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% A contribution to the proceedings of the Pralognan winter school
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% Title: Yarkovsky effect resurrected: a new light on the old problems
% Author: David Vokrouhlicky
% Version: 05/04/99
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\noindent{\Large \bf Yarkovsky effect resurrected: a new light on the old
 problems}
\bigskip\bigskip

\noindent{\Large David Vokrouhlick\'y}
\medskip

\noindent{\it Institute of Astronomy, Charles University,
 V Hole\v sovi\v ck\'ach 2, 18000 Prague 8, Czech Republic}
\bigskip

\hrule\bigskip

\noindent{\large \bf Abstract.}\hspace*{2mm}Bodies revolving around
radiation sources in the solar system (the Sun, planets, etc.) partly
absorb the radiation and later reemit this energy in the infrared band.
When the orbiting body is larger than a few centimeters (typically),
permanent temperature gradients on its surface may be established
as a result of a different orientation of its parts toward the
central object. The thermal radiation from the hotter parts on the
surface carries away more energy, and thus more linear momentum,
than from the cooler parts. The overall imbalance of the recoil effects 
of the thermal radiation is called the Yarkovsky effect/force.
We review the present state of art of the Yarkovsky force modelling
as well as some recent applications. We also give an outlook
of the forthcoming work where the Yarkovsky effect might play an
important role.
\bigskip

\hrule\bigskip\bigskip

\noindent{\large \bf 1. Introduction}
\medskip

\noindent Large bodies in the solar system (planets, their natural
satellites, etc.) are most often considered as the ideal
test bodies for the gravitational physics. As such, they even have a
capability to fruitfully probe the structure of the gravity theory,
which nowadays means the first post--Newtonian level of relativistic
theories (see, e.g., \cite{will}). On the contrary, the motion of small
solar system bodies (dust particles, artificial satellites, etc.) is
submitted to a number of non--gravitational forces that usually mask the
tiny details of the gravitational action exerted upon them by the Sun,
planets and other massive bodies. This conclusion is quantitatively
due to the fact that the majority of non--gravitational forces are
surface phenomena (absorption and emission of physical fields and/or
particles of the interplanetary medium) and thus scale with the second
power of the
body's size. On the other hand, it is a remarkable property of the
gravitational interaction that it rather depends on the volume (mass)
of the body and, hence, scales with the third power of its size.
The ratio of the strength of diverse non--gravitational forces to
that of gravity thus typically decreases with the body's size.

The accuracy of the available observations and the correctness of the
theoretical arguments determine whether in a particular case one may 
neglect the influence of the non--gravitational forces or not, since
these forces obviously act on the motion of even the biggest bodies.
For instance, the motion of comets was a purely gravitational problem
for astronomers at the beginning of the 19th~century, while the
precision of the late
19th~century observations already enabled to conjecture the dynamical 
(non--gravitational) action of the outgassing processes.
While comets are possibly a special case, the transition size of the
inactive solar system bodies for whose motion the non--gravitational
forces has to be considered seems to be an interesting, 
``epoch--dependent'' value. About a decade ago, centimeter to decimeter
sized bodies were at the edge of this transition. It is a purpose of
this review to demonstrate that understanding the dynamics
of bodies with sizes up to small asteroids ($\simeq$ 1-10~km), and that
of the Moon, requires an analysis of the non--gravitational force 
influence today.
%\bigskip
\newpage

\noindent{\large \bf 2. Yarkovsky effect(s) -- the principle and
 a qualitative discussion}
\medskip

\noindent Since the time of Maxwell we know that electromagnetic
radiation propagates energy as well as linear momentum (for
historical notes see, e.g., \cite{mig}). Absorption and/or emission
of the radiation means interchange of linear momentum between
the body and the radiation field by the obvious law of action and
reaction. Early astronomical applications of radiation pressure, 
and the derivation of its velocity dependent component (later
known as the Poynting--Robertson effect; for a final form see
\cite{rob}),
may be found for instance in \cite{poy}. A contemporary review
of the radiation force influence on the dynamics of solar system
bodies may be found in \cite{bls}.

A special kind of radiation pressure effect occurs when the surface
temperature of the body is non--uniform. Thermal photons emitted by
hotter regions on the surface carry away more energy, and thus more
linear
momentum, than the corresponding photons emitted by cooler regions
on the body's surface. As a consequence, the recoil action of the
thermal radiation is not averaged out if integrated over the whole
surface of the body and a net radiation force appears. This force
is by a curious historical tradition called the Yarkovsky force
(see, e.g., \cite{opik,v98a,vf98b}).
Though this is the basic principle behind the Yarkovsky effect, one
immediately faces a number of questions and problems. Here are a
few of them.

First, let us understand how a cosmic body may keep some part of its
surface at a higher temperature than other parts. There are
several possibilities. A particular case occurs when a cosmic
body has its own heat source that distributes the energy
anisotropically. The space probes Pioneer~10 and 11
containing asymmetrically located radioactive thermal generators
may be mentioned as an example. It seems that the recent observation
of a solar centered ``mysterious'' acceleration acting on these
probes \cite{anders}, that has been spuriously interpreted as a
violation of Newton's gravity, is due to the thermal (Yarkovsky)
force. Surface processes on active bodies (e.g. sublimation and
ejection of 
dust and gas on comets) may also result in temperature gradients.
However, the anisotropic absorption of external radiation
is the most common way of keeping temperature gradients on the 
surface of inactive cosmic bodies (e.g. asteroid fragments,
passive satellites). The radiation--exposed parts 
of the surface become hotter by a partial absorption
of the radiation energy. The most obvious source of radiation
is the Sun, but the infrared radiation of the planets may also act
as a source in particular cases (e.g. the artificial satellites).

Second, as the bodies get small the temperature differences
throughout their entire volume (and surface) naturally diminish due to
an efficient heat conduction. This conclusion is independent of the
particular mechanism of generation of these temperature differences.
When the small bodies approach the temperature equilibrium
the efficiency of the Yarkovsky effect on them decreases. We thus 
arrive at an important observation, i.e. that the Yarkovsky effect is 
most efficient in some range of sizes only (since for large
bodies its efficiency also decreases as explained in Sec.~1).
A precise evaluation of this range for bodies with particular values
of thermal constants, rotation speeds, orbit geometry etc. is
an important task for a quantitative modelling of the thermal 
effects. The simplest approach will be given in Sec.~3 below. 

Third, since the value of thermal conductivity is finite, there always
exists a time delay between the absorption of the external radiation
and its reemission. The extent of this time delay depends both on
the thermal parameters of the body and on the frequency of the
incoming flux changes on its surface. As it will be explained in detail 
below, this kind of inertia is essential for the dominant orbital 
effects (the semimajor axis drift, in particular).

Our final general comment is as follows: the true appeal of the 
Yarkovsky effect consists in the fact that it ``must exist'' and,
on top of that, it enables to naturally
explain some problems related to the dynamics of solar system
bodies (see Sec.~4). Put in another and sharper perspective: the
Yarkovsky effect was not ``invented'' as an artificial modification
of the existing physical laws in order to explain some observations.
Rather, it represents a truly natural physical and so far neglected
phenomenon. The central problem is then to compute/estimate
quantitatively the precise magnitude of the effect and to determine
its orbital consequences. The simplest model yielding quantitative
answers to the problems discussed above follows in the next section.
\bigskip

\noindent{\large \bf 3. Yarkovsky effect(s) -- the simplest solution}
\medskip

\noindent The problem of the Yarkovsky force estimation naturally splits
onto two steps: (i) determination of the surface temperature distribution,
and (ii) evaluation of the thermal radiation recoil force. The former
problem has been already studied in some detail, especially in the
context of the radiometry of asteroids (e.g. \cite{ls,sls}).
However, the theory of the Yarkovsky effect is characterized by two subtle
points. First, using the radiometric terminology, the Yarkovsky force
computation inevitably requires a thermophysical model (the so called
``standard'' radiometry model \cite{ls} is of no use here). On the
other hand, if the body's orbit is quasi--circular one may significantly
simplify the problem by assuming the temperature to be
close to a constant mean value. Then, a linearized theory may be
derived analytically, as it will be demonstrated below, without any need
of an involved numerical solution.
\medskip

\noindent{\bf 3.1. Assumptions}
\smallskip

\noindent The solution outlined briefly below is due to
Vokrouhlick\'y \cite{v99}. The reader is referred to this
paper for a more detailed discussion.

