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\noindent
\leftline {\large \bf MOMENT FORMALISM FOR THE RADIATIVE FORCE EVALUATION}
\vspace*{1.5cm}
\parindent 1in

{David Vokrouhlick\'y}
\vspace*{5mm}

{Observatoire de la C\^ote d'Azur, Dept. CERGA \par
  Av N Copernic, F-06130 Grasse, France
  \footnote{\rm On leave from Astronomical Institute, Charles University
   Prague, \v Sv\'edsk\'a 8, CZ-15000 Prague 5, Czech Republic.}}
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\noindent
{\bf ABSTRACT.} \hspace{3mm}We overview a theoretical scheme 
enabling evaluation of the
radiative force acting on different celestial bodies
(artificial satellites, interplanetary dust grains) based on
the moment formalism introduced in the theory of radiative
transfer. This approach allows an elegant and powerful
algorithm for the radiative force investigation in
complex situations (e.g. the Earth shadow penumbra transitions).
It accounts for the influence of general radiative field
generated by an extended source with arbitrary
surface emissivity. A number of tutorial computations
of the simplified configurations already met in the
literature are presented.

Recently, we have applied the moment formalism to the shadow
penumbra theory in context of the artificial satellite dynamics.
Here we follow several ideas generalizing previous model in different
routes: towards the (i) first order terms in presumably small
parameter $v/c$
($v$ stands for the object velocity, while $c$ for the
velocity of light) generally known as the Poynting-Robertson
terms, (ii) arbitrary distances from the
planet. 
\bigskip

\leftline{\bf 1. INTRODUCTION AND MOTIVATION}
\medskip

Celestial mechanics is generally thought to deal primarily with
the dynamics of an isolated system of gravitationally interacting
bodies. This scenario was predetermined by Newton's discovery
of dynamics in one line with gravitational law. Even now, celestial
mechanics considered from this viewpoint remains a crucial
problem of any theory of gravity (Damour \etal 1991).
Nevertheless, the idea of the non-gravitational forces influence
appeared progressively in context of the celestial bodies motion.
Cometary motion and secular acceleration of the Moon
are archetypes of those phenomena. More recently, dynamics of
interplanetary dust particles, planetary rings as well as the
motion of artificial satellites involve non-gravitational forces
as an important part.

Among all of them, radiative forces -- those consisting in
momentum exchange of a radiative field and considered body --
play a special role. They are often the most prominent
(e.g. direct radiative pressure in context of motion of solar
system dust particles, middle and high orbit artificial satellites)
or they give rise tiny secular phenomena (e.g. the
Poynting--Robertson effect for the dust particles, thermal thrust
in case of artificial satellites).

In this paper, we would like to avoid making an overview of
different situations where radiative force enters as an
important ingredient. Instead of that, we want to concentrate
on exposing an approach well suited for evaluation of the radiative
force even in case of complex configurations. We think that such
tool has been missing in celestial mechanics up to now. Most
of the previous contributions involving radiative forces were
based on simple treatment of the radiative fields while
concentrating on the dynamical aspects  (with only a few exceptions, 
e.g. Guess, 1962). As a result a detailed
theory of the planetary shadow penumbra effects (for example)
has not been elaborated up to recently.

In Sec.~2 we overview our approach to the radiative force evaluation.
In the last section we briefly overview the main results connected 
with the penumbra theory for the artificial satellites achieved by our
complex approach to the radiative force evaluation (Vokrouhlick\'y 
\etal 1993b, 1994a) and mention our plans to expand them for the
dynamics of the solar system dust grains (Vokrouhlick\'y \etal 1994b).
\bigskip

\leftline{\bf 2. RADIATIVE FORCE -- METHOD OF EVALUATION}
\medskip

In what follows we shall generally speak about a particle, having
in mind arbitrary body influenced by the radiative force (i.e.
dust grain, artificial satellite). However, our approach being
macroscopic excludes systems which would need quantum mechanical
treatment (e.g. molecules, atoms).

