Name: __________________

Lab Partner: __________________

Instructor's Signature: __________________

Lab Title: Kepler’s Third Law, and The Moons of Jupiter

Equipment:

Scientific calculator, color Macintosh computer, the Excel 98 program, Kepler’s Third Law, and the Contemporary Laboratory Experiences in Astronomy (CLEA) computer program Moons of Jupiter.

Purpose:

This lab is broken into two parts. The first part will help you to gain an understanding of Kepler’s Third Law of Planetary Motion. In the second part, you shall measure the orbital properties of Jupiter's moons, and analyze their motions using Kepler's Third Law in order to obtain the mass of Jupiter.

Requirements:

This lab is to be performed individually (or in groups of no more than two if there are not enough computers). If you must work in pairs, take turns manipulating the computer and taking the data. You should switch off from time to time so everyone gets a chance to use the computer. Each computer will use different initial data.

Although you may use the computer and the program with your partner to collect data, all calculations, graphing, and any narratives in your lab report must be your own original work! Your writeup for this lab will consist of two parts. The first will be the computer-generated report from the Kepler’s Third Law spreadsheet program (with any additional discussion that you wish to make), and the second will be the pages containing your work on the Moons of Jupiter program.

Introduction and Historical Background:

We can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them. In 1543, Nicolas Copernicus hypothesized that the planets revolve in circular orbits around the Sun. Tycho Brahe (1546-1601) carefully observed the locations of the planets and 777 stars over a period of 20 years using a sextant and compass. These observations were used by Johannes Kepler, a student of Brahe's, to deduce three empirical mathematical laws governing the orbit of one object around another.

In 1609, Galileo Galilei heard of the invention of a new optical instrument by a Dutch spectacle maker, Hans Lippershey. By using two lenses, one convex and one concave, Lippershey found that distant objects could be made to look nearer. This instrument was called a telescope.

Without even having seen an assembled telescope, Galileo was able to construct his own telescope with a magnification of about three. He soon perfected the construction of the telescope, and became famous as the builder of the world's best telescopes. His best telescopes had a magnification of about thirty.

Galileo immediately began observing celestial objects with his crude instrument. He was a careful observer, and published a small book in 1610 of his remarkable discoveries called the Sidereal Messenger. One can imagine the excitement these new discoveries caused in the scientific community. Suddenly, a whole new world was opened! Galileo found sunspots on the Sun, and craters on the Moon. He found that Venus had phases, much as the Moon has phases. He was able to tell that the Milky Way was a myriad of individual stars. He could see that there was something strange about Saturn, but his small telescope was not able to resolve its rings.

One of the most important discoveries was that Jupiter had four moons revolving around it. Galileo made such exhaustive studies of these moons that they have come to be known as the Galilean satellites. This "miniature solar system" was clear evidence that the Copernican theory of a Sun-centered solar system was physically possible.

Because he was developing a world view which was not easily reconciled with the religious dogma of his period, Galileo was compelled by the Inquisition to neither "hold nor defend" the Copernican hypothesis. Nevertheless, in 1632 he published his Dialogue on the Great World Systems which was a thinly disguised defense of the Copernican system. This led to his trial, his forced denunciation of the theory, and confinement to his home for the rest of his life.

In the second part of this lab, you are going to repeat Galileo's observations (without threat of government condemnation!). Today, however, we also know the size of Jupiter. Jupiter's diameter is 11 times Earth's diameter, or about 1.5 ´ 10⁵ kilometers. The data collected in this experiment, and this information allow us to determine the mass of Jupiter.

Before doing that, however, you need to gain some additional insight into Kepler’s Laws of Planetary Motion. You shall do this using the Excel 98 spreadsheet program, Kepler’s Third Law.

Kepler’s Third Law

Kepler’s Third Law is a computer-based lab; it is self-guided and self-contained. When the file starts, you will be directed through a series of frames that contain introductory material.

Do not skip the introductory material!

The labs contain procedures that require you to understand the introductory material. Read the material carefully. You have plenty of time, and you need it to understand what follows. If you skip it, you may not be able to finish the lab !

If you need to review any of the introductory material, simply select one of the introductory tabs, titled “Intro-X”, where “X” is the number of the introduction screen that you’d like to review, located at the bottom of the spreadsheet.

