Download - Orbital Mechanics Overview 2
GN/MAE155B 1
Orbital Mechanics Overview 2
MAE 155B
G. Nacouzi
GN/MAE155B 2
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
Periapsis
ApoapsisLine of Apsides
R
a c
V
Rpb
• Line of Apsides connects Apoapsis, central body & Periapsis• Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature)
S/C position defined by R & , is called true anomalyR = [Rp (1+e)]/[1+ e cos()]
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Elliptical Orbit Definition
• Orbit is defined using the 6 classical orbital elements:– Eccentricity, – semi-major axis, – true anomaly: position of
SC on the orbit– inclination, i, is the angle
between orbit plane and equatorial plane
– Argument of Periapsis (). Angle from Ascending Node (AN) to Periapsis. AN: Pt where S/C crosses equatorial plane South to North
- Longitude of Ascending Node ()~Angle from Vernal Equinox (vector from center of earth to sun on first day of spring) and ascending node
i
Vernal Equinox
AscendingNode
Periapsis
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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
• J2 term accounts for effect from oblate earth•Principal effect above 100 km altitude
• Other terms may also be important depending on application, mission, etc...
Reference: SpacecraftSystems Engineering,Fortescue & Stark
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Principal Orbital Perturbations
• Earth ‘oblateness’ results in an unsymmetric gravity potential given by:
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
GM
r1
2
n
ae
r
n
Jn
Pn
w( )
Note:J2~1E-3,J3~1E-6
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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 componentcauses a slight precession of the orbit plane.
The resulting orbital rotation is called regression of nodes andis approximated using the dominant gravity harmonics term, J2
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Regression of Nodes
• Regression of nodes is approximated by:
td
d
3 n J2
R2 cos i( )
2a2
1 e2 2
Where, ~ Longitude of the ascending node; R~ Mean equatorial radiusJ2 ~ 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 earthoblateness is similar to regression ofnodes. The phenomenon is caused bya higher acceleration near the equatorand a resulting overshoot at periapsis.This only occurs in elliptical orbits.The rate of rotation is given by:
td
d3n J
2 R
24 5 sin i( )
2 4a
21 e
2 2
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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 tracefrom 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