CELESTIAL MECHANICS. Advisor: Dr. Steve Surace Assistant: Margaret Senese

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1 CELESTIAL MECHANICS Ei Ce, Tyle Enst, Minqi Jing, Steve Kuei, Dniel Levine, Piy Mte, Jeemy Silve, Antony Svs, Stefn Tinte, Dmity Vgne, Stepnie Wng ABSTACT Adviso: D. Steve Sue Assistnt: Mget Senese Te gol of tis pojet ws to deive te lws of pysis wi goven te motion of te evenly odies. In ode to do tis, Keple s Lws of Plnety Motion wee deived fom simple pysis equtions nd wee ten elted to lws of geomety. Tese lws nd eltionsips wee ten pplied to te pysil, osevle univese y te use of speil tigonomety. Te elestil spee, te imginy spee wit te Et t te ente, ws used s model fo te osevle univese. All elestil ojets in te nigt sky ppe to exist on te spee. Mtemtil models n e used to pedit te lotion of ny plnet on ny given dy. It is lso possile to onstut sundil wi n tell te time to esonle degee of uy. INTODUCTION Te evenly odies ve een te sujet of study fo millenni nd in tt time ve quied level of unllengility, s sts nd plnets disppeed, eppeed, nd moved oss te nigt sky t egul intevls. Tese distnt ojets ve d pofound effet on mn s ultue nd development s iviliztion; in te mking of lend sesons, in te nvigting of te ses, nd in te fomtion of mytology. Humns ve sougt to ompeend te get ntul foes tt dive te elestil odies. Teoies wee put fot y Plto, Aistotle, Ptolemy, nd Copenius. Howeve, te key ly in te detiled notes of Tyo Be; neve efoe d nyone mde so mny lultions nd eods of te elestil odies. Fotuntely, fte Be s det, is notes went to is young ssistnt, Jonnes Keple. Fom tese eods, Jonnes Keple deived wt e now known s Keple s Lws of Plnety Motion. We now epitulte te fmous nlysis of Jonnes Keple nd its pplition to te el wold. GENEAL POLA FOMULA OF AN ELLIPSE Using Tyo Be s ute mesuements, Keple oseved tt te pt of elestil ody ppeed to e elliptil. Any ellipse wit semi-mjo xis, semi-mino xis, foi t x y (±, 0), nd eentiity e /, s genel eqution: + 1. If te ellipse is tnslted distne to te left so tt one fous is t te oigin, te fomul eomes: ( ) x + y + 1. Pol oodintes in tems of n ngle θ e moe useful fo tking te pt of n ojet out fixed point. Tus, tis eqution, wi is in etngul fom, must e onveted to pol fom. 6-1

2 Beginning wit te oizontl nd vetil omponents of pol funtion: x osθ nd y sinθ, nd te definition of eentiity: e, we n deive te following eltions of nd in tems of te viles nd e: ( 1 e ) Sustituting te omponents into te genel fomul gives: ( osθ + e) sin θ ( + os θ ( e ) + ( e( 1 e ) osθ ) ( 1 e ) 0 Tis is now qudti eqution, so we solve fo : ( 1 e )( eosθ ± 1) 1 eosθ eosθ ( )( ) Te dius must e positive fo te desied eqution, so we tke te positive oot, esulting in ( 1 e ) (1) eosθ ( ) Tus, (1) is te eqution of n ellipse in pol fom wit fous t te oigin. KEPLE S FIST LAW Keple s Fist Lw of Plnety Motion dittes tt ll plnets move in elliptil oits wit te Sun t one fous. To deive tis lw, egin wit Newton s Lw of Gvittion nd is Seond Lw of Motion: Mm F m G emoving te seond mss y division yields te eqution fo te eletion veto s funtion of oitl dius: G M Figue 1. Ellipse wit fous (te Sun) t te oigin. Te nd θ in tis figue epesent te sme quntities in te following deivtion. A genel veto my e oken into its x nd y omponents (Figue 1): x osθ y sinθ 6-

