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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.

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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

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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

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Innovators in Mathematics Carl Jacobi (1804-1851) § Hamilton-Jacobi theorem § Jacobi integral

Henri Poincaré (1854-1912) § Dynamical systems § Restricted 3-body problem

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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

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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

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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)

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Montgolfier Brothers (1740—1810) (1745-1799)

§ Joseph-Michel & Jacques-Étienne § Hot-air balloons § Public demonstration, 1783 § Flight to 3,000 ft, 25-minute duration

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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

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Mysorean & Congreve Rockets 1780-1870

2nd Anglo-Mysore Battle of Copenhagen War, 1780 1807

War of 1812

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More Early Rockets

Hale Rotary Rocket, Breeches Buoy Life-Saving Rocket 1844

Whaling Harpoon Rocket

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Locke’s Great Moon Hoax, 1835

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Franz von Gruithuisen, 1774-1852

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Signaling Other Worlds

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Would the Martians or Selenites See Lights on the Earth?

Would we see lights on the Moon? 16

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Ashen Light of Venus Unsolved mystery Hypothesized subtle glow during crescent phase

First reported by Riccioli, 1643

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Voyage à Venus, Achille Eyraud, 1865

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Canals on Mars Schiaparelli’s Canali, 1888

Lowell’s Channels, 1906

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1865

Jules Verne (1828-1905) 20

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Round the Moon

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Orbital Motion

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Two-Body Orbits are Conic Sections

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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 ⎦⎥ ⎣ ⎦ ⎣ ⎦

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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

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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

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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

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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

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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 1 0 ⎥ ⎣⎢ sinθ 0 cosθ ⎦⎥ ⎣⎢ sinθ 0 cosθ ⎦⎥ ⎡ cosθ 0 sinθ ⎤⎡ cosθ 0 −sinθ ⎤ = ⎢ ⎥⎢ ⎥ ⎢ 0 1 0 ⎥⎢ 0 1 0 ⎥ ⎣⎢ −sinθ 0 cosθ ⎦⎥⎣⎢ sinθ 0 cosθ ⎦⎥

⎡ 1 0 0 ⎤ = ⎢ ⎥ = I ⎢ 0 1 0 ⎥ ⎢ 0 0 1 ⎥ ⎣ ⎦ 34

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Spherical Model of the Rotating Earth Spherical model of earths surface, earth-fixed (rotating) coordinates

⎡ x ⎤ ⎡ cos L cosλ ⎤ ⎢ o ⎥ ⎢ E E ⎥ RE = ⎢ yo ⎥ = ⎢ cos LE sinλE ⎥ R ⎢ ⎥ ⎢ ⎥ zo sin LE ⎣ ⎦E ⎣ ⎦

LE : Latitude (from Equator), deg

λE : Longitude (from Prime Meridian), deg R : Radius (from Earth's center), deg

Earth's rotation rate, Ω, is 15.04 deg/hr 35

Non-Rotating (Inertial) Reference Frame for the Earth

Celestial longitude, λC, measured from First Point of Aries on the Celestial Sphere at Vernal Equinox

λC = λE + Ω(t − tepoch ) = λE + Ω Δt

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Transformation Effects of Rotation Rotation from inertial to rotating frame about North (z) Axis ⎡ ⎤ ⎡ cosΩΔt sinΩΔt 0 ⎤ ⎡ cosΩΔt sinΩΔt 0 ⎤ xo ⎢ ⎥⎢ ⎥ R = ⎢ sin t cos t 0 ⎥R = −sinΩΔt cosΩΔt 0 y E ⎢ − ΩΔ ΩΔ ⎥ I ⎢ ⎥⎢ o ⎥ 0 0 1 0 0 1 ⎢ ⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ zo ⎣ ⎦I

