Изворни код странице Problem dva tela

[[Датотека:orbit5.gif|мини|300px|desno|Dva nebeska tela sličnih masa okreću se oko zajedničkog [[baricentar|baricentra]] (centra mase) s eliptičnim putanjama (tipično za [[dvojne zvezde]]).]]
[[Датотека:orbit4.gif|мини|300px|desno| [[Baricentar]], [[težište]] ili središte mase, u astronomiji je tačka oko koje se kreću tela dvojnog ili višestrukog sistema. Na slici je sličan sistem kao [[Sunce]] - [[Zemlja]].]]
[[Датотека:orbit3.gif|мини|300px|desno|Baricentar sistema Zemlje i [[Mesec]]a nalazi se unutar Zemljine površine na udaljenosti 4670 [[metar|km]] od središta. Ta se tačka sistema, a ne središte Zemlje, kreće po [[Zemljina putanja|eliptičnoj stazi oko Sunca]].]]
[[Датотека:orbit2.gif|мини|300px|desno|Dva nebeska tela različitih masa okreću se oko zajedničkog [[baricentar|baricentra]] (centra mase). Na slici je sličan sistem kao [[Pluton]] - [[Haron (mesec)|Haron]].]]

'''Problem dva tela''', ili tаčnije rečeno '''gravitacijski problem dva tela''', osnova je [[nebeska mehanika|nebeske mehanike]].<ref>{{cite book|author=Curtis, Howard D.|title=Orbital Mechanics for Engineering Students, 2e|publisher=Elsevier|location=New York|date=2009|isbn=978-0-12-374778-5}}</ref><ref>{{cite book |last=Bate |first=Roger R. |author2=Mueller, Donald D. |author3=White, Jerry E. |title=Fundamentals of Astrodynamics |date=1971 |publisher=Dover Publications |location=New York |isbn=0-486-60061-0 |pages= |url=https://archive.org/details/fundamentalsofas00bate }}</ref> Primenjuje se kod [[Planetarna putanja|kretanja planeta oko Sunca]],<ref>{{Cite encyclopedia |url=https://www.britannica.com/EBchecked/topic/431123/orbit |title=orbit (astronomy) |encyclopedia=Encyclopædia Britannica |edition=Online |access-date=28 July 2008 |archive-date=5 May 2015 |archive-url=https://web.archive.org/web/20150505012919/https://www.britannica.com/EBchecked/topic/431123/orbit |url-status=live }}</ref><ref>{{Cite web |url=http://spaceplace.nasa.gov/barycenter/ |title=The Space Place :: What's a Barycenter |access-date=26 November 2012 |archive-date=8 January 2013 |archive-url=https://web.archive.org/web/20130108073405/http://spaceplace.nasa.gov/barycenter/ |publisher=NASA |url-status=live }}</ref> kretanja [[prirodni satelit|prirodnih satelita]], te [[dvojne zvezde|dvojnih zvezda]]. Kod proučavanja [[Универзални закон гравитације|Njutnovog zakona gravitacije]] (opšteg zakona gravitacije) prećutno se drži da je [[masa]] [[Природни сателит|satelita]] zanemariva u odnosu na masu središnjeg tela (-{''m ≪ M''}-). Takvo [[kretanje]] može se razmatrati kao '''problem jednog tela''', a njegovo tumačenje je, svakako, najjednostavnije. Pretpostavka nije ispunjena već u sistemu [[Zemlja|Zemlje]] i Meseca. Iako [[Mesec]] ima 81 put manju masu nego [[Zemlja]], njegov je uticaj na kretanje Zemlje oko Sunca merljiv. Problem dva tela je proučavanje kretanja u sistemu dva tela ako odnos njihovih masa nije beskonačan ili jednak nuli. Kod problema dva tela tačno vrede [[Keplerovi zakoni]].<ref name=":0">{{Cite web |title=Kepler's Laws |url=http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html |access-date=2022-12-13 |website=hyperphysics.phy-astr.gsu.edu}}</ref><ref name=":1">{{Cite web |title=Orbits and Kepler's Laws |url=https://solarsystem.nasa.gov/resources/310/orbits-and-keplers-laws |access-date=2022-12-13 |website=NASA Solar System Exploration}}</ref><ref>{{Cite book|ref=harv|last=Goldstein|first=H.|author-link = Herbert Goldstein |year=1980| title = [[Classical Mechanics (textbook)|Classical Mechanics]] | edition = 2nd. | publisher = [[Addison-Wesley]] | location = New York |isbn=978-0-201-02918-5}}</ref>

