Halley s Comet Project. Calculus III

Transcription

1 Hlle s Come Projec Clculus III Come Hlle from Moun Wlson, 1986 "The dvers of he phenomen of nure s so gre, nd he resures hdden n he hevens so rch, precsel n order h he humn mnd shll never be lcng n fresh nourshmen." ohnnes Kepler Ths erm we wll sud Hlle s Come, s poson s funcon of me, nd Kepler s Second Lw of plner moon. I wll hnd ou weel problems, whch I cll H problems. You wll hnd hese problems bc o me, he wll be grded, nd hnded bc o ou. You wll collec hese problems nd wll summrze he resuls he end of he erm n projec repor.

2 Hlle s Come Projec Clculus III Ths erm we wll sud he orb of Hlle s Come nd s poson s funcon of me. I wll hnd ou weel problems I wll cll H problems. We wll use power seres o esme he locons of he come vrous mes durng he 76 ers es o orb he Sun. You wll summrze he resuls of hese problems he end of he erm n projec repor. Edmond Hlle's Come In 175 Edmnnd Hlle predced, usng Newon s newl formuled lws of moon, h he comes seen n 1531, 167, nd 168 re ll he sme come nd would reurn n 1758 whch ws, ls, fer hs deh. The come dd ndeed reurn s predced nd ws ler nmed n hs honor. The verge perod of Hlle's orb s 76 ers. Come Hlle ws vsble n 191 nd gn n Is ne pssge wll be n erl 6. Comes, le ll plnes, orb he Sun n ellpc orbs, bu her orbs re ver eccenrc he mjor s s much lrger hn he mnor s. The pon where he come s closes o he Sun s clled perhelon, nd he pon where s he frhes s clled phelon see he fgure n he refresher shee ched. A phelon n 1948, he come ws 35.5 AU from he Sun, whle perhelon on Februr 9, 1986, ws onl.5871 AU from he Sun. An sronomcl un AU s he sem-mjor s for Erh, whch s bou 93 mllon mles. The ellpse s sem-mjor s s, whle s sem-mnor s s b. The eccenrc, he mesure of s elongon, s e nd s gven b e b 1, whch cn be solved for b o gve b 1 e. Eccenrc s beween nd 1. For crculr orb e, nd for ver elonged orb e s close o 1. The dsnce from he cener of he ellpse o eher focl pon s e. We wll le desgne Februr Wh hs convenon s Februr 6, nd 76 s Februr 6 when he come wll reurn o perhelon gn. The orbs of he Erh, Urnus, Nepune nd Hlle s Come Close up vew of he orb of Erh nd Hlle s Come

3 Refresher on Prmerc Equons of Conc Secons Prmerc equon of crcle r cener,, perod π sn cos Prmerc equon of n ellpse, mjor s, mnor s b, cener,, perod π sn cos b As bove, bu shf cener o, h h sn cos As bove, bu shf cener o, h b h sn cos As bove, bu chnge perod o B B h B sn cos π π As bove, bu chnge perod o B B b h B sn cos π π Prmerc equon of n ellpse, mjor s, mnor s b, eccenrc e, cener e, sn 1 cos B e e B π π b.e Aphelon Perhelon Plne moves slower Plne moves fser

4 Problem H1 Wre he prmerc equon of crculr orb wh rdus cenered he orgn wh prmeer, nd n orbl perod of. The plne s,.your nswer wll nvolve sne nd cosne funcons. b Wre he prmerc equon of n ellpc orb wh mjor s long he - s, mnor s b long he -s. The ellpse s cenered, wh prmeer, nd n orbl perod. The plne s poson should be, c Shf he ellpse n b lef so h he orgn s he rgh focl pon. Noe h he dsnce from cener o ech focl pon s e, where e s he eccenrc of he ellpse see Refresher. Wre he equon for hs orb. Your equons should be n erms of, b, e nd d The orb of Hlle s Come hs he followng vlues e AU ers, b 1 e AU AU s n sronomcl un whch s he verge dsnce from he Sun o he Erh for Erh.

