Appendix to Section 3: A brief overview of Geodetics MAE 5540 - Propulsion Systems Geodesy • Navigation Geeks do Calculations in Geocentric (spherical) Coordinates • Map Makers Give Surface Data in Terms !earth of Geodetic (elliptical) Coordinates • Need to have some idea how to relate one to another -- science of geodesy MAE 5540 - Propulsion Systems How Does the Earth Radius Vary with Latitude? k z ? 2 ? 2 REarth = x + y2 + z = r + z r R " Ellipse: y j ! r 2 + z 2 = 1 j 2 Req Req 1 - eEarth y x i "a" "b" r Prime Meridian ! x MAE 5540 - Propulsion Systems How Does the Earth Radius vary with Latitude? r 2 + z 2 = 1 2 Req Req 1 - eEarth ! 2 2 2 2 2 1 - eEarth r + z = Req 1 - eEarth ! z 2 2 2 2 z 2 r Req = r + 2 = r 1 + 2 1 - eEarth 1 - eEarth Rearth z " r 2 2 2 z 2 2 r = Rearth cos " r = tan " MAE 5540 - Propulsion Systems How Does the Earth Radius vary with Latitude? 2 2 Req 2 tan ! 2 = cos ! 1 + 2 = Rearth 1 - eEarth 2 2 2 1 - eEarth cos ! + sin ! 2 = 1 - eEarth Inverting .... 2 2 2 2 2 2 cos ! + sin ! - eEarth cos ! 1 - eEarth cos ! 2 = 2 1 - eEarth 1 - eEarth R earth ! = 2 Req 1 - eEarth 2 2 1 - eEarth cos ! MAE 5540 - Propulsion Systems Earth Radius vs Geocentric Latitude R earth ! = Polar Radius: 6356.75170 km 2 Req 1 - eEarth Equatorial Radius: 6378.13649 km 2 2 1 - eEarth cos ! 2 2 2 e = 1 - b = a - b = Earth a a2 6378.13649 2 6378.136492 - = 0.08181939 6378.13649 MAE 5540 - Propulsion Systems Earth Radius vs Geocentric Latitude (concluded) 6380. 6375. 6370. Radius, Km 6365. 6360. 6355. -100 -50 0 50 100 Geocentric Latitude, deg. MAE 5540 - Propulsion Systems Earth Radius … alternate formula • Earth radius as Function of Latitude MAE 5540 - Propulsion Systems What is the mean radius of the earth? RE = Req 2 ! 2 dA = " [ RE cos ( !) ] 1 - eEarth ! 2 2 1 - eEarth cos ! RE cos(!) • IAU Convention: ! Based on Earth's Volume RE d! cos (!) ! RE ! Sphere Volume: ! 3 4 " R = V 3 Emean E 2 RE cos(!) x RE d! cos (!) dV = " [ ] ! MAE 5540 - Propulsion Systems What is the Earth's Mean Radius? (continued) • Earth's (ellipsoid) Volume ! 2 V = R cos 3d = E ! E" " " -! 2 ! 2 1 - e2 3/2 R3 Earth cos3 d = eq ! 2 2 " " 1 - eEarth cos " -! 2 3 4 ! 2 R 1- e eq 3 earth MAE 5540 - Propulsion Systems What is the Earth's Mean Radius? (continued) • Based on Volume 3 Ellipsoid Volume: 4 ! 1- e2 R 3 eq 3 Sphere Volume: 4 ! R 3 sphere " R R = 1- 0.081819392 1/6 6378.13649 = 6371.0002 km sphere # mean • Mean Radius we have been using is for a Sphere with same volume as the Earth ME = $E VE "gravitational radius" MAE 5540 - Propulsion Systems Earth Radius vs Geocentric Latitude (concluded) “Gaussian Surface” 6380. 6375. 6370. Radius, Km 6365. 6360. 6355. -100 -50 0 50 100 Geocentric Latitude, deg. MAE 5540 - Propulsion Systems Geocentric vs Geodetic Coordinates Mean North (Celestial) Pole • Map makers Geocentric vs Geodetic C2 ooordinates define a new Req 1 - e(earth) latitude which is Mean Greenwich the angle that Meridian RE normal to the ! Earth's surface ! makes with the Geocenter R respect to the !' eq equatorial plane " Req • Geodetic latitude Equator MAE 5540 - Propulsion Systems Geocentric vs Geodetic Coordinates (contined) Mean •Since the Earth is North (Celestial) Elliptical only along the Pole z-axis ... geodetic and geocentric 2 h longitude are Req 1 - e(earth) identical Mean Greenwich Meridian RE • Altitude is an ! extension of the Geocenter ! line of latitude Req geodetic) !' " Req Equator MAE 5540 - Propulsion Systems Geocentric vs Geodetic Coordinates (contined) •Complex nonlinear Mean equations describe North (Celestial) relationship between Pole geocentric and geodetic latitude 2 h Req 1 - e(earth) Mean • Derivation requires Greenwich Meridian RE Extensive ! Knowledge of Spherical ! Trigonometry Geocenter R !' eq " Req Equator MAE 5540 - Propulsion Systems Geocentric vs Geodetic Coordinates (contined) Geocentric Cartesian Coordinates * rtarget xtarget = R' +h cos !' cos " !' h y = R' +h cos ' sin z target !' ! " 2 !' ztarget = R' 1 - e +h sin !' ! !' earth R' ' Req ! R' = !' 