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FRS 148 Seminar 2 2/13/19 Seminar 2 Early History & Fiction Orbital Motion FRS 148, Princeton University Robert Stengel Decades of the Great Dreams From the Earth to the Moon (Round the Moon), Jules Verne Equations of Motion Momentum and Energy Rockets, Missiles, and Men in Space, Ch 2 Round the Moon, excerpts Understanding Space, Ch 4 Copyright 2019 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/FRS.html 1 A Timeline of Events Critical to Space Travel – V Date Science Fact Science Fiction World Scene 17th-18th Gottfried Leibniz (1646-1716): Jonathan Swift, 1689: Peter the Great, cent Calculus (1675), founder of Gulliver's Travels Czar of Russia. 1690: "dynamics". Isaac Newton (1726), Battle of the Boyne. (1642-1727): Laws of Motion, 1729: Bach's St. Universal Gravitation, Principia Matthew Passion. (1687). 1746: PRINCETON IS FOUNDED! 18th-19th Pierre-Simon Laplace (1749- Voltaire, Micromégas AMERICAN REVOLUTION; cent 1827): Solar system (1752) FRENCH REVOLUTION; cosmology and dynamics. Carl AGE OF THE Friedrich Gauss (1777-1855): ENLIGHTENMENT; Orbit calculation, shape of the INDUSTRIAL Earth. "Balloonatics": REVOLUTION. 1781: Montgolfier brothers & Herschel discovers deRozier (1783), 1st manned Uranus. 1800s: flight; Madame Blanchard; Exploits of Napoleon. Major Turner; Glaisher & Steamships, railroads, Coxwell to 35K ft Babbage's Analytical Engine. 2 1 2/13/19 A Timeline of Events Critical to Space Travel – VI Date Science Fact Science Fiction World Scene Early 19th cent Karl Jacobi (1804- Edgar Allan Poe, The Bombardment of 1851): mathematics of Unparalleled Copenhagen (1807); dynamical systems. Adventure of One WAR OF 1812 (Star- William Congreve: Hans Pfaall (1835), Spangled Banner); Military & rescue The moon hoax (False Battle of Waterloo rockets. account of Herschel's research, 1835) Late 19th cent Henri Poincaré (1854- 1865: Jules Verne, AMERICAN CIVIL WAR; 1912): Restricted 3- From the Earth to the FRANCO-PRUSSIAN body problem. Kirchoff Moon, Alexandre WAR; VICTORIAN ERA. & Bunsen: Dumas, Voyage a la 1866: Nobel invents Spectroscope (1859) lune, Henri de Parville, dynamite. 1876: Bell Un Habitant de la invents telephone. planete Mars, Anon, 1895: Roentgen Voyage a la lune, discovers X-rays. English clergyman, Chrysostom Trueman (nom de plume), The History of a Voyage to the Moon, Camille Flammarion, Mondes imaginaires et mondes reels 3 Innovators in Mathematics Gottfreid Leibniz (1646-1716) § Invented “infinitesimal calculus” § Established notation for derivatives, integrals, and “chain rule”, e.g., 2 dy d ⎣⎡ f (x)⎦⎤ d y d ⎪⎧d ⎣⎡ f (x)⎦⎤ ⎪⎫ y = f (x); = ; 2 = ⎨ ⎬ dx dx dx dx ⎩⎪ dx ⎭⎪ Pierre Simon Laplace (1749-1827) § Laplace transform § Laplace’s equation § Shape of the Earth; spherical harmonics Karl Friedrich Gauss (1777-1855) • Method of “least squares” • “Gaussian distribution” 4 2 2/13/19 Innovators in Mathematics Carl Jacobi (1804-1851) § Hamilton-Jacobi theorem § Jacobi integral Henri Poincaré (1854-1912) § Dynamical systems § Restricted 3-body problem 5 Innovators in Computation Muhammad ibn Musa al-Khwarizmi (780-850) § Invented algebra § 1st algorithms for linear and quadratic equations Ismail Al-Jazari (1136-1206) § Programmable mechanisms Charles Babbage (1791-1871) § Difference Engine to tabulate polynomials (1822) § Analytical Engine with programmable