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\begin{document}
\begin{frontmatter}
\title{Thermal force effects on slowly rotating, spherical
 artificial satellites. II. Earth infrared heating}
\author[Prague]{David Vokrouhlick\'y} and
\author[Pisa]{Paolo Farinella}
\address[Prague]{Institute of Astronomy, Charles University,
     \v Sv\'edsk\'a 8, 15000 Prague 5,\\ Czech Republic}
\address[Pisa]{Gruppo di Meccanica Spaziale, Dipartimento di
     Matematica, Universit\`a di Pisa,\\ Via Buonarroti 2,
     56127 Pisa, Italy}
 
\begin{abstract}
We apply the theory developed in Farinella and
Vokrouhlick\'y (1996) for the thermal force effects affecting the
motion of LAGEOS--type artificial satellites to the specific case of
the the so--called Yarkovsky--Rubincam effect, due to anisotropic
heating of the satellite's surface by the Earth's infrared radiation.
This theory allows us to remove the usual assumption of a fast
rotational motion of the satellite, so as to allow for a misalignment
between the direction of the thermal force and the spin axis.  When
applied to the LAGEOS satellite, which has been strongly despun by
magnetic torques since its launch in 1976, our theory shows that in
the next decade the orbital perturbations associated with the
Yarkovsky--Rubincam effect will become much different than
those predicted by the simple rapid--rotation approach.  This is the
case both for the drag--like effect causing a semimajor axis decay,
and for the inclination changes.
\end{abstract}
\end{frontmatter}
 
\section{Introduction}
Among the nongravitational forces affecting the motion of passive,
geodynamical satellites such as LAGEOS, very important are those due
to thermal effects, i.e. the asymmetrical heating of the satellite
body by external radiation sources (the Sun and the Earth) and the
corresponding temperature gradients on the surface, resulting into a
net recoil force related to anisotropic emission of thermal radiation.
A first generation of theoretical models for these effects appeared in
the late 80's and early 90's (Rubincam, 1987, 1988, 1990; Slabinski,
1988; Afonso {\it et~al.}, 1989; Scharroo {\it et~al.}, 1991).
However, because of the complexity of the physical processes under
scrutiny, these models often adopted crude simplifications.  In recent
years, the need for more accurate and/or general models has been
highlighted by the increasing accuracy and time span of the LAGEOS
orbital data --- in particular the series of along--track acceleration
residuals, of the order of a few pm/s$^2$ and spanning two decades ---
and by the improved understanding of the rotation rate and spin axis
evolution of the satellite under magnetic and gravitational torques
[for a brief recapitulation of the subject, we refer the reader to the
introduction of Farinella and Vokrouhlick\'y (1996), hereinafter
referred to as paper~I; the notations and terminology introduced in
that paper will be kept here without significant changes].
 
In paper~I we have developed a new theory for the orbital
perturbations related to thermal effects on LAGEOS--type satellites,
modelled as spherical bodies with homogeneous physical and optical
properties.  The main improvement of this theory with respect to the
previous ones is that it is not restricted to the case when the
rotation of the satellite is fast with respect to its thermal inertia
and orbital period, and that as a consequence the recoil force in
general is not directed along the spin axis, but has nonzero
``equatorial'' components.  In other words, if we consider the
satellite as a small spinning planet, we account for ``diurnal''
effects as well as for ``seasonal'' ones.  In paper~I, this theory was
applied to the so--called Yarkovsky--Schach effect, caused by the
solar illumination of the satellite interrupted by passages through
the Earth's shadow, and a complete solution was derived for LAGEOS's
long--term orbital perturbations due to this effect.  Here, we plan to
perform a similar study but for the so--called Yarkovsky--Rubincam
effect, in which the satellite's heating due to the Earth's infrared
radiation is responsible for the surface temperature gradients and the
related perturbing recoil force.  Of course, in reality both effects
are present at the same time and cause orbital changes appearing in
the observed residuals of the orbital elements (in particular, the
semimajor axis).
 
