MIGRATION OF JUPITER-FAMILY COMETS AND RESONANT ASTEROIDS TO NEAR-EARTH SPACE
We estimated the rate of comet and asteroid collisions with the terrestrial planets by calculating the orbits of 13000 Jupiter-crossing objects (JCOs) and 1300 resonant asteroids and computing the probabilities of collisions based on random-phase approximations and the orbital elements sampled with a 500 yr step. The Bulirsh-Stoer and a symplectic orbit integrator gave similar results for orbital evolution, but sometimes give different collision probabilities with the Sun. A small fraction of former JCOs reached orbits with aphelia inside Jupiter's orbit, and some reached Apollo orbits with semi-major axes less than 2 AU, Aten orbits, and inner-Earth orbits (with aphelia less than 0.983 AU) and remained there for millions of years. Though less than 0.1\% of the total, these objects were responsible for most of the collision probability of former JCOs with Earth and Venus. Some Jupiter-family comets can reach inclinations $i$$>$$90^\circ$. We conclude that a significant fraction of near-Earth objects could be extinct comets that came from the trans-Neptunian region.
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\title{MIGRATION OF JUPITER-FAMILY COMETS AND RESONANT ASTEROIDS TO NEAR-EARTH SPACE}

\author{S. I. Ipatov\address{(1) NRC/NAS senior research associate, 
NASA/GSFC, Mail Code 685,
Greenbelt, MD 20771, USA (current address);
E-mail: siipatov@hotmail.com;
(2) Institute of Applied Mathematics,
Miusskaya sq. 4, Moscow 125047, Russia} and
J. C. Mather\address{NASA/GSFC, Mail Code 685, Greenbelt,
MD 20771, USA}}

\begin{document}

% typeset front matter
\maketitle

\begin{abstract}

We estimated the rate of comet and asteroid collisions with the 
terrestrial planets by calculating the orbits of 13000 
Jupiter-crossing objects (JCOs) and 1300 resonant asteroids and 
computing the probabilities of collisions based on random-phase 
approximations
and the orbital elements sampled with a 500 yr step. 
The Bulirsh-Stoer and a symplectic orbit integrator 
gave similar results
for orbital evolution, but sometimes give different collision
probabilities with the Sun.   
A small fraction of former JCOs reached 
orbits with aphelia inside Jupiter's orbit, and some reached  Apollo 
orbits with semi-major axes less than 2 AU, Aten orbits, and 
inner-Earth orbits (with aphelia less than 0.983 AU) and remained 
there for millions of years. Though less than 0.1\% of the total, 
these objects were responsible for most of the collision probability 
of former JCOs with Earth and Venus. Some Jupiter-family comets can 
reach inclinations $i$$>$$90^\circ$. We conclude that a significant 
fraction of near-Earth objects could be extinct comets that came from 
the trans-Neptunian region.

\end{abstract}

\section*{INTRODUCTION}

The main asteroid belt, the trans-Neptunian belt, and the Oort cloud 
are considered to be the main sources of the objects that could 
collide with the Earth. 
About  0.4\% of the encounters within 0.2 AU 
of the Earth are from periodic comets 
(http://cfa-www.harvard.edu/iau/lists/CloseApp.html), and 6 out of 20 
recent approaches of comets with the Earth within 0.102 AU  were due 
to periodic comets 
(http://cfa-www.harvard.edu/iau/lists/ClosestComets.html). 
The fraction of close encounters with the Earth 
due to active comets is $\sim$1\%. 
Reviews of the asteroid and comet hazard were given by Ipatov (2000, 
2001), and Bottke et al. (2002). Many scientists, e.g. Bottke et al. 
(2002), believe that asteroids are the main source of near-Earth 
objects (NEOs, i.e. objects with perihelion distance $q$$<$1.3 AU). 
However, Shoemaker {\it et al.} (1994) argued that active and extinct 
periodic comets may account altogether for about 20\% of the 
production of terrestrial impact craters larger than 20 km in 
diameter. There are about 40 active and 800 extinct Earth-crossing 
Jupiter-family comets with period $P$$<$20 yr and nuclei $\ge$ 1 km,
and about 140--270 active Earth-crossing Halley-family comets 
($20<P<200$ yr).

Duncan et al. (1995) and Kuchner (2002) investigated the migration of 
trans-Neptunian objects (TNOs)
to Neptune's orbit, and  Levison and Duncan (1997) studied the 
migration from Neptune's orbit to Jupiter's orbit. Ipatov and Hahn 
(1999) considered the migration of 48 Jupiter-crossing objects (JCOs) 
with initial orbits
close to the orbit of Comet P/1996 R2 and found that on average such 
objects spend $\sim$ 5000 yr
in orbits which cross both the orbits of Jupiter and Earth. Using 
these results and additional orbit integrations, and assuming that 
there are  $5\times10^9$ 1-km TNOs with 30$<$$a$$<$50 AU (Jewitt and 
Fernandez, 2001), Ipatov (2000, 2001) found that about $10^4$ 1-km 
former TNOs are Jupiter-crossers now and  10-20\% or more 1-km 
Earth-crossers could have come from the Edgeworth-Kuiper belt into 
Jupiter-crossing orbits. Note that previous estimates of the number 
of bodies with  diameter $d\ge1$ km and 30$\le$$ a$$ \le$50 AU were 
larger:  $10^{10}$ (Jewitt {\it et al.}, 1996) and $10^{11}$ (Jewitt, 
1999). In the present paper we use the estimates by Ipatov (2001), 
but now include a much  larger number of JCOs. Preliminary results 
were presented by  Ipatov (2002, 2003a-b), who also discussed the 
formation of TNOs and asteroids.


\section*{PROBABILITIES OF COLLISIONS OF NEAR-EARTH OBJECTS WITH 
PLANETS IN THE MODEL OF FIXED ORBITAL ELEMENTS}


As the actual collisions of migrating objects with terrestrial 
planets are rare, we use an approximation of random phases and 
orientations to estimate probabilities of collision for families of 
objects with similar orbital elements. We suppose that their 
semi-major axes $a$, eccentricities $e$ and inclinations $i$ are 
fixed, but the orientations of the orbits can vary. When a minor body 
collides with a planet at a distance
$R$ from the Sun, the characteristic time to collide, $T_f$, is a 
factor of $k =v/v_c = \sqrt{2a/R -1}$ times that computed with an 
approximation of constant velocity, where $v$ is the velocity at the 
point where the orbit of the body crosses the orbit of the planet, 
and $v_c$ is the velocity for the same semi-major axis and a circular 
orbit.  This coefficient $k$  modifies the formulas obtained by 
Ipatov (1988, 2000) for characteristic collision and close encounter 
times of two objects moving around the Sun in crossing orbits. These 
formulas also depend  on the synodic period and improve on 
\"Opik's formulas when the semi-major axes of the  objects are close 
to each other. As an example, at $e$=0.7 and $a$=3.06 AU,  we have 
$k$=2.26.

Based on these formulas, we calculated probabilities (1/$T_f$) 
for $\sim$ 1300 
NEOs, including  343 Venus-crossers, 756 Earth-crossers and 1197 
Mars-crossers. The  values of $T_f$ (in Myr), $k$, and the number 
$N_f$ of  objects considered are presented in Table 1. We considered 
separately the Atens, Apollos, Amors, and several Jupiter-family 
comets (JFCs). The relatively small values of $T_f$ for Atens and for 
all NEOs colliding with the Earth are due to several Atens with small 
inclinations discovered during the last three years. If we increase 
the inclination of the Aten object 2000 SG344 from $i$=$0.1^\circ$ to 
$i$=$1^\circ$, then for collisions with the Earth we find $T$=28 Myr 
and $k$=0.84 for Atens and $T$=97 Myr and $k$=1.09 for NEOs.  These 
times are much longer, and illustrate the importance of rare objects. 
%For Jupiter-family comets colliding with the terrestrial planets, 
%$k$$\ge$2. 

%For the three Halley-type comets with periods between 71 and 76~yrs,
%we obtain the characteristic times for the collisions with Earth and Mars
%far exceeding the age of the solar system.
%$T=350$ Gyr and $k=6$ (collision with Earth) and
%$T=3500$ Gyr and  $k=5$ (collision with Mars).

\begin{table}[h]
\begin{center}
\begin{minipage}{14.3cm}

\caption{Characteristic collision times $T_f$ (in Myr)
of minor bodies with planets, coefficient $k$,
and number $N_f$ of simulated objects for the set of NEOs known in 2001.}
$
\begin{array}{lccccc}
\hline
       & $Atens$     & $Apollos$    & $Amors$ &  $NEOs$ & $JFCs$ \\
\hline
$Planet$&T_f~~k~~N_f& T_f~~k~~N_f& T_f~~k~~N_f& T_f~~k~~N_f& T_f~~k\\
$Venus$ & 106~~ 1.2 ~~ 94 & 186~~ 1.7 ~~ 248& -  & 154~~ 1.5 ~~ 343& 
2900~~ 2.5 \\

$Earth$ &15~~ 0.9 ~~110  & 164~~ 1.4 ~~643& 211~~ 2.0~~1 & 67~~ 
1.1~756  &2200~~ 2.3\\

$Mars$  &475~~ 0.4~~6  & 4250~~ 0.9~~574 &5810~~ 1.1~~616 &4710~~ 1.0 
~1197& 17000~~ 1.8 \\
\hline

\end{array}
$
\end{minipage}
\end{center}
\end{table}

\section*{ORBITAL EVOLUTION OF JUPITER-FAMILY COMETS AND RESONANT ASTEROIDS }


As the next step in estimating probabilities, we calculated the 
orbits for thousands of test particles with orbits similar to known 
comets and asteroids, but having slightly different initial 
conditions.  The results confirm that most of the collision 
probability comes from a handful of very rare cases in which the test 
particle 
is Earth-crossing for an 
extended period of time.  They also show that the initial conditions 
matter more than the choice of orbit integrator.

