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Published on November 15, 2007

Author: cooper

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Orbit Particulars:  Orbit Particulars Lessons 4 thru 8 ORBITAL ELEMENTS:  ORBITAL ELEMENTS Classic Orbital Elements (COEs)/Orbital Element Set 6 quantities needed to describe orbit and spacecraft’s position within orbit Airplane: Latitude, Longitude, Altitude: Range Horizontal velocity, Heading, Vertical velocity: Range rate = velocity Satellite: Range: 3 – component vector Velocity: 3 – component vector KEPLER’S COEs:  KEPLER’S COEs Orbit size: Semi-major axis, a Circle: a = radius Ellipse: a = semi-major axis Parabola: a = ∞ Hyperbola: a < 0 2) Orbit shape: Eccentricity e [out-of-roundness] Circle: e = 0 Ellipse: 0 < e < 1 Parabola: e = 1 Hyperbola: e > 1 3) Orbit plane orientation: Inclination i [tilt of orbital plane wrt equatorial plane] Earth satellite: i = 0 or 180: equatorial orbit i = 90: polar orbit 0 < i < 90: direct or prograde orbit 90 < i < 180: indirect or retrograde orbit COEs (cont’d):  COEs (cont’d) 4) Orbit plane orientation: Right ascension [longitude] of the ascending node, Ω [0,360] Ascending node: Point where orbit crosses equatorial plane from below to above Descending node: point where orbit crosses equatorial plane from above to below Right ascension: Angle between vernal equinox direction and the ascending node [measured eastward] 5) Orbit orientation within plane: Argument of perigee, ω [0,360] Angle along orbital path in direction of motion between ascending node and perigee 6) Satellite location: Time of perigee passage, T or True Anomaly, ν [0,360] – angle between perigee and position vector at time T CALCULATIONS :  CALCULATIONS CALC (cont’d 1):  CALC (cont’d 1) CALC (cont’d 2):  CALC (cont’d 2) CALC (cont’d 3):  CALC (cont’d 3) SUMMARY:  SUMMARY ALTERNATE ORBITAL ELEMENT QUANTITIES (Figure 2.3-1):  ALTERNATE ORBITAL ELEMENT QUANTITIES (Figure 2.3-1) BACKGROUND:  BACKGROUND Newton: Needed 3 observations (position vectors) Halley: Mastered and refined Newton’s methodology Lambert: Geometrical arguments Lagrange: Developed mathematical basis for orbit calculations Laplace: Developed new methodology Gauss: Summarized, simplified and completed orbit determination work COORDINATE/REFERENCE FRAMES:  COORDINATE/REFERENCE FRAMES Need a reference frame to describe orbital motion Frame must be inertial [non-accelerating] for Newton’s Laws to apply Much of work describes earth orbiting spacecraft An important coordinate system/reference frame is earth oriented EARTH REFERENCE FRAME:  EARTH REFERENCE FRAME Origin of coordinate system – center of earth Fundamental plane: Earth’s equatorial plane Perpendicular to plane: North pole direction Principal direction: Vernal Equinox direction Vernal Equinox: I North Pole: K J in fundamental plane – I,J,K right-handed system see Figure 2.2-2, p.55 *** Geocentric-Equatorial Coordinate System *** SUN CENTERED FRAME:  SUN CENTERED FRAME Origin – center of frame Fundamental plane: earth’s orbital plane about sun - Ecliptic Fundamental direction: Vernal Equinox direction Vernal Equinox direction: Xє Yє: In ecliptic plane 90 deg from Xє in direction of earth’s motion Zє: Perpendicular to ecliptic plane, right-handed system See Figure 2.2-1, p. 54 *** Heliocentric-Ecliptic Coordinate System *** RIGHT ASCENSION-DECLINATION SYSTEM:  RIGHT ASCENSION-DECLINATION SYSTEM Origin – center of earth or point on surface of earth Not that important Fundamental plane: Earth’s equatorial plane extended to sphere of infinite radius – celestial equator Right ascension angle (): Angle measured eastward from vernal equinox in plane of celestial equator Declination angle (δ): Angle measured up (north) from celestial equator Primary use: Star catalog to help determine spacecraft position PERIFOCAL COORDINATE SYSTEM:  PERIFOCAL COORDINATE SYSTEM Origin – center of gravity of satellite Fundamental plane: Plane of satellite’s orbit Perpendicular to plane: Aligned with angular momentum vector (h) Principal direction: Xω pointing towards perigee Perigee direction: vector P 2nd vector in plane: 90 deg from Xω in direction of orbital motion (Yω), vector Q Zω along angular momentum