# Orbital Mechanics Overview 2

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```					Orbital Mechanics Overview 2

MAE 155B
G. Nacouzi

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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
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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...

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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
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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
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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;
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
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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

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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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