As the considered problem involves photons it is most appropriate
to use relativistic approach. On the other hand it is well known that
up to the first order in a small parameter $v/c$, where $v$ is
the particle velocity and $c$ light velocity, the problem is
essentially `classic' (Burns \etal 1979). However from
classical or semiclassical arguments lot of misinterpretation
has followed (see e.g. Lyttleton, 1976 and references therein). 
As a result we choose to solve the problem in rather
formal and strictly relativistic manner. Interestingly, it is
very similar to that used by Robertson (1937) apart of
generalizations mentioned later on.

The total radiative force essentially results from global momentum
budget of the ingoing and outgoing radiation in the particle's
local frame. One usually distinguish absorption, characterized
by the fact that all radiation removed from the incident beam
-- absorbed -- is reemited isotropically in the local frame, and
scattering, characterized by instantaneous anisotropic emission
of the removed radiation. We shall follow this scheme mainly
from pedagogical standpoint. Moreover, one can meet more
complicated situations when true absorption is followed with
anisotropic reemission in the local frame (e.g. Lyttleton, 1976).
The other example are Yarkovsky type effects (Burns \etal 1979;
Rubincam, 1988).

Our formalism, allowing to treat the situations with
arbitrary incident radiative field, is based on moment
decomposition of the specific intensity (e.g. Mihalas, 1970;
Thorne, 1981). This tool is widely used in the theory of
radiative transfer. Here, we shall expose only such material
which is necessary for understanding our approach.

In the following series of subsections we shall gradually develop a
general scheme for the force evaluation.
\smallskip

\noindent {\bf I. Reference frames introduced.} \hspace*{3mm}
We start with definition of the reference frames to be used later
on. First, we introduce global coordinate system $x^\alpha$
($\alpha = 1,2,3$) which serves for global description of the
particle motion (e.g. heliocentric, geocentric). We construct
a reference frames induced by the coordinate system $x^\alpha$
in a usual manner -- each of the frame base vectors is tangent to
one of the coordinates $x^\alpha$ -- at each point of the space.
We shall denote such frames generally by ${\cal S}$.

It should be noted that this introduction of the basic reference
frames is the most elementary. One often meets more general
situation, when the frames ${\cal S}$ are transported in the
space by preferred class of `observers' (e.g. expanding or
collapsing; see
Thorne, 1981), but for the purposes of applications in celestial
mechanics we shall restrict to `static' frames ${\cal S}$
introduced earlier.

Together with frames ${\cal S}$ we define a system of frames ${\cal S'}$
each of them being instantaneously comoving with the particle.
It should be understood that we do not have in mind absolutely
comoving frame with the particle, as this one will not be inertial,
while ${\cal S'}$ is supposed to be inertial (at least locally when
general relativistic point of view is applied). One concludes
from the preceding that the transformation between ${\cal S}$ and
${\cal S'}$ frames is simply given by a Lorentz boost with
instantaneous coordinate velocity ${\bf \vec v}$ of the particle.
In what follows we shall systematically denote physical quantities
measured in the comoving frames ${\cal S'}$ with prime.
\smallskip

\noindent {\bf II. Radiative field characterized.} \hspace*{3mm}
Essential quantity characterizing properties of the radiative
field is specific intensity (or just intensity) $I_\nu({\bf \vec n})$.
To be exact it represents a scalar field in the phase space
of photon states if polarization is disregarded (Mihalas, 1970;
Misner \etal 1973). Translated into more common language
$I_\nu({\bf \vec n})\times ({\bf \vec n} . {\bf \vec n}_{\rm s})$ gives
radiative energy passing (i) through unit surface area with normal
${\bf \vec n}_{\rm s}$ (ii) per unit of time (iii) in unit frequency
interval centered around $\nu$ (iv) into unit solid angle with the
axe given by ${\bf \vec n}$. Specific intensity depends on
(i) position $x^\alpha$ of the local frame ${\cal S}$ origin, (ii)
local direction $n^\alpha$ and (iii) frequency
$\nu$, so that extend notation will read $I(x^\alpha,n^\alpha,\nu)$.
\footnote{Note, that arguments $(n^\alpha,\nu)$ can be replaced with
photon momentum $p^\alpha$ as frequency $\nu$ is directly related to the
total momentum of considered photon.}
In agreement with widely used notation we shall suppress position
vector $x^\alpha$ and write the $\nu$ parameter as an index.

