Download Kristan Hemingway Planetary Motion If you are outside

Kristan Hemingway
Planetary Motion
If you are outside on a clear night it is almost inevitable to notice the always
changing array of lights in the sky. It has been an area of mystic from the beginning of
time. Whether you lived in 1500 BC, 1500 AD or year 2000 the night sky illuminates into
a canvas of mysterious lights. Throughout time very brilliant people have helped solve
this mystery and led the way to opening up deeper mysteries of the skies and Earth.
In ancient times people attributed things that happened on Earth to chance,
unforeseen forces or rather Gods acting out. The study of the night sky led to our
cosmological model and discoveries that explained things that happened on Earth. In a
sense, studying the sky eventually led to the technology revolution we all live in now.
Once the pattern of movement in the skies could be explained then the laws that govern
movement itself quickly followed. And once we have laws that govern motion we have
endless applications and discoveries that have the laws of motion as the foundation. For
example, today we enjoy vehicles. To have such technologies the understanding of
movement and forces is imperative. It simply explains how things work and gives us the
foundation of science and technology to make say the engine in the car.
Over years of experimenting with these discoveries has led us to the
technologies of today. We will take a look at how it started with the study of planetary
motion and how scientists took these ideas to explain our world.
Studying the sky throughout time has revolutionized the field of science. Until
modern times the stars seemed fixed in their positions in the sky so the early focus was
directed towards coming up with a science to explain the seemingly erratic behavior of
the planets that zoom through the sky. It wasn’t hard for early scientists to notice their
movement and begin to collect data that eventually led to a concrete way of explaining
and exploring planetary motion.
As with many sciences the ancient Greeks paved the way. It was still very
debatable before the 16th century whether or not the Earth was flat or round. A few
hundred years before A.D. times a famous Greek philosopher named Aristotle provided
two reasons as to why the Earth must be round. During a lunar eclipse the Earth’s
shadow on the moon was of circular shape. This is only possible with a spherical Earth.
The second argument involved the North Star. When headed North the North Star
would “sink” in the sky. If the Earth were flat, the stars positions would not change
based on a person’s location. These arguments were still not convincing enough for
many to change their ancient belief in a flat Earth. The overall acceptance of the idea of
a round Earth did not come until much later in time.
Aristotle had another idea that stuck for nearly 1,000 years. He along with many
before him believed that the Earth was at the center and that everything in the sky
revolved around it. Aristotle had nothing to arrive to the fact that the Earth was in motion
at all. His thought was that if it were moving then a ball thrown straight up would land
behind the person who threw the ball. So if the Earth was stationary and all the other
planets were in motion as seen in the night skies, the Earth must sit in the middle while
everything else revolved around it. This was common belief in ancient times, and it took
numerous scientific discoveries to prove otherwise. An ancient scientist, Ptolemy, first
published this Earth centered model in 150 AD. It is called the geocentric model.
Geocentric Model
But a new model was reintroduced by a Polish priest, Nicolaus Copernicus, in the
mid-16th century that revolutionized scientific thinking. The heliocentric, or Sun centered,
model allowed Copernicus to calculate the periods, relative distances and approximate
orbital shapes of all the known planets. There was one major problem with this
heliocentric model’s overall acceptance. The Church. Christian theology accepted the
geocentric model and in those times it was extreme and sometimes dangerous to
profess something that did not coincide with Christian philosophies. For instance,
scientist Giordano Bruno was charged of heresy and burned at the stake for teaching
heretical ideas including Copernicus’ heliocentric model. Those are serious
consequences.
Heliocentric Model
This model is very important and demonstrated cosmic order and allowed
Copernicus to more easily calculate distances of each planet to the Sun. For example,
let’s review his calculation of the distance from Venus to the Sun. With limited
technology and tools, Copernicus studied the planet’s brightness and position nightly to
determine when was the most accurate time to measure the distance. This occurs when
Venus is “half-lit” creating a 90 degree angle between Earth and the Sun as shown in a
following figure. Using the Astrononomical Unit, the distance from the Sun to the Earth
abbreviated AU, Copernicus could find the planet distances. With his model and
calculations he determined the then known planets distances with rather impressive
accuracy.
PLANET
Mercury
Venus
Earth
Mars
Jupiter
Saturn
COPERNICUS DISTANCE
0.376 AU
0.719
1.00
1.52
5.22
9.17
ACTUAL DISTANCE
0.387 AU
0.723
1.00
1.52
5.20
5.54
ܵ݅݊ሺ46 ݀݁݃‫ݏ݁݁ݎ‬ሻ =
‫ܦ‬௏௘௡௨௦
1 ‫ܷܣ‬
‫ܦ‬௏௘௡௨௦ = ܵ݅݊ሺ46 ݀݁݃‫ݏ݁݁ݎ‬ሻ
‫ܦ‬௏௘௡௨௦ = 0.719
Copernicus model used to find planet distances
But our journey to understanding planetary motion is far from done. Even with
Copernicus’ information the idea of the Sun at the center of our solar system was not
yet widely accepted. But further discoveries quickly attested to the heliocentric model. In
1610 the Italian scientist, Galileo Galilei, positioned his telescope to the sky at night and
for the first time in known history he saw that Jupiter had moons that orbited it. If the
geocentric idea was correct that all things orbited Earth then moons could not orbit only
Jupiter. But with the Christian Church still under strict control in Europe Galileo was also
reprimanded for his ideas and forced him to denounce his support for the heliocentric
model, and he was sentenced to house arrest for the last 8 years of his life.
