Kinsei Swmtwn sto Egg Diasthmik

Peribllon

K. G. Tsignh N. K. Sprou

Lktora Kajhght

Prqeire didaktik shmeisei gia to mjhma

Problmata tou Egg Diasthmiko Peribllonto

Jessalonkh, 2008

Eisagwg

Oi kinsei twn fusikn (plante, asteroeide) kai teqnhtn (dorufroi, dia-

sthmiko stajmo) ourniwn swmtwn kai o akrib prosdiorism th troqi tou

enai rrhkta sundedemna me th melth tou egg diasthmiko peribllonto. Oi

kinsei twn planhtn sqetzontai mesa me thn exlixh th bisfaira th Gh,

kaj ephrezoun shmantik thn troqi th kai ra ti klimatologik sunjke.

Den prpei epsh na xeqnome ti oi sugkrosei asteroeidn kai komhtn me

th Gh, kat to pareljn, qoun paxei ousiastik rlo sthn exlixh twn mbiwn

eidn, parembanonta baia sthn exeliktik diadikasa. Ap thn llh pleur, h

ekmetleush tou egg diastmato gia thlepikoinwniako, ereunhtiko, all

kai stratiwtiko skopo, sthrzetai sth dunattht ma na topojetome teqnh-

to dorufrou se gnwst troqi kai na sqedizoume me akrbeia diaplanhtik

taxdia, kti pou apaite polplokou kai exairetik akribe upologismo.

O kldo th Fusik pou asqoletai me th dunamik sumperifor twn oura-

nwn swmtwn onomzetai Ournia Mhqanik. Ousiastik, prkeitai gia to pedo

efarmog twn nmwn th Klasik Mhqanik se sustmata ulikn shmewn kai

steren swmtwn, pou allhlepidron kurw msw barutikn dunmewn. 'Etsi,

to basiktero prblhma th Ournia Mhqanik enai to barutik prblhma twn

do swmtwn, oi lsei tou opoou enai oi gnwst ma kwnik tom, alli

troqi tpou Kepler (kklo, lleiyh, parabol kai uperbol). Eidik gia thnknhsh mikrn swmtwn (asteroeide, teqnhto dorufroi), poll for qrei-

zetai na lboume upyh ma epiplon asjene dunmei, barutik fsew mh,

pw aut pou sqetzontai me ti palrroie to energeiak perieqmeno th

Hlisfaira, oi opoe prokalon diataraq sthn troqi.

O legqo kai oi prokajorismne metabol th troqi en diasthmiko

skfou enai ap ta shmantiktera problmata th diasthmik mhqanik, -

pw apokaletai enote o kldo th ournia mhqanik pou melet th dunamik

sumperifor twn teqnhtn dorufrwn. 'Allwste h topojthsh en teqnhto

dorufrou sthn telik tou troqi gnetai stadiak. H jhsh pou dqetai o do-

rufro se kje stdio kajorzei ti gewmetrik idithte th proswrin kai

th epijumht, telik troqi tou. Enai profan ti lgoi oikonoma enr-

geia (kausmwn) epiblloun ste oi epijumht metabol na gnontai me so to

dunatn mikrtero ksto. To dio isqei kai gia to sqediasm diaplanhtikn

diasthmikn apostoln. Tlo, o swst programmatism tou propologismo

kausmwn en dorufrou, me stqo th dunatthta ektlesh mikrn prosarmogn

th troqi kat th dirkeia th zw tou, apotele plon shmantik tmma kje

apostol, lgw kai tou diark auxanmenou kindnou sgkroush me diasthmik

skoupdia.

Sta paraktw keflaia ja asqolhjome diexodik me ta proanaferjnta pro-

blmata. Arqik ja meletsoume thn kinhmatik twn troqin twn planhtn tou

hliako sustmat ma kai ja anaferjome sti epiptsei pou qoun oi meta-

bol th troqi th Gh sti klimatalogik th sunjke. Sth sunqeia ja

asqolhjome me th melth tou problmato twn do swmtwn kai th majhmatik

perigraf twn kleistn troqin tpou Kepler (kklo, lleiyh) kai ja anal-

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soume to fainmeno th emfnish palirroiognwn dunmewn. Katpin ja melet-

soume ti troqi twn teqnhtn dorufrwn th Gh, perigrfonta thn kattax

tou me bsh th kinhmatik kai dunamik tou idithte. Qrhsimopointa ta

apotelsmata th anlush tou problmato twn do swmtwn, ja perigryoume

th diadikasa dirjwsh metjesh th troqi en dorufrou, upologzon-

ta parllhla to energeiak ksto gia kje epijumht metabol. Tlo, ja

perigryoume basik nnoie th jewra diataraqn, pou qrhsimopoietai gia th

melth th epdrash asjenn (barutikn mh) dunmewn sthn troqi en doru-

frou, dnonta orismna paradegmata diataraktikn dunmewn kai upologismo

twn antstoiqwn troqiakn metaboln.

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1. Oi troqi twn planhtn

Oi troqi twn planhtn all kai twn uploipwn mikrn swmtwn (asteroeide,

komte) tou hliako sustmat ma enai se prth prosggish elleyei, me ton

'Hlio na brsketai sth ma esta, pw dhlnei o prto nmo tou Kepler. Toeppedo th troqi kje planth enai diaforetik, me th sqetik gwna klsh

twn epipdwn knhsh twn planhtn na enai perpou i 27. Oi ekkentrthtetwn troqin twn planhtn enai en gnei mikr (e < 0.2). Axioshmewth exareshapotele o Plotwna, tou opoou h troqi qei klsh 17 w pro thn ekleiptikkai ekkentrthta 0.25, prosomoizonta tsi ti troqi twn mikrn swmtwnth znh Edgeworth-Kuiper. Shmeinoume ti, smfwna me prsfath, mh genikapodekt, apfash th Genik Sunleush th Diejno Astronomik 'Enwsh

(Agousto 2006), o Plotwna den jewretai plon planth, all to prtupo

mia na kathgora swmtwn pou onomzontai - pro to parn - nnoi plante

(dwarf planets).Sthn pragmatikthta, oi troqi twn planhtn den enai kleist elleyei,

ote qoun stajer sqma kai prosanatolism sto disthma. H bradea kai mi-

kro pltou metptwsh (precession) twn planhtikn troqin prokaletai kurwap thn barutik allhlepdrash twn planhtn (Neutneia sunistsa) kai deute-

reuntw ap thn barthta tou 'Hliou (Sqetikistik sunistsa). Paratata, oi

jsei kai oi taqthte twn planhtn mporon na problefjon me meglh akr-

beia, gia qronik diastmata th txh twn 10 100 ekatommurwn etn, me thqrsh th jewra diataraqn. Aut fanetai na enai kai to qronik rio proble-

yimthta twn planhtikn troqin kaj, pw deqnoun prsfate ereunhtik

melte, oi troqi tou enai asjen qaotik. Oi planhtik efhmerde, sti o-

poe katagrfontai oi jsei kai oi taqthte twn planhtn gia dedomnh qronik

stigm, kataskeuzontai ete me qrsh th jewra diataraqn me thn efarmog

arijmhtikn mejdwn oloklrwsh twn exissewn knhsh kai th bojeia H/U.

Ekt ap thn orgnwsh sqedou parathrsewn twn planhtn, oi efhmerde qrh-

simopoiontai gia th melth pijann susqetsewn, metax twn metaptsewn th

troqi th Gh kai llwn gewlogikn / gewfusikn dedomnwn, sta opoa qoun

katagrafe oi palaioklimatik sunjke th Gh.

Oi metaptsei th troqi kai kurw tou xona peristrof th Gh qoun

mesh susqtish me thn emfnish twn pagetnwn, kti pou prto paratrhse o

Milankovic th dekaeta tou 1920. Sugkekrimna, h metptwsh tou xona peristro-f, h metjesh tou perihlou th Gh kai oi metabol th ekkentrthta th

troqi th enai - se prth prosggish - periodik fainmena, o grammik sun-

duasm twn opown dnei diakrotmata. 'Etsi, to pos th hliak enrgeia pou

proslambnei h Gh an to parousizei mgisto kai elqista me perodo 100,000

etn. Axzei epsh na shmeiwje h epdrash th Selnh sthn peristrofik k-

nhsh th Gh. 'Opw prteine prsfata o Laskar, h mza m kai h apstash rth Selnh enai ttoie ste h metptwsh tou xona peristrof th Gh na

enai omal, me th lxwsh th ekleiptik na ektele talantsei mikro pltou

(perpou 3) grw ap mia msh tim, fainmeno gnwst w klnhsh tou xona.An oi tim twn m kai r tan diaforetik, o xona peristrof th Gh ja

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ektelose qaotik metaptsei kai h lxwsh th ekleiptik ja llaze diark

tim sto disthma (0, 90) kat tuqao trpo, me sunpeia na mhn enai dunat hanptuxh stajero klmato, kti pou ja eqe dramatik epiptsei sthn exlixh

th bisfaira th Gh.

Smera pisteoume ti oi mse apostsei twn planhtn ap ton 'Hlio den

tan pntote oi die me aut pou parathrome smera. O lgo enai ti, kat ta

prima stdia th exlixh tou Hliako Sustmato, tan h sunolik mza twn

mikrn swmtwn tou hliako sustmato tan perpou 103 104 for megal-terh ap' ti smera, h allhlepdrash twn planhtn me aut ta upolemmata th

dhmiourga tou odghse se ektetamnh metansteush twn planhtn. Parti to

fainmeno aut den qei akmh katanohje plrw, oi ereunht sumfwnon ti h

diadikasa th metansteush eqe dirkeia mikrterh ap 100 ekatommria qrnia.Epomnw, ed kai 4.5 di qrnia, to hliak ssthma qei ousiastik thn diaarqitektonik me autn pou parathrome smera.

Nmo twn Titius-Bode

Gia mikr qronik diastmata, oi troqi twn planhtn mpore na jewrhjon w

kleist elleyei me ametblhte diastsei kai stajer prosanatolism ston

adraneiak qro. 'Opw paratrhsan oi Titius (1766) kai Bode, oi tim toumeglou hmixona th troqi twn xi - tte - gnwstn planhtn (se astronomik

monde), ikanopoion thn apl majhmatik sqsh

ai = 0.4 + 0.3 2i (1)pou i = , 0, 1, 2, 4, 5. H anakluyh tou Ourano (a = 19.2 AU) to 1781 kaitou prtou asteroeido (Ceres, a = 2.8 AU) to 1801 jewrjhkan jramboi tounmou twn Titius-Bode, kaj oi jsei autn twn swmtwn antistoiqon staken troqiak th parapnw sqsh gia i = 3 kai i = 6. 'Omw, met ap thnanakluyh tou Poseidna (a = 30.1 AU) kai tou Plotwna (a = 39.4 AU), o nmo twn Titius-Bode qase thn aglh tou, afo oi tim tou ai gia i = 7, 8enai pol diaforetik ap ti parathromene tim gia tou duo auto plante.

Exllou, ote h aktna th troqi tou 'Arh (a 1.5 AU) dnetai ap thn parapnwprodo.

Enai genik apodekt ti oi apostsei twn planhtn, pw kai aut twn

dorufrwn twn meglwn planhtn, den enai tuqae all qoun prokyei ktw

ap polploke dunamik diergase, pou telik odghsan sthn eustjeia tou

sustmato. 'Omw, h eresh mia apl majhmatik sqsh, pou na perigrfei

to apotlesma lwn autn twn diergasin kai na sundei fusik posthte me

ti sqetik jsei twn planhtn, den qei epiteuqje. 'Allwste, h parxh mia

ttoia sqsh me genik isq den mpore na apodeiqte. Antjeta, qei apodei-

qte ti h parapnw sqsh twn Titius-Bode den enai statistik shmantik, kajdifore morf logarijmik gewmetrik prodou perigrfoun ti sqetik a-

postsei twn planhtn, pw kai twn dorufrwn twn meglwn planhtn, me

parmoia akrbeia.

