Planetary migration in protoplanetary disks
A. Del Popolo
In the current paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001) and extended to time-dependent planetesimal accretion discs in Del Popolo \& Ek\c{s}i (2002). We use a method developed by Stepinski \& Valageas (1996, 1997), that is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 {yr}$. The disc model is coupled to the migration model introduced in Del Popolo et al. (2001) in order to obtain the migration rate of the planet in the planetesimal disc. We find that in the case of discs having total mass of $10^{-3}-0.1 M_{\odot}$, and $0.1<\alpha<0.0001$, planets can migrate inward a large distance while if $M<10^{-3} M_{\odot}$ the planets remain almost in their initial position for $0.1<\alpha<0.01$ and only in the case $\alpha<0.001$ the planets move to a minimum value of orbital radius of $\simeq 2 {AU}$. The model gives a good description of the observed distribution of planets in the period range 0-20 days.
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\begin{document}
\title{Planetary migration in protoplanetary disks}
  \author{A. Del Popolo}
\affil{Dipartimento di Matematica, Universit\`{a} Statale di Bergamo,
Piazza Rosate, 2 - I 24129 Bergamo, ITALY}
%\author{Ima Co-Author}
\affil{Feza G\"ursey Institute, P.O. Box 6 \c Cengelk\"oy, Istanbul,
     Turkey}

\begin{abstract}

In the current paper, we further develop the model for the migration of
planets introduced in Del Popolo et al. (2001) and extended to time-dependent planetesimal 
accretion discs in Del Popolo \& Ek\c{s}i (2002). 
We use a method developed by Stepinski \& Valageas (1996, 1997), that is
able to simultaneously follow the evolution of gas and solid particles
for up to $10^7 {\rm yr}$. The disc model is coupled to 
the migration model introduced in Del Popolo et al. (2001) in order to 
obtain the migration rate of the planet
in the planetesimal disc. We find that in the case of discs 
having total mass
of $10^{-3}-0.1 M_{\odot}$, and $0.1<\alpha<0.0001$, planets can migrate inward a large distance
while if $M<10^{-3} M_{\odot}$ the planets remain almost in their initial 
position for $0.1<\alpha<0.01$ and only in the case $\alpha<0.001$ 
the planets move to a minimum value of orbital radius of $\simeq 2 {\rm AU}$.
The model gives a good description of the observed distribution of planets 
in the period range 0-20 days.

\end{abstract}

\section{Introduction}

In order to study the formation of planetary systems it is
necessary to study the global evolution of solid material which
constitutes, together with gas, the protoplanetary discs. 
In the following, we introduce a time-dependent accretion disc
model that shall be used in the next sections to study planets
migration.

It is usually assumed that the surface density in planetesimals is proportional to 
that of gas (Lin \& Papaloizou (1980, page 47); Murray et al. (1998))
and that the spatial structure and the time evolution of the
surface density, $\Sigma_{\rm s}(R,t)$, is such that $\Sigma_{\rm s}(R,t) \propto \Sigma(R,t)$.
Although this approximation has been and is widely used in literature, a better way to proceed is to 
directly calculate the evolution of solids in the disc.  
We know that the evolution of the surface density of gas is described 
by a diffusive-type equation, while that of solid particles is an advection-diffusion equation:
\begin{equation}
\frac{\partial \Sigma_{\rm s} }{\partial t}=\frac{3}{r}\frac{\partial }{\partial r}%
\left[ r^{1/2}\frac{\partial }{\partial r}(\nu_{\rm s} \Sigma_{\rm s}
r^{1/2})\right]+
\frac{1}{r}\frac{\partial }{\partial r}\left[ \frac{2r
\Sigma _{s}\langle \overline{v}_{\phi }\rangle _{s}}{\Omega _{k}t_{s}}\right]  
\label{advdiffusee}
\end{equation}
where $\nu_{\rm s}=\frac{\nu}{{\rm Sc}}$, the Schmidt number, Sc, is given by:
\begin{equation}
{\rm Sc}=\left( 1+\Omega _{\rm k}t_{\rm s}\right) \sqrt{1+\frac{\overline{\bf v}^2}{V_{\rm t}^2}}
\end{equation}
where ${\bf v}$ is the relative velocity between a particle and the gas, $V_{\rm t}$ the 
turbulent velocity, $\Omega _{\rm k}$ is the Keplerian angular velocity, $t_{\rm s}$ the so called stopping time.

Stepinski \& Valageas (1996, 1997) developed a method that, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles due to gas-solid-coupling, coagulation, sedimentation, and evaporation/condensation for up to $10^7 {\rm yr}$. 
%
%%Stepinski's model, goes only to times of $10^7$ years fundamentally 
%%because at such long times solids are mostly in large bodies (planetesimals)
%%and Stepinski's method, which neglect gravitational interactions between solids, is
%%not longer a reasonable approximation. 
%

The equation of gas evolution  
is solved by means of an implicit scheme and the evolution of gas
is computed independently from the evolution of particles. 
At every time step the quantities
needed for evaluating the change in the mass distribution
of solids are calculated and the change itself is computed
from Eq. (1) using the operator splitting method. In
such a method the advective term in Eq. (1) is treated by the
numerical method of characteristics, whereas an implicit
scheme is applied to the diffusion term. The obtained 
distribution of solid material is then modifed because of the
existence of the evaporation radius, and the mass distribu-
tion of the vapor is calculated using the implicit scheme.
Finally, the new particle size distribution is calculated be-
fore proceeding to the next time step.
%\section{Planets migration model}

We suppose that a single planet moves in a
planetesimal disc under the influence of the gravitational force
of the Sun. The equation of motion of the planet can be written as:
\begin{equation}
{\bf \ddot r}= {\bf F}_{\odot}
%+ {\bf F}_{planets}+{\bf F}_{tide}+
%{\bf F}_{GCM}+{\bf F}_{stars}
+{\bf R}
%+{\bf F}_{other}
\end{equation}
(Melita \& Woolfson 1996), where the term ${\bf F}_{\odot}$ represents
the force per unit mass from the Sun, while ${\bf R}$ is the dissipative
force (the dynamical friction term-see Melita \& Woolfson 1996). 

In order to take into account dynamical friction, we need a 
suitable formula for a disc-like structure such as the 
protoplanetary disc. 

\indent We assume that the matter-distribution is disc-shaped, then 
we have that:
\begin{equation}
{\bf R}=-k_{\parallel}v_{1 \parallel} {\bf e_{\parallel}}-
k_{\perp}v_{1 \perp}{\bf e_{\perp}}
\label{eq:dyn}
\end{equation}
where ${\bf e_{\parallel}}$ and ${\bf e_{\perp}}$ are two versors
parallel and perpendicular to the disc plane and 
$k_{\parallel}$ and $k_{\perp}$ are given in Del Popolo \& Ek\c{s}i (2002).