Before we start effectively the computation of the radiation force
on a particle, we have first to go through two preliminary steps,
not addressed in full detail here. They can be expressed by
the following problems:
\begin{itemize}

\item{} {\it How the radiative field is constructed provided one
knows emissivity of the radiative sources?} In order to solve
it one introduces the concept of {\it light rays}, which form important
guidelines along which intensity is transported. Light rays
correspond to the photon trajectories in considered medium
as far as methods of geometrical optics are physically accessible.
In celestial mechanics one often disregards refractive properties
of the background medium and considers light rays as straight
lines. Sometimes however, one faces to the problem of solving
nontrivial light rays (e.g. the case of the planetary shadow penumbra
effect, Vokrouhlick\'y \etal 1993b, 1994a). One usually employs Fermat
principle (e.g. Landau and Lifschitz, 1962) or sometimes solves directly
the dynamical equations of the photon motion (e.g.
Abramowicz \etal 1991; Vokrouhlick\'y and Karas, 1991).
It is worth noting that solving for the light rays is
essentially `initial problem' and {\it not} `boundary problem'.
A direction ${\bf n}$ in the local system ${\cal S}$
uniquely determines a light ray. On contrary, there may exist
several light rays connecting the same element of the radiative
source and origin of the chosen local frame ${\cal S}$. Such 
effects result in interesting focusing effects due to
the planetary atmospheres leading to the radiative
flux increase (to be discussed in detail in Vokrouhlick\'y \etal 1994b).

\item{} {\it How is the specific
intensity transported along the light rays?} This is the central
problem of the radiative transfer theory (e.g. Chandrasekhar,
1950; Mihalas, 1970). We shall not present here a complete
formulation of corresponding equations. It is sufficient to
know that there exist an unambiguous `shifting law' which
determines the value of the specific intensity transported along
given light ray provided its boundary value on the radiative
source is given (in most solar system cases this means in
the solar photosphere or on the planetary surface). In celestial
mechanics, one usually disregards extinction by the
medium in between the radiative source and considered particle
resulting in conservation of the specific intensity along
straight light rays. The other often met example is simple
exponential damping of the specific intensity, which
correspond to the formal solution of the radiative transfer
theory in case of absorbing media with constant absorption
coefficient (see Mihalas, 1970; Vokrouhlick\'y \etal 1993b).
\end{itemize}

Fig.~1 demonstrates the concepts introduced up to now.
\smallskip

\noindent {\bf III. Multipole moment formalism.} \hspace*{3mm}
Due to general theorems in analysis one can develop the
angular dependence of the specific intensity into multipole
series. Corresponding coefficients -- multipole moments --
then contain full information on the radiative field.
Moreover, low order moments have straightforward physical meaning
(e.g. Mihalas, 1970). This leads us
to use extensively moment formalism as basic mathematical tool.

Moments of the radiative intensity are defined according to the
following scheme (see Thorne, 1981)

$$ M_{\alpha_1 \alpha_2 \ldots \alpha_k} = \int_{(\theta,\phi)}
 d(\cos\theta) d\phi\; n_{\alpha_1} n_{\alpha_2}
  \ldots n_{\alpha_k} \,I(\theta,\phi)\; , \eqno(1)$$

\noindent where $\alpha_i$ are indexes spanning the values $(1,2,3)$
referring to the axes of the reference frame\footnote{We
shall generally use greek letters for spatial indexes, but
keeping letter $\nu$ for frequency and $\gamma$ for the obvious
relativistic factor (see bellow).}, and
$n_{\alpha_1}(\theta,\phi)$ are the components of the unit vector in
the direction $(\theta,\phi)$ [i.e., we have ${\bf \vec
n}^T(\theta,\phi) =$ ($\sin\theta\cos\phi$,
$\sin\theta\sin\phi$,$\cos\theta$)]. It is supposed 
that spatial events in the frame ${\cal S}$ are referred to
spherical coordinates $(\theta,\phi)$ introduced
in usual manner. Note, that we define moments of the
frequency integrated intensity $I = \int d\nu\, I_\nu$ and not
of the quantity $I_\nu$ itself (compare with Thorne, 1981). Moreover,
it can be shown (Thorne, 1981), that full information about the
radiative field is contained already in symmetric and trace-free
part of tensors (1). We shall extensively use only
moments up to order two ($k=2$). At this level the general
result mentioned is equivalent to trivial observation that
$M_{\alpha\beta}=M_{\alpha \beta}^{\rm STF}+{1 \over 3} M \delta_
{\alpha \beta}$, where $M_{\alpha \beta}^{\rm STF}$ symmetric
trace-free part of tensor $M_{\alpha \beta}$. Practically it means,
that instead the set $(M,M_\alpha,M_{\alpha \beta})$ necessary for
evaluation of the radiative force [see Eqs.~(12), (21) below], one
needs only the reduced set $(M,M_\alpha,M_{\alpha \beta}^{\rm STF})$
having one component less.
\smallskip

