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ASTR 2310: Chapter 3

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 3 rd Law Orbital Energy & Speed Virial Theorem. ASTR 2310: Chapter 3. Newton's Laws of Motion & Gravitation

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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)‏ • Can write in different forms, solve for angular speed at various positions • 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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