FINAL 1:THE LEARN'D ASTRONOMERS

This topic submitted by Austin Kleon, Ryan Lazowski, Laura Englehart, Miguel Padilla ((kleonaw@miamioh.edu, lazowsrd@miamioh.edu, englehle@miamioh.edu, padillmr@miamioh.edu)) at 7:26 pm on 12/6/01. Additions were last made on Wednesday, May 7, 2014. Section: Cummins

When I heard the Learn’d Astronomer

When I heard the learn’d astronomer;

When the proofs, the figures, were ranged in columns before me;

When I was shown the charts and the diagrams, to add, divide, and measure them;

When I, sitting, heard the astronomer, where he lectured with much applause in the lecture-room,

How soon, unaccountable, I became tired and sick;

Till rising and gliding out, I wander’d off by myself,

In the mystical moist night-air, and from time to time,

Look’d up in perfect silence at the stars.

- Walt Whitman


(From Kepler's Laws With Animation)

ABSTRACT

Our research project is all about understanding Kepler's Laws of Planetary Motion. The lab includes the observation and measurement of Jupiter and its moons' orbits and the usage of these measurements to find orbital distance from Jupiter, orbital period, and Jupiter's mass. Originally, we essentially wanted only to repeat the labs of the other classes that have done Kepler before. After much discussion, we decided that we wanted to do something that would benefit NS classes to come. Not only will we do the lab we originally planned to do, but we have provided an actual lab guide where students after us can repeat our experiment for themselves in their Natural Systems class. While preparing this lab guide we have make notes on the easiest and best ways of performing it, using our own experience. We have found an excellent reworking of Kepler's Third Law that allows easy calculation. We have also run staticstical tests that show what the best schedule of aquiring signifigant data is. Using our measurements, we have been able to calculate a mass of Jupiter that is very close to the given value. This lab is significally important to science students, because Kepler's Laws of Planetary Movement are fundamental to astronomy, as they explain how the whole system moves and works, and they are still used today (see our information on relevence to finding mass and black holes below!).


INTRODUCTION


Our purpose was to use Kepler's Third law to find the mass of Jupiter by observing the four major moons of Jupiter (Io, Europa, Ganymede, Callisto), and to form a discovery lab for other classes to follow. Our hypothesis was that the data we collect by using the telescope would help us determine orbital period for each moon and find the mass of Jupiter. This lab will helped us understand astronomy in a very significant way. The topic of heavenly bodies intrigued us. It seemed like an interesting challenge to explore a topic beyond the terrestrial realm. Due to the limited time and resources available to us for this lab, however, our experiment needed to be narrowly focused on one particular topic under astronomy. Using Kepler's Third Law was appropriate: it was a manageable task, but it was also very challenging, interesting and fun. Most of our lab was inspired by the work of Jeremy, Binnie, Dan, Lorraine, and Maxwell during their 1998 NS lab, but we have found a few methods and discoveries that have helped us form the ultimate discovery lab. We have improved on their lab by noting their errors, and coming up with easier equations. We have also expanded their lab to show relevence to today's science work by including Newton's reworking of Kepler's law to figure out masses of planets and black holes. In the process of executing the lab we have gained knowledge and experience of using the telescope and observing the planets. This is be important, because in the beginning we had little knowledge of how exactly the telescope works, and we needed this knowledge in order to obtain data and to write the discovery lab. We are now very good at using this great tool. Designing and executing the lab is an exercise of the scientific method, including: background research, hypothesis & predictions, methods & materials, data collection, data interpretation and conclusions -- and think of the lucky classes behind us that will benefit from our hard work of designing a Kepler discovery lab so they don't have to!


ASTRONOMY 101: THE BIG PLAYERS



Ptolemy

http://es.rice.edu/ES/humsoc/Galileo/Images/Port/ptolemy.gif


To understand Kepler and his ideas, one must know the history behind astronomy. It all starts with Ptolemy, who believed that the earth was in the center of a perfect divine universe and all the planets and the sun all formed in circles around it. Science back in those days was greatly influenced by religion, as no one in their right mind would go up against the church's teaching and say that the whole universe didn't really revolve around the Earth, the center of God's creation. When Copernicus came along, he asserted his idea that the sun was at the center of the universe (called the Heliocentric Model), quite controversial then, but now taken for granted. This is why until such pioneers as Copernicus, Kepler and Galileo, astronomy didn't take off in leaps and bounds. Their contribution is so important to our understanding of our world today. (Pribble et. al, 1998). The rest of the history of Kepler's Three Laws can be divided into the contributions of four men: Brahe, Galileo, Kepler himself, and Newton.