The next revolutionizing development in understanding planetary motion comes
from a German scientist, Johannes Kepler. By studying data provided by another
scientist on the orbit of Mars Kepler arrived at the realization that the orbits of the
planets were not “circles within circles” as described by Copernicus and Aristotle but
that they were in fact ellipses. Going from circles to ellipses set the stage for creating
the laws of planetary motion.
Ellipses have several properties that one needs to understand before being able
to tackle Kepler’s Laws of planetary motion.
There are two foci (plural of focus) where the sum of the
distances from a point on the ellipse to the foci is constant
a + b = constant
Eccentricity refers to the “flattening” of the
ellipse. Ellipses have eccentricity between 0
and 1, with a circle having 0 eccentricity.
The major axis is the long axis while the
minor axis is the shorter axis. The average
distance from the Sun to a planet can be
shown to be the length of the semimajor (half
of the major axis) orbit axis.
Now we can take a look at the laws that govern the motion of the planets derived by
Kepler and known as “The Three Laws of Planetary Motion” or Kepler’s Laws:
Law 1: The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
Law 2: The line joining the planet to the Sun sweeps out equal areas in equal times as
the planet travels around the ellipse.
Law 3: The ratio of the squares of the revolutionary periods for the two planets is
equal to the ratio of the cubes of their semimajor axes:
௉భమ
௉మమ
=
ோభయ
ோమయ
These laws were not proved with strenuous mathematics equations or with modern
proof methods. Kepler’s laws originated from raw planetary position data he had access
to from scientist Tycho Brahe. Brahe had been logging observations because he knew
to figure out what laws if any governed the planets it would take a lot of data. Kepler
spent years studying this “night sky data” and eventually came to his conclusions
above. Kepler knew there would be error and assumptions along the way but took those
into consideration when mapping the planets orbits. The following is a more in depth
look at each law.
Kepler’s First Law
The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
Each orbit of the planets is an ellipse with very low eccentricity. This is one problem that
made it very difficult for scientists. The orbital eccentricity of most of the planets is so
low that it nearly looks like a circle. Earth has an eccentricity of 0.0167 while Mercury
has the highest of all the planets at 0.2056. Kepler tried fitting many shapes to the data
he had. He used a certain number of observations to find a fitting curve and then used
the curve to find more positions that he could check with his existing data. His concrete
idea of orbital ellipses crushed thousands of years of circles in astronomy.
Example of planet orbit around Sun
Kepler’s Second Law
The line joining the planet to the Sun sweeps out equal areas in equal times as
the planet travels around the ellipse. At first Kepler had to use close approximations
for finding the areas. He added up areas of triangles with their vertex at the Sun and
very small vertical angles. He used vertical angles of one minute of arc. One minute of
arc equals 1/60th of one degree so this does make for very small triangles. This method
of adding up areas of very small triangles mimics calculus but that field of mathematics
came after Kepler’s discoveries.
The point where a planet is nearest to the Sun is called perihelion. The point
where a planet is farthest from the Sun is called aphelion. Because the sweeping areas
are the same for equal times the angular speed of the planet cannot be constant. This
law says that the planet moves fastest when it is near the perihelion point and slowest
when it is near the aphelion point. The following images express these ideas.
Kepler used very small triangles
to find areas within the ellipse.
planet as it orbits
Sun
The two shaded parts represent the
areas swept by the line from the Sun to
the planet in equal times. Therefore, for
the areas to be equal the planet must
move faster when it is nearing the
perihelion.
Kepler’s Third Law
The ratio of the squares of the revolutionary periods for the two planets is
equal to the ratio of the cubes of their semimajor axes:
ࡼ૛૚
ࡼ૛૛
=
ࡾ૜૚
ࡾ૜૛
P is the period of revolution for a planet
R is the length of its semimajor axis.
