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Space Engineering I – Part I Where are we?

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Where we going?

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Where are we relative to what?

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Johannes Kepler

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Issac Newton Newton'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.

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Coordinate Systems (X,Y,Z) (R, , ) (R, , ) Origin? Center of Earth Sun or a Star Center of a planetary body Others…. Reference Axes Axis of rotation or revolution Earth spin axis Equatorial Plane Plane of the Earth’s orbit around the Sun Ecliptic Plane Need to pick two axes and then 3 rd one is determined

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Ecliptic and Equatorial Planes Obliquity of the Ecliptic = ° Vernal Equinox vector - Earth to Sun on March 21st - Planes Equinox

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Inertial Coordinates

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Relationship between Coordinate Frames

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a - semi-major axis Ω - right ascension of ascending node e - eccentricity - argument of perigee i - inclination - true anomaly Orbital Elements

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a is the semimajor axis; b is the semiminor axis; r MAX = r a, r MIN = r p are the maximum and minimum radius-vectors; c is the distance between the focus and the center of the ellipse; e = c/a is eccentricity 2p is the latus rectum (latus = side and rectum = straight) p — semilatus rectum or semiparameter A = ab is the area of the ellipse Properties of Orbits

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Why do we need all this? Launch into desired orbit Launch window, inclination Ground coverage (ground track/swath) LEO/GEO Purpose of mission? Orbital Manoeuvers Feasible trajectories Minimize propulsion required Station keeping Tracking, Prediction Interplanetary Transfers Hyperbolic orbits Changing reference frames Orbital insertion Rendezvous/Proximity Operations Relative motion Orbital dynamics

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Bird’s Eye View A 3D Situational Awareness Tool for the Space Station

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Review of Orbital Elements Perigee 0° Apogee 180° e defines ellipse shape a defines ellipse size v defines satellite angle from perigee Semi-major axis True anomaly (angle) Eccentricity ( 0.0 to 1.0) 90° 120° a e v 150° e =0.8 vs e =0.0

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Inclination i Inclination (angle) Equatorial Plane ( defined by Earth’s equator ) Intersection of the equatorial and orbital planes (below) (above) Ascending Node Ascending Node is where a satellite crosses the equatorial plane moving south to north i

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Right Ascension of the ascending node Ω and Argument of perigee ω Vernal Equinox Perigee Direction Ω ω Ω = angle from vernal equinox to ascending node on the equatorial plane ω = angle from ascending node to perigee on the orbital plane Ascending Node

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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 time i = Inclination The angle between the orbital and equatorial planes Ω = Right Ascension (longitude) of the ascending node The angle from the Vernal Equinox vector to the ascending node on the equatorial plane = Argument of perigee The angle measured between the ascending node and perigee Shape, Size, Orientation, and Satellite Location.

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Computing Orbital Position Mean Anomaly Eccentric Anomaly True Anomaly Mean Motion

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Two Line Orbital Elements N ASA and NORAD Standard for specifying orbits of Earth-orbiting satellites I SS (ZARYA) U 98067A − Ref:

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Equations of Motion

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Equations of Motion (2)

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Equations of Motion (3) For a circular orbit, balancing the force of gravity and the centripetal acceleration Since or Conservation of Energy

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Equations of Motion (4) An orbit is a continually changing balance between potential and kinetic energy Potential Energy Kinetic Energy Using For any Kepler orbit (elliptic, parabolic, hyperbolic or radial), this is the Vis Viva equation

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P a 3 2 h = 160 n.mi P = 90 minutes “High” Earth Orbit H = 3444 n.mi P = 4 hours Geosynchronous Orbit h = 19,324 n.mi P = 23 h 56 m 4 s Orbital Period vs. Altitude

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h = 160n.mi V = 25,300 ft/s “High” Earth Orbit H = 3444 n.mi V = 18,341 ft/s Geosynchronous Orbit h = 19,324 n.mi V = 10,087 ft/s Orbital Velocity vs. Altitude

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h = 160 n.mi V = 33,320 ft/s h = 19,324 n.mi V = 5,273 ft/s Geosynchronous Transfer Orbit a = 13,186 n.mi e = Orbital Velocity vs. Altitude (Elliptical Orbits)

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e=0 -- circle e<1 -- ellipse e=1 -- parabola e>1 -- hyperbola e < 1 Orbit is ‘closed’ – recurring path (elliptical) e > 1 Not an orbit – passing trajectory (hyperbolic) Possible Orbital Trajectories

