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{\Large \bf Thermal force perturbations of the LAGEOS orbit: the
  albedo radiation part}
\bigskip\medskip

{\rm \large Gilles M\'etris}
\smallskip

{\it Observatoire de la C\^ote d'Azur, D\'epartement
 CERGA, Av N. Copernic,\\ 06130 Grasse, France}
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{\rm \large David Vokrouhlick\'y}
\smallskip

{\it Institute of Astronomy, Charles University, \v Sv\'edsk\'a~8,
 15000 Prague, \\ The Czech Republic}
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\noindent{\large \bf Abstract.}\hskip 3mm We clarify a role of the
Earth reflected radiation heating for the thermal effects
perturbing the motion of the LAGEOS satellite. A small, but not
negligible, part of the constant drag acting on the satellite and
detected via the orbit analysis is attributed to this phenomenon.
Contribution to the observed eccentricity excitations is
very small in contrast to the hopes raised in the recent analysis
by M\'etris et~al. [1995].
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\section{ Introduction}
Thermal effects, consisting in the anisotropic emission of the
thermal radiation by the satellite surface, have been recognized
as very important constituents of the empirically observed
long-term along-track residuals of the LAGEOS orbit [Rubincam, 1982,
1987, 1988; Afonso et~al., 1989; Scharroo et~al., 1991; Farinella
et~al., 1995]. More recently, M\'etris and coworkers [1995]
reported that similar effects contribute significantly to
analogous long-term eccentricity excitations of the LAGEOS
orbit.

Modeling of the thermal effects is highly complicated for
several reasons. Despite the simple geometrical form of the
LAGEOS satellite, heat transfer and related surface temperature
distribution is a rather intricate problem
due to the contribution of diverse constituents of the
satellite body and their differential sensitivity to the
visible and infrared heating. First models have been developed by
Rubincam [1987, 1988] and Afonso et~al. [1989].
To our knowledge, the recent work
of Slabinski [1995] is the most refined attempt to treat
this complex problem. Yet, this careful model still needs
empirical corrections when applied to the LAGEOS data
[Slabinski, 1995]. As we do not feel able to improve on this
subject, we shall follow in this paper a simpler, but qualitatively
acceptable, formulation by Rubincam [1987].
A second fundamental complexity of the
thermal effects modeling is due to different sources of the
satellite surface heating. In principle, one should account
for {\it the direct solar radiation}
[resulting in the version of the thermal effect often called {\it
Yarkovsky-Schach effect} in the literature related to LAGEOS;
Afonso et~al., 1989; Scharroo et~al., 1991], {\it the Earth
infrared radiation} [the corresponding thermal effect is
called {\it Yarkovsky-Rubincam
effect}; Rubincam 1987, 1988] and {\it the sunlight reflected
on the Earth surface} [hereafter called {\it the albedo radiation}].
Up to now the third heating source -- the albedo radiation --
has been neglected in the literature.

In this paper, we focus on understanding the essential nature of the
orbit perturbations due to the satellite thermal effects related to the
albedo radiation heating. Notice, that one can hardly reject the albedo
heating of the satellite surface a priori, because
the radiative flux due to the Earth reflected sunlight can
instantaneously exceed the Earth infrared flux by a factor greater
than $2$. However, the albedo radiative field as a heating source
is rapidly changing over the satellite revolution around the
Earth. The average albedo radiative flux is thus about one half of
the nearly constant infrared radiative flux [Afonso et~al., 1989].

Motivation to pursue this study is due to the surprising finding of
M\'etris et~al. [1995] that an empirical amplitude of the
Yarkovsky-Schach thermal effect should be about twice as large
as previously suggested values in order to explain anomalous
excitations of the LAGEOS eccentricity together with the
along-track residuals. Although the revised thermal model of
the LAGEOS satellite by Slabinski [1995] does suggest an increase
of the total amplitude of the effect, it generally fails to
explain the necessary factor $2$ (improving the factor of
theoretical mismodeling to about $1.7$; M\'etris et~al., 1995).
It was thus hoped that the missing albedo heating mechanism, both
in previous studies by Rubincam [1987, 1988] and Afonso et~al.
[1989] and a more recent study by Slabinski [1995], would, at least
partly, fill the gap between the empirically needed and the
theoretically expected values of the amplitude of the thermal
effects. The weight of the empirical arguments for increasing
the discussed amplitude of the thermal effects lies on side of the
eccentricity excitations. Thus, in the first instant we focus
on the determination of the eccentricity excitations. However,
as a side product of our analysis we shall also consider contributions
of the studied effects to the long-term along-track force and
inclination excitation.

