.\  
{\it Applied Mathematical Sciences,} New York, Springer.  
%% 
\bibitem[Laskar(1988)]{1988A&A...198..341L} 
Laskar, J.\ (1988) Secular evolution of the solar system over 10 million years.\  
{\it \aap, 198,} 341-362.
%
\bibitem{40}
Laskar, J. (1999)
Introduction to frequency map analysis.
In {\it Hamiltonian Systems with Three or More Degrees of Freedom} (C. Sim\'o, ed.),
pp. 134-150. Kluwer, Dordrecht.
%	
\bibitem[Liou and Malhotra(1997)]{1997Sci...275..375L} 
Liou J.~C., Malhotra R.\ (1997) 
Depletion of the outer asteroid belt.\  
{\it Science, 275,} 375-377.\  
% 
\bibitem[Message(1958)]{1958AJ.....63..443M} 
Message P.~J.\ (1958) 
Proceedings of the Celestial Mechanics Conference: The search for 
asymmetric periodic orbits in the restricted problem of three bodies.\  
{\it \aj, 63,} 443. 
%  
\bibitem{13}
Michtchenko, T.A., and Ferraz-Mello, S. (1995)
Comparative study of the asteroidal motion in the 3:2 and 2:1 resonances with Jupiter -- I.
{\it Astron. Astrophys., 303}, 945-963.
%
\bibitem{14}
Michtchenko, T.A., and Ferraz-Mello, S. (1997)
Escape of Asteroids from the Hecuba gap.
{\it Planet. Sp. Sci., 45}, 1587-1593.
%
\bibitem{15}
Migliorini, F., Michel, P., Morbidelli, A., Nesvorn\'y, D., and V. Zappal\`a (1998)
Origin of multikilometer Earth and Mars-crossing asteroids: A quantitative simulation.
{\it Science, 281}, 2022-2024.
%
\bibitem[Milani and Knezevic(1994)]{1994Icar..107..219M} 
Milani A., Kne\v{z}evi\`c, Z.\ (1994) 
Asteroid proper elements and the dynamical structure of the asteroid main belt.\  
{\it Icarus, 107,} 219-254.\  
% 
\bibitem[Milani et al.(1997)]{1997Icar..125...13M} 
Milani A., Nobili A.~M., 
Kne\v{z}evi\`c Z.\ (1997) Stable Chaos in the Asteroid Belt.\  {\it Icarus, 125}, 
13-31.\  
%
\bibitem[Milani and Nobili(1992)]{1992Natur.357..569M} 
Milani A., Nobili 
A.~M.\ (1992) An example of stable chaos in the Solar System.
{\it Nature, 357,} 569-571.  
%
\bibitem{35}
Moons M., (1993) 
On the resonant Hamiltonian in the restricted three-body problem. 
{\it Internal Report, 19}, Facult\'es Universitaires de Namur, Belgique.
%
\bibitem{16}
Moons, M., and Morbidelli, A. (1995) 
Secular resonances in mean-motion commensurabilities. The 4/1, 3/1, 5/2 and 7/3 cases.
{\it Icarus, 114}, 33-50.
%
\bibitem{17}
Moons, M., Morbidelli, A., and Migliorini, F. 1998, 
Dynamical structure of the 2/1 commensurability with Jupiter and the origin of the resonant asteroids.
{\it Icarus, 135}, 458-468.
%
\bibitem[Morbidelli(1996)]{1996AJ....111.2453M} 
Morbidelli A.\ (1996) The Kirkwood Gap at the 2/1 Commensurability With Jupiter: New Numerical 
Results.\  
{\it \aj, 111}, 2453. 
%
\bibitem{43}
Morbidelli, A. (2001)
Modern Celestial Mechanics. Aspects of the Solar System Dynamics.
in press.
%
\bibitem{18}
Morbidelli, A., and Moons, M. (1993)
Secular resonances in mean-motion commensurabilities. The 2/1 and 3/2 cases.
{\it Icarus, 102}, 316-332.
%
\bibitem{19}
Morbidelli, A., and Moons, M. 1995, 
Numerical evidences of the chaotic nature of the 3/1 mean-motion commensurability.
{\it Icarus, 115}, 60-65.
%
\bibitem{20}
Morbidelli, A., Zappal\`a, V., Moons, M., Cellino, A., and Gonczi, R. (1995)
Asteroid families close to mean-motion resonances: Dynamical effects and physical implications.
