Orbital speed: the vis-viva equation

Da yaushe jirgin sararin samaniya zai fara tafiya? Ka gaya mini yadda ya ke da nisa daga duniyar kuma yadda yake da girman orbit dinsa, kuma wani ma’auni mai zurfi yana amsa tambaya a duk inda take kan hanyarta. Masu shirya aikin suna amfani da shi a kowace rana.

Wani ma’ana mai gajeren lokaci yana ba da gudu a duk inda ya kasance a kan wani orbit.

Look

Duk wani abu a cikin orbit ne koyaushe ciniki tsawo ga gudu, kamar skateboarder a kan rabin-pipe.

• Kuma idan ya kusanci jikinsa, sai ya yi sauri.
• Kuma idan ya yi tsalle, sai ya yi sauri.

A cikin akwatin tsafi, alamar zira tana nuna gudu: alamar zira mai tsawo, sauri; alamar zira mai ƙaranci, dakata. Ka dũba yadda yake girma a lõkacin da jikinsa ke motsi cikin ciki, kuma ya na ƙarancin girma a lõkacin da yake fita.

Duniyar tana yin abu guda, kawai da sauri saboda orbit dinta yana kusa da kwance: tana motsa kusan 30.3 km/s a farkon watan Janairu da kuma kusan 29.3 km/s a farkon watan Yuli.

Understand

Equation vis-viva (“living force”, wata kalma da ta daɗe ana amfani da ita wajen nuna ƙarfi) tana ba da gudu a duk wani wuri:

v = √( μ · (2/r − 1/a) )

• v: speed
• μ: paramita na ma’auni na jiki mai tsakiya (GM)
• r: nisan yanzu daga tsakiyar jikin mai tsakiya
• a: maɓallin tsawo na ɓangaren

Check it:

• Circle (r = a): v = √(μ/a). That is the circular speed from the Newton lesson.
• A periapsis, r shine mafi ƙaranci, don haka 2/r shine mafi girma kuma v shine mafi sauri.
• Da nisa sosai a kan hanya mai gudu (a → ∞): v = √(2μ/r). Wancan ne saurin gudu, an ambace shi a cikin littafin nan mai zuwa.

Misali: Duniya a kusa da rana, μ = 1.327 × 10¹¹ km³/s², a = 149.6 miliyan km. A aphelion r = 152.1 miliyan km:

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

Master

Vis-viva shi ne tsarewar wutar lantarki a cikin ɓoye. The specific orbital energy (energy per kilogram) is

ε = v²/2 − μ/r = −μ / (2a)

Injin kinetic da kuma mai karfin lantarki suna daidai a kan orbit kuma suna dogara ne kawai akan a. Solve for v and you get vis-viva. Amfani:

• sign na ε yana rarraba orbit: mai kyau ga ellipses, sifar don parabola, mai kyau ga hyperbola.
• A cikin kõwace nĩsa wadda aka bai wa, gudun shi kaɗai ne ke ƙayyadẽwa a. Wannan shi ne dalilin da ya sa wani injiniya na guda mai zafi a wani wuri yana canza girman dukkan orbit, kuma me yasa karatun gaba zai iya magana game da delta-v bashin.

Apsis gudun bi daga vis-viva da r = a (1 ± e):

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

Suna da sakamakonsu v_p · v_a = μ/a = v_circ², don haka gudun kwance shi ne jiometriya na biyu, ba aritmetikin tsawo ba. Wannan zai amsa tambayar Yi kokarin.