From these ratios there are many calculations that can be done to piece together a
model of our solar system. This law implies that as a planet’s orbital radius increases so
does its period, or time it takes for the planet to orbit the Sun. For example, let’s
calculate the period of Mars based on its semimajor axis length. It is easiest to use
planet Earth as our other planet in the ratio because we can use the convenient unit of
measurement for the average separation of the Earth to the Sun, the AU. Also we can
use Earth years for our period unit. This keeps our calculation very simple. The length of
the semimajor axis for the orbit of Mars is 1.52 AU. Thus,
௉ಾೌೝೞ మ
ଵ ா௔௥௧௛ ௬௘௔௥ మ
=
ሺଵ.ହଶ ஺௎ሻయ
ሺଵ ஺௎ሻయ
ܲெ௔௥௦ ଶ = ሺ1.52 ‫ܷܣ‬ሻଷ
ܲெ௔௥௦ = 1.88 Earth years
So it takes Mars 1.88 Earth years (i.e. ~ 686 days) to make a complete orbit around the
Sun.
Kepler’s discoveries described or rather defined the motion of the planets. This
was a jumpstart to the field of science that defines motion itself. Isaac Newton took
Kepler’s work to a whole other level when he invented calculus and discovered the law
of universal gravitation.
It is safe to say that Isaac Newton was a genius. He was so gifted in
Mathematics that it took years for top scholars to understand it all. And his giftedness
seemed obvious at a young age. As a boy Newton had many inventions such as a kite
with an enclosed lantern which appeared as a “ghost” to the locals. He also created a
mill that ground wheat into flour using the power of a mouse!
Newton attended Cambridge University, and it is here that he was introduced to
the formal science of mathematics. He quickly mastered Euclid’s and Descartes’ work
and by the end of his second year at the university he took the place of his professor
who resigned in light of Newton’s superior skill. Now that is smart!
The physical principles that Newton discovered several hundred years ago can
be used to describe the motion of an aircraft today. Newton’s laws of motion again
revolutionized the field of sciences. He studied Kepler’s laws and the work of other
scientists such as Galileo and Descartes to form his laws that define motion. Newton
realized that all motion followed the same rules. The orbit of the planets and an apple
falling off of a tree follow the same principles of motion. In a sense Newton unified all
motion with three laws he discovered during the times of the Great Plague in England.
Newton’s First Law
Every body perseveres in its state of rest, or of uniform motion in a right line,
unless it is compelled to change that state by forces acting upon it. This is also
known as the definition of inertia. This law tells us that a moving object will not change
speed or direction unless an outside force acts on it. Also, an object at rest or having a
velocity of zero will not move unless an external force is applied causing the velocity
and direction of the object to change. We will follow an example involving a rocket to
exemplify this law. The net force is what causes the rocket to go and in which direction.
Newton’s Second Law
The alteration of motion is proportional to the motive force impressed; and is
made in the direction of the right line in which that force is impressed. Newton’s
calculus discovery plays heavily into this law. The law defines a force to be equal to the
change in momentum. Momentum is the mass of an object, m, times its velocity, v.
Force can be defined in differential form as
F=
ௗሺ௠௩ሻ
ௗ௧
OR F =
௖௛௔௡௚௘ ௜௡ ௠௢௠௘௡௧௨௠
௖௛௔௡௚௘ ௜௡ ௧௜௠௘
This law is more commonly known in its mathematical form when the mass is constant:
F = ma
where m is the object mass and ‘a’ is the acceleration of the object (i.e. change in
velocity with time). When an external force is applied to an object, the change in velocity
(keeping in mind that velocity includes speed and direction) depends on the mass of the
object. A force causes a change in velocity just as a velocity change generates a force.
Newton’s Third Law
To every action there is always opposed an equal reaction: or the mutual actions
of two bodies upon each other are always equal, and directed to contrary parts. In
other words, for every action (or force) in nature there is an equal and opposite reaction.
As described by Newton, “If you press a stone with your finger, the finger is also
pressed by the stone.” This can easily be seen in the following image where there is a
force that pushes the rocket forward and a “balancing” force that pushes exhaust
backward.
Newton also presented the idea of universal gravitation. He used this gravitation
as a study of his laws of motion. He claimed that all matter exerts a force that pulls other
matter towards its center. This force is what he called gravity. Gravity’s strength
depends on the mass of the object and the distance apart of the objects. The Sun
exerts a “pulling force” on the Earth as it travels which is what makes its path bend to
orbit the Sun instead of moving in a straight forward path. This gravitational theory also
explains the ocean tides as they are created by the gravitational pull of the Moon as it
orbits Earth.
Isaac Newton’s discoveries were the basis for all cosmological models and were
unchallenged until the 20th century when Albert Einstein presented his theory of
relativity in 1905. Newton revolutionized the field of mathematics with his discoveries
which led to modern day mathematical advances. All real world physics applications use
these laws as the foundation of motion. In other words, with understanding these motion
laws scientists can explain natural happenings such as the tides, explore aircraft
designs and put a man on the moon.
The study of planetary motion was the start to a revolution in the field of science
and mathematics. It led straight into understanding natural forces and how objects
moved. Once we understood how things moved and were able to use equations for this
old technologies and sciences were set up perfectly to evolve very quickly from the 19th
century through today.
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