\section{When can star impacting planetesimals be produced?}

During the migration of a major planet, mean-motion resonances
will be swept through the disk of planetesimals.
Though secular resonances are also capable driving
particles to extremely high eccentricities  (\cite{levisonb})
they may not necessarily be swept through the disk. 
We also cannot necessarily assume that secular
resonances are always strong in extra-solar systems (\cite{beust})
particularly as the innermost planet becomes more distant from
its neighboring planets.
So we concentrate here on mean motion resonances with one
major planet.

The maximum eccentricity reached by particles librating
in a resonance is extremely sensitive to the eccentricity
of the planet (\cite{beust}; \cite{yoshikawa90}; \cite{moons}).
We expand on the work of \cite{beust_} to determine
what range of eccentricities for a planet are required
to pump particle eccentricities to 1.  We created contour plots 
numerically from the Hamiltonian averaged over time
(as in \cite{beust} and \cite{yoshikawa90}).
For each resonance we then determined what minimum initial 
particle eccentricity is needed for a particle 
to later reach the star ($e=1$).  We estimated
this minimum eccentricity (shown in Fig.~1) for a range of
planet eccentricities, $e_p$.
These contour plots are only extremely weakly dependent on the planet
mass.   We see in Fig.~1 that past a planet eccentricity of 0.3
the 3:1, 4:1, 5:1, 5:2 and 7:2 resonances are all capable
of driving low eccentricity particles to extremely high 
eccentricities.  The eccentricities of the extrasolar planets 
are not restricted to extremely low values (\cite{marcy}).  
This implies that resonances which are capable of
causing star grazing or impacting planetesimals are likely
to exist in almost all of these systems.

\section{Simulation of particles in mean motion resonances 
during orbital migration}

To estimate the efficiency of production of high eccentricity
orbits we numerically integrate the orbits of particles
(using a conventional Burlisch-Stoer numerical scheme) 
during the slow migration of a major planet.
All particles are massless except for the star and
one planet with an eccentricity $\epsilon_p$,
which remains constant throughout the integration.
During the integration we
force the semi-major axis of the planet to drift inwards
at a rate given by the dimensionless parameter 
\begin{equation}
D_a = {da \over dt } {P \over a}
\end{equation}
for $P$ the period of the planet and $a$ its semi-major axis.
$D_a$ is fixed during the integration resulting in 
${da\over dt} \propto  \sqrt{a}$.
Particles are placed in the plane of the planet's orbit  
just within (a few resonance widths) 
the semi-major axis of either the 3:1 or 4:1 resonance.
For each particle the angle of perihelion 
and the mean anomalies were chosen randomly.
Massless particles were integrated until they 
were driven to high eccentricity  ($\epsilon > 0.995$)
and so impact the star,  
or crossed the Hill sphere radius of the planet and were
ejected to semi-major axes larger than the planet.
This took between a few times $10^5$ to $10^6$ periods
measured in units of the initial orbital period of the planet.
In Table 1 we note the initial conditions, migration rates, planet
masses and eccentricities (which remain fixed during the simulation), 
and final particle fates for a set of 10 particle integrations.
In Table 2 we note the resonances operating on the particles
in each simulation prior to impact or ejection.

A sample plot showing eccentricity and semi-major axes for
a run (denoted N8) are shown in Fig.~2.
Almost at all times particles are strongly affected by resonances.  
When a particle crosses the 3:1 or 4:1 resonance it may be 
trapped in a high eccentricity
region of the resonance.  Then the particle can be pumped to extremely
high eccentricities and impact the star.   We find that both the 3:1 and 4:1
resonances cause impacts.  However if the particle does not
remain trapped in the resonance it can later on be caught in
another resonance.   For example, we observe that particles not removed by 
the 3:1 may later on be caught in the 5:2 or 7:3 resonances and particles 
not initially affected by the 4:1 may subsequently be caught in the 
3:1, 7:2 or 8:3 resonances (see Table 2).
If the particle is trapped or strongly affected by a resonance
nearer to the planet (such as the 8:3 resonance) 
then it has a higher chance of being ejected than hitting the star.
In the slower migration rate simulations (N5, M5) we see that even 
minor resonances
such as the 11:3, 10:3, 11:4 cause jumps in the semi-major axis as
the particle crosses the resonance.  However only the 3:1 and 4:1 
are strong enough (and with large enough regions in phase space)
that particles are trapped in them for long periods of time.
These resonances are responsable for the majority of impacts.