\noindent {\bf IV. Radiative force evaluation.} \hspace*{3mm}
The essential scheme to evaluate the radiative force acting on
the dust particle consists in deriving this force in the
instantaneously comoving frame ${\cal S'}$, which is the
frame where unambiguous results are available and more meaningful
(e.g.
Mie scattering theory in case of the microscopic dust grains,
specular reflection or Lambert diffusion in case of artificial
satellites). Then, we have to transform the radiative force to
the static local frames ${\cal S}$, which serve as `a skeleton'
for the description of global motion of considered particle.

We found tutorial to treat separately cases of ideally absorbing
particle on one hand and purely scattering particle on the other
hand. However, it should be noted that the scheme is the same
and we shall finally put both results together to express
the radiative force acting on partially absorbing and partially
scattering particle.
\smallskip

\noindent {\bf V. Ideally absorbing particle.} \hspace*{3mm}
We start by focusing on the description of the situation on the
local comoving frame ${\cal S'}$. Here, we shall assume that
the particle is spherical resulting in isotropy of its optical
properties and we shall denote the effective cross-section for
removal of the incident energy due to the true absorption by
$C_{\rm ab} = AQ_{\rm ab}$, where $Q_{\rm ab}$ is the
efficiency factor and $A$ geometrical cross-section of the particle.
Both quantities $C_{\rm ab}$ and $Q_{\rm ab}$ could generally
show frequency dependence. In case of microscopic systems such as
interplanetary dust grains the efficiency factors are known from
the Mie theory (e.g. van de Hulst, 1957; Wickramasinghe, 1973),
while in case of macroscopic bodies such as artificial satellites
one usually accepts phenomenological approach built on
presumably unknown absorption coefficients (e.g. Milani \etal 1987,
Klinkrad \etal 1990). We shall also assume that the accumulated
thermal energy is reradiated isotropically in this system, so that
final momentum budget is only due to the absorbed incident
radiation. Resulting formula for the radiative force reads

$$ f'^{\rm (R)}_\alpha = {A \over c} \int d\omega' \int d\nu'
 \, Q_{\rm ab}(\nu')\, I_{\nu'}({\bf \vec n'})\, n'_\alpha \; ,
 \eqno(2) $$

\noindent where the two integrals are performed over: (i) all
spatial directions of the rays constituting the local radiative
field ($d\omega'$ stands for element of the solid angle),
(ii) all frequencies.

One should understand clearly at this point that although the
theory provides the value of $Q_{\rm ab}(\nu')$ we do not know
a priori the radiative field $I_{\nu'}({\bf \vec n'})$, as
this is `distorted' by aberration and the Doppler effect
when comparing to the radiative field $I_{\nu}({\bf \vec n})$
in the static frame ${\cal S}$ constructed by the procedure
of the previous sections. In order to transform
parameters of integration in (2) for `the unprimed' one
employs relations (see Thomas, 1930; Mihalas, 1970)

$$ I_{\nu'}({\bf \vec n'})\, d\omega'\, d\nu' = \gamma^2 \left(1-
 {n_\beta v_\beta \over c}\right)^2 I_{\nu}({\bf \vec n})\, d\omega\,
  d\nu \; , \eqno(3) $$
$$ n'_\alpha = \gamma^{-1} \left(1-{n_\beta v_\beta \over c}\right)^{-1}
 \left[n_\alpha + \left(\gamma-1\right){n_\beta v_\beta \over v^2} v_\alpha
 -\gamma {v_\alpha \over c}\right] \; , \eqno(4) $$

