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PHYS 3380 - Astronomy
Exam Wednesday September 21
The exam will cover anything that was covered in the
notes
The exam will be closed book.
You will be allowed to have one 8/12 X 11 sheet of
notes for the exam.
Good Luck!
PHYS 3380 - Astronomy
Hipparchus (c. 190 - 120 B.C.)
-placed Earth away from center of perfect spheres
- developed a star catalog
- method to predict positions of sun and moon
- discovered precession of earth’s axis of rotation - estimated
precession of equinoxes at 46”/year - actually 50.26”
Ptolemy (c. A.D. 100 - 170)
- combined all available astronomical information into 13 volume work,
the Almagest - a geometrical model of solar system with Earth in center
that predicted positions of Sun, Moon and planets - used for 1,500
years
- used idea of Apollonius (c. 240 - 190 B.C.) that planets move on small
circle that turns upon larger circle around the Earth explained
retrograde motion of planets - fully developed by Hipparchus accurately predicted planetary positions within a few degrees of arc
PHYS 3380 - Astronomy
Retrograde Motion of Planets
Apparent retrograde motion - reversal of planetary motion through
the zodiac from eastward to westward
Apparent retrograde motion of Jupiter
PHYS 3380 - Astronomy
Ptolemaic Model
Deferent - main circle of
planet path
Epicycle - small circles on
deferent along which
planet moved
PHYS 3380 - Astronomy
Ptolemaic Model Animation
PHYS 3380 - Astronomy
Ptolemy's model of the motions of the Sun, Moon,
and planets around Earth.
PHYS 3380 - Astronomy
Mars Retrograde Motion
Earth’s orbital speed is
faster than Mars’. When
Earth catches up with Mars,
Mars appears to move
backwards against the
background of the stars
PHYS 3380 - Astronomy
Mars Retrograde Motion Animation
PHYS 3380 - Astronomy
Heliocentric World View - The Copernican Revolution
Nicholas Copernicus (1473-1543)
- wanted better way to predict planetary positions adopted Sun-centered planetary system suggested by
Aristarchus
- worked out simple geometric relationships that allowed
calculation of planet’s true orbital period and distance from
Sun in terms of Earth-Sun distance (AU)
- still believed that heavenly motion must be perfect
circles - had to add circles upon circles as in the
Ptolemaic model - did not make substantially better
predictions and was not accepted for another 50 years
PHYS 3380 - Astronomy
Tycho Brahe (1546 - 1601)
- observed alignment of Jupiter and Saturn in 1563 - occurred
two days later than Copernicus prediction - decided to compile
precise measurements of stellar and planetary positions
- compiled best set of naked-eye observation ever made - to
within 1 arcminute (thickness of a fingernail at arm’s length)
- observed supernova of 1572 - proved it was farther away
than the Sun - called it a nova (“new star”)
- observed comet - proved was in the heavens
- believed planets must orbit Sun but never succeeded in
explaining planetary motion. Inability to detect stellar parallax
led him to believe that Sun orbited Earth while all other planets
orbited Sun
PHYS 3380 - Astronomy
A Complete Reformation of Astronomy
Johannes Kepler (1571 - 1630)
- hired by Tycho Brahe to explain observations
- years of calculations trying to find circular orbit for Mars that
matched Tycho’s observations - succeeded in matching Mars’
position in ecliptic - could not match position north or south of
ecliptic
- finally realized planetary orbits were not circular but elliptical developed three laws of orbital motion - accurately predicted
planetary motions and matched Tycho’s data
PHYS 3380 - Astronomy
Kepler’s Laws of Planetary Motion
:
Law 1: Law of elliptical orbits
Each planet moves in an elliptical orbit.
Law 2: Law of areas
The imaginary line connecting any planet to the sun
sweeps over equal areas of the ellipse in equal intervals of
time.
Law 3: Law of periods
The square of any planet's period of orbital revolution is
proportional to the cube of its mean distance from
the sun.
PHYS 3380 - Astronomy
Kepler’s First Law
The orbit of each planet around the Sun is an ellipse with the Sun at one focus
PHYS 3380 - Astronomy
(a)
Drawing a circle.
(b)
Drawing an ellipse.
(c)
Eccentricity
describes how much an
ellipse deviates from a
perfect circle -ratio
between distance from
the center of the ellipse
to the focus of the
ellipse and the semimajor axis.
PHYS 3380 - Astronomy
x2 y2
 2 1
2
a
b
or
x  acos t
y  bsin t
0  t  2
The eccentricity is :
2
b
c
e = 1- 2 
a
a
PHYS 3380 - Astronomy
Kepler’s Second Law
As a planet moves around its orbit, it sweeps out equal areas in equal times.
PHYS 3380 - Astronomy
Area and Time Animation
PHYS 3380 - Astronomy
Kepler’s Third Law
The square of any planet's period, P, of orbital revolution is
proportional to the cube of its mean distance, r, from the sun.
I.e., the more distant a planet, the slower it moves on average.
Example:
For earth, r E= 1 AU, PE = 1 year.
