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Zonal harmonic perturbations of an accurate reference orbit of an artificial satellite

1963; The National Institute of Standards and Technology; Volume: 67B; Issue: 4 Linguagem: Inglês

10.6028/jres.067b.016

2376-5283

John P. Vinti,

Space Satellite Systems and Control

The theor y developed in an earli er paper, for an accurate reference or bit of an artific ial satellite, is fi rs t slightly modified, so as to prepare the way for a treatment of zonal harmonic perturbations.Delaunay variables are next introduced , by means of certai n linear co mbin ations of th e action varia bles, along with their canonical conjugates.Appli cation of the von Zeipel method then permits th e calc ulation of the most important zonal harmoni c perturb ations.These ari se from the third , with coefficie nt 13, and the residual fourth, with coefficient 1. + 1~.The acc uracy of the sec ular and s hortperiodic effects is through terms of order 1~ and th at of th e long-periodi c e ffects is through ter ms of order h .Since the reference orbit itself, with its exac t sec ular terms, takes care of all but 0.5 per• ce nt of the de vi ati on of the earth 's gr avitational fie ld from spheri cal sy mm etry, th e overall sec ul ar accuracy of the final orbit surp asses that of other seco nd order theori es.T he res ults are co mp ared with those of Kozai .V' re pre sents exactly the zeroth harmonic -IL/r and the second harmonic and also gives higher e ve n harmonics , characterized by (1.05)In particular it gives J4 + J~ = 0, as compared with obs erve d values for the earth ranging from -(0.9)10-6 to (0.4) 10-6 (Kaula 1962; King-Hele, Cook, and Rees 1963).Consequently it accounts for about 99.5 percent of the deviation of V from the value -IL/r corresponding to spherical symme try.It thu s accounts almost completely for the flatte ning of the earth, leading to a geoid that never de parts by more than about 30 m from th e tru e sea-le vel s urface.For the drag-fre e motion of an artificial satellite the potential (1.00) le ads to a separable proble m, which has been worked out analytically [Vinti 1961[Vinti a,b , 1962]].This solution , holding for all angles of inclination and containing no critical inclination or long-periodic terms, gives secular terms exactly by me ans of rapidly converging infinite series and short-periodic terms correctly through order n.We call this orbit corresponding to (1.00) the reference orbit.For such a refere nce orbit error can ne ver accumulate , because of the exactn ess of th e secular terms, and the p e riodic te rm s can be in error only by amounts of the order J ~, i. e. , by about 1 part in 10 9 , since J~ = (1.08)10-: 1 for the earth. Zonal Harmonic PerturbationsFor a satellite of the earth, if its orbit is high enough so that drag is small and low enough so that the moon's effect is small, the above reference orbit ought to hold rather well for a good many revolutions.(I purposely choose vague words here, since numerical comparisons are still incomplete.)Eventually, however, the actual orbit will deviate more and more from such a reference orbit, because of the neglected forces.These include forces arising from drag, meteoritic impact, radiation, electromagnetic fields, the sun, and the moon, and the neglected part of the earth's gravitational potential, corresponding to (1.03) minus (1.00).Since the expansion of (1.00) in zonal harmonics is 00 V' =-lLr-1"'i(re/r)2m(-J2)mP2m(sin 8),(2.00) m = l this difference is V_VI=W-l[(~rJ3P3(Sin 8)+(~r(}4+J~)P4(Sin 8)+ (7YJ5P5(Sin 8) + (7 r(}6 -J;)P6(sin 8) + . . .] + tesseral harmonics. (2.01)Of these forces the most important, for any satellite with a large ratio of mass to area, are the forces corresponding to J3 and J4 + n in (2.01) and drag, which as determined empirically may include effects of meteoritic impact.For a double satellite [Langer and Vinti 1963] only (2.01) and the lunar-solar perturbation remain.The purpose of the present paper is to devise a method for correcting for the effects of any of the zonal harmonics in (2.01).The first example considered is the residual fourth harmonic, with coefficlent J4 + n• This harmonic leads not only to short-periodic effects and secular effects, but also to long-periodic effects depending on a resonance denominator ,1-5 cos 2 I" giving rise to a critical inclination 1=63.4°.The second example considered is the third harmonic, with coefficient J3.This gives rise only to short-periodic effects and to long-periodic effects without singularities, so that it is qualitatively less interesting.Because of its greater magnitude, however, J3 being about (-2.4)10-6 and IJ4 + n l being probably somewhat less than (0.5) 10-6 [Kaula 1962; King-Hele, Cook, and Rees 1963] , it leads to somewhat larger periodic effects.To compare accuraci es, we co nstruct th e following table, noting that the author's reference orbit accounts for about 99.5 perce nt of th e deviation of the earth's potential from spherical symme try.Thus my perturbation potential is only about 0.5 percent of Kozai's.