Here follows a list of the simplifying assumptions used in the
following:
\begin{itemize}
\item the body is spherical (for a generalization see
 \cite{v98b,vf98b}), \vspace{-2mm}
\item its surface emits thermally as a grey body with emissivity
 coefficient $\epsilon$ and according to the Lambert law,
 \vspace{-2mm}
\item the temperature $T({\bf r},t)$ at any position ${\bf r}$ in the
 body and time $t$ is close to a constant, mean value $T_{av}$,
 hence $T({\bf r},t)=T_{av}+\Delta T({\bf r},t)$ with $|\Delta
 T({\bf r},t)| \ll T_{av}$, \vspace{-2mm}
\item the body revolves around a radiation center on a circular
 orbit (eccentricity--related corrections neglected), and
 \vspace{-2mm}
\item the ratio $m$ between the rotation rate $\omega_{rot}$ and the
 orbital mean motion $\omega_{rev}$ is an integer number.
\end{itemize}
The third item above represents the most restrictive assumption,
since it excludes the analysis of the Yarkovsky effect on very
eccentric orbits (where large temperature variations
are to be expected along one revolution around the Sun).
An example of a full, non--linearized theory of the Yarkovsky effect
on large bodies has been developed in \cite{vf98b}. The last
assumption allows us only to express the incident radiation flux
on every surface element in a simple way. It can be easily removed
by a technique discussed in \cite{fv96}.
\medskip

\noindent{\bf 3.2. Basic equations}
\smallskip

\noindent The fundamental equations of the heat diffusion problem
are both energy conservation constraints; either applied to the
body's volume (called also the Fourier equation)
\begin{equation}
 \nabla.\left(K\nabla T\right) = \rho C {\partial T\over \partial t}
  \; , \label{one}
\end{equation}
or to its surface
\begin{equation}
 \left(K{\partial T\over \partial r}\right)_R + \epsilon \sigma T^4 =
  \alpha {\cal E} \; . \label{two}
\end{equation}
The latter equation appears in the problem as a boundary condition
for the temperature ($T$) distribution solution. The body is represented
as a continuum with the following parameters: $\rho$ the density, $K$
the thermal conductivity and $C$ the thermal capacity. All of these
quantities may be, in turn, temperature and location dependent.
In what follows, however, we
neglect these effects considering their values for the mean temperature
$T_{av}$. The surface absorptivity in the optical band is denoted by
$\alpha$ (one minus albedo) and the thermal emissivity by $\epsilon$.
Both of these parameters are typically close to unity. The incident
radiation flux on a given surface element is denoted by ${\cal E}$.

Neglecting the heat conduction term in (\ref{two}) and assuming a
circular orbit of the body we may define the mean temperature $T_{av}$
by: $4 \epsilon \sigma T_{av}^4 = \alpha {\cal E}_\star$. Notice the
factor $4$ in the left--hand side of the previous equation. It is a
consequence of the fact that the thermal energy is radiated by the
whole surface of the body ($4\pi R^2$) while absorption occurs by
the cross--section ($\pi R^2$) only. ${\cal E}_\star$ is the radiation
flux at the body's distance from the radiation center.

Before proceeding with the solution of the Eqs.~(\ref{one}) and
(\ref{two}), it is useful to introduce a set of new variables. As
mentioned above, we shall rather use $\Delta T = T - T_{av}$ since
this is a small quantity with respect to $T_{av}$. Moreover, a
suitable scaling of the variables removes constants and numerical
factors from the equations, compressing them into a single parameter
$\Theta = \sqrt{K\rho C} \sqrt{\omega_{rev}}/\epsilon \sigma
T_\star^3$, called the thermal parameter (here $T_\star$ is a subsolar
temperature defined by $\epsilon \sigma T_\star^4 = \alpha {\cal
E}_\star$). The final, non--dimensional variables are
\begin{itemize}
\item the radial coordinate $r$, measured from the center of the body,
 is to be scaled by $l_s = \sqrt{K/\rho C \omega_{rev}}$, i.e.
 $r \rightarrow r'=r/l_s$ (hereafter the non--dimensional variables
 will be primed), \vspace{-2mm}
\item the temperature change $\Delta T$ is to be scaled by the subsolar
 value $T_\star$, i.e. $\Delta T \rightarrow \Delta T' = \Delta T/
 T_\star$, \vspace{-2mm}
\item the variable part of the external radiation flux on a given surface
 element $\Delta {\cal E} = {\cal E} - {\cal E}_\star/4$ is to be
 scaled by the nominal value ${\cal E}_\star$ of the radiation flux
 at the body's distance from the source, i.e. $\Delta {\cal E}
 \rightarrow \Delta {\cal E}' = \Delta {\cal E}/{\cal E}_\star$, and
 \vspace{-2mm}
\item time $t$ is to be replaced by a complex variable $\zeta =
 \exp[i\omega_{rev} (t-t_0)]$.
\end{itemize}
The choice of the time origin $t_0$ has to do with the reference frame
used throughout the solution. Given the assumed symmetry of the body
we shall
use a system of spherical coordinates with the origin in the body's
center. The colatitude $\theta$ is measured from the body's spin
axis and the longitude $\phi$ from the equatorial axis $X$. The
system rigidly rotates with the body so that the axis $X$ points
toward the radiation center at an arbitrary time $t_0$ (one
easily proves that such an instant always exists on a circular
orbit).

Rewriting Eqs.~(\ref{one}) and (\ref{two}) into the new set
of variables, we obtain
\begin{equation}
 i \zeta {\partial \over \partial \zeta}
  \Delta T'(r';\theta,\phi;\zeta) =
   {1\over r'^2}\left\{{\partial \over \partial r'}
  \left(r'^2{\partial \over \partial r'}\right) + \Lambda\left(\theta,
  \phi\right)\right\}\, \Delta T'(r';\theta,\phi;\zeta) \label{three}
\end{equation}
replacing (\ref{one}) and
\begin{equation}
 \sqrt{2} \Delta T'+ \Theta\left({\partial \Delta T'\over \partial r'}
  \right)_{R'} + {\cal O}(2) = \Delta {\cal E}'\; ,\label{four}
\end{equation}
replacing (\ref{two}). The operator $\Lambda(\theta,\phi)$ in
Eq.~(\ref{three}) represents the angular part of the Laplacian and
reads
\begin{equation}
 \Lambda\left(\theta,\phi\right) = {1\over \sin\theta}\left[{\partial
  \over \partial \theta}\left(\sin\theta{\partial \over \partial \theta}
  \right)+ {1\over \sin\theta}{\partial^2 \over \partial \phi^2}\right]
 \; .\label{five}
\end{equation}
The ${\cal O}(2) \equiv {\cal O}[(\Delta T/T_{av})^2]$ term reminds
us that the non--linear terms in the presumably small parameter
$\Delta T/T_{av}$ are always omitted in our approach. Peterson \cite{pet}
developed a quadratic theory in this parameter and noticed that
his results differ by a small amount from the predictions of the linearized
theory, provided orbits of low eccentricity are assumed. This conclusion
has been later verified also within the complete non--linear theory
\cite{vf98b}.

