From the Family Tree of

Old School Astronomy

Lorraine, Dan, Maxwell, Binnie,

& Jeremy

"A Work of Art is Never Completed."

Introduction:

When our group was presented with the idea of creating a student-generated lab, we were baffled. What could our group do? The ideas mentioned to us seemed to have little interest to our group, and they did not seem like they would even hold that much challenge or fun. Then Hays came to our rescue. He proposed the idea of proving Kepler's laws of orbital motion. It seemed interesting, challenging, and it might even be fun. After heated debate, dominated by the all powerful Pribble, a functioning hypothesis was formed.

Hypothesis:

In this lab, we will prove Kepler’s Third law of planetary motion using the orbits of the moons of Jupiter as a model for all orbiting bodies.

So what does this mean? What exactly do we plan to accomplish with this lab? Beyond just the hypothesis, our group hopes to have an idea of how to go about creating a lab, and using experiments as well as using materials, resources to draw conclusions from data. We chose this project because learning about the stars; moons and planets, using a telescope of immense power and working late at night seemed like an interesting idea. It seemed that this research, and what we find would have an important impact on how we view the world around us. Kepler’s Laws have a huge historical significance, as they shaped the way we look at the universe today. That is the reason that this lab has such importance. Although it has been done before, this lab will help us appreciate what Kepler and his contemporaries did. We are using modern day technologically advanced equipment, and still we have problems, as Lorraine puts it, "He was looking through a cup, the guy is a genius. We are having a hard time, and we have a telescope. This just shows how much we can learn from history. So, lets learn…this is just a little glimpse at the man and the work behind it all.

Background Info:

In order to understand what led Kepler to formulate his laws of planetary motion, one must first understand the knowledge Kepler acquired as an astronomer from scientists before him. Moving in to 2AD, we recognize the Greek philosopher Ptolemy. His model of the heavens depicted the earth, still, as the center of an immaculate, divine universe. This "Ptolemaic system," put the earth at the center of eight concentric spheres with the moon, Mercury, Venus, the sun, Mars, Jupiter, Saturn, and the stars.

In this theory, it was believed that the eighth sphere of the stars was the only sphere that had been disclosed to man by God. Since the Christian church had adopted the Ptolemaic model, contrasting ideas of the universe as a heliocentric system (placing the sun at the center of the universe) were rejected consistently through the course of the next couple centuries. The expression of contrasting theories would lead to excommunication and exile, and subsequently, there were few voices that spoke loudly of alternate speculations. Even Copernicus, who was a priest as well as an astronomer, originally only posted his Heliocentric Theory anonymously, and only put his name to it shortly before his death.

After the death of Copernicus, his ideas of a heliocentric universe were quickly forgotten. Part of the reason for this was that other scientists of the day were still proponents of the geocentric model (earth is the center of the universe), and were turning out data that supported this theory. They were able to ignore the flaws in this model in favor of having a more perfect system. One of the most influential of these scientists was the notoriously meticulous, Tycho Brahe. Brahe’s work seemed to prove to him that the geocentric model was correct. Ironically enough, it would be Brahe’s data that Kepler would use to formulate his three laws, and prove beyond a doubt that geocentrism was wrong.

When Brahe died, he left his data to his assistant, Kepler. Kepler and Galileo were both proponents of Copernicus’ heliocentric model. Using Brahe’s data, and building on the data gathered by Galileo from his observations of Venus and the moons of Jupiter, Kepler derived his three laws. These laws are:

* All planets move in the shape of an ellipse with the sun at one focus,

* A line drawn from the planet to the sun sweeps out equal areas over equal time.

* The square of the planet’s period of revolution is proportional to the cube of the planet’s mean distance from the sun.

Kepler’s laws gained exceptence through the work of Galileo. Kepler had wanted to believe that the sun would be at the center, but that it would be surrounded by spheres, the perfect shape. Because of his devout Lutheran upbringing, Kepler thought that there could be a scientific explanation to the universe, but that it would also have a divine or perfect sense to it. When Kepler found that the planets in fact moved in the shape of an ellipse he was immensely relieved, as the ellipse was believed to be the next most perfect curve to the circle. Kepler’s third law took the longest to gain exceptance, as it could not be properly proven or explained until the advent of Newton and his laws of gravity.