For initial 
investigations of the migration of bodies under the gravitational 
influence of the planets, we used the integration package of Levison 
and Duncan, 1994.  In most cases we omitted the influence of Mercury 
and Pluto. Here and in Tables 2-3 
and Figs. 1, 2a-c, 3-5
we present the 
results obtained by
the Bulirsh-Stoer method (BULSTO code; Bulirsh and Stoer, 1966) with the integration step 
error 
less than
$\varepsilon$$\in$[$10^{-9}$-$10^{-8}$], and in the next 
section we compare them with those of BULSTO with 
$\varepsilon$$\le$$10^{-12}$ and  a symplectic method. 

\begin{table}[h]
\begin{center}
\begin{minipage}{17cm}

\caption{
Mean probability $P$$=$$10^{-6}P_r$ of a collision of an object with 
a planet (Venus=V, Earth=E, Mars=M)
during its lifetime,  mean time $T$ (in Kyr) during which 
$q$$<$$a_{pl}$, $T_c$=$T/P$ (in Gyr),
mean time $T_J$ (in Kyr) spent in Jupiter-crossing orbits, and ratio 
$r$ of times spent in Apollo and Amor orbits. Results from  BULSTO 
code with $\varepsilon$$\sim$$10^{-9}$-$10^{-8}$.
For $N$=7349, 2P runs were excluded.}

$ \begin{array}{lllllccccccccccc}

\hline

   & & &&&$V$ & $V$ & $V$ & $E$ & $E$ & $E$ & $M$ & $M$ & $M$ & &\\

\cline{6-14}

  & N& a& e&i&P_r & T &T_c& P_r & T &T_c& P_r & T &T_c& r & T_J \\

\hline

n1&1900& &&& 2.42 & 4.23 &1.75& 4.51 & 7.94 &1.76&  6.15 & 30.0 
&4.88& 0.7 & 119\\
$2P$& 501 &2.22 &0.85&12^\circ&141 &345 &2.45 & 110 & 397 &3.61&10.5& 
430 &41.0& 18.& 173\\
$9P$& 800&3.12 &0.52&10^\circ &1.34 &1.76 &1.31&3.72 & 4.11 
&1.10&0.71 & 9.73&13.7& 1.2 &96\\
$10P$& 2149&3.10&0.53&12^\circ &28.3 & 41.3&1.46& 35.6 & 
71.0&1.99&10.3 & 169.&16.4&1.6 &122\\
$22P$& 1000&3.47 &0.54&4.7^\circ &1.44&2.98&2.07&1.76&4.87&2.77&0.74 
& 11.0&14.9& 1.6 &116\\
$28P$& 750 &6.91&0.78&14^\circ & 1.7 & 21.8&12.8& 1.9 & 34.7&18.3& 
0.44 & 68.9&157&1.9 &443\\
$39P$& 750 &7.25&0.25&1.9^\circ & 1.06&1.72&1.62&1.19& 3.03&2.55& 
0.31 & 6.82&22 & 1.6 &94\\
\Sigma & 7349&&&& 9.52 & 16.2&1.70 & 12.6 & 27.9&2.21 &  4.89 & 
62.4&12.8 & 1.4 &112\\
\Sigma & 7850&&&& 17.9 & 37.7&2.11 & 18.8 & 51.5&2.74 &  5.29 & 
85.9&16.2 & 2.6 &116\\
\Sigma & 7852&&&& 130. & 72.6 & 0.56 & 84.5 & 95.7 & 1.13 & 13.5 & 
132.4&9.81 & 3.8 &116\\
$R2$& 24 & 3.79&0.31&2.6^\circ&0.53&0.6&1.13&2.84&1.6&0.56&6.97&14&2.0&&\\
3:1& 288 & 2.5&0.15&10^\circ &1286 &1886&1.47& 1889& 2747 &1.45& 
488&4173&8.55 & 2.7 &229\\
5:2& 288 & 2.82&0.15&10^\circ &101   &173&1.71 & 318 & 371 &1.16&209 
&1455&6.96 & 0.5 &233\\
\hline


\end{array} $
\end{minipage}
\end{center}
\end{table}


      In the first series of runs (denoted as $n1$) we calculated the 
evolution of 1900 JCOs moving in initial orbits close to those of 20 
real JCOs with period $5<P_a<9$ yr. In other series of runs, initial 
orbits were close to those of a single comet (2P, 9P, 10P, 22P, 28P, 
or 39P). For the 2P runs, we included Mercury in the integrations. We 
also investigated the orbital evolution of asteroids initially moving 
in the 3:1 and 5:2 resonances with Jupiter. For the JCOs we varied 
only the initial mean anomaly $\nu$.  The number of objects in one 
run usually was $\le$250. In most JCO cases  the time $\tau$ when 
perihelion was passed was varied with a step $d\tau$$\le$1 day
(i.e., $\nu$ was varied with a step $<$$0.2^\circ$). Near the $\tau$ 
estimated from observations, we used smaller steps. In most JCO cases 
the range of initial values of $\tau$ was less than several tens of 
days. For asteroids, we varied initial values of $\nu$ and the 
longitude of the ascending node from 0 to 360$^\circ$.
The approximate values of initial orbital elements
%($a$ in AU, $i$ in deg)
are presented in Table 2.
We initially integrated the orbits for  $T_S$$\ge$10 Myr. After 10 
Myr we tested whether some of remaining objects could reach inside 
Jupiter's orbit; if so,  the
calculations were usually continued. Therefore the results for orbits 
crossing or inside Jupiter's orbit were the same as if the 
integrations had been carried to the entire 
lifetimes of the objects. For Comet 2P and 
resonant asteroids, we integrated  until all objects were ejected 
into hyperbolic orbits or collided with the Sun. In some previous 
publications we have used smaller $T_S$, so these new data are more 
accurate.

      In our runs, planets were considered as material points so 
literal collisions did not occur.  However, using the formulas of the 
previous section, and the orbital elements sampled with a 500 yr 
step, we calculated the mean probability  $P$ of collisions. We 
define $P$ as  $P_\Sigma/N$, where $P_\Sigma$ is the probability for 
all $N$ objects of a
collision of an object with a planet during its lifetime,   the mean 
time $T$=$T_\Sigma/N$ during which
perihelion distance $q$ of an object was less than the semi-major 
axis $a_{pl}$ of the planet,
and the mean time $T_J$ during which an object moved in 
Jupiter-crossing orbits. The values of $P_r$=$10^6P$,
$T_J$ and $T$ are shown in Table 2. Here $r$ is the ratio of the 
total time interval when orbits are of Apollo
type ($a$$>$1 AU, $q$=$a(1-e)$$<$$1.017$ AU) at $e$$<$0.999 to that 
of Amor type ($1.017$$<$$q$$<$1.3 AU) and $T_c$=$T/P$ (in Gyr).
In almost all runs $T$ was equal to the mean time in planet-crossing 
orbits and 1/$T_c$ was a probability
of a collision per year (similar to 1/$T_f$).
%}
      The  results showed that most of the probability of collisions of
former JCOs with the terrestrial planets is due to a small 
($\sim$0.1-1\%) fraction that orbited for several Myr with aphelion 
$Q$$<$4.7 AU.
Some had typical asteroidal and NEO orbits and reached $Q$$<$3 AU for 
several Myr. 
Time variations in orbital elements of JCOs obtained by the BULSTO code are 
pre\-sen\-ted in Figs. 1, 2a-b. Plots in Fig. 1 are more typical than those in Fig. 2a-b, which were obtained  
for two JCOs with the highest probabilities of collisions with the terrestrial planets.
%for several JCOs and 
Fig. 2c shows the plots for an 
asteroid from the 3:1 resonance with Jupiter. 
The results obtained by a 
symplectic code for two JCOs are presented in Fig. 2d-e.
%The ratio of the mean probability of a collision of a JCO with $a$$>$1 AU
%with a planet to the mass of the planet was greater for Mars than
%that for Earth by a factor of several (
Large
values of    
$P$ for Mars in the $n1$ runs were caused by a single object with a 
lifetime of 26 Myr. 
%Results are presented in the Table below:

Total times spent by $N$ JCOs and asteroids during their lifetimes 
in orbits typical for
%Here we consider 
inner-Earth objects (IEOs, $Q$$<$0.983 AU), Aten 
($a$$<$1 AU and
$Q$$>$0.983 AU), Al2 ($q$$<$1.017 AU and 1$<$$a$$<$2 AU), Apollo, and 
Amor objects
are presented in Table 3.
These 
%short mean 
times for Earth-crossing objects were mainly due to 
a few tens of objects
   with high collision probabilities.
%Plots in Fig. 1 are more typical than those
%In Fig. 1a-b two objects with the highest probabilities of 
%collisions with the terrestrial planets are considered.
%%%
%We found that for 
Of
the JCOs with initial orbits close to those of 
10P and 2P, six and nine respectively moved into Apollo orbits with 
$a$$<$2 AU (Al2 orbits) for at least 0.5 Myr each, and five of them 
remained in such orbits for more than 5 Myr each. The contribution of 
all the other objects to Al2 orbits was smaller. 
Only one and two JCOs reached IEO and Aten orbits, respectively.
%%%
%
\begin{table}[h]
\begin{center}
\caption{
Times (in Myr) spent by $N$ JCOs and 
asteroids during their lifetimes, with results for first 50 Myr in  [ 
]
}
$ \begin{array}{lllllllll}

