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\documentstyle[11pt]{spacekap}
\hyphenation{non-gravi-tatio-nal}
\begin{opening}
\title{CLOUD COVERAGE ASPECTS OF THE ALBEDO EFFECT}
\author{DAVID VOKROUHLICK\'Y}
\author{LADISLAV SEHNAL}
\institute{Astronomical Institute, Charles University Prague\/ \\
\v Sv\'edsk\'a 8, 150 00 Prague 5, Czech Republic\/ \\
{\rm and} \\
Astronomical Institute of the Czech Academy of Sciences\/ \\
251 65 Ond\v rejov, Czech Republic\/}
\end{opening}
\begin{document}
\begin{abstract}
   We discuss some details of the cloud coverage aspects of
   the albedo effect -- the Earth reflected radiation pressure -- on
   the motion of an artificial satellite. Our care is focused on
   modeling of the Earth surface reflection and propagation of the
   radiation through the atmosphere. We adopt analytical models
   of these phenomena from the radiative transfer theory, in
   contrast to earlier developed approaches based on the fitting
   of the satellite photometry data. We perform several computations
   based on the accepted models for the ERS-1 and MACEK satellites
   to test the hierarchy of importance of the investigated effects.
   In the case of the mission MACEK (boarding the precise
   accelerometer) this information might be essential when
   interpreting the data. \\
\keywords
 Celestial mechanics -- Satellites -- Radiation mechanisms
 -- Earth atmosphere
\end{abstract}

\section{Introduction}
The nongravitational force perturbations of the motion of an artificial
satellite are the domain of interest of this paper. We concentrate our
discussion on one of them -- the effect of the sunlight reflected
by the Earth's surface (later on abbreviated as `the albedo effect' in
agreement with widely used terminology). First, let us briefly
describe the role of this effect in the context of the others.

During the last several years, the importance of the albedo effect together
with the atmospheric (both neutral and charged) drag and lift was
recognized in the context of the badly controlled nongravitational
perturbations of the motion of an artificial satellite. The albedo effect
is important
not only for the low and middle trajectory satellites, (our examples
in Sec.~3 belong to this category), but its significance is even strengthened 
for high altitude satellites, where the direct influence of the atmosphere
is negligible and radiative effects (generally) are the dominant poorly
modeled nongravitational phenomena.

As concerns the magnitude, the direct solar radiation pressure
is in most cases a comparable nongravitational effect to those quoted earlier.
However, there is an essential difference between the direct Solar
radiation pressure and the albedo effect -- the direct Solar radiative
field is nearly homogeneous, in contrary to the albedo effect radiative 
field. This substantially simplifies the treatment of the direct
Solar radiation pressure influence. Most often the main
part of the direct Solar radiation pressure can be expressed
analytically.
The remaining errors, arising from poor
knowledge of the optical characteristics of the satellite surface,
multiple reflections on the satellite surface segments (or other
geometrical complexities) or the transitional effects connected
with the satellite's entry into the Earth's shadow, can be in many
cases dealed with using numerical methods. Precise pre-launch tests,
clever Monte Carlo simulations (Klinkrad et~al., 1990) and approximate
models (e.g. Link, 1969; Kabel\'a\v c, 1988; Vokrouhlick\'y et~al.
1993) can suppress this error in the
modeling of the direct solar radiation pressure to the level of
errors concerning the albedo effect. All quoted sources of
errors work in the albedo effect case too, but still many
other errors disturb the precision of its modeling, as will be
listed later.

In particular situations, other nongravitational effects could play
a key role -- e.g., the Earth IR radiation (actually some authors
treat it together with the albedo effect as the theoretical
modeling is similar, e.g. Borderies, 1990), the effects
of thermal thrust (e.g. Rubincam, 1987; Afonso et~al., 1989;
Farinella et~al., 1990),
the Poynting-Robertson effect (e.g. Rubincam, 1982), the Schach effect (e.g.
Rubincam, 1982), interaction
of the charged satellite with the Earth's magnetic field (e.g.
Vokrouhlick\'y, 1989) etc. Some subject surveys, mainly inspired by
`the LAGEOS story', can be found (e.g. Rubincam, 1982). Let us stress
that the {\it magnitude} of these effects is of the same order as
the model errors of the dominant ones (atmospheric drag and lift,
direct solar radiation pressure and albedo effect). The following
question immediately arises from the discussion in the previous
paragraph~: What is the situation when the nondominant nongravitational
effects can play an important role, though their magnitude is always
lower than the magnitude of errors of the dominant ones? The
cumulative result of the `low-magnitude' nongravitational perturbations
coming from their {\it direction} is the natural answer to the
previous question. This is the case in the LAGEOS problem.

In this paper, we follow another situation where either no such
precision is required (the nongravitational effects act on
sufficiently small time spans, e.g. between trajectory corrections) or
one is interested in interpretation of the direct measurements
of the perturbative accelerations (e.g.
measurements of an accelerometer-type experiment). In these cases,
the restriction on the dominant nongravitational effects is
justified. 