3 Sine eletion is te seond deivtive of position, in tis ontext, te veto omponents e expessed tusly: d x osθ () d y sinθ () By eusive diffeentition, te veto omponents eome: d x d os( θ ) d sin( θ ) os( θ ) d y d d sin( θ ) + os( θ ) sin( θ ) d θ sin( θ ) d θ + os( θ ) Setting tese equtions equl to equtions () nd () espetively yields: d d osθ osθ sinθ osθ d d sinθ sinθ + osθ sinθ d θ sinθ d θ + osθ (3) (3) By, multiplying (3) y os θ nd (3) y sin θ nd dding te two equtions n e isolted: os sin d θ os d θ sin θ + d θ sinθ osθ os d sinθ osθ sin θ θ d θ sinθ osθ d θ + sinθ osθ Afte ddition: d (4) Multiplying (3) y sin θ nd (3) y os θ nd susequently sutting yields: d d sin( θ )os( θ ) sin( θ )os( θ ) sin d sin( θ )os( θ ) sin( θ )os( θ ) + d ( θ ) sin( θ )os( θ ) os ( θ ) sin( θ )os( θ ) sin d θ ( θ ) d θ + os ( θ ) 6-3

4 Afte sutting: d d θ 0 (5) Let p nd sustitute it into (5): 0 p dp p d d dp By integtion: ln p ln + Ten sustitute, te onstnt of integtion, into (4) d 3 (6) 1 Ten letu. Diffeentiting te esulting eqution yields: d du du du du u u u d d u d u d u u du (7) Sustituting (7) into (6) yields: u u d u u 3 enging nd sustituting u fo : d u u By inspetion: u B osθ + C sinθ + Tis ssetion n e oooted y diffeentiting twie. e-sustituting fo u yields: 6-4

5 1 B osθ + C sinθ + ( B osθ + C sinθ ) + 1 (8) equiing peielion (te point in te oit of losest ppo) to e on te positive x-xis mximizes te denominto D of (8): D ( B osθ + C sinθ ) + 1 dd ( Bsinθ + C osθ ) 0 ( Bsin 0 + C os0) ( C) 0 C 0 Tus te funtion is: B osθ + 1 (9) Note tt, G, nd M, e onstnts ( onstnt of integtion, te gvittionl onstnt, nd te mss of te Sun, espetively) nd tus te eqution mtes (1), te genel fomul fo n ellipse. Teefoe te plnets move in n elliptil oit wit te Sun t one fous. KEPLE S SECOND LAW Keple s Seond Lw sttes tt te e swept out y plnet duing its oit in given time peiod is te sme fo ll time peiods (Figue ). In ote wods, te te of nge of e is onstnt. Using te lws of diffeentition nd integtion nd ten sustituting : 1 A da da da d θ Tus, te nge in A ove time is onstnt. KEPLE S THID LAW Figue. Keple s Seond Lw 6-5

6 Keple s 3 d Lw sttes tt te sque of te peiod, T, of plnet s oit is popotionl to te ue of te semi-mjo xis,, of tt oit. da By si lws of integtion nd te ft tt is onstnt: T da da da A 0 0 T 0 da ( T ) T Afte enging: T A da Fom Keple s nd Lw, te deivtive of te e swept out y n oiting ody wit espet to time is. Tus, T A Sustitutingπ fo A: T π (10) Tis eqution is ten squed euse Keple s Tid Lw dels wit te sque of te peiod. 4π T Using te eltionsips etween,,, nd e (te elliptil onstnts): 4 4 (1 e ) T π As n e seen in Figue 1, te mximum vlue of te oitl dius is t + nd te minimum vlue is t. Tus, + B osθ + 1 Tis eqution wi employs (9) is ten used to solve fo t ot te minimum nd mximum 6-6

