Space Engineering I – Part I Where are we?

1. Space Engineering I – Part I Where are we?

3. Where are we?

4. Where we going?

5. Where are we relative to what?

6. Johannes Kepler 1571-1630

7. 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.

8. 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)

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

10. Inertial Coordinates

11. Relationship between Coordinate Frames

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

13. 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 Properties of Orbits

14. 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

15. Bird’s Eye ViewA 3D Situational Awareness Tool for the Space Station

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

17. 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

18. Right Ascension of the ascending nodeΩandArgument of perigeeω 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

19. The Six Orbital Elements a = Semi-major axis (usually in kilometers or nautical miles) e = Eccentricity (of the elliptical orbit) v = True anomalyThe angle between perigee and satellite in the orbital plane at a specific time i = InclinationThe angle between the orbital and equatorial planes Ω= Right Ascension (longitude) of the ascending nodeThe 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.

20. Computing Orbital Position Mean Anomaly Eccentric Anomaly True Anomaly Mean Motion

21. N ASA and NORAD Standard for specifying orbits of Earth-orbiting satellites Two Line Orbital Elements 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

22. Equations of Motion

23. Equations of Motion (2)

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

25. 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

26. Orbital Period vs. Altitude 3 a P = p m 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

27. 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

28. 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

29. 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)

30. Parabolic Trajectories Total Energy = 0

31. Hyperbolic Trajectories Total Energy > 0 As

32. State VectorsCartesian x, y, z, and 3D velocity

33. Integrating Multi-Body Dynamics

34. 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

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

36. Perturbations - J2000 Inertial Frame

37. 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

38. 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

39. 100 200 Typical 300 Shuttle Orbits 500 -6.7 1000 39 10 20 30 40 50 60 70 80 90 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. 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°) 0 180 170 160 150 140 130 120 110 100 Inclination, Degrees

40. Nodal Regression 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 GEO Geosynchronous Orbit TDRSS 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 orbit Used 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. Orbital Direction Changing Orbits - The Effects of BurnsPosigrade & Retrograde 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.

45. ( ) 1 2 V = m r a Changing Orbits - The Effect of BurnsRadial In & Radial Out 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

46. 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.

47. Homann Transfer Vc2 • 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

48. Homann Transfer Vc2 • 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

49. Homann Transfer Vc2 • 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

50. Homann Transfer Vc2 • 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