\hline

   & N& $IEOs$& $Aten$&  $Al2$& $Apollo$ & $Amor$ & a>5 $ AU$ \\

%\cline{6-11}
\hline
$JCOs$ & 7852 & 10 & 86 & 411 & 659 & 171 & 7100 \\

$JCOs without 2P$ & 7350 & 10 & 3.45 & 23 & 207 & 145 & 7000 \\
3:1 & 288   & 13  & 4.5 & 433 $ $[190] & 790 $ $ [540] & 290 $ $ 
[230] & 83 $ $[78] \\

5:2 & 288 & 0 & 0 & 17 $ $[2] & 113 $ $[90] & 211 $ $[90] & 253 $ $[230]\\

\hline

\end{array} $
\end{center}

\end{table}

    

One former JCO (Fig. 2a), which had an initial orbit close to that of 10P, 
moved in Aten orbits for 3.45
Myr, and the probability of its collision with the Earth from such 
orbits  was 0.344 (so $T_c$=10 Myr was even smaller than the values of $T_f$
presented in Table 1; i.e., this object had smaller $e$ and $i$
than typical observed Atens),  greater
than that  for the 7850 other simulated former JCOs during 
their lifetimes (0.15). It also moved for
about 10 Myr  in inner-Earth orbits before its collision with Venus, 
and during this time the
probability $P_V$=0.655 of its collision with Venus was greater 
($P_V$$\approx$3 for the time interval presented in Fig. 2a) 
than 
that  for the 7850 JCOs during their lifetimes (0.14).
At $t$=0.12 Myr orbital elements of this object jumped considerably
and the Tisserand parameter increased from $J$$<$3 to $J$$>$6, and $J$$>$10
during most of its lifetime.
Another object (Fig. 2b) moved in highly eccentric Aten orbits for 83 Myr, and 
its lifetime before collision
with the Sun was 352 Myr. Its probability of collisions with Earth, 
Venus and Mars during its
lifetime was 0.172, 0.224, and 0.065, respectively. These two objects 
were not included in Table 2 except for the entry for $N$=7852.
%%%
Ipatov (1995) obtained the migration of JCOs into IEO and Aten 
orbits using the approximate
method of spheres of action for taking into account the gravitational 
interactions of bodies with planets.
The mean time $T_E$ during which a JCO was moving in Earth-crossing 
orbits is $9.6\times10^4$ yr for
the 7852 simulated JCOs, and $\approx8\times10^3$ yr for the $n1$ case.
%%%
The data for Comet P/1996 R2 (line R2) were not included in the sums in Table 2.
In the present paper we consider only the integration into the future.
Ipatov and Hahn (1999) integrated the evolution of Comet P/1996 R2 both
into the future and into the past, in this case 
%for $N=48$, 
$T_E=5\times10^3$ yr.
%Earlier several scientists obtained smaller values of $P$ than those 
%presented in Table 2.
The ratio $P_S$ of the number of objects colliding with the Sun to 
the total number of
escaped (collided or ejected) objects was less than 0.015 
for the considered runs (except for 2P).
%small (except for 2P).

%+  return this text for UMD paper
% Need to place this table carefully to avoid being broken up
%\begin{table}[h]
  \begin{center}
%\begin{minipage}{17cm}
%%% \caption
{Ratio $P_S$ of objects colliding with the Sun 
  to those colliding with planets or ejected}
  $ \begin{array}{lllllll}
  \hline
  $Series$&n1 & 9P & 10P & 22P & 28P & 39P\\
  P_S & 0.0005 & 0 & 0.014 & 0.002 & 0.007 & 0 \\
  \hline
  \end{array}$
%\end{minipage}
  \end{center}
%\end{table}

Some former JCOs 
spent a long time in the 3:1 resonance with Jupiter
(Fig. 1a-b)
and with 2$<$$a$$<$2.6 AU. Other objects reached Mars-crossing orbits 
for long times. We conclude that JCOs can supply bodies to the 
regions which are considered by many scientists (Bottke et al., 2002) 
to belong to the main sources of NEOs, and that those rare objects 
that make transitions 
to typical NEO orbits
dominate the statistics.  
Only computations 
with very large numbers of objects can hope to reach accurate 
conclusions on collision probabilities 
with the terrestrial planets.


\begin{figure}
\includegraphics[width=91mm]{a95aq}
\includegraphics[width=91mm]{a95ei}
\includegraphics[width=91mm]{a146aq}
\includegraphics[width=91mm]{a146ei}
\includegraphics[width=91mm]{a115aq}
\includegraphics[width=91mm]{a115ei}
\includegraphics[width=91mm]{a38aq}
\includegraphics[width=91mm]{a38ei}
\caption{Time variations in $a$, $e$, $q$, $Q$, sin($i$) for a former JCO in 
initial orbit close to that of Comet 10P (a-f), or Comet 2P (g-h). 
Results from BULSTO code with $\varepsilon \sim 10^{-9}-10^{-8}$.} 
\end{figure}%




\begin{figure}

\includegraphics[width=60mm]{a51a}
\includegraphics[width=60mm]{a51e}
\includegraphics[width=60mm]{a51i}

\includegraphics[width=60mm]{a80a}
\includegraphics[width=60mm]{a80e}
\includegraphics[width=60mm]{a80i}

\includegraphics[width=60mm]{a135a}
\includegraphics[width=60mm]{a135e}
\includegraphics[width=60mm]{a135i}

\includegraphics[width=60mm]{a74a}
\includegraphics[width=60mm]{a74e}
\includegraphics[width=60mm]{a74i}

\includegraphics[width=60mm]{a197a}
\includegraphics[width=60mm]{a197e}
\includegraphics[width=60mm]{a197i}

%\includegraphics[width=91mm]{a51aq2}
%\includegraphics[width=91mm]{a51ei2}
%\includegraphics[width=91mm]{a80aq}
%\includegraphics[width=91mm]{a80ei}

%\includegraphics[width=91mm]{a135aq}
%\includegraphics[width=91mm]{a135ei}

%\includegraphics[width=91mm]{a74aq}
%\includegraphics[width=91mm]{a74ei}

\caption{Time variations in $a$, $e$, $q$, $Q$, 
% sin($i$) 
and $i$
for a former JCO 
in initial orbit close to that of Comet 10P (a), 2P 
%(b). 
(b, e), 9P (d), or 
an %? 
asteroid
at the 3/1 resonance with Jupiter (c). 
For (a) at $t$$<$$0.123$ Myr $Q$$>$$a$$>$1.5 AU. 
%These two objects are not included in Table 2. 
Results from 
BULSTO code with $\varepsilon \sim 10^{-9}-10^{-8}$ 
(a-c) and by a symplectic method 
with $d_s$=30 days (d) and with $d_s$=10 days (e).
%at $d_s$=10 days (c).
}

\end{figure}%



     In Fig. 3 we present the time in Myr during which objects 
had semi-major axes in an interval with a width of 0.005 AU 
(Figs. 3a-b) or 0.1 AU (Figs. 3c-d). At 3.3 AU (the 
2:1 resonance with Jupiter) there is a gap for asteroids that 
migrated from the 5:2 resonance and for former JCOs (except 2P). 
 



\begin{figure}
\includegraphics[width=91mm]{atl-c2}
\includegraphics[width=91mm]{ast} 
\includegraphics[width=91mm]{atalmc2a}
\includegraphics[width=91mm]{a100}
\caption{Distribution of migrating JCOs (a, c-d) and resonant asteroids
(b) with their semi-major axes.
The curves plotted in (c) at {\it a}=40 AU  are (top-to-bottom) for sum,
10P, n1, 39P, 22P, 9P, 28P, and 2P. For Figs. (a) and (c), designations are the
same. Results from BULSTO code with $\varepsilon \sim 10^{-9} - 10^{-8}$.} 
\end{figure}%

%\begin{figure}

%\includegraphics[width=60mm]{a197a}
%\includegraphics[width=60mm]{a197e}
%\includegraphics[width=60mm]{a197i}

%\caption{Time variations in $a$, $e$, $q$, $Q$, 
%sin($i$) 
%and $i$
%for a former JCO 
%in initial orbit close to that of Comet 2P. 
%Results from 
%a symplectic method at $d_s$=10 days.}

%\end{figure}%


\begin{figure}
\includegraphics[width=91mm]{aen10h}
\includegraphics[width=91mm]{ain10h}
\includegraphics[width=91mm]{aen10t}
\includegraphics[width=91mm]{ain10t}
\includegraphics[width=91mm]{aen2t}
\includegraphics[width=91mm]{ain2t}
\includegraphics[width=91mm]{aen39}
\includegraphics[width=91mm]{ain39}