2 1 - e sin2 !' !' earth rtargetr * We would be here all week if I try to derive this "Radius of Curvature" Req MWAhEit m55o4re0, S- tPeprohepnu Als.i, oand S Hyasetreinmgs, Edward A., Jr., FORTRAN Program for the Analysis of Ground Based Range Tracking Data--Usage and Derivations, NASA TM 104201, December, 1992 Geocentric vs Geodetic Coordinates (contined) Geocentric Polar Coordinates r 2 2 2 target R = x + y + z target target target target h z -1 ytarget target = tan " xtarget !' ! z -1 target R' !target = tan 2 2 !' xtarget + ytarget !' rtargetr "Radius of Curvature" Req MAE 5540 - Propulsion Systems Geocentric vs Geodetic Coordinates (concluded) Inverse Relationships, non-linear no direct solution 2 2 xtarget + ytarget h = - R' cos !' !' y = tan-1 target "target xtarget ztarget 1 ' = -1 # ! target tan 2 2 R' xtarget + ytarget 1 - e2 !' earth R' + h !' target MWAhEit m55o4re0, S- tPeprohepnu Als.i, oand S Hyasetreinmgs, Edward A., Jr., FORTRAN Program for the Analysis of Ground Based Range Tracking Data--Usage and Derivations, NASA TM 104201, December, 1992 Pulling it all together • Given geodetic coordinates -- compute geocentric i) Compute geocedtric cartesian coordinates Range, runway threshholds, radar antennae, beacon .... x = R' +h cos !' cos " target !' y = R' +h cos !' sin " target !' 2 ztarget = R' 1 - e +h sin !' !' earth Req R' = !' 2 1 - e sin2 !' MAE 5540 - Propulsion Systems earth Pulling it all together (continued) • Given geodetic coordinates -- compute geocentric ii) Compute Geocentric polar coordinates next 2 2 2 Rtarget = xtarget + ytarget + ztarget -1 ytarget target = tan ! xtarget -1 ztarget "target = tan 2 + 2 xtarget ytarget MAE 5540 - Propulsion Systems Pulling it all together (concluded) • Given geocentric (usually x,y,z) coordinates -- compute geodetic Req R' = ' 2 ! 1 - e sin2 !' GPS, INS, TLE's .... earth No explicit solution: 2 2 xtarget + ytarget requires h = - R' 1) series expansion solution, cos !' !' 2) numerical iteration, -1 ytarget 3) or a special solution "target = tan called "Ferrari's method"** xtarget z -1 target 1 !'target = tan # x 2 + y 2 R' target target 1 - e2 !' earth R' + h !' target **NASA Technical Paper 3430, Whitmore and Haering, FORTRAN Program forAnalyzing Ground-Based Tracking Data: Usage and Derivations, Version 6.2, 1995 MAE 5540 - Propulsion Systems Numerical Example • Edwards Air Force Base, Radar Site #34 Numerical example !' = 34.96081° " = –117.91150° h = 2563.200 ft • Find corresponding geocentric cartesian and polar coordinates MAE 5540 - Propulsion Systems Numerical Example (cont'd) • Compute Local Radius of Curvature Req R' = = !' 2 1 - e sin2 !' earth 6378.13649 km = 1 - 0.08181939 sin 34.96081 " # 2 180 6392.187109 km MAE 5540 - Propulsion Systems Numerical Example (cont'd) • Compute X and Y (geocentric) r = R' +h cos ' = target !' ! 6392.1871 + 2536.2 " 3. 048 " 10-4 cos 34.96081 " # = 180 5239.3131 km xtarget = rtarget cos $ = ytarget = rtarget sin $ = 5239.3131 km " -117.91150 " # = 5216.0074 km " sin -117.91150 " # = cos 180 180 -2452.5602 km -4629.83218 km MAE 5540 - Propulsion Systems Numerical Example (cont'd) • Compute z (geocentric) 2 ztarget = R' 1 - e +h sin !' = !' earth 6392.1871 km 0.081819392 2536.2 " 3. 048 " 10-4 sin 34.96081 " # = 1 - + 180 3638.7480 km MAE 5540 - Propulsion Systems Numerical Example (cont'd) • Compute Geocentric Polar Coordinates 2 2 2 Rtarget = 2452.5602 + 4629.83218 + 3638.7480 = 6378.94104km -1 ytarget = tan = !target xtarget 180 # -1 -4629.83218 = -117.9115° " tan -2452.5602 z -1 target = $target = tan 2 2 xtarget + ytarget 180 # -1 3638.7480 = 34.7803° " tan 5239.3131 MAE 5540 - Propulsion Systems Numerical Example (cont'd) • Compute Local Earth Radius and Geocentric Distance Above Geoid RE = Req = ! 2 1 - eEarth 2 2 1 - eEarth cos ! 6378.13649 km = 1 - 0.081819392 0.081819392 2 34.7083 " # 1 - cos 180 6364.23 km D = R - R = geoid target E! 6378.94104 - 6364.23 km = 14.7 km = 48228.25 ft MAE 5540 - Propulsion Systems Numerical Example (concluded) • Comparison Geodetic Geocentric !' = 34.96081° ! = 34.7803° " = –117.91150° " = –117.91150° h = 2563.200 ft Dgeoid = 48228.25 ft • Earth Oblateness is NOT trivial, and in the REAL World -- it must be accounted for MAE 5540 - Propulsion Systems Appendix II: Rigorous Derivation of Realizable Launch Inclination • Launch Initial Conditions North Pole Greenwich Meridian • Position: ", Latitude z #, Longitude (Inertial) Launch meridian h, Altitude Equator y Re " x Vernal Equinox !g 95 MAE 5540 - Propulsion Systems • Launch Initial Conditions North Pole Greenwich Meridian z Launch meridian • Launch Initial Conditions (Inertial Coordinates) Equator y Re " x Vernal Equinox !g Sidereal hour angle 96 MAE 5540 - Propulsion Systems Sidereal Hour Angle Ignore for now! 97 MAE 5540 - Propulsion Systems Computing the Hour Angle 98 MAE 5540 - Propulsion Systems.