Copernicus

http://www.th.physik.uni-frankfurt.de/~jr/gif/phys/kepler.jpg



Brahe
http://www.bun.kyoto-u.ac.jp/phisci/Gallery/tycho.html

Tycho Brahe was a Danish astronomer who lived from 1546 to 1601. Much of the funding for his work came from royal patronage; King Frederick of Denmark was one his most famous and wealthy sponsors, beginning in 1575. King Frederick provided Brahe his own research site, located on the island of Hven off the Danish coast. There Tycho and his assistants produced many precise instruments, including the Uraniborg Press, one of the world's first printing presses. These accomplishments pleased both Brahe and his sponsor: the latter wanted recognition and fame for sponsoring a scientific genius; the latter was achieving his goal of blending "fine arts, technology and science" in his research institution. This was one of Brahe's accomplishments: he made scientific investigation an interdisciplinary process, the products of which impressed people beyond only the scientific community. However, this new environment of scientific investigation created some problems for Brahe. There was much competition among scientists to make exciting discoveries and receive rewards from patrons. Plagiarism became more common, and the printing press allowed intellectual thieves to quickly lay false claims to another's work. Thus, Tycho was eager to publish one of his most famous works, a monograph on the comet of 1577. This document is of historical importance, for it established a precedent for the format and content of scientific publications. Brahe's document included such standards as: tables, diagrams, illustrations, a table of contents and chapters. Also important was a critical review of previously published literature on the subject. Unfortunately, although Brahe was systematic and thorough in procedure, he did not produce revolutionary thoughts or theories. In his flawed system explaining the laws of planetary motion, Brahe asserted that all the planets except the Earth revolved around the Sun; the Sun, in turn, revolved around the Earth. Later, Kepler, one of his research assistants on the island, would reinterpret the data and propose his famous laws of planetary motion. Brahe's data was so meticulous that it was invaluable to Kepler's theory (Christian, 2000).


Kepler.

Johannes Kepler was born in Weil der Stadt, Germany in 1571. He was educated at Maulbronn and the University of Tuebingen where he was in the Stift, a seminary for scholarship students. At Tubingen, Michael Maestlin taught Kepler astronomy using the Ptolemaic system in which the other planets revolved around the Earth. However, Michael Maestlin chose to also teach to a select group of students, including Kepler, about the new Copernican System in which all planets including Earth revolved around the Sun. Kepler quickly accepted the Copernican System because it had greater explanatory power than the Ptolemaic system. For instance, "the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun (they lie between Earth and the Sun) whereas in the geocentric theory there is no explanation of this fact" (pg. 3, Groups). In 1594, he taught mathematics at the Lutheran Stiftschule in Graz. In Graz he was later given the job of district mathematician and calendar maker where he made 5 calendars and did astrological nativities and prognostications for lords. He married a wealthy widow named Barbara Mueller in 1597. He was forced to leave Graz in 1598 and worked for a few months for Tycho Brahe before returning shortly to Graz. In 1600 he was permanently banished from the city and went to work for Tycho Brahe in Prague where he eventually took Brahe’s position when he died. When Brahe died, Kepler had to battle his family in order to get ahold of Brahe's detailed measurements that he needed in order to form his theory. Brahe really didn't trust Kepler, and thought that Kepler would exceed him, so while he was alive he gave him information on Mars to keep him occupied. It was this information that would help him form his three laws of planetary motion, which will be explained below.


Works Cited:


