Seminar 3! Precursors to Space Flight! Orbital Motion! FRS 112, Princeton University! Robert Stengel" Prophets with Some Honor" The Human Seed and Social Soil: Rocketry and Revolution" Orbital Motion" Energy and Momentum" Orbital Elements" Precursors to Space Flight:!Rocket, Missiles, and Men in Space, Ch 4 ER) " the Heavens and the Earth, Introduction, Ch 1" Orbital Motion:!Understanding Space, Ch 5" Copyright 2015 by Robert Stengel. All rights reserved. For educational use only.! http://www.princeton.edu/~stengel/frs.html! 1! Introduction, the Heavens and the Earth" 2!
Introduction, the Heavens and the Earth" 3! Introduction, the Heavens and the Earth" 4!
Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution! 5! Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution! 6!
Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution! 7! The Human Seed and Social Soil: Rocketry and Revolution! 8!
Early 20 th Century Rocket Vehicles" Valier Rocket Car" Opel RAK.3 Car" Opel RAK.1 Plane" Opel RAK.6 Automobile" https://www.youtube.com/watch? v=lmqiwptw-w8" Opel RAK.1 Airplane" https://www.youtube.com/watch? v=vsqg28y_s3s" 9! 10!
Konstantin Tsiolkovsky 1857-1935)" Russian father of spaceflight " High school teacher who wrote of rocket-propelled vehicles" The rocket equation 1897): vehicle speed change depends on" Rocket exhaust velocity" Ratio of vehicle s full-to-empty mass"!v = v exhaust ln m initial m final 11! Orbital Motion! 12!
Specific Energy " is the energy per unit of the satellite s mass"! Specific potential energy:" PE S =! m m! Specific kinetic energy:" KE S = 1 2 µ r =! µ r m m v2 = 1 2 v2! Specific total energy:" P 1 " P 2 " E S = PE S + KE S =! µ r + 1 2 v2 13! Vis Viva Living Force) Integral "! Velocity is a function of radius and specific energy drop subscript)" 1 2 v2 = µ r + E v =! 2 µ " r + E $ &! Specific total energy is inversely proportional to the semi-major axis see App. C.4, Sellers)" E =! µ 2a! Velocity is a function of radius and semi-major axis see http://en.wikipedia.org/wiki/visviva_equation)" v = " µ 2 r! 1 $ a& ' 14!
Orbital Period" From Kepler s 3 rd Law, Period of the Orbit, P, is see App. C.6, Sellers)" P = 2! a3 µ, min Thus, the orbital period is related to the specific total energy as" P =! "µ2 2E 3, min where E < 0 for an ellipse 15! Examples of Circular Orbit Periods for Earth and Moon" Altitude above " Surface, km" Period, min" Earth" Moon" 0" 84.5" 108.5" 100" 86.5" 118" 1000" 105.1" 214.6" 10000" 347.7" 1905" 16!
Angular Momentum of a Particle Point Mass)! h = r! mv) = m r! v) = m r!!r ) 17! Angular Momentum of a Particle! Moment of linear momentum of a particle" Mass times components of the velocity that are perpendicular to the moment arm" h = r! mv) = m r! v) Cross Product: Evaluation of a determinant with unit vectors i, j, k) along axes, x, y, z) and v x, v y, v z ) projections on to axes" r! v = i j k x y z = yv z " zv y )i + zv x " xv z v x v y v z )k ) j + xv y " yv x 18!
Dimension of energy?! Scalar 1 x 1)! Dimension of linear momentum?! Vector 3 x 1)! Dimension of angular momentum?! Vector 3 x 1)! 19! Cross Product in Column Notation! Cross product identifies perpendicular components of r and v" r! v = i j k x y z = yv z " zv y )i + zv x " xv z v x v y v z )k ) j + xv y " yv x Column notation" r! v = $ yv z " zv y ) zv x " xv z ) xv y " yv x ) & ' 20!
Can We Define a Cross- Product-Equivalent Matrix?" Cross product" yv z " zv & y ) r! v == zv x " xv z ) = xv y " yv x ) $ $ '????????? & ' $ Cross-product-equivalent matrix" v x v y v z & ' " $!r " $ $ 0!z y z 0!x!y x 0 ' ' ' & 21! Angular Momentum Vector is Perpendicular to Both Moment Arm and Velocity! h = mr! v = m $ = m $ 0 "z y z 0 "x "y x 0 yv z " zv & y ) zv x " xv z ) xv y " yv x ) ' & ' $ v x v y v z & = m!rv ' 22!
Specific Angular Momentum Vector of a Satellite! is the angular momentum per unit of the satellite s mass, referenced to the center of attraction" h S = m m r! v = r! v = r!!r It is constant and perpendicular to the orbital plane" How do we know that?" 23! Equations of Motion for a Particle in an Inverse-Square-Law Field " Recall" a t) =!v t) =!!r t) =! µ r 2 t) " $ t) r t) & ' =! µ r 3 t r I ) r t) or"!!r + µ r 3 r = 0 24!
