DEDUCTION OF THE GRAVITY LAW AND QUANTUM MECHANICAL MODEL OF DISCRETIZATION IN THE MACROSCOPIC GRAVITY SYSTEM FROM SOLAR SYSTEM DATA

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1 st Iteratioal Cogress of Serbia Society of Mechaics, 0-3 th April, 007, Kopaoik DEDUCTION OF THE GRAVITY LAW AND QUANTUM MECHANICAL MODEL OF DISCRETIZATION IN THE MACROSCOPIC GRAVITY SYSTEM FROM SOLAR SYSTEM DATA Aleksadar S. Tomić VI Gymasium, Milaa Rakića 33, 000 Belgrade Peoples observatory, Kalemegda, 000 Belgrade Abstract: The aalysis of the historical process of comig to kow the solar system from Koperik s measuremet of the plaetary distaces ad periods, ad Kepler laws of plaetary motio is applied for a recostructio of the path for a direct deductio of the law of gravity iteractio (differet from Newto s procedure) ad quatum mechaical model of the mass discretizatio i macroscopic gravity systems (Bohr-Sommerfeld s solutio), too. We used i cosideratio the ideas of Borelli, Kepler, Huiges, Boscovich, Titius ad Bode ad oe s ow cosideratio of the secod Kepler s law as a possible discrete distributio determied by atural umbers. The coclusios were: gravity costats by ature is kietic, kietic cocept demostrated the structural aalogies betwee gravity ad electrical force law with the same quatizatio rules. We extracted the possibility of uificatio of gravity ad electric force i kietic formalism. Structural aalogy of electric ad gravity iteractio, ad effors for uificatio of these physical fields, i this aalysis obtaied a importat support uificatio is possible, but o kietic level, istead used dyamic level. Key words: Newto s gravity law, Bohr s quatizatio, Solar system, epistemology, uificatio of forces. Itroductio Koperik published i the year 543 rd De revolutioibus orbium coelestium. His measuremet cofirmed cocept of world itroduced by Aristarch approximately 8 cetury ago, i which plaets revolvig aroud the Su, istead aroud the Earth. Trigoometric method has bee used by solvig the plae triagle for distace estimatio, simultaeously plaet to the Su ad plaet to the Earth. I the itervals bordered with accuracy of the measuremet Koperik obtaied permaet distaces to the Su ad very variable distaces to the Earth. With regard to the cosequeces produced o the theological doctries, usually were thought o the Koperik s astroomical revolutio [], or o the overthrow by which the Earth removed from the ceter of the world. Numerical data obtaied by Koperik presets also the same big surprise. Differece i the measured data ad adopted evaluatios util Koperik [] were remarkable. Solar system were much bigger tha it were assumed. Koperik itroduced a ew measure for distace astroomical uit, i absece of exact value for Earth s radius. Also, estimated sideral plaetary periods from itervals betwee oppositio (for exteral plaets) ad maximal elogatio (for iteral plaets), itroducig implicit a hypothesis o the summig of 63

2 A.S. Tomić: Deductio Of The Gravity Law Ad Quatum Mechaical Model Of Discretizatio From Solar System Data the agular velocities as the vector, what is the most ofte igored. As the result o this way were obtaied the possibility to calculate plaetary circular velocities, with importat detail - faster decreasig of the velocity by icreasig of the distace. Kepler held his attetio o this detail ad o the fact that seasos (determied precisely by solar positio o the celestial sphere) were ot equal i the duratio. He is lookig for cause, ito data for three decades permaetly Brahe s measuremet of plaetary positios, with the accuracy better tha each others. Result (o practically equally log calculatio) appears as Kepler s three law of plaetary motio aroud the Su, published i De harmoices mudi-liber V, 69 [3]. I Astroomia ova sive Physica coelestis [4] he proposed the mass of the celestial body as the origi of the attractio betwee celestial bodies. So were Kepler the first ma which uified celestial physics (plaetary motio) ad terrestrial physics (Galilea kietics). Both were cosidered i the frames of kietics, ad Kepler attributed to the plaets (freely movig through the space, but the most ot o the circles, ad ot uiformly) the same iertia. Lookig for explaatio appears as imperative. Galilei ad its followers regard that trasmissio of the iertia priciple to the plaets were just eough explaatio, ad igored Kepler s discovery. I these times Kepler marked ot that plaetary motio must have cetrifugal force as a compesatio i equilibrium. Borelli were the first which as scietist brig out the questio a quo movetur plaetae ad why plaets stad o these distaces.