7 vlues of, wen os θ -1 nd 1, espetively. Tis yields: ( ) min 1 B( ) nd ( + ) mx B( + ) Bot vlues of e sustituted into te eqution fo T. Tese equtions e ten dded nd divided y two. Tis llows te onstnt B to nel fom ot equtions. Agin using te elliptil onstnt eltionsips, te eqution is simplified to yield 3 4π T Tis eltionsip etween T nd 3 is te finl eqution tt Newton deived fom Keple s genel eltionsip etween T nd 3 of plnet s oit, nd sows tt te two e, in ft, elted. APPLYING KEPLE S LAWS TO THE OBSEVABLE UNIVESE elting n Ellipse to its Ciumsiing Cile It tuns out to e extemely diffiult to tk plnet s oit y using ngle θ lone, so new nd moe esily tele ngle E is found y insiing te ellipse in ile. Te new ngle E is fomed fom te ente of te ile to te point on te ile fomed y te intesetion of te ile nd te pependiul line extending fom te x-xis toug te plnets position on te ellipse (Figue 3). A useful eltionsip etween θ nd E is detemined fom Figue 3 using simple igt tingle tigonomety: + osθ os Ε Figue 3. Te ile iumsied ound te elliptil oit. Ten sustituting end simplifying, (1 e ) osε e + 1 osθ osθ + e e + osθ os Ε (11) eosθ 6-7

8 Ten, sine tis eltion is mu moe useful in tems of tngents nd lf-ngles, it is tnsfomed using te fomul fo te lf-ngle of tngent: Ε sin Ε tn os Ε Ε sin Ε tn e + osθ eosθ By onstuting igt tingle wit te two side lengts used to detemine te osine (te djent nd ypotenuse), nd ten solving fo te tid (opposite) side, one finds tt: sin Ε (1 e )(sin θ ) eosθ Ε tn sinθ (1 e ) eosθ (1 + e)(1 + osθ ) eosθ Ε tn Ε sinθ tn osθ (1 e e ) θ (1 + e)(1 e) (1 + e) tn e (1 + e) Ε 1 e θ tn tn e Keple s Lws nd Elliptil Geomety Te finl step in eting ptil pplition fo Keple s Lws is to detemine te position of plnet wit espet to time. Beginning wit Keple s Eqution, wi inludes some new quntity M: M E esin E (1) Tus, dm de de de eos E ( 1 eos E) (13) Diffeentiting nd enging eqution (11) gives: 6-8

9 de (1 e )sinθ (1 + eosθ ) sin E (14) Te tio of te y-oodintes t given vlue of x fo iumsiing ile nd n ellipse is yile given y y ellipse Using Figue 3 nd igt tingle tigonomety: y sinθ ellipse yile sinθ sinθ yile y ile sin E sinθ sin E (15) Sustituting eqution (15) into (14) nd (1) in fo gives: de ( eosθ ) epling wit nd gin using (1) fo fute simplifies to: de (1 + eosθ ) (16) 3 (1 e ) Next, sustitute (16) nd (11) into (13) to podue: dm (17) enging (10) to solve fo in tems of T,, nd nd sustituting it into (17) gives te esult: dm π onstnt T Now integting, t M π + C T Sine M (0) 0, te onstnt of integtion C

10 M t π T el Wold Applition Te next gol is to pply tese equtions to te pysil univese y detemining te vlues fo M (ommonly lled te men nomly ), E (ommonly lled te eenti nomly ),θ, nd. 11 Fo te plnet Et, T dys [1], m [], nd e [3]. Te dte of te peielion in 007 ws Jnuy 3, 0:00 Univesl Coodinted Time [4] o Jnuy 3, 3:00 P.M. lol time. On July 5, 007, t 03 dys (s ounted fom te peielion dte). π (03) M 3.49 dins Using e, tis vlue of M, te ft tt E is in te tid qudnt on July 5, nd (1) gives E dins. Using tis vlue of E nd (11), we find θ to e: dins. (Note: te vlues fo M, E, nd θ e vey simil euse te Et s elliptil oit is nely iul, e ) 11 Finlly, we use eqution (1) to find m. SPHEICAL TIGONOMETY Speil tigonomety is fundmentl tool in te development of te eltionsips involved in elestil menis. Speil tigonomety diffes fom pln tigonomety on mny fundmentl popeties. Using si popeties of speil ngles nd tingles, te speil Lw of Sines nd speil Lw of Cosines wee deived. Tese eltions wee ten uil tools in deiving eltionsips to tk te motion of elestil odies. Speil Lw of Cosines Figue 4. A speil tingle pojeted onto plne yields pln tingle. 6-10

