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Orbital Mechanics Overview 2

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					Orbital Mechanics Overview 2

          MAE 155B
          G. Nacouzi


            GN/MAE155B         1
  Orbital Mechanics Overview 2
• Summary of first quarter overview
  – Keplerian motion
  – Classical orbit parameters
• Orbital perturbations
• Central body observation
  – Coverage examples using Excel
• Project workshop
                     GN/MAE155B       2
    Introduction: Orbital Mechanics
• Motion of satellite is influenced by the gravity field of
  multiple bodies, however, two body assumption is usually
  sufficient. Earth orbiting satellite Two Body approach:
   – Central body is earth, assume it has only gravitational
     influence on S/C, assume M >> m (M, m ~ mass of
     earth & S/C)
       • Gravity effects of secondary bodies including sun, moon and
         other planets in solar system are ignored
       • Gravitational potential function is given by:
              = GM/r
   – Solution assumes bodies are spherically symmetric,
     point sources (Earth oblateness not accounted for)
   – Only gravity and centrifugal forces are present
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Two Body Motion (or Keplerian Motion)

• Closed form solution for 2 body exists, no explicit soltn
  exists for N >2, numerical approach needed
• Gravitational field on body is given by:
   Fg = M m G/R2     where,

   M~ Mass of central body; m~ Mass of Satellite
   G~ Universal gravity constant
   R~ distance between centers of bodies

   For a S/C in Low Earth Orbit (LEO), the gravity forces are:
   Earth: 0.9 g Sun: 6E-4 g Moon: 3E-6 g        Jupiter: 3E-8 g

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                  Elliptical Orbit Geometry &
                          Nomenclature

                                                          V

                          a                       c                       Periapsis
                                                          R
                                                              
                        Line of Apsides                       Rp
Apoapsis                                   b

                                                      S/C position defined by R & ,
     R = [Rp (1+e)]/[1+ e cos()]                      is called true anomaly


   • Line of Apsides connects Apoapsis, central body & Periapsis
   • Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature)

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                 Elliptical Orbit Definition
• Orbit is defined using the
  6 classical orbital
  elements:
    – Eccentricity,                                                     i
    – semi-major axis,                                                            
    – true anomaly: position
      of SC on the orbit                  Vernal
    – inclination, i, is the              Equinox
      angle between orbit                               
      plane and equatorial                                            Ascending
      plane                                                           Node
    – Argument of Periapsis
      (). Angle from
      Ascending Node (AN)      - Longitude of Ascending Node ()~Angle from
      to Periapsis. AN: Pt       Vernal Equinox (vector from center of earth to sun on
      where S/C crosses          first day of spring) and ascending node
      equatorial plane South
      to North                       GN/MAE155B                                       6
 Sources of Orbital Perturbations
• Several external forces cause perturbation to
  spacecraft orbit
  – 3rd body effects, e.g., sun, moon, other planets
  – Unsymmetrical central bodies (‘oblateness’
    caused by rotation rate of body):
     • Earth: Requator = 6378 km, Rpolar = 6357 km
  – Space Environment: Solar Pressure, drag from
    rarefied atmosphere
  Reference: C. Brown, ‘Elements of SC Design’

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      Relative Importance of Orbit Perturbations




                                                           Reference: Spacecraft
• J2 term accounts for effect from oblate earth            Systems Engineering,
                                                           Fortescue & Stark
    •Principal effect above 100 km altitude
• Other terms may also be important depending on application, mission, etc...

                                  GN/MAE155B                               8
   Principal Orbital Perturbations
• Earth ‘oblateness’ results in an unsymmetric
  gravity potential given by:
                            a  
                                    n            
                 GM 
                               e   Jn Pn( w) 
                                                     Note:
                  1
                  r          r                  J2~1E-3,
                      n 2                         J3~1E-6

  where ae = equatorial radius, Pn ~ Legendre Polynomial
        Jn ~ zonal harmonics, w ~ sin (SC declination)
• J2 term causes measurable perturbation which
  must be accounted for. Main effects:
   – Regression of nodes
   – Rotation of apsides
                         GN/MAE155B                             9
       Orbital Perturbation Effects: Regression of
                          Nodes
Regression of Nodes: Equatorial bulge causes component of gravity
vector acting on SC to be slightly out of orbit plane

                                  This out of orbit plane component
                                  causes a slight precession of the
                                  orbit plane.




    The resulting orbital rotation is called regression of nodes and
    is approximated using the dominant gravity harmonics term, J2
                               GN/MAE155B                         10
                Regression of Nodes
   • Regression of nodes is approximated by:
                                    2
                    3 n  J  R  cos ( i)
          d                     2
              
          dt
                       2a
                            2   1  e 2
                                            2

Where,
 ~ Longitude of the ascending node;
R~ Mean equatorial radius
J2 ~ Zonal coeff.(for earth = 0.001082)
n ~ mean motion (sqrt(GM/a3)), a~ semimajor axis

Note: Although regression rate is small for Geo., it is cumulative and
must be accounted for
                                                GN/MAE155B       11
Orbital Perturbation: Rotation of
          Apsides
             Rotation of apsides caused by earth
             oblateness is similar to regression of
             nodes. The phenomenon is caused by
             a higher acceleration near the equator
             and a resulting overshoot at periapsis.
             This only occurs in elliptical orbits.
             The rate of rotation is given by:

                 d
                      3n J  R
                                       
                                     2 4  5 sin ( i)   
                                                         2


                                                    
                 dt           2                          2
                                           2       2
                                       4a 1  e


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                     Ground Track
• Defined as the trace of nadir positions, as a function of
  time, on the central body. Ground track is influenced by:
    – S/C orbit
    – Rotation of central body
    – Orbit perturbations

Trace is calculated using spherical trigonometry (no perturbances)
   sin (La) = sin (i) sin ALa
   Lo =  + asin(tan (La)/tan(i))+Re

where: Ala ~ (ascending node to SC)
        ~ Longitude of ascending node
        I ~ Inclination
      Re~Earth rotation rate= 0.0042t (add to west. longitudes, subtract
  for eastern longitude)
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       Example Ground Trace




Ground trace
from i= 45 deg




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                Spacecraft Horizon
• SC horizon forms a
  circle on the spherical
  surface of the central
  body, within circle:
   – SC can be seen from
     central body
   – Line of sight
     communication can be
     established
   – SC can observe the
     central body




                            GN/MAE155B   15
       Central Body Observation




From simple trigonometry:
sin(h) = Rs/(Rs+hs) Dh = (Rs+hs) cos(h)
Sw~ Swath width = 2 h Rs
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