Rubincam (1982) was the first to point out the importance of this
particular thermal effect for the understanding of the observed
along--track residuals of LAGEOS's orbit.  It was an important first
step, because Rubincam clearly understood that a critical factor was
the ``seasonal'' character of the effect in the case of LAGEOS, in
contrast with the ``diurnal'' version of the Yarkovsky thermal force
usually described in previous literature about nongravitational forces
on natural solar system bodies (see for instance Radzievsky, 1952;
Peterson, 1975; Burns {\it et~al.}, 1979).  This was justified because
of the very fast rotational motion of LAGEOS at the time of launch
(about $1.6$ rotations per second), compared to the typical time
scales for the satellite's thermal inertia and orbital motion (which
are three to four orders of magnitude longer).  A quantitative
analysis of the along--track perturbations caused by this effect was
given by Rubincam (1987, 1988), who showed that such perturbations
could explain most of the previously unmodelled semimajor axis decay
of LAGEOS ($\approx 1$~mm/day).  Rubincam (1987) used a highly
simplified thermal model, while the thermal behaviour of the real
satellite is quite complex, as recently shown by Slabinski (1996).
However, his simplifying assumptions did not lead to quantitatively
important errors, and allowed him to introduce a fairly simple
mathematical formulation of the problem.  For these reasons, in
paper~I we have followed a similar approach.
 
The reason why a new theory accounting for the ``diurnal'' effects on
a slowly--spinning satellite is needed is that LAGEOS's rotation is
actually slowing down.  As suggested by both theoretical models and
observations (Bertotti and Iess, 1991), this despinning process ---
due to dissipation by eddy currents generated by the satellite's
motion through the geomagnetic field --- is an exponential one as a
function of time, with an $e$--folding characteristic time of $\approx
3$ years only.  As a consequence, by the end of this century LAGEOS's
rotational period will be of the order of $10^3$~s, namely comparable
to the typical thermal inertia time scale.  In such a situation strong
``diurnal'' temperature gradients will arise on the satellite's
surface, and the thermal force will cease to be aligned with the spin
axis.  The new ``equatorial'' components of the force will give rise
to qualitatively new perturbation patterns, and therefore it is clear
that the previously adequate fast--rotation approximation will be no
longer useful.  This is the main motivation of this work, in which we
are going to reformulate the Yarkovsky--Rubincam effect for arbitrary
values of the satellite's rotational period.
 
The remainder of this paper is organized as follows: in Sec.~2 we
recall the theoretical setting of the problem as given in paper~I and
adapt it for the analysis of the thermal effects caused by the Earth's
infrared heating.  Next, in Sec.~3 we apply the theory to the case of
the LAGEOS satellite and show that the effects due to the equatorial
thermal force components appearing in the slow--rotation case will
soon become detectable.  In Sec.~4 we summarize our conclusions and
sketch some future work on this subject.
 
\section{Theory}
\subsection{ Recalling the basic formulae}
Because we are going to use the same mathematical and physical
formulation of the problem as given in Sec.~2.1 of paper~I, we will
only give here the main formulae needed for understanding what
follows.  Given the constants specifying the material and geometric
properties of the satellite ($R$ its radius, $m$ the mass, $\alpha$
and $\epsilon$ the absorption and emission coefficients of the
surface, $C_p$ and $\kappa$ the specific heat and thermal
conductivity, $\rho$ the density, $T_0$ the average equilibrium
temperature), the perturbing thermal force per unit mass of the
satellite can be expressed as
\begin{equation}
 {\bf a}(t) = -{8\over mc} {\alpha\epsilon\sigma\over \rho C_p R}
 T_0^3 \int_{t'<t} dt' \, e^{-\nu_R(t-t')}\, {\bf h}\left(t,t';R
  \right) \; , \label{eq1}
\end{equation}
where $\sigma$ and $c$ are the Stefan--Boltzmann constant and the
velocity of light, respectively, whereas the ``kernel function'' ${\bf
h}(t,t';R)$ is defined by
\begin{equation}
 {\bf h}\left(t,t';R \right) = \int_{\Omega_\star(t)} dA(\theta,\phi)
  \, \Delta \Phi\left(t';R,{\bf n}'\right) {\bf n}(\theta,\phi)\; .
  \label{eq2}
\end{equation}
Here, $\Delta\Phi(t;R,{\bf n})$ is the scalar radiative flux due to
the external source(s) through the satellite surface element defined
by the unit vector ${\bf n}(\theta,\phi)$ at time $t$, $(\theta,\phi)$
being the spherical coordinates defined on the satellite surface.  The
integration domain $\Omega_\star(t)$ in the integral is implicitly
defined by the illumination condition $\Delta \Phi(t;R,{\bf n}) \geq
0$ (at time $t$).  We recall that the primed variables in
Eqs.~(\ref{eq1}) and (\ref{eq2}) are to be computed at the retarded
time $t'<t$.
 