Kepler’s first law clearly defines the shape of the planet’s orbit. An ellipse is a figure that can be defined geometrically or algebraically. To begin with, an ellipse must have two points within its boundaries known as foci. An ellipse is defined as the set of all points such that the distance from one focus to the edge of the ellipse and back to the other focus is always a constant number. Algebraically, an ellipse is defined by the equation x2/a2 + y2/b2 = 1. By Kepler’s first law, we see the sun as one of the foci in each planet’s elliptical orbit. By this definition of an ellipse, we can easily come to the derivation of the circle. Once the foci have moved to occupy the exact same point, the distance from that point to the outside will always be constant and we will have a circle. The orbits of planets are amazingly close to being circles, with the sun being very close to what would be the center of the orbit, however, it is off just enough that we can’t consider it to be a perfect circle.

Kepler’s second law governs the motion of the planets. In a circular orbit, one could determine that the planet would move at a uniform speed throughout its orbit. However, in an elliptical orbit this cannot be the case. The second law states that a line drawn from the planet to the sun sweeps out equal area over equal time. This implies that the planet must move faster when it is closer to the sun to form a shorter squatter wedge, and slower when it is farther away to form a longer, skinnier wedge. This shows that the motion of the planet simply cannot always be uniform.

Kepler’s third law is perhaps the most complex of the three. It deals with the period of time that the revolution of planet takes, compared with its mean distance from the sun. According to Kepler’s third law, p2 = km3 where p is the period of the planets revolution, m is the mean distance of the planet from the sun, and k is an arbitrary constant. This law can only be explained by gravity, which is why it was not commonly excepted until the 17th century when the work of Isaac Newton was beginning to become accepted as well.

Newton

Newton’s first and second laws of motion and his law concerning gravity are of vast importance when attempting to explain Kepler’s laws.

- The first law states that a body at rest will stay at rest until an outside force acts upon it, and conversely a body in motion will stay in motion until and outside force acts upon it. In space there is no friction, so there is no force that is stopping the motion of the planets, that is why they are in indefinite motion.

- The second law states that when a force acts upon a body, it will accelerate that body in the direction of the applied force. The magnitude of this acceleration is directly proportional to the magnitude of the force, and inversely proportional to the mass of the body. In the case of the planets, there are two forces acting of the planet. The first is centripetal force, that which pulls the body to fly in a straight line, and gravity, which is constantly pulling the body inward with a force equal to (GMmr)/r3. Where G is the gravitational constant of the system, M is the mass of the larger object, m is the mass of the smaller object, r3 is the distance between the two cubed, and r is the vector representation of the force. These two forces keep the planets traveling in their elliptical orbits. Newton’s second law explains this.

Jupiter

Jupiter is one of the most amazing planets in our solar system. It is a planet, which has stirred interest in the hearts and minds of humankind for years. People have been studying this mystifying planet of rings for years. So what is Jupiter really like? Why are people so fascinated with this red planet? To begin with one must consider the fact that Jupiter is the largest planet in our solar system measuring up to about 318 times the size of the earth. While in orbit Jupiter is 483 million miles from the sun and orbits the sun once every 11.9 earth years. The diameter of Jupiter is recorded as an astounding 89,000 miles, while it’s gravity is 2.64 times the gravity of the earth. This is just the simple measurement of the planet, and yet already one can see why so much study has taken place in regards to this planet. One of the most interesting and fascinating features of this planet is the rings that are visible around the planet. In actuality there are two sets of rings: light bands, and dark bands. The dark bands are thought to be lower layers of compounds of sulfur. The light bands are said to be a result of rising gas that mostly contains a gas form of ammonia. Both of these bands insinuate the idea of different layers of a dense atmosphere. Jupiter is also pretty famous for a phenomenon of oval features, especially a specific anomaly known as the Great Red Spot. It is believed that these oval features are really enormous columns of swirling gas, otherwise known as storms, in Jupiter’s atmosphere. The most famous of these, the Great Red Spot, is estimated to be an over 300-year-old storm on the surface of the planet. The last pertinent bit of information that we can instill on you is the fact that Jupiter is known as a brown dwarf. This name was given to the planet due to the fact that although its core temperature never exceeded 50,000 degrees, it happens that the energy that it receives from the sun is less than the energy that it produces. Even today though the iron core of Jupiter is only 25,000 degree’s. The final bit of information that makes Jupiter so fascinating concerns its four moons, Io, Europa, Ganymede, and Callisto. These are the four moons that Galileo observed to reinforce Kepler's Laws and therefore the idea of a Heliocentric system.

