Why do planets move on ellipses?
Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.
Saidia kutafsiri →Swali
Planets move around the Sun on ellipses, with the Sun at one focus. Why ellipses — and not circles, egg shapes or spirals?
Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.
Jaribio
Mikunjo ya kipolari
Kila pembe θ hutoa umbali r kutoka asili.
- orbit, Sun at the origin
- (0.242, 0.242) · r = 0.342
Vidokezo, ukivitaka
Kinachoendelea
Kinachoendelea
Johannes Kepler found the ellipses in 1609, by fitting Tycho Brahe's careful observations of Mars. He did not know why. Isaac Newton explained it in the Principia (1687): if the force pulling a planet towards the Sun has strength proportional to , then the planet's path is a conic section with the Sun at a focus. In polar coordinates centred on the Sun,
where is the eccentricity: gives a circle, an ellipse, a parabola and a hyperbola. Which one occurs depends on the planet's energy. Comets on long ellipses return; an object with too much energy passes once along a hyperbola and leaves — as the interstellar visitor 'Oumuamua did in 2017.
Ellipses are special. In 1873 Joseph Bertrand proved that among forces that depend only on distance, only two make every bounded orbit close up: the inverse-square law and a spring-like force proportional to . With any other law, orbits trace rosettes.
Real orbits are not perfect ellipses: the other planets tug, and the ellipse of Mercury slowly turns. Almost all of that turning is explained by Newton's law, but an extra arcseconds per century was explained only in 1915, by Einstein's general relativity.
Hii inaelekea wapi
- Shahada ya kwanza (mwanzo) · Somo linapatikanaClassical mechanicsNewton's laws as differential equations, conservation of energy and momentum, oscillators, central forces and Kepler's laws, and an introduction to Lagrangian mechanics.
- Mwaka wa kwanza wa uzamili · Somo linapatikanaThe geometry of spacetime: special and general relativityPhysics as geometry: Minkowski spacetime and the Lorentz group, proper time and causal structure, Lorentzian manifolds, the equivalence principle, Einstein's field equations, the Schwarzschild solution, geodesics of light and matter, and cosmological models.
- Njia za utafiti · Somo linapatikanaSymplectic geometry and topologyOptional research direction: the geometry of classical mechanics. Symplectic manifolds and Darboux's theorem, Hamiltonian flows, Moser's trick, moment maps, Gromov's non-squeezing theorem, the Arnold conjecture and Floer homology, and a glimpse of mirror symmetry.