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% Revised version, Febr. 4 1997
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\documentstyle[epsf]{l-aa}
\newcommand{\etal}{{\it et~al.}}
\newcommand{\etall}{{\it et~al. }}
\thesaurus{05.03.1; 07.13.2}
\title{Orbital evolution of asteroidal fragments into the
$\nu_6$ resonance via Yarkovsky effects}
\author{David Vokrouhlick\'y$^1$ and Paolo Farinella$^2$}
\institute{$^1$ Institute of Astronomy, Charles University,
    V Hole\v{s}ovi\v{c}k\'ach 2, CZ-180 00 Prague 8, Czech Republic\\
           $^2$ Gruppo di Meccanica Spaziale,
    Dipartimento di Matematica, Universit\`a di
    Pisa, Via Buonarroti~2, 56127 Pisa, Italy\\
    \hspace*{3mm}(Electronic mail: vokrouhl@mbox.cesnet.cz;
     paolof@dm.unipi.it)}
\offprints{D. Vokrouhlick\'y}
\date{\today}
\begin{document}
\maketitle
\markboth{D. Vokrouhlick\'y and P. Farinella: Asteroidal fragments
into the $\nu_6$ resonance}{}
\begin{abstract}
We analyze the dynamical evolution of asteroidal fragments released in
the Flora region, near the inner edge of the main asteroid belt, and
drifting into the $\nu_6$ secular resonance due to Yarkovsky
non--gravitational effects.  We find that fragments 5 to 20~m in size
evolve under the ``seasonal'' Yarkovsky effect which causes a secular
semimajor axis decay; they reach $\nu_6$ after a time shorter than
their collisional lifetime when they start within about $0.05$ to
$0.2$~AU out of the resonance.  Metal--rich fragments drift slower but
have have much longer lifetimes than stony ones, so they drift farther
from their formation site and sample a wider portion of the inner
belt.  Fragments around 100~m in size are mainly influenced by the
``diurnal'' Yarkovsky effect if their surface is covered by a (thin)
regolith layer; this causes a random walk in semimajor axis controlled
by impacts which reorient the spin axis.  Within their lifetime of
$\approx 100$~Myr these fragments can move throughout the inner part
of the asteroid belt, episodically crossing $\nu_6$.  Meter--sized
stony fragments, which probably deliver most meteorite falls, may also
drift into the resonance under the ``diurnal'' effect, provided their
surfaces have low thermal conductivities and/or their rotation is
unusually slow.  According to our dynamical model, which is truncated
to $15^{\rm th}$ degree in the fragment's orbital eccentricity,
$\nu_6$ resonance effects typically result into large eccentricity
increases, such that main--belt orbits rapidly become Earth--crossing
when the resonance is reached and/or crossed.  This confirms the idea
that the interplay of resonant dynamics and Yarkovsky--related
semimajor axis mobility is crucial in the transport of meteorites and
small near--Earth asteroids from the main asteroid belt to the
near--Earth space.
\keywords
{Celestial mechanics -- Minor planets, asteroids -- Meteoroids}
\end{abstract}
 
\section{Introduction} The Earth permanently experiences the infall of
interplanetary matter, a large fraction of which origins in the main
asteroid belt.  Because of the relatively frequent collisions there,
this source population spans all possible values of masses and sizes,
with a characteristic quasi--power--law distribution (Dohnanyi 1969,
Campo Bagatin \etal~1994).  Whereas most meteorites appear to be
delivered by pre--atmospheric bodies of the order of $0.1$ to $1$~m in
size, occasional bright bolides reach several meters, and according to
Ceplecha (1992, 1996) a clear maximum of the incoming mass versus size
curve is present at a size of $10$--$20$~m.  These objects have been
recently observed also in near--Earth space (Rabinowitz \etal~1993,
Rabinowitz 1994) and while exploding in the high atmosphere
(Tagliaferri \etal~1994).  For Tunguska--like impactors $50$--$100$~m
in diameter, only one event per $100$--$1000$~yr is expected, but
extensive disruption on the ground is engendered.  Therefore, for a
number of different reasons it is important to understand the
dynamical mechanisms which are involved in transferring from the main
belt objects of sizes between $\approx 0.1$ and $100$~m --- especially
if, as we will argue in this paper, these dynamical mechanism are
somewhat different from those relevant for larger asteroids.
 
The standard way of solving the problem of how material is transported
from the asteroid belt to the Earth resorts to the peculiar dynamical
evolution of bodies once they get close to or inside the main mean
motion and/or secular resonances with the planets.  Fragments from
asteroidal collisions undergo orbital velocity changes of the order of
$100$~m/s, and as a result the orbits of some of them are injected
into the resonances starting from the nonresonant orbits of their
parent bodies (Farinella \etal~1993a, 1994a).  Subsequently, as a
consequence of resonant gravitational perturbations by the giant
planets, the orbital eccentricity grows to values ($\approx 0.6$)
allowing Earth crossing within only a few Myr, and then the interplay
of resonant effects and planetary encounters drives the bodies to hit
the Sun or the planets, or to be ejected from the Solar System by
Jupiter (Farinella \etal~1993b, 1994b; Froeschl\'e \etal~1995; Jopek
\etal~1995; Migliorini \etal~1997; Gladman \etal~1997).
 
However, it has been realized recently that, at least for
meteorite--sized ($\approx 0.1$ to $10$~m) bodies, this cannot be the
whole story.  While in interplanetary space, meteorites are exposed to
cosmic radiation (provided they lay within about 1~m from the surface
of the body carrying them), and measurements of these cosmic--ray
exposure (CRE) ages give values between $5$ and $50$~Myr for most
stony meteorites, and about 10 times as long for iron--rich
meteorites.  Since these ages are much longer than the dynamical
transfer times through the resonances, and also than the typical
lifetimes of near-Earth objects ($\approx 10$~Myr, according to
Gladman \etal~1997), one must assume that asteroid fragments typically
spend relatively long times in nonresonant main--belt orbits before
being transferred to the near--Earth space.  As we have recently
pointed out (Hartmann \etal~1997, Farinella \etal~1997, 1998), a
non-gravitational dynamical mechanism allowing for such a slow drift
of small bodies through the asteroid belt before ``falling'' into the
resonances is known since a long time: it is the so--called {\it
Yarkovsky effect}.
 