Io is the closest of Jupiter's four major moons. Galileo discovered it in 1610 with the help from S. Marius. The name of Io comes from Greek mythology. The name Io is the name of a woman who loved Zeus and was victim to ugly jealousy by Zeus’s wife Hera. Io is known for its bright colors, and active volcanoes. The atmosphere of Io is made almost entirely of sulfur, although this changes depending on how active the volcanoes are. Currently Io is undergoing intense stress from Jupiter. It is this pressure which may be causing these volcanoes. The moon Io is only slightly larger than earth’s moon.

Europa on the other hand is slightly smaller than the Earth’s moon, and the smallest of the Galilean Moons. One very interesting aspect of this moon is that it has an icy surface, making it the smoothest object in the entire universe. The surface of this moon has five times the illumination of the Earth’s moon; in addition, it is an exceptionally flat satellite where nothing exceeds one meter in height. Scientists concur that there is an ocean on the surface of Europa, and that the lack of craters makes it very young. They also feel that what cause an ocean to exist on Europa, is the tidal pulling that results from the gravitational pull of Jupiter. Within this ocean of melting ice, scientists believe that life can exist similar to the life that exists at the bottom of the oceans on Earth. This, however, remains unproven. The ice that has been identified on Europa is cracked, forming geysers along the fault lines. It is these guesses the lead scientists to believe that Europa’s crust is 150 kilometers thick. Being that Europa has water and energy source, the possibility of life on the moon is heightened. Laboratory tools have detected complex molecules further assisting in the proving of this "life" theory. Europa is the only Galilean satellite on which life can exist.

Ganymede, named after one of Jupiter’s many lovers from Greek/Roman

Mythology, was discovered by Galileo in 1610. Out of Jupiter’s 16 moons, it is the seventh in sequence from nearest to farthest, being 670,900 km away. Much larger than the Earth’s moon, it is the largest moon in the solar system. The diameter of Ganymede is approximately equal to the distance across the United States, about 5 262 km. This moon is an ice covered satellite as well, however it lacks a grooved terrain. With no evidence of volcanism, scientists believe that there exists a type of surface motion or tectonic activity. Investigating the surface of Ganymede, scientists have discovered many faults and fractures, providing evidence of stress which the surface of this moon as sustained over time. There is also evidence of continental rifting, also called graben style faulting. Through the examination of Ganymede scientists have been able to discover that the progressive deformation of the surface of the moon results from enormous blocks of the moon’s crust being pulled apart. This process where large blocks are pulled apart is in effect, similar to terrestrial course of rifting. Even though there is no evidence of icy-volcanism, the being of surface extension and deformation of Ganymede’s crust implies that there has been some form of heating experienced by the moon. What scientists have thus far deduced through the Galileo spacecraft, is that Ganymede has three distinct layers consisting of an innermost layer (a metal core), a middle layer (rock), outer crustal layer (various phases of ice). In addition, scientists detected an extremely strong magnetic field, providing further reason to believe that Ganymede is no entirely frozen.

Callisto, Jupiter’s second largest moon being 1,883,000 km away, has generously similar properties to the Earth’s moon. Both moons possess a terrestrial surface and a frozen body of water. However, Callisto does have an extensively higher ratio of water to rock than the Moon. Callisto’s surface is comprised of half rock, half ice (water). In addition, Callisto is known as one of the most cratered bodies in our solar system. Callisto retains a luminary appearance due to craters created by meteorite impacts. These impacts cause bright rings around the spot of impact. Thus creating a luminous appearance. However, these bright rings could not possibly be derived from the moon itself, since the moon can only reflect the light from other heavenly bodies.

