lesson4

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