This effect is a recoil force due to radiation pressure, arising
whenever a spinning body re-emits anisotropically the absorbed solar
radiation.  As shown by a number of authors (\"Opik 1951; Radzievskii
1952; Peterson 1976; Afonso \etal~1995; Rubincam, 1995, 1998;
Farinella \etal~1998), it can cause significant long--term semimajor
axis effects on asteroid fragments in the size range from about $0.1$
to $100$~m (smaller bodies are more affected by other
non--gravitational forces, see Burns \etal~1979, while larger ones
have a too small area--to--mass ratio).  In particular Farinella
\etal~(1998) have recently shown that, depending on the size,
rotational state and thermal properties of the fragment, two different
variants of the Yarkovsky effect may play a dominant role: a
``diurnal'' effect, which is more important for slowly rotating,
low--obliquity, regolith--covered fragments; and a ``seasonal''
effect, favoured for high--obliquity, fast rotators with a lower
(bare--rock) surface thermal inertia.  Whereas the latter effect
always causes a secular, drag--like decrease of the orbital semimajor
axis, the former one may result into either positive or negative
semimajor axis changes, depending on the sense of rotation; in all
cases, the amplitude of the semimajor axis effect depends on the
obliquity angle.  If collisions change frequently enough and in random
way the orientation of the spin axis, the Yarkovsky--driven semimajor
axis drift also undergoes stochastic variations in rate.  These
peculiarities make a realistic modelling of the orbital evolution of
small asteroid fragments much more complicated than in the case of
larger bodies, for which gravitational forces alone are important.
 
In this paper we intend to start a detailed, realistic study of how
the Yarkovsky effect can interact with resonant N--body dynamics in
transporting small asteroid fragments to the near--Earth region.  We
will deal in particular with objects originating in the so--called
Flora region, in the inner part of the main asteroid belt, and
drifting into the $\nu_6$ secular resonance, which corresponds to the
inner edge of this zone at a semimajor axis slightly exceeding $2$~AU.
In the last decade several studies have addressed the purely
gravitational dynamics of bodies located in the $\nu_6$ resonance
(Froeschl\'e and Scholl 1987; Yoshikawa 1987; Scholl and Froeschl\'e
1991; Morbidelli and Henrard 1991; Morbidelli 1993; Valsecchi
\etal~1995) or injected into it as a result of collisions in the
neighboring Flora region of the main belt (Farinella \etal~1993a,b;
Morbidelli \etal~1994).  However, little is known about the
effectiveness of this resonance in pumping up the orbital eccentricity
in the presence of dissipative, non--gravitational perturbations such
as the Yarkovsky force.
 
Therefore, our aim here is that of presenting a relatively simple
model for the orbital evolution of asteroidal fragments in the region
of the orbital element space close to the $\nu_6$ resonance, under the
simultaneous influence of the gravitational perturbations by the major
planets and the Yarkovsky thermal effects.  In order to better
understand the significance of our results, we have decided to keep
our model analytical as far as possible for the time being.  Thus, we
average analytically the gravitational perturbing function and
integrate numerically the averaged system of Lagrange equations only.
Moreover, due to the complexity of the thermal effects, we consider
only their most important orbital effect, namely the secular changes
in semimajor axis, neglecting their influence on other orbital
elements.  This is certainly a crude approximation, as Rubincam (1995,
1998) and Vokrouhlick\'y and Farinella (1998) have showed that
Yarkovsky effects can lead to significant long--term changes in the
eccentricity and inclination as well.
 
The remainder of this paper is organized as follows.  Sec.~2 is
devoted to a description of our dynamical model, for both the
gravitational (Sec.~2.1) and non-gravitational (Sec.~2.2)
perturbations.  In Sec.~3 we present some tests of the corresponding
theory compared to direct numerical integrations and other results,
and then we discuss a number of runs for different model populations
of bodies, for which either the ``seasonal'' or the ``diurnal''
Yarkovsky effects play a dominant role.  In Sec.~4 we discuss the
significance of these results for our understanding of the delivery of
small asteroid fragments and meteorites to the near--Earth space.
 
\section{Dynamical model}
\subsection{Secular perturbations by the outer planets}
As a first step we are going to summarize our analytical model for the
secular influence of the major planets, Jupiter and Saturn, on the
fragment's orbit.
Following the formalism of planetary theories such as those of
Duriez (1977) and Laskar (1985), we introduce the pair of complex,
non-singular mean orbital elements $(\zeta, \xi)$ defined as
\begin{eqnarray}
 \zeta &=& k + \imath h = e \exp(\imath \varpi)\;, \label{2one}\\
 \xi   &=& q + \imath p = \sigma \exp(\imath \Omega)\;, \label{2two}
\end{eqnarray}
where $e$ is the mean eccentricity $e$,
$\varpi = \Omega+\omega$ the
mean argument of pericenter,
$\sigma = \sin I/2$, $I$ being the
mean inclination, and $\imath=\sqrt{-1}$. Then,
the secular evolution of the orbit is determined by the set of
differential equations
\begin{eqnarray}
 D \zeta &=& {1\over na^2} \left[2\eta {\partial R\over \partial {\bar
  \zeta}}+ {\zeta\over \eta} {\rm Re}\left(\xi{\partial R\over \partial
  \xi} \right)\right]\;, \label{2three}\\
 D \xi &=& {1\over na^2} \left[{1\over 2\eta} {\partial R\over \partial
  {\bar \xi}}+ {\xi\over \imath\eta} {\rm Im}\left(\zeta{\partial R\over
  \partial \zeta}\right)\right] \label{2four}
\end{eqnarray}
(see Laskar 1985), where $D = -\imath d/dt$, $\eta = \sqrt{1 - e^2} =
(1 - \zeta {\bar \zeta})^{1/2}$, $a$ is the mean semimajor axis, $n$
the mean motion ($n^2 a^3 = G m_\odot$, $m_\odot$ being the solar
mass).  Overbarred quantities are complex conjugates, while Re and Im
denote the real and imaginary parts of a complex quantity,
respectively.  The function $R$ in Eqs.~(3) and (4)
is the secular part of the planetary perturbing
function.  In our approximate model, we adopt the two following
simplifications:
 