The fundamental parameters controlling the thermal effects in our
model are: (i) $\sigma_R\equiv 2\pi\tau_R/T_{\rm rev}$, where $\tau_R$
is the thermal lag time characterizing the response of the satellite
surface to external heating and $T_{\rm rev}$ is the period of the
satellite's orbit around the Earth; and (ii) $r\equiv T_{\rm
rev}/T_{\rm rot}$, where $T_{\rm rot}$ is the satellite's rotational
period.  The basic set of time scales relevant for our problem is
thus: $(T_{\rm rot},T_{\rm rev},\tau_R)$.  We assume that all of these
time scales are much shorter than those over which the satellite's
spin axis direction and its orbit evolve.
 
The only (formal) constraint of our solution is $r\geq 1$, due to the
fact that the predicted evolution of LAGEOS's spin axis direction has
been derived from an averaged model (Farinella {\it et~al.}, 1996),
which cannot be applied in the transition through a spin--orbit
resonance (Habib {\it et al.}, 1994).
 
\subsection{The Yarkovsky--Rubincam effect}
Let us start by introducing a suitable reference frame, which will be
used throu\-ghout this paper: the $Z$--axis (with unit direction
vector ${\bf e}_Z$) is directed along the satellite's spin axis,
whereas the $X$--axis (unit direction vector ${\bf e}_X$) is chosen in
such a way that the satellite's geocentric position vector always lies
in the $XZ$--plane (with a {\it negative} $X$--component).  This
defines our system uniquely, with the exception of some singular
configurations (spin axis in the orbital plane), which do not require
a separate treatment.  Note that in general, this frame rotates in a
non--uniform way with respect to inertial space.  Contrary to the case
treated in paper~I (satellite heating caused by sunlight), the
position unit vector of the heat source --- the Earth in the current
case --- is not fixed in our reference system.  Actually, for the
colatitude $\theta_s(\lambda)$ of the heating source in the
$XYZ$--frame we obtain
\begin{equation}
 \cos\theta_s(\lambda) = -\sqrt{1-s_c^2}\, \sin\lambda\; ,\label{eq3}
\end{equation}
where $s_c\equiv {\bf s}\cdot{\bf c}$ is the cosine of the angle
defined by the unit vectors directed along the satellite's spin axis
(${\bf s}$) and the perpendicular to the orbital plane (${\bf c}$),
while $\lambda$ is the satellite's orbital longitude (like in paper~I,
we approximate the satellite's orbit as a circular one, so
$\lambda$ can be used instead of the time $t$ as an independent
variable).  We have arbitrarily chosen the origin of $\lambda$ so that
it corresponds to $\theta_s = \pi/2$; otherwise one should have
considered an additional constant $\lambda_0$ in the corresponding
angular argument of Eq.~(\ref{eq3}).  Since none of the final results
depend on this arbitrary quantity $\lambda_0$, we have preferred to
take directly $\lambda_0=0$.  Note also that the parameter $s_c$ is
the only geometrical quantity which plays a role in the formulation
and solution of our problem.
 
