Satellite motion in a non-singular gravitational potential

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1 DOI 0.007/s ORIGINAL ARTICLE Satellite motion in a non-gular gravitational potential Ioannis Haranas Spiros Pagiatakis Received: 6 September 009 / Accepted: 8 January 00 The Author(s 00. This article is published with open access at Springerlink.com Abstract We study the effects of a non-gular gravitational potential on satellite orbits by riving the corresponng time rates of change of its orbital elements. This is achieved by expanng the non-gular potential into power series up to second orr. This series contains three terms, the first been the Newtonian potential and the other two, here R (first orr term and R (second orr term, express viations of the gular potential from the Newtonian. These viations from the Newtonian potential are taken as sturbing potential terms in the Lagrange planetary equations that provi the time rates of change of the orbital elements of a satellite in a non-gular gravitational field. We split these effects into secular, low and high frequency components and we evaluate them numerically ug the low Earth orbiting mission Gravity Recovery and Climate Experiment (GRACE. We show that the secular effect of the secondorr sturbing term R on the perigee and the mean anomaly are /a, and /a, respectively. These effects are far too small and most likely cannot easily be observed with today s technology. Numerical evaluation of the low and high frequency effects of the sturbing term R on low Earth orbiters like GRACE are very small and untectable by current observational means. I. Haranas ( S. Pagiatakis Dept. of Physics and Astronomy, York University, 700 Keele Street, Toronto, ON MJ P, Canada ioannis@yorku.ca S. Pagiatakis Dept. of Earth and Space Science and Engineering, York University, 700 Keele Street, Toronto, ON MJ P, Canada spiros@yorku.ca Keywords Non-gular potential Lagrange planetary equations Disturbing potential Eccentricity functions Hansen coefficients GRACE Introduction A non-gular gravitational potential may take the following form (Williams 00 V(r GM p e λ/r, ( r where constant λ is fined as follows λ GM p c, ( G is Newton s gravitational constant, M p is the mass of the planetary body that produces the potential, c is the speed of light, and r is the raal stance of the satellite from the center of mass of the planetary body. The goal of this contribution is to examine the possibility of validating this non-gular potential by studying satellite orbit perturbations that might result from the viation of this gular potential from the Newtonian one. Various satellite effects can conveniently be expressed as orbital element time rates of change, which are observable by morn geotic techniques. In general, the well-known Lagrange planetary equations, as they are presented for instance in Kaula (000, link the orbital element time rivatives to their cause, a sturbing (or perturbing potential. Here, sturbing potential implies any viation of the total potential from a central Newtonian field. Accepting that ( holds true, we can write V(r as a central Newtonian potential plus other terms that constitute the sturbing components. These sturbing components can then be entered separately

