New Equation For Describing Time Dependence of Moon s Orbit Radius

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1 Nw Equation For Dscribing Ti Dpndnc of oon s Orbit adius ikrajuddin Abdullah Dpartnt of Physics, Bandung Institut of Tchnology Jalan Gansa 10 Bandung 4013, Indonsia IBE S&T Institut Jalan Sbrani 19 Bandung, Indonsia Abstract I drivd an quation to dscrib th dpndnc of oon s orbit radius around th Earth. I obtaind th radius changs with ti according to a scalling quation, t /. sing th quation I hav bn abl to prdict accuratly so quantitis that hav bn wll accptd. I. INTODCTION Th oon orbits th Earth at an lliptic orbit with th apog of 40,400 k and th prig of 36,600 k. Th avrag radius of th oon s orbit is 384,399 k [1]. It is blivd that th oon has bn cratd or that 30 a aftr th start of th Solar Syst []. Susprisingly, nw findings hav shown that th oon s orbit radius is gtting largr at a rat of about 37.8 /yar [3,4]. Th intrsting qustion thn, is thr any forula to dscrib th chang of oon s orbit radius with ti? How is th rat of chang of th orbit radius at th past and th futur? Th purpos of this short work is to driv an quation that dscribing th dpndnc of th oon s orbit radius with ti. Vry surprising that th drivd odl is abl to prdict diffrnt quantitis that wll accptd at th prsnt. II. ODELLING Sinc th Earth contains fluid coponnt (ocan and atosphr), th fraction of which ight b asily displacd by oon or Sun gravitation, th shap of th Earth, spcially on its surfac, will sligthly vibrat continuously (assud to b rando). This chang will disturbat slightly th strngth of th Earth s gravitational forc on th oon to iply th 1

2 distanc of th oon s orbit radius will chang slightly. Th rat of th oon s orbit radius chang will dpnd on th instantanous oon s orbit radius. As th oon s orbit gts larg, th fffct of Earth ass variation of th gravitational forc xprincd by th oon dcrass so that th rat of oon s orbit chang will dcras too. I siplify th odl by assuing that th doinant gravitational attraction xprincd by th oon originats only fro th Earth. Gravitational disturbancs that ight originat fro othr bodis such as th Sun or astroids ar nglctd. W also assud that th fluctuation in th Earth gravitational fild affcts only th orbital otion of th oon and dos not affct th oon s rotational orbit. For odling purpos, I propos th following assuptions. I assu th total ass of th Earth,, can b dividd into two coponnts: th fixd ass and th displacabl ass. Th fixd ass has a constant distanc fro th oon, and th displacabl ass has variabl distanc fro th oon, rando variabl, r r r, and nrgy of th oon du to Earth gravitation bcos r (oon s orbit radius) r with r is a is th Earth s radius. Th instantanous potntial G( ) G r G G r / G G r r (1) Thu th avrag potntial nrgy of th oon bcos G G G r r 3 () Sincr r is rando thn r 0 and Eq. () bcos G G r 3 (3) Thrfor, th prsnc of randoly Earth s ass displacnt rsultd in rduction in th oon potntial nrgy as uch as

3 G r 3 (4) This rduction in potntial nrgy can b copard to rduction in potntial nrgy du to fluctuation of charg distribution of ato or olculs as dscribd byth van dr Walls forc. In tr of ato whr th attaraction forc is th coulob forc, th rduction in th 6 potntial nrgy satisfis 1/ []. This potntial nrgy is th su of potntial nrgy by slightly sparatd positiv and ngativ chargs fro two atos. For th gravitational forc, sinc thr ar no gravitational chargs, th chang in th potntial nrgy varis according to invrs of qubical distanc. Du to chang in th potntial nrgy of gravitation, thr is a chang in th gravitational forc xprincs by th oon, satisfying d F d 3G r 4 () It is clar that th fluctuatd gravitational forc dcrass if th oon s orbit radius incrass, rsulting in dcras in th Earth-oon forc. sing th scond Nwton law of otion, th gravitational forc fluctuation will gnrat fluctuation in th radial acclration of th oon as d d F 3G 4 r 4 d 3G r (6) To obtain th solution of Eq. (6), lt us tporarily assu so that At (7) d ( 1) At 3

