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Chapter 5
Gravity and Motion: The Early History of Modern Astronomy
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Ancient people knew five planets – Mercury, Venus, Mars, Jupiter, and Saturn.
Observing these planets in the sky, we, as people long ago, notice that the positions of the planets
vary from night to night with respect to each other and with respect to the stars.
The word “planet” comes from the Greek word for “wanderer”.
Resulting from Earth’s rotation about its axis, both stars and planets move together across our sky
essentially once every 24 hours.
By the wandering of the planets we mean that the planets appear to move at a very slightly
different rate.
Over weeks and months, we notice them changing position with respect to the fixed stars.
The Earth Centered Astronomy of Ancient Greece
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Most of the time planets appear to drift slowly in one direction with respect to the background
stars.
This forward motion, moving slightly slower in the sky that the stars as they rise and set, is called
prograde motion.
Sometimes a planet drifts in the opposite direction with respect to the stars. This is called
retrograde motion.
The ancient Greeks made theoretical models of the Solar System in order to explain the motion of
the planets.
By comparing the length of the periods of retrograde motion of different planets, they were able
to discover the order of distance of the planets.
One of the earliest and greatest philosophers, Aristotle, lived in Greece in about 350 B.C.E. His
views of the Universe dominated thinking for 1800 years.
Aristotle believed that the Earth was the center of the Universe and that the planets, the Sun, and
the stars revolved around it.
In Aristotle’s model the Universe was made up a set of 55 celestial spheres that fit around each
other. Each sphere’s natural motion was rotation.
The planets were carried around by some of the spheres, and the motion of the spheres affected
the other spheres.
Some retrograde motion could be accounted for in this way.
The outermost sphere was that of the fixed stars. Outside this sphere was the “prime mover” that
couased the rotation of the stars.
Aristotle’s theory dominated scientific thinking until the Renaissance.
His theories were accepted so completely that they may have impeded scientific work that might
have led to new theories.
About 400 years after Aristotle, the Greek scientist Claudius Ptolemy presented a detailed theory
of the Universe that explained the retrograde motion.
Ptolemy’s model was geocentric – Earth centered – as was Aristotle’s.
To account for the retrograde motion, Ptolemy’s model had the planets traveling on small circles
that moved on the larger circles of the planets’ overall orbits.
The small circles are called epicycles and the larger circles are called deferents.
Since circles were thought to be “perfect” shapes, it seemed natural that planets should follow in
their motion.
Ptolemy’s views were very influential in studying astronomy for a long time.
His tables of planetary motion were accepted for nearly 15 centuries. His major work became
known as the Almagest (the Greatest).
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21. It contained not only his ideas but also a summary of the ideas of his predecessors.
22. Most of our knowledge of Greek astronomy comes from Ptolemy’s Almagest.
Ptolemaic Terms
Deferent – a large circle centered approximately on the Earth, actually centered midway between the
Earth and the equant
Epicycle – a small circle whose center moves along a deferent. The planets move on the epicycles
Equant – the point around which the epicycle moves at a uniform rate
Ptolemaic Model of the Universe: http://www.youtube.com/watch?v=faqjmAoXpM4&feature=related
Copernicus and the Sun Centered Universe
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Nicolaus Copernicus, a 16th-century Polish astronomer is credited with the modern view of the
Solar System and of the Universe.
Copernicus suggest a heliocentric (Sun centered) theory.
He explained the retrograde motion with the Sun rather than the Earth at the center of the Solar
System.
Copernicus heliocentric theory assumed that the planets orbited in perfect circles, though the
circles were not all centered on the Sun.
The notion that celestial bodies had to follow perfect shapes shows that Copernicus had not
entirely broken with old ideas.
Copernicus’s manuscript had the mechanisms that drove the planets housed in zones. The notion
of orbits did not yet exist.
Copernicus still invoked the presence of some epicycles in order to have his predictions agree
better with observations.
The detailed predictions that Copernicus computed were not in much better agreement with the
existing observations that tables based on Ptolemy’s models.
Copernicus often used Ptolemy’s tables.
They did not have the standards we now have of how to compare observations with theory.
The heliocentric theory appealed to Copernicus mostly on philosophical rather than on
observational grounds.
Aristarchus of Samos, a Greek scientist, had suggested a heliocentric theory 18 centuries earlier,
although we do not know how detailed a picture he presented.
His ancient heliocentric suggestion required the then apparently unaccepted idea that the Earth
itself moved.