\caption{Distribution of migrating objects in semi-major axes,
eccentricities, and inclinations for objects in initial orbits close to that of 10P (a-b),
2P (c), and 
%the resonance 3:1 (d).
39P (d). 
BULSTO code with $\varepsilon \sim 10^{-9} - 10^{-8}$.} 
\end{figure}%

\begin{figure}
\includegraphics[width=91mm]{ae31}
\includegraphics[width=91mm]{ai31}
\caption{Distribution of migrating objects in semi-major axes,
eccentricities, and inclinations for objects in initial orbits 
%close to that of 10P (a-b),
%2P (c), and 
at the 3:1 resonance with Jupiter ($e_o$=0.15, $i_o$=10$^\circ$).
%39P (d). 
BULSTO code with $\varepsilon \sim 10^{-9} - 10^{-8}$.} 
\end{figure}%

      For the $n1$ data set, $T_J$=0.12 Myr and, while moving in 
Jupiter-crossing orbits, objects had orbital periods $P_a$$<$$10$, 
10$<$$P_a$$<$20, 20$<$$P_a$$<$50, 50$<$$P_a$$<$200 yr for 11\%, 21\%, 
21\%, and 17\% of $T_J$, respectively. Therefore, there are three 
times as many  JCOs as  Jupiter-family comets (for which $P_a$$<$20 
yr).
      We also found that some JCOs, after residing in orbits with 
aphelia deep inside Jupiter's orbit,
transfer for tens of Myr to the trans-Neptunian region, either in low 
or high eccentricity orbits. We conclude that some of the main belt 
asteroids may reach typical TNO orbits, and then become 
scattered-disk objects having high eccentricities, and vice versa. 
The fraction of objects from the 5:2 resonance that collided with the 
Earth was only 1/6 of that for the 3:1 resonance. Only a small 
fraction  of the asteroids
from the 5:2 resonance reached $a$$<$2 AU (Fig. 3b).

The distributions of migrating former JCOs 
and resonant asteroids 
in $a$ and $e$ (left) and in $a$ and $i$ (right) are presented in Fig. 4-5. For each picture we considered 250 migrating objects
(288 for Fig. 5), 100 intervals for $a$, and about the same number of intervals for $e$ and $i$. 
Different designations correspond to different numbers $n$ of orbital elements 
(calculated with a step of 500 yr)  in one bin (in Fig. 4 $`$$<$=$`$ means $\le$). 
All the former JCOs  reached low eccentricity orbits 
very rarely with 2$<$$a$$<$3.5 AU and 11$<$$a$$<$28 AU. 
There were many positions of objects when their perihelia were close to a 
semi-major axis of a giant planet, mainly of Jupiter (Fig. 4a). 
Note that Ozernoy et al. (2000) considered the migration of Neptune-crossers and 
found that the main concentrations of perihelia were near Neptune's orbit. 
The pictures are different for different runs. For initial orbits close to that of Comet 2P, 
orbits often reached $90^\circ$$<$$i$$<$$180^\circ$ with various values 
of $a$$>$1.7 AU (Fig. 4c). 
For Fig. 4b (Comet 10P) the values $90^\circ$$<$$i$$<$$120^\circ$ were obtained only at 2.4$<$$a$$<$3 AU.



\section*{COMPARISON OF ORBIT INTEGRATORS}


To determine the effect of the choice of orbit integrators and 
convergence criteria, we made additional runs with BULSTO 
at $\varepsilon$=10$^{-13}$ and $\varepsilon$=10$^{-12}$ 
and with a 
symplectic integrator. 
The orbital 
evolution of 5400 JCOs was computed with the RMVS3 code. 
%For BULSTO we used $\varepsilon$=10$^{-13}$ 
%and $\varepsilon$=10$^{-12}$ instead of the $10^{-8} - 10^{-9}$ range 
%used in the previous section.  
For the  symplectic method we used an 
integration step $d_s$ of 3, 10, and 30 days.
%, and sometimes 3 days.  
We find 
that the results are not the same.
%, and conclude that the choice does make a significant difference.  
On the other hand, the differences 
between integrator choices 
(with $d_s$$\le$10 days)
are comparable to the differences between 
runs with slightly different initial conditions.  Our interpretation 
is that 1) very small numbers of particles contribute most of the 
collision probabilities
with the terrestrial planets, 2) runs with larger numbers of particles are 
more reliable, and 3) small differences in initial conditions or in 
the errors of the orbit integrators modify the trajectories 
substantially, especially for those particles making major changes in 
their orbits due to close encounters or resonances.  We conclude that 
for the purposes of this paper, the various choices of orbit 
integrator are sufficiently equivalent.
% that we should use those that 
% allow the most particles to be examined.

To illustrate these points, 
Tables 4-5  present the results obtained by
BULSTO with $\varepsilon$$\le$10$^{-12}$ and the symplectic method 
with $d_s$$\le$10 days.
%(if results were obtained for both values of $\varepsilon$, they are 
%presented in one line)
Most of the results obtained with these values of $\varepsilon$ and 
$d_s$ are statistically
similar to those obtained for $10^{-9} $$\le$$ \varepsilon$$\le$$ 
10^{-8}$. For example,
a few objects spent millions of years in Earth-crossing orbits 
inside Jupiter's orbit (Figs. 1-2), and their probabilities of collisions with 
the Earth were thousands of times greater than for more typical 
objects.
For series $n1$ the
%Comet P46 was included in the series $n1$ at $d_s$=10 days.
probability of a collision with Earth for 
one object with initial orbit close to that of Comet 46P
was 88.3\%
of the total probability for 1200 objects from this series, and the 
total probability for 1198 objects was only 4\%.
This object and the object presented in Fig. 2e and in the first line of Table 4
were not included in Table 5 with $N$=1199 for $n1$ and with $N$=250 for 2P, 
respectively.
For the 3:1 resonance  with $d_s$=10 days, 142 objects spent 140 and 
84.5 Myr in IEO and Aten orbits, respectively, even longer than
for $\varepsilon$$\sim$$10^{-9}$-$10^{-8}$.
Additionally, up to 40 Myr and 20 Myr were spent in such orbits by 
two other objects which had estimated probabilities of collisions 
with the terrestrial planets greater than 1. 
For the 2P runs with $\varepsilon$$\le$$10^{-12}$ and $N$=100, the 
calculated objects spent 5.4 Myr
in Apollo orbits with $a$$<$2 AU.
%The following table gives results for various categories of orbits.
%(the results obtained at $d_s$=10 days):

\begin{table}[h]

\begin{center}
\begin{minipage}{15.7cm}
\caption{Times (Myr) spent by three objects in various 
orbits, and probabilities of collisions
 with Venus ($p_v$), Earth 
($p_e$), and Mars ($p_m$)
during their lifetimes $T_{lt}$ (in Myr)}
$ \begin{array}{lllll lllll l}
\hline

   & d_s $ or $ \varepsilon&$IEOs$& $Aten$&  $Al2$& $Apollo$ & $Amor$ 
& T_{lt} &p_v&p_e&p_m\\
\hline
%\cline{6-11}
$2P$ & 10^d & 12 & 33.6 & 73.4 & 75.6 & 4.7 & 126 &0.18&0.68&0.07\\
%$2P$ & 10^{-12} &  &  &  &  &  &  &&ADD??&11-20\\
$46P$ & 10^d&  0 & 0    & 11.7 & 14.2 & 4.2 & 19.5 &0.02&0.04&0.002\\
%3:1 & 10^d &  & & & & & 253 &&&ADD\\
$resonance $ 3:1 & 10^{-12} & 0 & 0 & 20 & 233.5 & 10.4 & 247 
&0.008&0.013&0.0007\\

\hline

\end{array}$
\end{minipage}
\end{center}
\end{table}



%3/1 step 10 tx=190.46-50.46=140 my for 142 objects; aten 108.9-24.2=84.5

The values of $P_r$ presented in Table 5 are usually of the same order of
magnitude as those in Table 2, and the difference between the data 
presented in these tables is comparable to the differences
between different runs belonging to a series.
For Earth and Venus, the values of $P_r$ presented in both tables are
about 1--4 for 9P, 22P, 28P and 39P.  For 28P and 39P with the 
symplectic method
$P_r$ is about twice that for BULSTO. For 10P they are
several times larger than  for the above series, and 
for 2P they are several times larger than for 
10P. For the $n1$  run, $P_r$$>$4 for Earth.
The ratio of $P_r$ to the mass of the planet was typically several 
times larger for Mars than for Earth and Venus. The main difference 
in $P_r$ was found for the 3:1 resonance.
In this case greater values of $P_r$ were obtained for $d_s$=10 days. 
As noted above, a few exceptional objects dominated  the 
probabilities, and 
for the 3:1 resonance two objects, which had collision probabilities 
greater than unity for the terrestrial planets, were not included 
in Table 5.  These two objects can  increase the total value of $P_r$ 
for Earth by a factor of several.
%Time variations of orbital elements for Comet 2P obtained at $d_s$=10 days are 
%presented in Fig. 2e.

