Imithetho ka-Kepler

Ngo-1600, uJohannes Kepler wachitha iminyaka ezama ukwenza amaplanethi ahambe ngezinhlayiya eziphelele. Ukubuka kweMars akuzange kuhambelane. Uthola ukuthi kunemithetho emithathu eqhubeka ilawula yonke into esiyikhiphayo.

Amanombolo akazange avumeleke ukuba alungele izingqimba. Ngakho-ke uKepler washintsha amadayimane.

Bona

Ngo-1600, uJohannes Kepler wasebenza iminyaka eminingi yokubheka ngokucophelela iMars eyayenziwe nguTycho Brahe. Wonke umuntu wacabanga ukuthi iziplanethi zihamba ngezinhlayiya eziphelele. Amanombolo akazange alungele. Ngemuva kweminyaka yomsebenzi, uKepler wathola izimiso ezintathu ezifanele.

1. Izibhamu ziyi-ellipsis (izingqimba ezishisayo), nelanga elincane elingenaphakathi endaweni ebizwa ngokuthi yi focus.
2. Iplanethi ihamba ngokushesha eduze kwelanga futhi ihamba kancane kude nalo. Zoba umgwaqo kusuka elanga kuya kwiplanethi: isusa izindawo ezilinganayo ngezikhathi ezilinganayo.
3. Izijikelezo ezikhulu zithatha isikhathi eside kakhulu. Akufanele kube ngaphezu kwalokho ngoba indlela ingaphezulu, kodwa ngaphezu kwesilinganiso.

Kubhokisi le-sandbox phezulu, i-Halley’s Comet ifakwe. Bona ukuthi ihamba kanjani ngasogwini lwelanga bese uzulazula ngengxenye yangaphandle ye-orbit yayo. Iziqongo ezimnyama zonke ziqhutshwa ngesikhathi esifanayo: ezincane nezide eduze, ezincane nezibanzi eduze. Indawo efanayo, isikhathi esifanayo. Lolu wumthetho wesithathu osebenza.

Ukuqonda

I-ellipsis ibhalwe ngamanombolo amabili:

• ingxenye-yesimo esiyinhloko a: inhlamvu yesayizi ende kakhulu. Ihlela ubukhulu.
• I i-eccentricity e: ukuthi iphindwe kanjani. 0 yi-circle; ephakathi kwe-1 iphakeme kakhulu futhi inciphisa.

Ingxenye encane kakhulu yelanga yi- periapsis (i-perihelion yelanga), emgwaqweni a(1 − e). Ingxenye ehamba phambili yi apoapsis (i-aphelion), ku-a(1 + e).

Ingxenye	a	e
Iplanethi	1.000	0.017
i-Mars	1.524	0.093
I-Halley’s Comet	17.9	0.967

Imithetho yesithathu. Kunoma yini ejikeleza ilanga, isikwere sesikhathi silingana ne-cube yengxenye enkulu yomgwaqo. Ngonyaka nosuku:

T² = a³

Jupiter: a = 5.20 au, ngakho T = 5.20^1.5 = 11.9 iminyaka. Halley: a = 17.9 au inikeza T = 75.9 iminyaka, ezungeze unyaka wayo ovelele we-75 kuya ku-76.

Ibhokisi le-sandbox libonisa iT²/a³ eminyakeni² ngayinye ye-au³. Guqula i-orbit ejikeleze uLwandle Mgidlana njengoba ufuna: ihlala 1.

Umphathi

UNewton wathola zonke izimiso ezintathu kusuka kumthetho wakhe we-gravity. Imithetho yesithathu ngokugcwele injalo:

T = 2π · √(a³ / μ),   μ = G(M + m)

Ngakho T²/a³ = 4π²/μ ifana ngaso sonke isibhakabhaka esijikeleze isiqu esisodwa, kodwa sihlukile kulezi ziqu ezahlukene. Ushintshe isiqu sendawo ephakathi yebhokisi le-sandbox kumhlaba futhi isilinganiso sizoba ngama-9.9 × 10⁻⁵ s² km⁻³ endaweni ka-1 yr² au⁻³. I-m ku-μ iyisizathu sokuthi i-Kepler’s version incane kakhulu ngeJupiter, enesisindo esingu-0.1% selanga.

Imithetho yesibili yiyo egcina i-angle momentum. Ijubane lendawo yi dA/dt = h/2, lapho h = r²·dν/dt iyijubane elikhethekile le-angle, elihlala njalo nganoma yiluphi uhlangothi oluphakathi. Ku-periapsis ne-apoapsis ijubane lihamba ngokuqondile nge-radius, ngakho r_p·v_p = r_a·v_a: ukulingana kwejubane kulingana nokulingana komugqa oguqukayo. I-Halley, r_a/r_p = (1 + e) / (1 − e) ≈ 60, ngakho ihamba izikhathi ezingu-60 ngokushesha ku-perihelion kunaku-aphelion.

Ukubeka ilungu e orbitni layo ngesikhathi esinikeziwe, udinga ukuhluka okuphakathi M = n(t − t_p), nge-momentum ephakathi n = 2π/T, bese usebenzisa i-equation kaKepler, M = E − e sin E. Akunasinqumo esivaliwe; amathuluzi kule ndawo axazulula ngokuphinda-phinda kweNewton, okungukuthi, ukuhlangana kwezinqumo ezimbalwa ngisho noHalley’s e = 0.967.