Our method is fully analytical, which makes it appealing for
long-term studies. This is a somewhat surprising fact, because the
albedo radiation is generally thought to be a very complex
phenomenon requiring either a complicated analytical formulation
[e.g. Rubincam and Weiss, 1986; Borderies and Longaretti, 1990;
Borderies, 1990]
or a completely numerical setting. It has been, however, recently
shown by M\'etris et~al. [1995] that for the purpose of the
long-term eccentricity excitations, the direct albedo force
can be significantly simplified with only minor loss of
precision. We employ the same scheme for the thermal effects
studied in this work. The validity of our results
is then checked by comparison of the analytical results
with the completely numerical approach.

Finally, we note that this paper rests on the assumption of
LAGEOS fast rotation [see for discussion Farinella et~al., 1995].
In terms of physical assumptions it means that the thermal inertia
time-scale for all satellite components is significantly greater
than its rotation period. As a consequence, the satellite's fast
rotation averages out thermal force components not aligned
with the spin axis, leaving the sole non-vanishing component
directed along the spin axis. The LAGEOS~I rotation model suggests
that this essential assumption holds well for the first
$25$ years of the mission [Farinella et~al., 1995], the time
interval on which we currently dispose with orbital data.

The paper is organized as follows: in the next section, we derive
an estimation of the LAGEOS~I surface temperature variations due to
the albedo radiation heating and compute the corresponding thermal
force together with long-term orbit perturbations. Next, we
apply the theory to the LAGEOS~I satellite and discuss several
implications for interpretation of its observed orbital
excitations.

\section{ The albedo radiation heating of LAGEOS: formulation and
 dynamical consequences}
Mutual configuration of the satellite spin axis and the direction
towards the radiation source leads to differential heating of the
satellite surface and corresponding distribution of the surface
temperature. Once the temperature distribution is determined
one can easily compute the thermal force, related to differences in the
momentum carried away from the satellite by the thermal photons,
acting on the satellite [e.g. Afonso et~al., 1989]. Section
2.1 is devoted to the physical considerations yielding a simple
law for the temperature distribution on the satellite surface
and the resulting thermal force. In section 2.2 we give the
long-term perturbations of the satellite orbit.

\subsection{ Force modeling}
Principal simplifications adopted in our study can be listed as
follows:
\begin{itemize}
\item we neglect the finite width of the albedo radiation field near
 the satellite, referring to the radiation flux vector ${\bf F}$
 as its full characteristics [for terminology see Vokrouhlick\'y
 et~al, 1993];
\item we neglect ``latitudinal heat conduction'' (here the
 satellite's spherical coordinates are defined with respect to
its spin axis) relating
 the heat budget of a particular surface element (retroreflector,
 mounting rings etc.) to the equilibrium between absorbed/emitted
 radiation and the radial heat conduction in the thin surface
 shell.
\end{itemize}
We also recall implications of the satellite's fast rotation
discussed in the Introduction (the thermal force is oriented along the
spin axis). Our formulation is thus close to Rubincam's [1987]
treatment and we shall also follow his notation
and terminology. We note that this approach seems appropriate
for the treated albedo radiation thermal source because the
effect is smaller than the principal thermal effects.