arithmetic logic (1837) § Not built in his lifetime Ada King, Countess of Lovelace (1815-1852) § Wrote program for Analytical Engine § 1st computer algorithm 6 3 2/13/19 Computation Before Computers • Counting (35,000-25,000 BC) • Abacus (2,700-2,300 BC) • Multiplication (1,550 BC) • Multiplication/Division Tables, Lattices (13th c) • Logarithms (Napier, 1614) – Napier’s Bones 7 Calculators • Slide Rules (Oughtred, 1622) – Sliding sticks with rulings – Logarithmic scales • Multiplication, Division, Powers, Exponentials – Trigonometric Scales – Used well into the 1960s • Mechanical Calculators (1650) • Electrical Calculators (1934) 8 4 2/13/19 Montgolfier Brothers (1740—1810) (1745-1799) § Joseph-Michel & Jacques-Étienne § Hot-air balloons § Public demonstration, 1783 § Flight to 3,000 ft, 25-minute duration 9 The Herschels William (1738-1822) § Discovered Uranus § Built numerous reflecting telescopes § Double stars, deep sky survey § Surveyed stars of Southern Hemisphere in South Africa Caroline (1750-1848) § Worked with brother William § Discovered several comets § Catalog of stars and nebulas John (1792-1871) § Son of William § Polymath, astronomer, photographer § Invented blueprints 10 5 2/13/19 Mysorean & Congreve Rockets 1780-1870 2nd Anglo-Mysore Battle of Copenhagen War, 1780 1807 War of 1812 11 More Early Rockets Hale Rotary Rocket, Breeches Buoy Life-Saving Rocket 1844 Whaling Harpoon Rocket 12 6 2/13/19 Locke’s Great Moon Hoax, 1835 13 Franz von Gruithuisen, 1774-1852 14 7 2/13/19 Signaling Other Worlds 15 Would the Martians or Selenites See Lights on the Earth? Would we see lights on the Moon? 16 8 2/13/19 Ashen Light of Venus Unsolved mystery Hypothesized subtle glow during crescent phase First reported by Riccioli, 1643 17 Voyage à Venus, Achille Eyraud, 1865 18 9 2/13/19 Canals on Mars Schiaparelli’s Canali, 1888 Lowell’s Channels, 1906 19 1865 Jules Verne (1828-1905) 20 10 2/13/19 Round the Moon 21 Orbital Motion 22 11 2/13/19 Two-Body Orbits are Conic Sections 23 Newton’s 2nd Law § Particle of fixed mass (also called a point mass) acted upon by a force changes velocity with § acceleration proportional to and in direction of force § Inertial reference frame § Ratio of force to acceleration is the mass of the particle: F = m a d dv(t) ⎣⎡mv(t)⎦⎤ = m = ma(t) = F ⎡ ⎤ dt dt vx (t) ⎡ f ⎤ ⎢ ⎥ x ⎡ ⎤ d ⎢ ⎥ fx f ⎢ ⎥ m ⎢ vy (t) ⎥ = ⎢ y ⎥ F = fy = force vector dt ⎢ ⎥ ⎢ ⎥ ⎢ f ⎥ ⎢ f ⎥ ⎢ vz (t) ⎥ ⎢ z ⎥ ⎣⎢ z ⎦⎥ ⎣ ⎦ ⎣ ⎦ 24 12 2/13/19 Equations of Motion for a Particle Integrating the acceleration (Newton’s 2nd Law) allows us to solve for the velocity of the particle ⎡ ⎤ ⎡ ⎤ fx m ax dv(t) 1 ⎢ ⎥ ⎢ ⎥ = v! (t) = F = ⎢ fy m ⎥ = ay dt m ⎢ ⎥ ⎢ f m ⎥ ⎢ ⎥ ⎢ z ⎥ az ⎣ ⎦ ⎣⎢ ⎦⎥ T dv(t) T T 1 v(T ) = dt + v(0) = a(t)dt + v(0) = Fdt + v(0) ∫0 dt ∫0 ∫0 m 3 components of velocity ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ vx (T ) ax (t) vx (0) fx (t) m vx (0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ vy (T ) ⎥ = ⎢ ay (t) ⎥dt + ⎢ vy (0) ⎥ = ⎢ fy (t) m ⎥dt + ⎢ vy (0) ⎥ ⎢ ⎥ ∫0 ⎢ ⎥ ⎢ ⎥ ∫0 ⎢ ⎥ ⎢ ⎥ v T a t v 0 f t m v 0 ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ( ) ⎦⎥ 25 Equations of Motion for a Particle Integrating the velocity allows us to solve for the position of the particle ⎡ ⎤ ⎡ ⎤ x!(t) vx (t) dr(t) ⎢ ⎥ ⎢ ⎥ = r!(t) = v(t) = ⎢ y!(t) ⎥ = ⎢ vy (t) ⎥ dt ⎢ ⎥ ⎢ ⎥ ⎢ z!