Isivinini se-orbital: i-equation ye-vis-viva

Ingabe isindiza somhlaba sihamba ngokushesha kangakanani? Ngisho ukuthi ibanga liphi kusuka eplanethini nokuthi ikhulu kangakanani i-orbit yayo, futhi inqubekela phambili eyodwa encane izophendula imibuzo lapho iyiphi indawo ehamba ngayo. Abahlela imikhankaso basebenzisa le ndlela ngosuku ngalunye.

I-equation encane eyodwa inikeza isivinini nganoma yikuphi endaweni ejikelezayo.

Bona

Noma yini e-orbit ihlala ishintsha ubude bejubane, njengesikhashana esihamba ngebhayisikili.

• Ukuwa eduze kwengxenye ephakathi, ihamba ngokushesha.
• Ukunyuka eduze, kukhawulela.

Kubhokisi le-sandbox inkinobho emnyama ibonisa isivinini: inkinobho ende, ishesha; inkinobho emfishane, ihamba kancane. Bona ukuthi ikhula njengoba umzimba uqala ukujikeleza, futhi incane uma ikhuphuka.

Iplanethi iyenza into efanayo, kodwa ngokucophelela ngoba i-orbit yayo iyingxenye ye-circular: iyahamba ngamakhilomitha angama-30.3/s ekuqaleni kukaJanuwari futhi amakhilomitha angama-29.3/s ekuqaleni kukaJulayi.

Ukuqonda

I vis-viva equation (“ikhono eliphilayo”, igama elisha le-energy) linikeza isivinini nganoma yisiphi isikhundla:

v = √( μ · (2/r − 1/a) )

• v: isivinini
• μ: ipharamitha yesimo sezulu esiphakathi (GM)
• r: ubude obukhona kusuka engxenyeni yengxenye esephakathi
• a: isibhamu esincane-sikhulu se-orbit

Khangela:

• Isikhala (r = a): v = √(μ/a). Le yijubane elijikelezayo kusuka encwadini kaNewton.
• Ku-periapsis, r iyincane kakhulu, ngakho-ke 2/r ikhulu futhi v iyikhawulo.
• Kude kakhulu kundlela yokuyeka (a → ∞): v = √(2μ/r). Le yijubane lokuyeka, elibhalwe encwadini elandelayo.

Usibonelo esisebenza: Iplanethi ejikeleza ilanga, μ = 1.327 × 10¹¹ km³/s², a = amakhilomitha ayizigidi ezingama-149.6. Ku-aphelion r = 152.1 million km:

v = √(1.327 × 10¹¹ × (2/1.521 × 10⁸ − 1/1.496 × 10⁹)) ≈ 29.3 km/s.

Umphathi

Vis-viva yimpilo yokugcina amandla efihlakele. I-orbit ye-energy (i-energy ngayinye ngekilograma) yi

ε = v²/2 − μ/r = −μ / (2a)

Ugesi onamandla aphezulu we-kinetic uhlala njalo ngaphaya kwe-orbit futhi uxhomekeke kuphela ku-a. Kuxazululwe i-v bese uthola i-vis-viva. Imiphumela:

• I-iphawu le-ε lihlukanise i-orbit: e negativi ye-ellipsis, isibalo se-parabola, e positive ye-hyperbola.
• Kunoma iyiphi indawo enikeziwe, isivinini kuphela esichaza i-a. Le yisizathu sokuthi kungani i-injini eyodwa eqhuma endaweni eyodwa ishintsha ubukhulu be-orbit yonke, futhi kungani izifundo ezilandelayo zingakhuluma nge-delta-v budgets.

Ijubane le-Apsis lilandela kusuka ku-vis-viva nge-r = a(1 ± e):

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

Imikhiqizo yabo yi v_p · v_a = μ/a = v_circ², ngakho isivinini esijikelezayo yi geometric ephakathi kwezimbili, hhayi i-arithmetic ephakathi. Okuphendula umbuzo we-Try it.