Following Rubincam's [1987; Sec.~2] analysis one easily shows
that the temperature excess $\Delta T$ over the mean surface
value $T_0$ satisfies in the linearized theory
\begin{equation}
 \Delta T(t;\theta) = {3\varepsilon\over \rho C_p R_R}
  \int_{t' \leq t} dt'\, \Phi(t',\theta) e^{-\nu_R (t-t')} \; ,
\end{equation}
where $\Phi(t',\theta)$ is the radiation flux due to an external
source applied to the outer surface at satellite colatitude $\theta$
and instant $t'$. In addition, formula (1) contains several
material constants of the particular satellite surface element
(in agreement with Rubincam [1987] we shall speak about the
retroreflector): its emissivity $\varepsilon$, its density $\rho$,
 its specific heat $C_p$ and its radius $R_R$. The result (1)
corresponds well to physical intuition which suggests: (i)
a linear response to the heating $\Phi$ provided the effects are
sufficiently weak (nonlinear effects being neglected), and (ii)
exponential damping of the excited temperature difference
$\Delta T$. Theory suggests
\begin{equation}
 \nu_R = \tau_R^{-1} = {12\varepsilon \sigma T_0^3\over \rho
 C_p R_R} \;
\end{equation}
for the inverse of the characteristic damping time (also called
time-lag in context of the satellite thermal effects). Note that
we allow for a generally non-periodic (or at least not simply
periodic) heating signal $\Phi(t,\theta)$. On the contrary, Rubincam
[1987] assuming a sinusoidal function $\Phi_{\rm Rub}(t,\theta)
\simeq \exp(i\nu_{\rm Rub} t)$, obtained a simpler result
consisting merely in a constant
time shift between the heating of a given retroreflector
and its response expressed by the temperature increase.
Our formulation (1) is more general, and more complicated,
and reduces to Rubincam's formulae provided a simpler
heating signal $\Phi_{\rm Rub}(t,\theta)$ is assumed {\it and}
particular conditions on the frequency $\nu_{\rm Rub}$ are
satisfied.

Due to simplifying conditions listed at the beginning of
this chapter we relate the heating flux $\Phi(t,\theta)$
to the albedo radiation flux vector ${\bf F}$ by
\begin{equation}
 \Phi(t',\theta) = \left({\bf F}.{\bf s}\right) \cos\theta \; ,
\end{equation}
where ${\bf s}$ is the unit vector directed along the satellite
spin axis ($\theta$ is the satellite's colatitude as before). Armed
with the previous results we can easily compute the spin
axis directed component of the thermal acceleration ${\bf a}$
[see Afonso et~al., 1989, Eq.~(4)]
\begin{equation}
 {\bf a} = {16\over 3} {\varepsilon \sigma T_0^3\over mc}
 A_\perp \int_{-1}^1 d(\cos\theta) \Delta T(t;\theta) \cos\theta
 \; {\bf s} \;\; ,
\end{equation}
where $A_\perp \equiv \pi R_s^2$ is the satellite's geometrical
cross section, $m$ its mass and $c$ the light of velocity.