2 into the Lagrange planetary equations to study their effects on the satellite central field (Keplerian orbit, with the hope that we can see some measurable orbital element time rates of change and thus observationally verify or sprove (. The Lagrange planetary equations contain the rivatives of an appropriate sturbing potential R with respect to the orbital elements. Following Kaula (000 we can write the Lagrange planetary equations as follows: da na M, ( ( e e nea M nea ω, ( d e na e cos i na e i na e i cos i na e i i, (5 ω na e i, (6 i, (7 n ( e nea na a, (8 where a is the orbital semimajor axis, e is the orbital eccentricity, ω is the argument of the perigee, i is the orbital inclination, is the argument of the ascenng no, M is the mean anomaly, and n (GM p /a / is the mean motion of the satellite. In our study, the sturbing potential R R (a,e,ω,i,,m contains only the viations of the nongular potential (cf. ( from the central Newtonian. To obtain R, we use power series expansion of ( that allows expresg the non-gular potential as the sum of a central potential and its sturbing terms. The sturbing terms form R, which after appropriate transformations, it can be written as a function of the orbital elements and eccentricity functions (e.g. Kaula 000 so that its rivatives with respect to the orbital elements, as required by the planetary equations, can be taken (cf. ( (8. Ug the Lagrange planetary equations we can then numerically evaluate the time rates of change of the orbital elements due to R and thus, we will be able to assess whether the non-gular potential can or cannot be verified experimentally. The sturbing potential Without loss of generality, we consir a satellite that orbits the Earth at a certain raal stance r from the geocenter unr the influence of the non-gular potential given by (. For a more tailed finition of the orbital elements see Vallado (007. We can expand the exponential term of ( into power series and keeping terms up to second gree we obtain V(r GM p r GM pλ r GM pλ r. (9 Ug ( we can write (9 as follows V(r V N G M p c r G M p c r V N R R, (0 where V N is the central Newtonian potential, and the other two terms in the RHS of (0 express viations of the nongular potential from the central Newtonian and are noted as sturbing components R and R, respectively. The third term in the RHS of (0 is inversely proportional to r, with a similar raal pennce to the general relativistic potential that reads V GR GMh /c r (Murray and Dermortt 999, where h GMa( e is the angular momentum per unit mass of the primary body. In particular, ug the relativistic potential and substituting for the angular momentum h we can write R as a function of the relativistic potential as follows GM p R c a( e V GR. ( Theory prects that the relativistic potential causes secular perigee/perihelion variations in the orbit of a satellite (natural or artificial orbiting a massive body (Ghosh 000. The term /r l, where l is integer (see below, can be written as a function of the eccentricity functions G lpq (e and satellite orbital elements in the apparent right ascension system as follows (Kaula 000; p. 5: r l ( cos a l [(l p(ω f m( q G lpq (e ( cos [(l pω (l p qm m(, ( where f is the true anomaly, is the Greenwich sireal time, l is the gree and m is the orr of the spherical harmonic expansion of the potential, (p, q Z and 0 p l. The inces l, p, q, m intify the eccentricity function and also the trigonometric argument associated with a particular spherical harmonic term of gree l and orr m. These terms arise from the potential of the Earth when it is expressed in terms of spherical harmonics as given in Kaula [cf. (.; Kaula 000. Ug (, we write the two terms R and R as functions of the orbital elements in the following way: R G M p c [ a l q G lpq (e ( cos

3 [(l pω (l p qm m(, R G Mp [ c a l q G lpq (e ( cos [(l pω (l p qm m(. The secular sturbing potentials and time rates of change of the orbital elements ( ( Next, we examine only the secular terms resulting from R and R in ( and (, respectively. We can do this by eliminating the low frequency term ω from ( and ( by setting l p 0. Similarly, from ( and (, we eliminate the terms that are varying with high frequency, i.e., the terms that are functions of the mean anomaly M, and (. This can be achieved by setting their respective coefficients to zero, which results in (l p q 0, and m 0, which imply q 0 ce l p. These contions must hold simultaneously and finally, ( and ( become: R S G M p c a G,/,0, (5 R S G Mp c a G,,0, (6 where subscript S signifies secular. Clearly, in the case of R,wehavel (cf.( which implies p /. However, p must always be an integer (Vallado 007 and in adtion l harmonic is intically zero because the coornate system is geocentric. This incates that R is not physically meaningful and thus sregard from further consiration. In orr to proceed with the calculation of the secular time rates of change of the orbital elements due to R, we substitute (6 into the Lagrange planetary equations. The calculation of eccentricity function G lpq (e is not a trivial process because it requires the use of the so called Hansen coefficients X n,m k. Following Giacaglia (976 we have that G lpq (e X (l,(l p l pq, (7 and the corresponng eccentricity function G,,0 (e becomes (Kaula 000 G,,0 (e ( e /. (8 To monstrate the relation/fference between the general relativity and the non-gular potential effects on the above two orbital elements rived herein, we consir the following expressions for the prection of the secular rates of change of the perigee (Lucchesi 00 and mean anomaly, respectively (Schwarzschild 96. ( GRS (GM/ c ( e, (9 a5/ ( GRS (GM p / c a 5/ ( e. / (0 Ug R S we obtain ( (, ( R S 6n GR S ( n [ R S 6n ( e 7/ [( ( ( GR S where the subscript R S signifies secular changes caused by R. The low frequency sturbing potential and the time rates of change of the orbital elements Focug on the low frequency terms of R, we eliminate the terms from ( that vary with high frequency. This can be achieved by setting their respective coefficients to zero resulting to l p q 0 and m 0. For the /r term in ( we have that l and 0 p l and q p l, which implies that q [, therefore, ( becomes R L G M p c [ a q G,p,q (e cos[( pω. ( where subscript L incates Low frequency components. Substituting ( in the Lagrange planetary equations we obtain the following equations for the low frequency time rates of change of the orbital elements due to R, and the corresponng e terms will be zero. Therefore, the only nonzero time rates are G Mp e nc ea 5 ( pg,p,p (e [( pω, ( p0 G Mp e nec a 5 p0 G,p,p (e cos[( pω, (5

