ASTR 2310: Chapter 3

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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

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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

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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

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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

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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

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ASTR 2310 Chapter 3

• Orbit Equations
• R L2/(GMm2(1e cos theta))
• Circle (e0)
• Ellipse (0 lt e lt 1)
• Parabola (e 1)
• Hyperbola (e gt 1)

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ASTR 2310 Chapter 3

• Some terms
• Open orbits
• Closed orbits
• Axes and eccentricity e
• b2a2(1-e2)
• e(1-b2/a2)1/2
• ltrgta (those brackets mean average)?

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ASTR 2310 Chapter 3

• Special velocities
• Perhelion velocities
• (GM/a((1e)/(1-e)))1/2
• Aphelion velocity
• (GM/a((1-e)/(1e)))1/2

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ASTR 2310 Chapter 3

• General form of Kepler's third law
• P2 4 pi2 a3/G(Mm)
• M 4 pi2 a3/GP2
• Solar mass 1.93 x 1030 kg

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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 gt 1 ? hyperbolic
• When e1, parabolic
• vesc(r) (2GM/r)1/2
• Then e lt 1, elliptical

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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

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ASTR 2310 Chapter 3

• Virial Theorem
• Again, unfortunately, advanced math
• If you know vector calculus, check it out
• Bound systems in equilibrium
• 2ltKgt ltUgt 0