The essential step thus consists in approximating the albedo
radiation flux vector ${\bf F}$ from (3). Here, we shall rely
on the successful formulation by M\'etris et~al. [1995]
who used
\begin{equation}
 {\bf F}(\sigma) = {1\over 4} I_0 \bar{\cal A} \alpha^2 \left[
 \cos\psi(\sigma) \left({8\over 3}+3\alpha\right) {\bf r} -
 \alpha {\bf n}\right] \Gamma_{\rm alb}(\sigma) \; ,
\end{equation}
where $I_0$ is the solar constant, $\bar{\cal A}$ the averaged
Earth albedo, $\alpha \equiv R_e/a$ is the satellite parallax
($R_e$ the Earth radius and $a$ the satellite's semimajor axis)
and $\psi$ is the geocentric Sun-satellite angular distance. Unit
vectors ${\bf r}$, resp. ${\bf n}$, denote the geocentric positions
of the satellite, resp. the Sun. Notice that, for further use, we
prefer to parametrize the flux vector ${\bf F}$ by the
angle $\sigma$ defined as the difference between the satellite's
longitude in orbit $\lambda$ (measured from the orbital node)
and the solar longitude in orbit $\lambda_s$, i.e. $\sigma
\equiv \lambda - \lambda_s$ (see Fig.~1 for definitions). Finally,
``the albedo shadow function'' $\Gamma_{\rm alb}(\sigma)$ is simply
defined as the usual step function
\begin{eqnarray}
 \Gamma_{\rm alb}(\sigma) &= & 1 \qquad\quad {\rm for}\quad
  \sigma \in (-\pi/2,\pi/2)\; , \nonumber \\
 & = & 0 \qquad\quad {\rm for}\quad \sigma \in (\pi/2,3\pi/2)
  \;  \nonumber
\end{eqnarray}
[there is the obviously trivial relation $\sigma = 2\pi \nu t + \sigma_0$
between the angle $\sigma$ and the satellite's orbital
frequency $\nu$].  Defining the {\it instantaneous heating function}
$H_a(\sigma)$ and the {\it thermally relaxed heating function}
$h_a(\sigma)$ by the following relations
\begin{equation}
 F_S \equiv \left({\bf F}.{\bf s}\right) = {1\over 4} I_0
 \bar{\cal A} \alpha^2 H_a(\sigma) \;
\end{equation}
and
\begin{equation}
h_a\left(\sigma\right) \equiv {1\over \sigma_R}\int_{\sigma' \leq
 \sigma} d\sigma' \, H_a(\sigma') e^{-(\sigma-\sigma')/\sigma_R } \; ,
\end{equation}
[here $\sigma_R\equiv 2\pi \nu /\nu_R$]. The investigated thermal force
due to the albedo radiation heating of the satellite reads
\begin{equation}
 {\bf a}\left(\sigma\right) = \gamma h_a\left(\sigma\right)
  {\bf s} \;
\end{equation}
with
\begin{equation}
 \gamma \simeq {2\over 9} \varepsilon \bar{\cal A}
 \alpha^2 {A_\perp I_0\over mc} \; .
\end{equation}
Employing estimated values for the quantities in (9) [see
Rubincam, 1987, 1988; Slabinski, 1995], we expect $\gamma$
to lie in the interval $(1-5)\, {\rm pm/s}^2$. Similarly, a
reasonable range of the phase-lag $\sigma_R$ is $1.5-2.5$
radians.

An essential step towards the fully analytical formulation
of our problem consists in computing the relaxed heating function
$h_a\left(\sigma\right)$. Interestingly, it can be worked out
with only one minor restriction of the exact $2\pi$-periodicity
of the heating function $H_a(\sigma)$, thus neglecting changes
of the mutual configuration of the satellite orbit and the Sun.
More precisely, the following results hold if the
characteristic time scale on which the Sun-orbit configuration
changes is much greater than the thermal time-lag
$\tau_R$, a property which is well satisfied for LAGEOS. The
results then read
\begin{eqnarray}
 h_a(\sigma) & = & C_1 + D_1 \cos 2\sigma + D_2 \sin 2\sigma
  - (C_1 - D_1) {e^{-(\pi/2+\sigma)/\sigma_R}\over 1+e^{-\pi/
  \sigma_R}} \; , \nonumber\\ & & \qquad\qquad\qquad\qquad
  \qquad\qquad {\rm for}\quad  \sigma \in (-\pi/2,\pi/2)\; , \\
 & = & (C_1 - D_1){e^{(\pi/2 - \sigma)/\sigma_R}\over 1+e^{-\pi/
  \sigma_R}} \qquad\; {\rm for}\quad  \sigma \in (\pi/2,3\pi/2)
  \; ,
\end{eqnarray}
provided the following parameters are introduced
\begin{eqnarray}
 C_1 & = & {1\over 2} \left({8\over 3}+3\alpha\right) \left(
  n_x s_x + n_y s_y\right) - \alpha \cos\theta_r \; , \\
 C_2 & = & {1\over 2} \left({8\over 3}+3\alpha\right) \left(
  n_x s_x - n_y s_y\right) \; , \\
 C_3 & = & {1\over 2} \left({8\over 3}+3\alpha\right) \left(
  n_x s_y + n_y s_x\right) \; , \\
 D_1 & = & {C_2 - 2\sigma_R C_3\over 1+4\sigma_R^2} \; ,  \\
 D_2 & = & {C_3 + 2\sigma_R C_2\over 1+4\sigma_R^2} \; .
\end{eqnarray}
Here, $\cos\theta_r \equiv {\bf n}.{\bf s}$ in agreement with
the terminology of Scharroo et~al. [1991] and M\'etris et~al.
[1995]. Indexes $(x,y,z)$ correspond to the projection of the given
vector onto a direct triad of unit vectors $({\bf e}_x,{\bf e}_y,{\bf
e}_z)$: ${\bf e}_x$ is defined by the projection of Solar direction
${\bf n}$ onto the satellite orbit plane and ${\bf e}_z$ is in the
direction of the satellite angular momentum vector. Note also,
that the equality $\int d\sigma H_a(\sigma) = \int d\sigma h_a(\sigma)
(= \pi C_1)$ assures equilibrium between the absorbed radiative
energy and thermally emitted energy.