When you have completed all three procedures in this lab, you will be prompted for a filename in which to save your lab report. Be sure to use a unique filename and note the directory in which the file was saved. Your instructor may ask that you save the file as an HTML document. If he or she does, simply select the Options… button in the Save As… dialog box and select the HTML option.

Because your lab report is automatically generated, your score on this part of the lab will be based largely upon your comments and discussion, which we will use to indicate your level of involvement in and understanding of this part of the lab. Make certain that all of the comments that you intend to make are, indeed, included in your lab report, and make sure to add any comments that you want to beyond those allowed by the word limitation imposed by Excel 98. In addition, at the end of your lab report, you should add an additional page of comments and conclusions in your own words, as opposed to the rest of the comments in your report, which will be the same for everybody in your group.

Getting Started:

1) Open (double-click on): Astronomy 30 folder -> Kepler folder -> keplersthirdlaw file (it is a normal Microsoft Excel spreadsheet file icon).

2) Select “Enable Macros” when asked by Excel 98 (this is important).

3) Select “Medium Size” for monitor (small will be unreadable).

4) (VERY IMPORTANT) When done, select “Don’t Save” (from Don’t Save / Save / Cancel).

If you are restarting the lab, be sure to have the location of your previous Lab Report file handy. It is very important that you select Yes or OK when prompted to:

1) allow data to be pasted onto the clipboard,

2) allow data to be pasted into the lab report.

If at anytime you need to quit out of the lab, use the Exit button on either introductory page or the Quit now and save my Lab Report button in procedures 1 or 2.

Procedure 1

In this first procedure, you will build a few solar systems, starting with our own. The orbital radii for the nine planets in our solar system are listed at the top of the Procedure 1 spreadsheet. Enter those values, in order, in the grey input cells in the Orbital Radius in AU column.

Watch how the curves in the Orbital Data Plot change. Enter a solar system of your own design. Try different distances between the planets.

Note the following for your procedure summary:

1) The relationships of the different curves and what the different curves are.

2) What happens if each planet is five times farther than the last?

3) What happens if each planet is one AU farther than the last?

Procedure 2

In second procedure, you will explore the effects of mass on Kepler's third Law.

The Earth-Moon system distance is given in kilometers in Row 8 of the spreadsheet. You must first convert that value to AU using the Mileage Converter directly below it. You can use the mileage converter at any time to convert distance units.

Enter the distance in AU for Planet X in its grey input cell in the Orbital Radius in AU column. Next, enter the distance in AU for the Moon in its grey input cell in the Orbital Radius in AU column.

Note the following for your procedure summary:

1) How many miles is the Earth from the Moon? Is that more or less than you expected?

2) How does the orbital speed for Planet X, at that orbital distance from the center of the Sun, compare with the Earth's orbital speed? Its orbital period?

3) What is so different in the Earth-Moon system and the Planet-Sun systems that would create such different orbital data?

4) How does the orbital speed for the Moon, at that orbital distance from the center of the Earth, compare with orbital speed for Planet X at that distance from the Sun? How does the orbital period compare?

5) What would the Moon look like at this orbital radius from the Sun?

Procedure 3

In this last procedure, you are an astronomer who has just discovered an extra-solar planet. Your job now is to determine to orbital data for the planet so that you can issue a press release and get your name in the newspaper and a few physics journals, and impress your friends.

The Jupiter-sized planet is 103,000,000 km from its parent star, which weighs 3 solar masses. You must first enter the orbital radius in AU (use the converter) in the Orbital Distance column. Then you must devise a formula that calculates the orbital period in years.

Enter that formula in the grey input cell in the Orbital Period in Years column. All the information you need to complete your task is in the Procedure 3 spreadsheet.

In your procedure summary include an explanation of how Newton's revision to Kepler's law allows us to determine orbital data for any known system.

The Moons of Jupiter

KEPLER'S THIRD LAW (revisited): When one body such as a moon orbits around a much more massive parent body, Kepler's Third Law is:

where:

m is the mass of the parent body, in units of the mass of the Sun.

a is the length of the semi-major axis of the elliptical orbit in units of the mean Earth-Sun distance, 1 A.U. (Astronomical Unit). If the orbit is circular (as will be assumed in this lab), the semi-major axis is the same as the radius of the orbit.

p is the period of the orbit in Earth years. The period is the amount of time required for a moon to orbit the parent body once.