The general goal of the solution is to find the temperature $\Delta T'$
everywhere in the body $(r',\theta,\phi)$, with $r'$ ranging up to $R'$,
where the surface is located, and at any time $\zeta$. The solution is
determined by (i) the value of the thermal parameter $\Theta$, and
(ii) an explicit specification of the radiation flux term
$\Delta{\cal E}'$.
\medskip

\noindent{\bf 3.3. Radiation flux term}
\smallskip

\noindent The term $\Delta{\cal E}'$ is to be determined for any
surface element parametrized by the angular variables $(\theta,
\phi)$. Since it represents a function on a sphere, one may
seek a spherical function representation
\begin{eqnarray}
 \Delta {\cal E}' =\sum_{n\geq 1}\,\sum_{k=-n}^n a_{nk}(\zeta)\,
   Y_{nk}(\theta,\phi)\; . \label{six}
\end{eqnarray}
An advantage of the solution is due to the fact that only the dipole
($n=1$) coefficients are to be needed. The latter read
\begin{equation}
 a_{10}(\zeta) = \sqrt{{\pi \over 3}}\, \cos\theta_0\; , \qquad
 a_{1\pm 1}(\zeta) = \mp\sqrt{{\pi \over 6}}\, \sin\theta_0
 \, e^{\mp i \phi_0} \; ,  \label{seven}
\end{equation}
where $(\theta_0,\phi_0)$ are the colatitude and longitude of the local
position vector of the radiation center. As the body moves
on its orbit, these two quantities change. After some algebra one
finds
\begin{equation}
 \cos\theta_0 = -\sin\gamma \sin\lambda = {i\over 2}\sin\gamma
  \left(\zeta-\zeta^{-1}\right)\; , \label{eight}
\end{equation}
where $\gamma$ is the obliquity of the spin axis, and
\begin{equation}
 \sin\theta_0\, e^{\pm i\phi_0} = \sin^2{\gamma\over 2}\;
  \zeta^{\mp (m+1)} +\cos^2 {\gamma\over 2}\; \zeta^{\mp (m-1)}\; .
 \label{nine}
\end{equation}
This remarkably simple representation of the dipole part of the
insolation function can be recovered also by using more involved
expressions, introduced in paleoclimatic studies (e.g., \cite{rub94}).
\medskip

\noindent{\bf 3.4. Temperature dipole solution}
\smallskip

\noindent Having formulated the problem in detail, we can now embark
on solving the Eqs.~(\ref{three}) and (\ref{four}) with the
external flux term (\ref{six}). In particular, we shall be again
interested in obtaining the dipole part coefficients
$t'_{nk}(r';\zeta)$ of the spherical function representation
\begin{equation}
 \Delta T'(r';\theta,\phi;\zeta) = \sum_{n\geq 1}\,\sum_{k=-n}^n
  t'_{nk}(r';\zeta)\,Y_{nk}(\theta,\phi) \; . \label{ten}
\end{equation}
The most important simplification of the problem is due to the linearity
of the system (\ref{three}) -- (\ref{four}). The orthogonality of
the spherical harmonics $Y_{nk}(\theta,\phi)$ then results in a
complete decoupling of the equations for the amplitudes $t'_{nk}(r';
\zeta)$. In particular, they satisfy
\begin{eqnarray}
 i \zeta {\partial \over \partial \zeta}
  t'_{nk}(r';\zeta) = {1\over r'^2}\left\{{\partial \over \partial r'}
  \left(r'^2{\partial \over \partial r'}\right) - n\left(n+1
  \right)\right\}\, t'_{nk}(r';\zeta)\; ,\label{eleven}
\end{eqnarray}
with the boundary constraints
\begin{equation}
 \sqrt{2}\, t'_{nk}(R';\zeta) + \Theta\left({\partial t'_{nk}\over
 \partial r'}\right)_{(R';\zeta)} = a_{nk}(\zeta)
 \label{twelve}
\end{equation}
Since the operator $\zeta (d/d \zeta)$ ``recycles'' the Fourier modes
[$\zeta (d/d \zeta) \zeta^k = k \zeta^k$ for all integer values $k$], we
may also treat separately all the components of the $a_{nk}$--amplitudes
in the Fourier representation [see Eqs.~(\ref{eight}) -- (\ref{nine})
for the dipole terms]. We may thus easily separate the radial and
temporal parts in $t'_{nk}(r';\zeta)$. The radial functions then satisfy
a system of homogeneous spherical Bessel equations with complex
coefficients. The details of the solution may be found, for
instance, in \cite{v98a,v99}.

After some deal of algebra one can thus express the zonal dipole
coefficient $t'_{10}(r';\zeta)$ at the surface of the body $r'=R'$
as
\begin{equation}
 t'_{10}(R';\zeta) = -\sqrt{\pi\over 6}\, \sin\gamma \,
  {E_{R'} \sin(\lambda + \delta_{R'})\over 1+\chi}\; ,
 \label{thirteen}
\end{equation}
where $\lambda = \omega_{rev}(t-t_0)$ is the mean orbital longitude.
Here, we have introduced the amplitude $E_{R'}$ and the phase
$\delta_{R'}$ by
\begin{equation}
 E_{R'}\exp\left(i\delta_{R'}\right) =
 {A(x)+iB(x)\over C(x)+iD(x)} \; , \label{fourteen}
\end{equation}
with $x=\sqrt{2}R'$ and the auxiliary functions $A(x)$, $B(x)$, $C(x)$,
$D(x)$ reading
\begin{eqnarray}
 A(x) &=& -\left(x+2\right)- e^x\left[\left(x-2\right)\cos x - x
  \sin x\right]\; , \label{fifteen} \\
 B(x) &=& -x - e^x\left[x\cos x +\left(x-2\right)\sin x\right]\; ,
  \label{sixteen} \\
 C(x) &=& A(x)+ {\chi\over 1+\chi}\;\left\{3\left(x+2\right)+ e^x
  \left[3\left(x-2\right)\cos x+ x\left(x-3\right)\sin x\right]
  \right\}\; ,\label{seventeen} \\
 D(x) &=& B(x)+ {\chi\over 1+\chi}\;\left\{x\left(x+3\right) -e^x
  \left[x\left(x-3\right)\cos x- 3\left(x-2\right)\sin x\right]
  \right\} \; .\label{eightteen}
\end{eqnarray}
The parameter $\chi$ is defined by: $\chi = \Theta/(\sqrt{2}R')$.