As a second step, we have to specify the ``rotation law'' of the
satellite in the $XYZ$ reference system.  This means that we have to
describe the motion of each of its surface elements.  Because the
$Z$--axis coincides with the spin axis, the rotational motion of any
surface element changes its ``longitude'' $\phi$ and leaves unchanged
its ``colatitude'' $\theta$.  If we consider a surface element having
an initial longitude $\phi_0$ at an arbitrary starting time, we obtain
for its longitude $\phi$ as a function of $\lambda$:
\begin{equation}
 \phi(\lambda) = \phi_0 + (r-1) \lambda -f(\lambda) \; ,
 \label{eq4}
\end{equation}
where we have introduced the auxiliary function
\begin{equation}
 f(\lambda) = \arctan(s_c \tan\lambda) - \lambda\; . \label{eq5}
\end{equation}
Finally, we need to specify the external radiation field which heats
the satellite's surface.  In this respect, the Earth's infrared
illumination is more complicated than the sunlight case, because the
former is not unidirectional, but is better represented by a cone with
an $\approx 60^{\circ}$ aperture at LAGEOS's altitude.  However, we
shall neglect this geometrical complication, and approximate the
Earth's infrared radiation field at the satellite's altitude as a
homogeneous field of intensity $\Phi_0$ ($\approx 65\, {\rm W/m}^2$ in
the LAGEOS case).  Then, if we take the satellite's
surface element characterized by the spherical coordinates
$(\theta,\phi)$, we have
\begin{equation}
 \Delta \Phi(\lambda;R,{\bf n}) = \Phi_0 \left[\sin\theta_s(\lambda)
  \sin\theta\cos\phi +\cos\theta_s(\lambda)\cos\theta\right]\; .
 \label{eq6}
\end{equation}
The intermediate quantities that we need for computing the
perturbations on the orbital semimajor axis and inclination of the
satellite are the instantaneous values of the transverse
[$T(\lambda)$] and binormal [$W(\lambda)$] components of the thermal
force.  Simple geometry shows that they can be written as
\begin{equation}
 T(\lambda) = \left(1-s_c^2\right){\sin\lambda\cos\lambda\over
  \sin\theta_s(\lambda)} F_X -{s_c\over \sin\theta_s(\lambda)} F_Y
  + \sqrt{1-s_c^2} \cos\lambda \,F_Z  \label{eq7}
\end{equation}
and
\begin{equation}
 W(\lambda) = {\sqrt{1-s_c^2}\over \sin\theta_s(\lambda)} \left(
  s_c \sin\lambda \,F_X + \cos\lambda \,F_Y\right) + s_c F_Z
 \; , \label{eq8}
\end{equation}
where $(F_X,F_Y,F_Z)$ are the projections of the thermal force vector
onto the three unit vectors $({\bf e}_X,{\bf e}_Y,{\bf e}_Z)$.
Eq.~(\ref{eq3}) yields
\begin{equation}
 \sin\theta_s(\lambda) = \sqrt{\cos^2\lambda+s_c^2 \sin^2\lambda}
  \label{eq9}
\end{equation}
for the sine factor in Eqs.~(\ref{eq7}) and (\ref{eq8}).
 
Considering the rotation law (\ref{eq4}), one can easily show that
the vector ${\bf h}(\lambda,\lambda';R)$ appearing in Eq.~(\ref{eq2})
can be expressed as
\begin{equation}
 {\bf h}\left(\lambda,\lambda';R \right) = {2\pi\over 3} R^2 I_0
 \pmatrix{\sin\theta_s(\lambda') \cos\xi(\lambda,\lambda') \cr
          \sin\theta_s(\lambda') \sin\xi(\lambda,\lambda') \cr
          \cos\theta_s(\lambda') \cr}  \; , \label{eq10}
\end{equation}
where we have defined
\begin{equation}
 \xi(\lambda,\lambda') = (r-1)(\lambda-\lambda') + f(\lambda')-
  f(\lambda)\; . \label{eq11}
\end{equation}
Combining this result with formulae (\ref{eq1}) and
(\ref{eq2}), we can see that only the $Z$--component can be computed
analytically by simple quadratures. Thus, for the component of
the thermal force directed along the spin axis we obtain
\begin{equation}
 F_Z(\lambda) = -\gamma \sqrt{1-s_c^2}\,{\sin\lambda- \sigma_R
 \cos\lambda\over 1+\sigma_R^2}\; ,\label{eq12}
\end{equation}
with
\begin{equation}
 \gamma = - {16\over 3} {\pi R^2 \Phi_0\over mc} \tau_R T_0^3 {\alpha
 \epsilon\sigma\over \rho C_p R}\; .\label{eq13}
\end{equation}
$F_Z$ obviously vanishes for $s_c=1$.  Like in the case of the
Yarkovsky--Schach effect, discussed in paper~I, Eq.~(\ref{eq13}) for
the normalizing factor $\gamma$ provides just a good semi--quantitative
estimate for the intensity of the thermal force, but since real
satellites are not perfectly spherical and homogeneous, in practical
applications the precise value of $\gamma$ should be determined
empirically by a fit to the available data rather than calculated {\it
a priori}.
 
Introducing the previous result into Eqs.~(\ref{eq7}) and (\ref{eq8}),
we obtain the contribution of the $Z$ thermal force component to the
along--track perturbing acceleration (which, for small orbital
eccentricities, is just proportional to the time derivative of the
semimajor axis):
\begin{equation}
 T_Z = {\gamma\over 2} {\sigma_R\over 1+\sigma_R^2} (1-s_c^2)\ .
 \label{eq14}
\end{equation}
Gauss' perturbation equations also give the corresponding excitation
of the orbital inclination:
\begin{equation}
 \left({d I\over dt}\right)_Z = {\gamma\over 2na} {\sigma_R\over 1+
 \sigma_R^2} s_c \sqrt{1-s_c^2} \; . \label{eq15}
\end{equation}
 