This law applies to planets orbiting about the Sun (check: for the Earth orbiting around the Sun: a = 1 A.U. and p = 1 year, and we obtain for the mass of the Sun: m = 1 solar mass) or to any moon orbiting around its planet.

You will be determining a and p for the Galilean moons of Jupiter and then m_J, the mass of Jupiter, in solar masses.

Actually, in this computer simulation, you will first determine a in units of Jupiter's diameter (J.D.) and the period p will be in Earth-days. You will convert these units to A.U. and years at the end of the lab.

Jupiter's Moons

This lab can in principle be done by anyone with a set of binoculars or a small telescope. The computer simulation Moons of Jupiter replaces actual observing sessions at the observatory using the telescope. The computer simulation is based on the real orbital data for each satellite. As a matter of fact, if you were to set the simulation for today's date and time, you could verify the position of the Jovian moons by direct observation through a telescope at an observatory. You will note that the computer also provides you some of the pitfalls of actual live telescopic observations such as occasional cloudy nights!

You will obtain data from 18 clear observing sessions making observations twice per evening, spaced 12 hours apart.

We could do this lab for any one moon of Jupiter. If we did the experiment very accurately, the answer for Jupiter's mass should be the same whatever moon we used. But there will be errors, and we shall use data collected on all four Galilean moons of Jupiter. They are named Io, Europa, Ganymede and Callisto, in order of increasing actual distance from Jupiter. You can remember the order by the mnemonic "I Eat Green Carrots". We also refer to them in this exercise as moons I, II, III, and IV. If you looked through any small telescope, the picture might look like this:

The moons appear to be lined up because we are looking edge-on to the orbital plane of the moons around Jupiter. As time goes by, the moons will move about Jupiter. Thus, while the moons move in roughly circular orbits, we generally see only the apparent distance of each moon from Jupiter's center as projected in the east-west direction which is perpendicular to the line-of-sight between Jupiter and Earth.

On the computer screen, Jupiter and its moons will look much like Galileo's original sketches. Remember that west is to the right and east is to the left on the screen. This is the way the sky looks through a telescope. It will be necessary to record the apparent (east-west) distance of a moon from Jupiter's center in units of Jupiter diameters (J.D.). Lucky for you, the measurement mode of the computer equipment you will use provides a direct readout in J.D.

The computer simulation will be presenting data on the moons as they would be seen every 12 hours.

Such observations are possible only in the winter time when the nights are long. The observations are complete when you have obtained a total of at least 18 actual observations – NOT counting cloudy days. For each moon and for each of the 18 observing sessions, you are to measure the apparent distance of the moon from Jupiter. The data you will collect will be placed on the Data Table.

Procedure:

Your instructor will inform you about which computers on campus currently have the program installed on their hard drives. Position the mouse cursor arrow over the icon (picture) of the Mac's hard disk and double-click on the mouse button. A list of the contents of the hard disk will appear. Move the mouse cursor over the Astronomy 30 folder's icon and again double-click the mouse button to reveal the contents. Position the mouse cursor over the folder Galilean Satellites and double-click the mouse to see the file list. Position the cursor over the Moons of Jupiter icon and double-click to run the program.

The Moons of Jupiter program simulates the operation of an automatically controlled telescope with a Charge-Coupled Device (CCD) camera that provides a video image to a computer screen. It is a sophisticated computer program that allows convenient measurements to be made at the computer console, as well as adjusting the telescope's magnification. The computer simulation is realistic in all important ways, and using it will give you a good feel for how modern astronomers actually collect data and control their telescopes. Instead of using a telescope and actually observing the moons for many nights, the Moons of Jupiter computer simulation shows the moons to you as they would appear if you were to look through a telescope at the specified time.

Entering Student Accounting Information

After the display of a couple of introductory graphics, a Student Accounting dialog box appears. Enter the names of each student working at that particular computer. Do not use punctuation marks. Press the tab key after each name to skip to the next entry, or place the cursor in the next field by clicking on the mouse button. Also enter the Laboratory Table (Computer) Number you are seated at for this experiment. Carefully check all of the information you have entered. When all the information had been entered to your satisfaction, click OK to continue.

