The in-plane Motion of a Geosynchronous Satellite under the Gravitational Attraction of the Sun, the Moon and the Oblate Earth

Transcription

1 J Astrophys Astr (1990) 11, 1 10 The in-plane Motion of a Geosynchronous Satellite under the Gravitational Attraction of the Sun, the Moon and the Oblate Earth Κ Β Bhatnagar Department of Mathematics, Zakir Hussain College, Delhi Manjeet Kaur Department of Mathematics, Mata Sundri College, New Delhi Received 1989 June 20; accepted 1989 September 29 Abstract The in-plane motion of a Geosynchronous satellite under the gravitational effects of the sun, the moon and the oblate earth has been studied The radial deviation (Δr) and the tangential deviation (r c Δθ) have been determined Here r c represents the synchronous altitude It has been seen that the sum of the oscillatory terms in Δr for different inclinations is a small finite quantity whereas the sum of the oscillatory terms in r c Δθ for different inclinations is quite large due to the presence of the low-frequency terms in the denominator Key words: celestial dynamics artificial satellites 1 Notations C i amplitude of ith oscillatory component in Equation (1) D i amplitude of ith oscillatory component in Equation (2) r vector from centre of earth to the satellite t time α inclination of the satellite orbital plane to the reference plane α m inclination of the moon s orbital plane to the ecliptic α 1 inclination of the reference plane to the ecliptic α 0 steady-state value of α φ orbital angle of the earth around the sun θ orbital angle of the satellite around the earth θ m orbital angle of the moon around the earth ψ satellite orbital regression angle ψ m lunar regression angle ω i oscillatory frequency of ith component in Equations (1) or (2) G Universal gravitational constant = dyne cm 2 gm 2 J 2 coefficient due to the oblateness of the earth = M E mass of the earth = gms M m mass of the moon = gms M s mass of the sun = (332,9468 M E ) gms R 0 mean earth radius = km R distance between the centre of the sun to the centre of mass of the earth moon system = km ε obliquity = 23 27

2 2 Κ Β Bhatnagar & Manjeet Kaur 2 θ GM 0 E /r 3 c θ m moon s orbital rate = rad/solar day ψ m regression rate of the moon s orbital plane = rad/solar day ψ 0 steady-state regression rate of the satellite φ earth s angular rate = rad/solar day θ E earth s rotation rate = rad/solar day steady-state value of the synchronous altitude = km r c 2 Introduction Frick & Garber (1962, Part III) have discussed the in-plane perturbations of a Geosynchronous satellite under the gravitational effects of the sun, the moon and the oblate earth They have assumed that the sun, the moon, and the earth all lie in the plane of the ecliptic and the satellite s orbital plane and the reference plane coincide with the earth s equatorial plane Bhatnagar & Mehra (1986, 1987) have also studied the motion of a Geosynchronous satellite under the combined gravitational effects of the sun (including its radiation pressure), the moon and the oblate earth (including its ellipticity) They have studied only the motion of the orbital plane of the satellite It is shown that the orbital plane rotates with an angular velocity lying between 0042 /year and 0058 /year for a synchronous satellite It is further observed that the regression period increases both as the orbital inclination and the altitude increase Monica ( 1987) has shown that the major effect of the earth s equatorial ellipticity is to produce it change in the relative angular position Γ of the satellite as seen from the earth In the present paper, we are studying the in-plane motion of a Geosynchronous satellite under the effects of the sun, the moon and the oblate earth It has been assumed that the centre of the earth-moon system moves around the sun in a circle with constant angular velocity φ in the plane of the ecliptic; the moon moves about the earth in a circle with constant angular velocity θ m in a plane at an angle of 5 8' to the plane of the ecliptic and the earth is spinning about its axis Further, the nominal orbit is also supposed to be circular The nonlinear in-plane equations of motion have been made linear by applying the perturbation technique for determining the radial deviation Δr and the tangential deviation r c Δθ The amplitudes of the oscillatory terms in Δr and r c Δθ for different inclinations have also been determined 3 Equations of motion In Fig 1, S represents the sun, Ε the earth, Μ the moon, Ρ the satellite and G the masscentre of the earth-moon system Let SE = R E, SM = R m, SP = r s, SG = R, EP = r, EM = P 0, EG = P E, MP = r m The circular motions of the bary-centre relative to the sun and of the moon around the earth give