vector, vector W See Figure 2.2-4, Right-handed system Primary use: analysis associated with satellite TOPOCENTRIC-HORIZON COORDINATE SYSTEM:  TOPOCENTRIC-HORIZON COORDINATE SYSTEM Origin of system – point on surface of earth (topos) where sensor is located Fundamental plane: Horizon at sensor Coordinate directions: X points south, S Y points east, E Z points up, Z SEZ system: Sensor gives satellite elevation and azimuth (Az-El) Non-inertial system See Figure 2.7-1, p. 84 COORDINATE TRANSFORMATIONS:  COORDINATE TRANSFORMATIONS A vector may be expressed in any coordinate frame Must learn how to transform among coordinate frames Coordinate transformation: Changes the basis of a vector Magnitude remains the same Direction remains the same What it represents remains the same Transform using rotation [rigid body] Positive rotation: Right-hand rule – thumb + curl fingers SINGLE-AXIS ROTATION:  SINGLE-AXIS ROTATION Examine rotation about x-axis x,x’ y z y’ z’   ROTATIONS:  ROTATIONS SUCCESSIVE ROTATIONS:  SUCCESSIVE ROTATIONS EXAMPLE (cont’d):  EXAMPLE (cont’d) EXAMPLE 2 :  EXAMPLE 2 NED → XYZ: NED = North – East - Down x y z EX 2 (cont’d 1):  EX 2 (cont’d 1) POSITION AND VELOCITY FROM COEs:  POSITION AND VELOCITY FROM COEs Section 2.5 and 2.6.5 P 83 – Method leaves much to be desired Student read: Good background material SINGLE RADAR OBSERVATION:  SINGLE RADAR OBSERVATION POSITION AND VELOCITY:  POSITION AND VELOCITY P & V (cont’d 1):  P & V (cont’d 1) P & V (cont’d 2):  P & V (cont’d 2) EXAMPLE:  EXAMPLE EX (cont’d 1):  EX (cont’d 1) EX (cont’d 2):  EX (cont’d 2) EX (cont’d 3):  EX (cont’d 3) EX (cont’d 4):  EX (cont’d 4) 3 POSITION VECTORS (Gibbs Method):  3 POSITION VECTORS (Gibbs Method) OPTICAL SIGHTINGS:  OPTICAL SIGHTINGS Student read Preferred method later DIFFERENTIAL CORRECTION:  DIFFERENTIAL CORRECTION Tracking & Predicting Orbits – Big Picture Sensor site obtains range, azimuth, elevation Form R,V initial Form COEs initial perturbations predict COE future R, V future Sensor site Range, az, el DIFF CORR:  DIFF CORR DC (cont’d 1):  DC (cont’d 1) DC (cont’d 2):  DC (cont’d 2) PROCESS:  PROCESS n = # observations and p = # parameters Let: W be n X n diagonal matrix whose entries are square of confidence in observation measurements A be n X p matrix of partial derivatives b be n X 1 matrix of residuals z be p X 1 matrix of computed corrections Linear System Theory: p > n: no unique solution p = n: unique solution (linearly independent) p < n: no unique solution – develop least squares EXAMPLE (Text p 126):  EXAMPLE (Text p 126) EX (cont’d):  EX (cont’d) Now have y = -1 +x Check: x=2, y=1 x=3, y=2 Predictions = observations Expected since had linear Study Text example p 128 Y(x) is nonlinear, 4 points and 2 parameters, least-squares EX (cont’d):  EX (cont’d) Book example is for least-squares solution Real-world: Kalman Filter Works with 6 elements W is the variance/covariance matrix Iterative process based on differential correction GROUND TRACK (Circular Orbit):  GROUND TRACK (Circular Orbit) Ground track: The trace of a S/Cs path over the surface of the earth Track important for S/C looking down on earth Great Circle: A circle that cuts through the center of the earth [spherical trig] Orbits trace out a great circle Mercator Projection: A flat map projection of the spherical earth Orbits trace is a sine wave on Mercator Projection Earth rotates 360/24 = 15°/hour Orbit traces shift to the west GT (non-rotating):  GT (non-rotating) GT (rotating):  GT (rotating) GT (cont’d):  GT (cont’d) GT (rotating earth):  GT (rotating earth) GT – ELLIPTICAL ORBITS:  GT – ELLIPTICAL ORBITS Circular orbit has symmetrical ground track Elliptical orbit has non-symmetrical ground track S/C fastest at perigee – spreads out ground track S/C slowest at apogee – compresses ground track Molniya orbits – highly elliptical, position perigee over desired location S/C MISSIONS:  S/C MISSIONS Mission Orbit a(Km) P i e Communication Early Warning Geostationary 42,158 24 Hr 0 0 Nuclear Detonation Remote Sensing Sun-synchronous 6500 – 7300 90 min 95 0 Navigation Semi-synchronous 26,610 12 Hr 55 0 Comm/Intel Molniya 26,571 12 Hr 63.4 0

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