Malamulo a Kepler

Pafupifupi zaka 1600, Johannes Kepler anagwiritsa ntchito zaka zambiri pofuna kuti zigawo za dziko lapansi ziziyenda m’mapiri abwino. Kuyang’ana kwa Mars kunakana kugwirizana. Zimene anapeza m’malo mwake ndi malamulo atatu omwe amangoyendetsa zonse zomwe timatha.

Malinga ndi mfundo za m’Baibulo, anthu sayenera kuika maziko a ukwati. Koma Kepler anasintha mfundo zake.

Onani

Pafupifupi mu 1600, Johannes Kepler anagwira ntchito ndi zaka zambiri zoyeserera za Mars zomwe Tycho Brahe adachita. Anthu onse ankaganiza kuti maplaneti amayenda m’mapiri abwino. Malinga ndi mfundo za m’Baibulo, anthuwo sanakhulupirire. Pambuyo pa zaka zambiri za ntchito, Kepler anapeza malamulo atatu omwe amagwirizana.

1. Ma orbits ndi ma ellipses (kuyenda m’mapiri), ndi Sun pang’ono off-center pa gawo lotchedwa kulenga.
2. Planeti imayenda mofulumira pafupi ndi dzuwa ndipo imayenda mofulumira kuchoka pa izo. Zoberani mzere kuchokera ku dzuwa kupita ku planeta: imayenda mofanana nthawi zonse.
3. Ma orbits akuluakulu amatenga nthawi yayitali kwambiri. Sikuti nthawi yaitali chifukwa njira ndi yaitali, koma yochulukirapo.

Pafupi ndi mzindawu, pali malo otchuka a Halley’s Comet. Kuwonetsa kuti imayenda m’mphepete mwa dzuwa ndipo kenako imayenda m’mbali mwa gawo lakunja la orbit yake. Masamba onse amtunduwu amawoneka nthawi yomweyo: owala ndi okongola kwambiri kumbali ina, ofupika komanso osalala kumbali ina. Same malo, nthawi yomweyo. Izi ndi malamulo awiri m’chikhalidwe.

Kumvetsa

Ellipse amafotokozedwa ndi zigawo ziwiri:

• Semi-mawukulu axis a: the half of the longest width. Imakhala ndi kukula kwake.
• The excentricity e: momwe imapitirira. 0 ndi mzere; kuzungulira 1 ndi kwambiri yaitali ndi thinnest.

Mtundu wa mpweya wotentha kwambiri ndi mpweya wotentha kwambiri. periapsis (perihelion kwa dzuwa), pa mtunda a(1 − e). The mbali yaitali ndi apoapsis (aphelion), pa a (1 + e).

Zomwe zili	a (au)	e
Dziko	1.000	0.017
Mars	1.524	0.093
Halley’s Comet	17.9	0.967

Chilamulo chachitatu. Kwa chilichonse chomwe chimazungulira dzuwa, kagawo kanthawi ndi kofanana ndi kubowola kwa axis yayikulu. Mu zaka ndi au:

T² = a³

Jupiter: a = 5.20 au, so T = 5.20^1.5 = 11.9 years. Halley: a = 17.9 au gives T = 75.9 years, close to its famous 75 to 76-year return.

M’madera ena, nyengo ya chilimwe imayamba pa mphindi za m’mawa mpaka maola amadzulo. Mukhoza kusintha kutalika kwa mtunda pakati pa maplaneti monga mukufuna: imakhala 1.

Woyang’anira

Newton anapeza malamulo onse atatu a kuzungulira kuchokera palamulo lake la gravity. Ufulu wachitatu m’njira yokwanira ndi:

T = 2π · √(a³ / μ),   μ = G(M + m)

So T²/a³ = 4π²/μ is the same for every orbit around one central body, but different for different bodies. Kusintha kwa thupi lapakati pa sandbox kupita kudziko ndipo mfundo imakhala pafupifupi 9.9 × 10⁻⁵ s² km⁻³ potengera 1 yr² au⁻³. The m in μ is why Kepler’s version is slightly off for Jupiter, whose mass is about 0.1% of the Sun’s.

Mfundo yachiwiri ndi yoti magetsi amasunga mphamvu yake. Kutalika kwa mtunda pakati pa mapiri ndi dziko lapansi ndi 2h/km, komwe h ndi nthawi yoyenda ya dziko lapansi, ndipo m’malo mwake dn ndi mphindi za mtunda. Pa periapsis ndi apoapsis, kutalika kwa maulendo a ndege kumafanana ndi kuwala kwake, choncho r_p·v_p = r_a·v_a: kutalika kwa maulendo amalingana ndi kuzungulira kwa maulendo. Kuchokera pa mfundo imeneyi, tikupeza kuti: r_a/r_p = (1 + e)/(1 − e) ≈ 60, kotero kuti imayenda m’njira ya 60 nthawi zambiri kuposa momwe imayenda panthawi ya aphelion.

Kuti muike chinthu pa orbit yake nthawi ina muyenera kuganizira za m’mimba mwake anomaly M = n (t − t_p), ndi m’mimba mwake motions n = 2π/T, ndipo kenako Kepler’s equation, M = E − e sin E. Palibe njira yotsegulira; zida patsambali zimatha kuyambitsa ndi Newton iteration, yomwe imagwirizana m’madina angapo ngakhale kwa e ya Halley = 0.967.