Location of satellite, rotating and inertial frames

⎡ cos L cosλ ⎤ ⎡ cos L cosλ ⎤ ⎢ E E ⎥ ⎢ E C ⎥ rE = ⎢ cos LE sinλE ⎥(R + Altitude); rI = ⎢ cos LE sinλC ⎥(R + Altitude) ⎢ sin L ⎥ ⎢ sin L ⎥ ⎣ E ⎦ ⎣ E ⎦

Orbital calculations generally are made in an inertial frame of reference 37

Gravity Force Between Two Point Masses Magnitude of gravitational attraction

G : Gravitational constant = 6.67 ×10−11 Nm2 / kg2 st 24 Gm1m2 m1 : Mass of 1 body = 5.98 ×10 kg for Earth F = 2 m : Mass of 2ndbody = 7.35 ×1022 kg for Moon r 2 r : Distance between centers of mass of m1 and m2 , m

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Acceleration Due To Gravity

Gm1m2 µ1 F2 = m2a2 = 2 ! m2 2 r r “Inverse-square Law” Gm1 µ1 a2 = ! r2 r2

At Earth’s surface, acceleration due to gravity is µ a g E 9.8 m s2 g ! oEarth = 2 ! Rsurface

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Gravitational Force Vector of the Earth Force always directed toward the Earth’s center

⎡ x ⎤ µE ⎛ rI ⎞ µE ⎢ ⎥ F = −m = −m y (vector), as r = r g 2 ⎜ ⎟ 3 ⎢ ⎥ I I rI ⎝ rI ⎠ rI ⎢ z ⎥ ⎣ ⎦I (x, y, z) establishes the direction of the local vertical

⎡ x ⎤ ⎢ y ⎥ ⎢ ⎥ ⎡ cos L cosλ ⎤ ⎢ z ⎥ E I rI ⎣ ⎦I ⎢ ⎥ = = cos L sinλ 2 2 2 ⎢ E I ⎥ rI x + y + z I I I ⎢ sin L ⎥ ⎣ E ⎦ 40

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Equations of Motion for a Particle in an Inverse-Square-Law Field Integrating the acceleration (Newton’s 2nd Law) allows us to solve for the velocity of the particle

⎡ x ⎤ dv(t) 1 µE ⎛ rI ⎞ µE ⎢ ⎥ = v! t = F = − = − y ( ) g 2 ⎜ ⎟ 3 ⎢ ⎥ dt m rI ⎝ rI ⎠ rI ⎢ z ⎥ ⎣ ⎦ 3 components of velocity

⎡ ⎤ ⎡ 3 ⎤ ⎡ ⎤ vx (T ) x rI vx (0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ 3 ⎢ vy (T ) ⎥ = −µE ⎢ y r ⎥dt + ⎢ vy (0) ⎥ ∫0 I ⎢ ⎥ ⎢ 3 ⎥ ⎢ ⎥ vz (T ) z rI vz (0) ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ 41

Equations of Motion for a Particle in an Inverse-Square-Law Field Same as before. 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) ⎥ ⎣ ⎦ ⎣ ⎦ 3 components of position ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x(T ) vx (t) x(0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ y(T ) ⎥ = ⎢ vy (t) ⎥dt + ⎢ y(0) ⎥ ∫0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z(T ) vz (t) z(0) ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ 42

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Dynamic Model with Inverse- Square-Law Gravity § No aerodynamic or thrust force § Orbits in Equatorial plane § Neglect motions in the z direction Dynamic Equations Initial Conditions at Equator

3 v!x (t) = −µE xI (t) rI (t) 3 vx (0) = 7.5, 8, 8.5 km / s v!y (t) = −µE yI (t) rI (t) vy (0) = 0 x!I (t) = vx (t) x(0) = 0 y!I (t) = vy (t) y(0) = 6,378 km = R 2 2 where r t = x t + y t 43 I ( ) I ( ) I ( )

Trajectories Calculated with Inverse-Square-Law Model

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Equatorial Orbits Calculated with Inverse-Square-Law Model

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MATLAB Code for Spherical-Earth Trajectories Script for Numerical Solution