These results reproduce those of Rubincam (1987, 1990): in
Eq.~(\ref{eq12}) the coefficient $\gamma$ gives the amplitude of the
perturbing effect, the factor containing the $\sigma_R$ parameter
corresponds to Rubincam's thermal lag factor [$(\sin\delta)$, in his
notations], and the $(1-s_c^2)$ term is exactly the angular--dependent
piece of Rubincam's formula.  Whenever $s_c = \pm 1$ (spin axis normal
to the orbital plane), the $T_Z$ (``seasonal'') component of the
thermal drag--like effect vanishes.  On the contrary, it is maximum
for $s_c=0$, as expected.  Similarly, Eq.~(\ref{eq15}) corresponds to
the results of Farinella {\it et~al.} (1990), in particular their
Eq.~(18), provided one substitutes their ``delay angle'' $\theta$ and
amplitude factor $K$ by the relationships $\sin\theta \equiv
\sigma_R/\sqrt{1+\sigma_R^2}$ and $K \equiv
\gamma/\sqrt{1+\sigma_R^2}$.
 
The contribution of the thermal force components perpendicular to the
spin axis ($F_X$ and $F_Y$) to the along--track and out--of--plane
perturbations is more complicated.  We did not succeed in integrating
formulae (\ref{eq1}) and (\ref{eq2}) analytically and our results can
only be expressed as formal quadratures.  However, an efficient
numerical evaluation is always possible (some applications will be
discussed in the next section).  For this purpose, we define the
auxiliary quantities
\begin{eqnarray}
 I_X(\lambda) &= & \int_{\lambda-2\pi}^\lambda d\lambda'\,
  \sin\theta_s(\lambda') \cos\xi(\lambda',\lambda) e^{-(\lambda-
  \lambda')/\sigma_R} \; , \label{eq16} \\
 I_Y(\lambda) &= & \int_{\lambda-2\pi}^\lambda d\lambda'\,
  \sin\theta_s(\lambda') \sin\xi(\lambda',\lambda) e^{-(\lambda-
  \lambda')/\sigma_R} . \label{eq17}
\end{eqnarray}
Using this definition and applying the same averaging method we
explained in Sec.~2 of paper~I, we obtain for the mean value of the
along--track perturbing acceleration due to the $X$ and $Y$ thermal
force components:
\begin{eqnarray}
 T_{XY} &=& -{\gamma\over 2\pi\sigma} {\hat C}(r-1,{\bar e})
  \int_0^{2\pi}
  d\lambda\, \left[ I_X(\lambda) {\partial\over \partial\lambda}
  \sin\theta_s(\lambda) + I_Y(\lambda){s_c\over \sin\theta_s(
  \lambda)}\right] \nonumber \\
 & & +{\gamma\over 2\pi\sigma} {\hat S}(r-1,{\bar e})
  \int_0^{2\pi}
  d\lambda\, \left[ I_Y(\lambda) {\partial\over \partial\lambda}
  \sin\theta_s(\lambda) - I_X(\lambda){s_c\over \sin\theta_s(
  \lambda)}\right] , \label{eq18}
\end{eqnarray}
with
\begin{eqnarray}
 {\hat C}(r,{\bar e}) &\equiv & {1-{\bar e}\cos(2\pi r) \over
  1-2{\bar e}\cos(2\pi r)+{\bar e}^2} \; , \label{eq19}\\
 {\hat S}(r,{\bar e}) &\equiv & {{\bar e}\sin(2\pi r) \over
  1-2{\bar e}\cos(2\pi r)+{\bar e}^2} \label{eq20}
\end{eqnarray}
The total value of the thermal drag--like acceleration is of course
$T \equiv T_Z + T_{XY}$.
 
In general the asymptotic behaviour of $T_{XY}$ for large values of
the parameter $r= T_{\rm rev}/T_{\rm rot}$ (the rapid--spin
case) is obscured by the complicated structure of the previous
formulae.  We just mention that for large values of $r$ the arguments
of the trigonometric functions in the integrands of Eqs.~(\ref{eq16})
and (\ref{eq17}) change rapidly, leading to small mean values $I_X$
and $I_Y$.  In a few special cases, we can obtain an analytical
result.  When $s_c = \pm 1$, we have
\begin{equation}
 T_{XY}(r;s_c=1) = -\gamma\, {(r-1)\sigma_R\over 1+(r-1)^2\sigma_R^2}
  \; , \label{eq21}
\end{equation}
and $T_{XY}(r;s_c=-1) = T_{XY}(r+2;s_c=1)$.  Both values of $T_{XY}$
decay $\propto 1/r$ for large values of $r$.  On the other hand, it
can be shown by a straightforward but tedious computation that
$T_{XY}(r;s_c=0) \propto 1/r^2$.  Therefore, we expect that for
generic values of $s_c$ the averaged $T_{XY}$ component decays
$\propto 1/r^\alpha$, where $\alpha \in (1,2)$.
 
