PARAMETRIC ANALYSIS OF THE EARTH S SHADOW AND PENUMBRA

Transcription

1 ADVANCES IN SPACE DYNAMICS 4: CELESTIAL MECHANICS AND ASTRONAUTICS, H. K. Kuga, Editor, 33-4 (2004). Instituto Nacional de Pesquisas Espaciais INPE, São José dos Campos, SP, Brazil. ISBN PARAMETRIC ANALYSIS OF THE EARTH S SHADOW AND PENUMBRA Roberta Veloso Garcia Maria Cecília Zanardi Regina Elaine dos S. Cabette Group of Orbital Dynamics and Planetology Department of Mathematics, FE/UNESP, Guaratinguetá - SP rovgarcia@bol.com.br, cecilia@feg.unesp.br ABSTRACT The influence of the orbital elements and the positioning of the Sun in the duration of the Earth s shadow and penumbra is analyzed in this paper. The model for the shadow function considers geometric factors and three specific regions: shadow, penumbra and total illuminated. The mapping of the shadow function is shown for one orbital period, considering different sets of orbital characteristics for the satellite and positioning of the Sun in the Ecliptic. By the numerical results it is possible to observe that the orbital inclination, the major semi-axis and the longitude of the ascending node of the satellite orbit cause important variation on the duration of the penumbra and shadow of the Earth. INTRODUCTION The analysis of the influence of external forces and torques in the motion of artificial satellites is important to predict its trajectory and attitude in each instant of its useful life. The focus of this paper is the force and the torque due to the pressure of solar radiation, which are not zero when the satellite is totally or partially illuminated. The objective of this paper is to analyze the influence of the orbital elements and the positioning of the Sun in the duration of the Earth s shadow and penumbra on the orbit of an artificial satellite. The Kabelac model for the Earth s shadow function ( Kabelac,988) is used. In this model three regions are analyzed: total illuminated (where the shadow function ψ is equal ), the penumbra (where ψ assumes values between zero and ) and the shadow (where ψ is null). The model considers geometric factors, as mutual position Sun-Earth-satellite, the conical form of the shadow and refraction of the rays of light when crossing the terrestrial atmosphere. The mapping of the shadow function ψ is presented for one orbital period, considering different sets of orbital characteristics of satellite ( given by the major semi-axis (a), orbital eccentricity (e), orbital 33

2 inclinations ( I ), longitude of the ascending node ( Ω ), argument of the perigee (w)) and different characteristics positioning of the Sun in the Ecliptic ( given by celestial longitude of the Sun). By numerical results it is possible to evaluate the influence of each parameter on the duration of the shadow and the penumbra on the satellite orbit. EARTH SHADOW FUNCTION It is known that the force of solar radiation pressure cause disturbances in the orbit of the satellite, and it can cause a torque depending on the physical and geometric characteristics of the satellite. This solar radiation torque exists when the satellite is partially or totally illuminated. Therefore, to consider the discontinue effect of the solar radiation torque in the equations of the rotational motion it is usual to introduce a shadow function ψ ( Vilhena de Moraes & Zanardi, 997). In this paper the shadow function ψ has three phases of transition (Cabette, 200), which are represented in FIGURE : * transition for the dark region (umbra), when ψ = 0; * transition for the partially illuminated region (penumbra), when 0 < ψ < and * transition for the total illuminated region, when ψ =. illuminated umbra FIGURE : Transition Phases of the Earth Shadow Function In the mathematical model of Kabelac (988), the function shadow is defined by a physical function ψ f (that considers the influence of physical parameters) and a geometric function ψ g (that considers the influence of geometric parameters). In accordance with analyses developed by Cabette (200), the influence of the physical function ψ f is important only for the solar rays that pass through the Earth s atmosphere below 00 km before reaching the surface of the satellite. Thus in this paper will be considered ψ f =, so that ψ = ψ g. In Kabelac model the function shadow ψ g is given by (Cabette et al., 200): o 2 R 0 3? g = [?? sen?? cos?? + ( )? T ] p o () 3 a 0 where: ψ o and ψ T are auxiliary parameters defined by Kabelac (Kabelac,988); R o 0 and a o 0 are angles under which auxiliary sphere radius and Sun radius are observed at point P of the satellite orbit. The advantage of the Kabelac model is the inclusion of a transition region between the umbra and the total illuminated region, in which the shadow function assumes values between 0 and. 34