\begin{enumerate}
\item we neglect the inclinations of the perturbing planets ($\xi'
 =0$) and take into account only their eccentricities ($\zeta' \neq
 0$; planetary variables are primed);
\item we adopt a linear approximation for the averaged
 (secular) perturbing function, neglecting quadratic and higher
 order terms in the planetary masses.
\end{enumerate}
In this case, it is well known (see e.g.  Brouwer and Clemence 1961)
that the disturbing function $R$ is due to the direct part of the
planetary perturbation only, and reads
 
\begin{eqnarray}
 R &=& \left[{Gm'\over |{\bf r} - {\bf r'}|}\right]_{\rm sec}\nonumber \\
  &=& {Gm'\over a'}\sum_{i \in {\bf Z}^6} \phi_{\boldmath i}(\alpha)
  \xi^{2i_1}{\bar \xi}^{2i_2} \zeta^{i_3} {\bar \zeta}^{i_4} \zeta'^{i_5}
  {\bar \zeta}'^{i_6}\;, \label{2five}
\end{eqnarray}
where ${\boldmath i}$ denotes the vector of indexes $(i_1,i_2, ...
,i_6)$, $m'$ and $a'$ are the planet's mass and semimajor axis (a sum
over the two considered planets is implicit), and $\alpha$ is the
ratio between the fragment's and planet's semimajor axes.  Taking into
account the specific problem we are dealing with (fragments with
$a\approx 2$~AU and $e$ up to $\approx 0.7$), we have expanded the
perturbing potential (5) to a high order in the fragment's
eccentricity.  By using the algebraic manipulator MINIMS, developed
by M.~Moons at the University of Namur (Moons 1991), we have computed
all terms with indexes $|i_3|+|i_4|\leq 15$ and
$|i_1|+|i_2|+|i_5|+|i_6|\leq 3$. In other words, the
disturbing function has been developed up to the fifteenth degree in
the fragment's eccentricity.  The second condition means that we have
kept a lower number of terms in the fragment's inclination, namely
those up to $\sin^6 I/2$ (included) only.
 
The $\phi_{\boldmath i}(\alpha)$ factors in Eq.~(5) can be
expressed as functions of the Laplace coefficients and their
derivatives.  Because of the specific character of our application,
with secular changes in the fragment's semimajor axis, we have
calculated all these terms with particular care analytically.
Starting from the formul{\ae} for the lowest order Laplace
coefficients (Brouwer and Clemence 1961):
\begin{eqnarray}
 b^{(0)}_{1/2}(\alpha) &=& \frac{4}{\pi} K(\alpha)\;, \label{2six} \\
 b^{(1)}_{1/2}(\alpha) &=& \frac{4}{\pi} {K(\alpha)-E(\alpha)\over
  \alpha}\;, \label{2seven}
\end{eqnarray}
with $K(\alpha)$ and $E(\alpha)$ denoting the complete elliptic
integrals of the first and second kind, we have
\begin{eqnarray}
 b^{(j)}_{s+1/2}(\alpha) &=& \alpha^J w^{2s}(\alpha)\left[P^j_s(\alpha)
  b^{(0)}_{1/2}(\alpha)+Q^j_s(\alpha)
  b^{(1)}_{1/2}(\alpha)\right]\,, \nonumber \\
 & & \label{2eight}
\end{eqnarray}
where $J = {\rm min}(0,1-j)$ and $w(\alpha) = 1/(1-\alpha^2)$.
Recurrence formul{\ae} for the polynomials $P^j_s(\alpha)$ and
$Q^j_s(\alpha)$ are given by \v Sidlichovsk\'y and Melendo (1986) or
\v Sidlichovsk\'y (1989; alternatively, see Laskar 1991).  The
derivatives of the Laplace coefficients (8) can then be
computed by simple algebra.
 
The structure of the eccentricity equation (3) can be
schematically written as
\begin{eqnarray}
 D \zeta &=& g\left(a,e^2,\sigma^2\right)\zeta + f\left(a,e^2,\sigma^2
  \right) \zeta'\nonumber\\ & & +\; \mbox{\rm other nonlinear
  terms} \;, \label{2nine}
\end{eqnarray}
where $g$ and $f$ are known functions.  As for the long-term evolution
of the planetary orbits, we have used the Laplace--Lagrange secular
solution with numerical coefficients given by Kne\v zevi\'c (1986).
Jupiter's and Saturn's complex eccentricities are thus given by the
simple harmonic development
\begin{equation}
 \zeta' = \sum A_j \exp \imath\left(g_j t + \phi_j\right)\;, \label{2ten}
\end{equation}
with constant coefficients.  In our case, the most important of of
these constants is $g_6 = 27."360\;{\rm yr}^{-1}$; this value, derived
by Kne\v zevi\'c (1986), is in good agreement with numerical
determinations of the corresponding frequency (Laskar 1988; Nobili
\etal~1989); as we will see in Sec.~3.1, our results are not sensitive
to small changes in this value.  The structure of Eqs.~(9)
and (10) shows that the linear solution of these equations has
a singularity when $g\left(a, e^2,\sigma^2\right) = g_6$: of course
this is the well-known condition for the $\nu_6$ resonance, which is
located near the inner edge of the main asteroid belt.

We have not taken into account the influence of the inner planets for
several reasons.  On one hand, we have chosen to develop an analytical
model, and it is well known (see e.g. Hagihara 1971)
that classical developments of the planetary perturbing
potential in the form (5) suffer divergencies whenever the
minor body can cross the orbit of a planet.  As we shall see,
crossing the $\nu_6$ resonance typically leads to high--eccentricity
orbits, which cross those of both Mars and the Earth, and therefore
analytical developments of type (5) would be useless if we
wanted to include these planets.  On the other hand, we believe that
the main conclusions of our work would not be altered by the inclusion
of the gravitational influence of the inner planets, because (i) the
corresponding shift in the position of $\nu_6$ is small (Kne{\v
z}evi\'c \etal~1991); (ii) mean motion and secular resonances with the
inner planets can be effective only for semimajor axes smaller than
$2$~AU (Milani \etal~1989, Michel 1997); and (iii) close encounters
with Mars do not affect much the orbital evolution of bodies which are
being transported from the main asteroid belt to the near--Earth zone
(Gladman \etal~1997).