This has been a brief introduction to the background surrounding Kepler’s laws. We hope you have enjoyed it.

Relevance of Our Research:

To begin this project our group decided that we need a lot of information, not just on Kepler's laws, but on how he determined these laws. We wanted to find out exactly how he had gone about doing his research. So, we went to the library, looking up books on Kepler and his laws, and we did much work on the Internet looking up Kepler's laws. Finally we went and found out what other groups had done for their work on Kepler’s laws. We were able to find many articles dealing with the derivation of Kepler’s laws as well as the relationship between Kepler’s and Newton’s laws. Some good references we have found are listed at the end of the write-up.

Our research relates heavily to the outside world. This is because of gravity. Gravity connects us all. It effects everybody in all walks of life. Just because you’re poor doesn’t mean you fly off of the face of Earth, right? Of course not, silly. Our research will also teach us how to use a telescope. There’s a helluva skill, eh?

Materials:

- Telescope

- Custom Data Sheets

- Micrometer

- Flashlight

- Caffeinated beverages

- Writing utensils

Methods/ Schedule of Events:

September 2: The spawning of our generated lab idea

September 4: Group visit to the library to gather materials

September 8: Posted research ideas on web

September 9: Background research completed

September 12: Posted suggestions to peer ideas

September 15: Posted Kepler Progress report

September 21: First Jupiter observation

September 28: Second Jupiter observation

September 29: Posted Lab proposal

Posted peer progress reports

October 14: Third Jupiter observation

October 22: Posted Peer Progress Reports

October 23: Posted Final Student Generated Lab Packet

November 11: Fourth Jupiter Observation (the Big Mama of 5 hours)

November 21: Present project to a full classroom of astronomy affectionados.

Fifth and final (Allah be praised) observation, incorporating class

participation, music, dancing and a sumptuous feast

December 2: Present our results at the prestigious Boyd science facility

December 7: Final presentation of results in Leonard

December 11: Student Generated Lab due

Other Possible Dates will be included in the case of spontaneous Jupiter observation, Our class presentation, and our class participation exercise. This schedule is subject to change.

- As an introduction to this lab we conducted a discovery lab concerning the path of revolution of Jupiter's outermost Galilean satellite, Callisto. Using a computer printout from the Voyager Program that charted the path of Callisto over a years time. We carefully measured the peaks and valleys to find slight variations which would prove the elliptical nature of the Jovian system.

- Using a telescope we planned to observe Jupiter and Its four moons on various clear nights during October through mid November.

- Throughout the experiment we will assign two scientists to be the recorders of data, and three scientists to be observers of data to acquire more accurate results. There will also be two suckers who will pull tendons and bruise flesh attempting to carry the massive scope in it’s "user friendly" case.

- While observing we will first record the initial positions of Jupiter and the Galilean satellites, and then record the distances between Jupiter and the 4 moons at twenty minute intervals.

- We will record all data on the super high tech, custom data sheets created by the group.

- The next step in our method is to begin the calculations with which we will prove Kepler's Laws.

- We will do many confusing things which we cannot possibly explain right now. Your weak earthling minds simply are not ready for such information. Haven’t you ever heard of Pandora’s box?

- These above mentioned "confusing things" will include: using the equation p2 = k(a3), finding the value of "k", plugging the mean distances into the equation, and then comparing the calculated periods with the actual periods. Sound intense? Too much to handle? Hey we’ll walk you through it, come on tisn’t that bad, even Binnie understood it!