5 Kepler s Lw ses h he lne connecng he Sun o he plnes or comes sweeps equl res n equl me. The equon n c gnores hs lw nd wll, herefore, gve he correc orb, bu ncorrec locons for Hlle s Come. We wll see n Problem H how o fnd he correc posons. If Hlle s Come s perhelon Feb. 1986, fnd he ncorrec locon of hs plne usng he equon n c he gven mes below. Pu our nswer n ordered prs, nd use hree decml plces. Perhelon s when he plne s closes o he Sun for our problem hs s e, me n ers Incorrec locons Feb Feb Feb 6 e Grph he ellpc orb nd loce he bove locons on our grph. Use MAPLE, nd ch our grph. Ths s n emple of how ou cn plo he orb of plne nd plce he plne's posons on he orb usng MAPLE. > whplos > f->*cos*p*/1-*e; g->b*sn*p*/1; > 1.5 b1. e.6 13 > p1plo[f,g,..3],-3..3,-..,sclngconstrained, cmrs[-1,1],cmrs[-1,1] pponplo{[f.15,g.15],[f.5,g.5]},smbolcircle, colorblc,sclngconstrained dspl{p1,p};

6 Problem H ohnn Kepler n 169 dscovered h plnes nd comes orb he Sun n ellpc orbs nd h her orbl veloc s no consn bu vres. The followng summrzes Kepler s frs wo lws See Fgure 1 The plnes orb he Sun n ellpc orbs wh he Sun one of he focl pons. The lne jonng he Sun o plne sweeps ou equl res n equl me. Hs second lw smpl sd mens h plnes slow down when he re frher from he Sun, nd speed up when he re closer. Snce he lne jonng he Sun o he plne s shorer when he plne s closer, he lengh of he orb rveled b he plne n gven nervl of me would be lrger o me he res swep equl. Plne moves slower Plne moves fser b.e Aphelon Perhelon For crculr orb he eccenrc e s zero, bu s he orb ges more eccenrc elonged, e pproches 1. The pon of he orb closes o he Sun s clled perhelon, nd he pon frhes s clled phelon. To smplf he clculons for hs problem, whou loss of generl, we wll plce he orgn he focl pon where he Sun resdes, he -s long he mjor s. The lengh of he mjor s s, nd h of he mnor s s b. The cener of he ellpse s hen, e. We wll lso le me equl zero when nd he plne s perhelon. Wh hese ssumpons, he prmerc equons of he orb of plne re π cos e π b sn or π cos e π 1 e sn 1

7 Where s he orbl perod n Erh ers. Noe h when e, he bove equons urns no he prmerc equons of crcle wh cener he orgn. Equon 1 does no ccoun for Kepler s Second Lw I ssumes n lmos consn veloc. To ccoun for h Kepler develed he followng equon clled Kepler s Equon π E e sn E For gven me, ou frs solve for E from nd hen plug E n equon 1 nsed π of cos E e 1 e sn E 3 The vrble E s clled eccenrc noml, whle he epresson π s clled men noml. Noe h for crculr orb when e, hese wo re he sme, bu s e ges closer o 1, hese wo wll be dfferen. Equon 3 wll gve he correc poson of he come gven me. The onl problem wh hs s h becuse equon s n mplc equon n E, nd cnno be solved for E, ou mus solve for E usng numercl echnque. Forunel our TI clculor nd MAPLE hve SOLVE commnds o do hs for us solve equon n, for TI nd MAPLE We wll sud echnques o pprome E s funcon of n eplc form n problems H3 nd H4. Ths wll gve us E n epresson n whch we wll hen plug no 3 for E s n epresson. For problem H le equl he vlues n he ble below, solve for E from usng he solver commnd on our clculor or MAPLE me sure our clculor s n rdn mode. Now use 3 o fnd he correc locons for Hlle s Come. Wre he locon n ordered prs,, nd crr our resuls o hree decml plces.