We avoid a historical review of the development of the albedo effect
theory. An interesting contemporary survey can be found in Borderies
and Longaretti, 1990. Let us just remind that, basically, there are
two ways of considering the problem~:
\begin{itemize}
\item {\it analytical}, which requires principal simplifications
(the most promising progress can be found in Rubincam and Weiss, 1982
and Borderies, 1990, where under quite general assumptions, the
double integration evaluating the disturbing force components is reduced
to a single integration; despite this success, a huge analytical effort is
necessary, thus, the general priority of this approach is questionable),
\item {\it numerical}, which loses the subtlety of the previous
approach (one integrates a particular satellite trajectory)
but this approach enables one to directly take into account many
complex aspects.
\end{itemize}
The present study follows the latter framework, being purely numerical.

In previous paper (Vokrouhlick\'y and Sehnal, 1992 -- hereafter paper~I),
we briefly summarized
some of the subtle aspects of the albedo effect. Here is another list
of the details which together affect the result :
\begin{itemize}
\item physics of the sunlight reflection by the Earth's surface,
\item the role of scattering and absorption in the
atmosphere (including cloud coverage effects),
\item physics of the reflection by the satellite surface
(including geometrical effects),
\item other less important effects (some of them are described in paper~I).
\end{itemize}
In this paper, we concentrate mainly on the first and second items
leaving the others out of the discussion.

We continue the line of investigation of paper~I
and discuss in more detail the role of the cloud coverage
on the radiative field. We use a simple analytical
treatment of the first two items pointed out previously. Here,
our concept differs from the framework based on the
satellite photometric data reduction (e.g. Rubincam et~al., 1987;
Knocke, 1989). These data characterize the
reflection properties of the different `Earth surfaces' (e.g.
sea surface, grassed land etc.) and `atmospheric formations'
(e.g. clouds). Generally, one should expect
that, our framework being more simple, leads to less precise
results, while the latter concept being more complicated,
requires more computational time and memory. Comparison
of both concepts is left open. For more comments on these two
approaches see paper I.

Let us also stress that our aim is not to present all the
dynamical aspects of the discussed aspects (mainly of the
atmospheric phenomena in the albedo effect context) -- we are
predominantly interested (i) in the analytical approach for the
evaluation of the perturbative accelerations adopted from
the theory of radiative transfer, (ii) in proving, that
the studied phenomena could be important for trajectory analysis in case
of precise altimetric missions and even essential when interpreting the
accelerometer-type data. 

In Sec.~2, we introduce the
basic formulas and models. In Sec.~3, we apply previous results for
two satellites~: ERS-1 and MACEK.

\section{Basic formulas and models}
We accept the notation of paper~I as well as the basic equations
used there. We skip over the details of their derivation restricting
ourselves to the list of necessary formulas.
\begin{figure}
\vspace{76mm}
\caption[]{Geometry of the introduced frames and associated
coordinates. ${\cal P}$ stands for the chosen Earth surface
element, ${\cal S}$ for the position of the satellite.}
\end{figure}

Let us start with the discussion of the models describing the physical
processes of the solar radiation reflection by the Earth's surface.
We introduce the same scheme as in paper~I, in part a local frame joint with
the Earth's surface element ${\cal P}$ endowed with spherical coordinates
$(\theta,\varphi)$ -- see Fig.~1. As will be clarified later on, the choice
of the direction specifying $\varphi = 0$ does not intervene
in the formalism (and therefore is left unspecified) provided
that the z-axis of the local frame is normal to the surface
element ${\cal P}$. We assume illumination of ${\cal P}$ by the
Solar homogeneous radiative field of intensity $I_0$. Let the Solar
direction in the chosen frame be $(\theta_0,\varphi_0)$.
In what follows we will use $\mu =\cos \theta$, resp.
$\mu_0 =\cos \theta_0$, instead of $\theta$, resp. $\theta_0$.
Concerning the reflective modes of the sunlight on the
Earth's surface, we distinguish the following types.

{\bf I.} The surface element belongs to the {\it continent}. We
suppose diffusive reflection according to Lambert's law
\begin{equation}
I^{({\rm d})}_{{\rm refl}} = \frac{1}{\pi} I_0 a_0 \mu_0\/,
\end{equation}
where $a_0$ is the Earth albedo given by Sehnal's expansion in
spherical harmonics up to the order 6 (Sehnal, 1979). This corresponds
to the model (d,--) in paper~I.