\noindent where $\gamma = \left[1-\left(v/c\right)^2\right]^{-1/2}$
(summation of quantities with common index is assumed implicitly).
Truncating previous formulas at the first order of the
(presumably) small parameter $(v/c)$ and employing Doppler
shift law $\nu' \rightarrow \nu \left(1 - {n_\beta
v_\beta \over c}\right) + {\cal O}(2)$ [here ${\cal O}(n) = 
{\cal O}(c^{-n})$] we arrive at the following
formula for the force (2) on a spherical particle

$$ f'^{\rm (R)}_\alpha = A {Q_{\rm ab} \over c} \left[ M_\alpha -
 M {v_\alpha \over c} - M_{\alpha \beta} {v_\beta \over c}\right] -
 A {Q^{\rm (1)}_{\rm ab} \over c} M_{\alpha \beta} {v_\beta \over c}
 + {\cal O}(2) \; ,  \eqno(5) $$

\noindent where we introduced `frequency averaged' factors

$$ Q_{\rm ab} \equiv {\int d\nu\; Q_{\rm ab}(\nu) I_\nu \over
 \int d\nu\; I_\nu} \; , \eqno(6) $$

$$ Q^{\rm (1)}_{\rm ab} \equiv {\int d\nu\; \nu\left(\partial Q_{\rm ab}(\nu)
 /\partial \nu\right) I_\nu \over \int d\nu\; I_\nu} \; .\eqno(7) $$

\noindent It should be understood that in successive steps from formula
(2) to (5) we just changed parameters of integration in (2);
a priori `unknown' quantities $I_{\nu'}({\bf n}')$ and $n'_\alpha$
has been changed to construct radiative field linked to the
quantities $I_{\nu}({\bf n})$ and $n_\alpha$.

In the last step we have to transform the force components (5)
from the comoving frame ${\cal S'}$ to the static one ${\cal S}$.
We learn from the classical relativistic dynamics
(e.g. Landau and Lifschitz, 1962) that the following simple rule holds

$$ f^{\rm (R)}_\alpha = f'^{\rm (R)}_\alpha + {\cal O}(2) \; .\eqno(8) $$

\noindent We thus conclude that
Eq.~(5) gives already the final expression of the radiative force
acting on spherical particle up to specified order.

It may be interesting to discuss briefly features of the
right hand sides of formula (5). At first, we note that truncation to the
zeroth order in quoted parameter $(v/c)$ leads to

$$ f^{\rm (R)}_\alpha = A {Q_{\rm ab} \over c} M_\alpha
 + {\cal O}(1)\; . \eqno(9) $$

\noindent Formula (9) represents the radiative pressure exerted on 
a spherical particle as is well known from the literature
(see e.g. Vokrouhlick\'y \etal 1993a,b). We have used it extensively
for the description of the shadow penumbra effect in
context of the artificial satellite motion.

The last term in the right hand side of Eq.~(5) is usually omitted
in the
theory as the frequency dependence of the absorption
efficiency factor $Q_{\rm ab}(\nu)$ is either not great
or shows oscillations around the mean value, which
effectively average out its contribution. Second approximation
to be reminded here is incident the plane wave with
direction ${\bf \vec n}^{\rm (0)}$. Under such conditions
we may write $I_\nu({\bf \vec n}) = \Phi_\nu
\delta({\bf \vec n}-{\bf \vec n}^{\rm (0)})$ for the incident
radiative field with $\delta$ standing for Dirac's delta-function.
Realizing that

$$ M_{\alpha_1 \alpha_2 \ldots \alpha_k} = \Phi\; n^{\rm (0)}_{\alpha_1}
 n^{\rm (0)}_{\alpha_2} \ldots n^{\rm (0)}_{\alpha_k} \; , \eqno(10) $$

