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