In a similar way, the contribution of the $X$ and $Y$ thermal force
components to the averaged excitation of the orbital inclination
can be expressed as
\begin{eqnarray}
 \left({dI\over dt}\right)_{XY} &=& -{\gamma\sqrt{1-s_c^2}\over 2\pi na}
  {\hat C}(r-1,{\bar e}) \int_0^{2\pi}
  d\lambda\, {\cos\lambda\over \sin\theta_s(\lambda)} \Bigl[ s_cI_X(
  \lambda) \sin\lambda \nonumber \\
  & &\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + I_Y(\lambda)
  \cos\lambda\Bigr] \nonumber \\
 & & -{\gamma\sqrt{1-s_c^2}\over 2\pi na}{\hat S}(r-1,{\bar e})
  \int_0^{2\pi} d\lambda\,{\cos\lambda\over \sin\theta_s
  (\lambda)} \Bigl[ s_c I_Y(\lambda) \sin\lambda \nonumber \\
  & &\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad - I_X(\lambda)
  \cos\lambda\Bigr] \; . \label{eq22}
\end{eqnarray}
Again, we could not carry out the integrations appearing in
Eq.~(\ref{eq22}) analytically and, in the next section, we
will use a numerical method for their evaluation.
 
A brief comment is to be done about the eccentricity excitations,
which we have not mentioned so far.  As discussed by M\'etris {\it
et~al.} (1996), a simple order--of--magnitude estimate shows that the
corresponding Yarkovsky--Rubincam perturbing effects, in the
rapid--spin approximation, do not contribute in a significant way to
the observed eccentricity residuals for LAGEOS.  One can easily check
that the same holds also for the more general case including the $X$
and $Y$ force components, with which we are dealing here.  As a
consequence, we shall not discuss the eccentricity effects in the next
section.
 
\section{Applications}
The purpose of this section is to demonstrate that in the specific
case of the LAGEOS satellite the components of the thermal force
perpendicular to the spin axis, which are significant when the
satellite's rotation is slow enough, give rise to excitations of the
orbital semimajor axis ($T_{XY}$) and inclination [$(dI/dt)_{XY}$]
which cannot be neglected within an accurate orbit analysis and/or
determination.  As we will see, for LAGEOS the slow--rotation
condition will be met during the next decade.
 
\begin{figure}
%\begin{center}
%\begin{picture}(100,100)(145,-682)
%\special{em:graph sll1.pcx}
%\end{picture}
%\end{center}
\vspace{53mm}
\caption{The drag--like along--track component $T$ of the
Yarkovsky--Rubincam thermal force for LAGEOS vs. the $s_c$ parameter
(defined as the cosine of the angle between the satellite's spin axis
and the perpendicular to its orbital plane).  The dashed curve shows
the contribution of the force component directed along the spin axis,
whereas the solid curves show the contribution of the ``equatorial''
force components.  The latter components depend on the ratio $r$
between the orbital and rotational periods, so several cases have been
plotted, corresponding to different values of $r$.
\label{fig1}}
\end{figure}
 
In figure~\ref{fig1} we compare the results for $T$ in the case of
LAGEOS from the $Z$--component alone (rapid--spin approximation, curve
$Z$) to the contribution of the equatorial components, as a function
of the geometrical parameter $s_c \equiv {\bf s}\cdot{\bf c}$.
Because the ``diurnal'' effect associated with the equatorial
components depends on the ratio $r$ between the orbital and rotational
periods, we have plotted separately the curves corresponding to $r=1$
(curve~1), $r=2$ (curve~2), and so on.
 
The thermal inertia parameter $\sigma_R$ has been set to $2.4$~rad, in
agreement with M\'etris {\it et~al.} (1996), whereas the amplitude
parameter $\gamma$ has been chosen in such a way that:
$\gamma\sigma_R/2(1+\sigma_R^2) \simeq 3.08\, {\rm pm/s}^2$, which is
consistent with recent analyses of the LAGEOS along--track residuals
(for instance Scharroo {\it et~al.}, 1991; Farinella {\it et~al.},
1996).  This value is in good agreement with the approximate formula
(\ref{eq13}), once the constants appropriate for the LAGEOS satellite
are inserted in it.  This set of parameters is used throughout this
section.
 