4 cot ig Mp nc a 5 e ( pg,p,p (e [( pω, (6 p0 n ( e G Mp nec a 5 G,p,p (e cos[( p p0 G M nec a 5 G,p,p (e cos[( pω. (7 p0 Carrying out the summation in the above equations we obtain G Mp e nc ea 5 [(G,0, (e G,, (e (ω 0, (8 G M p e [ G,,0 (e nec a 5 ( G,0, (e G,, cos(ω G M c a 5 ( e, (9 cot ig Mp nc a 5 e [(G,0, (e G,, (e (ω0, (0 n ( e G Mp [( G,0, (e nec a 5 G,,(e cos(ω G M c nea 5 [(G,0, (e G,, cos(ωn. ( Ug the tabulated expressions of the eccentricity functions we have that (Kaula 000 G,0, (e G,0, (e 0, ( and ug (8, G,,0 e( e 5/. ( Equations (8 ( above give the low frequency variations of the orbital elements due to R from which the non-zero rates of change can be written as follows G M p nc a 5 ( e, ( n. (5 We see that from all the orbital elements, only the argument of the perigee is affected by the low frequency term due to R. This is a fraction of the secular variation given by general relativity calculated ug ( and therefore, we have that ω RLS 6 ω GR S. (6 5 High-frequency sturbing potentials and time rates of change of the orbital elements In orr to obtain the high frequency components of the sturbing function R, we simply eliminate the low-frequency terms in ( and we get R H G Mp [ c a m0 G,,q (e cos(qm m(, (7 q where subscript H signifies high frequency. Substituting (7 in the Lagrange equations we obtain the following high frequency variations of the orbital elements. We proceed with the rivation of the high frequency effects arig from R by summing over inx q for, when q>, the effects of R are O(0 8 on ȧ, O(0 on ė, O(0 on ω, and O(0 0 on i and O(0 on Ṁ. FortheR sturbing term we obtain the following non-zero time rates of change da G M p nc a m0 q ( e G M p nec a 5 m0 q G Mp e nec a 5 m0 q qg,,q (e [qm m(, (8 qg,,q (e [qm m(, (9 G,,q (e cos[qm m(, (0

5 G M p nc a 5 i e m0 q mg,,q (e [qm m(, ( n G Mp ( nc a 5 ( e 6e m0 q G,,q (e ( G,,q (e cos[qm m(. ( In the above equations we also need the following eccentricity functions (q. G,, (e e 7e 6, ( G,, (e 9e 7e, ( G,, (e 5e 6 9e5 56, (5 G,, (e 77e 6 9e6 60. (6 Substituting ( (6into(8 ( we get da G M p nc a ( e 7e 6 ( 9e 7e ( 0e 6 ( e 7e 6 5e 6 ( (M ( (6M (M 5e (M ( (M ( 77e ( ( 9e (8M 7e (M ( ( cos ( (M ( (7 M G p ( e enc a 5 ( ( e 7e (M 6 ( 9e ( 7e (M ( 0e 6 (6M ( 5e (8M ( e 7e (M ( 6 ( 9e 7e (M ( 5e 6 77e (M ( ( cos ( (M ( (8 G,, (e and G,, (e evaluated herein are intical to those given in Kaula (000andVallado(007.