{\bf II.} The surface element belongs to the {\it sea}. Here we
suppose partial diffuse and specular reflection.
Let us denote the diffusive coefficient by $D(\mu_0)$ and the
specular coefficient by $R(\mu_0)$ in agreement with Barlier et~al.,
1986. They describe the portion of the radiation reflected
diffusively resp. specularly by the Earth's surface element illuminated
by the Sun. The nominal albedo models (such as in Sehnal, 1979) are 
fitted to data from many satellite positions.
If we denote angular (with respect to $\mu_0$) and time
averaging by symbol $\langle .. \rangle$, we can mathematically formalise the previous statement as
\begin{equation}
a_0 = \langle D(\mu_0) + R(\mu_0) \rangle\,.
\end{equation}
The knowledge of the nominal albedo values $a_0$ and theoretical ideas 
on the $\mu_0$--profile
of $R(\mu_0)$ (see Barlier et~al., 1986) are not sufficient for the
determination of the reflected intensity splitting into diffusive and
specular parts. In other words, Eq.~(2) still allows many such splittings.
Here we accept two of them in order to see the importance of this phenomenon.
At first, \\
\[I^{({\rm s1})}_{{\rm dif}} = \frac{1}{\pi} I_0 a_0 \mu_0 (1-R(\mu_0))\/,\]
\begin{equation}
I^{({\rm s1})}_{{\rm spec}} = I_0 a_0 R(\mu_0) \psi_{\rm sc}(\mu_0)
\delta(\mu-\mu_0) \delta(\varphi-\varphi_0+\pi) \/,
\end{equation}
which is exactly our (s,--) model in notation of paper~I, i.e.
the model taking into account specular reflection on the Earth's
surface (note that $\psi_{\rm sc}(\mu_0)$
is a geometrical factor arising from the dilution of the specularly
reflected radiation by the Earth, for definition see Barlier et~al., 1986). 
Secondly,
\[I^{({\rm s2})}_{{\rm dif}} = \frac{1}{\pi} I_0 \mu_0 (a_0-{\bar R})\/,\]
\begin{equation}
I^{({\rm s2})}_{{\rm spec}} = I_0 R(\mu_0) \psi_{\rm sc}(\mu_0)
\delta(\mu-\mu_0) \delta(\varphi-\varphi_0+\pi) \/,
\end{equation}
where
\[{\bar R} = \int_0^1 R(\mu_0)\, d\mu_0 \/,\]
which in fact approximate closer the reality of the diffusive coefficient
independence from $\mu_0$ (which is reported in Barlier et~al., 1986).

{\bf III.} The surface element is covered by {\it clouds}. In this case, we assume
the reflecting law (see Chandrasekhar, 1950, Sec.~46-47)
\[I^{({\rm c})}_{\rm refl} = \frac{1}{\pi} \frac{\omega_0\mu_0 I_0}
{\mu+\mu_0} \bigl( H(\mu)H(\mu_0)\Psi(\mu,\mu_0)-\]
\begin{equation}
\qquad x(1-{\mu}^2)^{1/2}(1-{\mu_0}^2)^{1/2} H^{(1)}(\mu)
H^{(1)}(\mu_0) \cos (\varphi-{\varphi}_0)\bigr)\/,
\end{equation}
where
\[\Psi(\mu,\mu_0) = 1-c({\omega}_0)(\mu+\mu_0)-x(1-\omega_0)\mu\mu_0\/,\]
which solves the radiative scattering problem for the plane-parallel
atmosphere with an infinite optical thickness and scattering centers
characterized by the phase function
\begin{equation}
p(\Theta) = {\omega}_0 (1+x \cos \Theta),\quad 0\leq \omega_0 \leq 1
,\quad -1\leq x \leq 1.
\end{equation}
$H(\mu)$ and $H^{(1)}(\mu)$ are well known Chandrasekhar H-functions
(note their hidden dependence on the phase function parameters
${\omega}_0$ and $x$). Several algorithms for their evaluation can
be found (e.g. Chandrasekhar, 1950; van de Hulst, 1980). Three main
simplifications are associated with the model given by Eq.~(5) :
\begin{itemize}
\item infinite optical thickness of the atmosphere (i.e. clouds),
\item simplified phase function,
\item both vertical and horizontal homogeneity,
\end{itemize}
(the assumption of planarity is plausible since the geometrical
thickness of the atmosphere is much less than the Earth's radius).
The first item can be generalized to the case of the arbitrary
optical thickness (e.g. Chandrasekhar, 1950; van de Hulst, 1980),
but the resulting formulas are very complex. The real
phase function can be expanded in series of Legendre polynomials
(e.g. Chandrasekhar, 1950; van de Hulst, 1980), thus, formula
(6) represents the restriction on the first two terms. The remaining
terms can be added without difficulty but the resulting formulas
quickly become too long and difficult to handle. Following the previous
remarks, we accept the model given by Eqs.~(5)-(6). We skipped the
generalization of the third item as it requires complicated numerical
models (e.g. Irvine, 1975; van de Hulst, 1980).
\begin{figure}
\vspace{76mm}
\caption[]{Deviations of the accepted cloud scattering model from
the Lambert law. Solid curves correspond to the value $x = 1$ of the phase
function asymmetry parameter, dashed ones for
$x = 0$. The curves indicated by 1 stand for the model with
$\omega_0 = 1$, analogously 2 for $\omega_0 = 0.8$ and 3 for
$\omega_0 = 0.6$. We choose $\mu_0 = 0.8$ for the incident solar
radiation.}
\end{figure}

