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					 Chapter 5: Describing Orbits
   “Understanding Space”
           (Sellers)




(Mars Pathfinder Fact Sheet)
               Supplemental References
1. Notes on circular orbits and the elliptical orbits connecting them
http://www.umd.umich.edu/casl/natsci/physics/150/EllipticalOrbits.html

2. Thomson,W.T., “Introduction to Space Dynamics”, John Wiley
and Sons, 1961

3. Mars Pathfinder Fact Sheet
   http://mars.starlink.it/mpf/fact-sheet.html

4. Mars Pathfinder Cruise and Mars Approach Activities
   http://mars.sgi.com/mpf/mpfcruise.html

5. Polar Orbiting Satellites
   http://www-spof.gsfc.nasa.gov/Education

6. The Satellite Site--Lockheed-Martin
   http://www.thetech.org/hyper/satellite
                    Supplemental References
                          (Continued)
7. The Solar System, Department of Physics and Astronomy, University
   of Tennessee
   http://csep10.phys.utk.edu/astr161/lect/index.html
8. Hohmann Transfer
   http://www.sff.net/people/ckanderson/flight.htm
9. NSSDC Image Catalog Mission Index
   http://nssdc.gsfc.nasa.gov/imgcat/mission_index.html
10. NASA Images
    http://images.jsc.nasa.gov/images/pao/GT7/10074154.jpg
11. Landsat 7
   http://cne.gsfc.nasa.gov/warning.html
12. Types of Orbits
   http://marine.rutgers.edu/mrs/education/class/paul/orbits2.html#2
13. Molniya 1-91 - Information
   http://www.heavens-above.com/satinfo.asp?satid=25485
               Supplemental References (Concluded)
14. Thomson, William T., “Introduction to Space Dynamics”,
    John Wiley and Sons, Inc., 1961
15. Liftoff to Space Exploration
    http://liftoff.msfc.nasa.gov/toc.asp?s=Orbital%20Mechanics
16. Trajectories and Orbits
    http://www.hq.nasa.gov/office/pao/History/conghand/traject.htm
17. Coordinate Systems Overview (P. Dana)
    www.colorado.edu/geography/gcraft/notes/coordsys/coordsys.html
18. Orbital Description
    http://www.mindspring.com/~n2wwd/html/orbital_description.html
19. Orbit of Phase 3-D Spacecraft
    http://www.amsat.org/amsat/sats/phase3d/orbit.html#TRACK
20. Dialogue on Looping and Zig-zagging satellites
    http://www.catholicoutlook.com/dance1.php
21. Orbits
    http://zebu.uoregon.edu/~js/space/lectures/lec06.html
                      5.1 Orbital Elements
Orbital size, uses semimajor axis, a

Orbital shape, defined by eccentricity, e

Orientation of orbital plane in space, using
 inclination, i
 right ascension of the ascending node, 

Orientation of orbit within the plane defined
by argument of perigee, 

Spacecraft’s location in orbit represented
by true anomaly,                               (Orbital Description)
(P. Dana)
(P. Dana)
(P. Dana)
          Semimajor Axis and Specific Mechanical Energy

       
 
       2a



 apogee                                        focus
                       '
                   F                                       perigee
                           •                           •
                                     2c




                               2a=major axis

                 a=semimajor axis
    Eccentricity and Geocentric-equatorial Coordinate
                         System

          Conic Section   Eccentricity
           Circle          e=0
           Ellipse          0<e<1
           Parabola         e=1
           Hyperbola        e>1




(Orbital Description)
              Types of Orbits and Their Inclination
    Equatorial (i = 0° or 180°)             Polar (i = 90°)




  (The Satellite Site)                     (The Satellite Site)

Direct or Prograde (0°  i  90°)      Indirect or Retrograde (90° < i  180°)
                   Orbital Elements for Various Missions
                  Geostationary                      Sun Synchronous

•Comm                                   •Remote
•Early                                   sensing
 warning
•Nuclear
 detection


                (The Satellite Site)                  (Types of Orbits)