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Sqma 1: (pnw) Sqhmatik anaparstash th elleiptik troqi th Gh, me ton 'Hlio

na katalambnei th mia esta. (ktw) To stoiqeide embad pou diagrfei h epibatik

aktna, se qrno t.

Nmoi tou Kepler

Prin ap th jemelwsh th Neutneia Mhqanik, o Kepler, me bsh parathr-sei, diatpwse tou perfhmou trei nmou th knhsh twn planhtn. Prkeitai

gia nmou th kinhmatik, oi opooi ermhneoun thn gewmetra th troqi kai

thn fainmenh knhsh twn planhtn, qwr na apokalptoun to fusik atio th

knhsh (th dnamh th barthta), kti pou enai antikemeno th dunamik. Oi

nmoi tou Kepler orzoun ti

1. Oi troqi twn planhtn enai eppede elleyei, me ton 'Hlio na katalambnei

th ma esta.

2. H knhsh grw ap ton 'Hlio gnetai me stajer embadik taqthta

3. Ta tetrgwna twn peridwn perifor twn planhtn enai anloga twn k-

bwn twn meglwn hmiaxnwn th troqi tou.

O Netwna diatpwse to nmo th pagksmia lxew, deqnonta ti h ken-

trik, elktik, dnamh th morf 1/r2 enai h mnh pou ermhneei tautqronakai tou trei nmou th kinhmatik. H eppedh knhsh apodeiknetai ekola

ap thn parxh tou oloklhrmato th stroform, pou sunepgetai ti h k-

nhsh gnetai se na eppedo kjeto sto dinusma th stroform (bl. epmeno

keflaio). Orzonta polik suntetagmne (r, ) sto eppedo th knhsh, to

mtro th stroform an monda mza enai h = r2. Kat to qronik disthma

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(t, t + t) to stoiqeide embad A pou orzetai ap ti epibatik aktne r kair + r tou planth enai

A =1

2|r(r+ r)| = 1

2r(r + r) sin (2)

Diairnta me t kai parnonta to rio gia t 0 (opte sin = d)parnoume

dA

dt=

1

2r2d

dt=r2

2=h

2(3)

'Ara, h embadik taqthta enai stajer kai sh me to mis th stroform h. Giaelleiptik knhsh apodeiknetai ti

h2 = G(M +m)a(1 e2) = a(1 e2)pou G h stajer th pagksmia lxh, M kai m oi mze tou 'Hliou kai touplanth, antstoiqa, a o meglo hmixona kai e h ekkentrthta th lleiyh.H parmetro = G(M +m) onomzetai parmetro mza tou sustmato kaiorzetai w to ginmeno th stajer G kai th sunolik mza tou sustmato,gia kje ssthma do swmtwn. Shmeinoume ti, pw apdeixe o Netwna, h

sugkekrimnh morf tou nmou th pagksmia lxew enai h mnh pou topojete

to elktik kntro sthn esta th kwnik tom (ant tou kntrou th) kai tau-

tqrona ermhneei ton 3o (armonik) nmo tou Kepler (bl. paraktw). Antjetale oi kentrik dunmei (F = F (r)) odhgon se diatrhsh th stroform kaira se stajer embadik taqthta gia ti peratwmne troqi.

O trto nmo tou Kepler problpei mia rht sqsh metax th peridou pe-rifor en planth grw ap ton 'Hlio kai tou meglou hmixona th troqi

tou, T a3/2. H sqsh aut apodeiknetai ekola, me bsh th diatrhsh thembadik taqthta kai th gewmetra th lleiyh. Se ma perodo, T , to s-ma diagrfei mia plrh lleiyh, kalptonta embad so pro A = a b, pou

b = a1 e2. Smfwna me thn exswsh (3), to embad aut isotai me A = hT/2,

pou h2 = a(1 e2). Epomnw,

T 2 =42

a3 (4)

H parapnw sqsh apotele th majhmatik diatpwsh tou trtou nmou tou Kepler.Orzonta th msh suqnthta perifor tou smato msh knhsh n = 2/T ,o parapnw tpo parnei th morf,

n2 a3 = (5)

Jewrnta do plante tou Hliako Sustmato me mze m kai m kai efarm-zonta thn parapnw sqsh, parnoume

M +m

M +m=(a

a

)3 (T T

)2(6)

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Bbaia, sto Hliak Ssthma isqeim m M ki epomnw (a/a)3 (T/T )2.Shmeinoume ti h Astronomik Monda (AU, h msh tim tou meglou hmixonath troqi th Gh) orzetai ap thn parapnw sqsh, jtonta w monda tou

qrnou to na Gino to kai w monda mza th mza tou 'Hliou. Gia nan planth

mza m pou qei dorufro mza m m, o 3o nmo tou Kepler dnei th sqsh

m

M=(T

T

)2 (aa

)3(7)

pou T h perodo perifor kai a o meglo hmixona th troqi tou planthgrw ap ton 'Hlio kai T , a oi antstoiqe posthte gia thn troqi tou doruf-rou grw ap ton planth. H parapnw sqsh mpore na qrhsimopoihje gia ton

proseggistik upologism th mza tou en lgw planth.

Pardeigma 1: Brete ti peridou perifor twn Da (a = 5.2 AU) kai Plo-twna (a = 39 AU).

Qrhsimopointa th sqsh (6) gia th Gh (a = 1 AU, T = 1 yr) kai ton Dakai agnonta ti mze twn do planhtn, prokptei T1 = 11.86 yr gia ton Da.Omow, gia ton Plotwna, parnoume T2 = 243.6 yr.

Fainmenh knhsh twn planhtn

Gia nan parathrht sth Gh, h fainmenh knhsh en planth ston ouran kajo-

rzetai ap th sqetik tou knhsh w pro th Gh kai, pw fanetai sto Sqma

2, enai diaforetik gia katterou (anmesa sth Gh kai ton 'Hlio) kai anterou

(exwterik th G) plante.

Do plante enai se snodo tan brskontai sthn dia eujea me ton 'Hlio. Gia

nan kattero planth (Afrodth, Erm), qoume anterh snodo tan brsketai

psw ap ton 'Hlio kai katterh snodo tan brsketai mprost ap ton 'Hlio.

Gia tou anterou plante, mno ma snodo enai dunat. Sthn perptwsh pou

o 'Hlio, h Gh kai na antero planth enai sthn dia eujea, me th Gh na

brsketai anmesa stou llou do, tte lme ti o antero planth brsketai

se antjesh me th Gh.

H gwna pou sqhmatzei h jsh en planth me thn eujea Gh-'Hlio onom-

zetai apoq. Gia kje kattero planth uprqoun do jsei mgisth apoq

(Anatolik kai Dutik). Gia ton Erm kai thn Afrodth, oi tim th mgisth

apoq w pro th Gh enai 28 kai 47 antstoiqa. 'Etsi, oi do auto planteparathrontai ete lgo prin thn anatol lgo met thn dsh tou 'Hliou. Gia

anterou plante enai dunat na qoume tetragwnism me th Gh, tan h eujee

Gh-'Hlio kai Gh-planth sqhmatzoun orj gwna.

H astrik perodo en planth orzetai w to qronik disthma pou apaitetai

ste na breje sthn dia jsh ston ouran, w pro tou aplane astre. H

perodo th sqetik knhsh en planth w pro th Gh onomzetai sunodik

perodo kai isotai me to qronik disthma pou apaitetai ste ta do smata na

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Sqma 2: Sqetik knhsh w pro th Gh, gia nan kattero kai nan antero planth, se

elleiptik troqi. Diakrnontai oi do dnodoi (KS kai AS) gia nan kattero planth,

kaj kai oi jsei mgisth anatolik kai mgisth dutik apoq (MAA kai MDA).

Epsh, diakrnontai oi jsei sundou (S), antjesh (A) kai tetragwnismo (T1 kai

T2), gia nan antero planth.

epanljoun sthn dia sqetik jsh w pro ton 'Hlio (p.q. ap snodo se snodo).

An TE to astrik to th Gh kai TP to astrik to en planth, h sunodikperodo TS dnetai ap th sqsh

1

TE=

1

TP 1TS

(8)

pou to (+) antistoiqe stou anterou plante kai to () stou katterou.

Pardeigma 2: Brete ti oriak tim th apstash ap ton 'Hlio, gia ti

opoe TS > 2 yr. Poioi ap tou eswteriko plante ikanopoion thn parapnwsqsh;

Ap thn exswsh th sunodik knhsh qoume

1

TS=

1

TP 1TE

0.67 yr

gia katterou plante kai

1

TS=

1

TE 1TP

m2). To prblhma gkeitaisthn eresh th troqi twn do swmtwn, gia dedomne arqik sunjke.

H dnamh pou asketai sto sma i ap to j (i, j = 1, 2) dnetai ap ton tpo

1

Parti h sunrthsh r(t) den mpore na doje se kleist morf2

Aut to prblhma odghse ton Lagrange sth diatpwsh th mejdou th metabol twn stajern,gia thn eplush en mh omogeno sustmato D.E.

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Fi,j = Gmimjr3ij

rij (9)

pou r12 = r = r21 kai oi exissei knhsh enai

Ri = Gmjr3ij

rij (10)

Efson den uprqoun desmo th knhsh, to ssthma qei 6 bajmo eleujera,

dhlad qoume na ssthma 6 D.E. 2h txh.

Oloklhrmata th knhsh

Oi dunmei barthta enai sunthrhtik kai ra h olik mhqanik enrgeia tou

sustmato diathretai. Epsh, den askontai lle exwterik dunmei kai ra

diathretai h olik orm tou sustmato. Omow, den askontai exwterik rop

sta do smata kai ra diathretai h olik stroform tou sustmato w pro

O. H olik mhqanik enrgeia dnetai ap th sqsh

E =1

2

2i=1

mi|Vi|2 Gmimjrij

(11)

pou Vi = Ri. Antstoiqa, h olik orm, P, kai h olik stroform, L, tousustmato dnontai ap ti sqsei

P = m1 R1 +m2 R2 (12)

kai

L = R1P1 +R2P2 =2i=1

mi(RiVi) (13)

Knhsh w pro to kntro mza

To kntro mza tou sustmato orzetai w to shmeo K, tou opoou to dinusmajsh (w pro O) ikanopoie th sqsh

RK =m1R1 +m2R2

m1 +m2(14)

ap' pou paragwgzonta parnoume

VK = RK =m1 R1 +m2 R2

m1 +m2=P1 +P2m1 +m2

(15)

kai

RK =m1 R1 +m2 R2

m1 +m2=F1,2 + F2,1m1 +m2

(16)

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'Omw, ap ti Ex. (10) kai (16) sunepgetai

m1 R1 +m2 R2 = F1,2 + F2,1 = 0

kai ra

RK = 0 VK = P/(m1 +m2) = c1 RK = c1 t+ c2 (17)pou ci stajer diansmata. Ap ta parapnw gnetai faner ti, apousa exwte-

rikn dunmewn, to kntro mza tou sustmato kinetai eujgramma kai omal

w pro adraneiak ssthma anafor kai h orm tou (an upojsoume th mza tou

sustmato sugkentrwmnh sto K) isotai me thn olik orm tou sustmato.

