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Seminar 2! Early History & Fiction! Orbital Motion! FRS 112, Princeton University! Robert Stengel" Decades of the Great Dreams" From the Earth to 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 2015 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 17th-18th cent Science Fact Gottfried Leibniz (1646-1716): Calculus (1675), founder of "dynamics". Isaac Newton (1642-1727): Laws of Motion, Universal Gravitation, Principia (1687). Science Fiction Jonathan Swift, Gulliver's Travels (1726), 18th-19th cent Pierre-Simon Laplace (1749Voltaire, Micromégas (1752) 1827): Solar system cosmology and dynamics. Carl Friedrich Gauss (1777-1855): Orbit calculation, shape of the Earth. "Balloonatics": Montgolfier brothers & deRozier (1783), 1st manned flight; Madame Blanchard; Major Turner; Glaisher & Coxwell to 35K ft World Scene 1689: Peter the Great, Czar of Russia. 1690: Battle of the Boyne. 1729: Bach's St. Matthew Passion. 1746: PRINCETON IS FOUNDED! AMERICAN REVOLUTION; FRENCH REVOLUTION; AGE OF THE ENLIGHTENMENT; INDUSTRIAL REVOLUTION. 1781: Herschel discovers Uranus. 1800s: Exploits of Napoleon. Steamships, railroads, Babbage's Analytical Engine. 2! Gottfreid Leibniz! (1646-1716)! !" German mathematician and philosopher" !" Along with Newton, invented “infinitesimal calculus”" !" Established notation for derivatives, integrals, and “chain rule”, e.g.," dy d !" f ( x ) #$ = ; dx dx y = f ( x ); ! y dx = ! f ( x ) dx; d 2 y d %' d !" f ( x ) #$ )' = & * dx 2 dx (' dx +' ! y dx = ! y dx du du dy dy du1 du = i i!i n dx du1 du2 dx 3! Pierre Simon Laplace! (1749-1827)! !" !" !" !" !" !" Mathematical astronomy and statstics" The “French Newton”" Mécanique Céleste (1799-1825), 5 volumes" Laplace transform! Laplace’s equation! The “figure” of the Earth and its potential (TBD); spherical harmonics" " L [ x(t)] = x(s) = # x(t)e! st dt 0 s = $ + j% , ( j = i = !1) ! 2" = 0 4! Karl Friedrich Gauss! (1777-1855)! •" German mathematician who contributed to many fields" •" Determination of conic section orbits from observations" •" Method of “least squares”" •" “Gaussian distribution”" •" Discovered “Fast Fourier Transform” (Cooley, Tukey, 1965) before there was a Fourier Transform (1807)! •" Circuit laws, electromagnetics" 5! Gaussian probability density function of random variable, x 2 ( ) # 1 2 pr ( x ) = e 2" 2!" µ : Mean value of x; " : Standard deviation of x x# µ !" Describes variability of many physical symbols " !" Also called a “Normal Distribution”" !" Classic “bell-shaped curve”" 6! Charles Babbage! (1791-1871)! !" Conceived of Difference Engine to tabulate polynomials (1822)" !" Analytical Engine with programmable arithmetic logic (1837)" !" Neither machine was built in his lifetime" 7! Decades of the Great Dreams! 8! Near and Far Sides of the Moon" Clementine mission, 1994" Albedo! Height above Reference! 