[[Problem tri tela]] u nebeskoj mehanici, za razliku od problema dva tela, nema opšte analitičko rešenje. Ograničeni oblik problema razmatra kretanje tri tela, s time da je treće telo tačkasto i bez mase. Za treće je telo [[Joseph-Louis Lagrange|Žozef Luj Lagranž]] našao da može neporemećeno da opstane u sistemu, na položaju 5 tačaka u ravni u kojoj se sva tela kreću ([[Лагранжова тачка|Lagranžove tačke]]). Potvrda je toga postojanje [[Тројански астероид|Trojanskih planetoida]], koji se nalaze na [[Jupiter]]ovoj stazi, 60° ispred i iza Jupitera, a slično se ponašaju i neki planetni sateliti. Kako u [[Sunčev sistem|Sunčevom sistemu]] ima mnogo tela, ustanovljeno je da je staza svakoga tela poremećena ostalim telima, i to tim jače što je telo manje mase. Zato su Keplerovi zakoni samo približni. Otkloni su mali jedino zbog toga što su mase svih tela mnogo manje od Sunčeve. Nakon [[Isaac Newton|Isaka Njutna]], nebeska mehanika razvijala se u matematičkoj obradi poremećaja (perturbacija), kao otklona od [[matematika|matematičkog]] rešenja problema dva tela, što zapravo znači otklon od [[elipsa|elipse]]. Budući da su poremećaji mali, koristi se elipsa kojoj se parametri postupno menjaju; trenutna se elipsa naziva oskulirajućom. [[Diferencijalne jednačine]]<ref name="Betounes">{{cite book|title=Differential Equations|last=Betounes|first=David|url=https://books.google.com/books?id=oNvFAzQXBhsC&pg=PA58|isbn=978-0-387-95140-9|year=2001|publisher=Springer|pages=}}</ref> koje izražavaju vremenske promene svih parametara elipse izveo je Žozef Luj Lagranž (Lagranžove planetarne jednačine); one su tačne (egzaktne), ali mogu se rešiti jedino numerički, uzastopnim približavanjima (sukcesivnim aproksimacijama), i to za ograničeno vremensko razdoblje.<ref>''Nebeska mehanika'', [http://www.enciklopedija.hr/Natuknica.aspx?ID=43187] "Hrvatska enciklopedija", Leksikografski zavod Miroslav Krleža, www.enciklopedija.hr, 2014.</ref>

== Dva nebeska tela različitih masa ==

U najjednostavnijem slučaju dva se tela kreću koncentričnim [[kružnica]]ma. [[Centrifugalna i centripetalna sila|Centripetalno ubrzanje]] uzrokovano je [[gravitacija|gravitacionom silom]]. [[Универзални закон гравитације|Sila]] između masa -{''M<sub>1</sub>''}- i -{''M<sub>2</sub>''}- uzajamna je i jednaka:

::<math>F = G \frac{M_1 M_2}{r^2}\ </math>

[[Ubrzanje]] jednog tela je:

::<math>g_1 = G \frac{M_2}{r^2}\ </math>

a ubrzanje drugog tela je:

::<math>g_2 = G \frac{M_1}{r^2}\ </math>

Svako pojedino ubrzanje određeno je masom onog drugog tela. Sila između tela je stalna i oba su ubrzanja stalna, ako je razmak tela -{''r''}- stalan, što je ispunjeno kod koncentričnih kružnih staza, pa je tada i [[brzina]] svakog tela stalna. Tela u jednako vreme obiđu svako po svojoj kružnici. Kada toga ne bi bilo, jednom bi se tela susrela na bližim delovima staza, drugi put na udaljenijim, pa ni sila ne bi bila stalna. Zato se tela moraju uvek nalaziti na dijametralno suprotnim tačkama svojih staza i zbir poluprečknika staza jednak je razmaku tela:

::<math>r_1 + r_2 = r </math>

A kako tela obiđu staze u isto vreme, [[Ophodno vreme|ophodne brzine]] su u istom odnosu u kojemu su [[obim]]i ili [[poluprečnik|poluprečnici]] staza:

::<math>\frac{v_1}{v_2} = \frac{r_1}{r_2} </math>

Izjednačavanjem centripetalnih ubrzanja za svako telo s ubrzanjem gravitacijske sile, tako da je brzina izražena poluprečnikom i [[period]]om ophoda staze -{''P''}-, dobija se:

::<math>g_1 = \frac{v_1^2}{r_1}\ = \frac{4 \pi^2 r_1}{P^2}\ = \frac{G M_2}{r^2}\ </math>

::<math>g_2 = \frac{v_2^2}{r_2}\ = \frac{4 \pi^2 r_2}{P^2}\ = \frac{G M_1}{r^2}\ </math>

Odnos desnih jednakosti iskazuje veoma važnu činjenicu:

::<math>\frac{r_1}{r_2} = \frac{M_2}{M_1} </math>

Razmak tela od zajedničkog centra kruženja obrnuto je srazmeran masama tih tela. To je svojstvo koje pokazuje centar mase ili težište nekog sistema masa. Kada se neko složeno telo nalazi u gravitacijskom polju, kretanje tog tela kao celine odvija se kao da je sva masa postavljena u centar mase. U slučaju svemirskog sistema sastavljenog od dva tela njihov će zajednički centar mase ili mirovati, ili će se [[Jednoliko pravolinijsko kretanje|kretati jednoliko po pravcu]] (ako na njega ne deluju sile drugih [[nebesko telo|nebeskih tela]]). Sama tela će obilaziti oko centra mase. Spojnica dva tela uvek prelazi preko centra mase i ubrzanje je usmereno prema njemu. Lako se može osvedočiti da će sa povećanjem jedne mase na račun druge, centar mase stremiti prema većoj masi. Ubrzanje prvog tela postaje beznačajno, ako je masa drugog tela zanemariva prema masi prvog tela. To je bilo približenje (aproksimacija) učinjeno u problemu jednog tela. Tada se prvo telo, kao da je beskonačne mase, nalazilo u središtu kruženja. Tako se [[Sunce]] zamišlja u središtu kruženja planeta, a planete u središtu kruženja njihovih satelita. 

Zbog relativno velike Mesečeve mase, Zemlja ne obilazi oko Sunca po elipsi. Oko Sunca po elipsi ustvari putuje [[baricentar]]<ref>{{Cite web |title=Center of Gravity - an overview |url=https://www.sciencedirect.com/topics/earth-and-planetary-sciences/center-of-gravity |website=ScienceDirect Topics |quote=barycentre lies 1700 km below the Earth's surface (6370km–1700km)}}</ref> (centar mase) sistema [[Zemlja]] - [[Mesec]], a ne centar Zemlje. Ni Mesečeva staza oko Sunca nije [[elipsa]], ali Mesec na stazi nikada ne čini petlje (iako se tako crta na malim crtežima); štaviše, staza mu nikada nije izbočena (konveksna) prema Suncu. 

Ako zvezda ima tamnog pratioca (planetu velike mase), tada se u vlastitom kretanju zvezde mora javiti uticaj tog pratioca. Put zvezde će da krivuda oko linije kojom se kreće centar mase (baricentar).<ref>[[Vladis Vujnović]] : "Astronomija", Školska knjiga, 1989.</ref>

[[Датотека:Moon phases en.jpg|мини|center|800px|U trenutku [[Mesečeve mene|prve četvrtine]] Zemlja prethodi centru mase ([[baricentar]]), a [[Mesec]] zaostaje; u trenutku poslednje četvrtine [[Zemlja]] zaostaje na stazi za centrom mase, a Mesec mu prethodi.]]