Epomnw, to K mpore na oriste w h arq en epsh adraneiako sustma-to anafor Kxyz, me ti exissei kai ta prta oloklhrmata th sqetikknhsh w pro to kntro mza na dnontai ap ti sqsei

RK,i = mjr3ij

rij (18)

EK =1

2

2i=1

mi|VK,i|2 Gm1m2r

(19)

PK =2i=1

mi RK,i (20)

LK =2i=1

RK,iPK,i =2i=1

mi(RK,iVK,i) (21)

pou RK,i kai VK,i ta diansmata jsh kai taqthta twn do swmtwn, w pro

to kntro mza K.Oi lsei tou sustmato (18) onomzontai barukentrik troqi. Ap thn

exswsh tou kntrou mza (diatrhsh th olik orm) prokptei ti h knhsh

tou en ap ta do smata kajorzetai plrw an gnwrzoume thn knhsh tou

llou. Epomnw, o prosdiorism twn ci ap ti arqik sunjke meinei thn

txh tou diaforiko sustmato, dnonta na isodnamo prblhma 3 bajmn e-

leujera (b.e.). H diatrhsh twn E kai L prosdiorzei akmh 4 stajer. 'Etsi,brskoume 10 stajer th knhsh (ap ti sunolik 3 2 2 = 12). Epomnw,to prblhma angetai sth lsh ma bajmwt D.E. 2h txh, pou onomzetai i-

sodnamo monodistato prblhma, me ti tim twn do stajern th oloklrwsh

na kajorzontai epsh ap ti arqik sunjke.

Exswsh Sqetik Knhsh

Ap ti exissei (10) prokptei, me afaresh kat mlh, h exswsh th sqetik

knhsh tou m2 w pro to m1

r+G(m1 +m2)r

r3= 0 (22)

12

kai h plrh lsh tou problmato gkeitai plon sth lsh aut th dianusma-

tik diaforik exswsh. Gnwrzonta thn troqi th sqetik knhsh, upolo-

gzoume ekola ti barukentrik troqi twn do swmtwn, bsei twn tpwn

RK,1 = m2m1+m2 rRK,2 =

m1m1+m2

r (23)

Epomnw, oi barukentrik troqi enai moie me th sqetik troqi, me lgo

omoithta mj/(m1 +m2) gia to sma i. Epsh, oi parapnw sqsei deqnounti to kntro mza brsketai plhsistera sto bartero sma (r1/r2 = m2/m1).An m1 m2 (p.q. 'Hlio - planth, Gh - dorufro) mporome na agnosoume thmza m2, opte h exswsh sqetik knhsh parnei th morf

r+Gm1r

r3= 0 (24)

me to kntro mza na tautzetai me th jsh tou m1 kai th barukentrik troqitou m2 na tautzetai me thn sqetik troqi tou w pro to m1. Se autn thnprosggish, to hliokentrik (gewkentrik, k.o.k.) ssthma anafor orzetai w

adraneiak, prgma pou den isqei akrib. Tlo, anexrthta tou an agnosoume

th mza m2 qi, h exswsh sqetik knhsh tautzetai me thn exswsh knhshen smato monadiaa mza se pedo kentrikn dunmewn th morf k/r2

(pou k = G(m1 +m2) = GM = ), me to elktik kntro na brsketai sth jshtou m1. Dedomnou ti GM = , exswsh sqetik knhsh parnei th morf

r+ r

r3= 0 (25)

Ta oloklhrmata th sqetik knhsh orzontai sunjw me elafr diaforetik

trpo. 'Etsi, to oloklrwma th eidik mhqanik enrgeia, enrgeia an

monda mza, orzetai ap th sqsh

C =1

2v2

r(26)

pou v to mtro th sqetik taqthta. Prkeitai gia mia posthta me diast-sei enrgeia pou upologzetai sto mh adraneiak ssthma anafor, tautzetai

mw me to oloklrwma th enrgeia th upojetik monadiaa mza, pou kine-

tai sto kentrik pedo elktikn dunmewn k/r2. Analgw, orzoume thn eidikstroform (stroform an monda mza) tou sustmato w to dinusma

h = rr = rv (27)to opoo epsh tautzetai me to dinusma th stroform th monadiaa mza,

grw ap xona pou pern ap to elktik kntro. Gia ton lgo aut, sto efe-

x ja qrhsimopoiome tou rou enrgeia kai stroform, anafermenoi sti

posthte C kai h.

13

Apodeiknetai ekola ti h genik lsh th exswsh (25) enai h exswsh th

kwnik tom se polik morf,

r =p

1 + e cos( ) =h2

(1 + e cos )(28)

pou p = h2/ h hmiparmetro th tom, e h ekkentrthta th troqi, kai hgwna pou sqhmatzei h gramm twn aydwn (h eujea pou ennei ti do este) me

ton xona = 0. H gwna = dnei th jsh tou smato pnw sthn troqikai onomzetai alhj anwmala

3

. H metrtai ap thn elktik esta th kwniktom me arq th gramm twn aydwn kai auxnei kat th for th knhsh tou

smato.

Oi tsseri dunat kwnik tom qarakthrzontai ap ti tim tou meglou

hmixona th troqi, a, kai twn e kai p

a > 0 , e = 0 , p = a

a > 0 , 0 < e < 1 , p = a(1 e2)a , e = 1 , p = 2qa < 0 , e > 1 , p = a(e2 1)

(29)

pou antistoiqon se exswsh kklou, lleiyh, parabol kai uperbol. 'Opw

deqnoun oi parapnw sqsei, h parabol enai mia oriak perptwsh, th opoa

h hmiparmetro kajorzetai ap thn elqisth apstash, q, th troqi ap thnesta. Enai qrsimo na katatxoume ti troqi me bsh ti tim twn C kai h,pou enai kai ta basik, diathrsima, fusik megjh. Apodeiknetai ekola ti

a = 2C

, e =

1 +

2C h2

2(30)

ap' pou prokptei ti oi kleist troqi (kklo kai lleiyh) antistoiqon se

austhr arnhtik tim enrgeia (C < 0), en oi mh peratwmne troqi (pa-rabol kai uperbol) se jetik tim (C 0). O meglo hmixona th tro-qi exarttai mnon ap thn enrgeia, en h ekkentrthta exarttai kai ap th

stroform. Epomnw, se dedomnh tim enrgeia (kai hmixona a), antistoiqonpeire troqi diaforetik stroform (ekkentrthta).

Parabolik Knhsh - Taqthta Diafug

H elqisth taqthta, pou prpei na qei na sma se apstash r ap to elktikkntro ste na teje se parabolik troqi, diafegonta tsi ap th barthta

3

H qrsh th lxh anwmala qei na knei me to gegon ti h knhsh gnetai me stajer embadik

(kai qi grammik) taqthta, opte h gwna enai mh grammik sunrthsh tou qrnou gia le ti

kwnik tom, me exaresh ton kklo. Mlista, pw ja dome paraktw, h sunrthsh (t) den mporena doje se kleist morf, all mno se morf seir.

14

tou elktiko kntrou, onomzetai taqthta diafug parabolik taqthta kai

dnetai ap th sqsh

v =

2

r(31)

pou = GM kai M h sunolik mza tou sustmato. O dio tpo isqeikai gia sma mza m pou ektoxeetai ap thn epifneia th Gh, an jsoumer = R kai M = M +m M, pou R kai M h aktna kai h mza th Gh.Shmeinoume ti h taqthta diafug enai

2 1.414 for megalterh ap thn

kuklik taqthta, sthn dia apstash r. H idithta aut qrhsimeei sto sqediasmtroqin gia diaplanhtik apostol, pou sunjh praktik enai h topojthsh

tou skfou, arqik, se kuklik troqi grw ap th Gh, mqri thn katllhlh

qronik stigm, tan kai ja purodotsei ti mhqan tou gia na xekinsei thn

anoiqt, parabolik, troqi tou.

Jewrnta ton 'Hlio w to elktik kntro, enai faner ti la ta smata pou

ankoun sto Hliak Ssthma akoloujon troqi arnhtik enrgeia, alli ja

eqan diafgei sto peiro. To dio isqei kai gia ti troqi twn teqnhtn doru-

frwn th Gh. Paratata, uprqoun arketo komte se hmiparabolik troqi

(e 1). Epomnw, me mia mikr metabol th enrgeia C > 0, ja mporosanna diafgoun ap to hliak ssthma. Aut enai dunat na sumbe, an o komth

persei arket kont ap nan planth, ste h barutik tou allhlepdrash na

gnei shmantik. To fainmeno aut anafretai enote w fainmeno th sfentna

(slink-shot effect).

Sqma 4: H troqi tou diasthmoploou Cassini, ap th stigm th ektoxeus tou apth Gh (15/10/1997), mqri th stigm th sunnths tou me ton Krno (1/7/2004).

Parathreste ti, gia thn epteuxh tou taxidio, qreisthkan 4 kontin dielesei ap

tou plante Afrodth (2 for), Gh kai Da.

To fainmeno th sfentna brskei efarmog sto sqediasm diaplanhtikn

diasthmikn apostoln, gia ta opoa to energeiak ksto enai apagoreutik

15

(p.q. Voyager, Cassini). H barutik upobojhsh (gravity assist), pou prosfre-tai sto diasthmploio kat thn prosggish en planth, tou parqei (dwren)

enrgeia, ste na xekinsei na makr diaplanhtik taxdi. Poll for apai-

tetai o sqediasm diadoqikn proseggsewn me tou plante Afrodth kai Gh

- pnta me katllhlh gewmetra - ste, swrreutik, na kerdhje to aparathto

pos enrgeia. Endeiktik enai h troqi tou diasthmoploou Cassini (Sqma 4),pou to 2004 ftase epituq ston Krno.

Pardeigma 1: Dexte ti h taqthta diafug ap th Selnh enai mikrterh

ap to na ttarto th taqthta diafug ap th Gh.

Efarmzoume th sqsh (31) kai gia ta do smata. O lgo twn do taquttwn

enai

vMvE

=

M REE RM

pou oi dekte E kai M anafrontai sth Gh (Earth) kai sth Selnh (Moon),antstoiqa. O lgo mazn enai perpou M/E = 1/81 en o lgo twn aktnwntou enai RM/RE < 4. Antikajistnta sthn parapnw sqsh, prokptei

vM 0onomzetai anabibzwn sndesmo (AS) th troqi. To antidiametrik tou shmeo

28

Sqma 9: Orism twn stoiqewn th troqi dorufrou, sto adraneiak, gewkentrik,

ssthma anafor. Diakrnontai oi sndesmoi (AS kai KS) th troqi kai oi gwne kai i. Epsh shmeinontai oi paraplhrwmatik gwne twn kai .

w pro O onomzetai katabibzwn sndesmo (KS) kai h eujea pou sundei tado shmea onomzetai gramm twn sundsmwn. H gwna pou sqhmatzei o AS me

ton xona x onomzetai mko tou anabibzonto sundsmou, . Sto eppedo thtroqi, h gwna pou sqhmatzei h gramm twn aydwn (P'OP) me th gramm twn

sundsmwn orzei th gwna tou perigeou, . H gwna pou sqhmatzei h epibatikaktna tou kinhto me th gramm twn aydwn enai h alhj anwmala, . 'Etsi, hjsh tou dorufrou w pro to adraneiak ssthma anafor kajorzetai plrw

ap to snolo twn xi stoiqewn th troqi, (a, e, i,, , ). Sunjw ant th qrhsimopoiome th msh anwmala, M . Profan la ta stoiqea th troqienai stajer, ekt ap thn anwmala tou dorufrou. Epomnw, ta stoiqea th

troqi enai sunartsei twn oloklhrwmtwn th knhsh.