3 The in-plane motion of a geosynchronous satellite 3 Figure 1, Configuration giving the position vectors of the sun, moon, earth and satellite where φ = angular velocity of the earth around the sun; θm= angular velocity of the moon around the earth Proceeding as in Bhatnagar & Mehra (1986), the in-plane equations of motion of the satellite Ρ are given by (1) where (2) and All the notations used in the above equations are given in Section 1 Here r, θ determine the position of the satellite in the orbital plane shown in Fig 2 The amplitudes C i, D i are functions of any or all of the quantities α 0, α l, α m, r 0 and frequencies ω i are linear combinations of θ 0, θm, φ, ψ m and ψ 0 The values of the frequencies ω i and the amplitudes C i, D i (i =1 127) are not mentioned in this paper as these values cover about 45 pages Their values can be obtained from the author on

4 4 Κ Β Bhatnagar & Manjeet Kaur Figure 2 Parameters (r, θ, α, χ) representing the position of the satellite in space request Out of 127 frequencies, the frequencies corresponding to i = 1, 2, 23, 24, 32 are the low-frequency terms (Appendix 1); we have laid emphasis on these terms because of the presence of the factor 1/ω i which we obtain on integrating the equations of motion For a synchronous satellite, the steady-state value r o of r is replaced by r c in our subsequent study 4 Perturbation equations The linearized technique used by Frick (1962, Part II) while determining the motion of a Geosynchronous satellite is not applicable for Equations (1), and (2), since we obtain time-variable coefficients in the final linearized equations So we adopt a perturbation method as used by Frick (1962, Part III) Since for a synchronous satellite (Frick 1967), where θ E = earth s rotation rate = rad/solar day and ψ, = steady-state regression rate of the synchronous satellite = ( ) cos α 0, rad/solar day This equation enables us to determine the value of the synchronous altitude for different inclinations (0 α 0 90 ) Table 1 gives the dif f erences to the synchronous altitude, km, corresponding to inclination α 0 = 0 for different inclinations We may observe that the maximum change in the synchronous altitude is one and half kilometre at α 0 = 90 Table 1 The difference in the value of the synchronous altitude corresponding to inclination zero (r c = km)

5 The in-plane motion of a geosynchronous satellite 5 Now, the perturbation relative to a synchronous orbit can be defined by the equations: Putting these in Equations (1) and (2), we get (3) where (4) Equations (3) and (4) are two coupled linear differential equations with constant coefficients in variables Δr and θ After integrating Equation (4) we get where at t = t 0 we suppose that θ=θ 0, φ = φ 0, θ m = (θ m) 0, Δr=Δr 0, Δθ = 0 It is necessary to assume a nonzero initial value Δr 0 of Δr since the effects of the sun and the moon may give an additional bias in the value of the synchronous radius Now we solve Equation (5) for Δθ and substitute its value in Equation (3) we get (5) Suppose at t = t 0, D r = 0 then solving Equation (6), we have (6) (7)

6 6 Κ Β Bhatnagar & Manjeet Kaur Putting this value of r in Equation (5), we get Equation (8) for Δθ contains constant terms which would cause a steady-state increment in the angular rate relative to the desired synchronous conditions of the satellite So we put these terms equal to zero and determine the value of Δr 0 : (9) Hence the complete solution for Δr, Δθ are given by Equations (7) and (8) where Δr 0 is given by Equation (9) Equation (8) can be integrated and multiplied by r c to obtain the tangential displacement of the satellite from its desired synchronous position: (8) We observe from Equations (7), (9) and (10)that if the initial radius is corrected by an amount Δr 0 then the resulting variation in deviations Δr and rcδθ contain small it bias terms plus sinusoidal variations with angular frequencies of θ 0 and ω i (i = 127) However, as a result of the correction Δr 0 in the orbital radius there is no secular terms in the in-plane motion of the satellite Hence the deviations of the satellite from the desired synchronous position are bounded and the satellite will remain essentially synchronous (10)