We also did some symplectic runs with $d_s$=30 days. For most of the 
objects we got similar results, but
about 0.1\%  of the
objects    reached Earth-crossing orbits with $a$$<$2 AU for several 
tens of Myr
(e.g., Fig. 2d) 
and even IEO orbits. These few bodies increased the 
mean value of $P$ by a factor of more than 10. With $d_s$=30 days, 
four  objects from the runs $n1$, 9P, 10P had a probability of 
collisions
with the terrestrial planets greater than 1 for each, and for 2P 
there were 21 such objects among 251 considered.
For resonant asteroids, we also obtained much larger values than 
those for BULSTO for  $P$ and $T$ for RMVS3 with 
$d_s$=30 days, and similarly  for the 3:1 resonance even with $d_s$=10 
days. For this resonance it may be
better to use $d_s$$<$10 days.
Probably, the results of symplectic runs with $d_s$=30 days can be considered as 
such migration that includes some nongravitational forces.



%===========fig

\begin{table}[h]
\begin{center}
\begin{minipage}{17cm}

\caption{
%Values of $T$ and $T_J$ (in Kyr), $T_c$ (in Gyr), $P_r$, and $r$ 
%obtained using
%a symplectic integrator or the BULSTO code at small $\varepsilon$
%(Venus=V, Earth=E, Mars=M).
Probabilities of collisions with the terrestrial planets.
Designations are same as those for Table 2.
Results
from
the BULSTO code with $\varepsilon$$\le$$10^{-12}$ and  a symplectic method
with $d_s$$\le$10 days.}



$ \begin{array}{lllcc ccccc cccc}

\hline

    &&&$V$ & $V$ & $V$ & $E$ & $E$ & $E$ & $M$ & $M$ & $M$ & &\\

%\cline{4-12}

  &\varepsilon $ or $ d_s& N&P_r & T &T_c& P_r & T &T_c& P_r & T &T_c& 
r & T_J \\

\hline

n1&\le10^d&1200&25.4 & 13.8 &0.54& 40.1 & 24.0 &0.60&2.48 & 35.2 
&14.2& 3.0 & 117\\
n1&\le10^d&1199&7.88 & 9.70 &1.23& 4.76 & 12.6 &2.65&0.76 & 16.8 
&22.1& 2.8 & 117\\
$2P$&\le10^{-12}&100 &321&541&1.69&146&609&4.2   &14.8 & 634 &42.8 & 27.&20\\
$2P$&10^d & 251 &860&570& 0.66 & 2800 &788 & 0.28 &294 & 825 & 2.81& 22.&0.29\\
$2P$&10^d & 250 &160&297& 1.86 & 94.2 & 313 & 3.32 & 10.0&324 &32.3&35.1&0.29\\
$9P$&10^d & 400 &1.37&3.46&2.53&3.26&7.84&2.40&1.62&23.8&14.7& 1.1 &128\\
$10P$&\le10^d &450&14.9&30.4&2.04&22.4&41.3&1.84&6.42& 113.&17.6&1.5&85\\
$22P$&10^d & 250& 0.68 &2.87&4.23&1.39&4.96&3.57&0.60 & 11.5&19.2& 1.5 &121\\
$28P$&10^d & 250 & 3.87 & 35.3& 9.12& 3.99 & 59.0 &14.8& 0.71 & 109. 
&154&2.2 &535\\
$39P$&10^d & 250 & 2.30 &2.68& 1.17 &2.50& 4.22 & 1.69 & 0.45 & 7.34& 
16.3 & 2.2 &92\\
3:1&\le10^{-12}& 70 & 1162 &1943&1.67& 1511& 5901 &3.91& 587& 
803&1.37& 4.6 &326\\
3:1&10^d& 142 &27700&8617&0.31& 2725& 9177& 
3.37& 1136&9939&8.749& 16.4&1244\\
5:2&\le10^{-12}& 50 & 130   &113&0.87& 168 & 230 &1.37&46.2 &507 
&11.0& 1.4 &166\\
%3:1&10^d& 144? &for50 &1886&1.47& 1889& 2747 &1.45& 488&4173&8.55 & 2.7 &229\\
5:2&10^d& 144 & 58.6&86.8&1.48&86.7&174 &2.01&17.&354.8&20.9&1.73&224\\

\hline

\end{array} $
\end{minipage}
\end{center}
\end{table}



      In the case of close encounters with the Sun (Comet 2P and 
resonant asteroids), the probability
$P_S$ of collisions with the Sun  
during lifetimes of objects
was larger for RMVS3 than for BULSTO, % ','
and for
$10^{-13} $$\le$$ \varepsilon $$\le$$ 10^{-12}$ it was greater than 
for $10^{-9}$$ \le$$ \varepsilon $$\le$$ 10^{-8}$
($P_S$=0.75 for the 3:1 resonance % 3:1
with $d_s$=3 days). This probability is presented in Table 6 for several runs.

\begin{table}[h]

\begin{center}
\caption{Probability of collisions with the Sun}
$ \begin{array}{lllllll}
\hline
  & \varepsilon=10^{-13} & \varepsilon=10^{-12} & \varepsilon=10^{-9} 
& \varepsilon=10^{-8} & d_s=10 $ days$ & d_s=30 $ days$ \\
\hline
$Comet 2P$& 0.88 & 0.88 & 0.38 & 0.32& 0.99 & 0.8 \\

$resonance $ 3:1 &0.46& 0.5& 0.156 & 0.112 & 0.741 & 0.50 \\

$resonance $ 5:2 & &0.06& 0.062 & 0.028 & 0.099 & 0.155\\
\hline
\end{array} $

\end{center}
\end{table}

For Comet 2P the values of $T_J$ were much smaller for RMVS3 than 
those for BULSTO and they were smaller
for smaller $\varepsilon$; for other runs these values do not depend 
much on the method.
In our opinion, the most reliable values of $P_S$ and $T_J$ were 
obtained with $10^{-13} $$\le$$ \varepsilon$$ \le$$ 10^{-12}$.
In the direct integrations reported by Valsecchi et al. (1995), 13 of 
the 21 objects fell into the Sun,
so their value of $P_S$=0.62 is in accordance with our results 
obtained by BULSTO; it is less than
that for $\varepsilon$=$10^{-12}$, but greater than  for 
$\varepsilon$=$10^{-9}$.
Note that even for different $P_S$ the data presented in Tables 2 and 
5 usually are similar. As we did not calculate
collision probabilities of objects with planets by direct 
integrations, but instead calculated them with the random phase 
approximation from the
orbital elements, we need not make integrations with extremely high 
accuracy. Ipatov (1988b) showed that for BULSTO
the integrals of motion were conserved better and the plots of 
orbital elements for closely separated values of  $\varepsilon$ were
closer to one another with  $10^{-9} $$\le$$ \varepsilon $$\le$$ 
10^{-8}$. The smaller the value of $\varepsilon$, the more
integrations steps are required, so $\varepsilon$$\le$$10^{-12}$ for 
large time intervals are not necessarily better than those for 
$10^{-9} $$\le$$ \varepsilon $$\le$$ 10^{-8}$. Small $\varepsilon$ is 
clearly
necessary for close encounters. Ipatov and Hahn (1999) and Ipatov 
(2000) found that former JCOs
reached resonances more often for BULSTO than for RMVS3 with $d_s$=30 days. Therefore we
made most of our  BULSTO runs with  $10^{-9} $$\le$$ \varepsilon 
$$\le$$ 10^{-8}$.
For a symplectic method it is better to use smaller $d_s$ at a 
smaller distance $R$ from the Sun,
but in some runs  $R$ can vary considerably during the evolution.


%For RMVS3, former JCOs got resonant orbits less often than for BULSTO and more
%often collided with the Sun. For example, for Comet 2P Encke 
%$P_S$=0.99 at $d_s$=10 days instead of $P_S$=0.35 for BULSTO.
%Using a symplectic method, Levison and Duncan (1994) showed that 
%Comet 2P became sungrazing at $t$=0.05 Myr,
%and at the direct integrations by Valsecchi et al. (1995) 13 of the 
%21 such objects fell into the Sun.


\section*{MIGRATION FROM BEYOND JUPITER TO THE TERRESTRIAL PLANETS}


According to Duncan et al. (1995), the fraction $P_{TNJ}$ of TNOs reaching
Jupiter's orbit under the influence of the giant planets in 1 Gyr is
0.8-1.7\%. As the mutual gravitational influence of TNOs can play a 
lar\-ger role in variations of their orbital elements than collisions 
(Ipatov, 2001), we 
considered the upper value of
%estimated the probabilities  
$P_{TNJ}$.
% for these objects to reach other orbits. 
Using the total of $5\times10^9$ 1-km 
TNOs with $30$$<$$a$$<$$50$ AU,  and assuming
that the mean time for a body to move in a Jupiter-crossing orbit is 
0.12 Myr, we find that about $N_{Jo}$=$10^4$ 1-km former TNOs are now 
Jupiter-crossers, and 3000 are Jupiter-family comets.
Using the total times spent by 7350 (not including 
our 2P runs) and 7852 simulated JCOs in various orbits, we obtain the following 
numbers of 1-km former TNOs now moving in several types of orbits:
%\begin{table}[h]
\begin{center}
%\caption{  }
$ \begin{array}{cccccc}
\hline
$$N$$ & $IEOs$ & $Aten$ & $Al2$ & $Apollo$ & $Amor$ \\
\hline
$7350$ & 120 &  40 & 250 & 2500 & 1750\\
$7852$ & 110 & 950 & 4500 & 7200 & 1880\\
\hline

\end{array} $
\end{center}
%\end{table}

For example, the number of IEOs $N_{IEOs}$=$N_{Jo}t_{IEO}$/$(N_J t_J)$,
where $t_{IEO}$ is the total time during which $N_J$ former JCOs 
moved in IEO orbits, and $N_J t_J$ is the total time during which 
$N_J$ JCOs moved
in Jupiter-crossing orbits.
As we considered mainly the %?
runs with relatively high migration to the Earth, 
the actual above 
values are smaller by a factor of several, the actual portion
of IEOs and Atens % objects
can be smaller and that for Amors
  can be larger than those for our runs.
Even if the number of Apollo objects is an order of magnitude smaller
than the above value, it may still be comparable to the real number 
(750) of 1-km
Earth-crossing objects (half of them are in orbits with $a$$<$2 AU), 
although the
latter number does not include those in highly eccentric orbits.
%Eccentricities and inclinations of NEOs in our runs were larger than those for
%real NEOs, and probably most of 1-km former TNOs (extinct comets) 
%now moving in
%NEOs' orbits with high eccentricities and inclinations have not yet been
%discovered, as most of the time they move relatively far from the Earth.
%The values of $T_c$ for considered resonant asteroids and former 
%JCOs  (see Tables 2-3) didn't differ much.
%It means that they had similar $e$ and $i$ while moving in 
%Earth-crossing orbits.
%As such values of $T_c$ were greater than those for the observed 
%NEOs, the mean inclinations and eccentricities
%of unobserved NEOs can be greater than those for discovered NEOs 
%(such result was also obtained by Jedicke et al., 2003).