Shmeinoume ti, ekt tou proanaferjnto adraneiako sustmato anafo-

r, qrhsimopoietai epsh to gewkentrik peristrefmeno ssthma anafor, to

opoo peristrfetai grw ap to adraneiak ssthma me gwniak taqthta staje-

r kai sh me aut th Gh. Sto peristrefmeno ssthma, o anabibzwn sndesmo

th troqi qei mko = t, pou h gwniak taqthta peristrofth Gh. Sthn perptwsh pou o dorufro upkeitai se epiplon asjene dun-

mei, ta stoiqea th troqi tou den enai plen stajer all upkeintai se mikr

diataraq. O upologism tou gnetai me th qrsh th jewra diataraqn, pw

ja dome se epmeno keflaio.

Katanom twn troqin twn dorufrwn

H pleioyhfa twn teqnhtn dorufrwn th Gh akolouje troqi qamhlo you,

Hi = ri R

29

pou R h aktna th Gh kai to smbolo i antikajstatai ap ta p a antstoiqagia to pergeio (ri = rp = a(1e)) kai to apgeio (ri = ra = a(1+e)) th troqi.Oi troqi qamhlo you onomzontai troqi LEO (Low Earth Orbits) kai qa-rakthrzontai ap ton meglo arijm periforn th Gh pou ektele o dorufro

kat th dirkeia mia hmra (1016 perifor). Shmeinoume ti oi troqi LEOdqontai thn epdrash th aerodunamik trib, lgw tou ti brskontai kont

sthn anterh atmsfaira th Gh. Ekt ap aplo dorufrou, troqi tpou

LEO akoloujon (a) o Diejn Diasthmik Stajm (International Space Sta-tion, ISS, H 360 km) kai (b) To Diasthmik Thleskpio Hubble (Hubble SpaceTelescope, HST, H 500 km). O ISS apotele na ap ta korufaa degmata te-qnologik anptuxh tou anjrpou, kaj enai to megaltero (up kataskeu)

diejn ergastrio sto disthma. Antstoiqa, to HST enai sw to pio epituqh-mno optik thleskpio sthn istora th Astronoma, qonta odhgsei se pol

shmantik ereunhtik apotelsmata thn teleutaa dekaeta.

Perpou 15% twn dorufrwn akoloujon troqi meglou you, tsi stena ektelon ma do perifor th Gh an hmra. Oi troqi aut onomzontai

epomnw gewsgqrone (GEO) kai hmisgqrone, antstoiqa. Oi troqi tpouGEO pou brskontai kont ston ishmerin th Gh onomzontai gewstatik, meton dorufro na brsketai diark pnw ap sugkekrimno tpo th Gh. Oi

troqi endimesou you (MEO) enai elqiste.H meglh pleioyhfa twn troqin enai sqedn kuklik. Axioshmewth exare-

sh apotelon oi troqi tpou Molniya kai Tundra, twn opown oi ekkentrthteftnoun mqri thn tim e 0.8. Oi troqi auto tou tpou ( 15% tou su-nlou) anakalfjhkan kai qrhsimopoijhkan gia prth for th dekaeta tou '60

sthn tw Sobietik 'Enwsh. Prkeitai gia eidik kathgora hmisgqronwn kai ge-

wsgqronwn, antstoiqa, troqin, pou tautqrona qarakthrzontai ap stajer

klsh i = 63.4. Oi troqi aut qoun thn idithta na diathron to apgei toustajer pnw ap sugkekrimno tpo, uperniknta ti diataraq pou prokale

to ishmerin exgkwma th Gh (bl. epmeno keflaio). H iditht tou aut ti

kajist idanik gia thlepikoinwniak qrsh ap tpou meglou gewgrafiko

pltou. Perpou oi mis gewsgqrone kai hmisgqrone troqi enai tpou

Molniya kai Tundra, antstoiqa, en oi uploipe enai ishmerin polik.Mli to 10% twn dorufrwn akoloujon sqedn ishmerin troqi (i

15), sthn pleioyhfa tou kuklik troqi tpou LEO. Oi troqi me klshi 60 apotelon perpou to 75% tou katalgou. Ekt twn troqin tpou Mol-niya kai Tundra, meglo posost twn dorufrwn akolouje polik (i 8090)kai hliosgqrone (i 90110) troqi. Aut oi troqi qrhsimopoiontai giathn parakolojhsh kai katagraf gewlogikn, baruthmetrikn kai metewrologi-

kn dedomnwn th Gh, thn opoa kalptoun ex' oloklrou. Oi hliosgqrone

troqi qoun thn epiplon idithta na diathron stajer ton prosanatolism tou

anabibzonto sundsmou w pro ton 'Hlio. Aut epitrpei stajer fwtism twn

katptrwn tou dorufrou (kai tou stqou) pw kai thn diark parakolojhsh

tou 'Hliou gia episthmoniko lgou (p.q. o dorufro SOHO).

30

Sqma 10: Sqhmatik anaparstash th diadikasa parakolojhsh kai prosdiorismo

th troqi en dorufrou, ap ton stajm bsh (SB).

Prosdiorism kai Parakolojhsh Troqi

Kje dorufro brsketai se diark epikoinwna me nan perissterou staj-

mo bsh (SB), me ton opoo antallssei dedomna. Ekt th katagraf twn

parathrsewn, smfwna me tou stqou kje apostol, o SB enai epifortism-

no me th diark parakolojhsh, ton akrib prosdiorism kai thn diatrhsh th

epijumht troqi tou dorufrou. H diatrhsh th troqi enai epibeblhmnh,

kaj difore diataraq prokalon thn arg apomkrunsh tou dorufrou ap

thn epijumht troqi, kai epitugqnetai me thn apostol katllhlwn entoln ap

to upologistik kntro tou SB pro ton dorufro. H diadikasa parakolojh-

sh kai prosdiorismo th troqi tou dorufrou ap ton SB perigrfetai ap

to parapnw sqma.

Apodeiknetai ti trei diadoqik parathrsei tou dorufrou enai ikan gia

ton kajorism twn stoiqewn th troqi. Bbaia, h qrsh perissterwn para-

thrsewn (pou enai diajsime) odhge se pio akrib prosdiorism th troqi,

msw mia diadikasa polplokwn upologismn pou onomzetai diaforik dir-

jwsh. Sunjw o prosdiorism th troqi gnetai me mtrhsh th stigmiaa

apstash tou dorufrou ap ton SB kai tou rujmo metabol th . Autepitugqnetai me ton upologism tou qrnou diadrom pou apaitetai ste to s-

ma (fwtein palm laser dsmh radar) na fgei ap thn keraa tou stajmo,na anaklaste ap ton dorufro kai na epistryei sthn keraa tou SB. To sma

kalptei diadrom sh me 2 se qrno t, taxideonta me taqthta c, dhlad

=ct

2(61)

31

Tautqrona, lgw th sqetik knhsh tou dorufrou w pro ton SB, h su-

qnthta lyh tou smato ja enai lgo diaforetik ap th suqnthta ekpomp,

lgw metjesh Doppler, tsi ste

=f

2(62)

pou to ekpempmeno mko kmato kai f h metjesh Doppler th suqnthtaf , thn opoa ufstatai to sma, kaj dianei thn apstash . H parapnw sqshprokptei ap th basik exswsh tou fainomnou Doppler

v

c=

f

f

me antikatstash twn v = /(t/2) 2 kai c = f .'Eqonta ta stoiqea th troqi gia kpoia qronik stigm, mporome na ka-

taskeusoume efhmerde th mellontik jsh tou dorufrou, na upologsoume

ti qronik stigm anatol kai dsh kai thn oratthta tou dorufrou ap ton

SB, k.l.p. 'Etsi, gia kpoia qronik stigm (p.q. t4, bl. Sqma 10), mporomena gnwrzoume th gwna kat thn opoa prpei na stryoume thn keraa tou SB,

ste na parathrsoume ek nou ton dorufro kai na proume ta dedomna pou sto

metax qei sullxei. An h na jsh tou dorufrou enai shmantik diaforetik

ap thn problepmenh, tte upologzoume an kai kat pso apaitetai dirjwsh

th troqi.

Pardeigma 1: Ap ljo qeirismo, ta stoiqea th elleiptik troqi en

amelhta mza mh epandrwmnou teqnhto dorufrou th Gh (GM = ) qounqaje, en, lgw blbh, h mnh dunat epikoinwna me to dorufro gnetai se na

sugkekrimno, gnwst mko kmato pou ekpmpetai suneq ap ton pomp tou

dorufrou. Na exetaste p, me melth th fasmatik metjesh Doppler tousmato tou dorufrou, enai dunat o epanaprosdiorism twn stoiqewn th

troqi tou dorufrou (H dirkeia zw tou pompo tou dorufrou enai pepera-

smnh, qi megalterh olgwn periforn tou dorufrou).

H exswsh troqi tou dorufrou enai

r =p

1 + e cos

kai oi sunistse th taqthta enai

vr =dr

dt=h esin

p= e sin

h

kai

rd

dt=h(1 + e cos )

p=

h(1 + e cos )

32

Sunep, h aktinik taqthta kai ra kai h fasmatik metjesh

z =

=vrc, vr = cz

enai hmitonoeide sunartsei th alhjo anwmala . 'Opw fanetai kai stoSqma 2.5, gia knhsh ap to pergeio pro to apgeio th troqi qoume sin > 0kai ra vr > 0 kai z > 0, dhlad metjesh pro to erujr. Antjeta, gia knhshap to apgeio pro to pergeio th troqi qoume sin < 0 kai ra vr < 0 kaiz < 0, dhlad metjesh pro to kuan. 'Etsi, parathrome metjesh pro toerujr gia qronik disthma so me T/2, pou akoloujetai ap metjesh pro tokuan gia so qronik disthma. Profan, metrnta to qronik disthma kat

to opoo z > 0 ( z < 0) upologzoume thn perodo th troqi.'Etsi, me bsh ton trto nmo tou Kepler kai th gnwst plon ap parathrsei

periodikthta tou smato, enai dunat o upologism tou meglou hmixona th

troqi kai th eidik mhqanik enrgeia th troqi

a =(T

2

)2/31/3 , C =

2a

Exllou

e2 = 1 + 2C

(h

)2

Gia = 90 h fasmatik metjesh pro to erujr parnei th mgisth tim th

zmax = e

h c

Ap ti do teleutae sqsei kai me gnwst ta C kai upologzontai ta h kaie kai kat' epktash ta p kai b.

Shmewsh: An upojsoume ti to den enai gnwst, tte qrhsimopoiome epiplonth sqsh

h2

= p = a(1 e2)

en ap ti parapnw sqsei upologzoume antstoiqa ti posthte a/1/3, C(),e(, h) kai e(h, ). Ap ti do teleutae prokptoun ta kai h, katpin ta Ckai a kai tlo ta p kai b.

Dirjwsh kai Metjesh Troqi

H diadikasa topojthsh tou dorufrou se dedomnh troqi apaite meglo e-

nergeiak ksto. H apaitomenh jhsh tou dorufrou mpore na upologiste

me bsh thn arq tou puralou. H jhsh prokaletai ap thn ektxeush tou

33

prowjhtiko msou, msw en steno anogmato pou brsketai sthn krh th

dexamen kausmou. Oi perissteroi dorufroi qrhsimopoion qhmik iontik

projhsh, ektoxeonta, antstoiqa, aria kash inta. H apostol tou dia-

sthmiko skfou SMART sth Selnh apotlese thn prth apostol skfouth ESA, prowjomenou apokleistik me kinhtra intwn. H apostol telewseepituq to fjinpwro tou 2006, tan to SMART sunetrbh sthn epifneia thSelnh!

