LOOPS IN THE SUNS ORBIT

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1 Theoet.Appl.ech. TEOP7,Vol.4, No.1, pp , Belgade 213 LOOPS IN THE SUNS ORBIT ilutin ajanov doi:1.229/ta ath. Subj. Class.: 7F15; 72;799. Accoding to: Tib Jounal Abbeviations (C) athematical Reviews, the abbeviation TEOP7 stands fo TEORIJSKA I PRIENJENA EHANIKA.

2 THEORETICAL AND APPLIED ECHANICS Seies: Special Issue - Addess to echanics, Vol. 4(S1), pp , Belgade 212. LOOPS IN THE SUN S ORBIT UDC (45)=111 ilutin ajanov Beogad, 27 mata 39 mima@bvcom.net Abstact. Besides tanslation, spin aound its axis and otation aound cente of the ily Way, the Sun pefoms elative motion in the sola system Laplacian plane, also. This motion was anticipated by Newton himself, in his Pincipia. The fom of the Sun s obit is substantially diffeent fom the othe sola system bodies obits. Namely, the Sun moves along the path composed of the chain of lage and small loops [1, 2, 6, 9]. This chain is situated within the cicula outline with the diamete appoximately twice as lage as the Sun s is. Unde supposition that the sola system is stable, the Sun is going to move along it, in the same egion, fo etenity, neve eiteeiting the same path. It was also shown in this wo that velocity and acceleation of the Sun s cente of mass ae completely defined by the elative velocities and acceleations of the planets with espect to the Sun. Key wods: loops in the Sun s obit, velocity, acceleation. 1. INTRODUCTION As said, Isaac Newton was the fist who pointed to the fact that the Sun moves aound the cente of mass of the sola system (Philosophiae Natualis Pincipia athematica):...since that cente of gavity (the sola system mass cente) is constantly at est, the sun, accoding to the vaious positions of the planets, must continually move evey way, but will neve ecede fa fom that cente. If the sola system is teated as a stable, isolated system of the point mass paticles moving unde mutual gavitational inteactions, two dynamic consevation pinciples may be used fo the study of its motion: consevation of the momentum and of the angula momentum of the system. The consequence of the fist ule is unifom motion of the system s mass cente C, while the consequence of the second is motion of the system in one, Laplacian, o invaiant plane [3]. This plane is within, 5 of the Jupite s obital plane and may be egaded as the weighted aveage of all planetay obital planes. The point mass paticles sola system model, involves necessity of neglecting diffeences between the obital planes mainly

3 12 ILUTIN ARJANOV oiginated in the tansfe of the (small, but changeable) Sun s and planet s spin angula momenta to its total angula momentum. Neglecting otation of the sola system, as a whole, aound the cente of ou galaxy, this plane moves tanslatoy, togethe with the mass cente C though the space. 2. COORDINATE SYSTES Existence of the invaiant plane pemits intoduction of an "inetial" efeence fame xcy lying in it. Anothe Catesian coodinate system x y (x and y paallel with Cx and Cy) was adopted as the elative, that is, the heliocentic fame of efeence (Fig. 1). y R y' ω x' C x Fig. 1. Inetial and Relative Fame of Refeence. 3. POSITION, VELOCITY AND ACCELERATION VECTORS OF THE SUN It seems that, using astonometical data positioning the oute planets fom to 26., P. D. Jose was the fist one to detemine the Sun s obit, in his wo [1], This pape was abundantly cited late in the wos teating the sunspot activities. In his wo [2] R. Bitsch has detemined the Sun s obit integating diffeential equations of motions of this celestial body exposed to the esultant of the planets attacting gavitational foces. It was supposed that these heavenly bodies ae in cicula, unifom motions aound the Sun. The initial planet s configuation was adopted abitay, because the autho assumed that, in the long tem, it does not affect the shape of the obit. One would be tempted to use tem Keple s fo the model employed in the wo [1] and Copenicus fo the model in [2], but it would not be coect, since the Sun moves in both models, of couse. In fact, the choice of eithe model in detemining the tajectoy, velocity o acceleation of the Sun is completely ielevant, since the obit, velocity and acceleation of the Sun s cente of mass ae vey small compaed with the coespondent inematic paametes chaacteizing motions of the planets. Coectly defining the initial conditions is all that mattes. Fo that eason "Copenican" model is the adopted hee: the planets move at an aveage distance, with aveage angula velocity aound the Sun.