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Parabolic Trajectories Total Energy = 0

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Hyperbolic Trajectories Total Energy > 0 As

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State Vectors Cartesian x, y, z, and 3D velocity

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Integrating Multi-Body Dynamics

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Solar and Sidereal Time The Sun Drifts 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

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Ground Track Ground tracks drift westward as the Earth rotates below. 360 deg / 24 hrs = 15 deg/hr

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Perturbations - J2000 Inertial Frame

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Orbit Perturbations - Atmospheric Drag Effect is to lower the apogee of an elliptical orbit Perigee remains relatively constant Atmospheric density is a function of latitude, solar heating, season, land masses, etc. Drag also depends on spacecraft attitude

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Orbit Perturbations - Gravitational Potential Spherical Term Zonal Harmonics Tesseral HarmonicsSectorial Harmonics where =Universal Gravitational Constant X Mass of Earth r =Spacecraft Radius Vector from Center of Earth a e =Earth Equatorial Radius P()=Legendre Polynomial Functions =Spacecraft Latitude =Spacecraft Longitude J n = Zonal Harmonic Constants C n,m,S n,m = Tesseral & Sectorial Harmonic Coefficients Only 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.

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Orbit Perturbations - “The J2 Effect” h orbit J2 The Earth’s oblateness causes the most significant perturbation of any of the nonspherical terms. Right Ascension of the Ascending Node, Argument of Perigee and Time since Perigee Passage are affected. Nodal Regression is the most important operationally. Magnitude depends on orbit size (a), shape (e) and inclination (i). Posigrade orbits’ nodes regress Westward (0° < i < 90°) Retrograde orbits’ nodes regress Eastward (90° < i <180°) Inclination, Degrees h

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Nodal Regression Orbital planes rotate eastward over time. (below) (above) Ascending Node Nodal Regression can be used to advantage (such as assuring desired lighting conditions)

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Geosynchronous Orbit Coverage from GEO TDRSS Comm Coverage

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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 16 th orbit Used for reconnaissance (or terrain mapping – with a bit of drift).

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Molniya - 12hr Period ‘Long loitering’ high latitude apogee. Once used used for early warning by both USA and USSR

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Changing Orbits - The Effects of Burns Posigrade & Retrograde Orbital Direction A 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. hh hh VV VV

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Initial Velocity Resultant Velocity Radial In Burn Changing Orbits - The Effect of Burns Radial In & Radial Out ( ) 1 V = 2 r a EXAMPLE: Radial In Burn at Perigee Radial burns shift the argument of perigee without significantly altering other orbital parameters

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Orbital Transfers - Changing Planes V1 V2 VV Burn 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 V of over 470 ft/sec.

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Homann Transfer We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2

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We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2 Homann Transfer

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We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2 Homann Transfer

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We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Hohmann Transfer Orbit –Most Efficient Method LEO GEO r1r1 r2r2 V c1 V c2 Homann Transfer

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Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Leave LEO (r 1 ) with a total velocity of V 1 LEO GEO r1r1 r2r2 V c1 V1V1 V c2 GTO V1V1 Homann Transfer

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Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Leave LEO (r 1 ) with a total velocity of V 1 Transfer orbit is elliptical shape –Perigee located at r 1 –Apogee located at r 2 LEO GEO r1r1 r2r2 V c1 V1V1 V c2 GTO V1V1 Perigee Apogee Homann Transfer

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Arrive at GEO (apogee) with V 2 When arriving at GEO, which is at apogee of elliptical transfer orbit, must apply some V 2 in order to circularize: This is exactly the V that should be applied to circularize the orbit at GEO (r 2 ) –V c2 = V 2 + V 2 If this V is not applied, spacecraft will continue on dashed elliptical trajectory LEO GEO r1r1 r2r2 V c1 V1V1 V1V1 V2V2 V2V2 V c2 GTO Homann Transfer

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Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Coast to apogee and apply impulsive V2 : LEO GEO r1r1 r2r2 V c1 V1V1 V1V1 V2V2 V2V2 V c2 GTO Homann Transfer

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Super GTO 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. 2. Plane change plus initial Hohmann burn GEO Target Orbit 1. Launch to ‘Super GTO’ 3. Second Hohmann burn circularizes at GEO

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Orbital Rendezvous (Shuttle with ISS)

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Interplanetary Trajectories (Patched Conic Approximation)

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Gravity Assist

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Farewell

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