\subsection{Thermal perturbations of the fragment's orbit}
Besides the gravitational effects, and as a major novelty of this
work, we include in our model a simplified treatment of the thermal
Yarkovsky effects acting on small asteroidal fragments, by taking into
account the corresponding secular changes in semimajor axis.  A
complete theory for the orbital perturbations due to the Yarkovsky
force is fairly complex, since the intensity of the force at a given
time depends on the thermal state of the body, that is on the thermal
inertia and the insolation the body has received over a time of the
order of a rotation (for the ``diurnal'' effect) or orbital period
(for the seasonal one); hence, at least in the ``seasonal'' case, the
Yarkovsky force depends in a complex way on the orbital elements.  In
the current context, in order to keep the problem tractable, we just
simulate the secular semimajor axis effects by adding a simple linear
term in the semimajor axis of the fragment's orbit, that is assuming
that $a(t) \simeq a(0) + {\dot a}\, t$.  The value of the semimajor
axis rate ${\dot a}$ is in principle determined by a number of
different circumstances.  As we mentioned in Sec.~1, in the case of
the ``seasonal'' version of the Yarkovsky effect the sign of ${\dot a}$
is always negative, while in the case of of the ``diurnal'' version it
can be either positive or negative, according to the orientation of
the spin axis.  In quantitative terms, we are going to use the results
of Farinella \etal~(1998).  For the reasons discussed in that paper,
we assume that the rotation rate of asteroid fragments is inversely
proportional to their size, with 1~km diameter bodies spinning in
5~hr.

\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f1.eps}}
\end{center}
\vspace{-2.2cm}
\caption[]{Diffusion time $T_{cr}$ (in Myr) required for reaching
 the $\nu_6$ resonance at zero inclination vs. initial semimajor
 axis $a$ (in AU). Two types of objects are assumed to evolve under
 the Yarkovsky ``seasonal'' effect: (i) $R=5$~m
 basalt fragments (curve~1), and (ii) R=$10$~m iron--rich bodies
 (curve~2).
 \label{f1}}
\end{figure}

\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f2.eps}}
\end{center}
\vspace{-2.2cm}
\caption{Diffusion of regolith--covered stony objects 50~m in radius
under the ``diurnal'' Yarkovsky effect, taking into account the random
reorientation of spin axes caused by impacts (assumed to occur at
typical intervals $\tau_{rot}=24$~Myr). The initial semimajor
axes values (in AU) are given on the horizontal axis.  The full lines
give the average percentage of a collisional lifetime $\tau_{disr}$
required to reach $\nu_6$ and the dashed lines the percentage of
bodies reaching the resonance before being disrupted.  Curves ~1 and 2
were derived assuming $\tau_{disr}=140$~Myr (according to Farinella
\etal~1998) and $\tau_{disr}=4$~Byr, respectively.
 \label{f2}}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f3.eps}}
\end{center}
\vspace{-2.2cm}
\caption{The same as Fig.~2 but for objects 1~m in radius. Curves~1
correspond to stony bodies, with $\tau_{rot}=3.3$~Myr and
$\tau_{disr}=20$~Myr; curves~2 correspond to iron bodies, with
$\tau_{rot}=7$~Myr and $\tau_{disr}=1.4$~Byr (Farinella \etal~1998).
 \label{f3}}
\end{figure}


The ``seasonal'' effect is the dominant one for regolith--free
fragments of size of the order of 10~m. For this case, we adopt the
following empirical model: \begin{equation}
 {\dot a} \simeq -\,{\dot a}_{\rm max}(\Theta_n,R,a)\, {\sin^2
 \gamma} \;,\label{22one}
\end{equation}
where $\gamma$ is the obliquity of the fragment's polar axis (that is,
the angle between the spin axis and the normal to the mean orbit), $a$
is the semimajor axis and $\Theta_n$ is a thermal inertia
parameter related to temperature changes with periodicities of the
order of the orbital period.  The dependence of the amplitude ${\dot
a}_{\rm max}$ on $a$, on the body's (mean) radius $R$ and on the
$\Theta_n$ parameter is quite complex (see Farinella
\etal~1998, Rubincam 1998), so we just use the following typical
values, consistent with the results of the above--mentioned papers for
$R=5$~m and at an orbital semimajor axis $a\approx 2$~AU:
${\dot a}_{\rm max} \simeq
3.7\times 10^{-3}$~AU/Myr for stony fragments and ${\dot a}_{\rm max}
\simeq 4.2\times 10^{-4}$~AU/Myr for iron--rich objects.  If one
assumes a frequent enough collisional reorientation of the spin axis
(with all directions being equally likely), the average drift rate is
reduced by a factor $2/3$ with respect to ${\dot a}_{\rm max}$.
 
As for the ``diurnal'' Yarkovsky effect, which is probably the
dominant one for meter--sized bodies and for relatively large
($R\simeq 25-100$~m) regolith--covered fragments, we use
\begin{equation}
 {\dot a} \simeq {\dot a}_{\rm max}\, f\left(\Theta_\omega,R,a\right)\,
  {\cos\gamma}\;, \label{22three}
\end{equation}
where $\Theta_\omega$ is the thermal parameter related to temperature
changes over times comparable to the spin period.  In this case the
value of ${\dot a}_{\rm max}$ depends sensitively on the thermal
conductivity of the surface layer, as a very thin ($\approx 1$~mm)
layer of regolith--like low--conductivity material is enough to
strongly affect the ``diurnal'' component of the surface temperature
distribution.  Assuming thermal properties similar to those measured
for the lunar regolith (Rubincam 1995), Farinella \etal~(1998)
obtained in this case ${\dot a}_{\rm max} \simeq 2.65\times
10^{-3}$~AU/Myr for a $R=50$~m object and ${\dot a}_{\rm max} \simeq
4.52\times 10^{-2}$~AU/Myr for $R=1$~m.  On the other hand, for
bare--rock surfaces Farinella \etal~(1998) used the conductivity of
terrestrial basalt, which is close to the maximum values measured for
chondritic meteorites (Yomogida and Matsui 1983), and this gives the
much lower value ${\dot a}_{\rm max} \simeq 8.25\times 10^{-4}$~AU/Myr
at a radius of $1$~m.  For meter--sized, regolith--free iron
meteorites, using the higher conductivity of metallic iron, Farinella
\etal~(1998) obtained ${\dot a}_{\rm max} \simeq 7.29\times
10^{-5}$~AU/Myr.  These estimates were always obtained at
$a\approx 2$~AU.
 