3 NUMERICAL IMPLEMENTATION OF THE SHADOW FUNCTION Using the software MATLAB, the geometrical shadow function given by () can be mapped and then it possible to observe the duration of the passage through the umbra, penumbra and full illuminated region. Tables - 8 and figures 4 show the numerical results of some simulations for different set of orbital parameters satellites. Tables 8 2 and figures 8-2 show the numerical results for different positions of the Sun in the Ecliptic. The simulations are developed for one orbital period. NUMERICAL RESULTS FOR DIFFERENT ORBITAL PARAMETERS Tables 4 show the duration of a penumbra region, considering different orbital parameter. The mapping of shadow function are shown in the figures 2 and 3. In figure 3 and 5 it is possible to observe the penumbra region. Tables 5 8 present the comparison of the duration of the three regions. Table : Duration of a Region for different values of a and e, considering I = 0.5 rad ; w=0rad ;Ω = π /6 rad. Eccentricity (e) Major semi axis (a) * * * * * In these cases it is verified that the satellite radius vector is lesser than the Earth radius. 35

4 Table 2: Duration of a Region for different values of a and I, considering e = 0.02 ;w = 0 rad ;Ω = π /6 rad. Inclination (I) Major semi axis (a) ** ** ** ** ** ** ** ** In these cases the satellite is in total illuminated region and there isn t the penumbra region. Table 3: Duration of the Region for different values of a and w, considering I = 0.5 rad ; e = 0.02 ; Ω = π /6 rad. Argument of the perigee (w) major semi-axis 0 π/3 5π/6 23π/8 (a) Table 4: Duration of a Region for different values of a and Ω, considering I = 0.5 rad ; e= 0.02 ; w = 0 rad. Longitude of the ascending node ( Ω ) Major semi- π/6 8π/9 25π/8 6π/9 axis (a) * * * * *

5 Psi time(s) Figure 2: Shadow Function Evolution for a = 7000 km, e = 0.0, I = 0.5 rad ; w = 0 rad ; Ω = π /6 rad Psi time(s) Figure 3: Entrance in the for the satellite at Fig.2 37

6 Psi time(s) Figure 4: Shadow Function evolution for a = 2000 km, e = 0.0, I = 0.5 rad ; w = 0 rad ; Ω = π /6 rad Psi time(s) Figure 5: Exit of the for the satellite at Fig.4 38

7 Table 5 - Comparison of the three regions duration for the satellite: a = 7.000km; I = 0.3rad, e = 0.02 ;w = 0 rad ; Ω = π /6 rad ( in the table 2). Initial interval Umbra Illuminated Final interval Umbra Table 6 - Comparison of the duration of the three regions for the satellite: a = km; w = π/ 3 rad, I = 0.5 rad ; e = 0.02 ; Ω = π /6 rad ( in table 3).. UmbraInitial Interal of the Illuminated Umbra Final interval Table 7 - Comparison of the three regions duration for the satellite: a = km; Ω = 8π /9 rad; I = 0.5 rad ; e = 0.02 ; w = 0 rad ( in table 4) Initial interval Umbra Illuminated Final interval Umbra NUMERICAL RESULTS FOR DIFFERENT POSITIONS OF THE SUN Tables 8 0 show the duration of penumbra region when different values considered for the Sun position, eccentricity, argument of perigee and longitude of ascending node. Table 8: duration for different values of eccentricity and days of the year: I = 0.5 rad ; a = km; w = 0 rad ; Ω = π/6 rad. Day of the Year Eccentricity (e) 2/03 22/06 23/09 22/