Procedure for Initial Setup

The next dialog box to appear is called Start Date & Time. A set of startup values are needed by the computer to establish your initial observation session. The students at each computer will perform and analyze a different set of observing sessions. The Start Dates & Times for each computer are listed on a separate page that will be handed out to you before the lab. Fill in the table below by hand FIRST by consulting the handout page. The Observation Interval should be set to 12.00 hours.

Start Date & Time

Your Name:

Table (Computer) Number:

Year of Observations:

Month:

Day:

Universal Time: (24-hour clock)

Observation Interval: hours

Data is entered by first positioning the cursor (by moving the mouse around) over the item you wish to select, and double-clicking the mouse to select the entire contents of each box. You can then just type in the appropriate date and Universal Time for your particular computer number. The default is the current date.

Now you can enter the information into your computer in the Start Date & Time dialog box on your screen. When all of the information has been entered to your satisfaction, click OK to continue.

The Main Telescope Screen

After you have entered all of this information into the computer, it will display a screen similar to that shown below. If you wish to return to the original Start Date & Time screen at any time, go to the menu and select File ç Restart.

You control the observing session from this Main Telescope Screen. Notice that Jupiter is displayed in the center of your computer screen. To either side are the small point-like Galilean satellites. Sometimes a moon is behind Jupiter, so it cannot be seen. Even at high magnifications, the moons are very small compared to Jupiter. The current telescope magnification is shown in the upper left corner. The date, the UT (Universal Time -- the time in Greenwich, England), and the Julian Day (JD) are all displayed in the lower left-hand corner of the screen.

You may select Help from the menu to receive on-line help at any time while you are in the Main Telescope Screen. Within the Help menu are three choices that you can select:

Help ç How to Start advises you on how to set up the program before taking data,

Help ç How to Use Jupiter Lab explains how to manipulate the program in order to take data, and

Help ç About Help Windows explains how to use the Help windows.

To close the Help windows, click on the small close box located in the upper left corner of the window. You may leave the Help windows open while working if you prefer, in which case you will want to re-size the window. Position the cursor on the box located in the lower right corner of the window. Click and hold down the mouse button while re-sizing the window.

You can display the screen at four scales of magnification by clicking on the 100´, 200´, 300´, and 400´ buttons at the bottom of the screen. Try them now. To improve the accuracy of your measurement of a moon, you should always use the largest possible magnification which leaves the moon visible on the screen.

Be careful NOT to click the button marked Next just yet!

Now, position the mouse cursor over the screen and hold down the button. The measurement system turns on and displays the apparent perpendicular distance R (in J.D.'s) that the cursor is away from the center of Jupiter. Notice the edge of Jupiter is R = 0.5 J.D.

In order to measure the perpendicular distance of each moon from Jupiter, move the cursor until the moon is centered in the cross hairs and then hold down the mouse button. As you move the mouse, the current position of the cursor appears on the control panel. When the satellite is carefully centered, release the mouse button and information about the moon will appear at the lower right corner of the screen. This information includes the name of the selected moon, the X and Y pixel location on the screen, and the perpendicular distance R (in units of Jupiter's Diameter) from the Earth-Jupiter line-of-sight for the selected moon as well as E or W to indicate whether it is east or west of Jupiter. If the moon's name does not appear, you did not center the moon in the cross hairs exactly and you should try again! Record the reading, the date, and the identity of the moon.

To measure a moon, always switch to the highest magnification that still leaves the moon on the screen. It is important to use the highest magnification possible for each moon for best accuracy.

If a moon is behind Jupiter, record the apparent distance for that moon as ±0.5 since you don't know its location any better than this.

Now, begin the data collection process by recording the data for the first observations on the attached Data Table:

Column 1: Date

Column 2: Universal Time

Column 3: Day - number of the day (like 1.0, 1.5, 2.0).

Remember to keep incrementing the Day number even on cloudy days

Columns 4 – 7: Record each moon's position under the column for that moon. Count positions to the left (east) of Jupiter as negative and those to the right (west) as positive. If Europa were selected and had a R = 2.75 west of Jupiter, you would enter +2.75 in column 5.

When you have recorded the Universal Time and perpendicular distance for each moon, click the Next button and the image will advance by the amount of time you specified in the Observation Interval. Note that a certain percentage of observing sessions will be "cloudy".

If you encounter bad weather, just enter "Cloudy Day" in the space provided after you write in the Day number. Do NOT remeasure any of the Cloudy Days – you will just have to allow for any gaps in your data due to these days.