This idea seemed to contradict our senses as well as the theories of Aristotle. (If the Earth is
rotating, why aren’t birds and clouds left behind.
It was not until the 17th-century discovery by Isaac Newton of laws of motion that solved this
dilemma.
Copernicus published his theory in 1543 in the book De Revolutionibus (Concerning the
Revolutions).
The theory explained the retrograde motion of the planets as follows. First consider a planet that
is farther from the Sun than the Earth.
For Mars and other such planets, observe what happens when the Earth approaches the part of its
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orbit that is closest to Mars.
19. The farther away from the Sun the planet, the more slowly it revolves around the Sun.
20. As the Earth passes Mars, the projection of the Earth-Mars line outward to the stars moves
backward with the way it had been moving. Then, as the Earth and Mars continue around their
orbits, Mars appears to go forward again.
21. A similar analysis holds for Mercury and Venus, which orbit the Sun at smaller distances than
does the Earth.
22. The idea that the Sun was at the approximate center of the Solar System led Copernicus to two
additional important results.
a. First, he was able to work out the relative distances to the planets.
b. Second, he was able to derive the length of time the planets take to orbit the Sun from
observations of how long they take between appearances in the same place in our sky.
c. These two derivations played a large part in persuading Copernicus that his heliocentric
system was better than the geocentric model.
Ptolemaic Model vs. Copernicus/Galileo View of the Universe:
http://www.youtube.com/watch?v=GmwAr54L_pM&feature=related
Tycho Brahe
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In the last part of the 16th century, not long after Copernicus’ death, Tycho Brahe began to
observe Mars and the other planets.
Brahe set up an observatory on an island off the mainland of Denmark.
The telescope had not yet been invented, Tycho used giant instruments to make observations that
were unprecedented in their accuracy.
In 1597 Tycho lost his financial support and 2 years later arrived in Prague where Johannes
Keppler came to work with Tycho as a young assistant.
After Tycho’s death in 1601, Kepler was left to analyze all the observations Tycho and his
assistants had made.
Tycho Brahe and Johannes Kepler Video:
http://www.youtube.com/watch?v=xsX6BeBOorA&feature=related
Johannes Kepler
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Johannes Kepler studied with one of the first professors to believe in the Copernican view of the
Universe.
Kepler came to agree and made some mathematical calculations involving geometrical shapes.
His ideas were wrong, but his mathematical skills caught the attention of Tycho Brahe.
Tycho Brahe’s observational data showed that the tables of the positions of the planets then in use
were not very accurate.
When Kepler joined Tycho, he carried out detailed calculation to explain the planetary positions.
In the years after Tycho’s death, Kepler succeed in explaining the orbit of Mars.
But he could only do so by dropping the idea that the planets orbited in circles.
Kepler’s First Law
Kepler’s first law published in 1609, says that the planets orbit the Sun in ellipses, with the Sun at
one focus.
2. An ellipse is defined in the following way: Choose any two points on a plane – these points are
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called the foci (each is a focus).
From any point on the ellipse, draw two lines, one to each focus. The sum of these lengths of
these two lines is the same for each point on the ellipse.
The major axis of an ellipse is the line within the ellipse that passes through the two foci, or the
length of that line.
The minor axis is the perpendicular to the center of the major axis and the semimajor axis is half
the length of the major axis, and the semiminor axis is half the length of the minor axis.
A circle is the special case of an ellipse where the two foci are in the same place.
The shape of an ellipse will change if you change the distance between the foci.
By Kepler’s first law, the Sun is at one focus of the elliptical orbit of each planet. We say that
there is nothing at the other focus, we say that it is empty.
Kepler’s Second Law
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Kepler’s second law describes the speed with which the planets travel in their orbits.
It states that the line joining the Sun and a planet sweeps through equal areas in equal times.
It is thus also known as the laws of equal areas.
When a planet is at its greatest distance from the Sun as t follows an elliptical orbit, the line
joining it with the Sun sweeps out a long, skinny sector.
This sector has two straight lines starting at a focus and extending to the ellipse.
The curved part of the ellipse is relatively short for a planet at its great distance from the Sun.
On the other hand, by Kepler’s second law, the area of this long, skinny sector must be the same
as that of any short, far sector formed for the planet near perihelion, the closest point to the Sun in
a planet’s orbit.
Thus, the planet moves most quickly when it is near the Sun.
Since the time of the Greeks, the idea that the planets travelled at a constant rate had been
thought to be important.
Kepler’s second law replaced this old idea with the idea that the total area swept out change at a
constant rate.