(a cetury latterly it repeat Charles Boet. [9]) He cosidered [5],[6] this problem ad gives attetio to the missed compesatio. Cosequetly followig Aristotelia method coectio of the cosequece ad cause, Borelli cocluded that i the stable circular motio must be the equilibrium. He foud geometrical explaatio for the existece of elliptical orbit (by coical sectios), cosidered problem of the stability of the Solar system as the equilibrium of the cetrifugal ad cetripetal force (usig Galilea method of reductio celestial motio o the mechaical examples from terrestrial physics of fluid ad quasi magetic iteractio). Explicit coclusio were that it what movig is iteral priciple, ot itelliget exteral while. Celestial mechaics were fouded o the priciple of coservatio the motio. Borelli had atteded that startig force i the solar system is costat. O the oscillatig pedulum he demostrated that the motio by actio of the costat cetral force at bigger distace were slowly, how it has foud Koperik for plaetary motio. Accordig to Galileo the gravity betwee plaets ad the Su were active by origi of the plaets, ad plaets followed to fall to the Su up to the momet i which achieved its actual velocities. (Freely fall to the Su is possible for quiet plaet, ad i the ay sese Galileo is right.) He cosidered i terms of his kietics oly liear gravity actio, other words make the differece betwee this ad circular motio i the cause, too. Hooke ad Newto geeralized gravity actio to all bodies i the uiverse. Method of cosideratio used by Newto is well kow, ad we wish it repeat ot. We ca ow ask the aswer to the questio: Is possible the riches of cotets i Koperik s data, Kepler s laws ad Borelli s ideas utilize, ad obtai somewhat moreover? Specially, we bare o mid explaatio o the ecessary trasit from cosequetly cotiuity to discotiuity, i cosideratio by Boscovich of stable orbit i system of few (three) body with gravity force as iteractio [7],[8], ad Titius - Bode s rule for discrete plaetary distace [9]. With a small appedix we thik that it were possible, o followig way.. Iitial data ad assumed kowledge Our cocept is very simple. I the measuremet were foud the source of motile force by Galileo i the Earth, by Koperik ad Kepler i the Su, i.e. i the material bodies. Before tha it were made Newto, Kepler brought out the hypothesis o the mass of cetral body as the cause of plaetary motio. For stable plaetary motio the ecessity of velocity ad impulse coservatio, permaet presece of the cetral force as the cause of motio, ad the equilibrium of opposite 63

3 A.S. Tomić: Deductio Of The Gravity Law Ad Quatum Mechaical Model Of Discretizatio From Solar System Data orieted forces as the coditio of the stability itroduced Borelli. Newto accepted the iertial priciple of Galileo, itroduced hypothesis of the equivalece of the gravity acceleratio o the Earth s surface ad the cetripetal acceleratio to the Moo at the smallest possible distace - the Earth s surface as the cosequece of the same origi. He defied the acceleratio as the ratio actig force /mass of exposed body ad the equilibrium of actio ad reactio implicit as vectors. O the examples of two orbital motio, first the Earth aroud the Su, ad secod the Moo aroud the Earth, supposed the same origi of attractio i both cases (or equivalet gravity costats, as alterative), usig Picard s [0] measuremet of the Earth s radius (669-67), the third Kepler s law ad defiitio of cetripetal acceleratio, Newto derived the gravity force as fuctio mass of bodies ad distace. By retroactive cosideratio obtaied also the third Kepler s law i the form which determies the sese of the Kepler s costats as the multiple of gravity costats ad the mass of cetral body - if agular velocity is used: ω r 3 = γ M. Just it is cotets hypothesis of Kepler, for which Kepler were ot have had the time to make test of statemet. How big task it were, show Newto i his Pricipia. Kepleria ad Galilea physics stay kietic, other words defied for mass uit. The coservatios laws for isolated systems were formulated latter, as a geeralizatio of Newto s mechaics Our cosideratio of the secod Kepler s law as possible discrete distributio determied by atural umber (Table ) for the ext cosideratio were eedy too. A simple procedure i coectio of these facts gave very iterestig results. 3. Deductio of quatum mechaical discretizatio We start from the third Kepler s law i the form: v r = cost, () also from the secod Kepler s law i the evidetly discrete formulatio, followig from data give i Table : vr = ( vr ). () By itroductio of fuctioal multiplier f for the velocity discrete trasformatio by distace chage, which is ukow, if Eq. divides by Eq.: v r ( vr ) ( v f) = = v = v f. vr ( vr ) (3) ad dividig Eq. by squared Eq.: v r ( vr ) ( v f) f = = = ( vr) ( v r ) r r. (4) Now isert v = v f ad r = r / f i the third Kepler s law: v r = cost = v r = ( v f) ( r ) = v r f. (5) f It is satisfied oly for: f = /, (6) i.e. velocity ad stable distace trasformatio possess the form: v = v /, r = r. (7) Direct applicatio of formulae (7) o plaets i solar system, by aalogy with atom, is ot effective for Mercury order umber = is ot adequate. But, for umbers obtaied from quatizatio of agular momet (Table) it is just adequate. 633