The Fourier representation of the tesseral dipole coefficients
$t'_{1\pm 1}(r';\zeta)$ has a slightly more complicated form
\begin{equation}
 t'_{1\pm 1}(r';\zeta) = \tau^{\pm}_{1}(r')\, \zeta^{\pm (m+1)} +
  \tau^{\pm}_{-1}(r')\, \zeta^{\pm (m-1)}\; . \label{nineteen}
\end{equation}
The radial $\tau$--functions satisfy again a system of spherical
Bessel equations and can be found, for instance, in \cite{v99}.
\medskip

\noindent{\bf 3.5. Recoil force evaluation}
\smallskip

\noindent Having determined the basic features of the temperature
distribution on the body's surface, we may then proceed to compute
the recoil thermal (Yarkovsky) force. Given an oriented surface
element $d{\bf S}$ with temperature $T$, the recoil force per
unit mass of the body $d{\bf f}$ reads: $d{\bf f} = -(2/3)\,(\epsilon
\sigma T^4/m c)\, d{\bf S}$ (see, e.g., \cite{mnf}). The structure
of this formula is directly related to the assumed Lambert law for the
directional characteristic of the thermal radiation: (i) the force
is opposite to the surface orientation vector ($d{\bf f} \propto
-d{\bf S}$), and (ii) the factor $2/3$ is a consequence of the
radiation isotropy. Integrating over the whole surface of
the body (sphere), and keeping the linear approximation $T^4 \simeq
T_{av}^4 + 4 T_{av}^3 \Delta T + \ldots$, we obtain
\begin{equation}
 {\bf f}(\zeta) = -{2\sqrt{2}\over 3\pi} \alpha\Phi \int d\Omega\,
  \Delta T'(R';\theta,\phi;\zeta)\, {\bf n}\; ,\label{twenty}
\end{equation}
for the thermal recoil force per unit of mass of the body. Here,
$\Phi = ({\cal E}_\star \pi R^2/mc)$ is the usual radiation force
factor: $m$ is the body's mass, $c$ the velocity of light and ${\bf
n}$ the unit vector normal to the surface.

Since the surface normal vector ${\bf n}$ in the integrand of
(\ref{twenty}) can be expressed as a combination of the spherical
functions of degree $1$ only, and the spherical functions are
orthogonal, we obtain the following expression for the three
Yarkovsky force components
\begin{eqnarray}
 f_X(\zeta) + if_Y(\zeta) &=& -{8\over 3\sqrt{3\pi}}\, \alpha\Phi\,
  t'_{1-1}(R';\zeta)\; , \label{twentyone} \\
 f_Z(\zeta) &=& -{4\over 3}\sqrt{2\over 3\pi}\,\alpha\Phi\,
  t'_{10}(R';\zeta)\; . \label{twentytwo}
\end{eqnarray}
Obviously, these are again referred to the coordinate system rigidly
rotating with the body (introduced in Sec.~3.2). In particular,
the component $f_Z$ yields a projection of the Yarkovsky force
onto the spin axis direction, while $(f_X,f_Y)$ are the out-of-spin
components of the Yarkovsky force.

Notice that the spin--aligned component $f_Z$ -- (\ref{twentytwo})
and (\ref{thirteen}) -- does not depend on the rotation period
but it is uniquely determined by the revolution period around
the radiation center. As a consequence, it is called the seasonal
component of the Yarkovsky force. Interestingly, this component
has been entirely omitted in the early works \cite{opik,radz,pet}
but has been introduced only in the 80s in the context of the
LAGEOS dynamics \cite{rub87,slab}. Yet, we shall see below that
it may have important consequences for the delivery of stony meteorites
too.

The pioneering works of \"Opik and Radzievski \cite{opik,radz}
dealt with the effects of the out--of--spin Yarkovsky force
components $f_X$ and $f_Y$. With the help of explicit expressions
of the radial $\tau$--functions in (\ref{nineteen}) we obtain
\begin{equation}
 f_X+if_Y = -{4\over 9}\,{\alpha\Phi\over 1+\chi} \Bigl\{\sin^2
  {\gamma\over 2}\, E_{R_+'} \exp(-i\delta_{R_+'})\, \zeta^{-1}
  + \cos^2 {\gamma\over 2}\, E_{R_-'} \exp(-i\delta_{R_-'})\,
  \zeta\Bigr\}\,\zeta^{-m}\, . \label{twentythree}
\end{equation}
%\begin{eqnarray}
% f_X+if_Y\!\! &=&\!\! -{4\over 9}\,{\alpha\Phi\over
%  1+\chi} \Bigl\{\sin^2 {\gamma\over 2}\, E_{R_+'} \exp(
%  -i\delta_{R_+'})\, \zeta^{-1} \label{twentythree} \\
% & &\qquad\quad\;\;\, + \cos^2 {\gamma\over 2}\, E_{R_-'} \exp(-i
%  \delta_{R_-'})\, \zeta\Bigr\}\,\zeta^{-m}\, . \nonumber
%\end{eqnarray}
Here we have defined $R_\pm' = \sqrt{1\pm 1/m}\, R_m'$ ($R_m'=\sqrt{m}\,
R'$), while the remaining quantities are the same we used previously.
Since the force components in (\ref{twentythree}) depend on the
rotation frequency -- via the ``commensurability'' factor $m$ --
we call them diurnal.
\medskip

\noindent{\bf 3.6. Secular mobility of the semimajor axis}
\smallskip

\noindent The three Yarkovsky force components $(f_X,f_Y,f_Z)$ from
the previous Section are given in a coordinate system rigidly
rotating with the body. In order to exploit them to estimate
their orbital effects one has to transform them into the
orbit--oriented reference frame. The evaluation of the transverse
force component ${\cal T}$ is of a particular interest, since the
semimajor axis change is given by: $(da/dt)= 2{\cal T}/\omega_{rev}
+{\cal O}(e)$
\cite{bc}. In what follows we shall not consider the perturbations of
the other orbital elements, since they are of much lesser importance.

First, let us consider the effect of the spin--aligned Yarkovsky
force component $f_Z$. Averaging the corresponding Gauss
equation mentioned above over one revolution around the center
we obtain a mean drift rate of the semimajor axis in the form
\begin{equation}
 \left({da\over dt}\right)_s = \frac{4\alpha}{9}\,{\Phi\over
 \omega_{\rm rev}}
  \,{E_{R'}\sin\delta_{R'}\over 1+\chi}\, \sin^2\gamma\; .
 \label{twentyfour}
\end{equation}
Analogously, we may obtain the corresponding mean semimajor axis
drift--rate due to the diurnal Yarkovsky force components $(f_X,
f_Y)$ as
\begin{eqnarray}
 \left({da\over dt}\right)_d &=& -{8\alpha\over 9\omega_{\rm rev}}\,{
  \Phi\over 1+\chi} \Bigl\{\cos^4 {\gamma\over 2}\, E_{R_-'}
  \sin\delta_{R_-'} - \sin^4 {\gamma\over 2}\, E_{R_+'}
   \sin\delta_{R_+'}\Bigr\} \nonumber \\
   &\simeq & -{8\alpha\over 9}\,{\Phi\over \omega_{\rm rev}}
   \,{E_{R_m'} \sin\delta_{R_m'}\over 1+\chi}\, \cos\gamma \, .
 \label{twentyfive}
\end{eqnarray}
Here, we can note that in typical astronomical applications $m\gg 1$,
resulting in $R_\pm' \simeq R_m'$. We remind that $R_m'=\sqrt{m}\,
R'$, while $R' = R/l_s$ and $l_s$ is the penetration depth of the
seasonal thermal wave. We may also introduce the penetration depth
of the diurnal thermal wave $l_d=l_s/\sqrt{m}$, so that $R_m'=R/
l_d$.