\section{Results}

\indent Our model starts with a fully formed gaseous giant planet of
$1 M_{\rm J}$ at $5.2$ AU.
According to several evidences showing that the disc lifetimes range 
from
$ 10^5$ yr to $10^7$ ~yr (Strom et al. 1993; Ruden \& Pollack 1991), 
we assume that the disc has a nominal effective lifetime of $10^6$ 
years (Zuckerman et al. 1995). 
We integrated the model introduced in the previous section for
several values of disc
masses: $M_{\rm D}=0.1$, 0.01, 0.001, 0.0001 $M_{\odot}$, and several values of $\alpha$.\\

The results of the disc model are plotted in Fig.1.
Fig. 1a shows the evolution of the density of the gas component of the disc in the 
case $M=0.1 M_{\odot}$ and $\alpha=0.1$ (see figure).
%
%%The dashed line represents the gas density at $10^4 {\rm yrs}$, while 
%%the dotted and solid line that at 
%%$10^6 {\rm yrs}$ and $10^7 {\rm yrs}$, respectively. The plot shows how the 
%%gas in the disc is rapidly depleted 
%%when the viscosity is high. 
%
As shown in Fig. 1b, in agreement with Stepinski \& Valageas (1997) and Kornet et al. (2001),
the most important result of the low-mass models 
calculation (discs with $M<0.1 M_{\odot}$, and angular momentum $\simeq 10^{52} {\rm g cm^2/s}$) 
is that such models lead to the survival of solid
material, as can be seen from the evolution of the particle size,
or by the emergence of the converged, nonvanishing surface density distribution 
of solids. 

In the successive figures (2a-2d), we plot the evolution of
semi-major axis of the planet (see figure). 
Summarizing figures 2a-2d, according to the final distribution of planets distances, 
the present model predicts that, unless the disc mass is very small $M \simeq 0.0001 M_{\odot}$,
planets tend to move close to the central star to distances of the order of $0.03 {\rm AU}$.
It is also evident that it is possible to find a 
planet at any distance from their locations of
formation and very small distances from the parent star for peculiar values 
of the parameters $\alpha$ and $M$. 
 
In Fig. 3a, in order to show the predictions of the model for what concerns the distribution 
of planets in the inner part of the disc, we plot the fraction of planets in the orbital period 
range 0-20 days calculated using the model of this paper. Fig. 3b, represents the same distribution 
obtained with the data given in www.exoplanets.org (see also Kuchner \& Lecar 2002).

According to Kuchner \& Lecar 2002, disc temperature determines the orbital radii of 
the innermost surviving planets, similarly to our model. 
As streseed in DP1, DP2, 
the model has not the drawback of Murray et al. 1998 model, namely that of requiring a too large 
disc mass for migration and at the same time has the advantage of Murray et al. 1998 model of 
having an intrinsic natural mechanism that provides halting of migration.

\begin{thebibliography}{}
\bibitem[2001]{del popolo} Del Popolo, A., Gambera, M., E. N. Ercan, 2001, MNRAS, 325, 1402
\bibitem[2001]{del popolo1} Del Popolo, A., Ek\c{s}i, Y., E. 2002, MNRAS, 332, 485
\bibitem{} Kornet, K., Stepinski, T. F., R�zyczka, M, 2001, A\&A 378, 180  
\bibitem{} Kuchner, M.J., Lecar, M., 2002, astro-ph/0206232
\bibitem{} Lin D.N.C, Papaloizou J., 1980, MNRAS, 191 
\bibitem{} Melita M.D., Woolfson M.M., 1996, MNRAS 280, 854
\bibitem[1998]{murray} Murray, N., Hansen, B., Holman, M., Tremaine, S., 1998, Science 279, 69
\bibitem[1991]{RP} Ruden, S.P, Pollack, J.B., 1991, ApJ 375, 740
\bibitem[1996]{stepinski} Stepinski, T.F., Valageas, P., 1996, A\&A 309, 301
\bibitem{} Stepinski, T.F., Valageas, P., 1997, A\&A 319, 1007
%\bibitem{} Terquem C., Papaloizou J.C.B, Nelson, R.P, 2000, Space and Science Reviews, 1 
\bibitem[1995]{zucherman} Zuckerman, B., Forveille, \& Kastner, J.H., 1995, Nature 373, 494
\end{thebibliography}
\begin{figure}
\label{Fig. 1} \centerline{\hbox{(1a)
\psfig{figure=vsi0101g.ps,width=4.2cm,angle=270} (1b)
\psfig{figure=vsi0101d.ps,width=4.2cm,angle=270}
}}
\label{Fig. 1} \centerline{\hbox{(2a)
\psfig{figure=a1u.ps,width=5cm} (2b)
\psfig{figure=a2u.ps,width=5cm}
}}
\label{Fig. 1} \centerline{\hbox{(2c)
\psfig{figure=a3u.ps,width=5cm} (2d)
\psfig{figure=a4u.ps,width=5cm}
}}
\label{Fig. 1} \centerline{\hbox{(3a)
\psfig{figure=frac01.eps,width=4.6cm,height=4.6cm,angle=270} (3b)
\psfig{figure=figg.eps,width=4.6cm,height=4.6cm,angle=270}
}}
\caption[]{(a) Evolution of gas for a disc with
$M_d=0.1 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line). 
(b) Same as Fig. 1a but for solids.
}
\caption[]{(a) The evolution of the a(t) of a Jupiter-mass planet,
$M=1 M_{\rm J}$ in a planetesimal disc 
for $\alpha=0.1$ and several values of $M_{\rm d}$,
0.1  (solid line), 0.01 (dotted line),
0.001 $M_{\odot}$ (short-dashed line) and 0.0001
(long-dashed line). (b,c, d) Same as Fig. (6a) but with $\alpha=0.01, 0.001, 0.0001$
}
\caption[]{(a) Fraction of planets having orbital periods in the range 0-20 days, 
calculated according the model of this paper. 
(b) Fraction of planets having orbital periods in the range 0-20 days, 
calculated according to data (see www.exoplanets.org).}
\end{figure}
\end{document}

\section{Conclusions}

In the current paper, we further develop the model for the migration of
planets introduced in Del Popolo et al. (2001) and extended to 
time-dependent planetesimal accretion discs in Del Popolo \& Ek\c{s}i (2002). 
After releasing the assumption of Del Popolo \& Ek\c{s}i (2002)
that the surface density in planetesimals is proportional to that of gas, 
we used a method developed by Stepinski \& Valageas (1996, 1997), (+silicati)
using a series of simplifying assumptions, that is
able to simultaneously follow the evolution of gas and solid particles 
due to gas-solid-coupling, coagulation, sedimentation, and evaporation/condensation 
for up to $10^7 {\rm yr}$. Then we coupled the disc model to the migration model 
introduced in DP1 in order to 
obtain the migration rate of the planets
in the planetesimal disc and to study how the migration rate
depends on the disc mass, on its time evolution and on some values
of the dimensionless viscosity parameter $\alpha$. 
We found that in the case of discs having total mass
of $10^{-3}-0.1 M_{\odot}$, and $0.1<\alpha<0.0001$, planets can migrate inward a large distance
while if $M<10^{-3} M_{\odot}$ the planets remains almost in their initial 
position for $0.1<\alpha<0.01$ and only in the case $\alpha<0.001$ 
the planets move to a minimum value of orbital radius of $\simeq 2 {\rm AU}$.
The model was also used to calculate the distribution of planets in the inner part of the disc 
(0-0.14 AU). The result showed that the planets distribution obtained by means of 
the model of the present paper gives a good description of the observed distribution of planets 
in the period range 0-20 days.
\section*{Acknowledgments}
%%%%%%Work partially supported by funds ex-60\% 98.
%%V.A.-D. would like to thank Prof. Giuseppe Moncada.
We are grateful to E. N. Ercan for stimulating discussions during
the period in which this work was performed. A. Del Popolo would
like to thank Bo$\breve{g}azi$\c{c}i University Research
Foundation for the financial support through the project code
01B304.