http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/ECText?ch02_txt.htm#2


http://www.groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html



Galileo.
http://www.williams.edu/HistSci/resources/images/galileo/portrait.html


Galileo began to study medicine at the University of Pisa in 1581. But instead of finishing a degree in medicine, Galileo became intrigued with and decided to pursue mathematics. He taught mathematics at Pisa from 1589-1592, and became the Chair of Mathematics at Padua in 1592. Galileo made his first telescope in 1609. Although he was not the first to invent the telescope, Galileo increased the magnification of the design, which allowed for his discovery of Jupiter's four satellites. Galileo began his work with the telescope by observing the moon and its spots. He concluded that the moon's surface has mountains and valleys and so is not a perfect sphere, as previously believed. After completing his observations of the moon, Galileo turned his attention to the next brightest object in the night sky, Jupiter. He observed three little stars surrounding the planet that he believed were fixed. Continual, in-depth observation proved that these little stars changed their position with respect to Jupiter and each other, but did not leave Jupiter. He also verified that there were four of these stars, not just three as initially observed. Eventually, Galileo concluded that they were not fixed stars, but rather were planetary bodies that orbited Jupiter. This helped to advance the Copernican argument against geocentricism because it added another example of a central body with satellites of orbit. Thus, the earth was proven not to be central to all with Galileo's discovery of the four Jupiter satellites. (Van Helden, 1995) Galileo's discoveries are the whole inspiration for this lab, as we will be doing the exact thing that he did: look at Jupiter and its satellites. Galileo is mind blowing because he made all of these discoveries with very primitive instruments -- we can hardly even find Jupiter with our own advanced telescope! Kepler's ideas gained exceptance through Galileo's work.


What the Jovian System looks like. (From Ed Stepan's Excellent Animation Web Page)

A Sketch of Galileo's! (From The Art of Renaissance Science)


Newton.
http://www.math.fu-berlin.de/rd/ag/isaac/newton/newtn1_f.jpg

Sir Isaac Newton (1642-1727) made so many discoveries relative to math and science: he revolutionized physics and astronomy, and invented calculus almost single handedly. Newton did so many things it would be pointless to try and explain them all. From the Astronomy 161: The Solar System website, the poet Alexander Pope is quoted: "Nature and Nature's laws lay hid in night; God said, Let Newton be! and all was light". Certainly, our lab and Kepler's laws would be in the dark without Newton's contributions. Newton will be discussed in much length during the next section where we explain how Kepler's laws work. Kepler's laws actually influenced Newton's Three laws of motion and in turn, he proved Kepler's Third Law, which could never be done without his work with gravity. Kepler's laws explained planetary motion, but nothing about other motion, and really no one could properly explain why Kepler's laws even worked. So therefore, we move to....

 

ASTRONOMY 102: HOW THE LAWS WORK

Here we will attempt to illustrate Kepler's Three Laws and Newton's reworking of Kepler's Third law, based on his own three laws. The best way to understand Kepler's Three Laws of Planetary Motion is to observe the images that we have provided and pay attention to the description. Also, a great run-down and history of this information comes from the Astronomy 161: The Solar System website. Go there. It is very cool. Keep in mind, that with these models, we will substitute Jupiter for the sun, and one of the moons for the satellite or planet. Please bare with us:

 

KEPLER'S LAWS OF PLANETARY MOTION:

1. All planets move in the shape of an ellipse with the sun at one focus.


In this example, x and y are the planet's position on the ellipse and a and b are the axis of the ellipse. It was once thought that the orbits of the planet's were perfect circles. You call the flatness of the ellipse the eccentricity. The actual orbits of planets are not this eccentric, as they are very close to perfect circles. When observing Jupiter, we will be looking at this ellipse sideways.

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

As the planet goes around the sun, the further away it is, the slower it rotates. This is due to gravitational attraction and is explained by Newton's concept of gravity, which we will examine shortly.

3. The square of a planet's period of revolution is proportional to the cube of the planet's mean distance from the sun.

Simplified, the formula looks like this:

T^2 = k(R^3)

where T = the period of revolution of a planet, k = constant, R= mean distance of planet from sun

For our purposes, we will apply this to the Jovian moons. Jupiter will be substituted for the sun, and the moons will be substituted for the earth. By translating to the Jovian system, however, we will have to come up with a suitable figure for k. This was found out the hard way by the old Keplerytes, and thanks to their suffering, we don't have to. A workable k can be found by plugging in a known mean distance and revolution of one of the moons, and solving.

To find R, we will have to take the maximum extreme measurements of the elliptical moon orbit from Jupiter (remember, when we are looking at Jupiter, the ellipse will be flat, check out the Jovian system animation above)...

R= (X1+X2)/2 where X=extreme measurements on both sides of Jupiter. Ideally we would measure out the moon's furthest distance from the left of Jupiter, and from the right, as it makes it around the planet. However, this is not always easy, so a maximum distance found for each moon from Jupiter will work well for this purpose. To find period, you will use this "R" plugged back into the equation above.

It is essential to note Newton's discoveries. His theory of gravity was the only thing to finally prove this Third Law.