Formulas (10) and (11) [together with definitions (12) -- (16)]
complete our analytical approximation
for the thermal force related to the albedo radiation
heating of the LAGEOS satellite. As mentioned in the Introduction,
we shall check the validity of our approximate results by
employing a fully  numerical model and comparing its results with
those obtained by the simple the analytical formulation given before.
Practically it means that we shall compute components of the
albedo radiation flux vector ${\bf F}$ exactly [without
assuming simplification (5)] by employing
\begin{eqnarray}
 F_R &=& {I_0\over \pi}\,{\bar{\cal A}} \alpha^2 J_{x'}\left(
  \psi,\alpha\right) \; , \nonumber  \\
 F_T &=& -{I_0\over \pi}\, {({\bf n}.{\bf t})\over \sin\psi}
  {\bar{\cal A}} \alpha^3 J_{y'}\left(\psi,\alpha\right) \;
  , \\
 F_W &=& -{I_0\over \pi}\, {({\bf n}.{\bf w})\over \sin\psi}
  {\bar{\cal A}} \alpha^3 J_{y'}\left(\psi,\alpha\right)
  \; , \nonumber
\end{eqnarray}
for the radial, transverse and normal components of ${\bf F}$
in the orbital unit vector frame $({\bf r},{\bf t},{\bf w})$.
Formulation (17) has been given by Borderies and Longaretti
[1990] for the case of the Earth with constant albedo and diffuse
reflection on its surface (due to the small amplitude of the
studied effects, we shall not go beyond the assumption of constant
Earth albedo). Functions $J_{x'}(\psi,\alpha)$ and $J_{y'}(\psi,
\alpha)$ are given in Borderies and Longaretti [1990] in
the form of $1$-D integrals.
Once the components of the radiative flux vector are computed,
we can follow formulas (6 -- 9) to determine the thermal force
acting on the satellite. Obviously, in the second case all
quantities are computed numerically.

\subsection{ Long-term orbital perturbations}
Analysis of the long-term orbital perturbations is the proper task
of this study. We thus proceed to average the orbital
perturbations corresponding to the thermal force formulated
in the previous section over one satellite revolution around
the Earth. Because the effects are very small we limit
ourselves to direct computation of the mean values of the
perturbations. We also neglect the eccentricity of the satellite orbit.

Starting with the averaged along-track acceleration $T$ (related
to the semimajor axis excitation) we obtain
\begin{equation}
 T = {\gamma\over 3\pi} \left[\left(3C_1+ D_1\right) s_y -
 2D_2 s_x - 3\sigma_R\left(C_1 - D_1\right){s_x + \sigma_R s_y
   \over 1+\sigma_R^2}\right] + {\cal O}(e) \; .
\end{equation}
Similarly, excitations of the eccentricity vector components
$(h,k) \equiv (e\cos\omega,$ $e\sin\omega)$, given by the
corresponding set of the Gauss equations (e.g. Milani et~al.,
1987), read
\begin{eqnarray}
 {\dot h} & = & {\gamma\over 8na}\bigl[6C_1 s_b + \left(
  D_1s_y - D_2s_x\right) \cos\lambda_s \nonumber \\
  & & \qquad\;\; -\left(D_1s_x + D_2 s_y\right)\sin\lambda_s
  \bigr] + {\cal O}(e) \; , \\
 {\dot k} & = &{\gamma\over 8na}\bigl[-6C_1 s_a + \left(
  D_1s_x + D_2s_y\right)\cos\lambda_s \nonumber \\
  & & \qquad\;\; +  \left(D_1s_y - D_2s_x \right) \sin
  \lambda_s \bigr] + {\cal O}(e) \; ,
\end{eqnarray}
[overdot means time derivative] with notation
\begin{eqnarray}
 s_a &=& s_x \cos\lambda_s - s_y \sin\lambda_s \; , \nonumber \\
 s_b &=& s_x \sin\lambda_s + s_y \cos\lambda_s \; . \nonumber
\end{eqnarray}
Finally, we compute the long-term excitation of the inclination
$I$ of the satellite orbit. After straightforward algebra we
obtain
\begin{eqnarray}
 {d I\over dt} & = & {\gamma\over 3\pi na}\left({\bf s}.{\bf w}
  \right) \bigl[\left(3C_1+ D_1\right) \cos\lambda_s - 2D_2 \sin
  \lambda_s \nonumber \\
 & & \qquad\quad - 3\sigma_R\left(C_1 - D_1\right){\sin\lambda_s
   + \sigma_R \cos\lambda_s\over 1+\sigma_R^2}\bigr] + {\cal O}(e)
  \; .
\end{eqnarray}
In the following, we apply the previous results to the case of the LAGEOS~I
satellite.