8 Tme n ers Vlue of E Correc locons of he come Feb Feb Feb 6... b Plo he orb nd loce hese locons s ou dd n Problem H1. c Observe he dfference n hese locons nd h n Problem H1 nd summrze wh shor semen.

9 Problem H3 We sw n problem H h o fnd he correc locons of Hlle s Come we hd o solve he followng mplc equon for E eccenrc noml π E e sn E, 1 nd hen plug he vlue of E no he orbl equon for Hlle s Come gven b cos E e 1 e sn E Implc equons re no ver convenen when scens wn o predc he locon of plnes nd comes n he s, or wn o desgn spcecrf o lnd on or fl b hese celesl objecs. I s mporn o fnd n eplc epresson for E s funcon of me, E some epresson n, h we cn plug drecl n he rgumens of he cosne nd sne funcons n. In hs H problem nd he ne we wll sud power seres h wll pprome E s n eplc funcon of. Frs, we need o sud Bessel funcons before we cn proceed. Bessel funcons, le sn, cos, nd ln funcons, re clled rnscendenl funcons nd cn be presened eplcl onl b power seres. The re wren s, 1,, 3,... The subscrp gves he order of he funcon he bove re Bessel funcons of order, order 1, order, order 3,.. Bessel funcon of order s he soluon o he followng dfferenl equon,,1,, 3, For emple, s he soluon o. In Chper 7 we wll sud dfferenl equons, nd n secon 8.1 nd ler H problem we wll lern echnques o solve hese dfferenl equons. The soluons o hese dfferenl equons re gven b he power seres

10 ...!! 1 1!! 1! Wre generl power seres for Bessel funcon of order. Wre he frs four erms of he power seres of ech Bessel funcons n 4, n ec form, nd end ech wh o ndce nfne seres. Leve he denomnors n fcorl nd power form le 5!3! o show he perns DO NOT EXPAND THESE INTO LARGE NUMBERS 3 1

11 3 Turn he summons n equon 4 bove o prl sums, nd choose n for he upper lm of he sums n such h he prl sums wll gve Tlor polnomls T, T1, T, nd T 3 for, 1,, nd 3, respecvel n.. 4 Ener he Tlor polnomls T, T, T, nd T ppromons 1 3, 1,, nd 3 for, respecvel, no MAPLE worshee or our clculor [he commnd s sum,.. n; for MAPLE nd...,,, n for TI ]. Plo hese four funcons on he sme se of es on he wndow [,1], [ 1,1] nd ch our grphs. 5 MAPLE nows hese funcons s Bessel,, where s he order nd he ndependen vrble. For emple Bessel, s. Your clculors unforunel don hve Bessel funcons n her clogue. Use MAPLE o grph hrough 3 on he sme se of es nd on he sme wndows s n 4 nd ch he grphs. 6 Wre shor semen s o how he prl sum of he seres form of Bessel funcons nd MAPLE s Bessel funcons compre. Where re he smlr, where re he dfferen.

12 Problem H4 We sw n problems H nd H3 h o fnd he correc locons of Hlle s Come we hd o numercll solve he followng mplc equon for E eccenrc noml π E e sn E, 1 nd hen plug he vlue of E no he orbl equon for Hlle s Come gven b cos E e 1 e sn E. In order o vod hvng o solve he mplc equon 1 numercll, sronomers nd mhemcns hve develed soluon for he eccenrc noml E s n eplc funcon of, whch s power seres form gven b π e E π sn 1. 3 In 3, e s he Bessel funcon of order h we suded n H3 wh rgumens e, e, 3e,.... Noe h e self s rnscendenl funcon nd hs power seres epnson. You wll use MAPLE o do hs problem. See he noe below f ou would le o use our clculor. You cn ener hs power seres s wren n 3 no MAPLE usng Bessel, sn n MAPLE for e. Noe h s he order, nd s he rgumen, whch s e here. 1Wh e. 97 for Hlle s Come, use MAPLE o fnd he pprome decml vlues for he erms 1 e, e, 3 3e, 4 4e, nd wre π / 76 plus he frs four erms of he power seres for E n 3, hen end wh o ndce nfne seres. Leve he erm π s π 76, bu urn ll he coeffcens of he sne funcons no decmls. E......

