The effects of the Moon's shadow and pre-shadow on the artificial satellites motion

Khalil I. Khalil, Afaf M. Abd El-Hameed

National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, Cairo, Egypt

Correspondence: Afaf M. Abd El-Hameed, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, Cairo, Egypt. Tel. +20.2.25560645 Fax: +20.2.25548020. E-mail: afaf_m2000@yahoo.com

Key words: solar radiation pressure, Moon's shadow, pre-shadow, artificial satellites.

Received for publication: 9 February 2011.
Accepted for publication: 7 April 2011.

This work is licensed under a Creative Commons Attribution 3.0 License (by-nc 3.0).

©Copyright K.I. Khalil and A.M. Abd El-Hameed, 2011
Licensee PAGEPress, Italy
Astronomy Studies Development 2011; 1:e3
doi:10.4081/asd.2011.e3







Introduction

The artificial satellites, in their motion around the Earth, can sometimes enter the Moon's shadow.1,2 The probability of the satellite's entry the Moon's shadow can not be considered small,3,4 as it is equal to 5% for the near-equatorial orbits with semi-major axes of 10-15 radii of the Earth. The problem of pre-computing the Luni-Solar eclipses for the geostationary satellites was considered by several authors.5,6,7,8 The estimation of the influence of the solar eclipses on the Earth's satellites has been obtained by some previous works.9,10,11
The influence of the Moon's shadow and pre-shadow effects on the motion of the Earth's satellite will be studied and the corresponding estimation will be given.

Theoretical model

The perturbing acceleration

Let us suppose that the satellite is homogeneous and has a spherical surface and a constant mass. Then, the perturbing acceleration of such satellite due to solar light pressure can be given as:12

(1)

where k is the solar constant and k(≈4.5605x 10^–6 Newton/meter²);
A is the semi-major axis of the Earth's orbit;
is the vector of the heliocentric position of the satellite;
is the area of the largest section of the satellite (the middle area);
m is the satellite's mass;
CR is a constant, which value lies between 0 and 2, depending on the reflection properties of the satellite's surface.

Let

So, substituting these values into equation (1), taken into account the various values of Δs, the corresponding is obtained

It must be mentioned that equation (1) corresponds to the case when the satellite is lightened by the non-eclipsed Sun. If the solar disk is partly or fully eclipsed by the Moon or by the Earth, then, the perturbing acceleration due to the solar-light pressure may be written as:

(2)

Where the function f is the so-called shadow function,13 and is given by the equation (1). The value of f is within the limits of 0 and 1 and depends on both the shadow functions of the Earth fE, and of the Moon fL. Here we give formula for fL only.

Conditions for satellite to be in the Moon's shadow and semi-shadow

The Moon and the Sun are supposed to have spherical surfaces with radii RL and Rs, respectively. Let be the seleno-centric vector of the satellite position and also let us consider that . The angle between the directions from the satellite to the Sun and to the Moon is ψ. Therefore, ψ is determined by the formula:

(3)

The equations of the lightenment of the satellite are determined by the mutual positions of the disks of the Sun and the Moon as seen from the satellite. Let ωs and ωL be the angular radii of the Sun and the Moon respectively, as seen from the satellite, then we have:

(4)

Accordingly, three kinds of mutual positions can be distinguished:
i) The satellite is out of the shadow and pre-shadow of the Moon (Figure 1), it means that the Sun's disk as seen from the satellite is not covered by the Moon.
In this case we have:

(5)

ii) The partial solar eclipse is seen from the satellite. In this case the Moon's disk as seen from the satellite covers only a part of the solar one and we can write the following condition:

(6)

iii) The full or the ring eclipse of the Sun is seen from the satellite. In this case, we have:

(7)

Figure 1 illustrates all the three cases.
Let us suppose that the intensity of the solar radiation is constant over the whole solar disk. Then, according to the work of Kabelac,14 we can write fL in the form

(8)

Where σ* is the area of the solar disk when it is not covered by the Moon as seen from the satellite and σ is the area of the non-eclipsed part of the solar disk as seen from the satellite.
When the satellite is out of the shadow and pre-shadow of the Moon, then we have