9! A Timeline of Events Critical to Space Travel – VI! Date Early 19th cent Late 19th cent Science Fact Karl Jacobi (18041851): mathematics of dynamical systems. William Congreve: Military & rescue rockets. Science Fiction Edgar Allan Poe, The Unparalleled Adventure of One Hans Pfaall (1835), The moon hoax (False account of Herschel's research, 1835) Henri Poincaré (1854- 1865: Jules Verne, 1912): Restricted 3From the Earth to the body problem. Kirchoff Moon, Alexandre & Bunsen: Dumas, Voyage a la Spectroscope (1859) lune, Henri de Parville, Un Habitant de la planete Mars, Anon, Voyage a la lune, English clergyman, Chrysostom Trueman (nom de plume), The History of a Voyage to the Moon, Camille Flammarion, Mondes imaginaires et mondes reels World Scene Bombardment of Copenhagen (1807); WAR OF 1812 (StarSpangled Banner); Battle of Waterloo AMERICAN CIVIL WAR; FRANCO-PRUSSIAN WAR; VICTORIAN ERA. 1866: Nobel invents dynamite. 1876: Bell invents telephone. 1895: Roentgen discovers X-rays. 10! 1865! Jules Verne (1828-1905)! 11! Orbital Motion! 12! Orbits 101" Satellites! Escape and Capture (Comets, Meteorites)! 13! Two-Body Orbits are Conic Sections " 14! Orbits 102" (2-Body Problem) ! •" e.g., " –" Sun and Earth or" –" Earth and Moon or" –" Earth and Satellite" •" Circular orbit: radius and velocity are constant" •" Low Earth orbit: 17,000 mph = 24,000 ft/s = 7.3 km/s" •" Super-circular velocities" –" Earth to Moon: 24,550 mph = 36,000 ft/s = 11.1 km/s" –" Escape: 25,000 mph = 36,600 ft/s = 11.3 km/s" •" Near escape velocity, small changes have huge influence on apogee" 15! 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" dv ( t ) d = ma ( t ) = F !" mv ( t ) #$ = m dt dt ! fx # & % F = % fy & = force vector % f & %" z &$ ! v (t ) x # d m # vy ( t ) dt # #" vz ( t ) $ ! f $ & # x & & = # fy & & # & f z &% #" &% 16! 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 $ ! a $ & # x & # dv ( t ) 1 = v! ( t ) = F = # fy m & = # ay & m dt # f m & # & &% #" az &% #" z v (T ) = ! T 0 T T 1 dv ( t ) dt + v ( 0 ) = ! a ( t ) dt + v ( 0 ) = ! F dt + v ( 0 ) 0 0 m dt 3 components of velocity" ! v (T ) # x # vy ( T ) # #" vz (T ) $ ! a (t ) x & T# & = ' # ay ( t ) & 0# &% #" az ( t ) $ ! v (0) $ ! f (t ) m $ ! v (0) $ x x & & & & # # x T# & dt + # vy ( 0 ) & = ' # fy ( t ) m & dt + # vy ( 0 ) & & & 0# & & # # &% #" vz ( 0 ) &% #" fz ( t ) m &% #" vz ( 0 ) &% Equations of Motion for a Particle " Integrating the velocity allows us to solve for the position of the particle" ! x! ( t ) # dr ( t ) = r! ( t ) = v ( t ) = # y! ( t ) dt # #" z! ( t ) r (T ) = ! T 0 $ ! v (t ) & # x & = # vy ( t ) & # &% #" vz ( t ) $ & & & &% T dr ( t ) dt + r ( 0 ) = ! v dt + r ( 0 ) 0 dt 3 components of position" ! x (T ) # # y (T ) # #" z (T ) ! v (t ) $ x & T# & = ' # vy ( t ) & 0# #" vz ( t ) &% $ ! x (0) $ & & # & dt + # y ( 0 ) & & & # &% #" z ( 0 ) &% 18! 17! 