Amaphuzu we-Lagrange: ukulinganisela ezulu

I-spaceship edlula ilanga kuneplanethi kufanele ijikeleze kancane futhi idlule emuva. Kodwa-ke iJames Webb Space Telescope, engaphezu kukakhilomitha ayizigidi eziyizinkulungwane ezingu-1.5 eduze kweplanethi, igcina isivinini nathi ngonyaka wonke. Ihlala endaweni ye-Lagrange point, enye yezizindawo eziyisithupha lapho ukunyakaza kwezingxenye ezimbili nokujikeleza kwe-rounding kulinganise khona.

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

Bona

Yicabanga indawo lapho ubude busho ukuthi “iningi kangakanani amandla adingekayo ukuba lapha” ngento ejikeleza ilanga nezwe:

• I-wells ephakeme kakhulu kulanga nezwe.
• L1 ne L2: ama-mountain passes amabili (ama-saddles) ngaso sonke isiphetho sezulu, cishe 1.5 amakhilomitha ayizigidi ezide. Ihlanzekile, kodwa nje: cindezela isindiza somhlaba futhi sizohamba. Izisebenzi zikwenza ukulungisa okuncane ngalezi viki ezimbalwa.
• L3: enye inqwaba engxenyeni esenyakatho yelanga. Akukho nto esebenzayo edlula khona, ngoba ilanga liyivala.
• L4 ne L5: ama-hilltops amabili alula amadigri angama-60 ngaphambili kanye nomhlaba ongezansi. Uzolindele ukuthi izinto zihambe, kodwa iCoriolis effect yokujikeleza iguqula isikhala sibe yi-loop. Zihlala kahle, futhi ubuhle buzigcwalisa: iJupiter’s L4 ne L5 zigcina ngaphezulu kwamaphesenti ayishumi Trojan asteroids.

Ubani ohlala khona: I-SOHO ibheka uLwandle L1 kusukela ngo-1996; ibona iziphepho zelanga ngaphambi kokuba zifike kuthi. I-Webb isebenza ngaphesheya kwe-L2, lapho uLwandle, ilanga kanye nenyanga zihlala khona ngakwesobunxele, eduze kwe-sunshield yayo, zigcina iteleskope ku-230 °C.

Ukuqonda

Kulesi simo esijikelezayo neplanethi, into encane ibona ukunyakaza kwelanga, ukunyakaza kweplanethi kanye nomphumela wokujikeleza osuka phezulu. lapho zonke ezintathu zilahlekile, ingahlala ikhona kule phrojekthi, okungukuthi, ijikeleze uLwandle Mnyama ngonyaka owodwa.

Ku-L1 ne-L2 ubude kusuka kumuntu omncane bufana no Hill radius:

r ≈ a · ∛(m / 3M)

Ku-Sun ne-Earth: 149.6 million km × ∛(3.04 × 10⁻⁶ / 3) ≈ 1.5 million km. Ukuxazulula isibalo ngokucacile kunikeza amakhilomitha ayizigidi ezingama-1.49 ku-L1 namakhilomitha ayizigidi ezingama-1.51 ku-L2.

Umphathi

Kunkinga ye-circular ehlobene nengxenye ezintathu, ngebanga leyunithi yokuhlukaniswa kanye μ = m/(M+m), ikhono elisebenzayo liwu

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

Amaphuzu alinganayo axazulula ∂Ω/∂x = 0 ku-x-axis (i-quintic, ixazululwe ngokuzimele lapha); L4 ne-L5 zihlanganisa ama-triangles alinganayo neziqu ezimbili. Ucwaningo lwe-linear stability lubonisa ukuthi i-L1-L3 iyaqhubeka ingazinzi, futhi i-L4/L5 ihlala lapho i-μ < μ_Routh = ½(1 − √(23/27)) ≈ 0.0385. I-spacecraft eL1 ne-L2 indiza halo noma Lissajous orbits eduze kwengxenye endaweni yokuphumula.