 


NEWTON'S THREE LAWS OF MOTION


1. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

2. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma.

3. For every action there is an equal and opposite reaction.

 

GRAVITY ALWAYS WINS

from the Astronomy 161: The Solar System website (boy are we glad we found this:

For our purposes, R will be the distance between Jupiter and a Moon.

From the first law we can understand that planets are constantly in motion because objects move until a force is exerted on them. "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" (Pribble et. al). The planets are kept in their orbits because of the gravitational force of the sun. As the planets move around their orbits, they move faster when they are closer to the sun, because of the greater gravitational force that being closer allots, this proves the second law of Kepler.

From his third law, Newton decided that the sun is not only stationary, it also orbits around a center of mass, and the planets revolve around the sun. Center of mass will not affect our work and calculations directly, so a firm understanding is not necessary. However, he made a great change in Kepler's law because of this:

M = Mass of system or (m1+m2)

G = universal gravitational constant = 6.67x10^-11 (m^3)/(kg*s^2)

R = mean distance from Jupiter (in meters)

T = orbital period (in sec.)

Here, m1 = the mass of jupiter, m2 = a moon, and R=distance between them. This will be the formula we will be using for our calculations. The maximum distance we find from our measurements will be R, and Twill be our observed period. From this we can actually find the mass of Jupiter from the moons, as we will show later. "k" is the value found within the parenthesis.

Hopefully this isn't too confusing (ha!) we will clear up all questions in our class participation session, and e-mail us if you want to know more! (This information was gathered from Jennings, the website Astronomy 161: The Solar System, and Pribble et al.)

 

...AND NOW FOR SOMETHING PRACTICAL!!

One application of his third law is in the study of black holes. We will do these calculations in our class presentation. Imagine this scenario: through a telescope an astronomer observes a star that seems to be exhibiting rotational motion around some object. However, there is only darkness at the area around which the star rotates! One may predict a black hole exists in the central area of the rotation, whose gravitational force attracts the star and keeps it in orbit. Then one may want to estimate the mass contained in the black hole. A method of calculating an estimate is through the "Epsilon Aurigae" model. This model accounts for a star orbiting a black hole. The model's equation is below.

(P^2)M=R^3

With: P= orbital period in years R= average distance between star and black hole, measured in astronomical units M= total mass of system, measured in solar masses

With a telescope, one may measure the star's orbital period and its average distance from the black hole. Inserting these values into the calculation above, one may estimate the amount of matter concentrated in the black hole! Since the relative size of the star compared to the black hole is very small, we can count the total mass of the system as really the total mass of the black hole.

We will also use this method to approximate the mass of Jupiter by observing the moons. Say we take the same measurements as above with our telescope, only we replace the black hole with Jupiter and the star with a moon. The moon is also relatively tiny compared to Jupiter, so we can discount its mass and estimate Jupiter's mass as the total mass of the system!! (Shipman, 1976).

 

THE "EASY" MASS EQUATION

KEPLER'S THIRD LAW (Modified with Newton's laws for mass calculation):

Mj = Mass of Jupiter

G = universal gravitational constant = 6.67x10^-11 (m^3)/(kg*s^2)

R = mean distance from Jupiter (in meters)

T = orbital period (in sec.)

Jupiter's mass is immense compared to the mass of its moons. The moons are only a tiny fraction of the planet's mass, therefore we can estimate that the total mass of the system is close to Jupiter's actual mass, and eliminate the moon's mass from the equation. We can now rearrange the equation to look like this:

This equation is easiest to use because you can convert micrometer units to kilometers and then meters to use within the equations.

...THE BIG GAS BALL KNOWN AS JUPITER

Since we will be using the Jovian system for our lab, we need to know information about Jupiter and its moons.


Jupiter.
http://seds.lpl.arizona.edu/nineplanets/nineplanets/jupiter.html

Jupiter, often referred to as the "Giant Planet," is the largest planet in our solar system. It has a mass of 1.900*10^27 kg, a radius of 71,492 km, and a distance from the Sun that is more than five times greater than the Earth's distance from the Sun. Its greater distance from the Sun means that it revolves around the Sun once in about 12 Earth years. Also due to its distance, light, heat and other solar radiations that reach Jupiter's surface have 1/27 the intensity of those that reach the Earth's surface. Another interesting quality of Jupiter is the fact that the inclination of Jupiter's equator to the horizontal plane is small. Therefore, seasonal changes and effects are rarely observed.(Peek 1958).

