University of Michigan - Department of Astronomy

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Version: Introduction

Kepler's 3rd Law and the Moons of Jupiter

Introduction

Mensus eram coelos, nunc terrae metior umbras,
Mens coelestis erat, corporis umbra iacet.

Imeasured the skies, now I measure the shadows
Skybound ws the mind, the body rests in the earth.

--Kepler's epitaph

Overview

Introduction

In 1543 Nicolaus Copernicus hypothesized that the planets revolve in circular orbits around the sun. Tycho Brahe (1546-1601) carefully observed the location of the planets and 777 stars over a period of 20 years using a sextant and compass. Tycho actually sought to prove Copernicus wrong, but he needed help, so he hired Johannes Kepler to analyze the data. Kepler believed in the Copernican model, and after Tycho’s death in 1601 he set out to use Tycho’s data to prove it was correct.  This lead him before the end of the decade to deduce three empirical mathematical laws governing the orbit of one body around another. Using these laws, we can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them.

Kepler's third law for a moon orbiting a much larger parent body is Kepler's 3rd law where M is the mass of the primary body, in units of the solar mass, a is the length of the semi-major axis in astronomical units and P is the period if the orbit in Earth years. Altough Kepler was able to deduce these laws from his observations, he did not know why they worked.  In particular, they would only work with these particular units.  It would be another 80 years before Newton would develop a theory of gravitation that would show not only why Kepler’s laws worked, but also why the units chosen by Kepler were necessary.  Newton also figured out the conversion factors needed to use more common units -after all, length cubed over time squared does not normally equal mass!

By 1609 the telescope had been invented, allowing the observations of objects not visible to the naked eye. Galileo obtained a description of the instrument, and built a telescope of his own.  He was the first person to use a telescope to observe the sky and to publish his observations. Galileo believed that the Jupiter system was especially important because it looks like a miniature version of the solar system, right down to following Kepler’s laws. Galileo believed this provided clear evidence that Copernicus' heliocentric model of the solar system was correct. Unfortunately for Galileo, the inquisition took issue with his findings, or perhaps with his characterization of church doctrine as “simple-minded”.  He was tried and forced to recant, then spent the last few years of his life under house arrest.

Galileo first observed the moons of Jupiter on Jan 7 of 1610.  He wrote his observations on scrap paper, obviously just making a few notes with no real idea about what he was about the find.  A few days latter, he realizes what he is looking at, and starts making more detailed notes. You can see these notes in the University’s Special Collections library, on the 7th floor of the Grad library.  For more information on the special collections, see http://www.lib.umich.edu/special-collections-library.  An image and translation is available online at http://www.lib.umich.edu/special-collections-library/galileo-manuscript.

Using the software:

In this activity, you will be using starry Night to simulate the observations and measurements Galileo would have made of the four brightest moons of Jupiter.  There are two differences in your simulated observations and what early astronomers could have done: the computer will tell you what the angular separation is, and there are no cloudy nights.  You also have the options of turning off daylight and the horizon, so you can observe any time you want.

Starry Night uses files called "favorites" to store pre-set-up programs. The favorite for this activity is called Jupiter.snf.

Open the astroclass folder on the Dock (it will be the plain folder icon) by clicking on it. Right click on Jupiter.snf, choose "Open With >" and select Starry Night.

The favorite opens at 8 PM on January 7 of 1610 in Pisa, Italy, with a field of view of roughly 17’.  This is approximately the view Galileo would have had through his telescope had he been looking at 8 PM.
If at any time you get mixed up, you can close the favorite (File -> Close) and re-open it from the astroclass folder.  Do not save over it.  If you save over it, you will need to go to https://dept.astro.lsa.umich.edu/ugactivities/Labs/moons/Jupiter.snf and download a new copy (right-click and choose "Save Link As" or "Download linked file" and save it to the desktop).

Controls you need for this activity

The zoom control is on the right side of the toolbar.  You want the zoom to be between 10’ and 25’ when you are measuring the angular separation of the moons.  If you zoom out until you can see the horizon, you can see the cardinal points so you know what direction you’re facing.

img 6Make sure the angular separation tool is selected before measuring the angular separation. The tool selector is on the far left side of the toolbar (see image, right.)  For most of this activity, you will want to use the angular separation tool.

The Hand wil let you grab the screen and move around using the mouse. Moveing around this way will cause you to loos the lock on Jupiter. To get it back, go to the Find tab (on the left side), click the Find button button next to Jupiter, and select “Centre”.

The time control is right next to the tool selector. to change the time, click the part you want to change and use either the arrow keys to adjust time up and down, or type a number to jump to a different time.

A complete starry night quick reference is available at: https://dept.astro.lsa.umich.edu/ugactivities/Labs/Comp/index.html

The Starry Night User's Guide (a searchable pdf document) is under the help menu in Starry Night.

Activities and Resources

Discussion version

lab version

Starry Night Favorite


Last modified: 10/31/12 by SAM

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