Let us make some remarks on the cloud scattering model previously
described. Its two parameters are connected with the albedo of 
scattering centers in the atmosphere ($\omega_0$) and scattering
asymmetry ($x$). The effective albedo of the atmosphere
(`plane albedo', Irvine, 1975) due to the vertical radiative flux
\[F_{\bot}({\mu}_0) = {\mu}_0 F [1-q({\omega}_0,x)H({\mu}_0)]\/,\]
\noindent is given by
\begin{equation}
{\bar A}_{\rm eff}(\omega_0,x) = 1-q({\omega}_0,x){\alpha_0}({\omega}_0,x)
\/,
\end{equation}
where $q({\omega}_0,x)$ and ${\alpha_0}({\omega}_0,x)$ are connected with
the H-functions (e.g. Chandrasehar, 1950; van de Hulst, 1980).
Fig.~2 shows the deviations of the accepted model (Eq.~(5)) from
the Lambert formula in the plane given by the incident ray and
normal to the surface of element ${\cal P}$ for some values of
the parameters $\omega_0$ and $x$ using the quantity
\begin{equation}
\rho (\mu,\varphi) = \pi I^{({\rm c})}_{\rm refl}/(I_0 \mu_0
{\bar A}_{\rm eff}(\omega_0,x))\/,
\end{equation}
\noindent
which is closely related to the anisotropic factor defined in
Knocke, 1989.

Let us also remark that all the formulas (1), (3 -- 5) depend
on the difference $\varphi - \varphi_0$, rather than on
$\varphi$, $\varphi_0$ themselves. This proves that the
choice of $\varphi=0$ is irrelevant.

The algorithm deciding the character of the Earth surface element
${\cal P}$ remains to be clarified. We use the Earth continent mask
($2^{\circ} {\rm x} 2^{\circ}$ network with spline weighing for the chosen point
${\cal P}$). This mask is overlapped by the cloud coverage mask
($2^{\circ} {\rm x} 2^{\circ}$ network with spline weighing decision for noncenter
point). The latter can be `empty' for no cloud situation, `full' for
the total coverage by clouds or can reflect the actual distribution of
the clouds on the Earth's surface.

The Earth reflected radiative field in the location of the satellite is
given by integration of the elementary contributions from all the
points ${\cal P}$ visible from the satellite. The knowledge of the satellite
surface optical characteristics and its orientation are the last topics
for evaluating the perturbative acceleration. Denoting $\vec{n}^{(\zeta)}$,
($\zeta = \pm$), the normal vector of the `two sided' surface element
consisting of $n_{(\zeta)}$ segments each homogeneous in the optical
characteristics $(A^{(\zeta)}_{\rm i},\rho^{(\zeta)}_{\rm i},
\delta^{(\zeta)}_{\rm i})$, ${\rm i} = 1, ... ,n_{(\zeta)}$, where
$A^{(\zeta)}_{\rm i}$ is the area of the particular i-th segment,
$\rho^{(\zeta)}_{\rm i}$ and $\delta^{(\zeta)}_{\rm i}$ its coefficients of
specular and diffusive reflectivity, we can write for the radiation
pressure acceleration (see paper~I)
\begin{equation}
\vec{a} = {\it k} \sum_{\zeta = \pm } \Bigl[ \bigl( A^{(\zeta)}
{\bf E} + {\rho}^{(\zeta)}{\bf R}(\vec{n}^{(\zeta)})\bigr).\vec{J}_{1}
^{(\zeta)} + {\delta}^{(\zeta)}\vec{J}_{2}^{(\zeta)}\Bigr]\/,
\end{equation}
where
\[A^{(\zeta)} = \sum_{{\rm i}=1}^{{\rm n}_{({\zeta})}} A^{(\zeta)}_{\rm i}
\/ ,\quad \rho^{(\zeta)} = \sum_{{\rm i}=1}^{{\rm n}_{({\zeta})}}
\rho^{(\zeta)}_{\rm i} A^{(\zeta)}_{\rm i} \/ ,\quad
\delta^{(\zeta)} = {2 \over 3} \sum_{{\rm i}=1}^{{\rm n}_{({\zeta})}}
\delta^{(\zeta)}_{\rm i} A^{(\zeta)}_{\rm i}\/,\]
\[\vec{J}_{1}^{(\zeta)} = - \int_{\nu = (1-\alpha^2)^{1/2}}^1 \int_{\phi = 0}
^{2\pi} {\Pi}_{\vec{n}(\nu,\phi)}(\vec{n}^{({\zeta})})
I(\vec{n}(\nu,\phi))\,\tilde{\Theta}(\nu,\phi) \, d\nu d\phi\/,\]
\[\vec{J}_{2}^{(\zeta)} = - \int_{\nu = (1-\alpha^2)^{1/2}}^1 \int_{\phi = 0}
^{2\pi} {\Pi}_{\vec{n}^{({\zeta})}}(\vec{n}(\nu,\phi))
I(\vec{n}(\nu,\phi))\,\tilde{\Theta}(\nu,\phi) \, d\nu d\phi\/,\]
\[\tilde{\Theta}(\nu,\phi) =1-\Theta
(\vec{n}^{({\zeta})}.\vec{n}(\nu,\phi)) \quad,\quad k = c^{-1}m^{-1} \]
\[ \zeta = \pm \quad,\quad \alpha = R_{\rm e}/r \, ,\]
$\Theta(x)$ being the Heaviside function, $\Pi_{\vec{b}}(\vec{a})$ represents 
projection of vector $\vec{a}$ onto vector $\vec{b}$, i.e.
$\Pi_{\vec{b}}(\vec{a}) = \vec{a}\;(\vec{a}.\vec{b})$, and the matrix
{\bf R} is given by: ${\bf R}(\vec{n}^{(\zeta)}).\vec{J}_{1} = 
2\; \Pi_{\vec{n}^{(\zeta)}}(\vec{J}_{1}) - \vec{J}_{1}.$ Formula
(9) gives the components of the searched satellite acceleration in the coordinate
system tied to the satellite (see Fig.~1), where $(\vartheta,\phi)$
are spherical coordinates and $ \nu = \cos \vartheta$.

