Maitiro eKepler

Pashure pemakore 1600, Johannes Kepler akaedza kuita kuti maplanethi aite zvidimbu zvakaenzana. Kuongororwa kweMars kwakakanganwa kusangana. Chii akazowana panzvimbo ndechimwe chemaitiro matatu ayo anoenderera achitungamirira chero chikepe chepasi chedu.

Mazita akabvuma kuti aite sezvidimbu. So Kepler akachinja macircles.

Ona

Pashure pemakore 1600, Johannes Kepler akashanda nemakore ezvinyorwa zvemabhuku aTycho Brahe. Vanhu vese vakafunga kuti maplanethi aifamba mumigwagwa yakanyatsojeka. Mazita akabvuma kuti arege kuiswa. Pashure pemakore ebasa, Kepler akawana matatu mashoko akakodzera.

1. Orbits are ellipses (yakapetwa macircles), nezuva anenge off-center panzvimbo inonzi a focus.
2. Planet inofamba nekukurumidza pedyo nezuva uye inofamba zvishoma pedyo narwo. Dzokera mhiri kubva kuzuva kuenda kune planet: inofamba munzvimbo dzakaenzana munguva imwecheteyo.
3. Makuru orbits kutora nguva yakareba. Haisi chete kwenguva refu nekuti nzira iri zhinji, asi zvakasiyana-siyana kwenguva refu.

Muchirongwa ichi, Halley’s Comet inoonekwa seimwe yezviratidzo zvezuva. Ona kuti inofamba sei pamusoro pezuva uyezve ichitambanudza pasi pasi muchikamu chekunze cheobiti yayo. Zvidimbu zvakashongedzwa zvinofamba munguva imwechete: zvidiki uye zvakareba zviri kure, zvidiki uye zvakakura zviri pedyo. Same area, same time. Chii chinonzi pfungwa?

_Zvinyorwa

Ellipse inoratidzwa nemakumi maviri nemaviri ezvinyorwa:

• The semi-major axis a: half the longest width. Inoita kuti uone saizi.
• The eccentricity e: sei yakadzikama. 0 icho chidimbu chemativi maviri, uye chinodarika 1 chine hukuru hwakawanda.

Iyo nzvimbo iri pedyo nezuva ndeye periapsis (perihelion yezuva), panzvimbo a(1 − e). The farthest point is the apoapsis (aphelion), pa a(1 + e).

Chigadzirwa	a (au)	e
Earth	1.000	0.017
Mars	1.524	0.093
Halley’s Comet	17.9	0.967

Mutemo wechitatu. Kuti uone kuti zvinhu zvese zvinofamba mumigwagwa yezuva, square yenguva inoenderana necube ye semi-makuru axis. Mumakore uye au:

T² = a³

Jupiter: a = 5.20 au, saka T = 5.20^1.5 = 11.9 makore. Halley: a = 17.9 au inovimbisa T = 75.9 makore, pedyo neyakakurumbira 75-76-gore rekudzoka.

Iyo sandbox inoratidzwa T²/a³ mumakore² paau³. Iva nesimba rekuchinja kureba kwezuva sezvaunogona: inogara iri 1.

Munyori

Newton akawana zvese zvitatu zvemasimba kubva pakubata kwake. Chikamu chechitatu chemutemo mufananidzo wakadzama ndewekuti:

T = 2π · √(a³ / μ),   μ = G(M + m)

Saka T²/a³ = 4π²/μ imwechete kune ese ma orbits anenge ari pedyo nechiedza chepakati, asi inoenderana nechiedza chema body akasiyana. Kana tikachinja chimiro chepakati chesandbox kuenda kunyika, iyo yakafanana ichava 9.9 × 10⁻⁵ s² km⁻³ kwete 1 yr² au⁻³. M mu μ ndiyo imwe yezvikonzero nei Kepler’s version iri kubuda pachena nezveJupiter, iyo ine masendimita 0.1% ezuva.

Chirevo chekutanga chinonzi angular momentum. Iyo yepasi pevhu velocity ndeye dA/dt = h/2, uko h = r²·dν/dt ndiyo yakasarudzika angle momentum, yakaenzana kune chero simba repakati. Pa periapsis neapoapsis, simba remhepo rinogara rakazvimiririra nesimba remhepo, saka r_p·v_p = r_a·v_a: simba remhepo rinoenzaniswa nenguva yekutenderera. Kune Halley, r_a/r_p = (1 + e) / (1 − e) ≈ 60, saka inofamba nenguva 60 nguva nekukurumidza paperihelion kupfuura paaphelion.

Kuti uise chinhu panzvimbo yayo yeorbit panguva yapihwa iwe unofanirwa kushandisa iyo mean anomaly M = n(t − t_p), neyakaenzana kutenderera n = 2π/T, uyezve equation yeKepler, M = E − e sin E. Hapana chikonzero chekuvhurwa; maturusi ari pawebhusaiti anogadzirisa neNewton iteration, iyo inosangana munzira pfupi kunyangwe kune Halley’s e = 0.967.