In Fig.~\ref{fig1} we can observe several important features of the
Yar\-kov\-sky--Ru\-bin\-cam effect.  When the spin axis is
perpendicular to the orbital plane ($s_c = \pm 1$), the ``seasonal''
contribution vanishes while the ``diurnal'' contribution is maximum.
In agreement with Eq.~(\ref{eq21}), the latter contribution decays as
$1/r$ for increasing values of $r$.  On the contrary, when the spin
axis lies in the orbital plane, the ``seasonal'' contribution is
maximum while the ``diurnal'' contribution is relatively small, and
decays as $1/r^2$ for large values of $r$.  The arrow in the figure
points to the predicted value of $s_c$ when the $r=1$ spin--orbit
resonance will be met by LAGEOS (according to Farinella {\it et al.},
1996).  This shows that the ``diurnal'' contribution will be a
moderate one compared to its possible values for a high $s_c$, but as
discussed below even this limited effect will be large enough to be
observed.
 
\begin{figure}
%\begin{center}
%\begin{picture}(100,100)(145,-682)
%\special{em:graph sll1b.pcx}
%\end{picture}
%\end{center}
\vspace{53mm}
\caption{The same as Fig.~1, but for LAGEOS's inclination
excitation $(dI/dt)$ vs. $s_c$.
\label{fig1bis}}
\end{figure}
 
A similar plot can be made for the Yarkovsky--Rubincam inclination
effects.  Fig.~\ref{fig1bis} shows the time derivative (or excitation)
of LAGEOS's mean inclination $(dI/dt)$ as a function of $s_c$.  When
$s_c = \pm 1$ it is easy to see that the thermal force lies in the
orbital plane, and as a consequence the excitation of the inclination
vanishes.  The arrow again points to the value of the $s_c$ parameter
corresponding to the spin--orbit resonance for LAGEOS.  Note that the
contribution of the equatorial terms $(dI/dt)_{XY}$ is now quite
important, as it will be confirmed below.
 
\begin{figure}
%\vspace{17mm}
 
%\begin{center}
%\begin{picture}(100,100)(145,-682)
%\special{em:graph sll2.pcx}
%\end{picture}
%\end{center}
\vspace{53mm}
\caption{Predictions of the averaged along--track thermal force
component on LAGEOS due to the Yarkovsky--Rubincam effect vs. time (in
days), starting on Jan.~1, 2000.  The dashed curve~2 corresponds to
the rapid--spin approximation, whereas the solid curve~1 show the
prediction of the theory developed here, including the equatorial
force components as well.  The evolution of LAGEOS's rotational state
has been predicted according to the dynamical theory of Bertotti and
Iess (1991), following Farinella {\it et al.} (1996).
\label{fig2}}
\end{figure}
 
Fig.~\ref{fig2} shows the Yarkovsky--Rubincam drag--like along--track
perturbations predicted for LAGEOS after Jan.~1, 2000, assuming that
its rotation rate and spin axis direction evolve as predicted by
Farinella {\it et al.} (1996) on the basis of the theory developed by
Bertotti and Iess (1991).  The (dashed) curve obtained from the
rapid--spin approximation is almost constant, while the (solid)
curve corresponding to the total force, including the equatorial
components, shows significant changes.  We recall that the r.m.s.
deviation of the empirical along--track residuals with respect to
current dynamical models is of about $0.5\, {\rm pm/s}^2$,
considerably smaller than the difference between the two curves shown
in Fig.~\ref{fig2}.  Therefore a model accounting for the
Yarkovsky--Rubincam slow--rotation effects will become
essential to analyse the LAGEOS orbital residuals in the first decade
of the next century.
 
\begin{figure}
%\begin{center}
%\begin{picture}(100,100)(145,-682)
%\special{em:graph sll3.pcx}
%\end{picture}
%\end{center}
\vspace{53mm}
\caption{The same as in Fig.~3 but for LAGEOS's inclination excitation
$(dI/dt)$ vs. time, starting at launch on May 4, 1976.  The arrow
points to the epoch Jan.~1, 1997, when the influence of the equatorial
thermal force components will begin to be significant.
\label{fig3}}
\end{figure}
 