{\it Icarus, 118}, 132-154.
%
\bibitem{21}
Morbidelli, A., and Nesvorn\'y, D. (1999)
Numerous weak resonances drive asteroids toward terrestrial planets orbits.
{\it Icarus, 139}, 295-308.
%
\bibitem{22}
Murray, N., and Holman, M. (1997)
Diffusive chaos in the outer belt.
{\it Astron. J., 114}, 1246-1259.
%
\bibitem{23}
Murray, N., Holman, M., and Potter, M. (1998)
On the origin of chaos in the Solar System.
{\it Astron. J., 116}, 2583-2589.
%
\bibitem{24}
Nesvorn\'y, D., and Ferraz-Mello, S. (1997)
On the asteroidal population of the first--order resonances.
{\it Icarus, 130}, 247-258.
%
\bibitem{25}
Nesvorn\'y, D., and Morbidelli, A. (1998)
Three--body mean--motion resonances and the chaotic structure of the asteroid belt.
{\it Astron. J., 116}, 3029-3037.
%
\bibitem{26}
Nesvorn\'y, D., and Morbidelli, A. (1999)
An analytic model of three-body mean motion resonances.
{\it Celest. Mech. Dynam. Astron., 71}, 243-263.
%
\bibitem{34}
Nesvorn\'y, D., Morbidelli, A., Vokrouhlick\'y, D., Bottke, W, and Bro\v{z}, M. (2001)
The Flora family: a case of the dynamically dispersed collisional swarm?
submitted to {\it Icarus}.	
%
\bibitem[Petit et al.(1999)]{1999Icar..141..367P} 
Petit J., Morbidelli A., Valsecchi G.~B.\ (1999) 
Large scattered planetesimals and the excitation of the small body belts.\  
{\it Icarus, 141,} 367-387.\  
% 
\bibitem{27}
Roig, F., Nesvorn\'y, D., and Ferraz-Mello, S. (2002)
preprint.
%
\bibitem[{\v S}idlichovsk{\' y}(1999)]{1999CeMDA..73...77S} 
\v{S}idlichovsk\'y M. (1999) 
On Stable Chaos in the Asteroid Belt.\  
{\it Celestial Mechanics and Dynamical Astronomy, 73,} 77-86.\  
% 
\bibitem[Sidlichovsky and Nesvorny(1997)]{1997CeMDA..65..137S} 
\v{S}idlichovsk\`y M., Nesvorn\'y D.\ (1997) 
Frequency modified Fourier transform and its applications to asteroids.\  
{\it Celestial Mechanics and Dynamical Astronomy, 65,} 137-148.\
%  
\bibitem{28}
Wisdom, J. (1982) 
The origin of Kirkwood gaps: A mapping for asteroidal motion near the 3/1 commensurability.	
{\it Astron. J., 85}, 1122-1133.
%   
\bibitem{29}
Wisdom, J. (1983) 
Chaotic behaviour and the origin of the 3/1 Kirkwood gap. 
{\it Icarus, 56}, 51-74.
%
\bibitem[Wisdom(1980)]{1980AJ.....85.1122W} Wisdom J.\ (1980) 
The resonance overlap criterion and the onset of stochastic behavior in the restricted 
three-body problem.\  
{\it \aj, 85,} 1122-1133.\  
%
\bibitem[Wisdom(1985)]{1985Icar...63..272W} 
Wisdom J.\ (1985) 
A perturbative treatment of motion near the 3/1 commensurability.\  
{\it Icarus, 63,} 272-289.\  
% 
\bibitem{30}
Wisdom, J., and Holman, M. (1991)
Symplectic maps for the N-body problem.
{\it Astron. J., 102}, 1528-1538.