Class Participation

Purpose: The purpose of our laboratory class presentation is to educate our classmates on the fundamentals of Kepler's Laws. In this presentation we will delve considerably deeper into Kepler's Laws. In our presentation we will include information concerning Newton’s Laws as they apply to Kepler’s Laws, information about Jupiter and the Galilean Satellites, and lots of background information about how Kepler came to derive his laws. We will then go into an explanation of some of the equations we have derived that will be used to prove Kepler’s Laws. After that, we will present the class with some of our data, and ask them to figure out period, mean distance, and if those two values satisfy Kepler’s Third Law by using the aforementioned equations. Following that we will have a session teaching the class how to set up the telescope, and how to find heavenly bodies once the telescope has been erected. We will instruct the groups in a short crash course in mapping the moons of Jupiter. Then, on a given night, each group will come outside at a different time and help us collect data on the positions of the moons.

Results

Discovery Lab:

The results of our discovery lab were inconclusive. We were not able to find any negligible difference between the peaks and valleys of any given time period. Our conclusion? Simply this: We are attempting to reduce a gigantic orbit down to a couple of centimeters. There are a couple problems with this. Considering the fact that the orbits are close to circles anyway, how could the computer possible show such a slight variation, and if it could, how could we possibly measure it? Beyond this, Voyager is a simulation. Simulations of natural phenomenon are never perfect. We simply could not be precise enough to find any conclusive data. But fear not, we are not through with the fight.

"k"

There was one very important factor that we decided to ignore from the getgo. This turned out to be one of our biggest mistakes. What was this tiny little factor that almost ruined our entire lab? The answer is k. That seemingly benign letter very nearly sent us to the brink of insanity. When we first tried to synthesize some results out of our data, we decided to use the equation p2= a3. However, we soon learned that this was not going to be nearly enough. Let’s take Jupiter’s third moon, Ganymede for instance. This moon has a period of .0198551 years, and a mean distance from Jupiter of 1,070,000 km. After putting these numbers into the equation we come up with

(.0198551)2 = (1,070,000)3. Now, it doesn’t take a rocket scientist, or a couple of college freshman, to realize that this equation does not work out. We were baffled. What went wrong? Then we came to a horrifying revelation. When we went over Kepler’s laws in class, we were also presented with the equation p2 =ka 3 We forgot about k! But what exactly is k? We know that every body in the universe exerts a gravitational attraction on all other bodies in the universe. We also know that each body exerts a differing amount of gravitational attraction. K is an arbitrary constant that signifies the gravitational attraction of any given system. When doing calculations involving Kepler’s third law and our solar system, one doesn’t have to worry about k. Our solar system is the basis for Kepler’s laws, so any problem involving our solar system has a k that equals 1, so it can effectively be left out for simplification purposes. However, once one moves outside of our solar system to another system (say.... I don’t know...the Jovian system maybe?) one has to take into account a new gravitational constant that will not equal 1. We had forgotten this. It quickly became clear that in order to solve this problem, and to prove Kepler’s third law, we would have to find the k of the Jovian system.

So we picked our brains to figure out the best way to come up with k and quickly found the answer. Looking up the actual mean distance and period, we could very easily find k. Then we could formulate our results, compare them to the actual period and mean distance, and see if we were right! So we set up the following equation:

(0.198551)2 = k(1,070,000)3

or

3.8369 x 10-4 = k(1.225 x 1018)

After figuring this, it was child’s play to find k. Now all we had to do was some simple arithmetic.

(3.8369 x 10-4) = k

(1.225 x 1018)

After completing this division, we came up with

k= 3.13219 x 10-22

After finding k, we plugged it into our data and found our results.

Calculation Methods

Calculation and Analysis of the data collected followed the measurements taken. Just as Kepler himself did so many years ago, The Keplerites sat down to prove the equation P2= k (a3) with measurements of the distances between Jupiter and its four largest moons. We had a slight advantage over the big man in that we needed only to show that the famed third law worked not derive it. What follows are the steps necessary for greatness, our larger goal.

- To prove this we would begin with our measurements of Jupiter and its moons. The mean distance between each moon and Jupiter was found by averaging the measurements taken. The mean distance being the "a" in the third law’s equation.

- With all our data, and therefore the means, taken in units of the micrometer; the next step was to convert our mean distances into kilometers. A standard for conversion was selected based on the micrometer; our mean distances and the actual mean distances for the moons. This was found to be 102,758 kilometers per micrometer unit.