R = 6378; % Earth Surface Radius, km tspan = 6000; % seconds options = odeset('MaxStep', 10) xo = [7.5;0;0;R]; [t1,x1] = ode15s('RoundEarth',tspan,xo,options); for i = 1:length(t1) v1(i) = sqrt(x1(i,1)*x1(i,1) + x1(i,2)*x1(i,2)); r1(i) = sqrt(x1(i,3)*x1(i,3) + x1(i,4)*x1(i,4)); end

function xdot = RoundEarth(t,x) % x(1) = vx % x(2) = vy % x(3) = x % x(4) = y Function for mu = 3.98*10^5; % km^2/s^2 Numerical Solution r = sqrt(x(3)^2 + x(4)^2); xdot(1) = -mu * x(3) / r^3; xdot(2) = -mu * x(4) / r^3; xdot(3) = x(1); xdot(4) = x(2); xdot = xdot'; end 46

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Work § “Work” is a scalar measure of change in energy

§ With constant force,

In one dimension, W F r r 12 = ( 2 − 1 ) In three dimensions, W FT r r 12 = ( 2 − 1 ) § With varying force, ⎡ ⎤ r2 dx T ⎢ ⎥ W = F dr, dr = dy 12 ∫ ⎢ ⎥ r1 ⎣⎢ dz ⎦⎥

r2 = f dx + f dy + f dz ∫ ( x y z ) r1 47

Conservative Force

§ Assume that the 3-D force field is a function of position Force Emanating F = F(r) from Source

§ The force field is conservative if

r2 r1 ∫ FT (r)dr + ∫ FT (r)dr = 0 r1 r2

… for any path between r1 and r2 and back

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Gravitational Force Field § Gravitational force field

µE Fg = −m 3 rI rI

§ Gravitational force field is conservative because

r r 2 µ 1 µ − m E r dr − m E r dr = 0 ∫ r 3 I I ∫ r 3 I I r1 I r2 I … for any path between r1 and r2 and back

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Gravitational Force is Conservative

Because the force is the derivative (with respect to r) of a scalar function of r called the potential, V(r): µ µ V (r) = −m +Vo = −m 1 2 +Vo r (rT r)

⎡ ∂V ∂x ⎤ ⎡ x ⎤ ∂V (r) ⎢ ⎥ µ ⎢ ⎥ = ⎢ ∂V ∂y ⎥ = m 3 y = −Fg ∂r r ⎢ ⎥ ⎢ ∂V ∂z ⎥ ⎢ z ⎥ ⎣ ⎦ ⎣ ⎦

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Potential Energy Potential energy is defined with respect to a reference radius, r1 µ µ ΔPE ! V (r2 ) −V (r1 ) = −m + m r2 r1 (The term V cancels) o

ΔPE = 0 when r2 = r1

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Kinetic Energy Apply Newton’s 2nd Law to the definition of Work

dr dv = v; dr = vdt F = m dt dt

r2 t2 T T ⎛ dv ⎞ W = F dr = m v dt 12 ∫ ∫ ⎝⎜ dt ⎠⎟ r1 t1 1 t2 d 1 t2 d = m vT v dt = m v2 dt 2 ∫ dt ( ) 2 ∫ dt ( ) t1 t1

1 2 2 = m ⎡v (t2 ) − v (t1 )⎤ ! T2 − T1 ! ΔKE 2 ⎣ ⎦ T is the kinetic energy of the point mass, m 52

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Total Energy Potential energy depends only on gravitational force µ PE = −m r Kinetic energy depends only on velocity magnitude 1 T ! KE = mv2 2 Total energy is the sum E = PE + KE µ 1 = −m + mv2 r 2 53

Energy in a Conservative System Energy is constant on the trajectory µ 1 E = PE + KE = −m + mv2 = Constant r 2