7 The in-plane motion of a geosynchronous satellite 7 5 Results and discussion 51 Case 1: Frick Case From Equations (7), (9) and (10) Frick case can be deduced by putting so a modification to the initial radius r c is required due to the attractions of the sun and the moon on the satellite The magnitude of this correction depends on the initial position of the sun, the moon and the satellite relative to the earth and it can vary between km and km, being the maximum and minimum value of Δr 0 respectively Moreover, the in-plane perturbations of the satellite caused by the attraction of the sun and the moon are in the form of small amplitudes oscillations which result in a maximum deviation from the desired synchronous position of about 7266 km 52 Case 2: Orbit in any Plane From Equation (9) it is seen that because of the effects of the sun and the moon the initial value of r c must be altered by an amount Δr 0 to eliminate the steady-state value of Δθ The magnitude of this correction is a function of the initial geometry of the sunearth moon-satellite system as characterized by the angle φ 0, (θ m ) 0, θ 0, (ψ m ) 0 and ψ 0 For different orbital inclinations, the maximum possible negative value of Δr 0 occurs when θ 0 equals 90 and α 0, φ 0, (θ m ) 0, (ψ m ) 0 and ψ 0 are all zero provided Under these conditions Δr 0 is given by Δr 0 = km On the other hand for different orbital inclinations, the maximum possible positive value of Δr 0 occurs when θ 0 equals 0 0 and α 0, φ 0, (θ m ) 0,(ψ m ) 0 and ψ 0 are all zero Provided Under these conditions Δ r 0 is given by Δr 0 = km Here the maximum possible negative value of the maximum possible positive value are slightly different (0016 km) from Frick (1962) since he has not considered the inclination of the moon s orbital plane to the ecliptic (5 8 ) 53 Case 3: Oscillation Amplitudes The amplitudes of the various oscillatory components in Equation (7) for Δr for different inclinations α 0 (0 α 0 < 90 ) have been found The graph of the arithmetic sum of all the 129 amplitudes involved in Δr is given in Fig 3 for different inclinations of α 0 It shows that Δr 0 increases from km (α 0 = 0 ) to 8052 km (α 0 = 80 ) The amplitudes of the various oscillatory components in Equation (10) for the tangential deviation have been found for different inclinations α 0 (0 α 0 < 90 ) The graph of

8 8 Κ Β Bhatnagar & Manjeet Kaur Figure 3 Sum of amplitudes in Δr for different orbital inclinations Figure 4 Sum of amplitudes in r c Δθ for different orbital inclinations the arithmetic sum of all the 129 amplitudes involved in r c Δθ is given in Fig 4 for different inclinations of α 0 It shows that r c Δθ increases from km to km (α 0 = 80 ) Both Δr, r c Δθ are not defined when α 0 = 90 So the inplane perturbations of the satellite caused by the attraction of the sun and the moon and oblate earth are in the nature of large-amplitude oscillations because of the presence of low-frequency terms in the denominator and the orbital inclination α 0, whereas in Frick (1962) these perturbations are in the nature of small-amplitude oscillations which could result in a maximum deviation from the desired synchronous position of about 7266 km as he has studied only one case, ie, when α 0 = 0 and in that case all the low frequency terms vanish Appendix 1 Low frequency terms ω i (i =1, 2, 23, 24,, 32)

9 The in-plane motion of a geosynchronous satellite 9

10 10 Κ Β Bhatnagar & Manjeet Kaur where References Bhatnagar, K B, Mehra, M 1986, Indian J Pure Appl Math, 17, 1438, Bhatnagar, K B, Mehra, M 1987, Indian J Pure Appl Math, 18, 461 Frick, R H, Garber, T B 1962, R-399-NASA Frick, R H 1967, R-454-NASA Mehra, M 1987, PhD Thesis, Univ Delhi