As we explained in Sec.~1, an essential ingredient of the Yarkovsky
effects is the collisional reorientation of the fragment spin axes,
resulting into random changes of the obliquity angle $\gamma$.  To
take this into account, we can use the Farinella \etal~(1998)
estimates for the characteristic time between two collisions imparting
to a given target an angular momentum of rotation comparable to the
pre--existing one --- keeping in mind that estimates of this kind are
highly uncertain, owing to our poor knowledge of the small--size
projectile flux in the asteroid belt.  According to these estimates,
the reorientation time $\tau_{rot}$ scales proportionally to
$R^{1/2}$, and is $\approx 3.3$, $8$ and $24$~Myr for fragments $1$,
$5$ and $50$~m in radius, respectively.  For iron fragments, the
higher density results in time scales longer by a factor of about
$2.5$.
 
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f4.eps}}
\end{center}
\vspace{-2.2cm}
\caption{Mean eccentricity $e$ vs. the critical argument of the
$\nu_6$ resonance $\varpi- \varpi_6$ (with $\varpi_6=g_6 t+{\rm
constant}$), for initial conditions corresponding to Yoshikawa's
(1987) case A test body.  The plot shows the orbital evolution over a
time span of 1~Myr, computed by the analytical scheme described in
Sec.~2.
\label{f4}}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f5.eps}}
\end{center} \vspace{-2.2cm}
\caption{The same as Figure~3 but for Yoshikawa's
(1987) case B test body.
\label{f5}}
\end{figure}
 
In order to assess the extent of the overall semimajor axis decay in
the case of the ``seasonal'' Yarkovsky effect and of the corresponding
random walk for the ``diurnal'' effect, we have performed the
following tests.  Taking the initial semimajor axis in the range $2.1$
to $2.35$~AU, we have let the orbits evolve according to
Eqs.~(11) and (12).  We have considered separately
the cases of bodies dominated by the ``seasonal'' and ``diurnal''
effect: in the former case we simulated the orbital evolution of stony
and iron fragments, 5 and 10~m in radius, respectively, whereas in the
latter case we considered either ``large'' bodies, 50~m in radius, or
``small'', meter--sized fragments (both stones and irons).  For the
``diurnal'' runs we always assumed that the bodies are covered by
thin, regolith--like insulating layers, because otherwise the
random--walk evolution is very limited (at most a few hundredths AU).
In each case we performed $10^6$ test runs, always reorienting the
spin axis in a random direction at the typical intervals specified
above.
 
Figure~1 shows the mean time required for asteroid fragments drifting
under the ``seasonal'' effect to reach the value $a=a_6\simeq
2.06$~AU, which corresponds to the position of the $\nu_6$ resonance
for small--inclination orbits.  Statistically, a $5$-meter stony
boulder released at the middle of the Flora region (approximately
$2.25$~AU) reaches the resonance in about $120$~Myr, whereas a time
span a factor $10$ longer is necessary for iron--rich objects.  We
recall that, according to the estimates of Farinella \etal~(1998), the
typical collisional lifetimes of the two types of bodies in the main
belt are of $45$~Myr and $4.4$~Byr, respectively.  If these average
lifetimes are correct, our results imply that stony fragments can
reach the resonance before being shattered by impacts if they are
unusually long--lived and/or they start from a semimajor axis strip of
width $\approx 0.075$~AU along the resonance border, whereas iron
fragments may come from the whole Flora region.  Thus, although iron
fragments drift slower, they can drift much farther inside from their
formation site as a consequence of the longer collisional lifetime.
Note that the bodies discussed here are larger than the majority of
meteorites, which have pre--atmospheric sizes $\approx 0.1$--$1$~m;
nevertheless, we think that they are quite relevant not only in
themselves (as explained in Sec.~1), but also for meteorites, because
a significant fraction of their mass lies at depths such that
cosmic--ray irradiation is possible, and since many meteorites appear
to have undergone complex exposure histories (Wetherill 1980), it is
possible that they are just multi--generational fragments from objects
several meters across.

\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f6.eps}}
\end{center}
\vspace{-2.2cm}
\caption{Effects on the orbital eccentricity $e$ of $\nu_6$
 resonance crossings with different semimajor axis decay rates
 $da/dt$ (in AU/Myr). Higher decay rates result into a lower
 efficiency of the resonance in pumping up the eccentricity.
 \label{f6}}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f7a.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f7b.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f7c.eps}}
\end{center}
\vspace{-2.2cm}
\caption[]{Orbital evolution of a fictitious basalt fragment $5$~m in
radius under the ``seasonal'' Yarkovsky effect.  Panel (a) shows the
secular semimajor axis decrease (at a variable rate, depending on the
collisional changes in the obliquity of the spin axis), with the
dashed line marking the position of the $\nu_6$ resonance for the
instantaneous values of the eccentricity and inclination.  Starting
from $a=2.28$~AU, the resonance is reached in about $110$~Myr, in good
agreement with the statistical results shown in Fig.~1.  The mean
eccentricity [panel (b)] and inclination [panel (c)] undergo rapid
changes after crossing the resonance, and the eccentricity is pumped
up to about $0.6$.  The initial conditions correspond to the Flora
region asteroid 1981~WR.
\label{f7}}
\end{figure}

Figure~2 shows the results of our test runs for the larger,
regolith--covered fragments whose semimajor axis undergoes a kind of
random walk.  Of course, only a fraction of the test bodies reaches
the resonance within any given time.  The two dashed lines in the
figure give the fraction of bodies ending up into $\nu_6$ within
$140$~Myr (which is approximately the collisional lifetime of stony
objects of this size) and $4$~Byr.  Note that, while there is equal
probability of increasing and decreasing the semimajor axis at any
given instant, the probability of crossing the $\nu_6$-resonance is
greater than $50$~\% over $4$~Byr, because any excursion in semimajor
axis below the critical limit $a_6=2.06$~AU has been classified as a
resonance crossing, independently of the subsequent evolution.  The
full lines in Figure~2 show the average diffusion time as a function
of the starting semimajor axis. 
The ``jumps'' apparent in particular in the upper curve are related to
the discrete number of reorienting impacts undergone during the trip by
fragments starting at different distances from the resonance.
These results show that the $\nu_6$
resonance can collect regolith--covered fragments several tens of
meters in size from the whole inner part of the Flora region.
Possibly this might explain why bodies in this size range appear to be
overabundant in near--Earth space, compared to a power--law
extrapolation from km--sized objects (Rabinowitz 1993, 1997, 1998),
since the latter are not affected by non--gravitational forces in a
significant way.
 