The Eqs.~(\ref{twentyfour}) and (\ref{twentyfive}) giving the
seasonal and the diurnal mean mobility of the semimajor axis
resemble each other to a large extent apart from two important
differences: (i) although structurally they both contain the
$(E\, \sin\delta)$ factor,
the scaling of $R$ (radius of the body) is different in the
two cases (notice that the $\chi$ variable in the denominator is
scaling--invariant), and (ii) there is an entirely different dependence
on the obliquity $\gamma$. As far as the first issue is concerned, we
remind that $R_m'=\sqrt{m}\,R'$, where $\sqrt{m}\simeq 10^{2-4}$ in
the typical astronomical applications. Secondly, thanks to the
second power of $\sin\gamma$ in Eq.~(\ref{twentyfour}) the seasonal
effect always results in an orbit decay: $(da/dt)_s <0$ (notice
that the phase $\delta_R'$ is negative). On the other hand,
the diurnal semimajor axis mobility $(da/dt)_d$ may be either
positive or negative.
\begin{figure}[t]
\begin{center}
\epsfxsize=7.5cm
\mbox{\epsfbox{pra1a.eps}}
\epsfxsize=7.5cm
\mbox{\epsfbox{pra1b.eps}}
\end{center}
\vspace{-5mm}
\caption[]{The typical time $T$ (ordinate) for drifting by
 $0.05$~AU in semimajor axis for a body of radius $R$ (abscissa).
 The time has been averaged over an ensemble of objects with
 randomly distributed spin axis orientations. Spin axis re--orientation,
 modelled as a Poissonian process with a characteristic timescale
 $\tau_{rot}\simeq 3.3\,\sqrt{R}$~Myr, is taken into account in
 course of the evolution. Diurnal (left panel a)
 and seasonal (right panel b) effects treated separately. Bodies
 with different values of the surface conductivity are considered (the
 numerical value of the conductivity in W/m/K parametrizes the
 different curves). The dashed lines give an estimation of the
 collisional lifetime for a basalt body of given radius and a
 characteristic time for changing the orientation of its spin axis.
 \label{f1}}
\end{figure}

To get a first glimpse of the effect of the semimajor axis mobility
we performed a simple test (for more discussion see
\cite{hetal1,hetal2}). Let us assume a population of bodies (e.g.
asteroid fragments) at an approximate distance of $2.25$~AU
from the Sun. The initial orientation of the spin axes of the
bodies are randomly distributed in space. Then, the semimajor axis of
each of the bodies is numerically evolved until a total drift of
$0.05$~AU (a typical distance to the main--belt resonance locations)
is achieved. During the evolution we consider random
re--orientation events of the spin axis, modelled as a Poissonian
process with a characteristic timescale $\tau_{rot}\simeq
3.3\,\sqrt{R}$~Myr (see below; $R$ in meters). The influence of
both components of the Yarkovsky effect -- diurnal and seasonal --
is considered separately. The resulting dependence of the 
simualtion time on the size of
the bodies is shown in Figures~\ref{f1}. A couple of important
observations can be made with the help of these results.

First, notice a very strong dependence of the results on the
value of the surface conductivity $K$, which parametrizes each of the
curves in Figures~\ref{f1}. In the case of low conductivity surfaces
(the lowest value used in the simulation $K=0.0015$~W/m/K has been
reported for the lunar regolith \cite{lan}), the diurnal effect mobility
largely dominates the seasonal mobility. Such a low conductivity
is possible only if the surface is particulated and composed
of semi--detached small grains of dust. On the contrary, when a
high conductivity of the surface is assumed, the seasonal effect
starts to dominate the diurnal effect mobility.
The range of values $K=1-3$~W/m/K is typical for bare-rock meteorites
\cite{rub95}. The seasonal effect becomes inefficient at still
higher values of the conductivity -- e.g. $K=40$~W/m/K, a case
appropriate of iron--rich objects -- as it may be seen in panel (b).
However, iron--rich bodies are very resistive to collisional
disruptions, as discussed below, so that the low mobility is
compensated by extremely long lifetime of these objects. As a
result, the iron--rich bodies may very slowly drift over
nearly the whole width of the main asteroid belt during their
lifetime (see, e.g., \cite{hetal2,vf98a}). The transition of the
diurnal to seasonal effect dominance occurs for $K\simeq
0.01$~W/m/K, a value that corresponds to highly porous surfaces
for stony objects \cite{ym}.

Since no direct observations of the surface conductivity are
presently available, the above mentioned strong $K$--dependency of
the results remains the principal unknown factor in all
applications mentioned in Secs.~4 and 5. A limited knowledge
of this parameter is due to radiometry of large asteroids
\cite{sls,ls}, but we do not have any information about the
sub--dekameter objects including the particularly important
meter--sized meteorite parent bodies.

Though the above performed test includes one aspect of the
mutual collisions of the main--belt bodies, notably a rather
frequent reorientation of their spin axes and the corresponding
change of the obliquity, it neglects other aspects.
From the perspective of the Yarkovsky effect applications,
two categories of mutual collisions of 
main--belt objects are of concern (see, e.g., \cite{fvh}):
\begin{itemize}
\item sufficiently low--energy impacts that are not able to
 disintegrate the target but still exchange enough angular
 momentum that the target spin axis is significantly
 re--oriented (considered above), and \vspace{-2mm}
\item high--energy disrupting impacts (neglected above).
\end{itemize}
The latter events clearly constrain the time available for
drifting in semimajor axis. However, the former events are
also very important since a frequent reorientation of the spin
axis decreases the $a$--mobility, especially in the diurnal
effect case.

Farinella et~al. \cite{fvh} have estimated median statistical
timescales for both processes listed above and obtained (i)
$\tau_{rot}\simeq 3.3\,\sqrt{R}$~Myr ($R$ in meters) for the
characteristic
time to attain a sizable change in the orientation of the body's spin
axis, and (ii) $\tau_{disr}\simeq 16.8\, \sqrt{R}$~Myr for the
body's lifetime. A factor 2 error is possible in these estimations
since only a very simple ``Dohnanyi--like'' model \cite{doh} for the
stationary size distribution of the projectiles and a size
independent strength of the target were used.

Equipped with this knowledge, we may better understand the
Figures~\ref{f1}, where the estimated lifetime $\tau_{disr}$ and
the reorientation timescale $\tau_{rot}$ have been drawn as
dashed lines. Constraining the mean time to drift by
$0.05$~AU during the particle lifetime at the given size, we
notice that most fragments (apart from the large ones) with a very--low
conductivity surface manage to drift along this distance. Since
the necessary time is longer than the estimated
spin axis reorientation timescale, the $a$--mobility
is indeed altered by frequent changes in the obliquity.
Reminding that the diurnal effect can lead both to positive
and to negative $a$--drift, the evolution of such particles
should have the character of a random walk. On the other hand,
the seasonal effect can move bare--rock fragments (with $K\simeq
1-3$~W/m/K) by $0.05$~AU within their lifetime, provided their
size is larger than about $2-5$~meters. The orbits of these bodies
permanently decay toward the radiation center (the Sun), and
the obliquity--changing collisions affect the rapidity of this
drift only. However, 
the fact that individual bodies of a given size cannot be transported
by the Yarkovsky effects for more than a certain distance, for instance
high--conductivity fragments smaller than $1$ meter, does not mean
that sources (big asteroids) distant from the resonances cannot
feed them with bodies of this size. The collisional disruption of a 
first--generation body
releases a swarm of secondary fragments. The fate of their orbits
is again affected by Yarkovsky effects, so it might be only the
secondary, tertiary etc. generation of fragments that may
eventually reach the resonances. Resonance feeding by distant
sources of material (e.g. Vesta) is then a very complex
problem, whose solution needs an analysis of both collisional and
Yarkovsky evolution. A first step in building such an
involved model is presented in the following section.
\bigskip

\noindent{\large \bf 4. Applications -- worked examples}
\medskip

\noindent Since we are presently only starting to understand and/or 
exploit all possible applications of the Yarkovsky effect in
solar system dynamics, this section is intended to be a brief
overview of the results achieved so far. In the next section, we
shall attempt to envisage the future work.

