Sharuɗɗan Kepler

A kimanin shekara ta 1600, Johannes Kepler ya yi shekaru da yawa yana kokarin sanya taurãri su motsa cikin ɗaruruwan cikakkiyar. An ki yarda da ganin Mars ya yi daidai. Abin da ya samu a matsayinsa na uku dokoki da har yanzu ke jagorantar kowane jirgin sama da muka fara.

An ki yarda da su a cikin runduna. Sai Kepler ya musanya tãtsuniyoyi.

Look

A cikin 1600, Johannes Kepler ya yi aiki tare da shekaru masu yawa na binciken da Tycho Brahe ya yi na Mars. Kuma dukansu sunã zaton taurãri sunã gudãna a cikin ƙõramu. An ki yarda da su. Bayan shekaru da yawa na aiki, Kepler ya gano dokoki uku waɗanda suka dace.

1. Orbits are ellipses (stretched circles), with the Sun a little off-center at a point called a focus.
2. A planet accelerates up near the Sun and slows down far from it. Draw a line from the Sun to the planet: it sweeps out equal areas in equal times.
3. Mafi girma orbits dauki da yawa longer. Ba kawai mafi tsawo saboda hanyar ne mafi tsawo, amma da yawa mafi tsawo.

A cikin akwatin tsaunuka na sama, an loda Halley’s Comet. Ka dũba zuwa ga rãnã tanã gudãna zuwa ga wani sarari nãta. Kuma waɗanda suka yi ĩmãni, sunã da wata hujja bayyananna daga Ubangijinsu. Lalle ne sũ, sunã da wani abu mai kama da abin da aka sani game da su. Yanayin daidai, lokaci daidai. Wancan ne hukunci na biyu a cikin aiki.

Understand

An bayyana wani ellipse da lambobi biyu:

• Hafin-mai girman-axis a: rabin mafi tsawo. Yana daidaita girman.
• Excentricity e: yadda aka tsawaita shi. 0 wani daki ne; kusa da 1 yana da tsawo sosai kuma mai kauri.

The closest point to the Sun is the periapsis (perihelion for the Sun), at distance a(1 − e). A cikin wannan yanayin, apoapsis (aphelion) yana da a(1 + e).

Object	a	da
Duniya	1.000	0.017
Dakata	1.524	0.093
star name	17.9	0.967

Dokar uku. Ga duk abin da ke tafiyar hawainiya da rana, karfin lokaci yana daidai da kwance na maɓallin gefen babban. A cikin shekaru da au:

T² = a³

Jupiter: a = 5.20 au, don haka T = 5.20^1.5 = shekaru 11.9. Halley: a = 17.9 au yana ba da T = 75.9 shekaru, kusa da mai suna 75 zuwa 76-year dawowa.

The sandbox shows T²/a³ in years² per au³. Change the orbit around the Sun as much as you like: it stays 1.

Master

Newton ya samo dokoki uku daga dokarsa ta jujjuyawar jiki. Wannan shi ne hukunci na uku.

T = 2π · √(a³ / μ),   μ = G(M + m)

To T²/a³ = 4π²/μ shi ne daidai ga kowace orbit a kusa da wani jiki mai tsakiya, amma daban-daban ga wasu abubuwa. An canza jikin sandbox na tsakiya zuwa ƙasa kuma ma’aunin ya zama kusan 9.9 × 10⁻⁵ s² km⁻³ a matsayin 1 yr² au⁻³. Wannan ya sa kepler’s version na Jupiter, wanda yake da nauyin 0.1% na na rana, yana da nisa daga gaskiyar.

Dokar biyu ita ce ta tsarewar motsi na gefe. Yanayin da ake samu a cikin wannan yanayin shine: dA/dt = h/2, inda h = r²·dν/dt kenan motsin gefen, mai daidaituwa ga duk wani karfi na tsakiya. A periapsis da apoapsis, gudun yana daidai da radius, don haka r_p·v_p = r_a·v_a: juzu’in gudun ya daidaita da juzu’in nisan. Ga Halley, r_a/r_p = (1 + e) / (1 − e) ≈ 60, don haka yana motsa kusan sau 60 cikin sauri a perihelion fiye da aphelion.

Domin sanya wani abu a kan orbit dinsa a lokaci mai dacewa kana bukatar matsakaicin rashin daidaito M = n(t − t_p), tare da matsakaicin motsi n = 2π/T, sa’an nan kuma daidaiton Kepler, M = E − e sin E. Ba ta da mafita mai rufi; kayan aikin a wannan shafin suna warware shi ta hanyar sake maimaita Newton, wanda ke haɗuwa cikin matakai kaɗan har ma ga Halley’s e = 0.967.