Kepler’s second law is especially noticeable for comets, which have very eccentric (flattened)
elliptical orbits.
Kepler’s second law demonstrates why Halley’s Comet sweeps so quickly through the inner part
of the Solar System.
It moves much more slowly when it is farther from the Sun, since the sector swept out by the line
joining it to the sun is skinny but long.
Kepler’s Third Law
Kepler’s third law deals with the length of time a planet takes to orbit the Sun, which is its period
of revolution.
2. Kepler’s third law relates the period to some measure of the planet’s distance from the Sun.
3. It states that the square of the period of revolution is proportional to the cube of the semimajor
axis of the ellipse.
4. That is, if the cub of the semimajor axis of the ellipse goes up, the square of the period goes up by
the same factor.
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It is often easiest to express Kepler’s third law by relating values for a planet to values for the
Earth. If P is the period of revolution and R is the semimajor axis.
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PPlanet2PEarth2=RPlanet3REarth3
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We can choose to work in units that are convenient for us here on Earth.
We use the term 1 Astronomical Unit (1 A.U.) to mean the semimajor axis of the Earth’s orbit.
This semimajor axis can be show to be the average distance from the Sun to the Earth.
We use a unit of time based on the length of time the Earth takes to orbit the Sun: 1 year.
When we use these values, Kepler’s law appears in a simple form, since the numbers on the
bottom of the equation are just 1 and we have:
PPlanet2=RPlanet3 (distance in A.U.;period in Earth years
Example – We know from observation that Jupiter takes 11.86 years to revolve around the Sun. What is
Jupiter’s average distance from the Sun?
Answer: 11.862=RJupiter3
11. The period with which satellites revolved around bodies other than the Sun follows Kepler’s law
as well.
12. The laws also apply to artificial satellites in orbit around the Earth and to the moons of other
planets.
Kepler’s Laws Summary
To this day, we consider Kepler’s laws to be the basic description of the motions of the SolarSystem objects.
2. The planets orbit the Sun in ellipses, with the Sun at one focus.
3. The line joining the Sun and a planet sweeps through equal areas in equal times.
4. The square of the period of a planet is proportional to the cube of the semimajor axis of its orbit –
half the longest dimension of the ellipse (Loosely, the squares of the periods are proportional to
the cubes of the planets’ distances from the Sun.)
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Kepler’s Three Laws: http://www.youtube.com/watch?v=GcKiG-CuvtA&feature=related
Galileo Galilei
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Galileo Galiei worked in what is now Italy at the same time that Kepler was working in Prague.
Galileo began to believe in the Copernican heliocentric system in the 1590s and identified some
of our basic laws of physics in the following years.
In late 1609 or early 1610, Galileo was the first to use a telescope for systematic astronomical
study.
In 1610, Galileo reported in a book that with his telescope he could see many more stars than he
could with his unaided eye.
He saw individual stars that make up the Milky way, described views of the Moon, including the
discovery of mountains, craters and the relatively dark “seas.”
Very important was his discovery that small bodies revolved around Jupiter. This discovery
proved that not all bodies revolved around the Earth. Galileo showed that Aristotle and Ptolemy
had not been omniscient and opened the idea that more remained to be discovered.
Critical to the support of the heliocentric model was Galileo’s discovery that Venus went through
and entire series of phases.
The noncrescent phases could not be explained with the Ptolemaic system.
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After all, if Venus traveled on an epicycle located between the Earth and the Sun, Venus should
always appear as a crescent (or new). Thus, the complete set of phases of Venus provided the
fatal blow to the Ptolemaic model.
In 1612 Galileo discovered sunspots, more evidence that the celestial objects were not perfect. In
his book on sunspots, Galileo drew a series of photographs showing how sunspots looked from
day to day as the rotated with the Sun’s surface. Galileo’s contemporaries did not accept his ideas
or accept the use of the telescope as a legitimate scientific tool.
Galileo’s discoveries caused controversy with the Roman Catholic Church (Galileo was a Roman
Catholic), during which he was sentenced to house arrest.
In order to avoid being burned at the stake as a heretic like his contemporary Giordano Bruno for
his views about worlds beyond our Solar System.
The controversy still echoes to this day. People continue to search for evidence in the Vatican
archives for perhaps the real reasons behind the fight. There is now peace between the Roman
Catholic Church and scientists , and the Vatican supports a modern observatory and respected
contemporary astronomers.
