ASTR 2310: Chapter 3
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Orbital Mechanics
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Newton's Laws of Motion & Gravitation
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(Derivation of Kepler's Laws)
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Conic Sections and other details
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General Form of Kepler's 3rd Law
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Orbital Energy & Speed
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Virial Theorem
ASTR 2310: Chapter 3
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Newton's Laws of Motion & Gravitation
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Three Laws of Motion (review I hope!)
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Velocity remains constant without outside
force
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F = ma (simplest version)
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Every action has an equal and opposite
reaction
ASTR 2310: Chapter 3
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Newton's Laws of Motion & Gravitation
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Law of Gravitation (review I hope!)
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F = G Mm/r2
Also Optics, Calculus, Alchemy and
Preservation of Virginity Projects
ASTR 2310: Chapter 3
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Kepler's Laws can be derived from Newton
Takes Vector Calculus, Differential Eq.,
generally speaking, which is slightly beyond
our new prerequisites
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Will use some related results.
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If you have the math, please read
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Will see a lot of this in Upper-Level
Mechanics
ASTR 2310: Chapter 3
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Concept of Angular Momentum, L
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Linear version: L = rmv
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Vector version: L is the cross product of r and
p
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Angular momentum is a conserved quantity
ASTR 2310: Chapter 3
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Orbit Equations
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R = L2/(GMm2(1+e cos theta))
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Circle (e=0)
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Ellipse (0 < e < 1)
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Parabola (e =1)
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Hyperbola (e > 1)
ASTR 2310: Chapter 3
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Some terms
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Open orbits
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Closed orbits
Axes and eccentricity e
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b2=a2(1-e2)
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e=(1-b2/a2)1/2
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<r>=a (those brackets mean “average”)
ASTR 2310: Chapter 3
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Special velocities
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Perhelion velocities
•
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(GM/a((1+e)/(1-e)))1/2
Aphelion velocity
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(GM/a((1-e)/(1+e)))1/2
ASTR 2310: Chapter 3
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General form of Kepler's third law
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P2 = 4 pi2 a3/G(M+m)
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M = 4 pi2 a3/GP2
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Solar mass = 1.93 x 1030 kg
ASTR 2310: Chapter 3
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Orbital Energetics
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E = K + U = (½) mv2 – GMm/r
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More steps...vectors
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E = (GMm/L)2(m/2)(e2-1)
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e = (1 + (2EL2/G2M2m3))1/2
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Cases of e > 1 → hyperbolic
When e=1, parabolic
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vesc(r) = (2GM/r)1/2
Then e < 1, elliptical
ASTR 2310: Chapter 3
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Orbital Speed
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Lots here, not that simple
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Can write the “vis viva equation”
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•
•
•
•
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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
ASTR 2310: Chapter 3
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Virial Theorem
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Again, unfortunately, advanced math
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If you know vector calculus, check it out
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Bound systems in equilibrium:
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2<K> + <U> = 0