For Mars, r M= 1.52 AU, PM = 1.88 years
PM2 rM3
 3
2
PE
rE
1.88 2 1.523
 3
2
1
1
3.53  3.51
PHYS 3380 - Astronomy
A plot of the cube of the average
planetary distance vs the square
of the orbital period is a straight
line
The average orbital speed is
inversely proportional to the
average distance from the sun
PHYS 3380 - Astronomy
Semimajor
Axis (1010 m)
Period
(T yr)
T2/a3
(10-34 y2/m3)
Mercury
5.79
0.241
2.99
Venus
10.8
0.615
3.00
Earth
15.0
1
2.96
Mars
22.8
1.88
2.98
Jupiter
77.8
11.9
3.01
Saturn
143
29.5
2.98
Uranus
287
84
2.98
Neptune
450
165
2.99
Pluto
590
248
2.99
Planet
PHYS 3380 - Astronomy
Note:
- Kepler's third law needs to be modified when the orbiting
body's mass is not negligible compared to the mass of the body
being orbited.
- Kepler's laws assume a two-body system - particularly bad
approximation in the case of the Earth-Sun-Moon system
for calculations of the Moon's orbit, Kepler's laws are far
less accurate than the empirical method invented by
Ptolemy.
- Kepler's laws do not consider the emission of radiation or
relativity
- Because electrical forces, like gravity, obey an inverse square
law, Kepler's laws also apply to bodies interacting electrically.
Kepler did not understand why his laws were correct - Isaac Newton
discovered the answer more than fifty years later. For instance, the
second law also a statement of conservation of angular momentum.
PHYS 3380 - Astronomy
The Death of the Earth Centered Universe
Galileo Galilei (1564 - 1642)
- demonstrated that a moving object remains in motion unless
acted on by an outside force (Newton’s 1st law) - contradicted
Aristotle’s claim that the natural tendency of any moving object
is to come to rest. Birds, falling stones, clouds, etc.. would stay
with the Earth unless knocked away by some force
- used telescope that he built (invented by Hans Lippershey) - saw sunspots on the Sun, craters and valleys on the
moon - proved that the heavens were not perfect and
unchanging
- observed moons orbiting Jupiter
- observed the phases of Venus
- showed that Milky Way resolved into countless
individual stars - argued stars far more numerous and
distant than imagined - reason stellar parallax not
observed
- recanted before Church inquisition in Rome in 1633 - formerly
vindicated by the Church in 1992
PHYS 3380 - Astronomy
Major Discoveries of Galileo
• Moons of Jupiter
(4 Galilean moons)
(What he really saw)
• Rings of Saturn
(What he really saw)
PHYS 3380 - Astronomy
• Surface structures on the moon - shadows; first estimates of the
height of mountains on the moon
PHYS 3380 - Astronomy
Sun spots (proving that the sun is not perfect!)
PHYS 3380 - Astronomy
Phases of Venus (including “full Venus”) - In the Ptolemaic
system, phases would only range from new to crescent. Proved
that Venus orbits the sun, not the Earth!
PHYS 3380 - Astronomy
Phases of Venus Animation
PHYS 3380 - Astronomy
A New View of Nature
Sir Isaac Newton (1642 - 1727)
- followed Galileo’s lead - developed fundamental laws of motion
- revolutionized mathematics and science
- experienced moment of inspiration at 24 years old - saw apple
fall from tree and suddenly understood gravity
- published most famous book in science in 1687 - Philosophiae
Naturalis Principia Mathematica - Principia for short
- built first reflecting telescope
- invented calculus
- discovered three laws of motion
- discovered universal law of mutual gravitation
PHYS 3380 - Astronomy
Newton's Three Laws of Motion
First law:
A body remains at rest or moves along a straight line with constant
velocity so long as no external force acts upon it.
Second law:
A body (m) acted upon by a force (f) will accelerate (a) in the
direction of the applied force. The greater the force or the smaller
the mass, the greater will be the acceleration .
F =ma
Third law:
A body subjected to a force reacts with an equal counter force to the
applied force:
That is, action and reaction are equal and oppositely directed, but
never act on the same body.
PHYS 3380 - Astronomy
1st Law
A body moves
along a straight
line with constant
velocity so long
as no external
force acts upon it.
2nd Law
Force equals mass
times acceleration
3rd Law
For every force, there is
an equal and opposite
force
PHYS 3380 - Astronomy
Velocity and Acceleration
Newton showed that acceleration (a) is the
change of a body’s velocity (v) with time (t):
a
a = Dv/Dt
Differential calculus!
Velocity and acceleration are vectors.
v
Different cases of acceleration:
1. Acceleration in the conventional sense (i.e.
increasing speed)
2. Deceleration (i.e. decreasing speed)
3. Change of the direction of motion (e.g., in
circular motion)
PHYS 3380 - Astronomy
Galileo demonstrated that
acceleration of gravity is
independent of the mass of the
falling object - supposedly
dropped balls of different mass
from Leaning Tower of Pisa Newton showed why.
Feather and Hammer Movie