Isaac Newton
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Kepler discovered his laws of planetary orbits by trial and error and he had no physical
understanding of them.
Only with the work of Isaac Newton 60 years later did we find out why the laws existed. Newton
was born in England in 1642, the year of Galileo’s death.
He became the greatest scientist of his time and perhaps of all time.
His work in optics, his invention of the reflecting telescope, and his discovery that visible light
can be broken down into a spectrum, and his work on motion and on gravity.
Newton set modern physics on its feet by deriving laws showing how objects move on Earth and
in space, and by finding the law that describes the force of gravity.
Newton invented calculus. Newton long withheld publishing his work until he was urged to do so
by Edmond Halley (whose name is associated with Halley’s Comet). Newton’s Principia
appeared in 1687.
The Principia contains Newton’s three laws of motion.
a. The first is that bodies in motion tend to stay in motion, in a straight line with constant
speed, unless acted upon by a force. It is a law of inertia, which was really discovered by
Galileo.
b. Newton’s second law relates a force with its effect on accelerating a mass. A larger force
will make the same mass accelerate faster (F=ma, where m is the mass and a is the
acceleration).
c. Newton’s third law is often stated “For every action, there is an equal and opposite
reaction.” The Principia also contains the law of gravity.
d. One application of Newton’s law of gravity is the weighing of planets.
One of the most popular tales about Isaac Newton is that an apple fell on Newton’s head, leading
to the discovery of the concept of gravity.
a. No apple actually hit Newton’s head, the story Newton told himself, is that he saw an
apple fall and realized that just as the apple fell to Earth, the Moon is falling to Earth,
through its forward motion keeps it far from us.
b. When he calculated that the acceleration of the Moon fit the same formula as the
acceleration of the apple, he knew that he had the right method.
c. Newton was the first to realize that gravity was a force that acted in the same way
throughout the Universe.
d. Many pieces of the original apple tree have been enshrined. Newton’s house still stands
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in Woolsthrope, Lincolnshire, England which is now a museum.
Apple trees grow in front of the house, but whether they are descendents of the famous
tree is unknown.
Newton’s Version of Kepler’s Third Law
Newton derived the formula for Kepler’s third law (the period of a planet squared is proportional to its
distance from the Sun cubed (P2=constant x R3)) taking into account of his own formula for the law of
universal gravitation. Newton found that
P2=4π2Gm1+m2R2
where m1and m2 are the masses of the two bodies. For planetary orbits, if m1 is the mass of the Sun, then
m2 is the mass of the planet.
Newton’s formulation of Kepler’s third law shows that the constant of proportionality is determined by
the mass of the Sun. Thus, we can use the formula to find out the Sun’s mass. Newton’s version of
Kepler’s third law, as applied to any planet’s orbit around the Sun, also shows that the planet’s mass
contributes to the value of the proportionality constant, but its effect is very small because the Sun is so
much more massive. The effect can be
detected even today for only the most massive planets – Saturn and Jupiter. Newton therefore actually
predicted that the constant in Kepler’s third law depends slightly on the mass of the planet, being different
for different planets. The observational confirmation of Netwon’s prediction was a great triumph for
physicists and astronomers.
Kepler’s third law and its subsequent generalization by Newton, applies not only to planets orbiting the
sun but also to any bodies orbiting other bodies under the control of gravity. Thus it also applies to
satellites orbiting planets. We determine the mass of Earth by studying the orbit of our Moon, and
determine the mass of Jupiter by studying the orbits of its moons. Until recently, we were unable to
reliably determine the mass of Pluto because we could not observe a moon in orbit around it. In 1978
Pluto’s moon Charon was discovered and it allowed us to accurately determine Pluto’s mass. It was
discovered that our previous estimates which were based on Pluto’s now-discredited gravitational effects
on Uranus, were way off. The same formula can be applied to binary stars to find their masses.
Orbital Speed of Planets
The more distant a planet is from the Sun, the slower its orbital speed. We can derive this from Kepler’s
third law. Since most of the planets have nearly circular orbits, a reasonable approximation for our
purposes is to assume that the distance travelled by a planet in one full orbit is 2πR, the circumference of
a circle is radius R.
If the planet travels at a constant speed (which it would if the orbit were exactly circular), the distance
travelled would equal its speed multiplied by time: d=vt. Applying this to one full orbit, we find that
2πR=vP, where P is the planet’s period. Thus, P=2πRv, so
P ∝ Rv, where the symbol ∝ means “proportional to.”