%
\bibitem[Zappal{\` a} et al.(2000)]{2000Icar..145....4Z} 
Zappal{\` a} V., Bendjoya P., Cellino A., Di Martino M., Doressoundiram A., Manara A., 
Migliorini F.\ (2000) 
Fugitives from the Eos Family: First Spectroscopic Confirmation.\  
{\it Icarus, 145,} 4-11.\  
%  
\end{thebibliography}
%
\newpage
\begin{deluxetable}{rrrrrr} 
\tablewidth{420.pt}
\tablenum{1}
\tablecaption{Analytic results on the three--body MMRs. The respective columns are: 
resonance designation, resonant semimajor axis ($a_{\rm res}$), coefficient $\beta$ (eq. \ref{beta}),
resonance width ($\Delta a$ -- eq. \ref{rwidth}) and libration period ($T$ -- eq. \ref{rperiod}) for $e=0.1$, 
and the libration center ($\sigma_{\rm c}$). (from Nesvorn\'y and Morbidelli 1999)} 
\tablehead{ 
\colhead{MMR} & 
\colhead{$a_{\rm res}$ (AU)} & 
\colhead{$\beta\ (\times10^{-8})$} &
\colhead{$\Delta a$ ($10^{-3}$ AU)} & 
\colhead{$T$ ($10^3$ yr)} & 
\colhead{$\sigma_{\rm c}$}}
\startdata
4J:-1S:-1 & 2.2155 & $6.05e^2-2.22e^4$ & 0.37 & 52 & 0 \\
3J: 1S:-1 & 2.2994 & $3.78e^3-1.94e^5$ & 0.1 & 217 & 0 \\
4J:-2S:-1 & 2.3978 & $19.7e-3.95e^3$ & 2.4 & 9.9 & 0 \\
7J:-2S:-2 & 2.4479 & $46.0e^3-53.0e^5$ & 0.37 & 33 & 0 \\
7J:-3S:-2 & 2.5600 & $-4.35e^2+5.28e^4$ & 0.39 & 36 & $\pi$ \\
2J: 2S:-1 & 2.6155 & $6.03e^3+3.63e^5$ & 0.15 & 200 & 0 \\
6J:-1S:-2 & 2.6192 & $-17.6e^3+21.3e^5$ & 0.25 & 57 & $\pi$ \\
4J:-3S:-1 & 2.6229 & $-0.206-0.763e^2$ & 0.9 & 30 & $\pi$ \\
7J:-4S:-2 & 2.6858 & $0.255e-0.356e^3$ & 0.3 & 49 & 0 \\
3J:-1S:-1 & 2.7527 & $-7.99e+0.360e^3$ & 1.9 & 17.8 & $\pi$ \\
4J:-4S:-1 & 2.9092 & $0.0203e+0.0691e^3$ & 0.1 & 370 & 0 \\
5J:-1S:-2 & 2.9864 & $20.9e^2-21.3e^4$ & 1.1 & 19 & 0 \\
8J:-3S:-3 & 3.0166 & $-10.0e^2+24.0e^4$ & 0.8 & 19 & $\pi$ \\
3J:-2S:-1 & 3.0794 & $-3.06-15.5e^2$ & 4.5 & 9.9 & $\pi$ \\
6J: 1S:-3 & 3.1389 & $-22.4e^4+143e^6$ & 0.12 & 132 & $\pi$ \\
8J:-4S:-3 & 3.1421 & $0.357e-0.99e^3$ & 0.5 & 33 & 0 \\
3J: 3S:-2 & 3.1705 & $-25.8e^4+5.15e^6$ & 0.13 & 180 & $\pi$ \\
5J:-2S:-2 & 3.1751 & $45.6e-32.3e^3$ & 5.6 & 4.3 & 0 \\
7J:-2S:-3 & 3.2100 & $-115e^2+273e^4$ & 2.8 & 5.8 & $\pi$ \\
\enddata
\end{deluxetable}
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig1.ast3.ps,height=15.cm
%bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}}
\caption{Trajectories at the 3J:2 MMR -- below (a) and above (b) the Jupiter--crossing limit, 
3J:1 MMR (c), 5J:2 MMR (d), and 5M:9 MMR with $\varpi-\varpi_4=\pi$ (e) and $\varpi-\varpi_4=0$ (f).
On $x$-axis, $\sigma$ is defined by (\ref{threebody_var}).}
\label{fig.traj}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig2.ast3.ps,height=15.cm
%bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}}
\caption{Global structure of the MMRs in the asteroid belt. There are four different colors denoting the regions of 
resonant motion with Earth (light blue), Mars (red), Jupiter (yellow),
and Saturn (green). Each resonance corresponds to one V--shaped region except the large first--order 
MMRs with Jupiter which have particular shapes. Some of the resonances are labeled. For some Jupiter 
MMRs the projection of separatrices on $a,e$ plane is shown by black lines; for 2J:1, 3J:2, and 4J:3, these 
lines are bold. We also show the proper (dots - Milani and Kne\v{z}evi\'c (1994)) or orbital elements 
(crosses - Bowell et al. 1994) of asteroids with magnitudes up to the 
completeness level. In case of the group of small asteroids in the 2J:1 MMR (arrow indicates (3789) 
Zhongguo), the resonant elements are plotted (asterisks - Roig et al. 2002). Other resonant asteroids are 
the Hilda group in the 3J:2 MMR and (279) Thule in the 4J:3 MMR. The orbits above the dashed lines are 
planet--crossing.}
\label{fig.global}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig3.ast3.ps,height=15.cm
%bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}}
\caption{Sections of the dynamics of (\ref{nu5+nu6-mmres}) computed at $\sigma_6=0$ for the 3J:1 MMR. 