The original motivation behind the Yarkovsky effect was related
to the transport of bodies into Earth--crossing orbits
\cite{opik,pet}. However, since till the end of 70s nothing
precise was known about the role of the mean motion and the secular
resonances for the Earth--crosser dynamics, \"Opik \cite{opik} and even
Peterson \cite{pet} assumed a slow, permanent decay of
semimajor axes from from the typical values in the main belt
to $1$~AU (see also \cite{rub95}). Very long timescales and/or
unrealistically slow rotation periods were required (compare 
with results in Figs.~1 above), a situation that probably 
weakened the overall credibility of the Yarkovsky effect in late
70s. Both these assumptions casted a shadow of doubt that something
has been misunderstood or underestimated in the thermal models
of asteroid fragments. The perspective started to change with 
improved understanding of the fundamental importance of the mean--motion
and secular resonances for the meteoroids and NEAs delivery 
toward the Earth. The source regions of the fragments thus appeared
to be much closer to the escape hatches toward the Earth, so a
Yarkovsky dominated drift
of the semimajor axis of only a fraction of one Astronomical Unit 
(typically $0.05$ -- $0.15$~AU) is necessary for starting the
transport chain. Millions to tens of millions of years are then
sufficient for the Yarkovsky dominated period in the transport
scenario. Such a timescale becomes astronomically realistic,
since it is comparable to the estimated lifetime of the objects
and it also fits well to transport timescale as ``measured''
by the cosmic--ray exposure (CRE) ages of the meteorites
(e.g. \cite{mg}).

The orbital evolution under the mutual action of the Yarkovsky
thermal effects and the gravitational influence of the planets close
to the main--belt resonances is thus a particular sub--problem to be 
understood in the scenario outlined above. This task has been 
briefly studied both semi--analytically \cite{vf98a,vf98b} and 
numerically \cite{bott}.

The semi--analytical modelling in \cite{vf98a} confirmed the 
high efficiency of the Yarkovsky transport
of the Flora--region--born objects to the $\nu_6$ secular resonance.
Since the semimajor axis drift of the meter--sized and larger bodies
is sufficiently slow, the bodies typically spend enough time in the
resonance to significantly increase their orbital eccentricity and to
become Earth--crossers as outlined above. The results also
confirm that $10$ to $100$~Myr are statistically needed on
average to reach the resonance (corresponding thus well to the
analytic estimates given above). As a curiosity, Vokrouhlick\'y and 
Farinella \cite{vf98a} also observed orbits with semimajor
axis mobility driven by the diurnal Yarkovsky effect that cross the
$\nu_6$ resonance, but episodically return back to the main belt when
their obliquity is changed. In the course of this evolution the orbital
eccentricity undergoes large perturbations whenever the orbit
crosses the resonance. The real astronomical importance of these cases
may be, however, diminished by necessary long--timescales over which
the bodies typically suffer catastrophic disruption (see above) and
the neglected encounters with Mars.

The previous results, based on a simple semi--analytical model,
were confirmed by direct numerical integrations \cite{bott}. On top
of this confirmation, the latter also allowed to discover additional
subtle details about the interaction of the Yarkovsky driven orbits
with resonances. Most interestingly, Bottke et~al. \cite{bott} noticed 
that the $3:1$ resonance may be crossed/jumped
by a significant number of Yarkovsky drifting fragments originating 
in the Maria family. These authors also indicate an evidence for the
interaction of the slowly drifting large fragments with the high--order
and/or multiple resonances in the inner part of the main belt
(e.g. \cite{nm1}). Since this
issue is of particular interest for the origin of large NEAs,
as argued in \cite{fv99a}, a more detailed study whose objective would 
focus on statistics and timescales necessary for feeding these
complex resonances with large bodies is desirable.
\begin{figure}[t]
%\epsfxsize=\textwidth
\epsfxsize=13cm
\begin{center}
\mbox{\epsfbox{hcre.eps}}
\end{center}
\caption[]{Simulated distribution of the stony meteorite CRE ages
 $T_e$ (black curves) compared with the observed distribution for 240
 L--chondrites (shaded curve; data kindly provided by T.~Graf).
 CRE age on the abscissa and a normalized
 distribution on the ordinate. These results correspond to
 a permanent impact activity on (6)~Hebe that releases the initial
 population of ejecta handled by the Monte Carlo simulation in
 \cite{vf99}. Three values of the fragments surface conductivity assumed:
 (i) $K=0.0015$~W/m/K -- curve~3, (ii) $K=0.1$~W/m/K -- curve~2, and
 (iii) $K=1$~W/m/K -- curve~1. More details of the simulation in
 \cite{vf99}. \label{f2}}
\end{figure}

Whereas the previous references focused on the dynamical aspects of
the Yarkovsky effects (by investigating individual fragment orbits),
Vokrouhlick\'y and Farinella \cite{vf99} have modelled
in a statistical way the evolution of large ``swarms'' of
fragments released by catastrophic break--up events or impacts
on large asteroids in the main belt. Their dynamics is represented in
proper element space by a simple, two--component model, including
(i) the secular semimajor axis drifts (\ref{twentyfour}) and
(\ref{twentyfive}) due to Yarkovsky effects, and (ii) the effects
of random impact events resulting in the cascade--like
generation of new populations of fragments. The main features
of the model are as follows. The combination of the two studied 
phenomena can feed efficiently the main resonances with small
asteroid fragments from nearly all locations in the main belt,
implying that the transport mechanism of the meteorites and small
NEAs is less selective than thought before.
Direct injections, considered in the ``pre--Yarkovsky'' studies,
seem to dominate this feeding process only for sources close
to the resonances. The flux of the objects to the resonances is,
contrary to the previous scenario, spread over a long
timespan. As an example, we quote that some $50$ to $80$~\% of the mass
of the initial population of bodies released in the Flora--region
may be transported to the resonances (dominantly the $\nu_6$
resonance) over $0.5$ to $2$~Gyr. Another important quantitative
result from this model is that the distribution of accumulated
CRE ages in the population of fragments reaching
the Earth is in fair agreement with the observations
(e.g. \cite{mg}). Relatively old events are likely to generate the
background CRE age profiles (like in the case of L--chondrites)
peaked at $20$ -- $50$~Myr for stones and $200$ -- $500$~Myr for
irons, while comparatively recent events may create discrete peaks
in the CRE age distributions (such as the $7$ -- $8$~Myr
prominent peak for the H--chondrites). In the latter case, the bulk
of the original fragment population may still reside in the main
belt and will supply a significant flux of meteorites in the 
future (next Myrs) and alter the currently observed distribution of
their CRE ages. An example of the background CRE age
distribution simulation is given in Figure~\ref{f2}, where the
observed CRE age distribution of the L--chondrites is also shown
for comparison (shaded curve). Apart from a very good correspondence
(a similar simulation without Yarkovsky effects produces a
significant fraction of CRE ages shorter than $3$~Myr),
one can immediately notice a sensitivity of the results on the
surface conductivity value of the objects. This latter quantity
affects the thermal mobility of the semimajor axis as explained
and demonstrated above and, thus, regulates the efficiency and the
time dependence of the fragments flux to the resonances. Since the
surface conductivity of the objects seems to be the principal
unknown parameter, we hope that the forthcoming spacecraft data
(such as from the NEAR mission) may help in getting a real
constraint on this quantity.

