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1. www.astro.umd.edu · ~jph · A320_orbits_21.11. Parabolic Orbits - UMD

   www.astro.umd.edu · ~jph · A320_orbits_2

   

   In a hyperbolic orbit, the object approaches along one direction θin, asymptotic to a straight line, is deflected, and recedes asymptotically along another direction θout. These angles can be found by noting that r → ∞ if (1 + ecosθ) → 0. This can only occur for θ in the 2nd and 3rd quadrants, where cosθ is negative.

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2. assets.cambridge.org · sample · 9780521661676wsOrbitals in Chemistry A Modern Guide for Students

   assets.cambridge.org · sample · 9780521661676ws

   

   Orbitals are mathematical functions of the coordinates of each electron in atoms, molecules and other atomic aggregates; we will see that they carry the dimensions [length] ÿ3/2 .

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4. archive.aoe.vt.edu · lutze · Aoe5234Orbit Characteristics - Virginia Tech

   archive.aoe.vt.edu · lutze · Aoe5234

   

   The parabolic orbit serves as a boundary between the elliptic (periodic) orbits and the hyperbolic (escape) orbits. It is the orbit of least energy that allows escape.

5. phys.libretexts.org · Courses · Prince_Georges57.16: Parabolic Orbits - Physics LibreTexts

   phys.libretexts.org · Courses · Prince_Georges

   

   57.16: Parabolic Orbits. Page ID. Suppose we wish to calculate the position of a body that is in a parabolic or near-parabolic orbit ( e ≈ 1 ), as is the case with some comets in orbit around the Sun. The procedure is the same as outlined in Section 57.6, except for Eq. 57.6.1 through 57.6.5.

6. courses.physics.ucsd.edu · LECTURES · CH09Central Forces and Orbital Mechanics

   courses.physics.ucsd.edu · LECTURES · CH09

   

   An almost circular orbit has r(t) = r0 + η(t), where |η/r0| ≪ 1. To lowest order in η, one derives the equations d2η dt2 = −ω2 η , ω2 = 1 µ U′′ eff(r0) . (9.25) If ω2 > 0, the circular orbit is stable and the perturbation oscillates harmonically. If ω2 < 0, the circular orbit is unstable and the perturbation grows ...

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7. oer.pressbooks.pub · lynnanegeorge · chapterChapter 2 – Orbit Geometry – Introduction to Orbital Mechanics

   oer.pressbooks.pub · lynnanegeorge · chapter

   

   PARABOLIC ORBITS. Parabolic orbits are rarely found in use, but they are important and interesting because they are the borderline case between the elliptical (closed) orbit and the hyperbolic (open) orbit. Figure 10: Parabolic Orbit Visualization (Source: Brandir, 2006).

8. People also ask

   What is a parabolic orbit?

   • Parabolic Orbits In the case of zero total energy, = 0 , the orbit is parabolic. Since the eccentricity e = 1 while a = E ∞, the numerator of eqn (76), a(1 − e2), would seem undefined. But this product can remain finite: we can write the equation of the parabolic conic section as

   1.11. Parabolic Orbits - UMD

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   How do you calculate the speed needed to escape a parabolic orbit?

   • Theoretically, as the object’s distance from the earth approaches infinity, it’s speed approaches zero. This results in a total specific mechanical energy, ε = 0 for a parabolic orbit. We can then calculate the speed necessary to escape by writing the energy equation for this point: ε = V2 esc 2 − μ R = 0 Thus Vesc = √2μ R

   Chapter 2 – Orbit Geometry – Introduction to Orbital Mechanics

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   What is a hyperbolic orbit?

   • Hyperbolic Orbits > 1. When the object is at a great distance, the gravitational potential must go to zero, and all the energy is kinetic. Thus the velocity at large r is v∞ = √2E. (Recall that E is the energy per unit mass.) Looking at our previous equation for the ellipse (eqn 76), we see that since (1 − e2) is negative, if

   1.11. Parabolic Orbits - UMD

   www.astro.umd.edu/~jph/A320_orbits_2.pdf

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   How do you find the closest approach in a hyperbolic orbit?

   • We see that the closest approach occurs at θ = 0, for which rmin = (−a)(e − 1). In a hyperbolic orbit, the object approaches along one direction θin , asymptotic to a straight line, is deflected, and recedes asymptotically along another direction θout. These angles can be found by noting that r → ∞ if (1 + e cos θ) → 0.

   1.11. Parabolic Orbits - UMD

   www.astro.umd.edu/~jph/A320_orbits_2.pdf

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9. spacecraft.ssl.umd.edu · 791S16L02Orbital Mechanics - UMD

   spacecraft.ssl.umd.edu · 791S16L02

   

   Planetary Approach Analysis. Spacecraft has vh hyperbolic excess velocity, which fixes total energy of approach orbit. v2 μ = μ v2 = h. 2 r 2a 2. Vis-viva provides velocity of approach. 2 2μ. = vh +. r. Choose transfer orbit such that approach is tangent to desired final orbit at periapse.