The resulting perturbative acceleration of the whole satellite
is obtained by combining all accelerations of the particular surface
elements.

\section{Applications}
In this section, we apply the theory summarized in
Sec.~2 to the case of two satellites~: ERS-1 and MACEK. We test the importance of the
effects given by the various models of the Earth surface reflection
and the propagation of radiation through the atmosphere. On the
other hand, we do not take into account the effects caused by
inexact knowledge of the satellite surface optical characteristics
which we studied in more detail in paper~I. Nor do we consider 
effects due to the satellite's complex geometry (multiple
reflections, shadowing of particular segments etc.). In both cases,
we display the components of the perturbative acceleration as
functions of the mean anomaly during one revolution.

Let us also briefly justify the possible significance of the
studied phenomena in the chosen examples. Mainly, the ERS-1 case could
seem to be an ill chosen example to demonstrate the searched aspects
of the albedo effect. One might doubt, whether the results reflect
the pure effect of the studied phenomena or whether it is dimmed
by the satellite's complex geometry. However, comparing the 
results of ERS-1 and MACEK satellites (the latter being nearly spherical) we
conclude, that the essential properties of the cloud aspects within
the albedo effect are not changed by the complicated shape of 
ERS-1. Moreover, we have demonstrated in paper~I, that the influence of
the albedo effect is comparable to the atmospheric drag in the
ERS-1 case (see Fig.~4 in paper~I). This removes another doubt,
that the studied effects are fully overwhelmed by the errors of the other
nongravitational effects. Obviously, the role of the two effects
is different: (i) atmospheric drag is directed along-track, so it
causes mainly secular changes in the semimajor axis, (ii) albedo effect
causes predominantly radial and binormal perturbations, important
for short and long term effects of the other elements (e.g. inclination).
Moreover note, that ERS-1 is Sun-synchronous satellite which
might lead to cummulative phenomena connected with the albedo effect.
Both characteristics are important for very precise ERS-1
altimetric goal.

The case of the MACEK satellite is even more a pure example of
significance of the studied effects. This satellite boards a 
very precise accelerometer uniquely devoted to direct
measurements of nongravitational effects. In this case, our
results are directly applicable.

\subsection{ERS-1 satellite}
In the first example, we examine ERS-1 satellite and
continue the investigation of paper~I. We choose the same orbital
elements given by the epoch of June 21, 1989 (Wakker et~al., 1987).
ERS-1 is a Sun-synchronous satellite with the
Sun's direction approximately in the orbital plane. As
the shape of the satellite is rather complicated, we simplified it by
quasiplanar elements, some of them broken into several segments,
which are described in paper~I. Both, this model and the
orientation of the segments are based on Wakker et~al., 1987.

We already declared that we do not test in this paper the role of
the variation of the satellite surface optical characteristics, thus, we
fix for all the segments \hbox{$\rho^{(\zeta)}_{\rm i} = 0$, $\delta^{(\zeta)}
_{\rm i} = 1-\tau^{(\zeta)}_{\rm i}$,} where the absorption
coefficients $\tau^{(\zeta)}_{\rm i}$ are listed in paper~I, Tab.~1
(i.e. model (--,d) there).

We focus our attention on the role of the Earth surface cloud coverage.
Besides the cloud reflection model, we need information on its distribution
which is beyond the capability of meteorological prediction. On the other
hand, the input from Earth monitoring can be satisfactorily used for
accelerometer data reduction (e.g. the case of the MACEK satellite) or
`real-time' integration of the satellite orbit (e.g. ERS-1). In this paper,
having lower ambitions, we choose some extreme models for cloud
distribution on the Earth's surface~:
\begin{itemize}
\item no clouds on the Earth's surface (in the following figures
denoted $nc$),
\item total coverage of the Earth's surface by clouds ($atm$ stands
for this case in the following figures),
\item random distribution of the clouds with 60\% filling of
the Earth's surface (later on in the figures $m1$ model),
\item equatorial belt of clouds filling the region $-10^{\circ}
\leq\varphi_{\rm e}
\leq 10^{\circ}$, where $\varphi_{\rm e}$ is the geographical latitude
(in the figures denoted as $m2$ model).
\end{itemize}
The third item represents the random distribution extreme of the real
situation (60\% is chosen as it is the typical meteorological value of
this parameter, e.g. Stowe et~al., 1989) while the fourth one represents the ordered distribution
extreme. The limit situations suit us for testing the differences
in all the cases.
\begin{figure}
\vspace{76mm}
\end{figure}
\begin{figure}
\vspace{76mm}
\end{figure}
\setcounter{figure}{2}
\begin{figure}
\vspace{76mm}
\caption[]{Components of the albedo acceleration as a function
of the mean anomaly $M$ during one revolution ($10^{-8}{\rm m.s}^{-2}$
was chosen for unit on the vertical axis). The no cloud coverage
situation is displayed by the $nc$-curve while the full cloud coverage
situation is indicated by the $atm$-curve. The results of the random cloud
distribution with 60\% filling factor (see text) are bound in the dashed
region (denoted $m1$). The first model of the Earth surface reflection
(Eqs.~(3)) is accepted and the cloud reflection parameters (Eq.~(6))
are fixed to be $\omega_0 = 1$, $x = 1$. Note the decrease of the
acceleration in the free-cloud situation due to loss of specularly
reflected rays in the Alaska region.}
\end{figure}
\begin{figure}
\vspace{76mm}
\caption[]{The same as in Fig.~3a but here the second model of the
Earth surface reflection (Eqs.~(4)) is used.}
\end{figure}