4 A.S. Tomić: Deductio Of The Gravity Law Ad Quatum Mechaical Model Of Discretizatio From Solar System Data km plaet r (AU) v( ) s AU km r v ( ) v km ( ) s s Pluto 39,53 4,73 86,98,9 3 0,78 30,955 Neptue 30,0 5,43 63,50 4,74 7 0,048 6,97 Ura 9,8 6,80 30,68 3, 0,458,536 Satur 9,555 9,65 9, 46,56 5 0,00 5,75 Jupiter 5,03 3,06 67,95 85,8 0,0,3 Asteroids,709 8, 49,06 63,99 8 0,086 8,086 Mars,53 4,3 36,75 9,3 6 0,058 6,069 Earth,000 9,79 9,79 443,7 5 0,089 4,96 Veus 0,73 35,0 5,3 63,0 4 0,74 4,8 Mercury 0,389 47,87 8,5 45,77 3 0,055 3,059 ** 0,655 73,, 680,58 * 0,044 46,44 6,06 07,34 Table : Quatizatio of agular momet ad eergy of mass uit i the Solar system Here we have additioal argumet. If plaetary distaces express via geometric progressio, kow as Titius-Bode rule [9] the average distace ratio of successively plaets is equal []: r / =.695 r. (8) From quatizatio of agular momet for mass uit ad third Kepler s law ca be calculated distace ad velocity of the first two (missig) plaets, as: v r = γ M = cost (9.) v = ( v r) / ( v r), r = ( v r) / v, (9.) v = ( v r) / ( v r), r = ( v r) / v. (9.3) Obtaied values are icluded i Table. It is obvious that distace ratio icreasig if plaets comig earer to the Su: r / =.695 r, r 4 / r 3 =. 859, r 3 / r =. 350, r / r = (0) Expressed with first possible plaet distace r, determied by quatizatio of agular momet, istead of real earest plaet, we have beautiful agreemet: r = r =. 999 r (.) r3 =.350 r = 9.4 r = r (.) r4 =.859 r3 = 7.9 r = r (.3) r5 = r, etc. (.4) We ca coclude that the same formulas are really valid for electrical force i the atoms ad gravitatioal force i plaetary system, with oly differece that here exist vacat positios. The formulas (7) are well kow, but here obtaied from kietic cocept, while Bohr i his atomic model derived from dyamical cocept applied to the electric force. Electrical force law, aalogous i form with gravity force law, were obtaied experimetally a cetury after appeared Pricipia. For differece related to Galileo ad Kepler which start from measured data, Bohr used much more mathematical model kow as Newto s cocept of mechaics, ad coservatio laws for system i weak iteractio with eviromet. Applyig formulae for kietic ad potetial eergy ad coservatio of eergy, valid because of big distace to earest stars, i.e. good isolatio of solar system, it ca be obtaied formula for specific eergy (eergy per mass uit) of each plaet i kietic cocept, too: E E 634

5 A.S. Tomić: Deductio Of The Gravity Law Ad Quatum Mechaical Model Of Discretizatio From Solar System Data E E = (, () m m ) ad it s obvious from last colum of Table. We obtaied formulas for velocity, distace ad eergy (pro mass uit) i kow form, but here all quatity were kietic, ot dyamic. Other words all ca stay kietic. It is origial procedure, dyamical procedure is upgraded aex. Why is it importat? Gravity quatizatio ca be realized oly i this case. For electrical quatizatio it is preferable, ot critical. It is oe from reasos why through over eight decades all attempts by uificatio of electrical ad gravity force were ot gave result. A remark. The first useful attempt of coectig formulas for electric ad gravity iteractio made Ferado Saford, just after Bohr s model were accepted []. He attempt derive Bohr s equatios from data for plaets usig oly secod ad third Kepler s law, ad foud for plaetary distaces (r) ad periods (T) relatios: r / T = Q, (3) r = Q = 0.048[ AU ], (4) startig with =3 for Mercury. This were equivalet to our formulas () ad (7) for agular momet of mass uit discrete distributio (what Saford were ot recogized), ad for distaces. For each plaet Saford lookig for formula givig eergy, but ot for mass uit. How plaetary masses were differet, obtaied result were throw of, where editor ad reviewer were ot poited out that divisio by plaetary mass make all regular, but ow as eergy of mass uit ( /T =ν frequecy, k = cost ): E / m = k ν. (5) This attempt Saford made i the best momet. But, coceptual ommissio by author ad ot eough attetio by reviewers resulted as failure. (Saford's paper author obtaied by kidess of dr Siiša Igjatović, i this momet i Toroto, Caada, after ivestigatio were fiished.) 