Chapter 8: Predicting Orbits Understanding Space Sellers

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Chapter 8 Predicting OrbitsUnderstanding
Space (Sellers)
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References
1. Keplerian Elements http//marine.rutgers.ed
u/mrs/education/class/paul/orbits.html4 2. A
Guide to Computing Orbital Positions in the Solar
System http//www.met.rdg.ac.uk/ross/Document
s/OrbitNotes.html 3. Keplers Equation of
Elliptical Motion http//www.akiti.ca/KeplerEq
uation.html
4. Motion in the Plane of the Orbit
http//www.bruce-shapiro.com/pair/KeplersEquation.
pdf 5. Thomson,W.T., Introduction to Space
Dynamics, John Wiley and Sons, 1961 6.
Gyroscopic Flight Instruments,
http//avstop.com/AC/3-6.html 7. Molniya 1-91 -
Information http//www.heavens-above.com/sati
nfo.asp?satid25485 8. Types of Orbits
http//marine.rutgers.edu/mrs/education/class/paul
/orbits2.html2
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Keplers Equation and Time of Flight
TOF
Time of flight on a circular orbit
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Keplers Equation
Finding time of flight to a point on a circular
orbit is relatively easy Finding time of
flight to a point on an elliptical orbit is
difficult, since the velocity is not
constant Kepler solved the problem
geometrically by relating the position on the
elliptical orbit to one on a circular orbit
True anomaly Eccentric anomaly Mean
anomaly
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

x
ae
True Anomaly
a a
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Relation Between True, Eccentric, and Mean
Anomalies
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

x
ae
True Anomaly
a a
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Relation Between True, Eccentric, and Mean
Anomalies
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

x
ae
True Anomaly
a a
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Relation Between True, Eccentric, and Mean
Anomalies
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

x
ae
True Anomaly
a a
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Relation Between True, Eccentric, and Mean
Anomalies
Mean motion is average angular rate
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

For certain eccentricity and employment of
angular momentum invariance, we get
x
ae
True Anomaly
a a
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Relation Between True, Eccentric, and Mean
Anomalies
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

Total energy equation is
x
ae
True Anomaly
a a
Substituting in
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Relation Between True, Eccentric, and Mean
Anomalies
Focus
y
Eccentric Anomaly
Satellite
Center of orbit
a

R
E

?

x
ae
True Anomaly
a a
Keplers Equation
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Keplers Equation and Time of Flight
Time of flight on an elliptical orbit and
relating E to ?(?)
(KeplerEquation)
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Keplers Equation and Time of Flight
Time of flight on an elliptical orbit and
relating E to ?(?)
Minitial and Mfuture
(KeplerEquation)
Einitial and Efuture
Time of flight (tfuture-tinitial)
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Keplers Equation and Future Position
(KeplerEquation)
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Orbital Perturbations
Atmospheric drag For eccentric low-Earth
orbit, small ?Vs at perigee circularize
orbit Orbit eventually decays to point
where spacecraft re-enters Earths oblateness
(J2) Affects ? Affects ? See
figure 8-10, page 275, figure 8-11, page
276, and figure 8-12, page 275 Other
perturbations
(avstop)
1. Chapter 5. Space Environment and Orbital
mechanics http//www.fas.org/spp/military/doco
ps/army/ref_text/chap5im.htm
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Sun Synchronous Orbits
Allow a satellite to pass over section of Earth
at same time each day Since there are 365 days
per year and 360 in a circle, satellite has
to shift its orbit about 1 per day At a 700 -
800 km altitude, satellites use fact that Earth
not perfectly round but bulges in the
center Oblateness causes gravitational forces
to act on satellite, resulting in ? proceeding
or receding Proper allows spacecrafts
orbital plane to maintain same orientation to
Sun, which is great for photography
(Types of Orbits)

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Molniya Orbit
Inclination of 63.4 Perigee doesnt rotate
(fig 8-11, page 276) Spacecraft hangs over
Northern Hemisphere for nearly 11 hours out of
twelve hour period Orbit used because Russians
did not have launch vehicles large enough to
put communication satellites into
geosynchronous orbits at far north launch sites
(Molniya 1-91)
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Predicting Orbits in the Real World
(Keplerian Elements)
(Keplerian Elements)
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Predicting Orbits in the Real World
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Predicting Orbits in the Real World