The four panels correspond to increasing values of $J_\sigma$, the latter being related with 
the amplitude of oscillation of $a$. The label $q$ stands for $\varpi-\varpi_5$, while 
$N=J_\nu=\sqrt{a}(3-\sqrt{1-e^2})$. At the center of the 3J:1, the values 
$N=1.4, 1.5, 1.6, 1.7, 1.8, 1.9$ correspond to $e=0.2, 0.55, 0.72, 0.84, 0.92, 0.97$, respectively. 
In each panel, the lower limit on the $N$~axis is the value that identifies the separatrix of the 
MMR. Consequently, all the curves that seem to exit from the bottom border of the panels 
correspond to trajectories that hit the separatrix of the MMR during their 
secular evolution, and are therefore expected to be chaotic. (from Moons and Morbidelli 1995)}
\label{fig.nu5+nu6}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=namur4.eps,height=15.cm
%,bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}}
\caption{The charts of chaotic diffusion in the 2J:1 (left) and 3J:2 (right) MMRs. The plotted parameter is 
$\log_{10} |\delta \dot{\varpi}|$. The color coding is such that the starting positions 
of trajectories characterized by slow chaotic diffusion are shown in blue, while fast--diffusing 
trajectories are shown in red/yellow. The inset on top-left shows how the instabilities in the 2J:1 MMR
are enhanced when the period of $2\lambda_5-5\lambda_6$ is set to be 440 yr, a value possibly 
taken during the primordial phase of the planetary migration. 
(adapted from Nesvorn\'y and Ferraz-Mello 1997)}
\label{fig.charts}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig5.ast3.ps,height=15.cm
%,bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}} 
\caption{The estimate of the LCE computed for test bodies initially placed on a regular grid in semimajor 
axis, with $e=0.1$. In (a), the gravitational perturbations provided by the terrestrial planets were 
taken into account in addition to the perturbations of four giant planets. The figure shows many peaks 
which rise from a background level. The background value of about $10^{-5.3}\ {\rm yr}^{-1}$ is dictated by the 
limited integration timespan (2.2 My); increasing the latter, the background level generally 
decreases. The peaks reveal the existence of narrow chaotic regions associated with MMRs. The main resonances 
are indicated above the corresponding peaks. (from Morbidelli and Nesvorn\'y 1999)}
\label{fig.lyaps}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig6.ast3.ps,height=15.cm
%,bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}} 
\caption{The removal times for outer belt asteroids in a number of two--body MMRs with Jupiter. 
The solid symbols are the results of numerical integrations; the open symbols are analytic predictions from 
(\ref{removal_time}). The arrows represent lower limits obtained from numerical integration of bodies in 
MMRs characterized by extremely slow chaotic diffusion. Here, Jupiter Units are 5.2 AU. 
(from Murray and Holman 1997)} 
\label{fig.outer}
\end{figure}
%
\newpage
%
\begin{figure}
\centerline{\psfig{figure=fig7.ast3.ps,height=15.cm
%,bbllx=4cm,bblly=1cm,bburx=15cm,bbury=24cm
}} 
\caption{The evolution of orbital elements of the asteroid (41) Daphne.  The panels show: the 
semimajor axis (a), $\sigma = 9 \lambda_4 - 22 \lambda + 13\varpi$ (b), and 
$e^{13/2}$ (c). The evolution of (41) Daphne is characterized by the interaction with the 9M:22 MMR.
The width of a MMR is proportional to $e^{\rho/2}$ (see (\ref{rwidth}) and related discussion),
where here: $\rho = 13$. Consequently, the resonant width seen by the asteroid is largely
modulated with the parameter in (c). Resonant behavior occurs only when the eccentricity of
(41) Daphne is large. (adapted from \v{S}idlichovsk\'y 1999)}
\label{fig.virginia}
\end{figure}
%   
\end{document}