(9)

The most complex formula for fL takes place when the partial solar eclipse is seen from the satellite (the second case). In this case the angles ωL and ωs are small (<0.01 radius), so with error less than 1% the part of the unit sphere can be represented as a plane. The corresponding formula is:

(10)

Considering now that the Moon begins to cover the Sun, as seen in Figure 2, therefore, the angular radius ωs is continuously decreasing to zero value. Also the distance between the two centers of the Moon and the satellite is decreasing.
From the figure, we have the following relations describing these behaviors:

(11)

and

(12)

So, we have

When σ* and σ are known, then fL can be determined using equation (8). Finally, the third case, when the full eclipse of the Sun by the Moon is seen from the satellite, we have

(13)

During the solar eclipse the satellite is undergoing the radiation pressure in the direction from the effective center of the non-eclipse part of the solar disk to the satellite. Deflection of this direction from the center of the Sun to the satellite line is not more than the visible Sun's angular radius (<0.005 radians). The effect of this deflection in the perturbing acceleration of the satellite is no more than 1% of the direct solar pressure. This effect can be neglected with no harm to the accuracy of calculations.

Figure 1. Criteria for penumbra and umbra transits.

Figure 2. Depiction of the Sun, the Moon and the satellite with the line distances X, Y and Z.

Numerical estimation of the model

The model example has been taken in which the satellite moves around the Earth in an equatorial orbit with the semi-major axis A=1.227x10⁷ meter (Figure 1).
For the parameters in equations (1) and (4), the following values have been taken:

It is supposed that the Sun and the Moon move in the plane of the satellite's orbit along circular orbits, with the following radii:

with the angular velocity:

Three variants of the initial positions of the Sun (x⁰s, y⁰s) and of the Moon (x⁰L, y⁰L) have been taken as follows:

i) x⁰s=1.496x10¹¹m, y⁰s=6.4x10⁸m, x⁰L=3.843x10⁸m, y⁰L =8.5x10⁶m
ii) x⁰s=1.496x10¹¹m, y⁰s=6.4x10⁸m, x⁰L=3.843x10⁸m, y⁰L =6.7x10⁶m
iii) x⁰s=1.496x10¹¹m, y⁰s=6.4x10⁸m, x⁰L=3.843x10⁸m, y⁰L =3.8x10⁶m

The satellite's initial coordinates have been taken to be:

x⁰=0, y⁰= –1.227x10⁷m for all these variants.

These variants are distinguished by the conditions of passage of the Moon's shadow and pre-shadow at the first two revolutions of the satellite around the earth. The equations of motion of the satellite have been integrated numerically for these three variants. As for the perturbing forces due to the solar radiation pressure we considered four variants:
1. No perturbations due to the solar radiation pressure are presented.
2. No effects due to the Lunar shadow and semi-shadow are taken into account
3. The Lunar shadow has cylindrical form, and
4. The Lunar shadow and semi-shadow effects are taken into account according to the method given above.
In variants 2, 3, and 4, besides the Lunar shadow, also the Earth cylindric shadow was accounted for, the geopotential was presented by the first term of its development with respect to Legendre polynomials and functions. This term governs the unperturbed motion. Gravitational constant of the Earth has been taken as 3.986013x10¹⁴4m³/sec².
Table 1 shows the absolute values of differences of the position vectors of the satellite (geocentric positions) obtained for the above mentioned variants for the end of the satellite's second revolution around the Earth. The position vector of the satellite is designated as , where i is the number of the variant taking into account the solar pressure perturbing forces.

Table 1. Absolute values of differences of the position vectors of the satellite.

Conclusion

The obtained estimation allows us to make the following conclusions;
i) The influence of the Moon's shadow and semi-shadow must be taken into account when very accurate results are needed.
ii) The cylindrical lunar shadow can be considered only as a rough approximation to the stricter model that includes the pre-shadow.
iii) The effect of the cylindrical lunar shadow in the semi-major axes

References

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