== Treći Keplerov zakon za sistem dve mase ==
{{Glavni|Treći Keplerov zakon}}
[[Датотека:Sirius A and B Hubble photo.jpg|мини|desno|300px|[[Dvojna zvezda]] [[Sirius]] A i B.]]
Treba sabrati izraze za [[ubrzanje]] prvog i drugog tela, i to njihove desne jednakosti, ali tako da se uvede zbir [[poluprečnik]]a (-{''r<sub>1</sub>'' + ''r<sub>2</sub>'' = ''r''}-):

::<math>g_1 = G \frac{M_2}{r^2}\ </math>

::<math>g_2 = G \frac{M_1}{r^2}\ </math>

Proizlazi:

::<math>\frac{r^{3}}{P^{2}} = \frac{G}{4\pi ^{2}}\cdot (M_{1}+M_{2}) </math>

U konstanti trećeg Keplerova zakona nalazi se ukupna masa dvojnog sistema. 

U uopštenijem slučaju, tela se kretaću po eliptičnim stazama. Pritom su ispunjeni neki geometrijski uslovi. [[Ekscentricitet]] obe staze je jednak, smer velikih poluosa se podudara, a tela su na stazama uvek dijametralno suprotno onom [[žarište|žarištu]] u kojem je centar mase. Kod koncentričnih kružnica koje su razmatrane pre, razmak tela ima istu ulogu koju kod [[elipsa|eliptičnih staza]] ima srednji razmak (zbir velikih poluosa), dok brzina po putanji ima ulogu srednje brzine. 