The results based on the first three assumptions are shown in
Fig.~3 for parameters $\omega_0 = 1$, $x = 1$ of the cloud reflection
model (Eqs.~(5)-(6)) and the first Earth surface reflection model
(Eqs.~(3)). The acceleration is projected on the usual orbital frame -
$a_{\cal S}$ is the radial component, $a_{\cal T}$ is the orbital-plane
component normal to the radial direction (the `transversal' component) and
$a_{\cal W}$ is the component normal to the orbital plane (the `binormal'
component). As the third model is essentially probabilistic, we
performed 50 runs each time generating randomly the cloud distribution
mask. The dashed curves in Fig.~3 show the minimal and maximal values
obtained by this procedure, thus, all the results of the third experiment
are in the dashed region. We can observe some interesting
details. The results of the first three models are well separated
only in the radial component (Fig.~3a) and rather overlapped in the other two
components (Figs.~3b,3c). The acceleration radial component
dominates in magnitude the others which can be
easily understood from the albedo effect mechanism. Note also the sharp
decrease of all the perturbation components just after the satellite
leaves the Earth shadow. It is caused by the absence of the
specularly reflected component in the radiative field as the specular
reflection point goes over the Alaska region. The effect is
amplified in the near shadow configuration, where the satellite just begins
`to see' the illuminated Earth hemisphere. In that situation, nearly
tangential reflection on the Earth's surface appears and the specular
reflection dominates due to the behaviour of the reflectivity function
$R(\theta_0\rightarrow \pi/2)\rightarrow 1$.
\begin{figure}
\vspace{76mm}
\caption[]{The curves denoted by $nc$ and $atm$ are the same as in
Figs.~3a-4 (dashed for the second Earth surface reflection model).
The model with `equatorial cloud belt' (see text) is displayed by the
$m2$-curve. Here we choose $\omega_0 = 1$, $x = 1$ for the parameters
of the cloud reflection model. Ordinate units as before.}
\end{figure}
\begin{figure}
\vspace{76mm}
\caption[]{Comparison of the signals obtained from the `equatorial
cloud belt' morphology for three different effective albedos of
the clouds : curve 1 for ${\bar A}_{\rm eff} = 1$, curve 2 for
${\bar A}_{\rm eff} = 0.8$ and curve 3 for ${\bar A}_{\rm eff} = 0.6$.
In all the cases the phase function asymmetry parameter is $x = 1$.
Dashed curve stands for the full-cloud coverage, the bottom solid
one for the free-cloud situation. Ordinate units as before.}
\end{figure}

Fig.~4 displays the same situation as Fig.~3a for the second Earth surface reflection
model given by Eqs.~(4). All the other parameters remain unchanged.
Note a slightly different result for the cloud free
model. We do not display the other two components in this case as
the resulting graphs are very similar to the previous ones.

Fig.~5 relates to the fourth experiment. The increase of the perturbative
acceleration is clearly seen. Its maximum has nearly the same values
as the local value for the total coverage case. The reason is that the
Earth's cap visible for the satellite located above the equator is
almost fully filled with clouds. All the computations are
performed for $\omega_0 = 1$, $x = 1$ and both Earth surface reflection
models are compared.

In Fig.~6, a similar situation is illustrated but here clouds with
different effective albedos are taken into account. We can observe
a rather sensitive dependence on the cloud plane albedo. This makes us
conclude that the information on the cloud albedo is as
important as the information on its distribution. Obviously,
in the limit of zero effective cloud albedo, the results agree with
the cloud free model results.