The values of the characteristic time (usually $T_c$) for the 
collision of a former JCO or a resonant
asteroid with a planet (see Tables 2 and 5) are greater than the values 
of $T_f$ for NEOs in Table 1,
%(for the Earth $T_c=T/P\approx2.7$ Gyr for 7850 considered JCOs and 
$T_c$$\approx$1.1 Gyr for 7852 objects,
%while for observed Earth-crossing objects it is $\approx$100 Myr),
so we expect that the mean inclinations and eccentricities
of unobserved NEOs are greater than those for the NEOs that are 
already known.   Jedicke et al., (2003) found similar results.
% because our migrating JCOs and resonant asteroids moved in more 
%inclined and eccentric orbits than observed NEOs.
On average, the values of $T_c$ for our $n1$ series  and for most of 
our simulated JCOs were not greater than those for our calculated 
asteroids, and migrating Earth-crossing objects had similar $e$ and 
$i$ for  both former JCOs and resonant asteroids (see e.g. Fig. 4-5).
Former JCOs, which move in Earth-crossing orbits for more than 1 Myr, 
while moving in such orbits, usually had larger $P$ and smaller $e$ and $i$
(sometimes similar to those of the observed NEOs, see Figs. 1-2).
It is easier to observe orbits with smaller values of $e$ and $i$, and 
probably, many of the NEOs moving in orbits with large values of $e$
%(especially with $e$$\approx$1) 
and $i$ have not yet been discovered.
About 1\% of the observed Apollos cross Jupiter's orbit, and an 
additional 1\% of Apollos have aphelia between 4.7-4.8 AU,
but these Jupiter-crossers are far from the Earth  most of time, so 
their actual fraction of ECOs is greater
than  for observed ECOs. The fraction of Earth-crossers among 
observed Jupiter-family comets is about 10\%.
This is a little more than $T/T_J$ for our $n1$ runs, but less than 
for 7850 JCOs.
For our former resonant asteroids, $T_J$ is relatively large 
($\approx$0.2 Myr), and such asteroids can reach cometary orbits.

Comets are estimated to be active for $\sim10^3$--$10^4$ yr. Some 
former comets can move for tens or
even hundreds of Myr in NEO orbits,  so the number of extinct comets 
can exceed the number of active comets by several orders of magnitude.
The mean time spent by Encke-type objects in Earth-crossing orbits is 
about 0.4 Myr. This time corresponds to 40-400  extinct comets of 
this type. Note that the diameter of Comet 2P Encke
is about 5-10 km, so the number of smaller extinct comets can be much larger.

      The above estimates of the number of NEOs are approximate. For 
example, it is possible that the number of 1-km TNOs is several times 
smaller than $5\times10^9$,
while some scientists estimated that this number can be up to $10^{11}$. Also,
the fraction of TNOs that have migrated towards the Earth might be 
smaller. On the other hand, the above
number of TNOs was estimated for $a$ $<$ 50 AU, and TNOs from more 
distant regions can also migrate inward.
The Oort cloud  could also supply Jupiter-family comets. According to 
Asher et al. (2001), the rate of a cometary
object decoupling from the Jupiter vicinity and transferring to an 
NEO-like orbit is increased by a
factor of 4 or 5 due to nongravitational effects
(see also Fernandez and Gallardo, 2002). This would result 
in  larger values of $P_r$ and $T$
than those shown in Tables 2 and 5.
%We consider that collisions of JCOs with asteroids could increase 
%the number of
%objects migrating to the near-Earth space, especially to orbits with $a<2$ AU.
%     The less are the masses of objects, the more often they can 
%change their orbits due to collisions with smaller bodies.
%     Considering collision probability of an objects with radius of 
%1 km with a smaller object
%     to be $10^{-18}$ and $10^9$ impactors (probably, it is the 
%number of asteroids with $d$=25 m),
%     we obtain that the probability of such collision during 0.1 Myr 
%is 0.0001.
      Our estimates show that, in principle, the trans-Neptunian belt 
can provide a significant portion of the Earth-crossing objects, 
although many NEOs clearly came from the main asteroid belt. It may 
be possible to explore former TNOs near the Earth's orbit without 
sending spacecraft to the trans-Neptunian region.

      More than half of the close encounters of active comets with the 
Earth belong to long-period comets, which amount to about 80\% of the 
known population. Thus, though probabilities are smaller for larger 
eccentricities, the number of collisions of both long-period and 
short-period active comets with the inner planets can be of the same 
order of magnitude. According to our results, many
former Jupiter-family comets can have orbits typical of asteroids, 
and collide with the Earth from typical NEO orbits.

      Based on the estimated collision probability $P=4\times10^{-6}$ 
we find that  1-km former TNOs now collide with the Earth once in 
0.75 Myr. This value of $P$ is smaller
than that for our $n1$ runs, and is only 1/20 of that for our 7852 JCOs.
Assuming the total mass of planetesimals that ever crossed Jupiter's 
orbit is $\sim$$ 100m_\oplus$, where
$m_\oplus$ is the mass of the Earth (Ipatov, 1993, 2000), we conclude 
that the total mass of bodies that impacted the
Earth is $4\times10^{-4} m_\oplus$. If ices comprised only half of this mass,
then the total mass of ices $M_{ice}$ that were delivered to the 
Earth from the feeding zone of
the giant planets is about the mass of the terrestrial oceans 
($\sim2\times10^{-4} m_\oplus$).

      The calculated probabilities of collisions of objects with 
planets show that the fraction of the mass of the planet  delivered 
by short-period comets can be greater for Mars and Venus than for the 
Earth 
(compare the values of $P$$/$$m_{pl}$
using $P$ from Tables 2 and 5). 
This larger mass fraction would result in 
relatively large ancient oceans on Mars and Venus. On the other hand, 
there is the deuterium/hydrogen paradox of Earth's oceans, as the D/H 
ratio is different for oceans and comets.  Pavlov et al. (1999) 
suggested that solar wind-implanted hydrogen on interplanetary dust 
particles could provide the necessary low-D/H component of Earth's 
water inventory.

Our estimate of the migration of water to the early Earth is in 
accordance with Chyba (1989), but
is greater than those of Morbidelli et al. (2000) and Levison et al. (2001).
%Levison et al. (2001) concluded
%that the Earth has accreted only 5\% of its oceans from bodies that migrated
%from the feeding zones of Uranus and Neptune.
%Morbidelli et al. (2000) and Levison et al. (2001)
The latter obtained smaller
values of $M_{ice}$, and we suspect that this is because they did not 
take into account the migration of bodies into orbits with $Q$$<$4.5 
AU.
Perhaps this was
because they modeled a relatively small number of objects, and 
Levison et al. (2001) did not take into
account the influence of the terrestrial planets. In our runs the 
probability of a collision of a single object
with a terrestrial planet could be much greater than the total 
probability of thousands of other objects, so the statistics are 
dominated by rare occurrences that might not appear in smaller 
simulations.
The mean probabilities of collisions can  differ by orders of 
magnitude for different JCOs.
Other scientists considered other initial objects and smaller numbers 
of Jupiter-crossing objects, and
  did not find decoupling from Jupiter, which is a rare event.
We believe there is no contradiction between our present results and 
the smaller
migration of former JCOs to the near-Earth space that was obtained in 
earlier work,
including our own papers (e.g. Ipatov and Hahn, 1999), where we used 
the same integration package.