\begin{thebibliography}{}
\bibitem{} Barge, P., Sommeria, J., 1995, A\&A, 295, 1
\bibitem[1977]{binney} Binney J., 1977, MNRAS 181, 735
\bibitem[1986]{bodenheimer} Bodenheimer, P., Pollack, J.B., 1986, Icarus, 67, 391
\bibitem[1995]{boss1} Boss, A.P., 1995, Science 267, 360
\bibitem[1996]{boss2} Boss, A.P., 1996, Lunar \& Planetary Science 27, 139
\bibitem[]{} Butler R.P., Tinney C.G., Marcy G.W., Jones H.R.A., Penny A.J., Apps K., 2001, ApJ 555, 410
\bibitem[]{} Cassen, P.M., 1996, Meteor. Planet. Sci. 31, 793
\bibitem[1968]{chandrasekhar2} Chandrasekhar S., 1968, Ellipsoidal figures of equilibrium, Yale University Press.
\bibitem[]{} Del Popolo, A., Spedicato, E., Gambera, M., 1999, A\&A 350, 685
\bibitem[2001]{del popolo} Del Popolo, A., Gambera, M., E. N. Ercan, 2001, MNRAS, 325, 1402
\bibitem[2001]{del popolo1} Del Popolo, A., Ek\c{s}i, Y., E. 2002, MNRAS, 332, 485
\bibitem[1979]{goldreich} Goldreich P., Tremaine S., 1979, ApJ 233, 857
\bibitem[1980]{goldreich1} Goldreich P., Tremaine S., 1980, ApJ 241, 425
\bibitem{} Goldreich, P., Ward, W.R., 1973, ApJ 183, 1051
\bibitem{} Kornet, K., Stepinski, T. F., R�zyczka, M, 2001, A\&A 378, 180  
\bibitem{} Kuchner, M.J., Lecar, M., 2002, astro-ph/0206232
\bibitem[1981]{hayashi} Hayashi, C., 1981, Suppl. Prog. Theoretical Phys. 70, 35
\bibitem[1985]{hayashi} Hayashi, C., Nakazawa K., Nakagawa, Y., 1985, Formation of the Solar System. In Protostars and Planets II (D.C. Black and M.S. Mathews, eds.), pp. 1100-1153. Univ of Arizona Press, Tucson.
\bibitem[1990]{ida} Ida S., 1990, Icarus 88, 129
\bibitem[1992]{ida1} Ida S., Makino J., 1992, Icarus 98, 28
\bibitem{} Lin D.N.C, Papaloizou J., 1980, MNRAS, 191 
\bibitem[1986]{lin} Lin, D.N.C., Papaloizou, J., 1986, ApJ 309, 846
\bibitem[1993]{lin1} Lin, D.N.C., Papaloizou, J.C.B., 1993, in Protostars and Planets III, ed. E.H. Levy \& J.I Lunine (Tucson:Univ. Arizona Press), 749
\bibitem[1996]{lin2} Lin, D.N.C., Bodenheimer, P., Richardson, D.C., 1996, Nature, 380, 606
\bibitem[1993]{lissauer1} Lissauer, J.J., 1993, ARA\&A, 31, 129
\bibitem[]{} Marcy G.W., Butler R.P., Vogt S.S., 2000, ApJL 536, 43
\bibitem[1995]{mayor} Mayor, M., Queloz, D., 1995, Nature, 378, 355
\bibitem{} Melita M.D., Woolfson M.M., 1996, MNRAS 280, 854
\bibitem[1980]{mizuno} Mizuno, H., 1980, Prog. Theor. Phys., 64, 544
\bibitem[1989]{morfill} Morfill, G.E., Wood, J. A., 1989, Icarus 82, 225
\bibitem[1998]{murray} Murray, N., Hansen, B., Holman, M., Tremaine, S., 1998, Science 279, 69, (see also preprint SISSA astro-ph/9801138)
\bibitem{} \"Opik, E.J., Interplanetary Encounters, (Elsevier Scientific, Amsterdam 1976)
\bibitem{} Pollack J.B., Hollenbach D., Beckwith S., Simonelli D.P., Roush T., Fong W., 1994, ApJ 421, 615  
\bibitem[1996]{rasio} Rasio, F.A., Ford, E., 1996, Science, 274, 954
\bibitem[1991]{RP} Ruden, S.P, Pollack, J.B., 1991, ApJ 375, 740
\bibitem[1969]{safronov} Safronov, V.S., 1969. In Evoliutsiia doplanetnogo oblaka (Evolution of the protoplanetary cloud and formation of the Earth and planets), Moscow: Nauka press.
\bibitem{}  Sasselov, D.D., Lecar, M., 2000, ApJ 528, 995
\bibitem{} Schmitt, W., Henning, T., Mucha, R., 1997, A\&A 325, 569 
\bibitem[1993]{stepinski} Stepinski, T.F., Reyes-Ruiz, M., Vanhala H.A.T., 1993, Icarus 106, 77
\bibitem[1996]{stepinski} Stepinski, T.F., Valageas, P., 1996, A\&A 309, 301
\bibitem{} Stepinski, T.F., Valageas, P., 1997, A\&A 319, 1007
\bibitem[1993]{strom} Strom, S.E., Edwards, S., Skrutskie, M.F., 1993, in Protostars and Planets III, ed. E.H. Levy \& J.I Lunine (Tucson:Univ. Arizona Press), 837
\bibitem{} Tanga, P., Babiano, A., Dubrulle, B., Provenzale, A., 1996, Icar 121, 158
\bibitem{} Terquem C., Papaloizou J.C.B, Nelson, R.P, 2000, Space and Science Reviews, 1 (see also astro-ph/9905229)
\bibitem{} Trilling, D. E., Benz, W., Guillot, T., Lunine, J. I., Hubbard, W. B., Burrows, A., 1998, ApJ 500, 428 
\bibitem[]{} Vogt S.S., Marcy G.W., Butler R.P., Apps K., 2000, ApJ 536, 902
\bibitem[1986]{ward1} Ward W.R., 1986, Icarus 67, 164
\bibitem[1997]{ward3} Ward, W.R., 1997, Icarus 126, 261
\bibitem[1977]{weidenschilling} Weidenshilling, S.J., 1977, MNRAS 180, 57
\bibitem[1996]{weidenschilling} Weidenshilling, S.J., Marzari, F., 1996, Nature 384, 619
