Draft 1: THE RETURN OF KEPLER!!

This topic submitted by Austin Kleon, Ryan Lazowski, Laura Englehart, Miguel Padilla (kleonaw@muohio.edu, lazowsrd@muohio.edu, englehle@muohio.edu, padillmr@muohio.edu) at 9:43 am on 10/4/01. Additions were last made on Friday, April 19, 2002. Section: Cummins

(From Kepler's Laws With Animation)

1. INTRODUCTION

A. Purpose

Our purpose is to prove Kepler's
Third Law by observing the four major moons of Jupiter (Io, Europa, Ganymede,
Callisto). Our hypothesis is that the data we collect will help us prove the
Third Law.

B. Objectives

The primary goal of this lab is to
prove Kepler's Third Law. Most of our lab will be based on the work of Jeremy,
Binnie, Dan, Lorraine, and Maxwell during their 1998 NS lab. We hope to improve
on their lab by noting their errors, and also expand on it by using Newton's
reworking of Kepler's law to figure out masses of planets, and maybe
black holes. In the process of executing the lab we will gain knowledge and
experience of using the telescope and observing the planets. 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.

C. Motivational Interest

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. Proving Kepler's Third
Law is appropriate: it is a manageable task, but still promises to be challenging,
interesting and fun. Plus, we have been challenged by the old Keplerytes that
a better job cannot be done!

2. RELEVANCE

A. Historical Background

For the purposes of our explanation to the class, we will keep our historical
information brief. Back in the days of Ptolemy, it was believed that the earth
was the center of the universe. Despite flaws in this model, the geocentric
view was unchallenged for centuries. Copernicus challenge this view by claiming
that the earth and the other planets rotate about the sun, which is stationary
and central. Kepler agreed with Copernicus. Kepler's predecessor, Tycho Brahe,
was skeptical of both models. But it was Brahe's observations and data collections
that led Kepler to prove the heliocentric model correct and formulate his three
laws. Galileo's observations of the Jovian moons helped Kepler's laws to be
more commonly accepted. We are trying to keep Galileo as our inspiration, since
we'll be doing the same observations as he did. Here are some images of what
Galileo observed:

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)

Despite all this, Kepler's Third
Law could not be proven until Newton's gravitational laws were devised and the
Third Law was revised(Gore, Pribble et al.).

B. Background Information

Here we will attempt to illustrate Kepler's Three Laws, Galileo's observations
of the Jovian moons, 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. Please bare with us:

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

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

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:

P^2 = k(M^3)

where P = the period of revolution
of a planet, k = constant, M = 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. "k" is basically gravitational attraction.

To find M, we will have to take the
average of the extreme measurements of the elliptical moon orbit from Jupiter:

M= (X1+X2)/2

where X=extreme measurements on both
sides of Jupiter

To find the period, we will have
to use our timed observations of the different moons.

It is essential to note Newton's
discoveries. His theory of gravity was the only thing to finally prove this
Third Law. Newton also reworked the formula to look like this:

(m1+m2)P^2 = (d1 + d2)^3 = R^3

Here, m1 = the mass of jupiter, m2
= a moon, and R=distance between them. Using this formula, we are hoping to
be able to determine the mass of Jupiter by observing the moons orbit.

This is all very confusing, and hopefully
we will clear up all questions with our class participation session. (This information
was gathered from the website, Astronomy
161: The Solar System, and Pribble et al.)

4. MATERIALS AND METHODS

A. Materials

Telescope, micrometer eyepiece for
measurements, data sheets, writing utensils, hot chocolate

B. Timetable

We have accesed the USNO
Site and procured the rise, peak, and set times of Jupiter for observation.
On Monday, October 8, 2001 we will go out for the first time to learn how to
use the telescopes. Dan Pribble and Jeremy Anthony will show us how to use them
and how to take measurements. After familiarizing ourselves with the telescopes
on Monday the 8th, we will dedicate one night to mapping out the four moons
of Jupiter, just as did the great Galileo (this was Hays' idea).

We plan to go out and take our first
real measurements and observations of Jupiter on Friday the 12th in the early
morning. After that we plan to take measurements 2-3 times a week for a month.
We also plan to have a class participation session, where we will attempt to
explain the laws in greater detail, and observe Jupiter one night.

C. Methods

We will observe and record the position
of each of Jupiter's moons over ten minute time intervals for a period of two
hours a night using the telescope and micrometer. One person will observe and
another will record the data. Once all data is obtained, we will then insert
our data into Kepler's formula, in an attempt to prove the formula with each
moon. See above for how these calculations will be made. This will be the trick
to our whole lab, and the mathmatical calculations are mind boggling!

Click
here for our Data Sheet!

5. CONCLUSION

WE ARE PUMPED! We're ready to get
out there and start stargazing. Hopefully we can exceed Pribble and his Keplerytes!

6. REFERENCES:

Gore, Pamela J. W.

http://www.dc.peachnet.edu/~pgore/astronomy/astr101/kepler.htm

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

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

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

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

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

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

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