- With kilometers in hand and hearts pounding, we forged into the final step of calculation: substitution into the Third Law. We would use our mean distances in the equation: Period = The square root of (k times the average distance cubed), resulting in calculated periods. We would then compare these periods with the observed periods figured out by scientists in lab coats many moons ago.

The Numbers and Statistical Evaluation

Here are the actual figures that we came up with after imbibing in the process described above. After finding the periods and comparing them with industry standard, we then sent our poor distance measurements through yet another ringer. The figures were analyzed with respect to the orientation which we observed the moons. In all tests the different views were shown to be significantly different from one another, denoting that some views give more accurate views than others. The implications abound. Read on for the thrilling conclusion. The results are given moon by moon, judge us as you may, The Keplerites rest.

Io

Mean distance in Micrometer units: 1.010

Mean distance in Kilometers: 113,047

Calculated period in years: .0006727

Calculated period in days: .2457

Actual period: 1.769138

Difference: 1.52348

Percent error: 86.112

Europa

Mean distance in Micrometer units: 6.552

Mean distance in Kilometers: 670,037

Calculated period in years: .0097937

Calculated period in days: 3.57717

Actual period: 3.551181

Difference: .025989

Percent error: .7318

Ganymede

Mean distance in Micrometer units: 13.1428

Mean distance in Kilometers: 1.35207*10^6

Calculated period in years: .02741

Calculated period in days: 10.1627

Actual period: 7.154553

Difference: 3.008

Percent error: 42.045

Callisto

Mean distance in Micrometer units: 21.3948

Mean distance in Kilometers: 2.2001*10^6

Calculated period in years: .05779

Calculated period in days: 21.10764

Actual period: 16.68902

Difference: 4.41862

Percent error: 26.476

Possible Reasons Error

When we began this lab, we didn’t quite realize what we were getting into. When looking through the telescope, one can describe the image as four little dots and one big one. Using a bit of common sense we concurred that the big dot was Jupiter, and the four little dots were the four major moons of Jupiter. We saw a two dimensional figure. When looking at objects in orbit through the telescope, the viewer is denied the advantage of depth perception.

We feel that this is the main reason for the discrepancies found in our calculated mean distances, and thus the cause for error in our moon periods. As one may see in our Calculation Methods, we took the actual distance of a moon from Jupiter, Io for instance, and then compared that figure with a converted number which was the distance of Io from Jupiter we calculated. The calculated number we acquired, was derived from numbers that we obtained from our mean distances using the micrometer on the telescope. These distances we obtained, were taken by the distance of the moon form the right or left edge of Jupiter.

The problem with this is we do not have any sense of depth perception and thus can not tell, looking through the telescope exactly how far away from Jupiter the moon actually is. There are various scenarios that can occur when measuring the distance of a moon from Jupiter, for example, the moon can be situated in an ideal position where it is seen at its farthest distance from Jupiter. However, most of the time the moons will be seen closer to Jupiter than maximum distance away, and at times a moon may be unseen for it is in front or behind the planet.

This instance, of seeing the moon Io at a location in the heavens that is unfavorable brought about our most significant error. Our calculations for Io were greatly different than the actual distance.

Another reason that explains our error is the simple factor of time. We observed Jupiter on four separate occasions, three of which were for an hour. When Galileo studied these satellites he observed them every night for years. Had we had a couple more years to observe Jupiter and the Galilean satellites, we would have obtained more accurate calculation from our data.

Conclusion

After extensive research and synthesis of data, we feel that we can comfortably state that Kepler’s third law is indeed true. After collecting our data, figuring out k, plugging in our numbers in the necessary fashion and analyzing our findings, we have some pretty astonishing results. For three of the four moons, our percent error between the period of the moons we found and the actual period was below 43%. As a matter of fact, one was just under 27% and Europa was under 1%! We had a slightly larger error for Io, but please refer to the previous section for an explanation of that. We also found that some orientations provide more accurate measurements for the distances. We accomplished this by applying the same conversion factor used in the initial calculations to the mean distances for each view. For all moons except for Io%

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