4 Loops in Suns's obit 129 Taing the planets configuation on 21 ach 197 [5] as initial conditions and the model in which all planets, excepting ecuy (its obit is not stabilized in the sola system s invaiant plane yet) move unifomly aound the Sun,.ajanov [9] obtained, pactically, the same fom of the Sun s path as Jose had. This tajectoy coves the time inteval of fifty yeas: Since, in the inetial plane of efeence must be m = m v = m a = and m = 1 = 1 = 1 ( ), & + = + + =, + & R + m v m v R m a m a R = + m = 1 = 1 = 1 it follows out that m R = 1 =, v = = 1 & m R and a = = 1 & m R, whee = Thus, position, velocity and acceleation of the Sun s cente of mass ae completely defined by the elative positions, velocities and acceleations of the planets with espect to the Sun. Fig.2 shows the obit of the Sun fom to It, as mentioned, coesponds to the obit that got Jose /1/, but is substantially diffeent fom the one given by Bish /2/, as the latte had wongly assumed that initial conditions do not affect the fom of a path. The contou of the Sun is given as efeential and the dot mas denote yeas. Concening the influence of the inne planets on the fom of the path, Jose was ight: it is quite negligible. Thei influence would become visible only if a pat of seemingly smooth path was magnified thousand to ten thousand times. The influence of the inne planets actions is fa moe evident when speed and acceleation of the Sun ae consideed. The maximal distance fom Sun to the cente of mass of C m is obtainable when all planets ae lined up on the same side of the sta and the minimal, when Jupite is on one side and all the othe planets on the othe side of the Sun, in the same diection. Of couse, chances fo exactly such alignments of the celestial bodies ae educed to zeo /11/. The anticipated path of the Sun ove the next 2 yeas is epesented in the Figue 3. It is situated within the cicula outline with the diamete at least twice size of the Sun s. Again, the Sun s dis is given fo compaison. Povided that the sola system is stable, Sun is going to move in this egion fo etenity, neve eiteeiting the same path. = 1 m

5 13 ILUTIN ARJANOV 1,5 Sun' s Path : ,,5 mill.m., -1,5-1, -,5,,5 1, 1,5 -,5-1, -1,5 Fig.2. Sun s Path fom ORBIT 6. VELOCITY Sun's velocity fom to is epesented in the Figue 4. Hee we can see again that the oute, giant planets, especially Jupite and Satun have dominant influence on this inematical quantity. When these two planets ae in conjunction (~ evey 2 yeas), the speed of the Sun is maximal, and when in opposition, the velocity is minimal, about. 6. ACCELERATION The acceleation diagam fom to is given in the Figue 5. It shows much moe iegulaities and oughness than the pevious diagam and epesents, of couse, a measue of the esulting planets gavitational attactions. No need to say that the peas in this diagam coespond to the diffeent conjunctional combinations of two o moe planets.

Loops in Suns's obit 131 2 1,5 1,5-2 -1,5-1 -,5,5 1 1,5 2 -,5-1 -1,5 v = = 1 m R & -2 million m Fig. 3. Sun s Obit Ove 2 Yeas.

6 Loops in Suns's obit ,5 1,5-2 -1,5-1 -,5,5 1 1,5 2 -,5-1 -1,5 v = = 1 m R & -2 million m Fig. 3. Sun s Obit Ove 2 Yeas J & Sat Oute Planets 12 1 m / s Conj. J S 6 4 Opp. J S Sun Velocity : yeas Fig. 4. Sun s Velocity fom to

7 132 ILUTIN ARJANOV If all the planets fell in the same diection, on the same side of the Sun, theefoe, when they wee all in conjunction, the Sun would have a maximal acceleation. When Jupite is on one side and all the othe planets on the othe side of the Sun, in the same diection, acceleation of the Sun is minimal. a = = 1 m R &,3,25 J & Sat Oute Planets,2,15,1 2 µ / s Opp. J S Conj. J S,5 Sun Acceleati on : yeas, Fig. 5. Sun s Acceleation fom to Table: Aveage Sun - Planets & aximal Planet- Planets Inteactions Sun e V E a J S U N Sun 3,13 13,2,52,39 1,,9,34,16 e 3,13,,,,,,, V 13,2,,,,,,, E,52,,,,,,, a,39,,,,,,, J 1,,,,,,4,, S,9,,,,,4,, U,34,,,,,,, N,16,,,,,,,

