6 Johannes KeplerAstronomia Nova, Epitome of Copernican AstronomyPhysical basis for celestial motions based on nested solid shapesStudent of Tycho Brahe – analyzed data, made predictions of celestial events (solar transits)Isaac Newton’s Principia Mathematica (1687) derived the laws mathematically for a forcebased central law of gravity.
7 Issac NewtonNewton's laws of motion and universal gravitation dominated scientists' view of the physical universe for the next three centuries.Newton derived Kepler’s laws of planetary motion from his mathematical description of gravity, removing the last doubts about the validity of the heliocentric model of the cosmos.Isaac Newton’s Principia Mathematica (1687) derived the laws mathematically for a forcebased central law of gravity.
8 Coordinate Systems Origin? Center of Earth Sun or a Star Center of a planetary bodyOthers….Reference AxesAxis of rotation or revolutionEarth spin axisEquatorial PlanePlane of the Earth’s orbit around the SunEcliptic PlaneNeed to pick two axes and then 3rd one is determined(X,Y,Z)(R,q,l)(R,j,l)
9 Ecliptic and Equatorial Planes Obliquity of the Ecliptic = °Vernal Equinox vector- Earth to Sun on March 21st- Planes Equinox
12 Orbital Elementsa - semi-major axis Ω - right ascension of ascending nodee - eccentricity - argument of perigeei - inclination - true anomaly
13 Properties of Orbits a is the semimajor axis; b is the semiminor axis; rMAX = ra, rMIN = rp are the maximum and minimum radius-vectors;c is the distance between the focus and the center of the ellipse;e = c/a is eccentricity2p is the latus rectum(latus = side and rectum = straight)p — semilatus rectum or semiparameterA = ab is the area of the ellipse
14 Why do we need all this? Launch into desired orbit Orbital Manoeuvers Launch window, inclinationGround coverage (ground track/swath)LEO/GEOPurpose of mission?Orbital ManoeuversFeasible trajectoriesMinimize propulsion requiredStation keepingTracking, PredictionInterplanetary TransfersHyperbolic orbitsChanging reference framesOrbital insertionRendezvous/Proximity OperationsRelative motionOrbital dynamics
15 Bird’s Eye View A 3D Situational Awareness Tool for the Space StationIt’s the Mission Control Display that you see on TVDeveloped through student projects at Texas A&M, includingone of our post-graduates from Sydney Uni who came to theUS for a year.The partnership to do this involved a NASA contractor, USA,who then supported student projects at Texas A&M.Now it’s what NASA uses, and you can also see it on thebig screen at the CSA Control Center in Montreal, at ESA inGermany, in Japan, and in Geneva.
16 Review of Orbital Elements 120°150°90°Eccentricity (0.0 to 1.0)vTrue anomaly (angle)aApogee 180°Perigee 0°Semi-major axise=0.8 vs e=0.0e defines ellipse shape a defines ellipse size v defines satellite angle from perigee
17 Equatorial Plane ( defined by Earth’s equator ) Inclination iIntersection of the equatorial and orbital planesiInclination (angle)(above)(below)Ascending NodeEquatorial Plane ( defined by Earth’s equator )Ascending Node is where a satellite crosses the equatorial plane moving south to north
18 Right Ascension of the ascending node Ω and Argument of perigee ω Ω = angle from vernal equinox to ascending node on the equatorial planePerigee Directionω = angle from ascending node to perigee on the orbital planeωΩAscending NodeVernal Equinox
19 The Six Orbital Elements a = Semi-major axis (usually in kilometers or nautical miles)e = Eccentricity (of the elliptical orbit)v = True anomaly The angle between perigee and satellite in the orbital plane at a specific timei = Inclination The angle between the orbital and equatorial planesΩ = Right Ascension (longitude) of the ascending node The angle from the Vernal Equinox vector to the ascending node on the equatorial planew = Argument of perigee The angle measured between the ascending node and perigeeShape, Size, Orientation, and Satellite Location.