11 To deive eltionsip etween te sides nd ngles of speil tingles one n use known eltionsips out pln tingles, inluding te Pytgoen Teoem, Lw of Cosines, nd Lw of Sines. One must tke spee wit tingle on te sufe nd ple te spee on plne, tngent to one of te veties (in tis se, point C) of te speil tingle. One n ten extend te line etween te ente of te spee nd point A until it intesets te plne t point A. Doing te sme fo point B, tee now exists pln tingle A B C. By extending te dius OC tee now exist two igt tinglesoc A nd OC B (Figue 4). Using te fou pln tingles OC A, OC B, O A B, nd A B C, one n use te foementioned teoems nd lws to detemine eltionsips etween te sides in tems of te speil viles A, B, C,,,, nd. Te ngles of te tingles n e edefined in tems of te tey sweep out s: S θ wee S is te lengt, is te dius, nd θ is te ngle tt sweeps out te. Tus: A OC B OC Te tngents of OC B nd OC A, espetively wee used to find: ' tn (18) ' tn (18) Te Lw of Cosines s pplied to tingle A B C is: + Afte sustituting (18) nd (18) nd enging: ' tn + tn tn os C tn osc Hypotenuses of te igt tingles OC A ndoc B, espetively: x y + ' + ' 1 + tn 1 + tn se se (19) (19) Te Lw of Cosines s pplied to tingle Δ O A B is: 6-11

12 x + y xy os Afte sustituting (19) nd (19) nd enging: ' se + se se se os One n equte te two expessions fo nd use te identity, 1 + tn θ se θ, to yield: tn + tn tn tn osc tn + tn se se os Fom simple lgei mnipultion one gets: tn tn osc os se se Afte ewiting ll tigonometi funtions in tems of sine nd osine: sin sin osc os os os 1 os os Simplifying tis, one gets te speil nlogue fo te Lw of Cosines os os os + sin sin osc (0) Speil Lw of Sines One one s te eqution fo te Lw of Cosines, one n deive sin C fom os C using te identity os θ + sin θ 1. Stting wit (0) fte lgei mnipultion: os os os osc sin sin Using te Pytgoen Identity sin θ 1 os θ to eple os C wit sin C, yields: 6-1

13 sin C 1 os C os 1 os os os + os sin sin os Afte squing ot sides nd simplifying: sin sin os sin C + os os os sin sin Afte fute sustitution nd simplifition: sin + sin + sin + os sin C sin sin Dividing ysin yields: sin sin C sin + sin sin + sin sin os + os os sin os Te igt side of te eqution is symmeti in,, nd, teefoe it is onstnt fo given tingle nd one n equte te tee tems s: sin A sin B sin C sin sin sin os os os Fute simplifition yields te Speil Lw of Sines: sin A sin B sin C sin sin sin (1) Ptil Applition Distne Between Two Cities 6-13

14 An pplition of te Speil Lw of Cosines is finding te sotest distne etween two ities on te Et given tei ltitude nd longitude oodintes. One n ete speil tingle wit veties t te two ities nd pole (Figue 5). Te ngul lengt of te side etween given ity nd te pole is te omplement of te ltitude. Tis gives te vlue of two djent sides. Te ngle etween tem is te diffeene in te longitudes of te two ities. Given two djent sides nd n inluded ngle, one n find te tid side, te distne etween te ities, using te Speil Lw of Cosines. Tis will yield te ngul lengt; te line distne n e found y multiplying te ngle y te dius of te Et. An exmple would e finding te distne etween New Yok City nd London. Te oodintes of New Yok e N nd Figue 5. Te sotest distne etween two ities is te get ile onneting tem W, nd tose of London e N nd 0 07 W. Te diffeene in longitudes would e Wen te tingle is eted s ove (Figue 5), one gets one side to e 49 0, note side to e 39 30, nd te inluded ngle to e Using (0), one gets te inluded side to e.8834 dins, nd wen multiplied y te dius of te Et (3963 miles) te distne omes out to e 3501 miles. elting θ, λ, nd ω Te Univesl efeene As te oits of te plnets e diffeent, one must use univesl efeene fo ny lultions tt need to e done. Te Univesl efeene Line is te line pssing toug te Sun nd pointing to te onstelltion Aies. Lmd (λ) is defined s te ngle etween te Univesl efeene nd te line onneting te entes of te Et nd te Sun t ny given time. Tet (θ) is defined s te ountelokwise ngle fom te peielion to te Et wit te Sun s te vetex. In Figue 6. Digm of te Univesl efeene Line wit espet to Et nd its oit 6-14