\section{ Application to LAGEOS~I}
For a decade or so, the orbital analysis of the LAGEOS~I satellite
has provided superb ground for geodynamics and physics of the
Earth environment. Direct treatment of its orbit served for
setting the top models of the Earth gravitational field
together with its variations, Earth rotation monitoring,
tectonic plate motion etc. Yet, supplemental and
highly interesting information is hidden in the series of
empirical parameters necessary to be added to the basic dynamical model
to reach laser tracking precision of about $1-2$ centimeters.
They are usually given in the form of time derivatives
(called {\it excitations}) of the orbital elements.
A detailed overview of the observed LAGEOS~I orbit residuals
and/or orbital element excitations is given in Tapley et~al.
[1993]. It is generally believed that these excitations are due
to tiny physical effects mostly of nongravitational origin.
Their interpretation and link to particular physical phenomena
offers interesting problems at the frontier between physics and
dynamics.

The best understood among these orbit excitations are the
empirical series of the along-track acceleration (``semimajor
axis excitation''). Considerable effort has been paid to
their modeling in the last ten years. Much less
attention has been oriented towards understanding
excitations of the other orbital elements. Yet, parallel
understanding of several (or even all, if possible)
excitations yields, apart from overall consistency, also
the possibility of decorrelating some of the empirical parameters
involved in the suggested models for the non-gravitational
effects [see M\'etris et~al., 1995]. A recent attempt for a joint
solution of the along-track series and the eccentricity
excitations by M\'etris et~al. [1995] resulted in
the surprising conclusion that the empirically expected amplitude
of the Yarkovsky-Schach thermal effect is about $1.7$
times greater than its theoretical estimation. As mentioned
in the Introduction, one possible suggestion to explain
this discrepancy was the missing albedo heating
mechanism in the models used  for the thermal effects. We thus
start by considering the eccentricity excitations.
We also note that an important part of a correct determination
of the thermal effects is a firm knowledge of the satellite spin axis
orientation. In the case of LAGEOS~I this has been for a long time
a rather obscure issue. Recently, Farinella et~al. [1995] succeeded
in proving that the older theoretical model by Bertotti and
Iess [1991] is well suited to recover the main features of the
LAGEOS~I spin axis evolution. We thus employ the LAGEOS~I spin
axis evolution as derived by Farinella et~al. [1995].

