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
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.

7. 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.
Isaac Newton’s Principia Mathematica (1687) derived the laws mathematically for a force
based central law of gravity.

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 = °
Vernal Equinox vector
- Earth to Sun on March 21st
- Planes 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. 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

14. 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

15. 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.

16. 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

17. 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

18. 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

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 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.

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

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

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. Altitude3 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

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 Vectors Cartesian 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 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

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. 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

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 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. 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.

45. 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

46. Orbital Transfers - Changing PlanesV2 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 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

48. 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

49. 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

50. 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

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 V1
GTO
LEO r1
DV1
Vc1 r2 V1

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

53. 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

54. 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

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

56. Orbital Rendezvous (Shuttle with ISS)

57. Interplanetary Trajectories (Patched Conic Approximation)

58. Gravity Assist

59. Farewell