Space Engineering I – Part I Where are we?

Where are we?

Where we going?

Where are we relative to what?

Johannes Kepler 1571-1630 Astronomia Nova, Epitome of Copernican Astronomy Physical basis for celestial motions based on nested solid shapes Student of Tycho Brahe – analyzed data, made predictions of celestial events (solar transits) Isaac Newton’s Principia Mathematica (1687) derived the laws mathematically for a force based central law of gravity.

Issac Newton 1642-1727 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. Isaac Newton’s Principia Mathematica (1687) derived the laws mathematically for a force based central law of gravity.

Coordinate Systems 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 3rd one is determined (X,Y,Z) (R,q,l) (R,j,l)

Ecliptic and Equatorial Planes Obliquity of the Ecliptic = 23.44 ° Vernal Equinox vector - Earth to Sun on March 21st - Planes intersect @ Equinox

Inertial Coordinates

Relationship between Coordinate Frames

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

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 eccentricity 2p is the latus rectum (latus = side and rectum = straight) p — semilatus rectum or semiparameter A = ab is the area of the ellipse

Why do we need all this? Launch into desired orbit Orbital Manoeuvers 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

Bird’s Eye View A 3D Situational Awareness Tool for the Space Station It’s the Mission Control Display that you see on TV Developed through student projects at Texas A&M, including one of our post-graduates from Sydney Uni who came to the US 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 the big screen at the CSA Control Center in Montreal, at ESA in Germany, in Japan, and in Geneva.

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

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

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

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 w = Argument of perigee The angle measured between the ascending node and perigee Shape, Size, Orientation, and Satellite Location.

Computing Orbital Position Mean Anomaly Eccentric Anomaly True Anomaly Mean Motion

Two Line Orbital Elements N ASA and NORAD Standard for specifying orbits of Earth-orbiting satellites ISS (ZARYA) 1 25544U 98067A 08264.51782528 −.00002182 00000-0 -11606-4 0 2927 2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537 Ref: http://en.wikipedia.org/wiki/Two-line_element_set

Equations of Motion

Equations of Motion (2)

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

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

Orbital Period vs. Altitude 3 P = 2 p m 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 Velocity vs. Altitude 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 (Elliptical Orbits) h = 19,324 n.mi V = 5,273 ft/s h = 160 n.mi V = 33,320 ft/s Geosynchronous Transfer Orbit a = 13,186 n.mi e = 0.726

Possible Orbital Trajectories 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)

Parabolic Trajectories   Total Energy = 0  

Hyperbolic Trajectories Total Energy > 0     As  

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

Integrating Multi-Body Dynamics            

Solar and Sidereal Time The Sun Drifts east in the sky ~1° per day. Rises 0.066 hours later each day. (because the earth is orbiting) The Earth… Rotates 360° in 23.934 hours (Celestial or “Sidereal” Day) Rotates ~361° in 24.000 hours (Noon to Noon or “Solar” Day) Satellites orbits are aligned to the Sidereal day – not the solar day

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

Perturbations - J2000 Inertial Frame

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

Orbit Perturbations - Gravitational Potential where Spherical Term m = Universal Gravitational Constant X Mass of Earth r = Spacecraft Radius Vector from Center of Earth ae = Earth Equatorial Radius P() = Legendre Polynomial Functions f = Spacecraft Latitude l = Spacecraft Longitude Jn = Zonal Harmonic Constants Cn,m,Sn,m = Tesseral & Sectorial Harmonic Coefficients Zonal Harmonics 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. Tesseral Harmonics Sectorial Harmonics

Orbit Perturbations - “The J2 Effect” The Earth’s oblateness causes the most significant perturbation of any of the nonspherical terms. h orbit h J2 Right Ascension of the Ascending Node, Argument of Perigee and Time since Perigee Passage are affected. 100 300 200 500 1000 20 30 40 50 60 70 80 90 10 Typical Shuttle Orbits Nodal Regression is the most important operationally. -6.7 39 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°) 180 170 160 150 140 130 120 110 100 Inclination, Degrees

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)

Coverage from GEO Geosynchronous Orbit TDRSS Comm Coverage

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

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

Changing Orbits - The Effects of Burns Posigrade & Retrograde Orbital Direction Dh DV DV Dh 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.

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

Orbital Transfers - Changing Planes V2 DV V1 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 DV of over 470 ft/sec.

Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r1 and velocity Vc1 Desired GEO orbit has radius r2 and velocity Vc2 At LEO (r1), Vc1 = 7,724 m/s At GEO (r2), Vc2 = 3,074 m/s Could accomplish this in many ways GEO LEO r1 Vc1 r2

Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r1 and velocity Vc1 Desired GEO orbit has radius r2 and velocity Vc2 At LEO (r1), Vc1 = 7,724 m/s At GEO (r2), Vc2 = 3,074 m/s Could accomplish this in many ways GEO LEO r1 Vc1 r2

Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r1 and velocity Vc1 Desired GEO orbit has radius r2 and velocity Vc2 At LEO (r1), Vc1 = 7,724 m/s At GEO (r2), Vc2 = 3,074 m/s Could accomplish this in many ways GEO LEO r1 Vc1 r2

Homann Transfer Vc2 r1 Vc1 r2 We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r1 and velocity Vc1 Desired GEO orbit has radius r2 and velocity Vc2 At LEO (r1), Vc1 = 7,724 m/s At GEO (r2), Vc2 = 3,074 m/s Hohmann Transfer Orbit Most Efficient Method GEO LEO r1 Vc1 r2

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 V1 GTO LEO r1 DV1 Vc1 r2 V1

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 V1 Transfer orbit is elliptical shape Perigee located at r1 Apogee located at r2 GTO LEO r1 DV1 Vc1 r2 V1 Perigee

Homann Transfer Vc2 V2 DV2 r1 DV1 Vc1 r2 V1 GEO Arrive at GEO (apogee) with V2 When 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 + V2 If this DV is not applied, spacecraft will continue on dashed elliptical trajectory GTO LEO r1 DV1 Vc1 r2 V1

Homann Transfer Vc2 V2 DV2 r1 DV1 Vc1 r2 V1 Initial LEO orbit has radius r1 and velocity Vc1 Desired GEO orbit has radius r2 and velocity Vc2 Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee: Coast to apogee and apply impulsive DV2: DV2 V2 GEO GTO LEO r1 DV1 Vc1 r2 V1

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

Orbital Rendezvous (Shuttle with ISS)

Interplanetary Trajectories (Patched Conic Approximation)

Gravity Assist

Farewell