Figure 2 shows the real -- a) -- and the imaginary -- b) --
parts of the eccentricity excitations corresponding
to the albedo heated part of the thermal effects. The true
LAGEOS~I orbit is considered. We took $\gamma = 2.5\,
{\rm pm/s}^2$ for the amplitude of the effect and compared
results corresponding to several values of the phase-lags
$\sigma_R$ (in radians): $2.5$ solid line, $1.0$ dashed line and $0.5$
dashed-dotted line. Although there are small differences
according to the value of the phase-lag, the essential information
is given by the ordinate scale of both parts of Figure~2.
The eccentricity effect of the studied mechanism is much
smaller than that is needed for explaining the discrepancy
between the theoretical and empirical values of the
Yarkovsky-Schach effect discussed above.
We have checked that, as in the case of the direct albedo
radiation pressure [M\'etris et~al., 1995], the eccentricity
excitations computed analytically, with use of the
simplified formula (5) for the albedo radiative flux,
fit with surprising precision the corresponding numerical
solution with the complete computation (17) of the albedo
radiative flux. Notice also, that the real part of the signal (in
Fig.~2a) is in fact anticorrelated with the Yarkovsky-Schach
effect contribution and thus acts rather negatively in respect
of explaining the Yarkovsky-Schach thermal effect discrepancy
mentioned above. It has been recently communicated to us by
V.J. Slabinski [personal communication], that an intuitive
explanation of this feature may be due to the fact that the
Earth-albedo radiation preferentially illuminates the ``dark''
(anti-Sun) pole of the satellite.

Next, we turn to the possible contribution of the albedo
part of the along-track thermal thrust. Figure 3 shows a comparison of the
numerically determined thrust -- solid line -- and the
analytically modeled thrust by formula (18) -- dashed
line. As anticipated by analogy with the direct albedo
radiation pressure, the coincidence of the two signals
is weaker than in the case of the eccentricity excitations.
Nevertheless, the analytical formulation is still not
seriously corrupted, apart from occasional positive peaks.
It is interesting to note that the expected averaged
value of the drag is about $-(0.2-0.3)\, {\rm pm/s}^2$.
This fact has a direct implication on the value of the
adjusted constant along-track force which is to be
attributed to the neutral and charged particle drag at the LAGEOS
altitude [e.g. Afonso et~al., 1985; Barlier et~al. 1986].
Interestingly, an analogous, but positive, additional effect of
about $0.4\, {\rm pm/s}^2$ resulted from Slabinski's [1995]
model for LAGEOS~I thermal thrust. Slabinski interpreted
this effect in terms of the mutual coupling between the
Yarkovsky-Schach and Yarkovsky-Rubincam effects related
to the decrease of the satellite mean temperature during
the shadow crossing periods. The albedo radiation as a heat
source for the LAGEOS thermal effect thus seems
 to compensate for
at least half of Slabinski's correction.

Finally, we present a similar solution for the inclination
excitation in Figure~4. A completely numerical solution (solid
line) is again compared with the analytical solution
(21) (dashed line). Apart from good coincidence of the
two approaches, we note an average contribution to the
inclination excitation of about $0.1\, {\rm mas/yr}$.
This value is to be compared with the corresponding inclination
effect  of about $(0.7-1.0)\, {\rm mas/yr}$ due to the
Yarkovsky-Rubincam effect studied by Farinella et~al. [1990].
The observed value of the secular increase of the
LAGEOS inclination is of about $1\, {\rm mas/yr}$ [P. Exertier,
personal communication].

\section{ Conclusion}
The main results of this study can be summarized in
several items:
\begin{itemize}
\item the albedo radiation as a heating source for the
 LAGEOS thermal effects is unlikely to be responsible for
 the discrepancy between the theoretical and empirical
 estimated amplitude;
\item it is expected that the albedo heated thermal
 thrust contributes by about $-(0.2-0.3)\, {\rm pm/s}^2$
 to the secular part of the empirically observed
 along-track signal, partly absorbing the Slabinski's recent
 correction due to coupling between the Yarkovsky-Schach
 and Yarkovsky-Rubincam effects;
\item analogous secular contribution to the inclination
 excitation is of about $0.1$ ${\rm mas/yr}$.
\end{itemize}
The first result points to a persisting discrepancy between
the theoretical and empirical modeling of the LAGEOS~I
thermal effects. In the near future, some effort is still
needed for its understanding.
\smallskip

\noindent{\it Acknowledgements.}\hskip 2mm We are grateful to
 V.J. Slabinski who kindly communicated to us results of his
 LAGEOS thermal model and suggested several general ideas related
 to the albedo heated part of the thermal effects. F. Barlier and
 P. Exertier are thanked for helpful discussions, R.~Feldman and the
 unknown referees for numerous suggestions improving the English
 of the submitted version of this paper. One of us (D.V.)
 worked on this study when staying at the OCA/CERGA, Grasse (France)
 thanks to the H. Poincar\'e research fellowship.