H sunolik mza kausmou pou apaitetai gia thn topojthsh en dorufrou

se ishmerin gewsgqronh troqi enai kat kanna sh me autn tou skfou. Gia

th diatrhsh en dorufrou sthn epijumht troqi (p.q. tpou GEO), apaitetaimza kausmou perpou sh me 2% th mza tou skfou gia kje to, ste namporon na ekteleston oi aparathte diorjwtik kinsei. 'Etsi, tuqn ljo

upologism kat thn diadikasa topojthsh tou dorufrou sthn troqi tou,

pou ja odhgose se auxhmnh dapnh kausmou mli kat 2%, ja sterose nato zw ap ton dorufro.

H wstik dnamh F pou dqetai o dorufro kat th dirkeia th fsh pro-jhsh dnetai ap th sqsh

F = Vedm

dt+ AeP = Veff

dm

dt(63)

pou Ve h sqetik taqthta tou prowjhtiko msou w pro to skfo, m hsunolik mza dorufrou kai kausmou, Ae h epifneia tou anogmato th dexa-men kai P h diafor pesh metax th dexamen kai tou peribllonto. Todexi mlo th exswsh mpore na aplopoihje, orzonta thn upojetik sqetik

taqthta tou prowjhtiko msou w pro to skfo, Veff .Kje ssthma projhsh qarakthrzetai ap thn tim th stajer eidik

jhsh, Isp, h opoa dnetai ap th sqsh

Isp =F

g dmdt

(64)

pou g h epitqunsh th barthta. H Isp qei diastsei qrnou kai ekfrzeithn ikanthta axhsh th taqthta tou dorufrou, alli thn apodotikthta

kje sustmato projhsh, w pro thn metatrop th eswterik enrgeia

tou kausmou se kinhtik enrgeia tou skfou. H metabol th taqthta tou

dorufrou lgw projhsh dnetai ap th sqsh

V = t2t1

F

mdt (65)

pou t = t2 t1 h dirkeia th fsh projhsh. Antikajistnta ston para-pnw tpo thn F ap ton orism th Isp parnoume thn exswsh

V = g Isp

mfmi

dm

m(66)

th opoa h lsh onomzetai exswsh puralou

34

Sqma 11: Sqhmatik anaparstash th diadikasa allag tou you tou apogeou th

troqi O1

mf = mi exp

(Vg Isp

)(67)

kai sundei th metabol th taqthta tou skfou me thn arqik (mi) kai teliktim (mf ) th mza tou, gia dedomnh tim th eidik jhsh, Isp. H katanlwshkausmou m = mf mi dnetai ap thn kfrash

m = mi

[1 exp

(Vg Isp

)](68)

Gia kje tpo dirjwsh metjesh troqi, h prth ma knhsh enai o akri-

b upologism th apaitomenh metabol V . Anloga me ta qarakthristiktou sustmato projhsh, prokptei h katanlwsh kausmou m kai h dirkeiath fsh projhsh t. Enai faner ti, an apaitontai diadoqik metajseitroqi, pw p.q. gia thn ektlesh diaplanhtikn apostoln thn topojthsh

dorufrou se troqi tpou GEO, o sqediasm gnetai me bsh thn elaqistopoh-sh th sunolik katanlwsh kausmou. Shmeinoume, tlo, ti h dirkeia th

fsh projhsh t enai mikr, se sqsh me thn perodo perifor tou doruf-rou. 'Etsi, mporome na jewrsoume ti h metabol th taqthta tou dorufrou

gnetai stigmiaa, se kpoio dedomno shmeo th troqi tou. Sth sunqeia ja

perigryoume tou basiko tpou dirjwsh kai metjesh troqi.

Allag tou you tou perigeou/apogeou

H allag tou you tou perigeou ( tou apogeou) epitugqnetai, prokalnta

th metabol th taqthta tou dorufrou, kat th dibas tou ap to apgeio

(antstoiqa, to pergeio) th arqik troqi, O1. 'Opw fanetai sto Sqma 11, oido elleiptik troqi (O1 = (a, e) kai O2 = (a, e)) ja efptontai sto apgeio(antstoiqa, sto pergeio) th O1. O upologism tou V gnetai me bsh tontpo th eidik enrgeia

35

2a

= C =v2

2

r(69)

Sto apgeio th O1 h taqthta tou dorufrou dnetai ap th sqsh

v1 =

2(

ra rp1 + ra

)(70)

pou (ra,rp) sumbolzoun thn apstash tou apogeou kai tou perigeou antstoiqa,en oi dekte 1, 2 anafrontai sti troqi O1 kai O2 antstoiqa. Sthn parapnwsqsh kname qrsh tou ti, gia thn O1, 2a = rp1+ ra. Efson jloume na aux-soume to yo tou perigeou, ja prpei na auxsoume thn taqthta tou dorufrou

sto apgeio th troqi tou kat V = v2 v1 > 0, pou

v2 =

2(

ra rp2 + ra

)(71)

afo gia thn O2 isqei 2a = ra + rp2.Paromow, an jloume na auxsoume to yo tou apogeou th troqi tou

dorufrou, ja prpei na auxsoume thn taqtht tou kat th dibas tou ap to

pergeio th O1 (bl. Sqma 11). Oi antstoiqoi tpoi gia ti taqthte v1 kai v2enai

v1 =

2(

rp rp + ra1

)(72)

kai

v2 =

2(

rp rp + ra2

)(73)

pou kai pli isqei v = v2v1 > 0. Oi diorjsei auto tou tpou enai suqngia ti troqi LEO, kaj h aerodunamik trib me thn anterh atmsfaira thGh meinei diark to you tou perigeou.

Dirjwsh tpou (a1, e1) (a2, e2)Enote apaitontai tautqrone mikr diorjsei tso sthn tim tou meglou

hmixona so kai sthn tim th ekkentrthta th troqi tou dorufrou. Gia

na mporsoume na epitqoume sugkekrimne metabol twn stoiqewn (a1, e1) se(a2, e2) prokalome metabol th taqthta tou dorufrou, kat th dileus touap to apgeio th troqi tou.

Ap ton orism th eidik enrgeia C brskoume ti h taqthta tou dorufrousto apgeio th troqi O1 enai sh me

v21 = 2(1

ra1 1

2a1

)=

1 e1a1(1 + e1)

(74)

36

Sqma 12: Sqhmatik anaparstash th diadikasa allag twn stoiqewn (a, e) thtroqi O1

Gia thn troqi O2 kai sto shmeo allag th taqthta tou dorufrou isqei jar2 = ra1 = a1(1 + e1). Epomnw, o dorufro ja akoloujsei thn troqi O2, anh taqtht tou sto shmeo r2 = ra1 qei mtro

v22 = 2(1

r2 1

2a2

)= 2

(1

a1(1 + e1) 1

2a2

)(75)

kai diejunsh ttoia ste na efptetai sthn troqi O2. H tim th gwna brsketai ap ton upologism tou oloklhrmato th stroform h = r v cos .Dedomnou ti p = (h2/) = a2 (1 e22), prokptei h sqsh

cos2 = a2 (1 e22)v22 a

21 (1 + e1)

2(76)

kai h metabol th taqthta V qei mtro

V =[(v2 cos v1)2 + v22 sin2

]1/2(77)

kai prosanatolism

sin =v2 sin

V(78)

Prpei na shmeiwje ti den enai epitrept le oi metabol (a1, e1) (a2, e2).Oi periorismo ekfrzontai ap ti sqsei v22 0 kai cos2 1. Sto paraktwdigramma fanetai sqhmatik h epitrept perioq metabol tou a2, gia dedomnetim twn e2 kai (a1, e1). Tlo, shmeinoume ti h parapnw diadikasa dirjwshodhge se strof th gramm twn aydwn, h opoa den enai pnta epijumht. 'Etsi,

h diadikasa dirjwsh (a1, e1) (a2, e2) sunodeetai sunjw ap mia diadikasastrof tou .

37

Sqma 13: Epitrept perioq metabol twn (a, e) gia thn troqi O1

Sqma 14: Sqhmatik anaparstash th diadikasa metafor tpou Hohmann, anmesasti kuklik troqi O1 kai O2. Diakrnetai h elleiptik troqi metjesh (TO).

Metjesh tpou Hohmann

'Opw edame sthn prohgomenh pargrafo, apl diorjsei th troqi tou do-

rufrou mporon na epiteuqjon me ma mno purodthsh twn prowjhtikn pu-

ralwn. Antjeta, h metjesh tou dorufrou se na troqi, pou den qei kanna

koin shmeo me thn arqik, den enai dunat na gnei se ligtera ap do stdia.

'Etsi, o dorufro anagkzetai na akoloujsei proswrin mia troqi metjesh

(TO), h opoa efptetai tso sthn arqik so kai sthn telik tou troqi. Gia

metafor ap kuklik troqi aktna r1 se omkentrh kuklik troqi aktna r2,h elaqistopohsh th katanlwsh kausmou epitugqnetai akoloujnta th diadi-

kasa metjesh tou Hohmann.H metjesh Hohmann ap thn kuklik troqi O1 (aktna a1) sthn kuklik

troqi O2 (aktna a2) gnetai se do stdia, prokalnta (a) axhsh th taqth-ta tou dorufrou kat V1 ensw kinetai ep th O1, tsi ste na akoloujseithn elleiptik troqi metjesh TO, me apstash perigeou sh pro a1 kai ap-stash apogeou sh pro a2, kai (b) axhsh th taqthta tou dorufrou V2kat th qronik stigm pou o dorufro brsketai sto apgeio th TO, tsi ste

h taqtht tou na gnei sh me thn kuklik taqthta se apstash r = a2 kai na

38

akoloujsei thn telik troqi O2. Efson kai oi do metabol gnontai stiayde th TO, ta diansmata metabol Vi enai kjeta sth gramm twn aydwnth TO kai kat th for th knhsh tou dorufrou.

Kat thn prth fsh projhsh, h taqthta tou dorufrou auxnetai ap

v1 =/a1 se

vp =[2(1

a1 1a1 + a2

)]1/2=

(2 a2

a1 (a1 + a2)

)1/2(79)

Epomnw, h metabol th taqthta v1 enai sh pro

v1 = vp v1 =

a1

[2a2

a1 + a2 1

](80)

Omow, kat th deterh fsh projhsh tou dorufrou, h taqtht tou meta-

bletai ap

va =

[2

rpra (ra + rp)

]1/2=

[2

a1a2 (a1 + a2)

]1/2(81)

se v2 =/a2, ste h metabol v2 na dnetai ap th sqsh

v2 = v2 va =

a2

[1

2a1

a1 + a2

](82)

H olik metabol th taqthta tou dorufrou, bsei th opoa upologzetai h

katanlwsh kausmou enai V = V1+V2, afo kai oi do epimrou metabolprokalon axhsh th taqthta tou skfou.

Pardeigma 2: To diasthmploio Mars Odyssey 2000 kat thn fix tou ston'Arh (M,R) tjhke se elleiptik per autn troqi (a, e). Gia thn apodotikterhparatrhsh tou 'Arh apaitetai o Mars Odyssey 2000 na kinetai se kuklik tro-qi. Na perigryete pijano trpou pou aut mpore na epiteuqje kai na tou

sugkrnete oikonomik.