\bibitem[1993]{wuchterl} Wuchterl, G., 1993, Icarus 106, 323
\bibitem[1996]{wuchterl1} Wuchterl, G., 1996, BAAS, 28, 1108
\bibitem[1995]{zucherman} Zuckerman, B., Forveille, \& Kastner, J.H., 1995, Nature 373, 494
\end{thebibliography}
%\begin{figure}
%\label{Fig. 1} \centerline{\hbox{(1)
%\psfig{figure=fconf.ps,width=12cm,angle=270} 
%}}
%\caption[]{Comparison of the planetesimal density in the disc with that of gas  
%for a disc with
%$M_d=0.1 M_{\odot}$ and $\alpha=0.01$ at $t=10^4 {\rm yrs}$ (dashed line),
%$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line).}
%\caption[]{Same as Fig. 1 but with $\alpha=0.01$}
%Evolution of gas for a disc with
%$M_d=0.1 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
%$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line).}
%\end{figure}
\newpage
\begin{figure}
\label{Fig. 1} \centerline{\hbox{(1a)
\psfig{figure=dim.ps,width=8cm,angle=270} (1b)
\psfig{figure=dim1.ps,width=8cm,angle=270} 
}}
\caption[]{Evolution of the particle size in a disc with
$M_d=0.01 M_{\odot}$ and $\alpha=0.01$, panel (a), and $\alpha=0.0001$, panel (b), 
at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line).}
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(2a)
\psfig{figure=vsi0101g.ps,width=6cm,angle=270} (2b)
\psfig{figure=vsi0102g.ps,width=6cm,angle=270}
}}
\label{Fig. 1} \centerline{\hbox{(2c)
\psfig{figure=vsi0103g.ps,width=6cm,angle=270} (2d)
\psfig{figure=vsi0104g.ps,width=6cm,angle=270}
}}
\caption[]{ (a) Evolution of gas for a disc with
$M_d=0.1 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line). 
(b) Same as Fig. 2a but with $\alpha=0.01$. (c) Same as Fig. (2a) 
but with $\alpha=0.001$. (d) Same as Fig. (2a) but with $\alpha=0.0001$}
\label{Fig. 1} \centerline{\hbox{(3a)
\psfig{figure=vsi0401g.ps,width=6cm,angle=270} (3b)
\psfig{figure=vsi0402g.ps,width=6cm,angle=270}
}}
\caption[]{ (a) Evolution of gas for a disc with
$M_d=0.0001 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line). 
(b) Same as Fig. (3a) but with $\alpha=0.01$}
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(3c)
\psfig{figure=vsi0403g.ps,width=6cm,angle=270} (3d)
\psfig{figure=vsi0404g.ps,width=6cm,angle=270}
}}
{\bf Figure 3.} (c) Same as Fig. (3a) but with $\alpha=0.001$. 
(d) Same as Fig. (3a) but with $\alpha=0.0001$.
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(4a)
\psfig{figure=vsi0101d.ps,width=6cm,angle=270} (4b)
\psfig{figure=vsi0102d.ps,width=6cm,angle=270}
}}
\label{Fig. 1} \centerline{\hbox{(4c)
\psfig{figure=vsi0103d.ps,width=6cm,angle=270} (4d)
\psfig{figure=vsi0104d.ps,width=6cm,angle=270}
}}
\caption[]{(a) Evolution of solids for a disc with
$M_d=0.1 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line).
(b) Same as Fig. (4a) but with $\alpha=0.01$. (c) 
Same as Fig. (4a) but with $\alpha=0.001$.
(d) Same as Fig. (4a) but with $\alpha=0.0001$}
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(5a)
\psfig{figure=vsi0401d.ps,width=6cm,angle=270} (5b)
\psfig{figure=vsi0402d.ps,width=6cm,angle=270}
}}
\label{Fig. 1} \centerline{\hbox{(5c)
\psfig{figure=vsi0403d.ps,width=6cm,angle=270} (5d)
\psfig{figure=vsi0404d.ps,width=6cm,angle=270}
}}
\caption[]{(a) Evolution of solids for a disc with
$M_d=0.0001 M_{\odot}$ and $\alpha=0.1$ at $t=10^4 {\rm yrs}$ (dashed line),
$t=10^6 {\rm yrs}$ (dotted line) and $t=10^7 {\rm yrs}$ (solid line).
(b) Same as Fig. (5a) but with $\alpha=0.01$.
(c) Same as Fig. (5a) but with $\alpha=0.001$. 
(d) Same as Fig. (5a) but with $\alpha=0.0001$}
\label{Fig. 1} \centerline{\hbox{(6a)
\psfig{figure=a1u.ps,width=8cm} (6b)
\psfig{figure=a2u.ps,width=8cm}
}}
\caption[]{(a) The evolution of the a(t) of a Jupiter-mass planet,
$M=1 M_{\rm J}$ in a planetesimal disc 
for $\alpha=0.1$ and several values of $M_{\rm d}$,
0.1  (solid line), 0.01 (dotted line),
0.001 $M_{\odot}$ (short-dashed line) and 0.0001
(long-dashed line). (b) Same as Fig. (6a) but with $\alpha=0.01$}
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(6c)
\psfig{figure=a3u.ps,width=7cm} (6d)
\psfig{figure=a4u.ps,width=7cm}
}}
{\bf Figure 6.} (c) Same as Fig. (6a) but with $\alpha=0.001$. 
(d) Same as Fig. (6a) but with $\alpha=0.0001$.
\end{figure}