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" v x ( t ) = v x0 v!z ( t ) = !g x! ( t ) = vx ( t ) ( z positive up ) Analytic (Closed-Form) Solution" v x ( T ) = v x0 T z! ( t ) = vz ( t ) Initial Conditions" vz (T ) = vz0 ! " g dt = vz0 ! gT v x ( 0 ) = v x0 0 x ( T ) = x 0 + v x0 T T z (T ) = z0 + vz0 T ! " gt dt = z0 + vz0 T ! gT 2 2 vz ( 0 ) = vz0 x ( 0 ) = x0 0 z ( 0 ) = z0 19! Trajectories Calculated with Flat-Earth Model" vx ( 0 ) = 10m / s vz ( 0 ) = 100, 150, 200m / s x (0) = 0 z (0) = 0 20! Nike-Tomahawk 2-Stage Sounding Rocket" Nike-Ajax AntiAircraft Missile" 45-kg payload to 370-km altitude" Trajectory well approximated using flat-Earth model from 2nd stage burnout" 21! MATLAB Code for FlatEarth Trajectories" Script for " Analytic Solution" ! g t ! vx0 vz0 x0 z0 ! vx1 vz1 x1 z1 Script for " Numerical Solution" tspan = 40; % Time span, s! xo = [10;100;0;0]; % Init. Cond.! [t1,x1] = ode45('FlatEarth',tspan,xo);! = = 9.8; ! 0:0.1:40;! = = = = 10;! 100;! 0;! 0;! = = = = vx0;! vz0 – g*t;! x0 + vx0*t;! z0 + vz0*t - 0.5*g*t.* t;! Function for " Numerical Solution" function xdot = FlatEarth(t,x) ! % x(1) = vx! % x(2) = vz! % x(3) = x! % x(4) = z! g ! = 9.8;! xdot(1) = 0;! xdot(2) = -g;! xdot(3) = x(1);! xdot(4) = x(2);! xdot = xdot';! 22! end! Sub-Orbital (Sounding) Rockets 1945 - Present! NASA Wallops Island Control Center" Canadian! Black Brant XII! LTV! Scout! 23! Spherical Model of the Rotating Earth" Spherical model of earth s surface, earth-fixed (rotating) coordinates" ! xo $ ! cos LE cos ' E & # # R E = # yo & = # cos LE sin ' E # z & # sin LE " o %E " $ & &R & % LE : Latitude (from Equator), deg ! E : Longitude (from Prime Meridian), deg R : Radius (from Earth's center), deg Earth's rotation rate, !, is 15.04 deg/hr 24! Non-Rotating (Inertial) Reference Frame for the Earth" Celestial longitude, "C, measured from First Point of Aries on the Celestial Sphere at Vernal Equinox" !C = !E + "( t # t epoch ) = !E + " $t 25! Transformation Effects of Rotation" Transformation from inertial frame to rotating frame" $ cos !"t R E = & # sin !"t & 0 &% sin !"t cos !"t 0 0 0 1 $ cos !"t ' ) R = & # sin !"t ) I & 0 &% )( sin !"t cos !"t 0 0 0 1 ' $ xo ' ) )& y )& o ) )( & zo ) (I % Location of satellite, rotating and inertial frames" " cos LE cos ! E $ rE = $ cos LE sin ! E $ sin LE # % " cos LE cos !C ' $ ' ( R + Altitude ); rI = $ cos LE sin !C ' $ sin LE & # % ' ' ( R + Altitude ) ' & Orbital calculations generally are made in an inertial frame of reference" 26! Gravity Force Between Two Point Masses" Magnitude of gravitational attraction" Gm1m2 F= r2 G : Gravitational constant = 6.67 ! 10 "11 Nm 2 / kg 2 m1 : Mass of 1st body = 5.98 ! 10 24 kg for Earth m2 : Mass of 2 nd body = 7.35 ! 10 22 kg for Moon r : Distance between centers of mass of m1 and m2 , m Force between Earth and Moon = 1.98 x 1020N! 27! Acceleration Due To Gravity" Gm1m2 r2 Gm µ = 2 1 ! 21 r r F2 = m2 a1on2 = a1on2 “Inverse-square Law”! µ1 = Gm1 : Gravitational parameter of 1st mass At Earth’s surface, acceleration due to gravity is" ag ! goEarth = µE 2 Rsurface 3.98 ! 1014 m 3 s 2 2 = 2 = 9.798 m s ( 6, 378,137m ) What is the acceleration due to gravity at the Moon’s surface (R = 1,375,000 m)?" 