Figure~\ref{fig3} shows the excitation of the mean orbit inclination
of LAGEOS due to the Yarkovsky--Rubincam effect as a function of time
since the launch epoch.  We can again conclude that in the near future
the total effect (solid line) will become considerably different than
the predictions based on the rapid--spin approximation (dashed line).
Farinella {\it et~al.} (1990) studied this inclination effect assuming
a constant orientation of the satellite's spin axis, and concluded
that a secular value of about $-0.84\,{\rm mas/yr}$ had to be expected
(they also assumed a somewhat larger value of the phase--lag
parameter).  This result is in good agreement with the average value
we find for the first $15 - 20$ years after launch, but later on the
situation changes, due to the spin axis evolution.  The influence of
the equatorial thermal force components results in a fairly large
negative value of the inclination excitation, occurring some $33$
years after launch.  This feature should be easily observable by
accurate orbital determination.  The arrow in the plot points to
Jan.~1, 1997, an approximate date when the slow--rotation corrections
will start to be significant.
 
\section{Conclusions and future work}
\noindent The main results obtained in this paper can be
summarized as follows:
\begin{enumerate}
\item We have completed our theoretical treatment of the thermal
effects acting on the motion of LAGEOS--type geodynamics satellites by
taking into account the Earth's infrared heating (i.e., the
so--called Yarkovsky--Rubincam effect) after the solar heating
(Yarkovsky--Schach effect), which had been dealt with in paper~I. In
both cases our theory can be applied for arbitrary values of the
satellite's rotation rate.
 
\item In the case of LAGEOS, the slower and slower rotation due to
magnetic despinning and the corresponding ``diurnal'' temperature
gradients associated with the Earth's infrared flux will soon give
rise to significant components of the thermal force misaligned with
the spin axis.  As a consequence, within the next decade the
long--term drag--like semimajor axis decay caused by the
Yarkovsky--Rubincam effect will undergo significant modifications with
respect to the rapid--spin approximation adopted in previous
analyses.  The secular part of the orbital inclination excitation is
also going to show major changes.
\end{enumerate}
 
We stress that the theory developed in paper~I and applied again in
this paper is based on several approximations and simplifying
assumptions, and therefore further work should lead to significant
improvements from the quantitative point of view.  In particular, our
spherical and homogeneous thermal model is a rather primitive one even
for LAGEOS--type satellites.  The complex structure of real satellites
and the different interactions of their parts with the external
radiation fields and with each other (including their mutual
irradiation and shadowing effects) should be investigated and modelled
in some detail.  Slabinski (1996) performed a significant step forward
in this direction, but limited his analysis to the rapid--spin
case.  Removing this assumption and generalizing Slabinski's thermal
model to the situation investigated here appears as a promising area
for future work.
 
Other issues which we have neglected are the possible coupling between
the different thermal effects and the finite aperture of the Earth's
infrared radiation field.  As for the former problem, we have treated
separately the case when the satellite surface is heated by the Sun's
visible radiation (paper~I) and by the Earth's infrared radiation
(this paper).  On the other hand, Slabinski (1996) correctly pointed
that in reality there is some interaction between the two thermal
effects, because the average surface temperature $T_0$ depends on the
occurrence of eclipses.  The latter issue, arising from the finite
angular size of the Earth as seen from the satellite, probably results
into a slight decrease of the corresponding temperature gradients on
the satellite's surface, and consequently of the thermal force
perturbations.  Quantitative studies of both these effects appear
worth some effort.
 
Finally, we stress that in order to improve our capability of
predicting the perturbations due to thermal effects for LAGEOS--type
satellites  we need to better understand and model the evolution of
their rotation rate and spin axis direction. The theory of Bertotti
and Iess (1991) should be refined by taking into account new
perturbing effects (e.g. Vokrouhlick\'y, 1996), and the complete 
system of Euler's equations
should be studied numerically without the averaging assumption, in
order to deal with the spin--orbit resonance crossings and the
possible associated chaotic dynamics [see Habib {\it et al.} (1994)
for some work in this direction].
 
\noindent{\it Acknowledgements.}
We thank G. M\'etris for useful discussions and comments.
D.V. is grateful to the University of Pisa (Italy) for its kind
hospitality and to the Observatoire de la C\^ote d'Azur (Dept.  CERGA,
Grasse, France) where he could work to this project thanks the
``H.~Poincar\'e'' research fellowship.  P.F. acknowledges support from
the Italian Space Agency (ASI) and the Italian Ministry for University
and Scientific Research (MURST).
 
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\end{document}
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