The structure of Jupiter comprises two major parts. The easily observed gassy envelope that surrounds Jupiter, which contains mostly hydrogen and helium, contributes only about 95% of Jupiter's total volume. 95% of Jupiter's volume is in the core, which is composed mostly of heavy elements such as silicate. There are two common theories that explain the history of Jupiter's formation. In the core model, the core of Jupiter formed before the gassy ring did. Small bodies of matter randomly struck each other and stuck together, eventually coalescing into a definite core. The gravity exerted by this concentration of mass then attracted and condensed the ring of gas. In the gas model, the opposite process is postulated. Randomly interacting matter first formed the gas ring, which then attracted more matter to its center to create the core. (Morrison, 1982.)


Io.

http://seds.lpl.arizona.edu/nineplanets/nineplanets/jupiter.html

Io is the closest moon to Jupiter. Its temperature is -143 degrees Celsius (-230 degrees Fahrenheit). However, it has a large hot spot that measures 17 degrees Celsius (60 degrees Fahrenheit). Its radius is 1821km. It is named after a story from Greek Mythology in which Zeus fell in love with Io. He changed himself into the shape of a dark cloud to hide from his wife Hera. But Hera found out and as soon as she came to them, Zeus transformed Io into a white cow to hide her. But Hera was not deceived, and she sent the cow to her hundred-eyed servant, Argus, to watch over her. Zeus's servant Mercury, was sent to defeat Argus by telling him stories until he closed every one of his hundred eyes. But Hera discovered that Io was free and plotted to kill her. Zeus had to promise to no longer pursue Io for Hera to release her and make her human again. Io then became the first queen of Egypt (Windows to the Universe, 1995-1997, 2000). The most significant thing to note about the moon Io is that it has active volcanoes. Tidal forces cause this volcanic action. Tidal forces occur when Io's orbit is disrupted by Europa and Ganymede and then is pulled back into its regular orbit by the gravitational pull of Jupiter. This generates incredible amounts of energy that leads to volcanic eruptions. Io also creates energy as it orbits through Jupiter's magnetic field. One thousand kilograms of material is stripped from Io';s surface every second due to Jupiter's magnetic field. This material forms a torus, a doughnut shaped cloud of ions. Because the ions travel outwards, their pressure doubles the size of Jupiter's magnetic field. This generates almost twice as much heat as the earth despite the fact that Io is less than one third the size of the Earth (Hamilton 1997-2001).


Europa
.
http://www.solarviews.com

Europa is Jupiter's fourth largest moon out of seventeen, and is the sixth closest to Jupiter. Its mean distance from Jupiter is 670,900km. It is named after a story from Greek mythology in which Zeus came to the beautiful Europa in the form of a bull while she was gathering flowers. Due to his gentleness, she treated him kindly, and, as she climbed on his back to ride him, he carried her away and made her the Queen of Crete. Zeus reproduced the shape of the bull in the stars for her, which is still recognized today in the constellation Taurus (Windows to the Universe, 1995-1999, 2000). The most significant attribute of the moon Europa is its smooth and icy surface. The surface fractures along fault lines and ridges, creating eruptions of ice volcanoes. These ice crust movements prove that warmer ice and possibly even water is below the surface. There are even places on Europa that are similar to the ice flows in the Earth's polar regions. The possibility of the existence of liquid water on Europa could mean that life exists there. There is also a presence of oxygen in its atmosphere, formed by sunlight hitting the ice and creating water vapor. Oxygen is only present in a quantity that is one hundred billionth that of Earth, but its presence nonetheless indicates further that life could possibly exist on Europa (Hamilton, 1997-2001).


Ganymede.

http://www.solarviews.com

Ganymede has a mass of 1.48*10^23 kg, an orbital period of 7.154553 Earth days, and a mean distance from Jupiter that is about 1,070,000km. Its geologic history includes crater impacts, internal tectonic deformation and frequent ejection of materials from its surface. The terrain surface of Ganymede varies. Dark and densely clustered craters define almost half of Ganymede's surface. This appearance resembles that of Earth's moon. The dark area is speckled, however, by bright spots known as "palimpsests." Also in the dark area is the Gilgamesh Basin, which was created by the impact of a crater and is today the largest and clearly observable features on Ganymede. Lightly cratered and grooved terrain dominates the other half of Ganymede's surface. There are many light and dark ray craters that create the grooved appearance. Among these grooved areas, however, there are some patches of smooth plain.The thickness of Ganymede's regolith varies according to latitude and the distance from the peak of orbital motion. The regolith is the surface layer of Ganymede, which comprises rocky debris randomly strewn about by the impact of past meteor collisions. The fragmented quality of Ganymede's icy surface allows frequent and high levels of sputtering. This involves protons and helium ions that were once contained in the core of the moon, to eject themselves through the surface and release high amounts of energy in the process. They have the capacity to do this because Ganymede's core has an abundance of water, which permits the transfer of heat in low temperatures.


