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 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

ASTR 2310: Chapter 3 Newton's Laws of Motion & Gravitation Law of Gravitation (review I hope!)‏ – F = G Mm/r 2 Also Optics, Calculus, Alchemy and Preservation of Virginity Projects

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

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

ASTR 2310: Chapter 3 Orbit Equations – R = L 2 /(GMm 2 (1+e cos theta))‏ – Circle (e=0)‏ – Ellipse (0 < e < 1)‏ – Parabola (e =1)‏ – Hyperbola (e > 1)‏

ASTR 2310: Chapter 3 Some terms – Open orbits – Closed orbits Axes and eccentricity e – b 2 =a 2 (1-e 2 )‏ – e=(1-b 2 /a 2 ) 1/2 – =a (those brackets mean “average”)‏

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

ASTR 2310: Chapter 3 General form of Kepler's third law P 2 = 4 pi 2 a 3 /G(M+m)‏ M = 4 pi 2 a 3 /GP 2 Solar mass = 1.93 x kg

ASTR 2310: Chapter 3 Orbital Energetics – E = K + U = (½) mv 2 – Gmm/r – More steps...vectors – E = (GMm/L) 2 (m/2)(e 2 -1)‏ – e = (1 + (2EL 2 /G 2 M 2 m 3 )) 1/2 Cases of e > 1 → hyperbolic When e=1, parabolic – v esc (r) = (2GM/r) 1/2 Then e < 1, elliptical

ASTR 2310: Chapter 3 Orbital Speed – Lots here, not that simple – Can write the “vis viva equation” V 2 = 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

ASTR 2310: Chapter 3 Virial Theorem – Again, unfortunately, advanced math – If you know vector calculus, check it out – Bound systems in equilibrium: – 2 + = 0