⎛ µ 1 2 ⎞ ⎛ µ 1 2 ⎞ E2 − E1 = −m + mv2 − −m + mv1 = 0 ⎜ ⎟ ⎜ ⎟ ⎝ r2 2 ⎠ ⎝ r1 2 ⎠

⎛ µ µ ⎞ ⎛ 1 2 1 2 ⎞ P ⎜ −m + m ⎟ = ⎜ mv2 − mv1 ⎟ 1 ⎝ r2 r1 ⎠ ⎝ 2 2 ⎠

PE2 − PE1 = KE2 − KE1

Change in potential energy between any 2 points on the orbit

equals change in kinetic energy P2

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Elliptical Planetary Orbits

§ Assume satellite mass is negligible compared to Earth’s mass § Then § Center of mass of the 2 bodies is at Earth’s center of mass § Center of mass is at one of ellipse’s focal points § Other focal point is “vacant”

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Properties of Elliptical Planetary Orbits § Eccentricity can be determined from apogee and perigee radii

r − r r − r e = a p = a p ra + rp 2a

rp = a(1− e) ra = a(1+ e)

§ Semi-major axis is the average of the two r + r a = a p 2 56

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Properties of Elliptical Planetary Orbits

§ Semi-latus rectum, p, can be expressed as a function of a and e p = a(1− e2 )

§ Orbital radius, r, is given by p r = , m or km 1+ e cosθ θ : True Anomaly Angle from perigee direction, deg or rad

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Next Time: Precursors to Space Flight: Rocket, Missiles, and Men in Space, Prophets with Some Honor, Ch 4 (ER) … the Heavens and the Earth, Introduction, Ch 1 Orbital Motion: Understanding Space, Ch 5

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Supplemental Material

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Near and Far Sides of the Moon Clementine mission, 1994

Albedo

Height above Reference

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Conservation of Energy Energy is conserved in an elastic collision, i.e. no losses due to friction, air drag, etc. “Newton’s Cradle” illustrates interchange of potential and kinetic energy in a gravitational field

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Ellipse

As Conic Section As Tilted Circle

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Equations that Describe Ellipses

x2 y2 + = 1 a2 b2 b a : Semi - major axis, m or km y x b : Semi - minor axis, m or km a o

x(θ ) = a cos(θ ) y(θ ) = b sin(θ ) θ : Angle from x - axis (origin at center) rad

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Constructing Ellipses

F1P1 + F2P1 = F1P2 + F2P2 = 2a Foci (from center), ⎡ ⎤ x f ⎡ 2 2 ⎤ ⎡ 2 2 ⎤ ⎢ ⎥ = ⎢ − a − b ⎥, ⎢ a − b ⎥ ⎢ y f ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎣ ⎦1,2 ⎣ ⎦ ⎣ ⎦

String, Tacks, and Pen Archimedes Trammel Hypotrochoid

Therefore, an ellipse is an epicycle 64

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Ellipses

Semi - latus rectum ("The Parameter"), b2 p = , m a

b2 Eccentricity, e = 1− a2 b2 = 1− e; b = a 1− e a2

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% Initial Conditions vx0 = 10; vz0 = 100; MATLAB Code x0 = 0; for Math Review z0 = 0; % Analytic (Closed-Form) Solutions % FRS 148 Lecture 2: Orbital Motions vx1 = vx0; clear vz1 = vz0 - g * t; disp(' ') x1 = x0 + vx0 * t; disp('======‘) z1 = z0 + vz0 * t - 0.5* g * t.* t; disp('FRS 148 Lecture 2: Orbital ==Motions') disp('======') vz0 = 150; % New initial vertical velocity disp(' ') disp(['Date and Time are ‘, vx2 = vx0; num2str(datestr(now))]); vz2 = vz0 - g * t; disp(' ') x2 = x0 + vx0 * t; z2 = z0 + vz0 * t - 0.5* g * t.* t; % Flat-Earth Trajectories disp(' ') vz0 = 200; % New initial vertical velocity disp('Flat-Earth Trajectories') disp('======') vx3 = vx0; g = 9.8; % m/s^2 vz3 = vz0 - g * t; t = 0:0.1:40; x3 = x0 + vx0 * t; z3 = z0 + vz0 * t - 0.5* g * t.* t; 66