Figure~3 illustrates the corresponding results for meter--sized
objects.  Also at this size a large fraction of asteroid fragments can
reach the resonance from the Flora region within their collisional
lifetime, provided they are covered by a thin insulating layer.
The ``jumps'' are due to the same reason as for Fig. 2.
Again, iron bodies are more mobile owing to their longer lifetimes.
We stress that the ``diurnal'' Yarkovsky random walk becomes much less
effective assuming a surface conductivity typical of ``bare'' rock or
iron.  For real meteorites, the situation may be in between these two
extreme cases, because of the significant porosity of many meteorites
(Consolmagno \etal~1998) which is likely to lower their thermal
conductivity (Yomogida and Matsui, Fig.~10).  Also, we have neglected
the possibility that as a result of chance impacts some fragments end
up spinning at rates much lower than usual (say, with a $1$--hr spin
period).  Such slow rotators would have a ``diurnal'' ${\dot a}_{max}$
almost an order of magnitude larger than our nominal fast--rotating
bodies (see Farinella \etal~1998, Fig.~1), and therefore some
``steps'' in the random walk controlled by collisions may become much
larger than usual.

We note that in the models used to derive Figures 2 and 3 we have
neglected the presence of the 3:1 mean motion Jovian resonance near
$2.5$~AU, which would also rapidly eject any fragments inserted into
it.  Of course, taking into account this resonance would somewhat
decrease the percentages of $\nu_6$--reaching fragments starting from
the middle and outer portions of the Flora region.  On the other hand,
taking into account the relatively weak dependence of the Yarkovsky
effects on semimajor axis (Farinella \etal~1998), the results
illustrated in Figs.~2 and 3 can be applied more or less unchanged to
fragments starting near the 3:1 resonance and eventually falling into
it (instead of $\nu_6$).  However, we plan to investigate in more
detail the interplay of mean motion resonances and the Yarkovsky
effects in a future paper.

\section{Tests and results}
\subsection{Validation of the method}
Before discussing the results of our orbital evolution runs
including the Yarkovsky effects, we report on some checks
that we have made to test our analytical integration scheme
for the secular resonance effects.
 
\begin{figure*}[t]
\epsfxsize=18.cm
\begin{center}
\mbox{\epsfbox{f8.eps}}
\end{center}
\vspace{-1.cm}
\caption{The orbital evolution shown in Fig.~7 is shown here in the
eccentricity vs. critical argument ($\varpi -\varpi_6$) plane,
superimposed onto the the level curves of the averaged Hamiltonian.
The four plots refer to four successive stages of the evolution, in
which the semimajor axis and inclination are approximately constant.
As the semimajor axis is decreased, the level curves of the
Hamiltonian shift and change their topology, and as a result the orbit
evolves from a low--eccentricity circulation, to a libration around
$\varpi -\varpi_6 =180^\circ$ to a high--eccentricity circulation.
 \label{fig8}}
\end{figure*}

First, we repeated several integrations performed with a fully
numerical technique by Yoshikawa (1987).  Fig.~4 and 5
show our results over a $1$~Myr time span for Yoshikawa's
fictitious test particles A and B. Comparing these plots with
Yoshikawa's Figs.~9a and 9b, we observe an excellent agreement with
the numerical results, at least as far as the eccentricity evolution
is concerned.  Actually, the situation is somewhat worse for the
inclination evolution, but this had to be expected, because we have
neglected completely the planetary inclinations and have kept only a
small number of terms containing the asteroid's inclination in the
perturbing function (see Sec.~2). On the other hand, since we are
mostly interested in the way the $\nu_6$ resonance pumps up the
eccentricity, a very accurate model for the inclination effects is not
really needed.

By performing a number of tests of this kind we have become confident
that our results are realistic provided the eccentricity and the
inclination of the fragment's orbit do not exceed maximum values of
about $0.6-0.7$ and $10-15^\circ$, respectively.  For very high
eccentricities, the the results of the analytical model become
unreliable (for instance, we could not reproduce the Sun--grazing
dynamics described by Farinella \etal~1994b and Froeschl\'e \etal~1995).
Also, we failed to reproduce Yoshikawa's (1987) case C test
integration when the eccentricity exceeds $\approx 0.7$.  Actually, in
this case it is mainly the inclination that undergoes large excursions
(up to about $25^\circ$) near the peak of the eccentricity cycle, and
this appears to trigger an instability in our analytical results for
the eccentricity.  This behavior is probably related to the presence
of the $\eta$ factor, which becomes singular for $e\rightarrow 1$, in
the denominator of Eq.~(\ref{2four}).  Overall, however, these
limitations of the analytical model are not very important in the
current context, since fragments injected into $\nu_6$ become
Earth--crossing for eccentricities of about $0.5$, and most asteroids
in the Flora region have inclinations smaller than $10^\circ$.

As a second test, we have checked how the eccentricity evolution in
crossing the $\nu_6$ resonance is affected by different rates of the
semimajor axis decay.  The results are summarized in
Fig.~6.  The same initial conditions ($a=2.16$~AU,
$e=0.06$, $i=0$) have been taken in the three cases.  As the decay
rate $\dot{a}$ increases in magnitude (i.e., for faster crossings of
the resonance), the eccentricity evolution becomes less and less
sensitive to the resonance crossing.  Basically, this is due to the
fact that fast--decaying orbits spend shorter times in the
resonance region, where planetary perturbations are effective in
pumping up the eccentricity. This is a general property of resonance
crossing processes due to dissipative perturbations (see e.g.
Hamilton 1994, Liou and Zook 1997), and our model reproduces it very
well.

\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f9a.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f9b.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f9c.eps}}
\end{center}
\vspace{-2.2cm}
\caption[]{Orbital evolution of a fictitious regolith--covered
fragment $50$~m in radius under the ``diurnal'' Yarkovsky effect.  A
random walk--like evolution in semimajor axis is shown by the full
line in panel (a), and from time to time this leads the orbit to cross
the $\nu_6$ resonance (whose instantaneous location is shown by the
dashed line).  At the times of resonance crossing, the mean
eccentricity [panel (b)] and inclination [panel (c)] undergo rapid
changes, with the eccentricity reaching values $\approx 0.8$.  The
initial conditions correspond to a Flora region asteroid, (2580)
Smilevskia.
 \label{f9}}
\end{figure}

Finally, we have repeated the integrations of Fig.~6 but
replacing the Kne\v zevi\'c (1986) model for the evolution of
planetary orbits with that of Laskar (1988).  We have observed minor
changes, mainly due to a small difference in the value of the $g_6$
frequency and a much richer frequency spectrum in the evolution of
Jupiter's and Saturn's orbits. However, no qualitative change was
apparent in the results.
 