\noindent for all $k = 0,1, \ldots$ and $\Phi = \int d\nu\, \Phi_\nu$,
we get the final form of radiative force [the second term on
right hand side of (5) omitted]

$$ f^{\rm (R)}_\alpha = A {Q_{\rm ab} \over c} \Phi \left[n^{\rm (0)}_
 \alpha \left( 1- {n^{\rm (0)}_\beta v_\beta \over c}\right) -
 {v_\alpha \over c} \right] + {\cal O}(2) \; , \eqno(11) $$

\noindent well known for the expression of the Poynting--Robertson
effect (e.g. Robertson, 1937; Burns \etal 1979).
\smallskip

\noindent{\bf VI. Scattering particle.} \hspace*{3mm}
In this section we turn to the case of purely scattering particles.
The treatment is essentially similar to that in previous section,
but here the net momentum budget of the outgoing scattered radiation
is generally nonzero and should be accounted for.
The effective cross-section for removal of the incident
radiation by scattering will be denoted by $C_{\rm sc} = AQ_{\rm sc}$,
where $Q_{\rm sc}$ is the corresponding efficiency factor and
$A$ is geometrical cross-section of the particle. Next, we
denote the scattering phase function by $S_\nu(\Theta)$
which gives the probability distribution of the scattering in the direction
deflected by angle $\Theta$ from the incident plane wave mode
(e.g. van de Hulst, 1980).

As for the momentum budget of the removed radiation from the
incident beam we can apply the same procedure as in
previous section devoted to the absorbing particle
provided factors $Q_{\rm ab}$ and $Q^{\rm (1)}_
{\rm ab}$ are replaced with integral scattering efficiency factors

$$ Q_{\rm sc} \equiv {\int d\nu\; Q_{\rm sc}(\nu) I_\nu \over
 \int d\nu\; I_\nu} \; ,\eqno(12) $$

$$ Q^{\rm (1)}_{\rm sc} \equiv {\int d\nu\; \nu\left(\partial Q_{\rm sc}(\nu)
 /\partial \nu\right) I_\nu \over \int d\nu\; I_\nu} \; .\eqno(13) $$

Let us now turn to consider the scattered radiative field. As noted
previously, each of the "infinitesimal plane modes" $I_{\nu'}({\bf \vec
n'})$ with given ${\bf \vec n'}$, which  constitute the incident
radiative field, give rise to new radiative field $I_{\nu'}^{\rm sc}
({\bf \vec n''}) \equiv S_{\nu'}(\Theta) I_{\nu'}({\bf \vec n'})$, with
(${\bf \vec n''}.{\bf \vec n'}=\cos\Theta$), which is clearly
axially symmetric around the incident direction ${\bf \vec n'}$.
Notice, that the scattering redistribution in effectively only
spatial ($\Theta$) and not in frequencies ($\nu'$ is the same on both
sides of previous equation), which is well satisfied for
scattering on dust grains, but might become erroneous on
molecular level (see discussion in Mihalas, 1970). Next step consists
in integrating the total
scattering field over all incident infinitesimal plane modes
(one implicitly uses the assumption of spherical shape of the
particle) and substituting it into expression similar to Eq.~(2). Then
changing integration symbols one arrives at the following expression
for the radiative force due to the scattered radiative field

$$ f'^{\rm (R)}_\alpha = {A \over c} \int d\omega' \int d\nu'
 \, Q_{\rm sc}(\nu')\, g(\nu')\, I_{\nu'}({\bf \vec n'})\, n'_\alpha \; .
 \eqno(14) $$

\noindent where we define the phase parameter $g(\nu)$ by

$$ g(\nu) = \int_{-1}^1 d\left(\cos\Theta\right) \cos\Theta \; S_\nu\left(
 \Theta \right) \; . \eqno(15) $$

\noindent Note the sign minus (coming from derivative of cosine in
(15)) which expresses the fact that scattered
photons take away momentum from the system.

