lecture 4 - orbital mechanicsaeweb.tamu.edu/aero310/files/notes/lecture 4 - orbital...
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Orbital Mechanics
History
Geocentric model (Ptolemy)
Heliocentric model (Aristarchus of Samos)
Nicholas Copernicus (1473-1543)
In De Revolutionibus Orbium Coelestium
("On the Revolutions of the Celestial Orbs"),
which was published in Nuremberg in 1543,
the year of his death, stated that the Sun was
the center of the universe and that the Earth
orbited around this center.
His theory gave a simple and elegant explanation of the retrograde
motions of the planets (the annual motion of the Earth necessarily
projected onto the motions of the planets in geocentric astronomy)
and settled the order of the planets definitively.
Copernican Universe
Tycho Brahe (1546-1601)
•Tycho designed and built new instruments,
calibrated them, and instituted nightly
observations.
•Changed observational practice profoundly: earlier
astronomers observed the positions of planets and the Moon at
certain important points of their orbits (e.g., opposition,
quadrature, station), Tycho observed these bodies throughout
their orbits.
•As a result, a number of orbital anomalies never before
noticed were made explicit by Tycho. Without these
complete series of observations of unprecedented accuracy,
Kepler could not have discovered that planets move in
elliptical orbits.
Johannes Kepler (1571-1630)
•Using the precise data that Tycho had collected,
Kepler discovered that the orbit of Mars was an
ellipse.
•In 1609 he published Astronomia Nova, delineating his discoveries,
which are now called Kepler's first two laws of planetary motion.
• In 1619 he published Harmonices Mundi, in which he describes his
"third law."
• Kepler published the seven-volume Epitome Astronomiae in
1621. This was his most influential work and discussed all of
heliocentric astronomy in a systematic way. He was a sustainer of
the copernican system.
Isaac Newton1643 –1727
• Derived three laws of
motion
• Derived the Law of
Universal Gravitation
• Explained why
Kepler’s laws worked
2-Body Problem
=
=
Not Solving a Problem Can Get You a Prize!
The 3-Body Problem remained
a nagging problem until…
..in 1887, the King of Sweden
offered a prize for the answer
to the question: “Is the solar
system stable?”
Poincaré showed the impossibility of solution
AERO 660
Nonlinear Flight Dynamics
Instructor: Dr. T
Kepler’s 1st Law: Law of Ellipses
The orbits of the planets are ellipses with
the sun at one focus
Ellipses
FOCI
Period (T)
Semi-Major Axis (a)
Semi-Minor
Axis (b)
Kepler’s 2nd Law:
Law of Equal Areas
t0
t3
t1
t2
Area 1Area 2
t1-t0 = t3-t2
Area 1 = Area 2
The line joining the planet to the center of the sun
sweeps out equal areas in equal times
Kepler’s 3rd Law: Law of Harmonics
In Chinese:
The squares of the periods of
two planets’ orbits are
proportional to each other as
the cubes of their semi-
major axes:
T12/T2
2 = a13/a2
3
In English:
Orbits with the same semi-
major axis will have the
same period
Newton’s Laws
Law of Inertia: Every body continues in a state of
uniform motion unless it is compelled to change that
state by a force imposed upon it
Law of Momentum: Change in momentum is
proportional to the applied force
Action – Reaction: For every action, there is an equal
and opposite reaction
Universal Gravitation: Between any two objects there
exists a force of attraction that is proportional to the
product of their masses and inversely proportional to
the square of the distance between them
ORBIT CLASSIFICATION
�Location (equatorial, polar)
�Shape (circular, elliptical, parabolic, hyperbolic)
�Size/Period
ORBIT CLASSIFICATIONSize/Period
�Low Earth Orbit (LEO)
�High Earth Orbit (HEO)
�Semi-synchronous Orbit
�Geo-synchronous Orbit
LEOs are elliptical/circular orbits at a height of less than 2,000 km above the surface
HEOs typically have a perigee at about 500 km above the surface of the earth and an
apogee as high as 50,000 km.
a
ORBIT CLASSIFICATION
Shape (Conic Sections)
Apollonius of Perga ~BC 262 – 190Hypatia of Alexandria ~AD 370 - 415
ORBIT CLASSIFICATIONS
Circular Orbits
�Characteristics
– Constant speed
– Nearly constant altitude
�Typical Missions
– Reconnaissance/Weather (DMSP)
– Manned
– Navigational (GPS)
– Geo-synchronous (Comm sats)
ORBIT CLASSIFICATIONS
Elliptical Orbits
�Characteristics
– Varying speed
– Varying altitude
– Asymmetric Ground Track
�Typical Missions
– Deep space surveillance (Pioneer)
– Communications
– Ballistic Missiles
ORBIT CLASSIFICATIONS
Parabolic/Hyperbolic Trajectories
�Characteristics
– Escaped Earth’s gravitational
influence
– Heliocentric
�Typical Missions
– Interplanetary exploration (Galileo,
Phobos, Magellan)
ORBIT GEOMETRY
Apogee
cc
aEccentricity = c/a
Perigee
ORBIT CLASSIFICATIONSEccentricity
e = 0
0 < e < 1
e = 1
e > 1
ORBIT CLASSIFICATIONSEccentricity
e = 0
a
c = 0
0 < e < 1
c
a
Eccentricity = c/a
ORBIT CLASSIFICATIONSEccentricity
Eccentricity = c/a
e = 0.75
e = .45
e = 0
− Fg = mr − rθ2
0 = m2rθ + rθ
#
# ?
Angular momentum
HC/S = rC/S × mv = re r × mre r + rθe θ = mr2 θk
d
dtmr2 θ = m2rrθ + r2 θ = rm2rθ + rθ = 0
r2 θ = h
p ′′ + p = Gm S
h2
References
• Lecture notes/slides by Prof. Daniele Mortari and the Naval Postgraduate School's Digital Evaluation and Exploitation (DEEP) group