20 Computing Orbital Position Mean AnomalyEccentric AnomalyTrue AnomalyMean Motion
21 Two Line Orbital Elements N ASA and NORAD Standard for specifying orbits of Earth-orbiting satellitesISS (ZARYA)U 98067A −Ref:
24 Equations of Motion (3) Conservation of Energy For a circular orbit, balancing the force of gravity and the centripetal accelerationSinceor
25 Equations of Motion (4)An orbit is a continually changing balance between potential and kinetic energyPotential Energy Kinetic EnergyUsingFor any Kepler orbit (elliptic, parabolic, hyperbolic or radial), this is theVis Viva equation
26 Orbital Period vs. Altitude 3P=2pmh = 160 n.miP = 90 minutes“High” Earth OrbitH = 3444 n.miP = 4 hoursGeosynchronous Orbith = 19,324 n.miP = 23 h 56 m 4 s
27 Orbital Velocity vs. Altitude h = 160n.miV = 25,300 ft/s“High” Earth OrbitH = 3444 n.miV = 18,341 ft/sGeosynchronous Orbith = 19,324 n.miV = 10,087 ft/s
28 Orbital Velocity vs. Altitude (Elliptical Orbits)h = 19,324 n.miV = 5,273 ft/sh = 160 n.miV = 33,320 ft/sGeosynchronous Transfer Orbita = 13,186 n.mie = 0.726
29 Possible Orbital Trajectories e=0 -- circlee<1 -- ellipsee=1 -- parabolae>1 -- hyperbolae < 1 Orbit is ‘closed’ – recurring path (elliptical) e > 1 Not an orbit – passing trajectory (hyperbolic)
34 Solar and Sidereal Time The SunDrifts east in the sky ~1° per day Rises hours later each day.(because the earth is orbiting)The Earth…Rotates 360° in hours(Celestial or “Sidereal” Day)Rotates ~361° in hours(Noon to Noon or “Solar” Day)Satellites orbits are aligned to the Sidereal day – not the solar day
35 Ground tracks drift westward as the Earth rotates below. 360 deg / 24 hrs= 15 deg/hr
37 Orbit Perturbations - Atmospheric Drag Atmospheric density is a function of latitude, solar heating, season, land masses, etc.Drag also depends on spacecraft attitudeEffect is to lower the apogee of an elliptical orbitPerigee remains relatively constant
38 Orbit Perturbations - Gravitational Potential whereSpherical Termm = Universal Gravitational Constant X Mass of Earthr = Spacecraft Radius Vector from Center of Earthae = Earth Equatorial RadiusP() = Legendre Polynomial Functionsf = Spacecraft Latitudel = Spacecraft LongitudeJn = Zonal Harmonic ConstantsCn,m,Sn,m = Tesseral & Sectorial Harmonic CoefficientsZonal HarmonicsOnly the first 4X4 (n=4, m=4) elements are used in Space Shuttle software.J2 has 1/1000th the effect of the spherical term;all other terms start at 1/1000th of J2’s effect.Tesseral HarmonicsSectorial Harmonics
39 Orbit Perturbations - “The J2 Effect” The Earth’s oblateness causes the most significant perturbation of any of the nonspherical terms.horbithJ2Right Ascension of the Ascending Node, Argument of Perigee and Time since Perigee Passage are affected.1003002005001000203040506070809010TypicalShuttleOrbitsNodal Regression is the most important operationally.-6.739Magnitude depends on orbit size (a), shape (e) and inclination (i).Posigrade orbits’ nodes regressWestward (0° < i < 90°)Retrograde orbits’ nodes regressEastward (90° < i <180°)180170160150140130120110100Inclination, Degrees
40 Nodal Regression Ascending Node Orbital planes rotate eastward over time.(above)Ascending Node(below)Nodal Regression can be used to advantage (such as assuring desired lighting conditions)
41 Coverage from GEOGeosynchronous OrbitTDRSS Comm Coverage
42 Sun-Synchronous Orbits Relies on nodal regression to shift the ascending node ~1° per day.Scans the same path under the same lighting conditions each day.The number of orbits per 24 hours must be an even integer (usually 15).Requires a slightly retrograde orbit (I = 97.56° for a 550km / 15-orbit SSO).Each subsequent pass is 24° farther west (if 15 orbits per day).Repeats the pattern on the 16th orbitUsed for reconnaissance (or terrain mapping – with a bit of drift).