4 Substituting into Eq. (6) on has A 4 ( 1) t 3G r Equalizing powr and factor in both sids w hav 0 A ( 1) 3G r Th first quation rsulting and th scons quation rsulting G 1/ A 1.66 r Thrfor w obtain th ti dpndc of oon s orbit radius as 1/ / G r 1.66 t (8) III. DISCSSION Th qustion is, how to justify that Eq. (8) corrctly dscibs th chang of oon s orbit radius with ti. W hav a data that at prsnt th oon s orbit radius incrass with a rat of around 3 /yar = k/yar (so rport says 38 /yar). Th rat chang of oon s orbit radius is obtaind by diffrntiating of Eq. (8) as 1/ 3/ G r t d (9) W assu that th oon start to rvolv th Earth sinc its cratd for about 4.1 billion yars ago[6]. Starting fro that ti, w assu th oon s orbit radius startd to incras with ti. sing th data d / k/yar and t yars w find / 9 r / G to rsult 1/ r G (10) Thn Eq. (8) can b writtn as / 4.3t (11) 4

5 Again, using th prsnt ag of th oon, w stit th prsnt oon s orbit radius as / 4.37 = 39,0 k. Suprisingly, this valu is xcatly siilar th to prsnt avrag oon s orbit radius of 384,399 k [1] as a prov that Eq. (8) or (11) ight b corrctly dscrib th ti dpndnc of th oon s orbit radius. W can us th Eq. (10) to stiat th ass of th ocan displacd by th oon or th Sun gravitation. W us G = /kg s = k 3 /kg yar. W assud th displacnt of sa is approxiatdwith th high of sa tid, and hr w us th highst sa tid in Nova Scotia of around 16. so that r k [1]. sing Eq. (10) w gt th stiatd kg.th total ass of watr in th ocan is around kg [7,8]. Thrfor, th displacd watr ass accounts for approxiatly 0.0% of th total ass of th ocan. In y opinition, this figur is vry accptabl. Th avrag tidal hight in opn sa is around 0.6 [1]. Thus, th approxiatd displacd watd by th tid is around (4 r 0.6) 1, 000 = kg. Vry surprising that this valu is narly xactly siilar to our stiation for. Although th prsnt odl looks sipl, howvr, w hav bn abl to prdict accuratly th wll accptd data, spcially th prsnt oon s orbit radius and th ass of watr displacd by tid. CONCLSION Th proposd quation has bn abl to xactly stiatd diffrnt quantitis that hav bn wll accptd by scintists. Intrstingly, thr hav bn no fr paratrs usd to stiat such quantitis, instad all partr ar th asurd ons, to indicat that th proposd quation is in th right track. EFEENCES [1] n.wikipdia.org. [] D. J. Stvnson and A. N. Halliday, Th origin of th oon, Philos Trans A ath Phys Eng Sci. 37 (014). doi: /rsta [3] rtrivd Fb 4, 018. [4] rtrivd Fb 4, 018.

6 [] C. Kittl, Introduction to Solid Stat Physics, 7th d., Nw York: John Wily (1996), pp [6]. Barboni; P. Bohnk; B. Kllr; I. E. Kohl; B. Schon; E. D. Young; and K. D. ckgan, Early foration of th oon 4.1 billion yars ago, Sci. Adv. 3, (017). [7] Th Concis Colubia Elctronic Encyclopdia. 3rd d. Nw York: Colubia nivrsity Prss (1994). [8] CC Handbook of Chistry and Physics. Ed.. C. Wast. Nw York: CC Prss, F- 199 (1980). 6

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M