15 ddition, omeg (ω) is defined s te ngle etween te Univesl efeene nd te peielion (Figue 6). Oseving te eltionsip etween λ, ω, n, one finds tt: λ θ + ω π Te Celestil Spee Fo te puposes of stonomil lultions te univese n e modeled s elestil odies on te sufe of n enomous elestil spee enteed t te Et. Te elestil poles e ligned wit te Not nd Sout Poles of te Et. Similly, te elestil equto is get ile onenti wit te equto of te Et. Points on te elestil spee my e loted in tems of two quntities: igt sension, α, nd delintion, δ. Tese two e te elestil equivlents of teestil longitude nd ltitude espetively. Howeve, unlike longitude, te igt sension is mesued in tems of ous, wit 4 ous equivlent to 360 degees. Tis ngle is mesued ountelokwise wit espet to te Univesl efeene Line. Futemoe, s on te elestil spee e denoted y te entl ngle tt sweeps out tt. Tougout te ouse of ye, te Sun tvels on get ile, known s te elipti, wit mximum delintionε 3.5. Tis mximum delintion is equivlent to te Topi of Cne on Et. One n lso wite n eqution elting delintion δ to λ ndε. Applying (1) to te igligted speil tingle in Figue 7, one finds tt: π sin sin ε sinδ sin λ Figue 7. Te Celestil Spee Afte simplifying: sin δ sinε sin λ Applying (0) to te igligted tingle, one finds tt π os λ osα osδ + sinα sinδ os Figue 8. Digm of Sunise wit espet 6-15 to te elestil spee

16 Afte simplifying: Ptil Applition os α os λ os δ Sunise Times Using tese speil tigonometi lws on te Celestil Spee, one n detemine te ppoximte time of sunise. Let φ e te ltitude of teestil oseve. Using te speil tingle tt is igligted in Figue 8, one n use te Speil Lw of Cosines to set up eltionsip mongφ, δ, nd H, te ngul lengt etween te Sun s igest point in te sky nd sunise: π π π π π os os φ os δ + sin φ sin δ os H () π Using te ft tt osine nd sine e out of pse y, () n e ewitten: 0 sinφ sin δ + osφ osδ os H Afte simplifition: os H tnφ tnδ (3) In ode to test te model fo lulting sunise times, te sunise time ws detemined fo ptiul dy, July 30. Te delintion of te Sun fo tis dy ws [7], nd te ltitude of te lotion wee te sunise ws eoded ws 41. Using te vlues fo delintionδ nd ltitude φ in (3) we found vlue fo H. H is te ngul mesue etween te Sun s igest point in te sky nd sunise. Te Sun s igest point in te sky is nomlly noon, ut July 9 flls duing te dyligt svings time peiod so te Sun s igest point ous t 1:00 P.M. insted. Tis vlue is n ngul mesue tt must e onveted into ous. Te onvesion fto is ous. One onveted, te vlue must e sutted fom 1:00 P.M. os H tnφ tn δ os H tn 41 tn18 40' H os 1 [ tn 41 tn18 40' ] H Conveting tis into ous, one gets: H ous H 7 ous, 8 minutes, 0 seonds Afte sutting tis fom 1:00 P.M. te ppoximte sunise time is 5:51 A.M. CONSTUCTING A SUNDIAL In ode to onstut n ute sundil, te ngle of te sdow st y stik evey ou d 6-16