\begin{figure}
\label{Fig. 1} \centerline{\hbox{(7a)
\psfig{figure=frac01.eps,width=6.3cm,height=7.6cm,angle=270} (7b)
\psfig{figure=figg.eps,width=6.3cm,height=7.6cm,angle=270}
}}
\caption[]{(a) Fraction of planets having orbital periods in the range 0-20 days, 
calculated according the model of this paper. 
(b) Fraction of planets having orbital periods in the range 0-20 days, 
calculated according to data (see www.exoplanets.org).}

\end{figure}
\end{document}



%%\bibitem[1993]{aarseth} Aarseth, S.J., Lin, D.N.C., Palmer, P.L., 1993, ApJ 403, 351
%%\bibitem[1989]{anders} Anders, E., Grevesse, N., 1989, Geochim. Cosmochim. Acta 53, 197
%\bibitem{} Antonuccio-Delogu, V., Atrio-Barandela, F., 1992, ApJ 392, 403
%\bibitem{} Aumann H.H., Gillette F.C., Beichman C.A., et al., 1984, ApJ 278, L23
**\bibitem{} Barge, P., Sommeria, J., 1995, A\&A, 295, 1
\bibitem[1990]{beckwith} Beckwith, S.V.W., Sargent, A.I., Chini, R.S., G\"usten, R., 1990, Astron. J 99, 924
%\bibitem{} Beckwith G., Sargent A.I., 1993, Protostars and Planets III, Levy  E.H., Lunine J.I., (eds.) The University of Arizona Press. Tucson, p. 521
\bibitem[1996]{beckwith1} Beckwith, S.V.W., Sargent, A., 1996, Nature 383, 139
%\bibitem{} Brunini A., Fernandez J.A., 1996, A\&A 308, 988
**\bibitem[1977]{binney} Binney J., 1977, MNRAS 181, 735
%%%%%%%%%\bibitem[1987]{binney1} Binney J., Tremaine S., 1987, Galactic Dynamics,
%%%%%%%%%in Princeton Series in Astrophysics, Princeton University Press.
**\bibitem[1986]{bodenheimer} Bodenheimer, P., Pollack, J.B., 1986, Icarus, 67, 391
%\bibitem{} Boss, A.P., 1993,
%%\bibitem[1984]{boss} Boss, A.P., 1984, MNRAS 209, 543
**\bibitem[1995]{boss1} Boss, A.P., 1995, Science 267, 360
**\bibitem[1996]{boss2} Boss, A.P., 1996, Lunar \& Planetary Science 27, 139
**\bibitem[]{} Butler R.P., Tinney C.G., Marcy G.W., Jones H.R.A., Penny A.J., Apps K., 2001, ApJ 555, 410
%%\bibitem[1996]{butler} Butler, R.P., Marcy, G.W., 1996, ApJ 464, L153
%%\bibitem[1997]{butler1} Butler, R.P., Marcy, G.W., Williams, E., Hauser, H., Shirts, P., 1997, ApJ 474, L115
\bibitem[]{} Cannizzo, J.K., Lee, H.M. \& Goodman, J., 1990 ApJ 351, 38
\bibitem[]{} Cassen, P.M., 1994, Icarus 112, 405
**\bibitem[]{} Cassen, P.M., 1996, Meteor. Planet. Sci. 31, 793
\bibitem[1996]{chambers} Chambers, J.E., Wetherill, G.W., Boss, A.P., 1996, Icarus 119, 261
%%\bibitem[1942]{chandrasekhar} Chandrasekhar S., von Neumann J., 1942, ApJ 95, 489
%%\bibitem[1943]{chandrasekhar1} Chandrasekhar S., 1943, ApJ 97, 255
**\bibitem[1968]{chandrasekhar2} Chandrasekhar S., 1968, Ellipsoidal figures of equilibrium, Yale University Press.
%\bibitem{} Cochran A.L., Levison H.F., Stern S.A., Duncan M.J., 1995, AJ 455, 342
%%\bibitem[1997]{cochran} Cochran, W.D., Hatzes, A.P., Butler, R.P., Marcy, G.W., 1997, ApJ 483, 457
**\bibitem[]{} Del Popolo, A., Spedicato, E., Gambera, M., 1999, A\&A 350, 685
**\bibitem[2001]{del popolo} Del Popolo, A., Gambera, M., E. N. Ercan, 2001, MNRAS, 325, 1402
%%\bibitem[1993]{donner} Donner K.J., Sundelius B., 1993, MNRAS 265, 88
**\bibitem[2001]{del popolo1} Del Popolo, A., Ek\{c}si, Y., E. 2002, MNRAS, 332, 485
\bibitem[]{} Dubruelle, B., 1993, Icarus 106, 59
%\bibitem{} Duncan M.J., Quinn T., Tremaine S., 1988, ApJ 328, L69
%\bibitem{} Duncan M.J., Levion H.F., Budd S.M., 1995, AJ 110, 373
%\bibitem{} Edgeworth K.E., MNRAS 1949, 109, 600
%%\bibitem[1997]{eggleton} Eggleton, P., Kiseleva, L., 1997, The 10th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun
%\bibitem{} Fernandez J.A., 1980, MNRAS 192, 481
\bibitem[1984]{fernandez} Fernandez, J.A., Ip, W.H., 1984, Icarus 58, 109
\bibitem[]{} Filipov, L.G., 1984 Adv. Space Res. 3, 305
\bibitem[]{} Filipov, L.G., Shakura, N.I. \& Lyubarskii, Y. 1988 Adv. Space Res. 8,163
\bibitem[1999]{ford} Ford, E.B., Rasio, F.A., Sills, A., 1999, ApJ 514, 411
%%%%%%%%\bibitem[1975]{genkin} Genkin, I.L., Chumak, Z.N., 1975, AZh, 52, 1329
%%%%%%%%(English translation in SvA 19, 792
\bibitem[1995]{frank} Frank, J., King, A.R., Raine, D.J., 1985, in Accretion Powers in Astrophysics. Cambridge Univ. Press, Cambridge, UK
\bibitem[1993]{gladman} Gladman, B., 1993, Icarus 106, 247
**\bibitem[1979]{goldreich} Goldreich P., Tremaine S., 1979, ApJ 233, 857
**\bibitem[1980]{goldreich1} Goldreich P., Tremaine S., 1980, ApJ 241, 425
**\bibitem{} Goldreich, P., Ward, W.R., 1973, ApJ 183, 1051
\bibitem[1997]{gonzales} Gonzales, G., 1997, MNRAS 285, 403
\bibitem[1998a]{gonzales1} Gonzales, G., 1998a, A\&A 334, 221
\bibitem[1998b]{gonzales2} Gonzales, G., 1998b, in Brown Dwarfs and Extrasolar Planets, ed. R.Rebolo, (Provo: ASP Conference Proceedings), in press
%%\bibitem[1996]{guillot} Guillot, T., Burrows, A., Hubbard, W.B., Lunine, J.I., Saumon, D., 1996, ApJ 459, L35
%%\bibitem{} Haghighipour, 1999, MNRAS 31, 504
**\bibitem{} Kornet, K., Stepinski, T. F., R�zyczka, M, 2001, A\&A 378, 180  
**\bibitem{} Kuchner, M.J., Lecar, M., 2002, astro-ph/0206232
**\bibitem[1981]{hayashi} Hayashi, C.,....................................
\bibitem[1977]{hayashi} Hayashi, C., Nakazawa K., Adachi, I., 1977, PASJ, 29, 163
**\bibitem[1985]{hayashi} Hayashi, C., Nakazawa K., Nakagawa, Y., 1985, Formation of the Solar System. In Protostars and Planets II (D.C. Black and M.S. Mathews, eds.), pp. 1100-1153. Univ of Arizona Press, Tucson.
%\bibitem{} Hills J.G., 1981, AJ 86, 1730