But Kepler’s law states that P2 ∝ R3. When we substitute the expression P ∝ Rv for P into Kepler’s law,
we get Rv2 ∝ R2. Rearranging gives v ∝1R, and hence
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v∝1R12
In words, the speed of a planet is inversely proportional to the square root of its orbital radius, distant
planets move more slowly than those near the Sun. For example, a planet whose orbital radius is 9 times
that of the Earth’s has an orbital speed that is 1912=13 as fast as the Earth’s orbital speed, which itself is
about 30 km/sec.
This “this inverse-square-root” relationship is characteristic of systems in which a large central mass
dominates over the masses of orbiting particles.
The Rotation and Revolution of the Planets
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Studying the orbits of the planets shows that the Solar System has some basic properties.
a. For example, all the planets move in the same direction around the Sun and are more or
less in the same plane.
Considering these regularities gives important clues as to how the Solar System may have
formed.
As we study other planetary systems being discovered, it will be of interest to see how many
things about our own Solar System are truly fundamental, and how many may be specific to our
case.
The motion of a planet around the Sun in its orbit is its revolution.
The spinning of a planet is its rotation.
a. The Earth, for example, revolves around the Sun in one year and rotates on its axis in one
day.
b) According to Kepler’s third law, planets far from the Sun have longer periods
of revolution (orbital periods) than planets close to the Sun. The distant planets also move
more slowly.
The orbits of the planets take up a disk rather than a full sphere.
The inclination of a planet’s orbit is the angle that the plane of its orbit makes with the plane of
the Earth’s orbit.
The fact that the planets all orbit the Sun in roughly the same plane is one of the most important
generalities about the Solar System.
Spinning or rotating objects have a property called angular momentum.
a. This angular momentum is larger if an object is more massive, or if mass is farther from
the axis of spin.
b. One important property of angular momentum is that the total angular momentum of a
system doesn’t change (it is conserved), unless the system interacts with something from
outside that brings in or takes up angular momentum.
c. Thus, if one body were to spin more and more slowly, or were to begin spinning in the
opposite direction, the angular momentum would have to be taken up by another body.
We will apply this idea to see how the planets undoubtedly all revolve around the Sun in the same
direction because of how they were formed.
The Formation of the Solar System
We can accurately date the formation by studying the oldest object we can find in the Solar
System and allowing a little more time.
2. For example, astronauts found rocks on the Moon older than 4.4 billion years. It is accepted that
the Solar System formed about 4.6 billion years ago.
3. The best current idea is that 4.6 billion years ago a huge cloud of gas and dust in space collapsed,
pulled together by the force of gravity. What triggered the collapse isn’t known.
4. The Solar System began to spin faster as it collapsed. The original gas and dust may have had
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some spin, and this spin is magnified in speed by the collapse, because the total angular
momentum doesn’t change.
Objects that are spinning tend to fly off, and this force eventually became strong enough to
counteract the effect of gravity pulling inward.
Thus the Solar System stopped collapsing in one plane. Perpendicular to this plane, there was no
spin to stop the collapse, so the solar nebula would be a disk.
In the disk of gas and dust, we think that material began to clump.
Smaller clumps joined together to make larger ones, and eventually planetesimals, bodies a few
hundred kilometer across, were formed.
Gravity subsequently pulled many planetesimals together to make protoplanets orbiting a
protosun.
The protoplanets then contracted and cooled to make the planets we have today, and the protosun
contracted to form the Sun.
Some of the planetesimals may still be orbiting the Sun, that is why we are interested in studying
small bodies of the Solar System, like comets, meteorites, and asteroids.
Most of the unused gas and dust was blown away by a strong solar wind when the Sun was young
and very active.
In one of the models for the formation of the outer planets, the solar nebula first collapsed into
several large blobs. These blobs then became the outer planets.
If this model is correct, the outer planets should all contain about the same relative amounts of the
different elements as each other and as the space between the stars from which the gas came.
The gravity of this core then attracted the gas from around it. For this second model, the relative
amounts of elements in the rocks and in the gasses could be different from planet to planet.
a. The Voyager spacecraft found that Jupiter, Saturn, Uranus, and Neptune have different
relative amounts of some of the elements in their atmospheres.
b. Also, the atmospheres of Jupiter and Saturn are very much more massive than the
atmospheres of Uranus and Neptune.
c. On the other hand, their cores all contain about the same amount of mass – 10 to 20 times
the mass of the Earth.
Thus it seems that the outer planets assembled in several stages, with the cores forming first.