Farinella and Vokrouhlick\'y \cite{fv99a} have also noted that 
Yarkovsky
effects are capable of providing some semimajor axis mobility even to
km--sized small asteroids in the main belt. Typically, bodies in the
$1$ to $10$~km diameter range may move in semimajor axis by $\approx
0.01$~AU within their collisional lifetimes of $0.1$ to $1$~Byr.
Interestingly, this result is almost independent of the thermal
properties of the surface material. This mobility may be a key
mechanism for feeding the high--order and/or multiple resonances in the
inner asteroid belt, which have been recently identified as the most
likely dynamical routes for multi--km sized Mars--crossers and NEAs
\cite{metal}. Other likely consequences are the eventual fall
into the main resonances of fragments generated ``on the brink''
\cite{mf,kmf} and the gradual spreading in semimajor axis of the small
members of the most compact asteroid families. A gradient in
sizes of the bodies lying on the very edge of the resonances
(with smaller objects getting closer to the center of the resonance)
would be another consequence, though other mechanisms may produce this
effect too \cite{nak}.
\bigskip

\noindent{\large \bf 5. Outlook and conclusions}
\medskip

\noindent Hereafter, we briefly comment on the open/unsolved problems
related to the Yarkovsky effects that we expect to be investigated
in the near future. Obviously, understanding details of the dynamics of
the meter-- to kilometer--sized bodies in the solar system may
reveal other situations where the Yarkovsky effects may play an 
interesting role.

As far as the thermal physics modelling is concerned we may mention
that a number of improvements and/or consistency checks would be
desirable in the near future. The models we are using today, some
of them have been mentioned above, are in many respects only
approximations. The assumptions of the fragments homogeneity
and the temperature independence of their thermal constants
are surely not satisfied exactly. The corresponding refinement of the
models might help to better understand the Yarkovsky effect
contribution in some particular situations.
For instance, in a recent work Vokrouhlick\'y and Bro\v z \cite{vb99}
have solved exactly the seasonal variant of the Yarkovsky effect
for structured objects with a hard--rock core covered with a thin
insulating layer (``regolith''). Their results indicate that the
previous estimation of the seasonal mobility of the semimajor axis
has been underestimated by a factor of $2$ to $10$ for this class
of objects (depending on the thickness of the surface low conductive
layer). 
 
The evidence of the Yarkovsky effect in all applications discussed 
above was mostly based on investigating the statistical properties
of large samples of objects (CRE age distributions) or
on qualitative arguments (origin
of large NEAs). It would surely be interesting to prove the
existence of Yarkovsky effect by detecting orbital perturbation
of an individual object. Though we succeeded to do so in the case
of the artificial satellite motion around the Earth \cite{rub87},
a similar evidence has not been obtained yet for the 
dynamics of natural bodies. On one hand, we do know nearly
nothing about the orbital dynamics of meter-- to deka--meter sized
objects, for which the effects are maximum; on the other hand the
Yarkovsky force becomes rather small for large asteroids. Both
aspects make the proof of the direct dynamical evidence of the 
Yarkovsky effect difficult. In this respect the motion of the 
Apollo--type asteroid (1566) Icarus might be the most promising 
case (G.~Valsecchi, personal communication), since its orbit
approaches very closely the Sun ($q=0.187$~AU; \cite{yeo}). This 
observation, together with its smallness ($R\simeq 450$~m;
\cite{vee}), is obviously favourable for maximizing the thermal
effects. Indeed, Sitarski \cite{sit} does report an unexplained
secular drift of Icarus' semimajor axis of about 
$7.5\times 10^{-4}$~AU/Myr, a value which seems to fit fairly
well the expected Yarkovsky drift. Using the orbital elements
of (1566) Icarus from \cite{yeo} and the orientation of the spin
axis from \cite{dean}, we one can estimate a mean drift of
the semimajor axis. By tuning the surface conductivity value to
$K=0.01$~W/m/K one obtains a nearly perfect agreement with 
Sitarski's observed value (W.F.~Bottke, personal communication).
However, a detailed determination of the (1566) Icarus orbit,
including the error analysis, would be very important since it
might reveal an interesting test of the Yarkovsky effect.

In Sec.~4 we have reported on modelling the CRE age distribution
for the stony meteorites and the role of the Yarkovsky effects
therein. The investigation of the CRE age distribution of the 
iron--rich meteorites has been excluded from that discussion because this
latter problem is more difficult. The slower drift in semimajor axis
for the iron--rich bodies (see, e.g., \cite{fvh}) is consistent with
the longer transport timescale to the principal resonances in the main
belt (and thus longer observed CREs), but it also means that
the principal resonances may not be the only routes that participate
on their delivery. In particular, the recently discovered high--order
and/or multiple resonances \cite{metal,nm1} might be of importance.
A rather complex information is necessary for including these
subtle resonances into Monte Carlo simulations similar to \cite{vf99}
and awaits thus a further effort.

The Yarkovsky effects may also provide a natural mechanism for
explaining the observed overabundance of $10$--$100$~m bodies
among NEAs (e.g. \cite{rab1,rab2}). The Yarkovsky mobility of a
population of bare--rock (or iron--rich) fragments, dominated by
the seasonal variant of the Yarkovsky effect, is maximum for bodies
of size comparable to the penetration depth of the seasonal thermal
wave (about $10$~m for stones and $20$~m for irons) -- see Figure~1b.
Thus, we may
expect that these bodies are preferentially removed from the main
belt, and eventually show up in a relative overabundance within the
population of Earth--crossing objects (e.g. \cite{vf98a,hetal2}).
Moreover, their removal from the main--belt population would imply a
longer collisional lifetime for the bodies of about $100$~meters in
size (which are typically fragmented by impacts of the $10$~meter
bodies), allowing them to drift over a wider portion of the belt
and eventually feed the resonances. Details of the intimately
connected size distribution statistics of main--belt and near--Earth
objects can be thus significantly influenced by the Yarkovsky
effect and requires a quantitative study.

We have already mentioned our belief that many of the Yarkovsky
effect applications in the solar system dynamics still await
their ``discovery''. Among the potential candidates of such
problems we may quote the existence and fate of the
possible intra--mercurian objects, the dynamics of the meter--sized
objects in the planetary rings, the stability of asteroid binaries,
the Kuiper--belt objects dynamics
etc. We thus give here some useful internet addresses where the 
interested reader may check further developments in the theory
and applications of the Yarkovsky effect: 
\begin{itemize}
\item http://astro.mff.cuni.cz/davok/davok.htm, \vspace{-2mm}
\item http://tycho.dm.unipi.it/{\~{ }}paolof/homefarinella.html,
 \vspace{-2mm}
\item http://astrosun.tn.cornell.edu/staff/bottke/index.html,
 \vspace{-2mm}
\item http://www.kolej.mff.cuni.cz/{\~{ }}mbro4036/mp/.
\end{itemize}
\smallskip

\noindent{\bf Acknowledgements.}\hspace{1mm}The author thanks
 P.~Farinella, whose ideas have been at the origin of many of the
 applications discussed above, for discussions and improvements of the
 final version of this text. He also acknowledges
 hospitality of the organizers of the winter school at Pralognan la
 Vanoise that enabled him to attend the meeting.