%\newpage
\subsection{MACEK satellite}
\begin{table}
\caption[]{The expected orbital elements of the MACEK satellite}
\begin{center}
\vspace{1mm}
\doublerulesep 5mm
\begin{tabular*}{118pt}{|@{~~~~~}c@{~~~~~}|@{~~~~~}c@{~~~~~}|} \hline
a \rule[0.ex]{0mm}{3ex} \rule[-1.2ex]{0mm}{3ex} & $7\,200\,$km \\ \hline
e \rule[0.ex]{0mm}{3ex} \rule[-1.2ex]{0mm}{3ex} & $0.07$ \\       \hline
I \rule[0.ex]{0mm}{3ex} \rule[-1.2ex]{0mm}{3ex} & $80^{\circ}$ \\     \hline
$\Omega$ \rule[0.ex]{0mm}{3ex} \rule[-1.2ex]{0mm}{3ex} & $100^{\circ}$ \\
  \hline
$\omega$ \rule[0.ex]{0mm}{3ex} \rule[-1.2ex]{0mm}{3ex} & $200^{\circ}$ \\   \hline
\end{tabular*}
\end{center}
\end{table}

The MACEK mission is a project developed by the Astronomical Institute
of the Czechoslovak Academy of Sciences (see e.g. Pe\v rest\'y and
Sehnal, 1992).
An accelerometer is the satellite's central board device.
It can exactly measure the nongravitational accelerations along
the satellite orbit;
$5 \times 10^{-10}{\rm m.s}^{-2}$ is the supposed precision. As
follows from the goal of the mission, our results might be 
directly applicable. However, the interpretation of the data
would need more advanced models of the cloud distribution and
physical properties. In this section, having lower ambitions, we
want to demonstrate the real significance of the discussed
aspects (keeping in mind the supposed precision of the
device).

As the orbital elements are still uncertain, we accept the expected
ones which are listed in Tab.~1. We choose the angular elements
$\Omega,\omega$ so that the apogee is situated near the center of the
illuminated Earth hemisphere. This configuration is advantageous
for measuring the details of the albedo effect since the apogee
height is about 1300 km. In fact the orbit was designed according
to the experience with the French accelerometric mission CACTUS
(e.g.~Barlier et~al., 1977). As the significance of the 
atmospheric drag decreases approximately exponentially with
increasing distance from the Earth, while the radiative pressure
arising from the albedo effect only {\it polynomially}, 
the role of atmospheric drag is less important near the apogee,
while the contrary is true near the perigee (about 300 km). This choice of
the orbit gives then the possibility to study both the aspects
of the atmospheric drag (with the goal of improving the current
total atmospheric models for the satellite purposes) and the
aspects of the radiative forces (generally).
\begin{figure}
\vspace{76mm}
\caption[]{Radial acceleration component due to the albedo effect
as a function of the mean anomaly (units are chosen to be
$10^{-9}{\rm m.s}^{-2}$). The $atm$-curve is the result for
total cloud coverage of the Earth, the remaining two for
the free-cloud situation. The curve denoted by 1 stands for
the first Earth surface reflection model (Eqs.~(3)) while the
dashed curve 2 for the second model (Eqs.~(4)). The `apogee vicinity'
interval (where the instantaneous height above the Earth's surface is greater
than 800 km) is marked by the vertical lines.}
\end{figure}

The geometry of this satellite is much simpler than in the case of
ERS-1. It is nearly spherical with diameter $D_{\rm MAC} = 0.51\,{\rm  m}$
and mass of about 20 kg. We assume diffusive reflection on the
satellite surface (unfortunately no prelaunch tests are available).
We employ the same numerical experiments as in the ERS-1 case.

In Fig.~7, the results of experiments (i) and (ii) are drawn.
Similarly to the ERS-1 outputs we observe small difference between
the two Earth surface reflection models. Let us also briefly
comment on the easily understood fact that the acceleration magnitude
maximum in the case of the total cloud coverage model with unit
albedo ($\omega_0 = 1$, $x = 1$) is approximately three times greater
than the maximum of the no cloud model (see Fig.~7 and previous
figures as well). The approximate average value of the Earth's albedo
is about 0.3, hence the accepted cloud effective albedo is about three
times greater (the deviations from the Lambert law are not globally
important for this kind of argument). It is natural to expect the same
factor in the resulting perturbations. The vertical lines
in Fig.~7 mark the interval where the height of the satellite above
the Earth surface is greater than $800\,km$. 
\begin{figure}
\vspace{76mm}
\caption[]{Comparison of the radial albedo effect accelerations
for cloud reflection models with different asymmetries of the
scattering centers phase function. Dashed curves for the symmetric
scattering $x = 0$, solid curves for the asymmetric one $x = 1$.
The 1,2 and 3 sequence of curves displays the results of the
`equatorial cloud belt' model with cloud effective albedos
1,0.8 and 0.6. The Earth total cloud coverage signal is drawn
by the $a$-curves : $a1$ for $x = 0$, $a2$ for $x = 1$.}
\end{figure}

We compare the results of the equatorial cloud belt experiment for
different asymmetry parameters of the scattering center phase
function in Fig.~8. The dashed lines relate to the symmetric
scattering ($x=0$), the solid ones to the extremely asymmetric ($x=1$).

In each of these cases, the studied cloud coverage aspects of the
albedo effect are significant (comparing to the expected
precision of the mission device). Even though we performed here
only very simplified (and partially extreme) tests, we
think that the same conclusion remains for the real
situation.

\section{Conclusion}
We have studied some aspects associated with the role
of the atmosphere in the context of the albedo effect. Some
attention has been paid to different views on the radiation
reflection from the Earth's surface. Most of the care was
put into adopting the analytical models of the planetary atmosphere
radiative transfer to satellite motion theory purposes.
In this detail, our study differs from the previous ones
devoted to the cloud aspects in context of the albedo effect
(e.g. Rubincam et~al., 1987; Lucchesi and Farinella, 1992).