Callisto.

http://www.solarviews.com

Callisto, one of the four main moons of Jupiter, was discovered on January 7, 1610 by Galileo Galilei. It is the second largest moon of Jupiter and the third largest in the solar system. Its diameter is about 4,806 km. Callisto has a rotational and orbital period of 16.68902 days and a mean distance from Jupiter of 1,833,000 making it the farthest moon from Jupiter of the four big moons. Its orbit is almost a perfect circle. The surface is twice as bright as Earth’s moon and its heavily cratered ice and rocky dust surface hasn’t changed for about 4 billion years making it "the oldest landscape in the solar system" (page 2, nasa.gov). The most prominent cratered region is known as Valhalla which is about 300 km across with concentric ridges extending 1,500 km from the center. The interior is made up of ice and rock and has the lowest density (1.86gm/cm cubed) of the Galilean moons. Callisto’s subsolar Temperature is 168 K and its Equatorial Subsurface Temperature is 126 K.

Useful Stats (www.solarviews.com)


Jupiter

Mass= 1.900*10^27 kg

Radius= 71,492 km


Io

Mass= 8.94*10^22 kg

Orbital Period= 1.769 Earth days

Mean distance from Jupiter= 421,600 km


Europa

Mass= 4.8*10^22 kg

Orbital Period= 3.55 Earth days

Mean distance from Jupiter= 670,900 km


Ganymede

Mass= 1.48*10^23 kg

Orbital Period= 7.154553 Earth days

Mean distance from Jupiter= 1,070,000 km


Callisto

Mass= 1.08*10^23 kg

Orbital Period= 16.689 Earth days

Mean distance from Jupiter= 1,883,000 km



SO HOW IN THE HELL DID YOU GO ABOUT THIS MESS???


The materials of our madness weren't that extensive: Telescope, micrometer eyepiece for measurements, data sheets, writing utensils, hot chocolate, and dear lord don't forget the flashlight. There were two huge parts of our lab: one was gathering the data, and the other was plugging the data into the equations and proving the laws. The first part involved finding a time when Jupiter was visible using the USNO Site and SkyMag online (usually only at times when we'd rather have been sleeping), setting up the telescope (which is no easy feat) finding Jupiter (Austin tackled this one), and using a micrometer eyepiece to record the positions of each moon every ten minutes for a period of one hour. The exact measurements were made from the middle of Jupiter to the moon. One person observed while the other will recorded the measurements on our super-cool data sheet.

For the first part, our methods of observation were similar to those that the Keplerites of 1998 used. We used the measured distances between Jupiter and each moon with the telescope micrometer and then converted the micrometer values to actual distances in kilometers. What we then did was find the mean distance in micrometers out of all five observational nights using Statview. We then used the actual radius of Jupiter from the books and converted our measurement of Jupiter in micrometers to kilometers. We then had a ratio to work with. It turns out that the average distances were not very accurate, so we used the maximum distances, and this worked out far better. We compared the calculations made by these average and maximum distances for each moon in our results section. Regardless of which measurements we used however, we were able to calculate a good mass for Jupiter! These results are shown in the RESULTS section, how novel.

A problem that the old Keplerites found was depth perception: as the Keplerites mentioned in their report, there was difficulty discerning how far back or how far in front a moon might be in relation to Jupiter. This depth is a dimension that determines the distance from the moon to Jupiter. Unfortunately, we are also unable to account for this dimension. Still, we may reason, as the Keplerites, that the varying vertical components of the moons' distances from Jupiter will essentially cancel out as multiple data samples are compared and averaged. As a matter of differing from their design, we accounted for the mysterious "k" value that hoodwinked the Keplerites in 1998.