There is additional evidence from the classification of objects based 
on their albedos. From measured albedos, Fernandez et al. (2001) 
concluded that
the fraction of extinct comets among NEOs and unusual asteroids is significant
(at least 9\% are candidates).
The idea that there may be many extinct comets among NEOs was 
considered by several scientists.
Rickman et al. (2001) believed that comets
played an important and perhaps even dominant role among all km-size 
Earth impactors.
In their opinion, dark spectral classes that might include the 
ex-comets are severely underrepresented
(see also Jewitt and Fernandez, 2001).
Our runs showed that if one observes
former comets in NEO orbits, then it is  probable that they have already
moved in such orbits for millions (or at least hundreds of thousands) 
years, and only a few of them have been
in such orbits for short times (a few thousand years).
Some former comets that have moved in typical NEO orbits for millions 
or even hundreds of millions of years,
and might have had
multiple close encounters with the Sun
(some of these encounters can be very close to the Sun, e.g.
in the case of Comet 2P at $t$$>$0.05 Myr), could have lost their mantles,
which causes their low albedo, and so change their albedo
and would look like typical asteroids.
Typical comets have larger rotation periods than typical NEOs 
(Binzel et al., 1992, Lupishko and Lupishko, 2001),
but, while losing considerable portions of their masses, extinct 
comets can decrease these periods. In future we plan to consider a 
larger initial number of objects initially located beyond Jupiter
in order to better estimate their
probabilities of migration to a near-Earth space.


\section*{CONCLUSIONS}

Collision statistics for the terrestrial planets
are dominated by very small numbers of bodies 
that reach orbits with high collision probabilities, so it is 
essential to consider very large numbers of particles. The initial 
conditions for the orbit integrations appear to matter more than the 
choice of orbit integrator.  Some Jupiter-family comets can reach 
typical NEO orbits and remain there for millions of years. While the 
probability of such events is small
(about 0.1\% in our runs and perhaps smaller for other initial data), 
nevertheless the majority of collisions of former JCOs with the 
terrestrial planets are due to such objects.  The amount of water 
delivered to the Earth during planet formation could be about the 
mass of the Earth oceans. From the dynamical point of view there 
could be (not `must be`) many extinct comets among the NEOs. For better 
estimates of the portion of extinct comets among NEOs
 we will need orbit
integrations for many more TNOs and JCOs, and wider analysis of observations and craters.  

This paper is an extended version of the paper submitted to {\it Advances in Space Research}. 

\section*{ACKNOWLEDGMENTS}

      This work was supported by NRC (0158730), NASA (NAG5-10776), 
INTAS (00-240), and RFBR (01-02-17540).
      For preparing some data for figures we used some subroutines 
written by P. Taylor.
We are thankful to W. F. Bottke, S. Chesley, and H. F. Levison for 
helpful remarks.

\begin{thebibliography}{}


\item Asher, D. J., M. E. Bailey, and D. I. Steel, The role of 
non-gravitational forces in decoupling orbits from Jupiter, in {\it 
Collisional Processes in the Solar System}, edited by M.Ya. Marov and 
H. Rickman, ASSL, {\bf 261}, 121-130, 2001.

\item Binzel, R. P., S. Xu, S. J. Bus, and E. Bowell, Origins for the 
near-Earth asteroids, {\it Science}, {\bf 257}, 779-782, 1992.

\item Bottke, W. F., A. Morbidelli, R. Jedicke, et al., Debiased 
orbital and absolute magnitude
distribution of the near-Earth objects, {\it Icarus}, {\bf 156}, 399-433, 2002.

\item Bulirsh, R., and J. Stoer, Numerical treatment of ordinary differential
Equations by Extrapolation Methods, {\it Numer. Math.}, {\bf 8}, 1-13, 1966.

\item Chyba, C. F., Impact delivery and erosion of planetary oceans 
in the early inner solar system, {\it Nature}, {\bf 343}, 129-132, 
1989.

\item Duncan, M. J., and H. F. Levison, A disk of scattered icy 
objects and the origin of Jupiter-family comets, {\it Science}, {\bf 
276}, 1670-1672, 1997.

\item Duncan, M. J., H. F. Levison, and S. M. Budd, The dynamical 
structure of the Kuiper belt, {\it Astron. J.}, {\bf 110}, 3073-3081, 
1995.

%\item Eneev, T. M., On Possible Structure of Outer (Trans-Neptunian) 
%Regions of the Solar System, {\it Sov. Astron. Letters}, {\bf 6}, 
%pp. 295-300 in Russian %edition, 1980.

%\item Harris, N. W., and Bailey, M. E., Dynamical Evolution of
%Cometary Asteroids, {\it Mon. Not. R. Astron. Soc.}, {\bf 297}, 
%1227-1236, 1998.

\item Fernandez, J. A. and T. Gallardo, Are there many inactive 
Jupiter-family comets among
the near-Earth asteroid population, {\it Icarus}, {\bf 159}, 358-368, 2002.

\item Ipatov, S. I., Evolution times for disks of planetesimals, {\it 
Soviet Astronomy}, {\bf 32 (65)}, N 5, 560-566, 1988a.

\item Ipatov, S. I., The gap problem and asteroid-type resonant orbit 
evolution, {\it Kinematics
and Physics of Celestial Bodies}, {\bf 4}, N 4, 49-57, 1988b.

\item Ipatov, S. I., Migration of small bodies to the Earth, {\it 
Solar System Research}, {\bf 29}, 261-286, 1995.

\item Ipatov, S. I., Migration of trans-Neptunian objects to the
Earth, {\it Celest. Mech. Dyn. Astron.}, {\bf 73}, 107-116, 1999.

\item Ipatov, S. I., {\it Migration of Celestial Bodies in the Solar 
System}, Editorial URSS Publishing Company, Moscow, 2000. (In Russian)

\item Ipatov, S. I., Comet hazard to the Earth, {\it Advances in 
Space Research}, Elsevier, {\bf 28}, 1107-1116, 2001.

\item Ipatov, S. I., Migration of matter from the Edgeworth--Kuiper 
and main asteroid belts to the Earth, in {\it
%Proc. of the IAU Colloquium No 181 and COSPAR Colloquium No. 11
Dust in the Solar System and
%\underline{
Other Planetary Systems},
%(April 10-14, 2000, Canterbury, UK)},
edited by S. F. Green, et al.,
%I. P. Williams, J. A. M. McDonnell and N. McBride,
COSPAR Colloquia Series, Pergamon, {\bf 15}, 233-236 
(http://arXiv.org/format/astro-ph/0205250), 2002.

\item Ipatov, S. I., Formation and migration of trans-Neptunian 
objects, in press, in %\underline{Proc. of the intern. conference}
{\it Scientific Frontiers in Research of
Extrasolar Planets},
%(18-21 June 2002, Washington D.C., USA)},
edited by D. Deming and S. Seager, ASP Conference Series 
(http://arXiv.org/\-format/astro-ph/0210131), 2003a.

\item Ipatov, S. I., Formation and migration of trans-Neptunian 
objects and asteroids, in press, in
%\underline {Proc. of the conference}
{\it Asteroids, Comets, Meteors, 2002}
%(July 29 - August 2, 2002, Berlin)},
(http://arXiv.org/format/astro-ph/0211618),
2003b.

\item Ipatov, S. I. and G. J. Hahn, Orbital evolution of the P/1996 
R2 and P/1996 N2 objects, {\it Solar System Research}, {\bf 33}, 
487-500, 1999.

\item Jedicke R., A. Morbidelli, T. Spahr, et al., Earth and 
space-based NEO survey simulations:
prospects for achieving the Spaceguard Goal, {\it Icarus}, {\bf 161}, 
17-33, 2003.

\item Jewitt, D., Kuiper belt objects, {\it Ann. Rev. Earth. Planet. 
Sci.}, {\bf 27}, 287-312, 1999.

\item {Jewitt, D., Luu, J., and Chen, J. }, The Mauna
Kea--Cerro--Tololo (MKCT) Kuiper belt and Centaur survey, {\it 
Astron. J.}, {\bf
112}, 1225-1238, 1996.

\item Jewitt, D., and Y. Fernandez, Physical properties of planet-crossing
objects, in {\it Collisional Processes in the Solar System}, edited 
by M. Ya. Marov and H. Rickman, ASSL, {\bf 261},  143-161, 2001.

\item Kuchner, M. J., M. E. Brown, and M. Holman, Long-term dynamics 
and the orbital inclinations of the classical Kuiper belt objects, 
{\it Astron. J.}, {\bf 124}, 1221-1230, 2002.

\item Levison H. F. and M. J. Duncan, The long-term dynamical 
behavior of short-period comets, {\it Icarus}, {\bf 108}, 18-36, 1994.

\item Levison H. F. and M. J. Duncan, From the Kuiper belt to 
Jupiter-family comets: The spatial distribution of ecliptic comets, 
{\it Icarus}, {\bf 127}, 13-23, 1997.

\item Lupishko D. F. and T. A. Lupishko, On the origins of 
Earth-approaching asteroids, {\it Solar System Research},
{\bf 35}, 227-233, 2001.

\item Morbidelli, A., J. Chambers, J. I. Lunine, et al.,
Source regions and timescales for the delivery of water to the Earth, 
{\it Meteoritics \& Planetary Science}, {\bf 35}, 1309-1320, 2000.

%\item Morbidelli, A., and D. Nesvorny, Numerous weak resonances 
%drive asteroids %Toward Terrestrial Planets Orbits, {\it Icarus}, 
%{\bf 139}, 295-308, 1999.

\item Ozernoy, L. M., N. N. Gorkavyi, and T. Taidakova, Four 
cometary belts associated with the orbits of giant planets: a new 
view of the outer solar system's structure emerges from numerical 
simulations, {\it Planetary and Space Science}, {\bf 48}, 993-1003, 
2000.

\item Pavlov, A. A., A. K. Pavlov, J. F. Kasting, Irradiated 
interplanetary dust particles as a possible solution for the 
deuterium/hydrogen paradox of Earth's oceans, {\it Journal of 
Geophysical research}, {\bf 104}, No. E12, 30,725-30,728, 1999.