17 to e lulted. Tese lultions wee pefomed using model of dome to epesent te sky in te viinity of te stik nd te stik tilted t n ngle of ϕ (equl in vlue to te ltitude t wi te stik is onstuted) towds te not (Figue 9). In ode to e le to te te sunys pts to te gound nd detemine wee te sdow of te stik will e t e ou, te digm ws plotted onto oodinte system, wit te x nd y xes long te plne of te Sun s pt, nd te z xis long te stik. Te Sun ises nd tvels oss te sky, eing its igest point t 1:00 P.M. due to Dyligt Svings Time. At tis time, te stik s sdow points dietly not. It is given tt te Sun tvels 15 in te sky e ou. Tus, t 1:00 PM, te Sun is 15 to te est of its igest point. Te oodintes of te Sun (s) nd te tip of te stik (T) t tis time wee detemined. Te oodintes wit espet to te oiginl xes: T:(0, 0, L) s:(sin15, os15, 0) Figue 9. Te Sundil wit espet to te sky in its viinity. s epesents te Sun. Aftewds, te entie oodinte plne ws otted so tt te x nd y xes ly long te plne of te gound (Figue 10). Te oodintes wit espet to te otted xes: T:(0, -Losϕ, Lsinϕ ) s:(sin15, os15 sinϕ, os15 osϕ ) Te eqution of te line tt onnets te two points s nd T in tee-dimensionl spe ws detemined nd te point t wi it inteseted te x y oodinte plne (lso known s te gound) ws lulted in ode to find te ngle θ of te stik s sdow t 1:00 P.M. Note tt time t is mesued in ous. Figue 10. Te Sundil nd its viinity wit otted oodinte xes Te line is found to e: x t sin15 y L osϕ + t( os15 sinϕ + L osϕ) z L sinϕ + t( os15 osϕ L sinϕ) If it intesets te x-y plne, z 0, teefoe, 6-17

18 t Lsinϕ os15 osϕ Lsinϕ Sustituting t, y Lsinϕ sin15 x os15 osϕ Lsinϕ Los ϕ os15 + L osϕ sinϕ L os15 sin ϕ L os15 osϕ Lsinϕ sinϕ osϕ Divide y x tn θ tn - Lsinϕ sin15 L sinϕ sin15 θ L os ϕ os15 L os15 sin ϕ L os15 (os ϕ + sin ϕ) tnθ sinϕ tn15 Tis eomes te genel eqution to find te ngle of te stik s sdow t ny ou of te dy: tn θ sdow sinϕ tn θ Sun. Te following tle (Tle 1) ws lulted fo evey ou wit ϕ 41 : CONCLUSION Tle 1: Sundil Angles Duing Dyligt Hous Time θ Sun θ sdow 1 PM 0 0 PM/1 PM PM/11 AM PM/10 AM PM/9 AM PM/8AM PM/7AM Te pupose of tis ese ws to exploe te motion of evenly odies nd pply mtemtil nd pysil fomule to tk tei oits. Keple s Lws eme te foundtion of tis pojet, s tey explin te fundmentls of plnety oits. Tese lws wee deived using lulus. Elliptil geomety ws ten used to onfim tt tese lws wee, in ft, Keple s Lws. Aftewds, new Lw of Sines nd Lw of Cosines wee deived fo use on speil tingles, so tt plnety oits ould e viewed s tey e fom ou etly pespetive. Wit tese findings, sevel elted polems n e solved. Te Lw of Cosines ws used to detemine sunise time, s well s te distne etween two ities. Speil nd Ctesin geomety ws lso used to onstut sundil tt now utely tells te time of dy, one gin onfiming Keple s Lws. 6-18

19 EFEENCES [1] Simon Cssidy. Te Topil nd te Anomlisti Ye <ttp:// Aessed 007 Jul 5. [] Enylopædi Bitnni. <ttp:// Aessed 007 Jul 5. [3] [UCA] Univesity Copotion fo Atmospei ese. 005 Deeme 16. Univesity of Miign. <ttp:// ntiity.tml&eduig>. Aessed 007 Jul 5. [4] U.S. nvl osevtoy wemste. 003 Ot 30. Et s sesons, et. <ttp://.usno.nvy.mil/dt/dos/etsesons.tml>. Aessed 007 Jul 5. [5] Wete Dy Dt. 003 Ot 17. <ttp://nsuet.ome.msi.om/edution/ssf/otoe/owddt.tml>. Aessed 007 Jul 9. [6] Cuent lol time in London. <ttp:// Aessed 007 Jul 9. [7] Tle of te Delintion of te Sun. <ttp:// Aessed 007 Jul