13 Ener equon 3 n MAPLE usng he frs 5 erms..5, usng he funcon noon for E [hs wll loo le E ->sum. ]. Ener he followng vlues of n he ble below o fnd he vlues of E. Plug hese vlues of E no equon o fnd he, locons of Hlle s Come nd fll he ble below Tme n ers Vlue of E Approme locon of he come Feb Feb Feb Wre shor semen s o how hs compres wh our correc locons ou go n problem H.

14 *Noe You cn use our clculor o do hs problem, bu snce our clculor does no now Bessel funcons, ou need o use he power seres for e 1 e e,!! nd plug h n 3 o ge E π 1 1 e!! π sn. The clculor wll gve smlr nswers o MAPLE f ou use he frs ff erms for boh of he seres prl sums. The clculor s, however, ecrucngl slow. If ou do hs, s bes o sore he numbers n, nd e nd ener hs equon wh nd e smbols nd no numbers. Sore he funcon s f, nd hen ener f, f.5 f1,. o ge vlues for E.

15 Problem H5 In prevous H problems we used Bessel funcons o model he orb nd he locon of Hlle s Come. In hs ls H problem we wll cull solve dfferenl equon o fnd he power seres of one of hese Bessel funcons s n emple of how he power seres for Bessel funcons re derved. I wll hnd ou gudelne o s ou o summrze he resuls of problems H1 hrough H5 no projec repor ne wee Bessel funcon of order, s we hve seen before, s he soluon o he followng dfferenl equon,,1,, 3,.... For emple, s he soluon o. If we le, nd choose prer nl condons, he soluon o he nl vlue problem, 1, 1 s he Bessel funcon of order h s. Solve he bove nl vlue problem 1 usng he power seres echnque. Me sure ou show ll our seps nd pu he fnl nswer n form. b Fnd he nervl of convergence of. c Assumng h ou cn solve he dfferenl equon for n Bessel funcon, fnd he nervl of convergence of he generl Bessel funcon. The form for ws found n H3. d Grph severl Tlor polnomls for unl ou rech one h loos le good ppromon o over he nervl [-5, 5]. Presen he grphs nd he Tlor polnoml h does hs ppromon Ths problem wll be grded on he use of good mhemcl noon nd complee wre up of our wor.

16 Wrng Your Projec Repor You re now red o presen our scenfc wor on Kepler s Lws s ppled o Hlle s Come. Here s gudelne for our presenon for he resuls of problems H1 hrough H5. Plese do no ch or refer o n of he H problems n our repor. You cn cu nd pse resuls he re nel done, bu use our own words. Wre our repor s f someone who does no now nhng bou he H problems, bu nows mh s redng our repor. Your repor should be word processed. b You wll summrze ll he nformon h ou hve lerned n he H problems n our repor. Your repor 1 Inroducon Summrze Kepler s Lws nd wh he projec wll presen he ec mplc nd he pprome eplc equons for E jus n words No equons n Inroducon. Mn Presenon Orgnze he nformon n w h wll me sense o n ousde reder. Frs presen Kepler s frs wo lws. Ne presen Hlle s Come nd s orbl elemens, e,, b. Then he mplc equon h Kepler derved, nd fnll he pprome eplc form h were ler develed, nd compre he resuls. You lso need o presen Bessel funcons n here s well. Emphsze Power Seres for E nd Bessel Funcons h we hve lerned n hs clss. Include ll he bles nd he formuls nd he grphs h we hve develed n he H problems. The soluon o ou found n H5 cn be presened here or s n ppend n he bc. If ou presen s n ppend n he bc, menon here h s n emple we wll presen he soluon for he frs Bessel funcon n Append A. 3 Summr Summrze he resuls of hs projec nd ll h ou hve lerned jus n words. The summr wll be jus n words wh no equons or grphs or bles.