28! Gravitational Force Vector of the Earth" Force always directed toward the Earth’s center" ( x + µ E " rI % µE * Fg = !m 2 $ ' = !m 3 * y - (vector), as rI = rI rI # rI & rI *) z -, I (x, y, z) establishes the direction of the local vertical" rI = rI ! x $ & # # y & ! cos LE cos ' I #" z &% # I = # cos LE sin ' I x I2 + yI2 + zI2 # sin LE " $ & & & % 29! 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 ) µ E " rI % µE * 1 = v! ( t ) = Fg = ! 2 $ ' = ! 3 * y m dt rI # rI & rI *) z -, v (T ) = ! T 0 T T 1 dv ( t ) dt + v ( 0 ) = ! a ( t ) dt + v ( 0 ) = ! F dt + v ( 0 ) 0 0 m dt 3 components of velocity" ! v (T ) # x # vy ( T ) # #" vz (T ) $ ! x r3 I & T# & = ' µ E ( # y rI 3 0 & # 3 &% #" z rI ! v (0) $ $ & # x & & dt + # vy ( 0 ) & & # & #" vz ( 0 ) &% &% 30! 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 ) # dr ( t ) = r! ( t ) = v ( t ) = # y! ( t ) dt # #" z! ( t ) r (T ) = ! T 0 $ ! v (t ) & # x & = # vy ( t ) & # &% #" vz ( t ) $ & & & &% T dr ( t ) dt + r ( 0 ) = ! v dt + r ( 0 ) 0 dt 3 components of position" ! x (T ) # # y (T ) # #" z (T ) ! v (t ) $ x & T# & = ' # vy ( t ) & 0# #" vz ( t ) &% $ ! x (0) $ & & # & dt + # y ( 0 ) & & & # &% #" z ( 0 ) &% 31! Dynamic Model with InverseSquare-Law Gravity" !" No aerodynamic or thrust force" !" Neglect motions in the z direction" Dynamic Equations" v!x ( t ) = ! µ E x I ( t ) rI 3 ( t ) Initial Conditions at Equator" vx ( 0 ) = 7.5, 8, 8.5 km / s v!y ( t ) = ! µ E yI ( t ) rI 3 ( t ) vy ( 0 ) = 0 y! I ( t ) = vy ( t ) y ( 0 ) = 6, 378 km = R x! I ( t ) = vx ( t ) where rI ( t ) = x I2 ( t ) + yI2 ( t ) x (0) = 0 32! Trajectories Calculated with Inverse-Square-Law Model" 33! Equatorial Orbits Calculated with Inverse-Square-Law Model" 34! 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 for " Numerical Solution" function xdot = RoundEarth(t,x)! % x(1) = vx! % x(2) = vy! % x(3) = x! % x(4) = y! mu = 3.98*10^5; % km^2/s^2! 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! 35! Work" !" “Work” is a scalar measure of change in energy" !" With constant force," In one dimension, W12 = F ( r2 ! r1 ) = F"r In three dimensions, W12 = FT ( r2 ! r1 ) = FT "r !" With varying force," " dx % ' $ W12 = ! F dr, dr = $ dy ' r1 $# dz '& r2 r2 ( T = ! f x dx + fy dy + fz dz r1 ) 36! Conservative Force" !" Assume that the 3-D force field is a function of position" Iron Filings! Around Magnet! F = F(r) !" The force field is conservative if" r2 r1 Force Emanating from Source! T F r dr + F ( ) ! ! ( r ) dr = 0 T r1 r2 … for any path between r1 and r2 and back" 37! Gravitational Force Field" !" Gravitational force field" Fg = !m µE rI rI 3 !" Gravitational force field is conservative because" r2 r 1 µE µ ! " m 3 rI drI ! " m E3 rI drI = 0 rI rI r1 r2 … for any path between r1 and r2 and back" 38! How Do We Know that 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):" V ( r ) = !m µ µ + Vo = !m 1 2 + Vo T r (r r) " !V !x % " x % ' !V ( r ) $ µ$ ' = $ !V !y ' = m 3 $ y ' = (Fg !r r $ !V !z ' $# z '& & # This derivative is also called the gradient of V with respect to r" 39! Potential Energy" Potential energy is defined with respect to a reference radius, r1" µ µ +m r2 r1 (The term Vo cancels) !PE ! V ( r2 ) " V ( r1 ) = "m !PE = 0 when r2 = r1 Example:" m = 10kg µ = 3.98 ! 