In Fig.~2a we see that particles trapped in the 3:1 and 4:1 resonances can 
make multiple close approaches to the star.
During a close approach the surface of a planetesimal will graze
the star and so be sublimated it.
Thus we would predict that a migrating planet would cause
continuing production 
of `falling evaporative bodies',  as proposed to explain the transient
absorption lines observed against beta Pictoris and other stars
(e.g., \cite{beust}; \cite{crawford}; \cite{lagrange96}; \cite{grady}).
We see in our simulations that 
more than one resonance can cause star grazers.
If star grazers are produced by more
than one resonance then particles could approach
the star from different angles with respect to the planet's angle of 
perihelion.  This might provide an alternative explanation for
the occasional blue-shifted event on beta Pictoris (\cite{blueshift}).


Even though the 3:1 and 4:1 resonances can pump
eccentricities to 1.0, in every simulation (see Tab.~1) we find 
particles which pass
through these resonances that are not pumped
to high eccentricities and so removed from the system
by an impact with the star.
These particles can later be ejected by the planet.
While the 3:1 and 4:1 resonances can reduce the surface density
in a disk of planetesimals, they do not create a hole as they
are swept through the disk.  
If the density of planetesimals is
high enough, a planet migrating as a result of
ejection of planetesimals can migrate to within
its original (at formation) 3:1 or 4:1 mean motion resonances.
This would allow a planet to migrate a substantial
fraction of the planet's semi-major axis via ejection of 
planetesimals.  

Particles which impact the star loose all their angular momentum
to the planet which would reduce the planet's
eccentricity.  During the time they are trapped in a resonance
their semi-major axes decreases.  This implies
that the planet will gain energy.
However the decrease in semi-major axis was typically less than $30\%$
of the particle's initial semi-major axes,   
so the energy gained by the planet from trapped particles
should be small compared to that lost from ejected particles.

We did not find that the fraction of impacts was strongly dependent on 
the planet migration rate,  
initial particle conditions, or planet eccentricity. 
However more particles should be integrated to verify this.
We would have expected that slower migration rates, 
more massive planets, lower eccentricity initial
particle eccentricities, and higher planet eccentricities 
would result in an 
increase in the efficiency of trapping particles in resonances
and so in producing impacts.  However the number of resonances
operating on each particle makes it difficult to predict the final
states.  For example when the migration rate
is fast or the planet eccentricity is lowered then  we found that
weaker resonances such as the 4:1 or 7:2 did not affect the particles much,
however the 3:1 was still strong enough to cause impacts.  

\subsection{Survival until impact}

In our integrations we can estimate  
the timescale for the eccentricity to reach $\gtrsim 0.995$. 
While some particles impact the star
on a very short timescale (e.g., particle 1 in Fig.~2), others
are slowly pumped to high eccentricities (e.g., particles
5,6,7,and 8 in Fig.~2).
For the slower approaches the particle or planetesimal
could make $\sim 10^4 - 10^5$ close passages to the star before impact.
When the migration rate was slower 
particles typically experienced larger numbers of close passages
before impact.  We have estimated the mass loss from
a rocky body during a free fall time at solar
radius from the sun to be $\sim 30$ cm.
If the planetesimal makes $10^4$ of such close passages then 
a km body will be completely evaporated by a solar type star.  
For particles making multiple close approaches
only large bodies $\gtrsim 1$km will survive until
impact.  Notably the size distribution of
asteroids subsequent to Gyr timescale collisional evolution 
(\cite{davis}; \cite{greenberg})
is expected to be such that most of the integrated disk {\it mass} is contained
in the largest bodies.  
When migration is relatively quick, 
this mechanism could be a way to increase the metallicity of the star, 
despite the fact that the lower mass bodies may not survive until impact.
Smaller bodies which will completely evaporate could
manifest themselves as transient absorption
features, a phenomenon which is observed on beta Pictoris 
and other stars (e.g., \cite{crawford}; \cite{lagrange96}; 
\cite{grady}).  