The design of this lab was not easy to undertake. Finding Jupiter and looking at things millions of miles away is prone to error. When we went out with Jeremy and Dan to check out the telescope for the first time, we really saw how tough this will really be. However, the concept of the design is statistically sound: we made measurements, we converted them to numbers we could use, we plugged them into our equations, then used these formulas to try and figure out Jupiters mass using only our measurements, and compared our data, finding the best times for viewing, and what the best moons were for calculations.

Click Here for the Past Data collected by the last group of Keplerites.

Click here for our Data Sheet!


NO ONE IS SAFE!

We included the class in our lab by having them help us get our actual measurements while observing Jupiter, and then trying to help them understand the theory of the Laws. The class observed the moons one early morning after our five hour bonanza and then we had a teaching session afterwards.

A simulation of what we saw that night!

Here you can find our killer handout for Jupiter's mass calculation. In our class section, we went over the history, the actual laws along with Newton's contributions, and calculations of Jupiter's mass using the known values of period and distance for Io. We also talked about black holes. We thoroughly enjoyed the company outside in the freezing COLD!

Timeline?

We accesed the USNO Site and procured the rise, peak, and set times of Jupiter for observation. Below is our timeline:

Oct.8: We had our first encounter with the formidable telescope we'll be using, with Dan and Jeremy as our guides. The instrument is VERY heavy and complex. It took some energy to lug it and its cumbersome case to the field between Boyd and Peabody. Once there, though, Pribble and Jeremy showed us how to construct and use the telescope. We had to assemble the tripod, mount the telescope on that and attach the eyepiece to the telescope. These tasks were somewhat difficult without the aid of light; next time we go out, we'll have a flashlight. Unfortunately, not many stars or planets were visible, but Jeremy found for us a star to look at.

Nov. 1: First observation of Jupiter and the four major moons. Hay's annual Jupiter viewing.

Nov. 5: First time without the assistance of a "veteran." Didn't find Jupiter, couldn't assemble eyepiece.

Nov. 7: First succesful observation of Jupiter w/ measurements. Used SkyMag to find it. Revamped Data Sheet to match micrometer measurements.

Nov. 8: Second successful observation. Health starts deteriorating.

Nov. 9: Third successful observation.

Nov. 11: Fourth observation.

Nov. 12: Fifth observation. Five-hour bonanza ending with a class observance.

Nov. 28: Analyzed data using statview and Exphrem

Dec. 2: Final analyzation of data. Found distances, orbital period, and Jupiter mass.

Dec. 4: Powerpoint final presentation to class.

 


RESULTS:


Click here for a spreadsheet of our raw data measurements.

Entering our raw data measurements into statview, we came up with the following stats:



Converting Micrometer Units:


Jupiter = 4 Micrometer Units Across (observed)

Jupiter's Radius = 71, 492 km (given) = 2 Micrometers (observed)

71,492 km / 2 = 1 Micrometer

35, 746 km = 1 Micrometer


Using this ratio, we were able to convert the micrometer measurements from above into kilometers, and these kilometer values are found below in our "Kepler Calculations" table below.

 

Finding "k"

We found a workable "k" by using this equation and method from above:

T^2 = k(R^3)


k = the figures in parethesis

G = 6.67x10^-11 (m^3)/(kg*s^2) = converted for our measurements using km and earth days = 4.979 x 10^-10 (km^3)/(kg*d^2)

M = 1.900 x 10^27 kg

k = (4 pi^2) / ( 4.979 x 10^-10 (km^3)/(kg*d^2) ( 1.900 x 10^27 kg) )

k = 4.176 x 10^-17 (days^2) / (km^3)

 

Using the Third Law to Find Orbital Period and Jupiter's Mass:


T^2 = k(R^3)


Using the values of distance shown in the "Kepler Calculations" table below, we were able to plug these in as "R". Inputing k as the value from the "finding 'k'" section, we could then find the period, using the equation above. Using our calculations of distance ("R") and orbital period ("T"), we could find the mass for Jupiter using the equation below:


 

Here are our results from these calculations:


The Given values for distance from Jupiter, orbital period, and the Mass of Jupiter are shown in the first group. Our calculations from using the average observed measurement from Jupiter for each moon are shown in the second group, and the calculations from the maximum observed distance from Jupiter are shown in the third.

As you can see, using the maximum distance from Jupiter for each moon to do calculations provided us with more accurate results. A comparison of these results is shown below in our graphs, the bars showing percent of the actual value given.