8 Loops in Suns's obit SOLAR SYSTE BODIES GRAVITATIONAL INTERACTIONS Pevious diagam will be moe undestandable if one loos at the following Table of the of the mean Sun - planets and the maximal planet- planets inteactions. The exposed table contains two inds of data. The fist ow and the fist column epesent, that is, the aveage Sun planets inteactions. These foces ae aveage because of the adopted "Copenicus " and not of the "Keple s" model. Of couse, the Sun-Jupite inteaction is the geatest one and it was taen to be the efeential: its value is 1. An inteesting fact is that the second one is the Sun - Venus inteaction and that the Sun-Eath is of the same ode of magnitude as the Sun-Satun inteaction, although the Satun s mass is 95 times geate than the Eath s. All the othe values in the Table 1 epesent : maximal, that is, conjunctional inteactions of the planets. In the ange of the hee adopted five digit pecision, the only existing is the one between Jupite and Satun. It is evident that motions of the inne planets, especially that of Venus and Eath ae the main cause of the fluctuations aound a smooth acceleation cuve epesenting the influence of the oute planets, and above all, the influence of Jupite Satun.. CONCLUSION The Sun and the planets in the pape wee simulated by the model of stable, isolated system of the point mass paticles, moving unde mutual gavitational inteactions in the aveaged Laplacian plane. Shape of the sola path is entiely diffeent fom the obits of the othe bodies moving aound it. Namely, the Sun moves along the path in the fom of chain consisting of lage and small loops. This chain is situated within the cicula outline with the diamete appoximately twice as lage as the Sun s is. Unde supposition that the sola system is stable, the Sun is going to move along it, in the same egion, fo etenity, neve eiteeiting the same path. Then the velocity and acceleation diagams of cente of mass of the Sun wee given. At last, fo bette undestanding of the obtained esults, Table of the of the mean Sun - planets and the maximal planet- planets inteactions was contibuted. REFERENCES. [1] P. D. Jose: Sun s otion and Sunspots, The Astonomical Jounal, Vol. 7, N. 3, [2] R. Bitsch: The Sun oves! attn@sunobit.net, 23. [3]. ilanović: Osnovi nebese mehanie, Naučna njiga, Beogad, 19. [4] T. Anđelić: Uvod u astodinamiu, atematiči institut, Beogad, 193. [5] 197 Yeaboo of Astonomy, Ed. P. ooe, W.W.Nothon&Company Inc, New Yo. [6] K. Coswell: Planet Quest, Oxfod Univesity Pess, Oxfod, [7] Heliocentic Ephemeis 2 Achena UK io. [] ajanov.: Gavitational Resonances, Bull. Aston., Belgade, 156, 9 19, [9] ajanov.: The Sun s Obit and Centes of Gavitation of the Sola System, Lectue on the athematical Institute SANU, Depatment of echanics, ach 25.

9 134 ILUTIN ARJANOV [1] ajanov.: On the Cause of Resonant otions of Celestial Bodies, Facta Univesitatis, Seies: echanics, Automatic Contol and Robotics, vol. 4, N o 17, 25. [11] ajanov.: Pythagoas Numbes and Quasipeiodicity of the Sola System, Zboni adova Gadjevinsog faulteta u Subotici, 26. [12] ajanov.: One Solution of the Newton s Thee Body Poblem, Lectue on the athematical Institute SANU, Depatment of echanics, Dec. 2. PETLJE U SUNČEVOJ PUTANJI ilutin ajanov Poed tanslatonog etanja, otacije oo centa lečnog Puta i oo svoje ose, Sunce obavlja i elativno etanje u Laplasovoj avni Sunčevog sistema. Na to etanje je uazao još Njutn, u svome delu Pincipia. Obli Sunčeve putanje bitno se azliuje od obita ostalih tela oje oo njega uže. Sunce se, naime, eće duž putanje u fomi lanca sačinjenog od veliih i malih petlji. Taj lanac je smešten u ovi užne ontue čiji je pečni pibližno dva puta veći od pečnia Sunca. Uz petpostavu da je Sunčev sistem stabilan, Sunce će se večno etati duž njega, u istoj oblasti, niada ne ponavljajući isti put. Zatim su dati dijagami bzina i ubzanja centa mase Sunca. Na aju je piložena tablica posečnih inteacija Sunce planete, ao i masimalnih inteacija među planetama, s tačnošću od pet bojčanih jedinica. Ona je doba poazatelj zbog čega na obli obite, ao i na dijagam bzina najviše utiču spoljne, džinovse planete, do na izazito neujednačenost fome dijagama ubzanja utiču, pe svega, unutašnje planete. Ključne eči: petlje u Sunčevoj obiti, bzina i ubzanje centa mase Sunca Submitted on Apil 29, accepted on June 212 DOI : 1.229/TA12S1127 ath.subj.class.: 7F15; 72;799;

Determining solar characteristics using planetary data

Determining solar characteristics using planetary data Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation