GAUSS PLANETARY EQUATIONS IN A NON-SINGULAR GRAVITATIONAL POTENTIAL

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1 GAUSS PLANETARY EQUATIONS IN A NON-SINGULAR GRAVITATIONAL POTENTIAL Ioannis Iaklis Haanas * and Michal Hany# * Dpatmnt of Physics and Astonomy, Yok Univsity 34 A Pti Scinc Building Noth Yok, Ontaio, M3J-P3, Canada, ioannis@yoku.ca # 84 Noth 700 Wst, Plasant Gov, Utah, 8406, USA Michal.hany@signaldisplay.com Abstact W study th ffcts of a non-singula gavitational potntial on satllit obits by calculating th cosponding changs of its obital lmnts, using Gauss plantay quations. W div two non-zo xpssions fo th changs of th agumnt of th pig and th man anomaly, and w compa thm to thos of th gnal lativity. Using th GRACE satllit systm, w obtain numical sults fom which w conclud that th ffct of such a potntial, on th pig cannot b spaatd fom that of gnal lativity. Futhmo, w conclud that th ffct on th man anomaly can pobably b obsvd by today s tchnology. Ky Wods: Gavitation, clstial mchanics Gauss quations, GRACE, obital lmnts. Intoduction A nw non-singula gavitational potntial appas in th litatu that has th following fom []: GM V () () wh th constant appaing in th potntial abov is dfind as follows: GM () c and G is th Nwtonian gavitational constant, M is th mass of th main body that poducs th potntial, and c is th spd of light. In this pap w wish to invstigat th motion of a satllit in such a potntial using Gauss plantay quations of clstial mchanics. Th goal of this contibution is to xamin th possibility of validating this non-singula potntial by studying its ffct on th changs of th obital lmnts of a satllit. Vaious satllit ffcts can convnintly b xpssd as obital lmnt tim ats of chang, which a obsvabl by modn goic tchniqus. In gnal, th wll-known Gauss-plantay quations, as thy a psntd fo instanc in Blanco and McCusky [], link th obital lmnt tim divativs to thi caus, a distubing (o ptubing) foc p unit mass. H, ptubing foc p unit mass implis any dviation of th total acclation of a cntal Nwtonian fild. Accpting that Eq. () holds tu, w can wit V() as a cntal Nwtonian acclation plus oth tms that constitut th total distubing acclation. Ths distubing acclation componnts can thn b ntd spaatly into th Gauss -plantay quations to study thi ffcts on th satllit cntal fild (Kplian) obit, with th hop that w can s som masuabl obital lmnt tim ats of chang and thus obsvationally vify o dispov Eq. ().

2 In a tatmnt dvlopd by Gauss, th ptubing focs acting on a satllit a solvd into a th mutually ppndicula componnts []: FX R F (4) Y F Z wh R is a ptubing function, and F X ppndicula to th obital plan, positiv towads th noth pol, F Y ppndicula to th adius vcto in th obital plan, positiv in th diction of incasing longitud, and F Z is th diction of th adius vcto, positiv in th diction of incasing adial distanc and thfo Gauss quations can b wits as: da a FZ sin f + F Y n (5) d na F + cos f + + cos f + cos f F Z sin f Y (6) dω dω F cos f + + F sin f cosi Z Y (7) na a di cos na a ( ω + F X (8) dω dm ( ω + sin F X (9) na a sin i n + f FZ FY sin f na cos a na + a( (0) wh, a is th smi-majo axis of th obit, i, a th inclination and ccnticity of th obit Ω, th agumnt of th ascnding nod, and ω is th agumnt of th pig, and M is th man anomaly of th satllit dfind as Mn(t-T) and n π/p GM/a -3/ and f is th tu anomaly th angl btwn th pig and th adial vcto of th satllit. Equations (5) (0) a convnint bcaus thy allow us fo th influncs of th th componnts F X, F Y, F Z to b spaatly studid. W can s that th influnc of F X consists in changing th obital ointation o th lmnts i and Ω. Nxt F Y changs th smi-majo axis assuming <<, and it is impotant fo th satllit s manuvs. Th ptubing function Nxt, w obtain an xpssion th ptubing acclation p unit mass du to th non-singula potntial to b:

3 R NS V ( ) GM () which bcoms: R GM NS. () Fom Eq.() w s that th fist tm in th RHS is th Nwtonian gavity multiplid by th facto -/ (-/). This foc p unit mass has only a adial componnt and F X F Y 0 simplifis Gausss quations considably. In obital scnaios sinc << th foc function in Eq. () can b to fist od appoximatd by: GM F Z RNS substituting with Eq. () w obtain GM GM R NS c... (4) This is th adial ptubing potntial componnt to b in Gausss obital quations. Nxt, substituting Eq. (4) into Eqs. (5)-(0) w obtain th non zo tim ats of chang associatd with this non-singula distubing potntial to b: (3) da GM GM sin n c f d GM GM sin f na c (5) (6) dω GM GM cos f na c dm n na cos f (7) GM GM (8) a c 3 Solving th obital quations To solv qs. (5)-(8), w valuat thm on th unptubd Kplian llips, assuming that th obit dos not dviat to much fom that of a Kplian llips, and that a Kplian llips constituts a good appoximation. Thfo, w us that: ( a + cos f (9)

4 w also us th tansfomation xpssing tim in tms of th tu anomaly, and thfo w hav [3] df. (0) n a Substituting Eqs. (9) and (0) w obtain that 3/ ( df. () n ( + cos Thfo Eq. (5) bcoms: GM sin f da n a GM c cos a df. () To find th chang in on volution w intgat fom 0 to π and thfo w hav: cos a sin π GM f GM a 0 df. (3) n a 0 c Similaly Eq. (6) ov on volution givs cos a π GM sin f GM df 3 n a 0 c similaly cos a 0, (4) π GM GM ω 3 n a c 0 cos fdf (5) and Eq. (5) bcoms G M ω 4 5 c a n which could also b wittn as follows: ( ac GM ) πg M GM ( ) ( ) ω. (7) 4 n a c c a Equation (7) can b also wittn as a function of th paamt of th non-singula potntial in th following way: π ( ) ( ) ω. (8) a a To compa w say that gnal lativity pdicts a pig chang that is givn by [Taff, 985]: ω GR ( (6) 6πGM, (9) c a

5 using Eq. (7) and (8) w obtain that ωns 3 a Finally Eq. (0) bcoms M π 0 3/ cos GM 3 n a ω GR GM c cos a Intgating fom 0 to π th intgal abov simplifis to: 6πG M M 4 n c a ) GM 3c a / cos f cos df which can b also wittn as a function of th non-singula potntial paamt as follows: 6 π M. (33) a 3a Th man anomaly chang that gnal lativity pdicts in a ya is givn by [Schwazschild, 96] 3/ ( GM ) p 5/ 3 M / t. (34) c a Compaing to Eq. (34) fo t ya w can wit that: M π 3a NS M GR (30) (3) (3). (35) 4 Numical sults To calculat th changs p volution of th two non-zo obital lmnts w us th obital paamts of th Gavity Rcovy and Climat Expimnt GRACE. GRACE-A satllit has a km, and , and thfo n ad/s 5.3 v/d, i , ω , Ω , M [ Substituting ths valus in Eqs. (7) and (3) w obtain th cosponding changs on ω and M du to th non-singula potntial to b: ω /d (3) M /d. (33) Thfo, fo GRACE satllit using Eq. (3) and (33) w calculat an annual chang of th pig to b qual to 4.66/a and similaly fo th man anomaly w obtain a ngativ -4./a. Compaing with Eq. (9) w calculat th chang of th pig attibutd to gnal lativity to b 4.05/a. This is appoximatly th tims lag, than th on pdictd by th non-singula

6 potntial and most likly it would not b obsvd. Fo th man anomaly, chang gnal lativity pdicts a positiv chang of appoximatly 4.5/a. W conclud, that fo th cunt stat of tchnology such a dcas in man anomaly of 4./a could b asily ctd. 5 Conclusions W usd Gauss plantay quations, in od to validat th non-singula potntial givn by Eq.() using satllit obit ptubations. W hav divd th non-zo obital changs fo th pig and th man anomaly, and w hav compad thm to th ons pdictd by gnal lativity. W conclud that fo such a potntial th pig ffct will not b asily spaatd by that of gnal lativity, wh th yaly ffct of th man anomaly could b pobably obsvd with today s tchnology. Rfncs [] Williams, P., E., Mchanical Entopy and its Implications, Entopy, 3, pp. 76-5, 00. [] Blanco V., M., McCusky S. W., Basic Physics of th Sola Systm, Addison and Wsly, p.77-78, 96 [3] Roy., A., E., Obital Motion, Adam Hilg, pp. 36, 988. [4] Taff Launc, G., Clstial Mchanics A Computational Guid fo th Pactition, John Wily & Sons, 985, p.50.