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History of Modern Astronomy
Chapter 5
Topics
• Major characters in the development of our
understanding of the motions of planets.
• Kepler’s three laws of planetary motion
• Newton’s three laws of motion and the law
of gravitation
A little drama
• Characters in the great drama
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Claudius Ptolemy (140)
Nicolaus Copernicus (1473-1543)
Tycho Brahe (1546-1601)
Johannes Kepler (1571-1630)
Galileo Galilei (1564-1642)
Isaac Newton (1642-1727)
Aristotle’s shoes
• It seems natural that our first hypothesis
regarding the “structure” of the Solar
System would be geocentric.
• Being more philosophical and less
empirical, we would hope to see harmony
and perfection in the heavens that fit our
philosophy--thus, the motions of bodies are
perfect circles.
Then came Ptolemy
• A good theory should explain what is observed
and be able to make predictions.
• Planets move in circles called epicycles.
• The center of the epicycle moves in a circle called
a deferent.
• To make theory match prediction, Earth isn’t
exactly at the center of the deferent.
• The test of all knowledge is measurement
– Ptolemy’s theory explained the retrograde motion of the
planets
– Predicted future locations of the planets
1400 years later heliocentric idea
• Copernicus, for philosophical reasons, sought to
explain the retrograde motion of planets using a
heliocentric solar system. (animation)
• He still assumed perfect circles for the orbits of
planets (with the Sun at the center of the orbits).
• He could calculate
– the relative distances to the planets
– the orbital periods of the planets
• Predictions of the future positions of planets were
not much better than those from the Ptolemaic
model
We need better data!
• Tycho Brahe had ideas for a new model but
recognized the need for more precise
measurements.
• He devoted his life to making more precise
measurements of the positions of stars and planets.
• He built the first modern observatory
• He amassed records of planetary positions from
1576 to 1591
• His observations were 2.5 times more accurate
than any previous records
Finding a needle in a haystack
• Kepler believed the Copernican model and sought to prove
that it was correct using Brahe’s data for the positions of
the planets.
• He found that
– Planets orbit in elliptical paths (not circles!) with the Sun at one
focus of the ellipse.
– A line from the Sun to a planet will sweep out the same area in a
certain time interval, regardless of where the planet is in its path.
– The ratio of the (period)2 to (semi-major axis)3 was the same for
every planet.
• He described the planets’ orbits, but could they be
explained? Kepler answered “What?” but didn’t know
“Why?”
Standing on the shoulders of
giants
• Isaac Newton formulated three laws of motion and a law of
gravitation.
• This model for understanding motion (how motion is
related to forces) and gravitation explained Kepler’s three
laws.
• When “Why?” matches “What?” (theory matches
observation), we must reexamine our dearly held beliefs.
• This happened again in 1911 with Einstein’s publication of
the General Theory of Relativity
– an entirely different explanation of gravity
– explained phenemena that Newton’s law of gravitation could not
explain.
– has been verified by experiment to this day
Kepler’s laws of planetary
motion
details
Kepler’s first
law
• planet’s orbit the Sun in
ellipses, with the Sun at
one focus.
• the eccentricity of the
ellipse, e, tells you how
elongated it is.
• e=0 is a circle, e<1 for all
ellipses
e=0.02
e=0.4
e=0.7
Experiment and theory
eccentricity of the planets
Mercury
0.206
Saturn
0.054
Venus
0.007
Uranus
0.048
Earth
0.017
Neptune
0.007
Mars
0.094
Pluto
0.253
Jupiter
0.048
Kepler’s second law
• The line joining the Sun and a planet
sweeps out equal areas in equal time
intervals.
• As a result, planets move fastest when they
are near the Sun (perihelion) and slowest
when they are far from the Sun (aphelion).
• simulation 1
If it sweeps out equal areas in equal times, does it
travel faster or slower when it is far from the Sun?
If is sweeps out equal areas in equal times, does it
travel faster or slower when far from the Sun?
If is sweeps out equal areas in equal times, does it
travel faster or slower when far from the Sun?
Same Areas
Kepler’s Third Law
• Period of a planet, P
• Average distance from the Sun (semimajor axis of ellipse),
R
• P2/R3 = 4p2/(G(m1+m2))
• Approximately, P2earth/R3earth = P2planet/R3planet
• Sometimes we use Earth-years and Earth-distance to the
Sun (1 A.U.) as units.
• The constant of proportionality depends on the mass of the
Sun--and that’s how we know the mass of the Sun.
• We can apply this to moons (or any satellite) orbiting a
planet, and then the constant of proportionality depends on
the mass of the planet.
Practice
• While gazing at the planets that are visible with
the naked eye, you tell a friend that the farther a
planet is from the Sun, the longer its solar year is.
Your friend first asks what a solar year is. After
explaining that it’s the time required for a planet to
return to its same position relative to the Sun, your
friend then asks, “Why does it take longer for the
outermost planets to orbit the Sun?” What is your
reply?
Practice
• What is the best method for determining the
mass of Astronomical objects?
• Kepler’s Third Law
• For distant stars, this doesn’t work very
well. Fortunately, there is a relationship
between mass and brightness that will help
us out.
Newton’s laws of motion
• Newton’s first law
– an object will have a constant velocity (constant speed,
moving in a straight line) unless a net force acts on it
• Newton’s second law
– the acceleration of an object is proportional to the net
force on the object divided by its mass
• Newton’s third law
– if object A exerts a force on object B, B exerts a force
of equal magnitude back on A
Newton’s law of gravitation
• mass attracts mass
• the magnitude of the force of attraction is
proportional to the product of their masses
and the inverse of the square of the distance
between them
ForceB on A
mA
ForceA on B
mB
Gravitational force and distance
• If the bodies are twice as far apart, the
gravitational force of each body on the other is 1/4
of their previous values.
• This is called an “inverse-square law.”
ForceB on A
mA
ForceA on B
mB
Practice
• The Earth exerts a gravitational force on an
orbiting satellite. Use Newton’s third law to
compare the force of the satellite on the
Earth. Draw a picture similar to the ones I
drew for object A and object B.
• According to Newton’s second law,
compare the accelerations of the satellite
and Earth as a result of their interaction.
Deriving Kepler’s laws
• Newton’s law of gravitation, and Newton’s
second law (net force = mass x acceleration)
can be used to derive Kepler’s three laws of
planetary motion.
Summary
• Understand the importance of experiment.
– when theory does not explain measurements, a new hypothesis
must be developed; this may require a whole new model (a way of
thinking about something).
– know why the geocentric view was abandoned.
– know what experiments verified the heliocentric view.
• Understand the roles of the “characters” in the revolution
from a geocentric to a heliocentric model.
• Understand Kepler’s three laws of planetary motion
– these described the planet’s motions
• Understand Newton’s law of gravitation and the three laws
of motion
– these explain why Kepler’s three laws are “true”