ASTR 2310 Chapter 3 Orbital Mechanics Newtons Laws

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 – Angular momentum is a conserved quantity Vector version: L is the cross product of r and p

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 <r>=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 1030 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 + (2 EL 2/G 2 M 2 m 3))1/2 • • Cases of e > 1 → hyperbolic When e=1, parabolic – • vesc(r) = (2 GM/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) 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 Virial Theorem – – Again, unfortunately, advanced math If you know vector calculus, check it out Bound systems in equilibrium: 2<K> + <U> = 0