1014 m 3 s 2 r1 = 6, 378,000 m r2 = 10,000,000 m 1 $ #1 ' !PE = 3.98 " 1015 & 7 + % 10 6.378 " 10 6 )( = 2.26 " 10 8 kg i m 2 / s 2 = 2.26 " 10 8 joules 40! Kinetic Energy" Apply Newton’s 2nd Law to the definition of Work" dr = v; dr = vdt dt r2 t2 T " dv % W12 = ! F dr = m ! $ ' v dt # dt & r t 1 t2 T 1 t 1 d T 1 2 d 2 = m ! ( v v ) dt = m ! ( v ) dt 2 t1 dt 2 t1 dt F=m dv dt 1 2 2 1 = mv = m "# v 2 ( t 2 ) ! v 2 ( t1 ) $% 2 2 t1 t = 1 m "# v22 ! v12 $% ! T2 ! T1 ! &KE 2 Work evaluated as the integral from 1st to 2nd time" T is the kinetic energy of the point mass, m! 41! Total Energy" !" Potential energy of mass, m, depends only on the gravitational force field " !" Kinetic energy of mass, m, depends only on the velocity magnitude measured in an inertial frame of reference " !" Total energy is the sum of the two:" PE = !m T ! KE = µ r 1 2 mv 2 E = PE + KE µ 1 = !m + mv 2 r 2 42! Energy in a Conservative System" Energy is constant on the trajectory" E = PE + KE = !m µ 1 2 + mv = Constant r 2 E2 ! E1 = 0 " µ 1 2% " µ 1 2% $# !m r + 2 mv2 '& ! $# !m r + 2 mv1 '& = 0 2 1 " µ µ% " 1 2 1 2% !m + m = $ mv2 ! mv1 ' $# & 2 r2 r1 '& # 2 P1! PE2 ! PE1 = KE2 ! KE1 In a potential force field (e.g., inverse-square-law gravity), change in potential energy between any 2 points on the orbit equals change in kinetic energy" P2! 43! 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" 44! Ellipse" As Conic Section! As Tilted Circle! 45! 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”" r= p , m or km 1+ e cos! ! : True Anomaly Angle from perigee direction, deg or rad 46! Properties of Elliptical Planetary Orbits" !" Eccentricity can be determined from apogee and perigee radii" e= ra ! rp ra ! rp = ra + rp 2a rp = a (1! e ) ra = a (1+ e) !" Semi-major axis is the average of the two" a= ra + rp 2 47! Properties of Elliptical Planetary Orbits" !" Semi-latus rectum, p, can be expressed as a function of a and e! p = a (1! e2 ) !" Semi-minor axis, b, can be expressed as a function of ra and rp! b = ra rp !" Area of the ellipse, A, is! A=! ab = ! a 2 (1" e) , m 2 48! 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! ! 49! !upplemental Ma"ria#l 50! Equations that Describe Ellipses" x 2 y2 + =1 a2 b2 a : Semi - major axis, m or km b : Semi - minor axis, m or km a! b! y! x! o! x (! ) = a cos (! ) y (! ) = b sin (! ) ! : Angle from x - axis (origin at center) rad 51! Constructing Ellipses" F1P1 + F2 P1 = F1P2 + F2 P2 = 2a Foci (from center), ! xf # # yf " $ ! 2 2 & = # ' a 'b & 0 %1,2 #" String, Tacks, and Pen! $ ! &, # &% #" $ a2 ' b2 & 0 &% Archimedes Trammel! Hypotrochoid! 52! Ellipses" Semi - latus rectum ("The Parameter"), b2 p= , m a b2 Eccentricity, e = 1! 2 a 2 b = 1! e; b = a 1! e a2 53!