Day 4
Chapter 2
Gravitation
and the Motion
of the Planets
Science is the key to understanding
• Science: a body of knowledge and a process of learning about
nature (called the scientific method).
• Knowledge is acquired by observations and experiments.
• Scientific method is a process for gaining more knowledge, that
can be tested and accepted by everyone.
• Scientific theory is an explanation of observations or
experimental results that can be described quantitatively and
tested.
• The theory must make testable predictions that can be verified
by new observations or experiments, and can possibly be
refuted.
• Theories can be modified and should be the simplest version
that explains the observations (Occam’s razor).
• Observe, hypothesize, predict, test, modify, economize.
The Copernican Revolution
• The model of the Greeks (attributed primarily to
Ptolemy) had the Earth at the center of everything.
• Copernicus proposed a model of the solar system
with the Sun at the center (hence a solar system).
• Galileo made a telescope and used it to view the sky,
and saw that the phases of Venus refuted the
geocentric model of the Greeks.
• Tycho Brahe and Johannes Kepler developed a more
detailed heliocentric model with elliptical orbits.
Retrograde motion of a planet occurs over several weeks,
and involves motion to the west, as compared
to prograde (direct) motion, which is to the east
(relative to the stars of the ”celestial sphere”).
The Geocentric Model of planetary motion (Greek philosophy)
The Geocentric Model does explain retrograde motion,
using concepts like deferent and epicycle. However,
it does not predict the motion with much accuracy, and
does not predict phases of Venus (seen with a telescope).
Ptolemy’s Model of planetary motion used deferents (big circles)
and epicycles (little circles centered on a point that moves on the
deferent). This involved up to 80 circles to describe 7 objects!
Occam’s
Razor says:
Simplify this!
Nicholas Copernicus and his Heliocentric model of the Solar
System explained this in a simpler way with the Sun at the center.
Retrograde motion is seen in this model,
using Earth and Mars as the example.
Retrograde Motion of Mars as seen from Earth
Galileo Galilei and the Birth of Modern Astronomy
Galileo built a telescope in 1609 and looked at the sky.
Four objects:
The Moon
The Sun
Jupiter
Venus
(and much more)
Galileo looked at the Moon and saw
mountains, craters, valleys, and topography
like you might find on the Earth.
The Moon was perhaps an object like the Earth!
By projecting an image of the Sun,
he could see imperfections on the Sun.
Sunspots could be seen to move from
east to west on the Sun and he deduced
that the Sun rotated about once a month.
Galilean Moons of Jupiter
Small point of light could be
seen near Jupiter. By observation
during several weeks he deduced
that these were moons and that
they revolved around Jupiter.
Perhaps this planet was like
the Earth, with several moons
of its own. It also seemed like
a miniature model of the
heliocentric solar system.
Venus Phases in the Heliocentric model
These are consistent with the observations in a telescope.
Venus Phases in the Geocentric model are obviously wrong as
soon as you observe with a telescope. This refutes Ptolemy!
Geocentric vs. heliocentric theories
Both described the positions and movement of the Sun,
Moon, and 5 visible planets, as seen without a telescope.
The geocentric theory was too complicated (80 circles!).
(Occam’s razor could be invoked to seek a simpler way.)
Once the telescope was used to observe Venus, the
geocentric theory could not explain the phases of Venus.
The heliocentric theory of Copernicus explained many of
Galileo’s observations, but also used circular orbits.
More accurate measurements did not agree with the simple
theory of Copernicus (circles had to be replaced by ellipses
in the newer theory of planetary motion).
After Copernicus
and Galileo,
two major figures
changed the way
we come to
understand the
Universe:
Kepler’s laws
of planetary
motion
Newton’s laws
of mechanics
Further development of the heliocentric theory
More detailed observations were made by Tycho
Brahe (commonly called Tycho, 1546 - 1601).
He made observations of a supernova in 1572
which convinced him that it was a distant star.
He received an island and built an observatory to
measure planetary motion to high accuracy over
a period of more than 20 years.
His observations were inherited by an assistant,
Johannes Kepler, when Tycho died in 1601.
Tycho Brahe
obtained data over
a period of 21 years
that were later used
by his assistant
Johannes Kepler
to determine that
planetary orbits
are NOT circles,
but are ellipses.
Johannes Kepler and the Laws of Planetary Motion
Kepler used decades of Tycho’s
observations in his mathematical
calculations, to determine the shape of the
planetary orbits, and the speed of the
planets as they went around the Sun.
This massive effort resulted in three major
statements about the characteristics of
planetary orbits:
Kepler’s three laws of planetary motion.
Kepler’s three laws of planetary motion
• Orbital paths of the planets are ellipses.