(t) ⎥ ⎢ vz (t) ⎥ ⎣ ⎦ ⎣ ⎦ T dr(t) T r(T ) = dt + r(0) = v dt + r(0) ∫0 dt ∫0 3 components of position ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x(T ) vx (t) x(0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ y(T ) ⎥ = ⎢ vy (t) ⎥dt + ⎢ y(0) ⎥ ∫0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z T v t z 0 ⎣⎢ ( ) ⎦⎥ ⎣⎢ z ( ) ⎦⎥ ⎣⎢ ( ) ⎦⎥ 26 13 2/13/19 Trajectories Calculated with Flat-Earth Model § Constant gravity, g, is the only force in the model, i.e., no aerodynamic or thrust force § Can neglect motions in the y direction Dynamic Equations Analytic (Closed-Form) Solution v t v x ( ) = x0 v!z (t) = −g (z positive up) v T v x ( ) = x0 x!(t) = vx (t) T z!(t) = vz (t) vz (T ) = vz − gdt = vz − gT 0 ∫ 0 0 Initial Conditions x T x v T ( ) = 0 + x0 v 0 = v x ( ) x0 T 2 vz (0) = vz 0 z(T ) = z0 + vz T − gt dt = z0 + vz T − gT 2 0 ∫ 0 0 x(0) = x0 27 z(0) = z0 Trajectories Calculated with Flat-Earth Model vx (0) = 10m / s vz (0) = 100,150, 200m / s x(0) = 0 z(0) = 0 28 14 2/13/19 MATLAB Code for Flat- Earth Trajectories Script for Script for Numerical Solution Analytic Solution tspan = 40; % Time span, s g = 9.8; xo = [10;100;0;0]; % Init. Cond. t = 0:0.1:40; [t1,x1] = ode45('FlatEarth',tspan,xo); vx0 = 10; Function for vz0 = 100; x0 = 0; Numerical Solution z0 = 0; function xdot = FlatEarth(t,x) % x(1) = vx vx1 = vx0; % x(2) = vz vz1 = vz0 – g*t; % x(3) = x x1 = x0 + vx0*t; % x(4) = z z1 = z0 + vz0*t - 0.5*g*t.* t; g = 9.8; xdot(1) = 0; xdot(2) = -g; xdot(3) = x(1); xdot(4) = x(2); xdot = xdot'; 29 end Sounding (Sub-Orbital) Rockets 45-kg payload to 370-km altitude Trajectory well approximated using flat-Earth model from 2nd stage burnout 30 15 2/13/19 Matrix and Its Transpose • Matrix: – Usually bold capital or capital: F or F – Dimension = (m x n) • Transpose: ⎡ a b c ⎤ – Interchange rows ⎢ ⎥ d e f and columns A = ⎢ ⎥ ⎢ g h k ⎥ ⎡ a d g l ⎤ ⎢ ⎥ ⎢ ⎥ AT b e h m ⎣⎢ l m n ⎦⎥ = ⎢ ⎥ ⎢ c f k n ⎥ 4 x 3 ⎣⎢ ⎦⎥ 3 x 4 31 Matrix Products Matrix-vector product transforms one vector into another Row-by-column multiplication and addition ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ y1 a b c x1 ax1 + bx2 + cx3 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ y = ⎢ y2 ⎥ = Ax = ⎢ d e f ⎥⎢ x2 ⎥ = ⎢ dx1 + ex2 + fx3 ⎥ ⎢ y ⎥ ⎢ g h i ⎥⎢ x ⎥ ⎢ gx + hx + ix ⎥ ⎣ 3 ⎦ ⎣ ⎦⎣ 3 ⎦ ⎣⎢ 1 2 3 ⎦⎥ (n ×1) = (n × m)(m ×1) Matrix-matrix product produces a new matrix Row-by-column multiplication and addition ⎡ ⎤ ⎡ a a ⎤ ⎡ b b ⎤ (a1b1 + a2b3 ) (a1b2 + a2b4 ) A = ⎢ 1 2 ⎥; B = ⎢ 1 2 ⎥; AB = ⎢ ⎥ ⎢ a3 a4 ⎥ ⎢ b3 b4 ⎥ ⎢ a b + a b a b + a b ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ( 3 1 4 3 ) ( 3 2 4 4 ) ⎦ (n × m) = (n × l)(l × m) 32 16 2/13/19 Orientation of One Axis Frame with Respect to Another Transformation x2 = Ax1 ⎡ x ⎤ ⎡ cosθ 0 −sinθ ⎤ ⎡ x ⎤ ⎢ y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ 0 1 0 ⎥ ⎢ y ⎥ ⎢ z ⎥ ⎢ sinθ 0 cosθ ⎥ ⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦ ⎣ ⎦1 −1 Inverse Transformation x1 = A x2 ⎡ x ⎤ ⎡ cosθ 0 sinθ ⎤ ⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ y = ⎢ ⎥ y ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎢ z ⎥ ⎢ −sinθ 0 cosθ ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦ ⎣ ⎦2 33 Identity Matrix and Matrix Inverse ⎡ 1 0 0 ⎤ ⎢ ⎥ I = ⎢ 0 1 0 ⎥ ⎢ 0 0 1 ⎥ ⎣ ⎦ x = A−1x = A−1Ax = Ix 1 2 1 1 −1 ⎡ cosθ 0 −sinθ ⎤ ⎡ cosθ 0 −sinθ ⎤ A−1A = ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ 0
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