In the last step the algebra is the same as in the case
of the absorbing particle to transform the force components
(14) into the static frame ${\cal S}$. Final expression reads

$$ f^{\rm (R)}_\alpha = A {Q_{\rm sc} \over c} g
 \left[ M_\alpha - M {v_\alpha \over c} - M_{\alpha \beta} {v_\beta \over
  c}\right] - A {Q^{\rm (1)}_{\rm sc} \over c} g^{\rm (1)} M_{\alpha \beta}
   {v_\beta \over c} + {\cal O}(2) \; ,  \eqno(16) $$

\noindent where we introduce frequency integrated versions of the phase
parameter

$$ g \; (=\; <\cos\theta>) \equiv {\int d\nu\; Q_{\rm sc}(\nu) g(\nu) I_\nu
 \over \int d\nu\; Q_{\rm sc}(\nu) I_\nu} \; ,\eqno(17) $$

$$ g^{\rm (1)} \equiv {\int d\nu\; \nu\left[\left(\partial Q_{\rm sc}(\nu)
 /\partial \nu\right) g(\nu) + \left(\partial g(\nu)/\partial \nu\right)
 Q_{\rm sc}(\nu)\right] I_\nu \over \int d\nu\;
\left(\partial Q_{\rm sc}(\nu)/\partial \nu\right) I_\nu} \; .\eqno(18) $$

Putting together results (8) and (15) we finally obtain expression
for the total radiative force acting on the particle partially
absorbing (with efficiency $Q_{\rm ab}$) and partially scattering
(with efficiency $Q_{\rm sc}$ and phase function $S_{\nu}(\Theta)$)
the incident radiation accurate
to the first order in small parameter $(v/c)$ in form

$$ f^{\rm (R)}_\alpha = A {Q_{\rm ab}+Q_{\rm sc}\left(1+g\right) \over c}
 \left[ M_\alpha - M {v_\alpha \over c} - M_{\alpha \beta} {v_\beta \over
  c}\right] - A {Q^{\rm (1)}_{\rm ab}+Q^{\rm (1)}_{\rm sc}\left(1+g^{\rm
 (1)}\right) \over c} M_{\alpha \beta} {v_\beta \over c}
 + {\cal O}(2) \; .  \eqno(19) $$

Formula (19) is valid for arbitrary extended source of radiation
and partially absorbing and scattering particle.

It is worth noting that up to zeroth order in the parameter
$(v/c)$ we again get simple expression

$$ f^{\rm (R)}_\alpha = A {Q_{\rm ab}+Q_{\rm sc}\left(1+g\right) \over c}
  M_\alpha  + {\cal O}(1) \; ,  \eqno(20) $$

\noindent for the usual radiation pressure.

Let us demonstrate, its validity on the known case of polished sphere,
reflecting all the radiation specularly, met in the theory of the
artificial satellites. In that case $Q_{\rm ab}=0$ and $Q_{\rm sc}=1$.
Moreover, one can easily show that
$S_\nu(\Theta) \approx 1$ (see e.g. Landau and Lifschitz, 1960)
and thus $g=0$. As a result $f^{\rm (R)}_\alpha = {A \over c} M_\alpha$
which corresponds to earlier results (e.g. Vokrouhlick\'y \etal
1993a).

In Eq.~(19) the bracket in the first term and part $ M_{\alpha
\beta} {v_\beta \over c}$ in the second term depend on the local
radiative field (i.e. expressed in ${\cal S}$ frame) and particle's
velocity with respect to this frame. Hence they are independent
on the optical properties of the particle itself. On contrary,
terms involving efficiency parameters ``$Q$" depend on the
frequency distribution of the incident radiative field and the
optical features of the particle. It can be also said, that
they represent "inertiality parameters" characterizing resistance
of the particle to the dynamical changes under the influence of
the radiative field similarly to the inertial mass in the
theory of gravitation. We would like to stress, that they
might not be perfectly constant due to their dependence on the
frequency spectrum of the incident radiation. One generally
takes into account solar spectrum out of the atmosphere to
evaluate integral efficiency factors of absorption $Q_{\rm ab}$
and scattering $Q_{\rm sc}$ (e.g. Lamy, 1974; Burns \etal 1979).
This is perfectly justified in most part of the particle
trajectories around planets. But it may not be exactly met
in the period of planetary shadow penumbra penetration, when
the solar spectrum is significantly altered due to the passage
through the planetary atmosphere.
\medskip