\section{ References}

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 9381-9398, 1985.
\smallskip

\noindent{\bf Afonso, G., F. Barlier, M. Carpino, et~al.} Orbital
 effects of
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 501-514, 1989.
\smallskip

\noindent{\bf Barlier, F., M. Carpino, P. Farinella, et~al.}
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\noindent{\bf Borderies, N.} A general model of the planetary
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\noindent{\bf Borderies, N., and P.Y. Longaretti.} A new treatment
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 and of a spherical satellite, {\it Celest. Mech.,} {\it 49,} 69-98,
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\smallskip

\noindent{\bf Farinella, P., A.M. Nobili, F. Barlier, and F. Mignard.}
 Effects of thermal thrust on the node and inclination of LAGEOS,
 {\it Astron. Astroph.,} {\it 234,} 546-554, 1990.
\smallskip

\noindent{\bf Farinella, P., D. Vokrouhlick\'y, and F. Barlier.}
 A consistent theoretical model for the rotation of LAGEOS, {\it
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\noindent{\bf M\'etris, G., D. Vokrouhlick\'y, J.C. Ries, and R.J.
 Eanes.} Non-gravi\-ta\-ti\-onal effects and the LAGEOS
 eccentricity excitations, {\it J. Geoph. Res.,} submitted, 1995.
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\noindent{\bf Milani, A., A.M. Nobili, and P. Farinella,} {\it
 Non-gravi\-ta\-ti\-onal perturbations and satellite geodesy},
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\noindent{\bf Rubincam, D.P.} On the secular decrease in the semimajor
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\noindent{\bf Rubincam, D.P.} LAGEOS orbit decay due to infrared
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\noindent{\bf Rubincam, D.P.} Yarkovsky thermal drag on LAGEOS,
 {\it J. Geoph. Res.,} {\it 93,} 13805-13810, 1988.
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\noindent{\bf Rubincam, D.P., and N.R. Weiss.} Earth albedo and
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\noindent{\bf Scharroo, R., K.F. Wakker, B.A.C. Ambrosius, and R. Noomen.}
 On the along-track acceleration of the LAGEOS satellite, {\it J. Geoph.
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\noindent{\bf Slabinski, V.J.} A numerical solution for LAGEOS
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\noindent{\bf Tapley, B.D., B.E. Schutz, R.J. Eanes, J.C. Ries,
 and M.M. Watkins,}
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\noindent{\bf Vokrouhlick\'y, D., P. Farinella, and F. Mignard.}
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\smallskip

\newpage

\bigskip\bigskip

\noindent{\bf  \Large Figure captions:}
\bigskip\bigskip

\noindent {\bf Figure 1.}\hskip 3mm Orbital parameters and
 related quantities introduced in the text.
\bigskip

\noindent {\bf Figure 2.}\hskip 3mm LAGEOS eccentricity
 excitations due to the albedo heated thermal effect vs. time
 since launch: {\bf a)} the real part and {\bf b)} the
 imaginary part. Assumed value of the amplitude $\gamma$ is
 $2.5\, {\rm pm/s}^2$. Different curves correspond to different
 values of the phase-lag $\sigma_R$ (in radians): (i) $2.5$ solid 
 curve, (ii) $1.0$ dashed curve and (iii) $0.5$ dashed-dotted curve.
\bigskip

\noindent {\bf Figure 3.}\hskip 3mm Along-track thermal thrust
 due to the albedo heated part vs. time since launch. Solid
 curve corresponds to the completely numerical setting, while
 dashed curve to the fully analytical solution. Parameters:
 $\gamma = 2.5\, {\rm pm/s}^2$ and $\sigma_R = 2.5$~rad.
\bigskip

\noindent {\bf Figure 4.}\hskip 3mm Inclination excitation vs.
 time since launch. Plotted curves as in the previous figure:
 solid line for numerical solution, dashed line for the
 analytical solution. Parameters:  $\gamma = 2.5\, {\rm pm/s}^2$
 and $\sigma_R = 2.5$~rad.

\end{document}