43 Molniya - 12hr Period‘Long loitering’ high latitude apogee. Once used used for early warning by both USA and USSR
44 Changing Orbits - The Effects of Burns Posigrade & Retrograde OrbitalDirectionDhDVDVDhA posi-grade burn will RAISE orbital altitude.A retro-grade burn will LOWER orbital altitude.Note – max effect is at 180° from the burn point.
45 Changing Orbits - The Effect of Burns Radial In & Radial Out ()1V = m2raEXAMPLE: Radial In Burn at PerigeeRadial In BurnResultant VelocityInitial VelocityRadial burns shift the argument of perigee without significantly altering other orbital parameters
46 Orbital Transfers - Changing Planes V2DVV1Burn point must be intersection of two orbits (“nodal crossings”)Extremely expensive energy-wise:For 160 nmi circular orbits, a 1° of plane change requires a DV of over 470 ft/sec.
47 Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEOInitial LEO orbit has radius r1 and velocity Vc1Desired GEO orbit has radius r2 and velocity Vc2At LEO (r1), Vc1 = 7,724 m/sAt GEO (r2), Vc2 = 3,074 m/sCould accomplish this in many waysGEOLEOr1Vc1r2
48 Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEOInitial LEO orbit has radius r1 and velocity Vc1Desired GEO orbit has radius r2 and velocity Vc2At LEO (r1), Vc1 = 7,724 m/sAt GEO (r2), Vc2 = 3,074 m/sCould accomplish this in many waysGEOLEOr1Vc1r2
49 Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEOInitial LEO orbit has radius r1 and velocity Vc1Desired GEO orbit has radius r2 and velocity Vc2At LEO (r1), Vc1 = 7,724 m/sAt GEO (r2), Vc2 = 3,074 m/sCould accomplish this in many waysGEOLEOr1Vc1r2
50 Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEOInitial LEO orbit has radius r1 and velocity Vc1Desired GEO orbit has radius r2 and velocity Vc2At LEO (r1), Vc1 = 7,724 m/sAt GEO (r2), Vc2 = 3,074 m/sHohmann Transfer OrbitMost Efficient MethodGEOLEOr1Vc1r2
51 Homann Transfer Vc2 r1 DV1 Vc1 r2 V1 GEO Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee:Leave LEO (r1) with a total velocity of V1GTOLEOr1DV1Vc1r2V1
52 Homann Transfer Vc2 Apogee r1 DV1 Vc1 r2 V1 Perigee GEO Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee:Leave LEO (r1) with a total velocity of V1Transfer orbit is elliptical shapePerigee located at r1Apogee located at r2GTOLEOr1DV1Vc1r2V1Perigee
53 Homann Transfer Vc2 V2 DV2 r1 DV1 Vc1 r2 V1 GEO Arrive at GEO (apogee) with V2When arriving at GEO, which is at apogee of elliptical transfer orbit, must apply some DV2 in order to circularize:This is exactly the DV that should be applied to circularize the orbit at GEO (r2)Vc2 = DV2 + V2If this DV is not applied, spacecraft will continue on dashed elliptical trajectoryGTOLEOr1DV1Vc1r2V1
54 Homann Transfer Vc2 V2 DV2 r1 DV1 Vc1 r2 V1 Initial LEO orbit has radius r1 and velocity Vc1Desired GEO orbit has radius r2 and velocity Vc2Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee:Coast to apogee and apply impulsive DV2:DV2V2GEOGTOLEOr1DV1Vc1r2V1
55 Super GTO 3. Second Hohmann burn circularizes at GEO GEO Target Orbit Initial orbit has greater apogee than standard GTO.Plane change at much higher altitude requires far less ΔV.PRO: Less overall ΔV from higher inclination launch sites.CON: Takes longer to establish the final orbit.1. Launch to ‘Super GTO’2. Plane change plus initial Hohmann burn
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