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