Sto apkentro th troqi isqei

v2A =

a

1 e1 + e

en h kuklik kai h parabolik taqthta se aut th jsh (r = a(1 + e))enai

v2CA =

a(1 + e)v2PA =

2

a(1 + e)= 2v2CA

Epomnw

39

v2Av2CA

= 1 e < 1

v2A = (1e)v2CA < v2CA . Antjeta, sto perkentro th troqi isqoun oi sqsei

v2 =

a

1 + e

1 ekai

v2C =

a(1 e) v2P

=2

a(1 e) = 2v2C

ki epomnw

v2v2C

= 1 + e > 1

v2 = (1 + e)v2C

> v2C .

Allag troqi sto apkentro: Epeid sto apkentro v2A < v2CA

, to mtro th

taqthta tou skfou sto A ja prpei na auxhje ap vA se vCA , me tautqronhdiatrhsh th katejuns th. H apstash ja paramenei stajer kai sh me

a(1 + e) = a, pou ja enai h aktna th na kuklik troqi. An h vA auxhjese vCA < v

A < vPA , ja prokyei na elleiptik troqi me perkentro to A. Tte,

h = vAa(1 + e) = vAa

(1 e)kai

C = 2a

=v

2A

2 a(1 + e)

Allag troqi sto perkentro: Epeid sto perkentro v2 > v2C, to mtro th

taqthta tou skfou sto P ja prpei na meiwje ap v se vC , me tautqronhdiatrhsh th katejuns th. H apstash ja paramenei stajer kai sh me

a(1 e) = a, pou ja enai h aktna th na kuklik troqi. An h v meiwjese v < vC , ja prokyei na elleiptik troqi me apkentro to P. Tte,

h = va(1 e) = va(1 + e)kai

C = 2a

=v

2

2 a(1 e)

H enrgeia pou ja dapansoume gia na epitqoume th dirjwsh th troqi

enai, se kje perptwsh, sh me th diafor th kinhtik enrgeia pou qei o

40

dorufro (sth jsh metabol th v) sthn elleiptik kai thn kuklik troqi.Gia ti do periptsei, enai

E1 =

(v2C2 v

2A

2

)e

2a(1 + e)

kai

E2 =

(v22 v

2C

2

)e

2a(1 e)antstoiqa. Epomnw

E2 > E1

To parapnw apotlesma shmanei ti, gia na epitqoume kalterh diakritik i-

kanthta sti parathrsei ma (mikrterh aktna troqi) apaitetai megalterh

katanlwsh enrgeia.

Allag troqi me th bojeia th aeropdhsh: Sthn perptwsh aut to diasth-

mploio dirqetai msa ap thn atmsfaira tou 'Arh kai ufstatai atmosfairik

trib, lgw th opoa h taqtht tou meinetai (bl. epmeno keflaio). Sunep,

to fainmeno aut mpore na qrhsimopoihje sthn deterh (kai pio dapanhr) ap

ti parapnw periptsei. Eidiktera, h mewsh th taqthta v epitugqnetaime th dibash tou diasthmoploou ap thn (anterh) atmsfaira tou 'Arh, tan

to diasthmploio brsketai kont sto perkentro th troqi tou. H aeropdhsh

mpore na epanalhfje ste na epiteuqje h mewsh th taqthta kai endeqomnw

kai to edo th na troqi, qwr katanlwsh kausmou.

Topojthsh dorufrou se troqi GEO

Mqri stigm meletsame diorjsei kai metajsei anmesa se suneppede tro-

qi. Sti perisstere twn periptsewn, mw, apaitetai metjesh anmesa se

troqi me diaforetik tim klsh, w pro to ishmerin eppedo th Gh. Qara-

kthristik pardeigma enai h topojthsh dorufrou se ishmerin, kuklik, gew-

sgqronh troqi (GEO), thn opoa ja meletsoume se autn thn pargrafo.H ektxeush tou dorufrou gnetai ap kpoio shmeo th Gh me gewgrafik

plto 6= 0. Kat sunpeia, h elqisth dunat tim th klsh tou epipdou thtroqi tou w pro ton ishmerin ja enai i = . Oi HPA sunjw ektoxeounteqnhto dorufrou ap to akrwtrio Kennedy th Florida, pou brsketai segewgrafik plto = 28.5. Antjeta, o Eurwpak Organism Diastmato(European Space Agency, ESA) qrhsimopoie th bsh ektxeush sto Kourou thGallik Guyana, me = 5.2. An jloume o dorufro na metapsei se ishme-rin troqi, ja prpei na prokalsoume strof tou diansmato th stroform

tou, ste na gnei kjeto pro to eppedo tou ishmerino. Aut epitugqnetai

an stryoume to dinusma th taqthta tou dorufrou kat gwna i, th stigmpou aut dirqetai ap to ishmerin eppedo. Enai profan ti h sugkekrimnh

41

Sqma 15: (pnw) Sqhmatik anaparstash th diadikasa topojthsh dorufrou se

ishmerin troqi tpou GEO. To apgeio th GTO brsketai sto epijumht eppedo thtelik troqi. (ktw) Anaparstash th metabol tou diansmato th taqthta (a)

kat to prto stdio th diadikasa metjesh se do bmata (mhdenism th klsh)

kai (b) kat th diadikasa sunduasmnh metjesh se na bma.

metjesh troqi enai ligtero dapanhr gia dorufrou th ESA par gia doru-frou twn HPA. Ekt ap th strof tou diansmato th taqthta, to mtro

th ja prpei epsh na metablhje, ste h telik troqi tou dorufrou na enai

kuklik. Shmeinoume ti mia tupik troqi ektxeush enai pol kkentrh, me

apstash perigeou rp 200 km.H topojthsh dorufrou se ishmerin troqi (GEO) apaite prosektik sqe-

diasm th troqi metjesh, GTO. Sugkekrimna, h GTO sqedizetai tsi steto apgei th na brsketai pnw sto ishmerin eppedo (en gnei, sto epijumht

eppedo th GEO) kai se apstash sh me ra = 42, 164.2 km = R - to yoth kuklik, gewstatik, troqi. H strof tou diansmato th taqthta

sunepgetai metabol

V1 = 2 Va sin(i/2)

pou Va = Vi = Vf to mtro th taqthta tou dorufrou, sto apgeio thGTO. Akmh kai met ap thn parapnw dirjwsh, h ekkentrthta th ishmerintroqi enai meglh, kaj to apgei th qei tim ra = 42, 164.2 km = R, ento pergei th qei tim rp = 200 km. H troqi tou dorufrou gnetai kuklik(metjesh GTOGEO), me axhsh th taqtht tou kat th dileus tou apto apgeio th troqi (bl. prohgomenh pargrafo). H apaitomenh metabol

th taqthta dnetai ap ton tpo

42

V2 =

R

(1

2rp

R + rp

)

kai h olik metabol th taqthta enaiV = V1+V2. Enai ekolo na apodei-qje ti enai dunat na epiteuqje sunduasmnh metjesh (mhdenism th klsh

me tautqronh axhsh tou perigeou) me ma mno purodthsh tou sustmato pro-

jhsh, diadikasa pou odhge se elaqistopohsh th katanlwsh kausmou. An

VGTO enai h taqthta tou dorufrou sto apgeio th troqi GTO (tan tmneito ishmerin eppedo) kai VGEO enai h taqthta th kuklik knhsh se ishmeri-n troqi you R, h olik metabol th taqthta tou dorufrou se na bmadnetai ap ton tpo

V 2 = V 2GTO + V2GEO 2 VGTO VGEO cos i

pw prokptei ap me efarmog tou tpou tou sunhmitnou gia to trgwno tou

Sqmato 15b.

Pardeigma 3: Dorufro sunolik mzams = 2, 000 kg ektoxeetai se tro-qiGTO me apstash perigeou rp = 200 km, apstash apogeou ra = 42, 164.2 km =R kai klsh i = 7. H stajer th eidik jhsh tou sustmato projhshenai Isp = 300 sec. Upologste ti tim twn V kai to antstoiqo energeiakksto gia metptwsh se troqi GEO (i) se do fsh kai (ii) se ma fsh.

Smfwna me ti sqsei th prohgomenh paragrfou, o mhdenism th kl-

sh antistoiqe se V1 = 194.97 m/sec, en o mhdenism th ekkentrthtaantistoiqe se V2 = 1, 477.76 m/sec. 'Etsi, h olik metabol th taqthta enai

V = 1, 672.73 m/sec

Ap th sqsh (68) prokptei ti qreizetai na katanalsoume m = 867 kgkausmou. Antjeta, h sunduasmnh metjesh (se ma fsh) antistoiqe se

V = 1, 502.4 m/sec

kai katanlwsh m = 800 kg kausmou. Epomnw, h sunduasmnh metjeshodhge se exoikonmhsh 67 kg kausmou, dhlad 3.3% th sunolik mza toudorufrou, gegon pou dnei th dunatthta partash th apostol tou kat

1.5 to.

Asksei

1. Dorufro topojetetai se troqi LEO me Hp = 500 km kai e = 0.04. An,met ap arket kair, diapistsoume ti to yo tou perigeou qei meiwje

kat Hp = 25 km (lgw trib me ta antera strmata th atmsfaira)

43

kai jloume na epanafroume ton dorufro sthn arqik tou troqi, na bre-

je (i) poiou tpou metjesh ja qrhsimopoisoume, (ii) poia h tim tou Vpou prpei na epiteuqje kai (iii) poio to ksto (se mza kausmou), an to

ssthma projhsh qei Isp = 250 sec.

2. Diasthmik skfo, sunolik mzams = 104 kg, kinetai se kuklik troqi

tpou GEO. Upologste thn tim tou V kai to energeiak ksto pouprpei na katabloume (gia Isp = 300 sec) ste to skfo na odhghje sena, epsh kuklik, troqi prosggish me th Selnh (a 350, 000 km).

3. Upologste th diafor energeiako kstou gia topojthsh dorufrou m-

za ms = 2, 000 kg se troqi tpou GEO, an h ektxeush gnetai (i) ap toKourou (i 7) (ii) ap to akrwtrio Kennedy (i 30). Jewrste ti htroqi metafor (GTO) antistoiqe se apstash perigeou rp = 250 km kaih stajer th eidik jhsh tou sustmato projhsh enai Isp = 250 sec.Upologste ti tim twn V kai m gia topojthsh (i) se do fsei kai(ii) se ma fsh.

44

4. Diataraq sthn knhsh dorufrou

'Opw qoume dh anafrei, oi troqi twn dorufrwn - pw kai aut twn pla-

nhtn tou Hliako Sustmato - den enai tleie elleyei. O lgo enai ti,

par ton prwtarqik rlo th barutik lxh th Gh, o dorufro dqetai thn

epdrash epiplon dunmewn, barutikn mh, oi opoe prokalon en gnei mikr

diataraq sthn elleiptik tou knhsh. Sti barutik diataraq sugkatalgontai

(a) h dnamh pou asketai lgw asummetra tou barutiko pedou th mh-sfairik

Gh (kurw ap to ishmerin th exgkwma), (b) oi parlxei th Selnh, tou

'Hliou kai twn llwn planhtn kai (g) oi palrroie. Oi krie mh barutik (mh

sunthrhtik) dunmei pou askontai sto dorufro enai (a) h aerodunamik trib

me thn anterh atmsfaira th Gh (kurw gia troqi LEO) kai (b) h pesh thhliak aktinobola. Mikrtere diataraq prokalontai ap (a) sqetikistik

fainmena, (b) thn asummetra anmesa sti dieujnsei aporrfhsh kai epanek-

pomp th hliak aktinobola ap ton dorufro (jermik fainmeno Yarkovsky)kai (g) ti qronik metabol tou barutiko pedou th Gh, pou ofelontai sthn

knhsh aerwn mazn, thn metaknhsh gkwn pgou stou plou, thn hfaisteiak

kai seismik drasthrithta, k.a. Gia diasthmploia pou brskontai se megle a-

postsei ap th Gh, diataraq prokalontai epsh ap thn allhlepdras tou

me to ulik th Hlisfaira.