• An imaginary line connecting the planet with
the Sun sweeps out equal areas of the ellipse
in equal intervals of time.
• The square of a planet’s orbital period is
proportional to the cube of its semi-major
axis.
• Kepler published this in 1609, the same year that
Galileo built his first telescope.
Kepler’s laws of planetary motion
Kepler’s first law: The orbital paths of the
planets are elliptical, with the Sun at one focus.
Kepler’s second law: An imaginary line
connecting the Sun to any planet sweeps out
equal areas of the ellipse in equal intervals of
time.
Kepler’s third law: The square of the planet’s
orbital period is proportional to the cube of its
semimajor axis.
An Ellipse can be drawn with string and TWO foci
For an ellipse,
r1 + r2 = 2a
The eccentricity
is defined as:
e = c/a
A circle results
when e = 0
GeoGebra demonstration:
http://people.ucalgary.ca/~louro/geogebra/ellipse.html
Some Properties of Planetary Orbits
Kepler’s laws of planetary motion
Kepler’s first law: The orbital paths of the
planets are elliptical, with the Sun at one focus.
Kepler’s second law: An imaginary line
connecting the Sun to any planet sweeps out
equal areas of the ellipse in equal intervals of
time.
Kepler’s third law: The square of the planet’s
orbital period is proportional to the cube of its
semimajor axis.
Kepler’s Second Law: equal areas in equal time
This also means higher speed at closer distances.
Another graphic on Kepler’s Second Law:
The Astronomical Unit is about 150,000,000 km
Kepler’s laws of planetary motion
Kepler’s first law: The orbital paths of the
planets are elliptical, with the Sun at one focus.
Kepler’s second law: An imaginary line
connecting the Sun to any planet sweeps out
equal areas of the ellipse in equal intervals of
time.
Kepler’s third law: The square of the planet’s
orbital period is proportional to the cube of its
semimajor axis.
Kepler’s Third Law: P2 (in years) = a3 (in a.u.)
Basically, it means that large orbits have long periods.
Real orbits have the
center of mass
as one focus
For the Sun and
planets, this is
not a large effect.
For binary stars,
the center of mass
may be near the
middle of the line
connecting them.
Let’s review Kepler’s Laws.
Review: see if you can tell what
these are simulating:
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_kepler2
.html
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_period
s_sim.html
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_solar_s
im.html
Newton’s Laws of Physics
•
•
•
•
•
•
First law: inertia
Second law: F = ma
or acceleration = force / mass
Third law: Action and Reaction
means that forces occur in pairs.
These can be used to show that orbits
should obey Kepler’s 3 laws.
Isaac Newton developed a quantitative and
explanatory theory of mechanics, explaining
the motion of objects resulting from forces.
Newton’s First Law: The law of inertia.
An object will continue in it’s motion
without change of velocity unless
it is acted on by a net external force.
Newton’s Second Law: F = ma
The acceleration of a mass is proportional
to the total force acting upon it, and
inversely proportional to the mass of the object.
Newton’s Third Law: Action-reaction
For every force acting upon an object (action),
there is a force acting on another object (reaction)
which has the same magnitude (size) but
points (acts) in the opposite direction.
Newton also developed the
universal law of gravity.
Gravitational force varies with
the distance between the objects.
It depends on the product
of the two masses, i.e.,
m1 x m2
and on the inverse of the square of
the distance between the masses
(assuming they are small
compared with the distance).
1/r2
The Sun’s gravity causes planets to move on a path
called an orbit. These orbits obey Kepler’s Laws.
Newton’s Laws explain Kepler’s Laws
• Newton’s Laws account for all three of Kepler’s Laws.
• The orbits of the planets are ellipses, but it is also possible to have
orbits which are parabolas or hyperbolas. (conic sections)
• Edmond Halley predicted a comet would return in 1758 and every
76 years after that. (seen in 1910, 1986, and will return in 2061)
Halley’s comet has an elliptical orbit extending out past Neptune.
• William Herschel discovered Uranus in 1781 by accident.
• After 50 years it was seen to deviate from an elliptical orbit, and a
calculation led to the discovery of Neptune in 1846.
• To be precise, elliptical orbits would only occur if there were only
the Sun and one planet. There are 8 planets and other objects
which cause deviations from the perfect elliptical orbit.
The first exam is on Thursday, Feb. 4 (next week!)
We will have about 30 minutes of class before the exam.
Then you will take the exam (which uses a Scantron).
The exam is multiple choice and true/false questions.
Coverage is Chapters 1 and 2 in your textbook.
To review, look at the chapter summaries, my day notes,
and a study guide that I will post this weekend.