4. Gravity law Itroduced hypothesis o the mass of cetral body as the cause of plaetary orbital motio showed as right. Oly this quatity is the same i the iteractio with each plaet. I the third Kepler's law, the costats which as multiplyer of the mass equalize quatities ad uits, is gravity costats, itroduced by liear proportioality: v r = cost = γ M. (6) Cetripetal acceleratio is equal to the ratio of squared velocity ad distace. If simple divide by r obtaies acceleratio to the cetral body: v / r = γ M / r = a c, (7) eough for descriptio of plaetary motio. Coectio of cetripetal acceleratio ad plaetary mass ito gravity force made Newto. Usig here oly Newto's the secod priciple it follows directly the gravity law: F = m ac = γ M m / r. (8) The previous cosideratio gave a possibility to defie gravity costate i solar system as Kepler's costate o mass uit of cetral body (the most correct - at miimal possible distace from the Su) as follows from Eq.: v r / M = v r / M = γ. (9) How we see, it presets also kietic quatity, ad gravity costats realy belogs to Kepler's cocept. Mass stay as a cause of motio, but determied (or harmoized) by kietic parameters. I the pairs of body Earth-Moo ad other system plaet-satelite, the same umerical value were 635

6 A.S. Tomić: Deductio Of The Gravity Law Ad Quatum Mechaical Model Of Discretizatio From Solar System Data obtaied, so that ca be word o the uiversality of this costats. Dimezioaly: 3 [ γ ] = kg s m, what ca be expressed also as claim that a multiple of mass ad squared time is iverse equivalet (o the uit circle) to the space, i.e. to volume, if the gravity costats treated as dimesioless quatity. I the same status is Coulomb's costats, defied pro uit of electricity of the cetral body. O this way it ca be word o the mass (or electricity) as origi of the space, ad the time as the measure of the chage (or evolutio) i the distributio of the mass / electricity. These categories were ot so clea defied i published papers. 5. Coclusios The historical process of how to kow the solar system from Koperik, Kepler, Huiges ad Newto is permaet preset part of scietific heritage i cotemporary educatio. With additio of Borelli idea's (ed of 6.cetury), Kepler's hypothesis o the cause of orbital motio, ad Boskovic's deductio of the trasformatio of the cotiuity ito discotiuity i gravity systems [7],[8], it appears as cosequetly logic coected with our cosideratio of the d Kepler's law as the discrete distributio determied by atural umbers - other words, with fractal structure of solar system [3]. By usig methaphysic most strog arms - logical deductio, we show that i the last etape existece of methaphysics i sciece were omited obvious possibility for short deductio of (Newto's) gravity law ad Bor- Somerfeld's solutio for discretizatio i the macroscopic gravity systems. The first atempt of coectio obtaied formulas for electrical ad gravity field, made by Saford i the year 9 were estimated as failure. Structural aalogy of electric ad gravity iteractio, ad effors for uificatio of these physical fields, i this aalysis obtaied a importat support uificatio is possible, but o cietic level, istead o used dyamic level. Refereces [] Koyre A., The astroomical revolutio, Herma (Paris) Methue (Lodo)- Corell Uiv. Press (Ithaca-NewYork), 973, [] Patricius F., Nova de uiverses philosophia / Nova sveopća folozofija, Ferrara / Zagreb, 59/97. [3] Kepler J., De harmoices mudi, Liber V, Wuetemberg, 69. [4] Kepler J., Astroomia ova sive Physica coelestis, Wuetemberg, 65. [5] Borelli J.A., Theoricae Mediceorum plaetarum ex causis physicis deductae, Flores, 666. [6] Borelli, J.A., De motioibus aturalibus a gravitate pedetibus, Bologa, 670. [7] Boskovich R., Theoria philosophiae aturalis, Veetia, Teorija prirode filozofije, Zagreb, 763/974. [8] Tomić A.S., p i Grujić P.- Ivaović M. (Eds) Epistemological problems i the sciece, (Lex uica virium i atura of Rudjer Bošković) I serbia, IKSI, Beograd, 004. [9] Tomić A., VASIONA, 4, -3, (Why plaets are where they are; I Serbia), 993. [0] Picard J., Mesure de la Terre, 67. [] Tomić A.S., Flogisto, 7, 5-68 (Plaetary distaces as the golde sectio; I Serbia), 998. [] Saford F., Popular Astroomy, 9, ( Quatum equatio i the Solar system), 9. [3] Tomić A, S., Proceed. 8 th SAUM, p. 8- Fac. Mech. Egi. Belgrade, (Fractal hierarchical structure i solar system arragemet),