Kuthamanga kwa orbital: kuyerekezera kwa vis-viva

Kodi ndege yonyamula anthu ikuyenda bwanji? Ndiuzeni kuti ndi mbali iti yomwe ili kudzera pa dziko lapansili komanso kuti ndi yotani, ndipo yankho la funso limeneli limapezeka m’mafunso angapo. Mission planners kugwiritsa ntchito izo tsiku lililonse.

M’madera ena, kuwala kwa dzuwa kumatha kuwonekera m’malo onse a orbit.

Onani

Chomwecho chilichonse m’mapiri nthawi zonse chimasinthana ndi kuwala kwa maola, monga skateboarder pa half-pipe.

• Kugwa pafupi ndi thupi lapakati, imafulumira.
• Kukwera kuchoka, kumathamanga.

M’chipinda chosagwira, mzere wofiira umasonyeza liwiro: mzere wautali, wamphamvu; mzere wamfupi, wothamanga. Onani kuti imakula ngati thupi limayenda, ndipo limachepetsedwa monga momwe likugwirira ntchito.

Dziko limachitanso zimenezi, koma mofulumira chifukwa choyenda m’mphepete mwa nyanja: imayenda ndi liwiro la makilomita 30.3 pa ola limodzi kumapeto kwa Januware ndipo liwiro lake limafika makilomita 29.3 pa ola limodzi kumapeto kwa July.

Kumvetsa

The Vis-viva equation (“living force”, mawu oyamba a mphamvu) amapatsa mtunda panthawi iliyonse:

v = √( μ · (2/r − 1/a) )

• v: liwiro
• μ: paramita ya gravity ya thupi lapakati (GM)
• r: tsopano kuwala kuchokera m’mphepete mwa thupi lachikulu
• a: axis yayikulu ya orbit

Onani:

• Mzere (r = a): v = √(μ/a). Izi ndi zofanana ndi mfundo za Newton za kuwala.
• Ku periapsis, r ndi wamng’ono kwambiri, choncho 2 / r ndi wamkulu kwambiri ndipo v ndi wamphamvu kwambiri.
• Kudekha kwambiri panjira yotuluka (a → ∞): v = √(2μ/r). Izi ndi escape galimoto, anafotokoza mu nkhani ina.

Worked example: Earth around the Sun, μ = 1.327 × 10¹¹ km³/s², a = 149.6 million km. Pa aphelion r = 152.1 miliyoni km:

v = √(1.327 × 10¹¹ × (2/1.521 × 10⁸ − 1/1.496 × 10⁸)) ≈ 29.3 km/s.

Woyang’anira

Vis-viva ndi kupulumutsa mphamvu m’njira. Kugwiritsa ntchito mphamvu ya orbital (enerji pa kilogram) ndi

ε = v²/2 − μ/r = −μ / (2a)

Kinetic ndi mphamvu zofunikira ndi nthawi zonse m’mphepete mwa orbit ndi kutengera okha pa a. Kuchotsa kwa v ndi inu mupeza vis-viva. Zochitika:

• The mfundo Kuphatikiza apo, mfundo ya ε imagawa ma orbits: yoipa kwa ellipses, zero kwa parabola, yopindulitsa kwa hyperbola.
• Pa nthawi iliyonse yapitayo, the speed yekha amatsimikizira a. Izi ndichifukwa chake injini imodzi yokha imawononga panthawi imodzimodziyo imasintha kukula kwa orbit yonse, ndipo chifukwa chiyani maphunziro otsatira anganene za ndalama za delta-v.

Apsis kutalika kutsatira kuchokera vis-viva ndi r = a (1 ± e):

v_p = √(μ/a · (1+e)/(1−e)),   v_a = √(μ/a · (1−e)/(1+e))

Their product is v_p · v_a = μ/a = v_circ², so the circular speed is the geometric mean of the two, not the arithmetic mean. Izi amafunsa funso Try it.