%\bibitem{} Holman M., Wisdom J., 1993, AJ 105, 1987
%%\bibitem[1996]{holman} Holman, M., Murray, N., 1996, Astron. J. 112, 1278
%%\bibitem[1997]{holman1} Holman, M., Touma, J., Tremaine, S., 1997, Nature, 386, 20
%\bibitem{} Holtsmark, P.J.: (1919), Phys. Z. 20, 162
%%\bibitem[1985]{horedt} Horedt G.P., 1985, Late stages of planetary accretion, Icarus 64, 448
\bibitem[1985]{hornung} Hornung P., Pellat R., Borge P., 1985, Icarus 64, 295
**\bibitem[1990]{ida} Ida S., 1990, Icarus 88, 129
%%%%%%%%%%%\bibitem{} Ida, S., 1991, Icarus, 88, 129
**\bibitem[1992]{ida1} Ida S., Makino J., 1992, Icarus 98, 28
\bibitem[1993]{ida2} Ida, S., Makino, J., 1993, Icarus 106, 210
%%\bibitem[1993]{ida3} Ida S., Kokubo E., Makino J., 1993, MNRAS 263, 875
%\bibitem[2000]{ida4} Ida, S., Bryden, G., Lin, D.N.C., H. Tanaka, 2000,
%preprint
\bibitem[2000]{ida4} Ida S., Bryden G., Lin D.N.C., Tanaka H., 2000, ApJ 534, 428
%%submitted to ApJ, (preprint available from
%%http://www.geo.titech.ac.jp/nakazawalab/ida/DG.html)
%\bibitem{} Ipatov, S.I., 1998, Migration of Kuiper-belt objects inside
%%the Solar system, "Planetary systems- the long view", Proc. 9th
%%Rencontres de Blois (June 22-28 1997), Editions Frontieres, Gif. sur
%%Yvette Cedex, Celnikier, L.M., and Tran Thanh Van, J., (eds.), in press
%\bibitem{} Ipatov, S.I., 1999, preprint
%\bibitem{} Jewitt D., Luu J.X., 1993, Nature, 362, 730
%\bibitem{} Jewitt D., Luu J.X., 1995, AJ 109, 1867
%\bibitem{} Kandrup, H.E.: (1980), Phys. Rep., 63, 1
%%\bibitem[1997]{kiseleva} Kiseleva, L., Eggleton, P., 1997, The 10th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun
%%\bibitem[1993]{kley} Kley, W., Papaloizou, J.C.B., Lin, D.N.C., 1993, ApJ 416, 679
%\bibitem{} Kokub E., Ida S., 1995, Icarus 114, 247
%\bibitem{} Kokubo E., Ida S., 1998, preprint
%\bibitem{} Kuiper G. P., 1951, in Astrophysics: A Topical Symposium, edited by J. A. Hynek (McGraw Hill, New York), pp. 357-424
%%\bibitem[1984]{larson} Larson, R.B., 1984, MNRAS 206, 197
%\bibitem{} Levison H.F., 1996, in ASP Conf. Proc. 107, Completing the Inventory of the Solar System, ed. T.W. Rettig \& J.M. Hahn (San Francisco: ASP), p. 173
%\bibitem{} Levison H.F., Duncan M.J., 1990, AJ 100(5), 1669
%\bibitem{} Levison H.F., Duncan M.J., 1993, ApJ 406, L35
**\bibitem{} Lin D.N.C, Papaloizou J., 1980, MNRAS, 191 
\bibitem[1986]{lin} Lin, D.N.C., Papaloizou, J., 1986, ApJ 309, 846
\bibitem[1993]{lin1} Lin, D.N.C., Papaloizou, J.C.B., 1993, in Protostars and Planets III, ed. E.H. Levy \& J.I Lunine (Tucson:Univ. Arizona Press), 749
**\bibitem[1996]{lin2} Lin, D.N.C., Bodenheimer, P., Richardson, D.C., 1996, Nature, 380, 606
%%\bibitem[1997]{lin3} Lin, D.N.C., Ida, S., 1997, ApJ 477, 781
\bibitem[1974]{lynden} Lynden-Bell D., Pringle J.E., 1974, MNRAS 168, 603
\bibitem[1997]{liou} Liou, J.C., Malhotra, R., 1997, Science 275, 375
%%\bibitem[1992]{lissauer} Lissauer, J.J., Stewart, G.M., 1992, in Protostars and Planets III, ed. E. Levy and M. Mathews, (Tucson: Univ. Arizona Press) in press
**\bibitem[1993]{lissauer1} Lissauer, J.J., 1993, ARA\&A, 31, 129
\bibitem[1952]{Lust} L\"ust, R, 1952, Z. Naturforsch. 7a, 87
%\bibitem{} Luu, J., Marsden B.G., Jewitt D., Trujillo C.A., Hergenrother C.W., Chen, J., Offutt W.B., 1997, Nature, 387, 573
\bibitem[]{} Lyubarskii, Y.E. \& Shakura, N.I., 1987, Soviet Astronomy Letters, 13, 386L
\bibitem[1993]{malhotra} Malhotra R., 1993, Nat. 365, 819
%%\bibitem[1995]{malhotra} Malhotra R., 1995, AJ 110, 420
%%\bibitem[1996]{marcy} Marcy, G.W., Butler, R.P., 1996, ApJ 464, L147
%%\bibitem[1997]{marcy1} Marcy, G.W., Butler, R.P., Williams, E., Bildsten, L., Graham, J., Ghez, A., Jernigan, F., 1997, ApJ 481, 926
**\bibitem[]{} Marcy G.W., Butler R.P., Vogt S.S., 2000, ApJL 536, 43
**\bibitem[1995]{mayor} Mayor, M., Queloz, D., 1995, Nature, 378, 355
%%\bibitem[1996]{mazeh} Mazeh, T., Krymolowski, Y., Rosenfeld, G., 1996, ApJ, 466, 415
\bibitem[1994]{mccaughrean} McCaughrean, M.J., Stauffer, J.R., 1994, Astron. J., 108, 1382
**\bibitem{} Melita M.D., Woolfson M.M., 1996, MNRAS 280, 854
\bibitem[]{} Mineshige, S., 1991, MNRAS 250, 253
\bibitem[]{} Mineshige, S., Nomoto, K. \& Shigeyama, T., 1993, A\&A 267, 95
**\bibitem[1980]{mizuno} Mizuno, H., 1980, Prog. Theor. Phys., 64, 544
**\bibitem[1989]{morfill} Morfill, G.E., Wood, J. A., 1989, Icarus 82, 225
**\bibitem[1998]{murray} Murray, N., Hansen, B., Holman, M., Tremaine, S., 1998, Science 279, 69, (see also preprint SISSA astro-ph/9801138)
**\bibitem{} \"Opik 1976......................
%%\bibitem[1997]{noyes} Noyes, R.W., Jha, S., Korzennik, S.G., Brown, T.M., Kennelly, E.J., Horner, S.D., 1997, ApJ 483, L111
%%\bibitem[1997]{noyes1} Noyes, R.W., Saurabh, J., Korzennik, S.G., Krockenberger, M., Nisenson, P., 1997, SISSA preprint, astro-ph/9704248
%\bibitem{} Oort J.H., 1950, Bull. Astr. Inst. Netherlands, 11, 91
%%\bibitem{} Palmer P., Lin D.N.C., Aarseth S.J., 1993, ApJ 403, 336
%\bibitem{} Pesce E., Capuzzo-Dolcetta R., Vietri M., 1992, MNRAS 254, 466
\bibitem[1942]{peek} Peek, B.M., J. Brit. Astronom. Assoc. 53, 23
\bibitem[]{} Perna, R., Hernquist, L. \& Narayan, R., 2000, ApJ, 541, 344
%%\bibitem[1996]{pollack} Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y., 1996, Icarus, 124, 62
**\bibitem{} Pollack J.B., Hollenbach D., Beckwith S., Simonelli D.P., Roush T., Fong W., 1994, ApJ 421, 615  
\bibitem[1974]{pringle} Pringle, J.E., 1974, Ph.D. thesis. Cambridge Univ.