6 ( ( 8e (M 6 ( ( 9e 7e (M ( ( 59e ( 965e cos ( (6M 6 56 G Mp ( e 77e ( 87e5 (8M 80 nec a 5 ( 8e cos(m ( 6 ( 9e 7e cos(m ( ( 59e 965e cos(m ( 6 56 ( 77e 87e5 cos(m ( 80 ( ( e G Mp csc i 7e 9e cos(m 6 7e cos(m nc a 5 e 5e 77e cos(m cos(m 6 6 ( e 7e 6 ( ( ( e 6e 8e (M 6 ( 9e 7e ( ( ( e 9e 6e 7e (M ( 5e n G Mp 6 9e5 56 ( ( e 59e 965e 6e 6 56 nc a ( (6M ( 77e 6 9e6 60 ( ( e 77e 87e5 6e 80 ( (8M ( cos ( Astrophys Space Sci (9 (50. (5 6 Numerical results We calculate the secular orbital element changes (cf. (5 (6 specifically for the Gravity Recovery and Climate Experiment GRACE mission, ug the orbital parameters of GRACE-A satellite that has a km, and e , and therefore n rad/s 5. rev/d, i , ω 0., 5.79, M [ grace/newsletter/archive/august00.html. Because all rived (9 ( are inversely proportional to fferent powers of the semimajor axis, the secular rates of change of the orbital elements due to general relativity minish rapidly for higher altitu satellites thus, the choice of GRACE mission (low orbit. Substituting these values in (9 and (0 we obtain the corresponng secular general relativistic effects on ω and M as follows (.0/a, (5 GR s ( GR s.0/a. (5

7 Similarly, ug ( and ( we calculate the corresponng secular rates of change of ω and M due to R for which we obtain ( /a, (55 R S ( /a. (56 R S The low frequency maximum effect on the perigee is and far too small to be observed with today s technology. Finally, for the numerical calculation of the high frequency effects of R H on the orbital element time rates of change we choose to calculate only the maximum effect because (7 (5 contain many e waves of various frequencies. This can be done by setting all trigonometric terms equal to unity implying that all constituent waves are in phase. The maximum effects on a, e, ω, i, and M are m,.00 0,5.0 0, , and.9 0, respectively, whereas the effect on is zero. Apparently, these maximum variations are far too small to be observed with today s technology. Next, we calculate the secular effects of R and, we find that the corresponng time rates of change of the perigee and mean anomaly are extremely small, namely ω RS /a, and Ṁ RS /a, leaving the time rate of change of the mean anomaly practically unchanged, and equal to that of the Newtonian field. With reference to GRACE-A satellite only, these rates of change of the perigee are by far smaller than any technology can measure today, and require very long orbiting times that far exceed the sign lifetime of low Earth orbiters. For natural satellites or planets like Mercury that is the closest planet to the massive Sun, there might actually be a possibility to obtain measurable effects. For Mercury with a semimajor axis a km and eccentricity e (Vallado 007 we obtain ω RS /a, and Ṁ RS /a, still much too small to accumulate to a measurable effect in time-scales of centuries in a way similar to the relativistic effect of the perihelion of Mercury. 7 Conclusions We used Kaula s approach to transform and validate the nongular potential given by ( ug satellite orbit perturbations. Examining the high frequency terms we found that their corresponng effects are far too small to be tected. Similarly, we found that the low frequency effect of R on the perigee is far too small to be observed with today s technology. In adtion, and for GRACE mission, the calculated secular changes related to R were found to be extremely small, and impossible to observe with current technology. In conclusion, ( cannot be verified ug low Earth orbiters, at least with the current technology. Acknowledgements This research was financially supported by the Natural Sciences and Engineering Research Council (NSERC of Canada. We thank the two anonymous reviewers for their thoughtful comments and suggestions that significantly improved the original manuscript. Open Access This article is stributed unr the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, stribution, and reproduction in any meum, provid the original author(s and source are creted. References Ghosh, A.: Origin of Inertia. Apeiron, Montreal (000, p. 9 Giacaglia, G.E.O.: A note on Hansen s coefficients in satellite theory. Cel. Mech., 55 5 (976 Kaula, W.: Satellite Geosy. Dover, New York (000 Lucchesi, D.M.: LAGEOS II perigee shift and Schwarzschild gravitoelectric field. Phys. Lett. A 8, 0 (00 Murray, C.D., Dermortt, S.F.: Solar System Dynamics. Cambridge (999 Schwarzschild, K.: Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit. Sitzungsber. K. Preuss. Akad. Wiss., (96 Vallado: Fundamentals of Astrodynamics and Applications, rd edn. Space Technology Library (007 Williams, P.: Mechanical entropy and its implications. Entropy, 76 5 (00