\noindent {\bf VII. `Cooking book' for the radiative force evaluation.}
\hspace*{3mm}We thus suggest the following algorithm for the 
evaluation of the radiative force on celestial bodies sufficiently
general even for complex situations:
\begin{itemize}
\smallskip

\item{} Choice of the global coordinate system, which (i) serves for
 the description of the global motion of the particle under consideration,
 (ii) induced local frames ${\cal S}$ (e.g. heliocentric cartesian
 system);
\vspace{-1mm}
\item{} choice of the radiative sources together with description
 of their emissivity (e.g. solar photosphere models);
\vspace{-1mm}
\item{} construction of the radiative field in each of the local
 systems ${\cal S}$ using the concept of the light rays (Fermat
 principle) and theory of the specific intensity transport along
 given ray (radiative transfer equations);
\vspace{-1mm}
\item{} computation of the radiative intensity moments (1) up to
 order 2 in local systems ${\cal S}$;
\vspace{-1mm}
\item{} computation of the radiative force using one of the
 formulas (5), (9), (16), (19) or (20) (depending on the considered
 problem).
\end{itemize}
\bigskip

\leftline{\bf 3. SOME PREVIOUS RESULTS}
\medskip
In this section, we shall mention a few applications of the
previous theory in celestial mechanics. Let us remark that
according to our opinion moment formalism allows to
treat these complex problems with maximal possible
clarity. 

Recently, we have elaborated detailed theory of the Earth shadow penumbra
theory aimed for the artificial satellite dynamics based
on previous formalism (Vokrouhlick\'y \etal 1993b). Importance
of such theory is given by the fact that solar radiation
pressure is the dominant non-gravitational effect perturbing
satellite orbits at altitudes higher than about $400$ km.
Uncertainties of modeling of this effect crucially affect
the orbit predictions (e.g. Fliegel \etal 1992; Ries \etal 1992).

It is interesting to note that eclipsing of the solar disk by
the Earth body plays only minor role on forming the
penumbra decrease of the radiative perturbations. As a result
previous studies based mainly on this `geometrical' approach
could not reach high precision. The atmospheric effects
-- refraction and extinction -- dominantly influence the penumbra
effect. We have identified several tiny effects of interesting
physical origin which constitute decrease of the radiative flux
when passing through the shadow penumbra. Let us note for
example refractive attenuation of the solar incident flux
of pure geometrical origin, which was also discussed as early
as in the beginning of the century in context of the theory
of lunar eclipses (e.g. Link, 1933).

Probably the most interesting result obtained so far concerns
long periodic perturbations of the LAGEOS semimajor axis. We shall
not remind here extensive literature on this appealing puzzle
of the satellite orbital motion (for introduction to the problem
see e.g. Smith and Dunn, 1980; Barlier \etal 1986). We have
found that additional perturbations due to exact treatment
of the shadow penumbra phenomena could account under plausible
assumptions on the north/south asymmetry of the meteorological
conditions for about $10-20$ \% of the observed residuals
(see Vokrouhlick\'y \etal 1994a). This is rather unexpected
result regarding to the fact that the penumbra transitions
cover maximally $5$ \% of the orbital period (about $3.75$ hours).

We have also investigated the radiative perturbations for the
dust grains captured to the Earth vicinity (Vokrouhlick\'y 
\etal 1994b). Here arbitrary
distances from the Earth have to be considered. This leads
to much richer set phenomena, such as occurrence of higher order
solar images, `atmospheric flux lensing' etc. The possible
dynamical consequences of those effects remain to be established.

\noindent
{\bf Acknowledgements.} \hspace*{4mm}
The author thanks F. Mignard for careful reading of the manuscript and
valuable comments. The staff of CERGA (Grasse, France) is
acknowledged for its kind hospitality. This work has been supported
in part by EC grant 2836.

\vspace*{3cm}
\parindent 0pt

{\bf FIGURE CAPTIONS:}
\medskip

{\bf Figure 1}.\hspace{3mm} Geometrical objects introduced in the
text: global coordinate system, induced local (static) frame
${\cal S}$, local (instantaneously) comoving frame ${\cal S}'$,
light ray emitted by the radiative source characterized by
local directional vector ${\bf n}$.

\newpage
\parindent=-10pt

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\medskip

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\end{document}
