Lagrange points: daidaita a sama

Wani jirgin sararin samaniya wanda yake kusa da rana fiye da na ƙasa ya kamata ya yi tafiyar sa ta sauri kuma ya ragu. Amma kuma James Webb Space Telescope, kilomita miliyan 1.5 daga ƙasarmu, yana ci gaba da tafiya tare da mu duk shekara. Yana zaune a Lagrange point, daya daga cikin wurare biyar inda jan jiki biyu da motsin tafiyar da ke juyawa suna daidaita.

Five parking spots in the sky, and only two of them are free of charge.

Look

Ka yi tunanin wani yanki na ƙasa inda tsayi ke nufin “ina yawan ƙarfin da yake buƙata don zama a nan” ga wani abu wanda ya juya kusa da rana tare da ƙasa:

• Two deep wells at the Sun and at Earth.
• L1 da L2: tsaunuka biyu (saddles) a kan kowane gefen duniya, kusan kilomita miliyan 1.5 daga juna. Mai daidaituwa, amma kawai: ka nutsa jirgin sararin samaniya kuma zai yi gudu. Masu aikin nan suna yin gyaran kai a kowace mako.
• L3: wani saddle a gefen da ke kusa da rana. Bãbu wani abu mai amfãni yanã tafiya a can, dõmin hasken rãnã yã hana shi daga gare mu.
• L4 da L5: tsaunuka biyu masu kyau digiri 60 gaba da baya ga Duniya. Za ka yi tsammanin abubuwa za su yi gudu, amma tasirin Coriolis na juyawa yana mayar da shãlan cikin wani layin. Sun kasance masu daidaituwa, kuma halitta ta cika su: Jupiter’s L4 da L5 suna da fiye da dubu goma Trojan asteroids.

Wane yake zaune a can: SOHO ya lura da rana daga L1 tun 1996; yana ganin guguwar rana kafin su isa mana. Webb yana aiki a kusa da L2, inda rana, ƙasa da wata duka ke zama a gefe guda, bayan bangon ta, yana kiyaye telescope a kusan -230 °C.

Understand

A cikin tsarin da ke juyawa tare da ƙasa, wani abu mai ƙaranci yana jin matsin rãnã, matsin rãnar ƙasa da kuma tasirin centrifugal na waje. A inda dukan uku suka canza, za ta iya zama a cikin wannan tsari, wato, orbits rana a cikin daidai shekara daya.

Ga L1 da L2, nisan daga jikin ƙaramin abu yana kusa da radius ɗin Hill:

r ≈ a · ∛(m / 3M)

Ga rana da ƙasa: 149.6 miliyan km × ∛(3.04 × 10⁻⁶ / 3) ≈ 1.5 miliyan km. An yi wannan daidaitacce ne don samar da kilomita miliyan 1.49 ga L1 da kuma kilomita miliyan 1.51 ga L2.

Master

A cikin matsalolin jiki uku masu ƙunshe da ƙananan, tare da nisan a cikin rundunonin rabawa da μ = m / (M + m), ƙarfin aiki shine

Ω(x, y) = −(1−μ)/r₁ − μ/r₂ − ½(x² + y²)

An yi amfani da wannan tsari ne wajen samar da taswirar duniyarmu, inda aka sanya ∂Ω/∂x = 0 a kan x-axis (a quintic, an warware shi ta hanyar lamba a nan); L4 da L5 suna yin triangles na daidaitattun kusurwa tare da jikin biyu. Linear stability analysis shows L1-L3 are always unstable, and L4/L5 are stable when μ < μ_Routh = ½(1 − √(23/27)) ≈ 0.0385. Spacecraft a L1 da L2 tashi halo ko Lissajous orbits a kusa da mataki fiye da zama a kanta.