Having examined the diverse detailed aspects of the albedo effect
we continued the investigation in the same spirit as explained in paper~I,
classifying the aspects influencing the albedo effect.
This information could be essential when interpreting the
measurements of the accelerometer MACEK mission.

Though we considered only a few situations, we believe them to be representative.
The corrections due to the cloud coverage itself could be
of the same order as the basic albedo effect. The result is
strongly influenced by the effective cloud albedo and actual
cloud distribution (large cloud fields could manifest themselves strongly
in the results). Changes of the other cloud reflection model
parameters (e.g. scattering asymmetry factor) seem to influence
the results at a lower level which could, however, still be important
in particular situations. More care has to be given to the Earth
surface reflection mechanisms as well.

In case of precise altimetric missions (generally; the ERS-1 is
an example) demonstrated cloud aspects in context of the albedo effect
could be an important source of short and long term trajectory errors.
Keeping in mind the interpretation of the MACEK accelerometer data,
the information about the cloud distribution and the physical
properties (at least plane albedo) seems to be necessary.

Application of our framework in the case of the LAGEOS satellite
is under preparation.

\acknowledgements
One of us (D.V.) expresses his thanks to the IBM corp. for kind
support based on the IBM Academic Initiative in Czech Republic.
We are grateful to Dr. P.\/Farinella for valuable comments and
Dr. Ph.\/Knocke for considerable improving of our English.

\begin{thebibliography}{}
\article{Afonso, G., Barlier, F., Carpino, M., Farinella, P., Mignard, F.,
 Milani, A., Nobili, A.M.}{1989}{Ann.~Geoph.}{7}{501}
\article{Barlier, F., Boudon, Y., Falin, J.\/L., Futaully, R., Villain, J.\/P.,
 Walch, J.\/J., Mainguy, A.\/M., Bordet, J.\/P.}{1977}{Space~Research}
 {XVII}{341}
\article{Barlier, F., Carpino, M., Farinella, P., Mignard, F., Milani, A.,
 Nobili, A.M.}{1986}{Ann.~Geophys.}{4}{193}
\article{Borderies, N.}{1990}{Celest.~Mech.}{49}{99}
\article{Borderies, N. and Longaretti, P.-Y.}{1990}{Celest.~Mech.}{49}{69}
\article{Farinella, P., Nobili, A.M., Barlier, F., Mignard, F.}{1990}{A\&A}{234}{546}
\book{van de Hulst, H.C.}{1980}{Multiple Light Scattering : Tables, Formulas,
 and Applications}{Academic Press}{New York}
\book{Chandrasekhar, S.}{1950}{Radiative Transfer}{Oxford Univ. Press}{Oxford}
\article{Irvine, W.\/M.}{1975}{Icarus}{25}{175}
\article{Kabel\'a\v c, J.}{1988}{Bull.~Astron.~Inst.~Czechosl.}{39}{213}
\article{Klinkrad, H., Koeck, Ch., Renard, P.}{1990}{ESA Journal}{14}{409}
\bibitem{} Knocke, P.: 1989, {\it Earth Radiation Pressure Effects on
 Satellites}, Dissertation, The University of Texas at Austin
\book{Link, F.}{1969}{Eclipse Phenomena in Astronomy}{Springer-Verlag}{Berlin}
\article{Lucchesi, D. and Farinella, P.}{1992}{JGR}{97}{7121}
\bibitem{}Pe\v rest\'y, R. and Sehnal, L.: 1992, {\it In-Orbit 
 Microaccelerometric Experiment}, 43rd Congress of the IAF (IAF-92-0782),
 Washington, DC
\article{Rubincam, D.P.}{1982}{Celest.~Mech.}{26}{361}
\article{Rubincam, D.P.}{1987}{JGR}{92}{1287}
\article{Rubincam, D.P. and Weiss N.R.}{1986}{Celest.~Mech.}{38}{233}
\article{Rubincam, D.P., Knocke, P., Taylor, V.R., Blackwell, S.}{1987}
 {JGR}{92}{11 662}
\article{Sehnal, L.}{1979}{Bull.~Astron.~Inst.~Czechosl.}{30}{199}
\article{Stowe, L.L., Yeh, H.Y.M., Eck, T.F., Wellermeyer, Ch.G., Kyle, H.L.}
 {1989}{Jour.~Climate}{2}{671}
\article{Vokrouhlick\'y, D.}{1989}{Celest.~Mech.}{46}{85}
\bibitem{} Vokrouhlick\'y, D. and Sehnal, L.: 1992, {\it Celest.~Mech.},
 to appear (paper~I) 
\bibitem{} Vokrouhlick\'y, D., Farinella, P. and Mignard, F.: 1993, in
 preparation
\bibitem{} Wakker, K.F., Ambrosius, B.A.C., Zandbergen, R.C.A., van Geldorp, G.H.M.:
 1987, {\it Report ESA Contract} 6140/84/D/IM,TU Delft, The Netherlands
\bibitem{}
\end{thebibliography}
\end{document}