                         Molniya                      Space Shuttle / ISS
•Comm/                                  •Science/
 intelligence                            other




                                                            (NASA)
                       (Molniya 1-91)
            Section 5.2, Computing Orbital Elements
   V2 
   
   2 R
Where
V=magnitude of spacecraft’s
   velocity vector (km/s)
=gravitational parameter (km³/s²)
 =3.986x105 km³/s² for Earth
R=magnitude of spacecraft’s position
   vector (km)
                
Knowing V and R , can solve for
semimajor axis                            (Orbital Description)
      
a
      2
Eccentricity is
  1  2       
e  V   R ( R V )V 
           R         
         Section 5.2, Computing Orbital
                    Elements                          
Inclination, i                       K=magnitude of K =1
                                                    
                                   h=magnitude of h (km²/s)
 A B  AB cos
                 
             A B
  cos1 (       )
              AB
             
        1  K  H 
 i  cos 
            KH   
                  
Where

 i = inclination (deg or rad)
K = unit vector through North Pole      (Orbital Description)

h = specific angular momentum vector (km²/s)
                   Section 5.2, Computing Orbital Elements
Right Ascension of Ascending Node, 
        
n  K xh
                   
          I n
          1
  cos (      )
           In
Where
=right ascension of the ascending
  node (deg or rad)

I =unit vector in principal direction

n   =ascending node vector (km²/s,
     points at ascending node)
                     
               I
I=magnitude of  =1                          (Orbital Description)
n=magnitude of n (km²/s)
             Section 5.2, Computing Orbital Elements
Argument of Perigee, 
                 
          n e
  cos (1
               )
           ne
Where
=argument of perigee (deg / rad)

n =ascending node vector (km²/s,
    points at the ascending node)


e =eccentricity vector (unitless,
    points at perigee)                 (Orbital Description)
                      
n = magnitude of n (km²/s)
                      
e = magnitude of      e (unitless)
                 Section 5.2, Computing Orbital Elements
Finding True Anomaly, 
                
          e R
  cos (
        1
               )
           eR
where
=true anomaly (deg or rad)

e = eccentricity vector (unitless,

    points at perigee)
R = position vector (km)
                  
e = magnitude of e (unitless)
                     
R =magnitude of R (km)                    (Orbital Description)
     Section 5.3 Spacecraft Ground
                 Tracks
                     Landsat 7 Orbit



             3
         a
P  2
         
Landsat 7 Revisit Pattern
Landsat 7 Revisit pattern
         Ground Tracks for Three
           Different Satellites
•INMARST 3-F2

•MARISAT 3

•BRASILSAT 1




                         (catholicoutlook)
                    INMARSET 3-F2
•Orbit is almost perfectly aligned
 with equator, and satellite always
 appears directly above equator

•Satellite’s orbit is almost perfectly
 circular, moves at a nearly
 constant velocity, in synch with
 rotation of the earth

•Appears to hover over a single
 spot on the equator
                                         (catholicoutlook)
                              MARISAT 3
•Orbit significantly inclined
 with respect to equator
•Goes above and below
 equator every day
•Because it orbits in synch
 with earth’s rotation, it appears to
 be moving straight up and down
•Orbit isn’t perfectly circular, but
 slightly elliptical, so satellite will
 speed up and slow down slightly
 as it moves toward and away from
 earth
•When temporarily faster than earth
 is rotating, moves slightly to east      (catholicoutlook)
•When temporarily slower than earth
 is rotating, moves slightly to west
•Traces out figure 8
                            BRASILSAT 1
•Orbit significantly inclined with
 respect to equator
•If in synch with earth’s rotation,
 it would trace out a straight line
 up-and-down every day
•Satellite not in a truly
 geosynchronous orbit, but
 orbiting slightly farther from
 earth
•Moving just slightly slower than
 earth is rotating
•As satellite traces out up-and-
 down motion, constantly lagging
 behind earth’s rotation, moving to
                                          (catholicoutlook)
 west
•Makes a zigzag path
    Ground Track of Highly
       Elliptical Orbits




(www.amsat.org)     (www.amsat.org)

				
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