Enai profan ti h akrib gnsh tou apotelsmato twn parapnw diatara-

qn apaite akribe metrsei polln fusikn posottwn. Antstrofa, h akrib

gnsh th jsh tou dorufrou kje qronik stigm, dhlad twn apoklsewn a-

p thn epijumht troqi, epitrpei ton prosdiorism twn timn twn sugkekrimnwn

fusikn megejn. H ragdaa anptuxh th thlemetra me th qrsh desmn la-ser qei odhgsei sth dunatthta prosdiorismo th troqi en dorufrou meakrbeia th txh tou 1 cm! H epistmh pou asqoletai me thn anlush doru-forikn dedomnwn kai ton prosdiorism twn en lgw fusikn megejn kai twn

qronikn metaboln tou onomzetai (doruforik) fusik gewdaisa. Oi gew-

daitik diasthmik apostol ta epmena qrnia, anamnetai na apodsoun nan

terstio gko plhroforin, anaforik me fusik diergase pou sumbanoun sthn

epifneia, to eswterik, thn atmsfaira kai to egg diasthmik periblon th

Gh.

Genik diataraq - exissei Gauss

A upojsoume ti, ekt th epitqunsh th barthta, o dorufro dqetai

epitqunsh p lgw kpoia diataraktik dnamh. H diataraq mpore na enaisunthrhtik mh. An to mtro th p enai mikr se sqsh me thn epitqunsh thbarthta k, tte mporome na genikesoume ton orism twn stoiqewn th tro-qi, par to gegon ti h alhj troqi tou dorufrou, O, den enai pia kleist.Sugkekrimna, gia kje qronik stigm t, kat thn opoa o dorufro brsketaisth jsh S, orzoume thn stigmiaa troqi O (osculating orbit), w thn elleiptiktroqi tpou Kepler pou efptetai me thn alhj troqi sto S, kai thn opoa jaakoloujose o dorufro, an xafnik mhdeniztane oi diataraq (bl. Sqma 16).

45

Sqma 16: Orism twn dianusmtwn (R,S,W) kai th stigmiaa troqi (stikt gram-m).

Ta stoiqea th stigmiaa elleiptik troqi onomzontai stigmiaa stoiqea th

troqi (osculating elements) kai den apotelon stajer th knhsh. Gia sun-thrhtik diataraq mikro pltou, ta stoiqea th troqi metablontai arg,

ektelnta mikr talantsei grw ap kpoia msh tim. Aut oi mse tim

sunjw onomzontai dia stoiqea th troqi (proper elements). Oi metaboltwn stoiqewn th troqi mporon na upologiston me bsh to formalism tou

problmato twn do swmtwn, upologzonta ti metabol twn oloklhrwmtwn

th knhsh. H mjodo aut odhge se na ssthma diaforikn exissewn 1h

txh, ti exissei tou Gauss.Orzoume na trisorjognio ssthma anafor me monadiaa diansmata (R,S,W)

pw fanetai sto sqma (W = R S), to opoo kinetai maz me ton dorufro.Se aut to ssthma anafor, to dinusma th epitqunsh p grfetai

p = rR+ s S+ wW (83)

Xekinnta ap to jerhma metabol th kinhtik enrgeia, mpore kane na

apodexei ti o rujm metabol th eidik mhqanik enrgeia C isotai me thnisq th diataraktik epitqunsh

dC

dt= v p (84)

Qrhsimopointa th sqsh pou sundei thn enrgeia me thn tim tou meglou

hmixona, parnoume ton rujm metabol tou a

da

dt=

2a2

v p (85)

To dinusma th taqthta analetai sto ssthma (R,S,W), dnonta

v = v sin R+ v cos S (86)

46

pou = (v,S) h gwna ptsh. Oi sunistse th taqthta, vR kai vS, taut-zontai me ti vr kai v (aktinik kai epitrqia), antstoiqa

v sin = vr , v cos = v (87)

pou . Antikajistnta ti ekfrsei twn vr kai v (sqsei 35-37) sthsqsh (85), prokptei h telik sqsh gia ton rujm metabol tou a

da

dt=

2

n1 e2 [e sin r + (1 + e cos ) s] (88)

pou ta stoiqea th troqi sto dexi mlo th exswsh jewrontai stajer.

Qrhsimopointa to basik jerhma th Klasik Mhqanik pou sundei to

rujm metabol th stroform en sustmato me thn rop pou asketai se

aut

dh

dt= r p (89)

kaj kai thn exswsh th lleiyh (28) kai thn exswsh tou Kepler (49), prok-ptoun oi exissei tou Gauss kai gia ta uploipa stoiqea th troqi. Gia ta ekai i oi antstoiqe exissei enai

de

dt=

1e2na

[sin r + (cosE + cos ) s]

di

dt= 1

na1e2

racos( + ) w (90)

Sunthrhtik diataraq - Exissei Lagrange

Sthn eidik perptwsh pou oi diataraq enai sunthrhtik (p = U), oi exi-ssei metaboln twn stoiqewn th troqi mporon na grafon w sunartsei

twn paraggwn tou dunamiko th diataraq, U . Oi exissei diataraqn pouprokptoun onomzontai exissei tou Lagrange. Qarakthristik twn exissewnautn enai ti oi metabol twn (a, e, i) exartntai mno ap ti paraggou touU w pro ti gwne (, ,M)

(a, e, i) = f(U

M,U

,U

)

en oi metabol twn (, ,M) exartntai mno ap ti paraggou tou U wpro ta (a, e, i)

(, , M) = g(U

a,U

e,U

i)

Shmeinoume ti o parapnw formalism mpore na qrhsimopoihje sthn per-

ptwsh pou oi diataraq prokalontai ap (a) palrroiogne dunmei, (b) thn

47

epdrash trtou smato kai (g) thn asummetra tou barutiko dunamiko th mh

sfairik Gh. Antjeta, h pesh th hliak aktinobola kai h aerodunamik

trib enai mh sunthrhtik diataraq kai den mporon na perigrafon ap ti

exissei Lagrange. Sti epmene paragrfou ja asqolhjome ektenstera meti basik diataraq pou askontai stou dorufrou.

Dunamik th peplatusmnh Gh

'Opw gnwrzoume, h Gh ma den qei sfairik sqma (ote kan elleiyoeid), dh-

lad h katanom th mza th Gh den enai sfairik (ote axonik) summetrik.

Epomnw h qrsh tou tpou U = /r gia to dunamik th Gh enai apl miaprosggish, qi kai tso kal tan qoume na knoume me knhsh dorufrou se

yo H = 200 50, 000 km. Shmeinoume ti h asummetra tou pragmatiko du-namiko th Gh prokale ti pio shmantik diataraq sti troqi twn teqnhtn

dorufrwn.

To pragmatik dunamik th Gh enai mia sunrthsh trin metablhtn, U(r, , ),pou (, ) oi gewgrafik suntetagmne tou tpou th Gh, ston opoo enai k-jeto to dinusma jsh tou dorufrou, r. H pio sunhjismnh anaparstash th

U(r, , ) enai to anptugm th se sfairik armonik, mia apeiroseir th mor-f

U(r, , ) = r+ B(r, , ) (91)

pou

B(r, , ) =

r{n=2

(Rer

)nJn Pn(sin)

+n=2

(Rer

)nPn,m(sin) [cn,m cos(m) + sn,m sin(m)]}

pou Re h msh aktna th Gh ston ishmerin, (Jn, cn,m, sn,m) stajero suntele-st pou perigrfoun th sqetik suneisfor kje rou kai Pn,m ta polunumaLegendre txh n kai bajmo m. O prto ro tou ajrosmato perigrfei tiarmonik twn ishmerinn zwnn, pou exartntai mno ap to gewgrafik pl-

to . O detero ro perigrfei ti armonik twn meshmbrinn atrktwn, pouexartntai kai ap to gewgrafik mko . Enai qrsimo na shmeiwje ti oisuntelest (Jn, cn,m, sn,m) den meinontai shmantik kaj auxnontai ta (n,m),all to plto kje trigwnometriko rou meinetai shmantik, lgw tou pa-

rgonta (Re/r)n. 'Etsi mporome na gryoume proseggistik to dunamik me th

morf

U r(U0 + U2 + . . .) (92)

pou

48

U0 = 1 , U2 = 12

(Rer

)2J2 (3 sin

2 1)

O ro U2 perigrfei to ishmerin exgkwma th Gh (peplatusmnh Gh) kai enaio pio shmantik diorjwtik ro tou U . H tim tou suntelest J2 mpore nametrhje me meglh akrbeia, an analsoume thn parathromenh troqi tou do-

rufrou (thlemetrik dedomna) se seir Fourier kai metrsoume to plto tou

rou pou antistoiqe sth suqnthta f2 = . Sth sunqeia, afairnta aut thngramm ap to fsma Fourier th troqi kai analonta thn aplopoihmnh seirprosdiorzoume to plto tou U3 k.o.k. Oi apodekt tim twn suntelestn twnprtwn rwn th B(r, , ), smfwna me to pagksmio gewdaitik ssthma WGS84, enai

J2 = 1082.6 106 , J3 = 2.53 106 , J4 = 1.61 106c2,1 = s2,1 = 0 , c2,2 = 1.57 106 , s2,2 = 0.9 106

An antikatastsoume sthn parapnw kfrash gia to U2 thn kfrash twn (r, )w pro ta stoiqea th troqi kai ektelsoume ti trigwnometrik prxei, par-

noume th na kfrash tou U2

U2 = J2

a

(Rea

)2 (12 3

4sin2 i

)(1 e2)3/2 (93)

Ap thn parapnw sqsh fanetai ti to dunamik th diataraq pou prokaletai

ap to ishmerin exgkwma th Gh den exarttai ap ta (, ,M). Epomnw,smfwna me ti exissei Lagrange, ta stoiqea (a, e, i) paramnoun stajer katthn knhsh tou dorufrou. Antjeta, oi gramm twn sundsmwn kai twn aydwn

metapptoun arg, me suqnthte

= 3nJ2 cos i2(1 e2)2

(Rea

)2

= 3nJ2(1 5 cos2 i)

4(1 e2)2(Rea

)2(94)

kai h msh knhsh tou dorufrou allzei ste

M = n+3nJ2(3 cos

2 i 1)4(1 e2)3/2

(Rea

)2(95)

dhlad, o 3o nmo tou Kepler ((M)2a3 = ) den isqei akrib. Oi parapnwexissei deqnoun ti o prosanatolism tou epipdou th troqi en doruf-

rou allzei me to qrno. Aut mpore na prokalsei poll problmata kaj,

sunjw, jloume o dorufro na pern periodik pnw ap sugkekrimno tpo

49

th Gh to eppedo th troqi tou na diathre stajer gwna me th diejunsh

Gh - 'Hlio. Profan, to ksto se kasima gia diatrhsh th troqi tou doru-

frou nanti th diataraq J2, mpore na enai apagoreutik meglo. Gia to lgoaut sqedizoume eidik troqi, pou upernikon ti diataraq. Qarakthristik

paradegmata enai oi troqi tpou Molniya kai oi hliosgqrone troqi.