== Reference ==
{{reflist|}}

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*{{cite book|last1=Lindsay|first1=Robert Bruce|title=Physical Mechanics|url=https://archive.org/details/physicalmechanic0000lind|url-access=registration|edition=3rd|location=Princeton|asin=B0000CLA7B|oclc=802752879|publisher=D. Van Nostrand Co.|year=1961}}
*{{cite book|last1=Meirovitch|first1=Leonard|title=Methods of Analytical Dynamics|url=https://archive.org/details/methodsofanalyti0000meir|publisher=McGraw-Hill Book Co.|year=1970|isbn=978-0-07-041455-6|location=New York}}
*{{cite book|last1=Meyer|first1=Kenneth Ray|last2=Hall|first2=Glen R.|title=Introduction to Hamiltonian Dynamical Systems and the {{mvar|n}}-body Problem|publisher=[[Springer Science & Business Media]]|year=2009|isbn=978-0-387-09724-4|location=New York}}
*{{cite journal|last1=Mittag-Leffler|first1=G.|title=The {{mvar|n}}-body problem (Prize Announcement)|journal=Acta Mathematica|date=1885–1886|volume=7|pages=I–VI|doi=10.1007/BF02402191|doi-access=free}}
*{{cite book|last1=Moulton|first1=Forest Ray|title=An Introduction to Celestial Mechanics|url=https://archive.org/details/introductiontoce0000fore|publisher=[[Dover Publications]]|year=1970|isbn=978-0-486-62563-8|location=New York}}
*{{cite book|oclc=915353069|author-link=Isaac Newton|last=Newton|first=Isaac|title=Philosophiae Naturalis Principia Mathematica|url=https://archive.org/details/philosophiaenatu00inew|location=Londini [London]|language=Latin|publisher=Jussu Societatis Regiæ ac Typis Josephi Streater. Prostat apud plures Bibliopolas.|year=1687}} Also English translation of 3rd (1726) edition by I. Bernard Cohen and Anne Whitman (Berkeley, CA, 1999). {{Isbn|978-0-520-08817-7}}.
*{{cite journal|last1=Ram|first1=Parikshit|date=2009|first2=Dongryeol|last2=Lee|first3=William B.|last3=March|first4=Alexander G.|last4=Gray|url=http://www.fast-lab.org/papers/ram2010ltaps.pdf|title=Linear-time Algorithms for Pairwise Statistical Problems|journal=NIPS|pages=1527–1535|access-date=2014-03-28|archive-url=https://web.archive.org/web/20170421013700/http://www.fast-lab.org/papers/ram2010ltaps.pdf|archive-date=2017-04-21|url-status=dead}}
*{{cite book|last1=Rosenberg|first1=Reinhardt M.|title=Analytical Dynamics, of Discrete Systems|location=New York|chapter=Chapter 19: About Celestial Problems|pages=364–371|publisher=Plenum Press|year=1977|isbn=978-0-306-31014-0}}
*{{cite book|author=Gallant|first=Roy A.|title=The Nature of the Universe|publisher=[[Doubleday (publisher)|Doubleday]]|year=1968|location=Garden City, NY|oclc=689289|agency=In partnership with [[Society for Science|Science Service]]}}
*{{cite journal|last=Sundman|first=K.&nbsp;F.|title=Mémoire sur le problème de trois corps|journal=Acta Mathematica|volume=36|date=1912|pages=105–179|doi=10.1007/bf02422379|url=https://zenodo.org/record/2440705|doi-access=free}}
*{{cite journal|last1=Tisserand|first1=François Félix|url=https://iris.univ-lille.fr/bitstream/handle/1908/4228/Q11840-3.pdf?sequence=4|title=Traité de Mécanique Céleste|journal=Lilliad - Université de Lille - Sciences et Technologies|volume=III|location=Paris|publisher=Gauthier-Villars Et Fils|year=1894|page=27|hdl=1908/4228|oclc=951409281}}
*{{cite journal|last1=Trenti|first1=Michele|last2=Hut|first2=Piet|date=2008|title={{mvar|n}}-body simulations|journal=Scholarpedia|volume=3|issue=5|page=3930|doi=10.4249/scholarpedia.3930|bibcode=2008SchpJ...3.3930T|doi-access=free}}
*{{Cite book|last1=Truesdell|first1=Clifford|title=Essays in the History of Mechanics|url=https://archive.org/details/essaysinhistoryo0000true|publisher=[[Springer Science+Business Media|Springer-Verlag]]|year=1968|isbn=978-3-642-86649-4|location=Berlin; Heidelberg}}
*{{cite book|last1=Van Winter|first1=Clasine|contribution=The {{mvar|n}}-body problem on a Hilbert space of analytic functions|oclc=848738761|title=Analytic Methods in Mathematical Physics|editor1-first=Robert P.|year=1970|editor1-last=Gilbert|editor-link=Roger G. Newton|editor2-first=Roger G.|editor2-last=Newton|location=New York|pages=569–578|publisher=Gordon and Breach}}
*{{cite journal |author-link=Qiudong Wang |last=Wang |first=Qiudong |bibcode=1991CeMDA..50...73W |title=The global solution of the {{mvar|n}}-body problem |url=https://archive.org/details/sim_celestial-mechanics-and-dynamical-astronomy_1991_50_1/page/73 |journal=[[Celestial Mechanics and Dynamical Astronomy]] |issn=0923-2958 |volume=50 |issue=1 |year=1991 |pages=73–88 |doi=10.1007/BF00048987 | mr=1117788 |s2cid=118132097 }}
*{{cite journal |first=Zhihong |last=Xia |title=The Existence of Noncollision Singularities in Newtonian Systems |journal=[[Annals of Mathematics]] |volume=135 |issue=3 |pages=411–468 |date=1992 |jstor=2946572 |doi=10.2307/2946572}}
{{Литература крај}}

== Spoljašnje veze ==
{{Commonscat|Two-body problem}}
* [http://scienceworld.wolfram.com/physics/Two-BodyProblem.html Two-body problem] at [[ScienceWorld|Eric Weisstein's World of Physics]]
* [http://web.njit.edu/~gary/430/assets/physics430_lecture18.ppt Two-body Central Force Problems] by D. E. Gary of the [[New Jersey Institute of Technology]]
* [http://www.dartmouth.edu/~phys44/lectures/Chap_4.pdf Motion in a Central-Force Field] {{Wayback|url=http://www.dartmouth.edu/~phys44/lectures/Chap_4.pdf |date=20180921034511 }} by A. Brizard of [[Saint Michael's College]]
* [http://ads.harvard.edu/books/1989fcm..book/Chapter5.pdf Motion under the Influence of a Central Force] by G. W. Collins, II of [[Case Western Reserve University]]
* [http://videolectures.net/mit801f99_lewin_lec22/ Video lecture] by W. H. G. Lewin of the [[Massachusetts Institute of Technology]]

{{нормативна контрола}}

{{СОРТИРАЊЕ:Problem dva tela}}
[[Категорија:Фундаментални концепти физике]]
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[[Категорија:Класична механика]]