We now consider whether large bodies are likely to fragment
upon close approach.  If the object is strengthless then 
it is likely to fragment at periapse
only if the density of the object is lower
than the mean density of the star 
(e.g., \cite{asphaug}; \cite{sridhar}).
The mean density of the sun is $\rho \sim 1.4$ g cm$^{-3}$
so that on a solar type star all but the least dense asteroids 
should not fragment and so should survive
until impact.  On lower mass main sequence stars (which are denser), 
however, denser objects
could be fragmented during close passages subsequent to impact.  
On higher mass stars, such as beta Pictoris, even cometary material 
will not be fragmented by the star during close passages.  



\section{Summary and Discussion}

We have presented a series of numerical integrations
of particles initially at low eccentricities which
pass through mean motion resonances with
a major moderate eccentricity migrating planet.
We confirm that the 3:1 and 4:1 resonances can pump
the particles eccentricities to 1.0 and so can cause particles
trapped in them to impact the star or be evaporated by it. 
As a planet migrates through a disk of planetesimals we would
expect continuing production of bodies undergoing close
approaches to the star.
This provides us with a possible observational test.
A recent study finds that beta Pictoris may be quite young ($2\times 10^7$ years;
\cite{agebetapic}).  If orbital migration occurs commonly
during this timescale then a multi-object (or multi-fiber) survey
in young clusters should detect
transient absorption features due to evaporating bodies 
similar to those seen in beta Pictoris and other stars.

Our integrations show that many particles which pass
through these resonances will not be pumped
to high eccentricities and so removed from the system
by evaporation or by impact with the star.
These particles can subsequently be ejected by the planet.
This implies that a planet can migrate 
a significant fraction of its initial semi-major axis via
ejection of planetesimals.

For the faster migration rates,
we estimate that $\gtrsim 1$ km sized rocky bodies will survive
heating from a solar type star during multiple close passages
and so can become incorporated into the convection
zone of the star.  Because we expect that most of the mass 
will be in the most massive bodies,
this migration process may be capable of increasing the 
metallicity of the star.
Planet migration should occur on a $10^7$ year timescale 
(\cite{murray}) so we do not expect
the star to be fully convective during migration.
Metals dumped into the star should remain
in the convection zone of the star.
This scenario therefore offers a plausible explanation
for the metallicity enhancements observed in stars
with extrasolar planets (\cite{gonzalez99}).

To migrate a significant fraction of its semi-major axis
the planet must eject on the order of its mass (\cite{murray}) in 
planetesimals.  If the material ejected is rocky then the original 
proto-stellar disk would have had $\gtrsim 30$ times this mass in
gas and volatiles.  It is not inconceivable that this amount of material
was left in and interior to a Jupiter mass planet after formation.
However planetesimals exterior to the planet forced to 
high eccentricity by a secondary planet may also
be ejected by a planet and so cause its
migration.  Some fraction of these particles will also impact the star
(e.g., as seen in simulations of short period comets, \cite{levison}). 
This suggests another possible link between star grazers and 
impactors and orbital migration.


\acknowledgments

This work could not have been carried out without
suggestions, discussions and correspondence 
from N.~Murray, C.~Pilachowski, R.~Strom, M.~Sykes, D.~Davis, R.~Greenberg,  
J.~Raymond,
D.~Trilling, D.~Garnett, B.~Livingston, R.~Kudriski, and J.~Lunine.
We also thank G.~Rivlis and K.~Ennico for
donations of computer time and support.
We acknowledge support from 
NASA project numbers NAG-53359 and NAG-54667. 

\clearpage

