Orbital speed: the vis-viva equation

Ndeipi nguva yakareba sei iyo spacecraft ichifamba? Ndipe ruzivo rwekuti ndeipi nzvimbo iri kure neplaneti uye kuti ndeipi nzvimbo yakakura yeobiti yayo, uye imwe equation inotsanangura kuti chii chiri kuitika. Mission vatungamiriri kushandisa izvi zuva nezuva.

Mufananidzo uyu unopa mukana wekuziva kuti mhepo iri sei muzvisikwa zvakasiyana-siyana.

Ona

Chii chero chipi zvacho chiri muorbit chinoramba chichichinja urefu nesimba, sezvakaita skateboarder pahafu-pipe.

• Kusvika pakubuda kwemvura mumuviri, inowedzera kuomarara.
• Kukwira kure, inononoka.

Mu sandbox, arrow yakasviba inoratidza kukwira kwechikepe: arrow yakareba, yakanyanya; arrow pfupi, yakaderera. Ona kuti inokura sei sezvaifamba mumuviri, uye inodzikira sezvainokwira kunze.

Icho chinhu chimwechete chinoitwa nenyika, asi zvakaomarara nekuti orbit yayo inenge iri yechikuru: inofamba ne30.3 km/s muna January uye ne29.3 km/s muna July.

_Zvinyorwa

Equation yevis-viva (“chisimba chinorarama”, izwi rakare rekuti simba) inotiudza kuti kunogona kunge paine nguva yakareba sei panzvimbo imwecheteyo:

v = √( μ · (2/r − 1/a) )

• v: speed
• μ: chimiro chepakati chekumanikidza parameter (GM)
• r: ikozvino distance kubva kumashure kwechikwata chepakati
• a: semi-makuru axis yeorbit

Cheka:

• Circle (r = a): v = √(μ/a). Chikamu ichi chine simba rakakura kupfuura simba re Newton.
• Pa periapsis, r ndeyakanaka kwazvo, saka 2/r ndiyo yakakura uye v ndiyo inonyanya kuoma.
• Pane nzira yekubuda (a → ∞): v = √(2μ/r). Chii chinonzi escape speed?

Mufananidzo wakashanda: Iyo Yepasi inogara pasi pezuva, μ = 1.327 × 10¹¹ km³/s², a = 149.6 million km. Pa aphelion r = 152.1 miriyoni km:

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

Munyori

Vis-viva ndiyo kuchengetedzwa kwesimba muzvifananidzo. Iyo yakasarudzika orbital simba (simba pakilograma) ndeye

ε = v²/2 − μ/r = −μ / (2a)

Kinetic pamwe chete nesimba rekuita simba rinogara pasi peorbit uye rinoenderana chete nea. Solve for v uye iwe unowana vis-viva. Zvikonzero:

• The kuratidza ye ε inoparadzira orbit: yakaipa kune ellipses, zero kune parabola, yakanaka kune hyperbola.
• Panzvimbo ipi neipi, kukwira kwemhepo kunopa nguva. Izvi ndizvo nei imwe injini burn panzvimbo imwe magadzirirwo upamhi hwese orbit, uye nei zvinotevera dzidzo anogona kutaura nezve delta-v mari.

Apsis mhepo dzinotevera kubva vis-viva ne r = a (1 ± e):

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

Zvigadzirwa zvavo zvinosanganisira v_p · v_a = μ/a = v_circ², saka kutenderera kwesimba ndekwe geometric pakati pezviviri izvi, kwete pakati pe arithmetic. Chii chinopindura mibvunzo ye Try it.