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MATLAB Code for Math Review figure subplot(2,2,1) plot(t,vx1,t,vx2,t,vx3),grid, title('Analytic Flat-Earth Trajectory') xlabel('Time, s'), ylabel('Horizontal Velocity, m/s') legend('Case 1','Case 2','Case 3') subplot(2,2,2) plot(t,vz1,t,vz2,t,vz3),grid xlabel('Time, s'), ylabel('Vertical Velocity, m/s') subplot(2,2,3) plot(t,x1,t,x2,t,x3),grid xlabel('Time, s'), ylabel('Horizontal Position, m') subplot(2,2,4) plot(t,z1,t,z2,t,z3), grid xlabel('Time, s'), ylabel('Vertical Position, m') figure plot(x1,z1,x2,z2,x3,z3),grid, title('Analytic Vertical vs. Horizontal Position') xlabel('Horizontal Position, m'), ylabel('Vertical Position, m') legend('Case 1','Case 2','Case 3')

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MATLAB Code for Math Review % Numerical Solutions tspan = 40; xo = [10;100;0;0]; [t1,x1] = ode45('FlatEarth',tspan,xo); xo = [10;150;0;0]; [t2,x2] = ode45('FlatEarth',tspan,xo); xo = [10;200;0;0]; [t3,x3] = ode45('FlatEarth',tspan,xo); figure subplot(2,2,1) plot(t1,x1(:,1),t2,x2(:,1),t3,x3(:,1)),grid, title('Numerical Flat-Earth Trajectory') xlabel('Time, s'), ylabel('Horizontal Velocity, m/s') legend('Case 1','Case 2','Case 3') subplot(2,2,2) plot(t1,x1(:,2),t2,x2(:,2),t3,x3(:,2)),grid xlabel('Time, s'), ylabel('Vertical Velocity, m/s') subplot(2,2,3) plot(t1,x1(:,3),t2,x2(:,3),t3,x3(:,3)),grid xlabel('Time, s'), ylabel('Horizontal Position, m') subplot(2,2,4) plot(t1,x1(:,4),t2,x2(:,4),t3,x3(:,4)), grid xlabel('Time, s'), ylabel('Vertical Position, m') figure plot(x1(:,3),x1(:,4),x2(:,3),x2(:,4),x3(:,3),x3(:,4)),grid, title('Numerical Vertical vs. Horizontal Position') xlabel('Horizontal Position, m'), ylabel('Vertical Position, m') legend('Case 1','Case 2','Case 3’)

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MATLAB Code for Math Review % Numerical Solutions R = 6378; % Earth Surface Radius, km tspan = 6000; % seconds options = odeset('MaxStep', 10); xo = [7.5;0;0;R]; [t1,x1] = ode15s('RoundEarth',tspan,xo,options); for i = 1:length(t1) v1(i) = sqrt(x1(i,1)*x1(i,1) + x1(i,2)*x1(i,2)); r1(i) = sqrt(x1(i,3)*x1(i,3) + x1(i,4)*x1(i,4)); end xo = [8;0;0;R]; [t2,x2] = ode15s('RoundEarth',tspan,xo,options); for i = 1:length(t2) v2(i) = sqrt(x2(i,1)*x2(i,1) + x2(i,2)*x2(i,2)); r2(i) = sqrt(x2(i,3)*x2(i,3) + x2(i,4)*x2(i,4)); end xo = [8.5;0;0;R]; [t3,x3] = ode15s('RoundEarth',tspan,xo,options); for i = 1:length(t3) v3(i) = sqrt(x3(i,1)*x3(i,1) + x3(i,2)*x3(i,2)); r3(i) = sqrt(x3(i,3)*x3(i,3) + x3(i,4)*x3(i,4)); end 69