Cross Products of Radius and Radius Rate! Then" " r!!!r + µ r r $ 3 & ' = 0 r! r = 0!r!!r = 0 because they are parallel" d dt Chain Rule for Differentiation" r!!r ) =!r!!r ) + r!!!r ) = r!!!r ) 25! Specific Angular Momentum! " r!!!r + µ r r $ 3 & ' = r!!!r ) + µ r! r) r 3 = d dt r!!r ) = dh S dt Consequently" h S = Constant = 0!h S " r!r ) Perpendicular to the plane of motion) Orbital plane is fixed in inertial space" 26!
Eccentricity Vector!!!!r + µ r r $ " 3 & ' h =!!r ' h + µ r r ' h = 0 3!!r! h = " µ r r! h = " µ 3 r r! r!!r ) 3 With triple vector product identity see Supplement)"!!r! h = " µ r r! r!!r ) = " µ!rr " r!r ) = µ d 3 r 2 dt Integrating" "!!r! h)dt =!r! h = µ r $ r + e & ' $ r & r ' e = Eccentricity vector Constant of integration) 27! Significance of Eccentricity Vector! )!r! h " µ r $ r + e &, + '. * - T h = 0!!r " h) T h µrt h µe T h = 0 r!"µe T h = 0 0" 0"! e is perpendicular to angular momentum,"! which means it lies in the orbital plane"! Its angle provides a reference direction for the perigee" 28!
Classical Orbital Elements" Dimension and Time" a : Semi-major axis e : Eccentricity t p : Time of perigee passage Orientation"! :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ": Argument of Perigee 29! In-plane Parameters of an Elliptical Orbit" Dimensions of the orbit" p = h2 µ = Semi-latus rectum h = Magnitude of angular momentum E =! µ r + 1 2 v2 =! µ = Specific total energy 2a e = 1+ 2 Ep µ = Magnitude of eccentricity vector p a = = Semi-major axis 2 1! e r a = a 1+ e ) = Apogee radius ) = Perigee radius r p = a 1! e µ E = 3.98!10 5 km 3 s 2 R E = 6,378 km 30!
In-plane Parameters of an Elliptical Orbit" Position and velocity of the satellite" p r = 1+ ecos! = h 2 µ 1+ ecos!! = True anomaly v = µ ar 2a " r ) Period of the orbit" P = 2! a3 µ 31! Orientation of an Elliptical Orbit" 32!
Position and Velocity Following Launch Determine Orbital Elements" Identical major axes = " Identical orbital periods" 33! Planar Orbit Establishment from Measured Radius, Angular Rate, Velocity, and Time" 2 Given r h = r o!! o o,!! o,v o,t o p= h 2 µ!! o = v o cos" o r o ) E = v 2 o " 2 " µ o : Flight Path Angle r from Local Horizontal o µr a = o 2 ) 1 p &, 2µ " r! o = cos "1 o v o + "1 e $ r o '. * - e = 1" p a 2 a " r / o = cos "1 o & $ ae ' : Eccentric Anomaly FPS-16 Radar" t p = a3 r p = a 1" e ) µ / o " esin/ o ) 34!
Effect of Launch Latitude on Orbital Parameters" Typical launch inclinations from Wallops Island" Latitude = 38 )" Launch latitude establishes minimum orbital inclination" Time of launch establishes line of nodes" Argument of perigee established by" Launch trajectory" On-orbit adjustment" 35! Typical Satellite Orbits " GPS Constellation" 26,600 km" Sun- Synchronous Orbit" 36!
Geo-Synchronous Ground Track " Geo- Synchronous Ground Track" 42, 164 km" Marco Polo-1 & 2" 37! Orbital Lifetime of a Satellite" Aerodynamic drag causes orbit to decay" dv dt =! C D"V 2 S /2!B * "V 2 S /2 m B* = C D S /m Air density decreases exponentially with altitude"! =! SL e "h/h scale! SL = air density at sea level; h scale = atmospheric scale height Drag is highest at perigee" Air drag circularizes the orbit" Large change in apogee" Small change in perigee" Until orbit is ~circular" Final trajectory is a spiral" 38!
Orbital Lifetime of a Satellite" Aerodynamic drag causes energy loss, reducing semi-major axis, a" da dt =! µab * " SL e! a!r ) / h scale Variation of a over time" a e! a!r ) / h s " da =! µb * SL " dt a a 0 Time, t decay, to reach earth s surface a = R) from starting altitude, h 0 " t 0 t decay = h scale e h 0 / h scale µrb *! "1 ) SL 39! NRL Starshine 1 Orbital Decay 2003)" ISS Altitude = 330-435 km" http://www.azinet.com/starshine/descript.htm! 40!
Next Time:! Early Space Age!" the Heavens and the Earth, Ch 2 to 5; "! Launch Dynamics & Staging!" Understanding Space, Sec 9.1, 14.1 pp. 535-542), 14.3" 41! Supplemental Material 42!
Prophets with Some Honor" 43! Long-Distance Communication" 44!
Industrial Revolution" 45! Electric Power Storage, Generation, and Transmission" 46!
Pangaea and Eusthenopteron" 47! Industrial Revolution and Government Science" 48!
Government, Technology, and War" 49! Background Math! Triple Vector Product Identity! a! b! c) " a i c)b a i b)c = a T c)b a T b)c Dot Product of Radius and Rate! r i!r = r T!r = r dr dt 50!