Having gained confidence in our integration scheme we are going to
discuss, in the next two subsections, the results of two sets of runs
performed with different samples of bodies crossing the $\nu_6$
resonance as a consequence of the Yarkovsky effects.  First, we use a
sample of regolith--free bodies $5$ to $20$~m in size, both stony and
iron--rich.  In this case, as we discussed earlier (Sec.~2.2), the
``seasonal'' Yarkovsky effect is probably the dominant one and
Eq.~(\ref{22one}) can be used for the semimajor axis evolution.
Secondly, we consider two populations of asteroidal fragments, $50$~m
and $1$~m in radius, with regolith--like surface properties, and in
this case we use Eq.~(\ref{22three})
% and (\ref{22four})
to model the dominant ``diurnal'' Yarkovsky effect.
 
\subsection{Regolith--free bodies evolving under the ``seasonal''
effect}
Figure~7 shows the evolution of the mean orbital elements which
is typical for this class of objects.  We have chosen initial mean
elements corresponding to asteroid 1981 WR, that is $a=2.280061$~AU,
$e=0.0889699$, $i=2^{\circ}.82532$ (from the Milani and Kne{\v
z}evi\'c database, see Milani \etal~1994). This asteroid is located
more or less in the middle of the Flora region.

\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f10a.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f10b.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f10c.eps}}
\end{center}
\vspace{-2.2cm}
\caption{The same as Figure~9, but with different initial conditions,
corresponding to the mean elements of asteroid 3996 1988 XG$_1$.  At
the end of the integrated evolution, the eccentricity has been pumped
to a very high value (greater than $0.9$).  The inclination also shows
relatively large oscillations. At this point our integration scheme
gives spurious results, as discussed in the text.
\label{f10}}
\end{figure}

Panel (a) of the figure shows the secular semimajor axis decay, with a
variable rate due to random changes in the obliquity $\gamma$.  The
time required to reach the resonance matches the time scale shown in
Fig.~1.  When the $\nu_6$ resonance is crossed, the mean
eccentricity [shown in panel (b)] undergoes a sudden increase, and
then keeps oscillating around a mean value of about $0.62$.  Thanks to
this high eccentricity, the mean inclination [shown in panel (c)] also
undergoes large perturbations.  The way the resonance crossing works
in pumping up the eccentricity is shown clearly in Fig.~8: as the
semimajor axis decreases, the topology of the level curves of the
averaged Hamiltonian is drastically changed and the orbit is
``dragged'' from a low-- to a high--eccentricity circulation of the
critical argument through an intermediate phase of resonant libration
around $\varpi-\varpi_6 = 180^\circ$.  This behavior matches well the
topology of the phase space at the $\nu_6$ resonance according to the
theory of Morbidelli and Henrard (1991, Fig.~1), which shows
moderate--eccentricity librations around $\varpi-\varpi_6 = 180^\circ$
at mean inclinations between $3^\circ$ and $5^\circ$.  Note that the
integrated orbit follows only approximately the level curves, with
``wiggles'' caused by Jupiter's nonresonant perturbations.  We
have performed a large number of similar runs for both stony and iron
bodies, and found in all cases the same kind of qualitative results,
namely the transition to a high--eccentricity circulation mode after
crossing the resonance.
 
Of course, the real orbital evolution is expected to be more
complicated and should be investigated by using fully numerical
integrations (we plan to do this in the next stage of our work on this
issue).  After crossing the $\nu_6$ resonance, the fragments soon
encounter the $4:1$ mean motion resonance with Jupiter and the
$\nu_{16}$ secular resonance, and after this other resonances with the
inner planets can affect the orbital elements in a significant way
(Michel and Froeschl\'e 1997, Michel 1997).  However, we believe that
the most important result reported above --- the effectiveness of the
$\nu_6$ resonance in pumping up the eccentricity of orbits crossing it
under the Yarkovsky effect --- is a robust one, and will be confirmed
by more complex dynamical models.  The same applies to the results
described in the next subsection.
 
\subsection{Regolith--covered bodies evolving under the ``diurnal''
effect}
Figure~9 shows an example of orbital evolution for a
regolith--covered body $50$~m in radius in the Flora region, under the
effect of the ``diurnal'' Yarkovsky effect.  The initial conditions
correspond to the mean elements of asteroid (2580) Smilevskia
($a=2.18255$~AU, $e=0.19527$, $i=0^{\circ}.32962$).
 
Panel (a) shows that the semimajor axis undergoes a random walk with
varying slopes, corresponding to collisional changes in the obliquity.
In two occasions the orbit crosses the resonance, and at these times
the eccentricity undergoes sudden and drastic changes, and the
inclination is also affected. In this particular case the
eccentricity, after staying at high peak values for a few tens of Myr
at the resonance crossings, returns to more or less the previous
moderate ($\approx 0.3$) values when the orbit is driven back into the
Flora region.

\addtocounter{figure}{1}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f12.eps}}
\end{center}
\vspace{-0.5cm}
\caption{The same as in Fig.~11 but for the interval preceding the end
of the integration.  Here the orbit gets trapped into a
large--amplitude, high--eccentricity libration about $\varpi -\varpi_6
=0$, which gradually shifts upwards as the semimajor axis is
increased.
\label{fig12}}
\end{figure}
 
However, this is not always the case.  Figure~10 shows another
example of the orbital evolution driven by the ``diurnal'' Yarkovsky
effect.  The initial conditions now correspond to the mean elements of
asteroid 3996 1988 XG$_1$ ($a=2.25933$~AU, $e=0.103693$,
$i=0^{\circ}.89691$).  Several features are similar to those of the
previous case, but here at the end of the $400$~Myr run the mean
eccentricity exceeds $0.9$, and after the first resonance crossing the
orbit spends most of the time in the high--eccentricity state.  We
acknowledge that for such extreme values of the eccentricity the
$15^{\rm th}$ degree truncation of the perturbing function in the
asteroid's eccentricity adopted in our dynamical model (see Sec.~2.1)
is certainly not sufficient to yield an accurate orbital evolution,
and therefore we may get spurious results.
 