Pardeigma 1: Oi troqi tpouMolniya sqedisthkan me skop thn kalterhepikoinwna dorufrou - Stajmo Bsh, se megla gewgrafik plth (ra, me

meglh tim tou i). Gia na paramenei o dorufro pnw ap megla plth giameglo mro th peridou perifor tou, ja prpei (i) na qei arket meglh

ekkentrthta kai (ii) to apgei tou na brsketai diark pnw ap ton epijumht

tpo. Enai dunat na epiteuqje h en lgw gewmetra;

Gia na diathre stajer tim to apgeio th troqi tou dorufrou, ja prpei

h suqnthta metjesh tou perigeou na mhdeniste. H sqsh (94) deqnei ti aut

ja sumbe an h klsh th troqi tou ikanopoie thn exswsh 1 5 cos2 i = 0 pouqei lsh

ic = 63.4

Aut h tim onomzetai krsimh tim th klsh kai antistoiqe se mia troqi pa-

gwmnou apogeou. An epiplon jloume h suqnthta perifor tou dorufrou

na enai sh (troqi Tundra) mis (troqi Molniya) ap aut th Gh, h sqsh(95) ma deqnei ti, anloga me thn tim tou a, h ekkentrthta th troqi jaenai th txh tou e 0.7.

Pardeigma 2: Oi hliosgqrone troqi apotelon eidik upokathgora twn

polikn troqin (i 90) kai qoun efarmog sth suneq paratrhsh olklhrhth epifneia th Gh, me stqo thn katagraf gewdaitikn, metewrologikn k.a.

dedomnwn. Epomnw, mia tupik polik troqi prpei na qei mikr yo (kal

diakritik ikanthta) kai mikr ekkentrthta (stajer diakritik ikanthta). Enai

dunat na epitqoume, epiplon, stajer fwtism tou stqou kat th dirkeia th

apostol;

Gia na qoume stajer fwtism, ja prpei to eppedo th troqi tou doru-

frou na strbei me ton dio rujm pou metablletai h orj anafor tou msou

'Hliou, ste h eujea Gh - 'Hlio na periqetai sto eppedo th troqi tou do-

rufrou. Epomnw, h suqnthta metptwsh th gramm twn sundsmwn th

troqi tou dorufrou, ja prpei na enai sh me

= 360/yr

Gia mia tupik polik troqi me apgeio se yo Ha = 1000 km kai pergeio seyo Hp = 500 km h ekkentrthta enai sh me e = 0.03057. Qrhsimopointa thn

prth ap ti exissei (94) kai jtonta = 360/to, prokptei ti h klshtou epipdou th troqi prpei na qei tim

50

iss = 98.368

pou shmanei ti h hliosgqronh troqi den enai akrib polik (90). H tim thiss exarttai ap ti tim twn a kai e.

Diataraq lgw palrroia

'Opw edame sto Keflaio 2, h epdrash th Selnh (kai tou 'Hliou) sth Gh

prokale palrroie. Epomnw, h katanom mza th Gh, pw fanetai ap ton

dorufro, metablletai arg me to qrno, kaj oi wkeano diogknontai kat

th diejunsh Gh -Selnh. Mlista, epeid h troqi th Gh qei klsh 5 wpro ton ishmerin th Gh, o xona summetra th palrroia sqhmatzei gwna

5 me ton xona peristrof th Gh. Aut h metabol th katanom mza thGh prokale mikr diataraq sthn troqi tou dorufrou. A jewrsoume mia

sqedn sfairik Gh, h opoa apoteletai ap sfairik purna kai na klufo pou

paramorfnetai lgw palrroia (wkeano). H exswsh th epifneia isorropa

th Gh brsketai exisnonta thn palirroiogno dnamh Ft pou aske h Selnh me

th dnamh th barthta th Gh se apstash R ap to kntro th. An = 90(bl. Sqma 7) enai h gwna pou sqhmatzei to dinusma jsh en tpou sthn

epifneia th Gh me thn eujea Gh -Selnh (xona summetra th palrroia),

to topik yo th epifneia isorropa dnetai ap th sqsh

R() = R0

[1 +

g

2(3 cos2 1)

]

pou to plto th palrroia kai R0 h msh aktna th Gh. Gia na shmeo(r, , ) pou brsketai xw ap thn katanom mza th Gh, p.q. sth jsh toudorufrou, to dunamik th paramorfwmnh Gh dnetai ap th sqsh

V(r, ) = 4GR30

3

[1

r+3

5 g(3 cos2 1)

](96)

pou h msh puknthta th Gh. H parapnw sqsh deqnei ti to dunamikth Gh sth jsh tou dorufrou enai so me to jroisma tou msou dunamiko

/r kai tou palirroiognou dunamiko Vt. To sunolik dunamik den enai piakentrik, kaj exarttai ap thn polik gwna , dhlad, th gwna pou sqhmatzeito dinusma jsh tou dorufrou me ton xona pou enai kjeto sto eppedo th

troqi th Selnh.

Aerodunamik Trib - Aeropdhsh

Gia troqi tpou LEO, pou diapernon thn anterh atmsfaira th Gh, h epdra-sh th aerodunamik trib enai shmantik. 'Estw h puknthta tou ara stosugkekrimno yo, thn opoa jewrome stajer gia mikr metabol tou you.

H atmosfairik trib pou epibradnei ton dorufro dnetai ap th sqsh

51

FT =1

2 v2CTS (97)

pou v h taqthta tou dorufrou, S h kjeth (w pro v) diatom th epifneitou kai CT o suntelest aerodunamik trib tou uliko tou dorufrou. Hepibrdunsh (an monda mza) lgw trib ja dnetai ap th sqsh T = FT/ms,pou ms h mza tou dorufrou. An h troqi tou dorufrou enai kuklik kaijewrsoume ti paramnei kuklik kat th dileus tou ap ta antera strmata

th atmsfaira, tte ap ti exissei tou Gauss prokptei ti

v T = v3CTS

2ms

kai o rujm metabol th aktna th troqi dnetai ap th sqsh

da

dt= a CTS

ms(98)

H arq th aeropdhsh sthrzetai akrib sthn epdrash th aerodunamik

trib. 'Estw na dorufro pou ektele diaplanhtik apostol kai epijumome

na prosedafiste sthn epifneia tou 'Arh. An kaj plhsizei sta ria th (arai-

) areian atmsfaira, stryei ta ktoptr tou kjeta pro th diujunsh th

taqtht tou, tte h energ diatom tou, S, auxnei shmantik, me apotlesmah aerodunamik trib na ton epibradnei (aeropdhsh). O rujm metabol th

aktna th troqi tou exarttai ap th gwna strof twn katptrwn (dhlad

thn tim tou S), thn opoa mporome na metablloume ste na epitqoume omalprosgewsh.

H aeropdhsh brskei epsh efarmog ston upologism th troqi jan-

tou en ginou dorufrou. 'Otan na dorufro se troqi LEO oloklhrseithn apostol tou, tte ja tou qei apomenei elqisth enrgeia, ikan mli gia

na strafon ta ktoptr tou pro kpoia epijumht diejunsh. Mporome e-

pomnw na upologsoume thn epijumht gwna strof, tsi ste h metabol

th aktna th troqi tou na enai ttoia pou ja ton odhgsei se sgkroush

me kpoio asfal shmeo sthn epifneia th Gh - p.q. ston Eirhnik wkean.

Qarakthristik pardeigma ttoiou upologismo apotele h troqi jantou tou

diasthmiko stajmo MIR, o opoo eqe teje se troqi ap thn prhn Sobieti-k 'Enwsh. 'Otan epibebaijhke ti o MIR ja psei sth Gh, mia diejn omdaepisthmnwn upolgise th diaspor taquttwn twn meglwn jrausmtwn, ta o-

poa ja dhmiourgontan ap thn epdrash tso th aerodunamik trib so kai

th Ginh palrroia sto skfo tou MIR. Oi elafr diaforetik arqik ta-qthte ja odhgosan ta jrusmata se ptsh ent mia perioq th Gh me

diastsei 6.000 ep 200 qilimetra! Smfwna me tou diou upologismo, h pe-

rioq aut brskontan ston Eirhnik wkean, anmesa ap ti akt th Qil

kai th Wkeana. 'Opw apodeqthke sth sunqeia, oi upologismo tan, eutuq,

swsto...

52

Pesh hliak aktinobola

H pesh th hliak aktinobola FR ofeletai sth metabol th orm twn fw-

tonwn pou prospptoun kjeta sthn epifneia tou dorufrou. H dnamh aut

asketai kat th diejunsh th gramm 'Hlio - dorufro kai qei mtro

|FR| = FeACPc

(99)

pou A h probol th diatom tou dorufrou kjeta sth gramm 'Hlio - dorufro,CP o suntelest aporrfhsh tou uliko tou skfou (0 < CP < 2) kai c h ta-qthta tou fwt sto ken. Gia plrw aporrofhtik ulik (mlan sma) isqei

CP = 1 en, gia plrw anaklastik ulik, isqei CP = 2. To smbolo Fe ana-parist th fwtein ro (isq an monda epifneia) pou ftnei sthn apstash

th Gh kje mra kai dnetai ap thn kfrash

Fe =1358

1.0004 + 0.0334 cosDW/m2 (100)

pou D h hliak fsh th Gh, metromenh ap th jsh tou afhlou th tro-qi th (4 Ioulou gia to to 2004). H msh tim tou lgou P = Fe/c enaiP = 4.5106 kg/(ms2). Sthn parapnw anlush agnosame th suneisfor twnintwn tou hliako anmou, kaj h sunolik tou orm sth geitoni th Gh enai

perpou 100 1000 for mikrterh ap aut twn fwtonwn kai twn energhtiknswmatidwn.

Asksei

1. An w wflimh embleia en dorufrou orsoume to qronik disthma kat

to opoo h gwnidh apstash tou dorufrou ap to Stajm Bsh enai

mikrterh ap = 45, upologste thn wflimh embleia (i) gia mia ku-klik troqi tpou GEO kai (ii) gia mia troqi tpou Molniya me e = 0.7.Gia thn deterh perptwsh, jewrste ti o SB brsketai akrib ktw ap

to apgeio th troqi.

2. Teqnht dorufro, mza 2, 000 kg kai diatom S = 2 m2, kinetai sekuklik troqi tpou LEO, you H = 400 km, up thn epdrash aerodu-namik trib. Jewrnta ti h troqi paramnei diark kuklik, brete

to qronik disthma pou apaitetai ste to yo th troqi na meiwje sto

mis th arqik tim tou, w sunrthsh th puknthta th atmsfaira

kai tou suntelest trib, CT .

53

Qrsimh bibliografa

Gia perisstere plhrofore pnw se jmata Ournia kai Diasthmik Mhqani-

k kai geniktera sth Fusik tou hliako sustmato, protenontai ta paraktw

suggrmmata:

Q. Brboglh, N. Sprou kai B. Mparmpnh, Problmata Astronoma, Ek-dsei PHGASOS 2000, Jessalonkh.

I. Qatzhdhmhtrou, Jewrhtik Mhqanik, Tmo A', Ekdsei Giaqodh,Jessalonkh, 2000.

R.G. Madonna, Orbital Mechanics, Krieger Publishing Company, Inc., 1997. C.D. Murray, S.F. Dermott, Solar System Dynamics, Cambridge UniversityPress, 1999.

D.A. Wallado, Fundamentals of Astrodynamics and Applications, KluwerAcademic Publishers, 2001.

M.J. Sidi, Spacecraft Dynamics & Control, Cambridge University Press,1997.

B. Bertotti, P. Farinella and D. Vokrouhlicky, Physics of the Solar System,Kluwer Academic Publishers, 2003.

I. de Patter and J.J. Lissauer, Planetary Sciences, Cambridge UniversityPress, 2001.