 



By comparing the graphs above, we determined that Io is the best moon to observe in order to do these calculations.

Observance Schedule:

We ran a t-test on the data from each moon to see if the measurements from one night to the other were significant. Significance from one night to the other would help us form an ideal observance schedule. Our t-tests showed that 60 % of the nights that we viewed Io and Callisto were significant, 80% of the nights viewed we viewed Europa and Ganymede were significant. From this we determined that most of the information we gathered over the period of a week was significant, so it is essential to view Jupiter over a few days, not just one. Another observance we found by taking down data during our observations was that the moons don't move very much at all during the course of an hour. Therefore, it is not essential that you freeze out in the cold for an hour to take a measurements. One measurement will do for each moon each night. However, we recommend that a 5 hour bonanza is experienced in order to see movement and to get the "full experience".

 

DISCUSSION:

From our observations, trial and error, calculations, and data, we have successfully been able to come up with a discovery lab for other students to follow. We would now recommend that only maximum values are used from the measurements. It worked this way because when viewing the moons and Jupiter on the elliptical plane, depth cannot be observed, therefore the furthest out you observe a moon would be closer to the true orbital distance.

Our numbers are off from the given values for a few reasons. One, it is very hard to get accurate measurements from using a telescope and a micrometer. Two, we used a conversion method that was based on what we observed to be Jupiter's diameter in micrometer units. If we had known the actual value in kilometers of a micrometer unit, our numbers would be even closer.

Io turned out to be the best moon to use for accuracy because of it's fast orbital period. It revolves around Jupiter in less than two Earth days, so it is the easiest to observe and find it's maximum orbital distance, as it moves the most out of all the moons. We were able to find a accurate mass for Jupiter from all our measurements simply
because Jupiter is so massive.

Our results made sense with the work done by other groups. We were happy that our caculations ended up relatively close to the given values. Io's calculations were the most helpful in finding accurate mass. Some further work could be done by exploring the topic of black holes. We touched upon this subject with our lab, but didn't dive into it deeply. We feel confident that following our discovery lab that we have designed will allow future classes to repeat our lab and exceed it using their own research. Kepler would be proud.

REFERENCES:


Astronomy 161: The Solar System. http://csep10.phys.utk.edu/astr161/lect/index.html

C.D. Murray & S.F. Dermott. Solar System Dynamics. 1999

Christian, John Robert. On Tycho's Island. Cambridge University Press, Cambridge, 2000.

Ed Stephan's Excellent Animation Web Page. http://www.ac.wwu.edu/~stephan/

Gore, Pamela J. W. http://www.dc.peachnet.edu/~pgore/astronomy/astr101/kepler.htm

Hamilton, Calvin J. http://www.solarview.com/eng/io©1997-2001.

Hetherington, Norriss S, editor. Encyclopedia of Cosmology. 1993

http://observe.ivv.nasa.gov/nasa/education/reference/orbits/orbit_sim.html

Ian Ridpath and John Woodruff, editors. Cambridge Astronomy Dictionary. 1995

Jenning, George A. .Modern Geometry with Applications. 1994

Kepler's Laws With Animation. http://www.cvc.org/science/kepler.htm

Kozhamthadam, Job. Discovery of Kepler's Laws. 1994

Morrison, David (Editor). Satellites of Jupiter. University of Arizona Press, Tuscon, Arizona, 1982.

Murdin, Paul, Editor-in-Chief. Encyclopedia of Astronomy and Astrophysics, Vol. 2. 2001.

Peek, Bertrand Meigh. The Planet Jupiter. Faber and Faber, London, 1958.

Pribble, et al. "Final: From the Family Tree of Old School Astronomy" http://jrscience.wcp.miamioh.edu/nsfall98/finalarticles/final.fromthefamilytreeof.html

Price, Fred W. The Planet Observer's Handbook. 1994

Shipman, Harry L. Black Holes, Quasars and the Universe. Houghton Mifflin Company, Boston, 1976.

The Art of Renaissance Science. http://www.pd.astro.it/ars/arshtml/arstoc.html

USNO Astronomical Applications Department Website. http://aa.usno.navy.mil/

Van Helden, Albert. The Galileo Project. Rice University. http://es.rice.edu/ES/humsoc/Galileo 1995.

Windows to the Universe. http://www.windows.ucar.edu. University Corporation for Atmospheric Research ©1995-1999, 2000. The Regents of The University of Michigan.

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