MATLAB Code for Math Review figure subplot(3,2,1) plot(t1,x1(:,1),t2,x2(:,1),t3,x3(:,1)),grid title('Numerical Round-Earth Trajectory') xlabel('Time, s'), ylabel('Horizontal Velocity, km/s') legend('Case 1','Case 2','Case 3') subplot(3,2,2) plot(t1,x1(:,2),t2,x2(:,2),t3,x3(:,2)),grid xlabel('Time, s'), ylabel('Vertical Velocity, km/s') subplot(3,2,3) plot(t1,x1(:,3),t2,x2(:,3),t3,x3(:,3)),grid xlabel('Time, s'), ylabel('Horizontal Position, km') subplot(3,2,4) plot(t1,x1(:,4),t2,x2(:,4),t3,x3(:,4)), grid xlabel('Time, s'), ylabel('Vertical Position, km') subplot(3,2,5) plot(t1,v1,t2,v2,t3,v3), grid xlabel('Time, s'), ylabel('Velocity, km/s') subplot(3,2,6) plot(t1,r1,t2,r2,t3,r3), grid xlabel('Time, s'), ylabel('Radius, km')

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MATLAB Code for Math Review figure plot(x1(:,3),x1(:,4),x2(:,3),x2(:,4),x3(:,3),x3(:,4)),grid, title('Vertical vs. Horizontal Position') xlabel('Horizontal Position, m'), ylabel('Vertical Position, m') legend('Case 1','Case 2','Case 3'), axis equal % Point-Mass Gravity Field % ======disp(' ') disp('Acceleration Magnitude Plots') disp('======') % 1-D Plot of Acceleration Magnitude vs. Radius mu = 3.98*10^14; % Earth's Gravitation Parameter, m^3/s^2 r = 6378137:100:10*6378137; % Radius Range to be Plotted f = mu./(r.*r); % Magnitude (positive number) figure plot(r,f, -r,f),grid, xlabel('Radius, km'), title('Acceleration Magnitude') ylabel('Gravitational Acceleration, m/s^2') % 3-D Plot of Acceleration Magnitude vs. x and y M = zeros(100,100); % Initialize matrix with zeros MatrixSize = size(M) % Identify dimensions of the matrix for i = 1:100 for j = 1:100 r = 6378137 * 0.1 * sqrt(0.5*((i-50)^2 + (j-50)^2)); f = log10(mu / r^2); M(i,j) = f; % Compute M values end end % Surface, Mesh, and Contour Plots of M figure mesh([1:100],[1:100],M),grid on, view(-217,30), title('Acceleration Magnitude') % Oblique "3-D" view of M zlabel('Log10(mu)/r^2, m/s^2'),xlabel('x, km'),ylabel('y, km') 71

Columbiad Code

CannonLength = 274; % m GunCotton = 61; % m LaunchLength = 274 - 61; % m ExitVelocity = 16500; % m/s ExitMach = ExitVelocity/340; StaticTemp = 273 + 20; % Kelvin StagTemp = StaticTemp*(0.2*ExitMach^2); % Kelvin MeltTempAlum = 273 + 660; StagPresRatio = (0.2*ExitMach^2)^(1.4/0.2); % - tGun = 2*LaunchLength/ExitVelocity; % s AvAccel = (ExitVelocity^2)/(2*LaunchLength);% m/s^2 AvAccelG = AvAccel/9.8; % g

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Particulars of the Columbiad Launch

• Launch Length = 213 m • Exit Velocity = 16,500 m/s • Exit Mach = 48.5 • Time in Gun = 26 ms • Acceleration G = 65,210 g • Acceleration = 639,085 m/s^2 • Stagnation Temperature at Exit = 138,000 K • Aluminum Melting Temperature = 933 K • Stagnation Pressure Ratio = 5.144e+18

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