\item Rickman, H., J. A. Fernandez, G. Tancredi, J. Licandro, The 
cometary contribution to planetary impact craters, in {\it 
Collisional Processes in the Solar System}, edited by M. Ya. Marov 
and H. Rickman, ASSL, {\bf 261}, 131-142, 2001.

\item   Shoemaker, E. M., P. R. Weissman, and C. S. Shoemaker, The flux of
periodic comets near Earth, in {\it Hazards Due to Comets and 
Asteroids}, edited by T. Gehrels, pp. 313-335, The University of 
Arizona Press, Tucson \& London,  1994.

\item Valsecchi, G. B., A. Morbidelli, R. Gonczi, et al., The 
dynamics of objects in orbits resembling that of P/Encke, {\it 
Icarus}, {\bf 118}, 169-180, 1995.


\end{thebibliography}
\centerline{}

%E-mail address of S. I. Ipatov:  siipatov@hotmail.com, sipatov@gmu.edu

%Manuscript received ......; accepted ......

\end{document}








astro-ph0303219.extracted_bib
\begin{thebibliography}{}


\item Asher, D. J., M. E. Bailey, and D. I. Steel, The role of 
non-gravitational forces in decoupling orbits from Jupiter, in {\it 
Collisional Processes in the Solar System}, edited by M.Ya. Marov and 
H. Rickman, ASSL, {\bf 261}, 121-130, 2001.

\item Binzel, R. P., S. Xu, S. J. Bus, and E. Bowell, Origins for the 
near-Earth asteroids, {\it Science}, {\bf 257}, 779-782, 1992.

\item Bottke, W. F., A. Morbidelli, R. Jedicke, et al., Debiased 
orbital and absolute magnitude
distribution of the near-Earth objects, {\it Icarus}, {\bf 156}, 399-433, 2002.

\item Bulirsh, R., and J. Stoer, Numerical treatment of ordinary differential
Equations by Extrapolation Methods, {\it Numer. Math.}, {\bf 8}, 1-13, 1966.

\item Chyba, C. F., Impact delivery and erosion of planetary oceans 
in the early inner solar system, {\it Nature}, {\bf 343}, 129-132, 
1989.

\item Duncan, M. J., and H. F. Levison, A disk of scattered icy 
objects and the origin of Jupiter-family comets, {\it Science}, {\bf 
276}, 1670-1672, 1997.

\item Duncan, M. J., H. F. Levison, and S. M. Budd, The dynamical 
structure of the Kuiper belt, {\it Astron. J.}, {\bf 110}, 3073-3081, 
1995.

%\item Eneev, T. M., On Possible Structure of Outer (Trans-Neptunian) 
%Regions of the Solar System, {\it Sov. Astron. Letters}, {\bf 6}, 
%pp. 295-300 in Russian %edition, 1980.

%\item Harris, N. W., and Bailey, M. E., Dynamical Evolution of
%Cometary Asteroids, {\it Mon. Not. R. Astron. Soc.}, {\bf 297}, 
%1227-1236, 1998.

\item Fernandez, J. A. and T. Gallardo, Are there many inactive 
Jupiter-family comets among
the near-Earth asteroid population, {\it Icarus}, {\bf 159}, 358-368, 2002.

\item Ipatov, S. I., Evolution times for disks of planetesimals, {\it 
Soviet Astronomy}, {\bf 32 (65)}, N 5, 560-566, 1988a.

\item Ipatov, S. I., The gap problem and asteroid-type resonant orbit 
evolution, {\it Kinematics
and Physics of Celestial Bodies}, {\bf 4}, N 4, 49-57, 1988b.

\item Ipatov, S. I., Migration of small bodies to the Earth, {\it 
Solar System Research}, {\bf 29}, 261-286, 1995.

\item Ipatov, S. I., Migration of trans-Neptunian objects to the
Earth, {\it Celest. Mech. Dyn. Astron.}, {\bf 73}, 107-116, 1999.

\item Ipatov, S. I., {\it Migration of Celestial Bodies in the Solar 
System}, Editorial URSS Publishing Company, Moscow, 2000. (In Russian)

\item Ipatov, S. I., Comet hazard to the Earth, {\it Advances in 
Space Research}, Elsevier, {\bf 28}, 1107-1116, 2001.

\item Ipatov, S. I., Migration of matter from the Edgeworth--Kuiper 
and main asteroid belts to the Earth, in {\it
%Proc. of the IAU Colloquium No 181 and COSPAR Colloquium No. 11
Dust in the Solar System and
%\underline{
Other Planetary Systems},
%(April 10-14, 2000, Canterbury, UK)},
edited by S. F. Green, et al.,
%I. P. Williams, J. A. M. McDonnell and N. McBride,
COSPAR Colloquia Series, Pergamon, {\bf 15}, 233-236 
(http://arXiv.org/format/astro-ph/0205250), 2002.

\item Ipatov, S. I., Formation and migration of trans-Neptunian 
objects, in press, in %\underline{Proc. of the intern. conference}
{\it Scientific Frontiers in Research of
Extrasolar Planets},
%(18-21 June 2002, Washington D.C., USA)},
edited by D. Deming and S. Seager, ASP Conference Series 
(http://arXiv.org/\-format/astro-ph/0210131), 2003a.

\item Ipatov, S. I., Formation and migration of trans-Neptunian 
objects and asteroids, in press, in
%\underline {Proc. of the conference}
{\it Asteroids, Comets, Meteors, 2002}
%(July 29 - August 2, 2002, Berlin)},
(http://arXiv.org/format/astro-ph/0211618),
2003b.

\item Ipatov, S. I. and G. J. Hahn, Orbital evolution of the P/1996 
R2 and P/1996 N2 objects, {\it Solar System Research}, {\bf 33}, 
487-500, 1999.

\item Jedicke R., A. Morbidelli, T. Spahr, et al., Earth and 
space-based NEO survey simulations:
prospects for achieving the Spaceguard Goal, {\it Icarus}, {\bf 161}, 
17-33, 2003.

\item Jewitt, D., Kuiper belt objects, {\it Ann. Rev. Earth. Planet. 
Sci.}, {\bf 27}, 287-312, 1999.

\item {Jewitt, D., Luu, J., and Chen, J. }, The Mauna
Kea--Cerro--Tololo (MKCT) Kuiper belt and Centaur survey, {\it 
Astron. J.}, {\bf
112}, 1225-1238, 1996.

\item Jewitt, D., and Y. Fernandez, Physical properties of planet-crossing
objects, in {\it Collisional Processes in the Solar System}, edited 
by M. Ya. Marov and H. Rickman, ASSL, {\bf 261},  143-161, 2001.

\item Kuchner, M. J., M. E. Brown, and M. Holman, Long-term dynamics 
and the orbital inclinations of the classical Kuiper belt objects, 
{\it Astron. J.}, {\bf 124}, 1221-1230, 2002.

\item Levison H. F. and M. J. Duncan, The long-term dynamical 
behavior of short-period comets, {\it Icarus}, {\bf 108}, 18-36, 1994.

\item Levison H. F. and M. J. Duncan, From the Kuiper belt to 
Jupiter-family comets: The spatial distribution of ecliptic comets, 
{\it Icarus}, {\bf 127}, 13-23, 1997.

\item Lupishko D. F. and T. A. Lupishko, On the origins of 
Earth-approaching asteroids, {\it Solar System Research},
{\bf 35}, 227-233, 2001.

\item Morbidelli, A., J. Chambers, J. I. Lunine, et al.,
Source regions and timescales for the delivery of water to the Earth, 
{\it Meteoritics \& Planetary Science}, {\bf 35}, 1309-1320, 2000.

%\item Morbidelli, A., and D. Nesvorny, Numerous weak resonances 
%drive asteroids %Toward Terrestrial Planets Orbits, {\it Icarus}, 
%{\bf 139}, 295-308, 1999.

\item Ozernoy, L. M., N. N. Gorkavyi, and T. Taidakova, Four 
cometary belts associated with the orbits of giant planets: a new 
view of the outer solar system's structure emerges from numerical 
simulations, {\it Planetary and Space Science}, {\bf 48}, 993-1003, 
2000.

\item Pavlov, A. A., A. K. Pavlov, J. F. Kasting, Irradiated 
interplanetary dust particles as a possible solution for the 
deuterium/hydrogen paradox of Earth's oceans, {\it Journal of 
Geophysical research}, {\bf 104}, No. E12, 30,725-30,728, 1999.

\item Rickman, H., J. A. Fernandez, G. Tancredi, J. Licandro, The 
cometary contribution to planetary impact craters, in {\it 
Collisional Processes in the Solar System}, edited by M. Ya. Marov 
and H. Rickman, ASSL, {\bf 261}, 131-142, 2001.

\item   Shoemaker, E. M., P. R. Weissman, and C. S. Shoemaker, The flux of
periodic comets near Earth, in {\it Hazards Due to Comets and 
Asteroids}, edited by T. Gehrels, pp. 313-335, The University of 
Arizona Press, Tucson \& London,  1994.

\item Valsecchi, G. B., A. Morbidelli, R. Gonczi, et al., The 
dynamics of objects in orbits resembling that of P/Encke, {\it 
Icarus}, {\bf 118}, 169-180, 1995.


\end{thebibliography}