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ASTR 2310: Chapter 3

This chapter explores the fundamental concepts of orbital mechanics rooted in Newton's Laws of Motion and Gravitation, leading to the derivation of Kepler's Laws. It delves into conic sections, the general form of Kepler's 3rd Law, orbital energy, and speed, along with the Virial Theorem. Additionally, the chapter provides a comprehensive review of angular momentum, orbit equations, and special velocities associated with elliptical, parabolic, and hyperbolic orbits, making essential connections to more advanced topics in mechanics.

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ASTR 2310: Chapter 3

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  1. ASTR 2310: Chapter 3 • Orbital Mechanics • Newton's Laws of Motion & Gravitation • (Derivation of Kepler's Laws)‏ • Conic Sections and other details • General Form of Kepler's 3rd Law • Orbital Energy & Speed • Virial Theorem

  2. ASTR 2310: Chapter 3 • Newton's Laws of Motion & Gravitation • Three Laws of Motion (review I hope!)‏ • Velocity remains constant without outside force • F = ma (simplest version)‏ • Every action has an equal and opposite reaction

  3. ASTR 2310: Chapter 3 • Newton's Laws of Motion & Gravitation • Law of Gravitation (review I hope!) • F = G Mm/r2 • Also Optics, Calculus, Alchemy and Preservation of Virginity Projects

  4. ASTR 2310: Chapter 3 • Kepler's Laws can be derived from Newton • Takes Vector Calculus, Differential Eq., generally speaking, which is slightly beyond our new prerequisites • Will use some related results. • If you have the math, please read • Will see a lot of this in Upper-Level Mechanics

  5. ASTR 2310: Chapter 3 • Concept of Angular Momentum, L • Linear version: L = rmv • Vector version: L is the cross product of r and p • Angular momentum is a conserved quantity

  6. ASTR 2310: Chapter 3 • Orbit Equations • R = L2/(GMm2(1+e cos theta)) • Circle (e=0) • Ellipse (0 < e < 1) • Parabola (e =1) • Hyperbola (e > 1)

  7. ASTR 2310: Chapter 3 • Some terms • Open orbits • Closed orbits • Axes and eccentricity e • b2=a2(1-e2) • e=(1-b2/a2)1/2 • <r>=a (those brackets mean “average”)‏

  8. ASTR 2310: Chapter 3 • Special velocities • Perhelion velocities • (GM/a((1+e)/(1-e)))1/2 • Aphelion velocity • (GM/a((1-e)/(1+e)))1/2

  9. ASTR 2310: Chapter 3 • General form of Kepler's third law • P2 = 4 pi2 a3/G(M+m) • M = 4 pi2 a3/GP2 • Solar mass = 1.93 x 1030 kg

  10. ASTR 2310: Chapter 3 • Orbital Energetics • E = K + U = (½) mv2 – GMm/r • More steps...vectors • E = (GMm/L)2(m/2)(e2-1) • e = (1 + (2EL2/G2M2m3))1/2 • Cases of e > 1 → hyperbolic • When e=1, parabolic • vesc(r) = (2GM/r)1/2 • Then e < 1, elliptical

  11. ASTR 2310: Chapter 3 • Orbital Speed • Lots here, not that simple • Can write the “vis viva equation” • V2 = GM ( 2/r – 1/a) • Other forms possible • Can solve for angular speed with position • Concept of transfer orbits (e.g. Hohmann)‏ • See example page 77 for Mars • Related concept of launch windows

  12. ASTR 2310: Chapter 3 • Virial Theorem • Again, unfortunately, advanced math • If you know vector calculus, check it out • Bound systems in equilibrium: • 2<K> + <U> = 0

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