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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Earth penumbra effects on AES motion taking into
account the refraction and the extinction of the light
in the atmosphere
A. M. Fominov
Institute of Applied Astronomy, St. Petersburg, Russia
The perturbing acceleration F of the Artificial Earth Satellite (AES) due to
the solar radiation pressure can be described as follows:
F = f \Delta D \Delta N;
where D --- the expression for the absolute quantity of the acceleration, N ---
the unit vector along the resultant of the radiation pressure forces, f --- the
shadow function, with f = 0 when the satellite is in Earth shadow (umbra);
f = 1 when the satellite is in sunlight; 0 ! f ! 1 when the satellite is in partial
shadow (penumbra). The penumbra is postulated as an enlarged near--earth space
area in which the solar radiation acting on the satellite surface undergoes the
refraction and the extinction in the earth's atmosphere. The shadow function
can be described as follows:
f = J 1
J 0
; J 0
=
Z Z
s
I 0 ds; J 1
=
Z Z
s 1
I s ds;
where I 0
--- the nominal radiation intensity of the solar disk element ds referred
to the unit celestial sphere with the centre in the satellite position, s --- the
nominal area of the solar disk, I s
--- the observed radiation intensity of the solar
disk element ds taking into account the extinction of the light in the earth's
atmosphere, s 1
--- the observed area of the solar disk (or the part of the solar disk
eclipsed by the Earth) taking into account its deformations due to the refraction
of the light in the earth's atmosphere.
The complete theory of the direct solar radiation pressure perturbations acting
on AES orbits was presented by some authors [1]--[3]. We have developed the
more simple approach to investigating the problem of the refraction and the
extinction of the light in the Earth's atmosphere [4]. The relationship for the
calculation of the refraction angle is obtained by assumption that the air density
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depends exponentially on the height. The relationships for the evaluation of the
light extinction are derived taking into account its dependence on the light wave
length. The variations of the sun energy flow with the wave length were considered
as well [5]. The direct use of these integral formulas needs a lot of computer time.
The evident computer time reduction may be reached when the satellite is moving
on a near--circular orbit. In this case the computation of the shadow function is
most efficiently done by a two--step procedure. In step 1, the table of the shadow
function values with the angle distances d between the solar disk and the earth
disk edge may be calculated. This numerical dependence f = f(d) is considered
as a starting one for each specific near--circular orbit. Step 2 of the computation
is the approximation of function f = f(d) by some elementary function. The
latter dependence is used for calculating f­values. We have used this algorithm
to study the penumbra effects in the motion of 12--hour satellites ``Navstar''. The
main results obtained in this paper can be summarized as follows: (1) the effect
of displacement of unit vector N from the solar--satellite direction is too small
[6] and can be neglected; (2) the refraction and the extinction of the light in the
atmosphere must be taken into account for estimating the shadow function when
the trajectory measurements have the accuracy of several centimetres.
References
1. Vokrouhlick'y D., Farinella P., Mignard F. Solar radiation pressure pertur­
bations for Earth satellites. I. A complete theory including penumbra tran­
sition. Astron. and Astrophys., 1993, 280, 295--312.
2. Vokrouhlick'y D., Farinella P., Mignard F. Solar radiation pressure pertur­
bations for Earth satellites. II. An approximate method to model penumbra
transitions and their long­term orbital effect on LAGEOS. Astron. and As­
trophys., 1994, 285, 333--343.
3. Vokrouhlick'y D., Farinella P., Mignard F. Solar radiation pressure pertur­
bations for Earth satellites. IV. Effect of the Earth's polar flattening on the
shadow structure and the penumbra transitions. Astron. and Astrophys.,
1996, 307, 635--644.
4. Murray C. A. Vectorial Astrometry. Bristol: A. Hilger, 1983.
5. White O. The Sun Energy Flow and Its Variations. M.: Mir, 1980 (in Rus­
sian).
6. Hlali Ya. E., Batrakov Yu. V., Fominov A. M. Moon penumbra effects on
Earth satellites motion when solar brightness weakens to disk edge, IAA
Trans., 1999, 4, 300--309 (in Russian).
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