%%\bibitem[1981]{pringle} Pringle, J.E., 1981, ARA\&A, 19, 137
\bibitem[1994]{prosser} Prosser, C.F., et al., 1994, ApJ, 421, 517
**\bibitem[1996]{rasio} Rasio, F.A., Ford, E., 1996, Science, 274, 954
%%\bibitem[1996]{rasio1} Rasio, F.A., Tout, C.A., Lubow, S.H., Livio, M., 1996, ApJ 470, 1187
\bibitem[1995]{RR} Reyes-Ruiz, M., Stepinski, T., 1995, ApJ 438, 750
%\bibitem[1991]{ruden} Ruden, S.P., Pollack, J,B., 1991, ApJ 375, 740
%%%%%%%%%\bibitem[1972]{rybicky} Rybicky, G.B., 1972, in Lecar M., ed.,
%%%%%%%%%Gravitational N-body Problem. Reidel, Dordrecht, p.22
%\bibitem{} Quinn T.R., Tremaine S., Duncan M.J., 1990, ApJ 355, 667
\bibitem[1986]{ruden} Ruden S.P, Lin D.N.C., 1986, ApJ 308, 883
**\bibitem[1991]{RP} Ruden, S.P, Pollack, J.B., 1991, ApJ 375, 740
\bibitem[]{} Ruden, S.P., 1993, in Planets around Pulsars, (J.A.Phillips, S.E. Thorsett, and S.R. Kulkarni, Eds.), pp. 197-215, ASP Conference Series, Vol. 36
%%\bibitem[1992]{ryu} Ryu, D., Goodman, J., 1992, ApJ 388, 438
**\bibitem[1969]{safronov} Safronov, V.S., 1969. In Evoliutsiia doplanetnogo oblaka (Evolution of the protoplanetary cloud and formation of the Earth and planets), Moscow: Nauka press.
**\bibitem{}  Sasselov, D.D., Lecar, M., 2000, ApJ 528, 995
**\bibitem{} Schmitt, W., Henning, T., Mucha, R., 1997, A\&A 325, 569 
%%\bibitem[1998]{sandquist} Sandquist, E., Taam, R.E., Lin, D.N.C., Burkert, A., 1998, SISSA preprint astro-ph/9808128
\bibitem[1973]{SS} Shakura, N.L., Sunyaev, 1973, A\&A 24, 337
%%\bibitem[1990]{skrutskie} Skrutskie, M.F., Dutkevitch, D., Strom, S.E., Edwards, S., Strom, K.M., Shure, M., 1990, AJ 99, 1187
%Murray, Hansen, Holman, Tremaine, 1998, Science 279, 69).
%\bibitem{} Smith B.A., Terrile R.J., 1984, Sci, 226, 1421
**\bibitem[1993]{stepinski} Stepinski, T.F., Reyes-Ruiz, M., Vanhala H.A.T., 1993, Icarus 106, 77
**\bibitem[1996]{stepinski} Stepinski, T.F., Valageas, P., 1996, A\&A 309, 301
**\bibitem{} Stepinski, T.F., Valageas, P., 1997, A\&A 319, 1007
\bibitem[1998]{stepinski1} Stepinski, T.F., 1998, Icarus 132, 100
%\bibitem{} Stern S.A., 1995, AJ 110(2), 856
%\bibitem{} Stern S.A., 1996a, A\&A 310, 999
%\bibitem{} Stern S.A., 1996b, AJ 112(3), 1203
%\bibitem{} Stern S.A., Colwell J.E., 1997a, ApJ 490, 879
%\bibitem{} Stern S.A., Colwell J.E., 1997b, AJ 114(2), 841
%%\bibitem[1980]{stewart} Stewart G.R., Kaula W.M., 1980, A gravitational kinetic theory for planetesimals. Icarus 44, 154
\bibitem[1988]{stewart1} Stewart G.R., Wetherill G.W., 1988, Icarus 74, 542
%%%%%%%%%%\bibitem[1996]{stone} Stone, J.M., Balbus, S.A., 1996, ApJ 464
**\bibitem[1993]{strom} Strom, S.E., Edwards, S., Skrutskie, M.F., 1993, in Protostars and Planets III, ed. E.H. Levy \& J.I Lunine (Tucson:Univ. Arizona Press), 837
**\bibitem{} Tanga, P., Babiano, A., Dubrulle, B., Provenzale, A., 1996, Icar 121, 158
%%\bibitem[1996](take) Takeuchi T., Miyama S.M., Lin D.N.C., 1996, ApJ 460, 832
**\bibitem{} Terquem C., Papaloizou J.C.B, Nelson, R.P, 2000, Space and Science Reviews, 1 (see also astro-ph/9905229)
**\bibitem{} Trilling, D. E., Benz, W., Guillot, T., Lunine, J. I., Hubbard, W. B., Burrows, A., 1998, ApJ 500, 428 
%%\bibitem[1968]{thorne} Thorne, R.M.,  1968, ApJ 151, 671
%\bibitem{} Tremaine S., 1990, In Baryonic Dark Matter, edit by D.
%%%%%%%Lynden-Bell and G. Gilmore (Lluwer Dordrecht), p. 37
**\bibitem[]{} Vogt S.S., Marcy G.W., Butler R.P., Apps K., 2000, ApJ 536, 902
\bibitem[1943]{vW} von Weizs\"acker, C.F., 1943, Z. Astrphys. 22, 319
\bibitem[1948]{vW1} von Weizs\"acker, C.F., 1948, Z. Naturforsch 3a, 524
%%\bibitem[1982]{ward} Ward, W.R., 1982, Lunar and Planetary Abstracts., 13, 831
**\bibitem[1986]{ward1} Ward W.R., 1986, Icarus 67, 164
%%\bibitem[1989]{ward2} Ward, W.R., Hourigan, K., 1989, ApJ 347, 490
**\bibitem[1997]{ward3} Ward, W.R., 1997, Icarus 126, 261
**\bibitem[1977]{weidenschilling} Weidenshilling, S.J., 1977, MNRAS 180, 57
**\bibitem[1996]{weidenschilling} Weidenshilling, S.J., Marzari, F., 1996, Nature 384, 619
%\bibitem{} Weidenschilling S.J., Spaute D., Davis D.R., Marzari F., Ohtsuky K.,  1997, Icarus 128, Issue 2, 429
%\bibitem{} Weissman, P.R., A'Hearn , M,F., McFadden, L.A., Rickman, H., 1989, Asteroids II, Tucson, : Univ. Arizona Press, Binzel, R.P., Gehrels, T., Matthews, M.S., (eds.), 880-919
%\bibitem{} Weismann P.R., 1995, ARA\&A 33, 327
%\bibitem{} Weismann P.R., Levison H.F., 1997, in Pluto and Charon, edited by S.A. Stern \& D.J. Tholen (University of Arizona Press, Tucson) (in press)
%\bibitem{} Wetherill, G.W., 1988, Icarus, 76, 1
%%\bibitem[1989]{wetherill} Wetherill, G.W., and Stewart, G.R., 1989, Accumulation of a swarm of small planetesimals. Icarus 77, 330
%\bibitem{} Wetherill, G.W., 1989, Meteoritics, 24, 15
%%\bibitem[1990]{wetherill1} Wetherill, G.W., 1990, Annual Rev. Earth Planet Sci., 18, 205
%\bibitem{} Wiegert P., Tremaine S., 1997, preprint SISSA astro-ph/9705153
%%\bibitem[]{wolsczan} Wolsczan, A., Frail, A., 1992, Nature 355, 145
%%\bibitem[1994]{wolsczan1} Wolszczan, A., 1994, Science, 264, 538
**\bibitem[1993]{wuchterl} Wuchterl, G., 1993, Icarus 106, 323
**\bibitem[1996]{wuchterl1} Wuchterl, G., 1996, BAAS, 28, 1108
%\bibitem{} Yamamoto T., Mizutani H., Kadota A., 1994, PASJ, L5
**\bibitem[1995]{zucherman} Zuckerman, B., Forveille, \& Kastner, J.H., 1995, Nature 373, 494



