8 Table 9: duration for different values of the argument of the perigee and of days of the year: I = 0.5 rad; a = km; e = 0.02 ;Ω = π / 6 rad. Day of the Year Argument of pericentro (w) 2/03 22/06 23/09 22/ π/ π/ Table 0: duration for different values of the longitude of the ascending node and of days of the year: I = 0.5 rad; a = km; e = 0.02; w = 0 rad. Day of the Year Longitude (Ω) 2/03 22/06 23/09 22/2 π/ π/ π/ ANALYSIS OF THE RESULTS By the results presented in the previous figures and tables for the same day it is possible to observe the following influences of the orbital elements in the duration of the penumbra, umbra and illuminated regions: Major semi-axis: the increase of the major semi-axis causes the increase of the penumbra and the reduction of the umbra. It is observed a significant increment in the illuminated region when the major semi-axis increases; Orbital inclination: the satellite remains more time in the illuminated region than in the umbra when the orbital inclination increases, while the penumbra is practically unchanged; Argument of the perigee: the illuminated region and the umbra have significant variation with the variation of the argument of the perigee. When the duration of the illuminated region increases (decreases), the umbra duration decreases (increases). For the penumbra the variation is not significant; Longitude of the ascending node: the illuminated region and the umbra have no significant variations with the variations in the longitude of the ascending node. The penumbra has significant variations when the longitude of the ascending node is changed; Eccentricity: the umbra and the illuminated region have significant variations with the different values for the eccentricity. When the duration of the illuminated region increases (decreases), the umbra duration decreases (increases). The penumbra duration are not affected by variation in the orbital eccentricity. 40

9 By analyzing the results for different days of the year, it is possible to observe that: Eccentricity: the penumbra is practically the same independently of the day of the year for each eccentricity. The umbra and the illuminated region have significant variations for each day and eccentricity; Argument of the perigee: with the increase of the argument of the perigee, the duration of the illuminated region decreases and the umbra increases for the same date of the year. When only the position of the Sun is changed there is no alterations in the umbra, penumbra and illuminated region; Longitude of the ascending node: there are modifications in the umbra and illuminated regions for different Sun positions and there is no significant variation in the penumbra; Major semi-axis and orbital inclination: there are no modifications in the umbra, penumbra and illuminated regions for different Sun positions. CONCLUSION By the presented analysis which is developed by different positions of the Sun in the ecliptic, it is observed small variation in the penumbra duration, while the duration of the umbra and illuminated regions are modified. The variation on the umbra and illuminated region are significant for variations in the longitude of the ascending node, argument of the perigee and eccentricity. For the same time of the year, the major semi-axis, longitude of the ascending node and inclination cause significant variations in the penumbra, umbra and illuminated region. The argument of the perigee also influences the umbra and illuminated region. For some simulations for high altitudes satellites (major semi-axis bigger than 2000 km) it is observed that the penumbra and umbra do not exist, it means that the satellite is always illuminated. The presented results will be useful to analyze the influence of the solar radiation force and the torque on the orbit and attitude of the satellite, respectively. REFERENCES Cabette, R. E. S.; Zanardi, M. C.; Vilhena de Moraes, R. (200) The Earth s Shadow and the Spacecraft s Attitude Propagation. IN: 6 o CONGRESSO BRASILEIRO DE ENGENHARIA MECÂNICA, Uberlândia, Anais, CD-ROM, p Cabette, R. E. S.; Zanardi, M. C.; Vilhena de Moraes, R. (2002) Shadow Function of the EarthIN: ADVANCES IN SPACE DYNAMICS, O C. Winter, A F. B A Prado, Editors,Vol. 3, p.60-69, INPE, São José dos Campos. Garcia, R. V. (2002) Relatório Técnico de Bolsa PIBIC/ CNPq, 64p, UNESP Campus de Guaratinguetá. Kabelac, J. (988) Shadow Function Contribution to the Theory of the Motion of Artificial Satellite, Bull. Astron. Inst. Czechos., v. 39, p