On the other hand, the same ``phase space diagrams'' we used in Fig.~8
are useful to understand the dynamical mechanisms at work in this case
as well.  Panels (a) and (b) of Fig.~11 correspond to two intervals of
a few Myr centered at about $283$ and $320$~Myr after the beginning of
the integration shown in Fig.~10, respectively.  In the former
case, the orbit passes directly from the low-- to the
high--eccentricity circulation mode, without being captured into a
``libration island'' as in the case of Fig.~8.  Possibly this is
simply due to the higher rate of the semimajor axis change in this
case.  The opposite happens in the panel (b) interval, when the
eccentricity is rapidly decreased by the resonance crossing.  Finally,
Fig.~12 shows what happens just before the end of the evolution shown
in Fig.~10: here the orbit gets trapped into a large--amplitude,
high--eccentricity libration about $\varpi -\varpi_6 =0$, which shifts
to higher eccentricities as the semimajor axis grows due to the
``diurnal'' Yarkovsky effect (see Figs.~10a and 10b).  Of course, this
kind of mechanism should be explored by using a better dynamical
model, maintaining a good accuracy at very high eccentricities.

\addtocounter{figure}{-2}
\begin{figure*}[t]
\epsfxsize=19.5cm
\begin{center}
\mbox{\epsfbox{f11.eps}}
\end{center}
\vspace{-6.5cm}
\caption{The same as in Fig.~8 but for the orbital evolution shown in
Fig.~10, near a time of rapid eccentricity growth [panel (a)] and
decrease [panel (b)]. In either case the orbit switches from low-- to
high--eccentricity circulation or {\it vice versa}, without being
captured in a libration zone.
 \label{fig11}}
\end{figure*}
\addtocounter{figure}{1}
 
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f13a.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f13b.eps}}
\end{center}
\vspace{-2.2cm}
\end{figure}
\begin{figure}[t]
\epsfxsize=8.7cm
\begin{center}
\mbox{\epsfbox{f13c.eps}}
\end{center}
\vspace{-2.2cm}
\caption{The same as Figure~10, but with the parameter values
appropriate for a 1--meter body.
\label{f13}}
\end{figure}
 
Finally, Fig.~13 shows the result of an evolution starting from the
same initial conditions used for Fig.~10 but taking the $\dot a_{max}$
and $\tau_{rot}$ value appropriate for a 1--meter body. The main
difference with respect to Fig.~10 is that the time step for the
random walk in semimajor axis is shorter. On the other hand, the
behaviors of the eccentricity and the inclination are very similar
to those of Fig.~10.
 
\section{Conclusions}
The main results of this paper can be summarized as follows:
\begin{enumerate}
 
\item We have estimated by numerical simulations the characteristic
time scales for transporting asteroid fragments into the $\nu_6$
resonance through diffusion of semimajor axes caused by the Yarkovsky
effects.  Regolith--free bodies 5 to 20~m in size can drift into the
resonance through the ``seasonal'' semimajor axis decay within their
collisional lifetime from a region of the inner
asteroid belt of width ranging from several hundredths to a few tenths
of AU, depending on the stony vs. metal--rich composition.  Larger,
100--m sized bodies can also random walk into the resonance from the
inner part of the Flora region provided their surface is covered by a
thin regolith. For meter--sized bodies, the effectiveness of the
``diurnal'' Yarkovsky effect in changing their semimajor axis depends
sensitively upon the thermal conductivity of their surface layer and
their rotation rate. Further data on these physical properties are
needed to assess whether or when the Yarkovsky effect is important in
delivering them to the resonances.
 
\item The bodies whose orbital evolution is dominated by the
``seasonal'' version of the Yarkovsky effect always undergo large
increases of eccentricity (enough for their orbits to become
Earth--crossing) when they cross the resonance at semimajor axes
$\approx 2.1$~AU.
 
\item When the ``diurnal'' effect is the dominant one, the semimajor
axis evolution has a random walk--like character, allowing these
bodies to cross the $\nu_6$ resonance in both senses; such episodic
resonance--crossing events are typically accompanied by large
``jumps'' in the eccentricity and lesser perturbations in the
inclination.  The Yarkovsky--driven evolution into $\nu_6$ may explain
the observed overabundance of ``Small Earth Approachers'' (SEAs)
compared to larger near--Earth asteroids (Rabinowitz 1994, 1998).
However, it seems unlikely that the subgroup of SEAs with $a\approx
1$~AU and small eccentricities (Rabinowitz \etal~1993, Bottke
\etal~1996) can come from the main asteroid belt thanks to Yarkovsky
effects, as proposed by Rubincam (1995).  The reason is that according
to our results the $\nu_6$--crossing episodes always lead to large
eccentricity increases (even disregarding other resonances), and the
Yarkovsky force is not effective enough in circularizing the orbits
while their semimajor axes are decreased (Rubincam 1998,
Vokrouhlick\'y and Farinella 1998).  Of course Earth encounters could
bring down the eccentricity, but this is not likely to occur
frequently enough.
\end{enumerate}
 
Further work on the issues addressed in this paper is needed in
several directions.  In particular, we plan to carry out fully
numerical simulations of fragment orbits including the gravitational
forces of both the outer and inner planets besides the Yarkovsky
effects, and taking into account the stochastic obliquity changes
related to impacts.  As far as analytical models are concerned, it
would be very useful to develop them, in analogy to what we have done
here for $\nu_6$, for the Jovian mean motion resonances, such as the
3:1 and 4:1 resonances which are probably very important for meteorite
transport.
 
\medskip \noindent{\bf Acknowledgements}\hskip 2mm We dedicate this
paper to the memory of Mich\`ele Moons, who had kindly provided us with
her algebraic manipulator MINIMS.  We are grateful to B.~Gladman,
Z.~Kne{\v z}evi\'c, F.~Marzari and A.~Morbidelli (in his capacity as a
reviewer) for useful comments and discussions.  P.F. acknowledges
support from the Italian Space Agency (ASI) and the Italian Ministry
for University and Scientific Research (MURST).
 
\bigskip
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