\subsection{Comparison of the model results with observations}

The configuration of observed planets can be reproduced by means
of a combination of $\alpha$ and $M_{\rm d}$. For example, the
parameters of 47 UMa b ($a=2.11$ AU) can be explained, for
example, with $\alpha=0.01$ and $M_{\rm d} \simeq 0.0001$ or
$\alpha=0.1$ and $M_{\rm d} \simeq 0.01$, while planets like
$\tau$ Bootis b or 51 Peg b, having very small semi-major axis,
can be reproduced with $\alpha=0.001$ and $0.1 M_{\odot}<M_{\rm
d}<0.0001 M_{\odot}$ or $\alpha=0.01$ and $0.1 M_{\odot}<M_{\rm
d}<0.0035 M_{\odot}$. Configurations similar to that of 55 Cnc b
($a=0.11$ AU), $\rho$ CrB b ($a=0.23$ AU) are obtained for
$\alpha=0.01$, $M_{\rm d} \simeq 0.0015 M_{\odot}$ and
$\alpha=0.01$ $M_{\rm d} \simeq 0.0012 M_{\odot}$, respectively.
%or disc evolution (e.g., $M_{\rm D}=10^{-2} M_{\odot},
%$t_{evol}=10^8$ yr)
The semi-major axis of planets like 70 Vir b ($a=0.43$, $e_{\rm
p}=0.4$; $M \sin{i_{\rm p}} \sim 6.6 M_J$) and HD 114762 b
($a=0.3$; $e_{\rm p}=0.25$; $M \sin{i_{\rm p}} \sim 10 M_{\rm J}$)
can be explained by radial migration, as shown, while their high
value of eccentricities can be explained as described in Del
Popolo et. al (2001). \footnote{If a planet having mass $M \ge 3
M_{\rm J}$ moves in a planetesimal disc during interactions,
planetesimals scattered from their Hill sphere can be ejected with
$|\frac{\Delta E}{\Delta L}|<1$, (where $\Delta E$ and $\Delta L$
are respectively the energy and angular momentum removed from a
planet by the ejection of a planetesimal), and the eccentricity
$e_{\rm p}$ tends to increase.}
\indent Our Solar System could have been subject
%%not have been immune
to giant planet migration. For example, a shrinkage of the orbit 
of Jupiter of 0.1-0.2 AU could naturally explain the depletion of 
the outer asteroid belt (Fernandez \& Ip 1984; Liou \& Malhotra 1997). 
Some of our model runs produce a 1 $M_J$ planet that move from 5.2 AU 
inwards for a fraction of AU. The planetesimal disc enabling this 
small migration has a lifetime $\simeq 10^7$ yr, so that the disc 
gas must have disappeared soon after Jupiter fully formed 
%%soon after Jupiter grew to its
%%present mass,
%%well before Jupiter could spiral into the Sun
(Boss 1996). \\
\indent This last result is obviously strictly valid only for a 
single planet orbiting around the Sun because in presence of several 
planets, migration becomes more complex. A close example is that of 
the solar system. In this case two planets, Uranus and Neptune were 
subject to outward migration, which is the opposite of what expected. 
Several models have been proposed to explain this outward migration. 
A first model is connected to gravitational scattering between
planet and residuals planetesimals (Malhotra 1993; Ida et al. 2000).











astro-ph0304041.extracted_bib
\begin{thebibliography}{}
\bibitem[2001]{del popolo} Del Popolo, A., Gambera, M., E. N. Ercan, 2001, MNRAS, 325, 1402
\bibitem[2001]{del popolo1} Del Popolo, A., Ek\c{s}i, Y., E. 2002, MNRAS, 332, 485
\bibitem{} Kornet, K., Stepinski, T. F., R�zyczka, M, 2001, A\&A 378, 180  
\bibitem{} Kuchner, M.J., Lecar, M., 2002, astro-ph/0206232
\bibitem{} Lin D.N.C, Papaloizou J., 1980, MNRAS, 191 
\bibitem{} Melita M.D., Woolfson M.M., 1996, MNRAS 280, 854
\bibitem[1998]{murray} Murray, N., Hansen, B., Holman, M., Tremaine, S., 1998, Science 279, 69
\bibitem[1991]{RP} Ruden, S.P, Pollack, J.B., 1991, ApJ 375, 740
\bibitem[1996]{stepinski} Stepinski, T.F., Valageas, P., 1996, A\&A 309, 301
\bibitem{} Stepinski, T.F., Valageas, P., 1997, A\&A 319, 1007
%\bibitem{} Terquem C., Papaloizou J.C.B, Nelson, R.P, 2000, Space and Science Reviews, 1 
\bibitem[1995]{zucherman} Zuckerman, B., Forveille, \& Kastner, J.H., 1995, Nature 373, 494
\end{thebibliography}