\begin{thebibliography}{}
\bibitem{anders} Anderson, J.D., et~al.: 1998, {\it Phys. Rev. Lett.}
 {\bf 81}, 2858.\vspace{-2mm}
\bibitem{bott} Bottke, W.F., Burns, J.A. and Rubincam, D.P.: 1999,
 Dynamical evolution of main belt meteoroids: Numerical simulations
 incorporating planetary perturbations and Yarkovsky thermal forces,
 {\it Icarus}, submitted.\vspace{-2mm}
\bibitem{bc} Brouwer, D. and Clemence, G.: 1961, {\it Methods of
 Celestial Mechanics} (Academic Press, New York). \vspace{-2mm}
\bibitem{bls} Burns, J.A., Lamy, P.L. and Soter, S.: 1979, {\it Icarus}
 {\bf 40}, 1. \vspace{-2mm}
\bibitem{dean} De Angelis, G.: 1995, {\it Planet. Space Sci.} {\bf 43},
 649. \vspace{-2mm}
\bibitem{doh} Dohnanyi, J.W.: 1969, {\it J. Geophys. Res.} {\bf 74},
 2531. \vspace{-2mm}
\bibitem{fv96} Farinella, P. and Vokrouhlick\'y, D.: 1996, {\it Planet.
 Space Sci.} {\bf 44}, 1551. \vspace{-2mm}
\bibitem{fv99a} Farinella, P. and Vokrouhlick\'y, D.: 1999, {\it Science}
 {\bf 283}, 1507.\vspace{-2mm}
\bibitem{fvh} Farinella, P., Vokrouhlick\'y, D. and Hartmann, W.K.: 1998,
 {\it Icarus} {\bf 132}, 378. \vspace{-2mm}
%\bibitem{getal} Gladman, B.J., et~al.: 1997, {\it Science} {\bf 277}, 197.
% \vspace{-2mm}
\bibitem{hetal1} Hartmann, W.K., et~al.: 1997, in Proc. Lunar and Planet.
 Sci. Conf. XXVIII (Houston: Lunar Planet. Inst.), p.~517. \vspace{-2mm}
\bibitem{hetal2} Hartmann, W.K. et~al.: 1999, {\it Meteoritics Planet.
 Sci.} {\bf 34}, in press. \vspace{-2mm}
\bibitem{kmf} Kne{\v z}evi\'c, Z., Milani A. and Farinella, P.: 1997,
 {\it Planet. Space Sci.} {\bf 45}, 1581.\vspace{-2mm}
\bibitem{lan} Langseth, M.G., Keihm, S.J. and Chute, J.L.: 1973, in:
 {\it Apollo~17 -- prelimnary Science Report}, NASA-SP-330. \vspace{-2mm}
\bibitem{ls} Lebofsky, L.A. and Spencer, J.R.: 1989, in: {\it
 Astroids~II}, eds. R.~Binzel, T.~Gehrels and M.S.~Matthews (Arizona
 Univ. Press, Tuscon), p.~128. \vspace{-2mm}
\bibitem{mg} Marti, K. and Graf, T.: 1992, {\it Ann. Rev. Earth Planet.
 Sci.} {\bf 20}, 221. \vspace{-2mm}
\bibitem{metal} Migliorini, F., et~al.: 1998, {\it Science} {\bf 281},
 2022. \vspace{-2mm}
\bibitem{mig} Mignard, F.: 1992, in: {\it Interrelations between physics
 and dynamics for minor bodies in the solar system}, eds. D.~Benest and
 C.~Froeschl\`e (Editions Fronti\'eres, Gif-sur-Yvette), p.~419.
 \vspace{-2mm}
\bibitem{mf} Milani, A. and Farinella, P.: 1995, {\it Icarus} {\bf 115},
 209.\vspace{-2mm}
\bibitem{mnf} Milani, A., Nobili, A.-M. and Farinella, P.: 1987,
 {\it Non-Gravitational Perturbations and Satellite Geodesy}
 (A.~Hilger, Bristol). \vspace{-2mm}
\bibitem{nak} Nakamura, T.: 1994, in: {\it Seventy-five Yaers of Hirayama
 Asteroid Families}, eds. Y.~Kozai, R.P. Binzel and T. Hirayama
 (Astron. Soc. Pacific Conf. Series, Vol.~63, San Francisco), 
 p.~52.\vspace{-2mm}
\bibitem{nm1} Nesvorn\'y, D. and Morbidelli, A.: 1998, {\it Astron. J.}
 {\bf 116}, 3029. \vspace{-2mm}
\bibitem{opik} \"Opik, E.J.: 1951, {\it Proc. Roy. Irish Acad.} {\bf 54},
 165.\vspace{-2mm}
\bibitem{pet} Peterson, C.: 1976, {\it Icarus} {\bf 29}, 91.\vspace{-2mm}
\bibitem{poy} Poynting, J.H.: 1903, {\it Phil. Trans. Roy. Soc. A}
 {\bf 202}, 525.\vspace{-2mm}
\bibitem{rab1} Rabinowitz, D.L.: 1993, {\it Astrophys. J.} {\bf 407}, 412.
 \vspace{-2mm}
\bibitem{rab2} Rabinowitz, D.L.: 1994, {\it Icarus} {\bf 111}, 364.
 \vspace{-2mm}
\bibitem{radz} Radzievskii, V.V.: 1952, {\it Astron. Zh.} {\bf 29}, 162.
 \vspace{-2mm}
\bibitem{rob} Robertson, H.P.: 1937, {\it Mon. Not. R. Astr. Soc.}
 {\bf 97}, 423. \vspace{-2mm}
\bibitem{rub87} Rubincam, D.P.: 1987, {\it J. Geophys. Res.} {\bf 92},
 1287. \vspace{-2mm}
\bibitem{rub94} Rubincam, D.P.: 1994, {\it Theor. Appl. Climatol.}
 {\bf 48}, 195. \vspace{-2mm}
\bibitem{rub95} Rubincam, D.P.: 1995, {\it J. Geophys. Res.} {\bf 100},
 1585. \vspace{-2mm}
\bibitem{rub98} Rubincam, D.P.: 1998, {\it J. Geophys. Res.} {\bf 103},
 1725. \vspace{-2mm}
\bibitem{sit} Sitarski, G.: 1991, {\it Astron. J.} {\bf 104}, 1226.
 \vspace{-2mm}
\bibitem{slab} Slabinski, V.J.: 1997, {\it Celest. Mech. Dyn. Astron.}
 {\bf 66}, 131. \vspace{-2mm}
\bibitem{sls} Specer, J.R., Lebofsky, L.A. and Sykes, M.V.: 1989,
 {\it Icarus} {\bf 78}, 337. \vspace{-2mm}
\bibitem{vee} Veeder, G.J., et~al.: 1989, {\it Astron. J.} {\bf 97},
 1211.\vspace{-2mm}
\bibitem{v98a} Vokrouhlick\'y, D.: 1998a, {\it Astron. Astrophys.}
 {\bf 335}, 1093.\vspace{-2mm}
\bibitem{v98b} Vokrouhlick\'y, D.: 1998b, {\it Astron. Astrophys.}
 {\bf 338}, 353.\vspace{-2mm}
\bibitem{v99} Vokrouhlick\'y, D.: 1999, {\it Astron. Astrophys.}
 {\bf 344}, 362.\vspace{-2mm}
\bibitem{vb99} Vokrouhlick\'y, D. and Bro\v z, M.: 1999, {\it Astron.
 Astrophys.}, submitted.\vspace{-2mm}
\bibitem{vf98a} Vokrouhlick\'y, D. and Farinella, P.: 1998a, {\it Astron.
 Astrophys.} {\bf 335}, 351.\vspace{-2mm}
\bibitem{vf98b} Vokrouhlick\'y, D. and Farinella, P.: 1998b, {\it Astron.
 J.} {\bf 116}, 2032.\vspace{-2mm}
\bibitem{vf99} Vokrouhlick\'y, D. and Farinella, P.: 1999, to be
 submitted.\vspace{-2mm}
\bibitem{will}C.M. Will, C.M.: 1981, {\it Theory and Experiment in
 Gravitational Physics}, (Cambridge University Press, Cambridge);
 second edition 1993.\vspace{-2mm}
\bibitem{yeo} Yeomans, D.: 1991, {\it Astron. J.} {\bf 101}, 1920.
 \vspace{-2mm}
\bibitem{ym} Yomogida, K. and Matsui, T.: 1983, {\it J. Geophys. Res.}
 {\bf 88}, 9513.
\end{thebibliography}
\end{document}