The observations are complete when you have obtained a total of 18 actual observations. Stop collecting data after you have 18 non-cloudy observations and when you have completely finished filling out the Data Table, select File ç Quit.

Remember, once you Quit the program altogether, you cannot continue where you left off!

Reduction Of The Data (Computer not needed from here on.)

You will use the data to obtain a graph that looks like this (the data points shown in the graph below are for an imaginary moon named Clea and is different from the moons in the lab).

Moon Clea

p = 14 days a = 3.0 J.D.

p = 0.0383 years a = 0.00286 A.U.

Each dot in the figure is one observation of Moon Clea. Note the irregular spacing of dots, due to clouds or poor weather and other observing problems on some nights. The curve drawn through the points is the smooth curve that would be made by Clea if you had enough observations. The shape of the curve is called a sine curve.

You will need to determine the sine curve that best fits your data in order to determine the orbital properties of each moon. Here are a few hints:

(i) the orbits of the moons are regular, that is, they don't speed up or slow down from one period to the next, and

(ii) the actual radius of each orbit does NOT change from one period to the next.

The sine curve that you draw through your data points should therefore also be REGULAR and SMOOTH. It should go through all of the points, and NOT be higher at the maxima in some places than others. It should also NOT be wider in some places than others.

Using the data from Moon Clea shown in the above graph, it is possible to determine both the radius, a, and the period, p, of the orbit for each moon. The period is the time it takes to get back to the same point in the orbit. Thus:

(i) the time between two consecutive maxima (or minima) is the period,

(ii) the time between two consecutive crossings at 0 J.D. is equal to 1/2 the period because this is the time it takes to get from the front of Jupiter to the back of Jupiter (or vise versa) which is 1/2 way around in its orbit,

(iii) the time between a crossing at 0 J.D and the nearest maximum or minimum is equal to 1/4 the period.

For some moons, you may not get enough observations for a full period, so these points may be of use to you in determining the period, even though the moon has not gone through a complete orbit.

On the other hand, if you have enough observations for several cycles, you can find a more accurate period by taking the time it takes for a moon to complete, say, 4 cycles, and then dividing it by 4. Your period must then be converted to units of years by dividing by 365.25 days. Remember, it is important to stay in the correct set of units if you are going to use Kepler's Third Law.

You can determine the semi-major axis (the radius), a, for each moon by measuring the maxima and minima in your smooth sine curves. When a moon is at the maximum position eastward or westward, it is at the largest apparent distance from the planet. Remember that the orbits of the moons are nearly circular, but since we see the orbits edge on, we can only determine the actual radius of the orbit when the moon is at its maximum position eastward or westward from Jupiter.

Repeat Steps 1–4 below for each of the four Jovian moons that you measured:

Step 1. Enter the data for the four moons on the graphs provided for each one. Along the horizontal scale, write day numbers starting on the left with the number of the first day on which you have data. Calibrate the horizontal scale of each graph to make your data as stretched out as possible. (The graph for Io, in particular, will benefit from this stretching). The vertical scale is already marked. Each day's measurement of a moon's apparent separation from Jupiter should give you one dot on the graph for that moon. Since each observing day has two sessions, be sure that each point on the graph on the horizontal axis represents only one session. Remember NOT to plot any points for Cloudy Days since you don't know where any of the moons are on those days.

Step 2. For each moon, draw a smooth sine curve through the points. Mark all maxima and minima on the curve by crosses. They need not fall on one of the grid lines. The curve should be symmetric about the horizontal line corresponding to zero apparent separation. The maxima and minima should have the SAME values, except for their sign.

Step 3. Read off the period, p, and the semi-major axis, a, from your figure, in the manner shown in the earlier example for Moon Clea. These units will be days for p and J.D. for a.

Step 4. To obtain p in years, divide your result in days by 365.25 since there are 365.25 days in a year. To obtain a in A.U., divide your result in J.D. by 1050, since there are 1050 Jupiter diameters (J.D.) in one A.U. Enter your converted values in the spaces provided at the bottom of each graph.

You now have all the information you need for each of the four moons to use Kepler's Third Law, to determine the mass of Jupiter, m_J, as:

where: p has units of years,

a has units of A.U.'s, and

m_J has units of solar masses.

Data Table

Example:

(1) Date	(2) Time	(3) Day	(4) Io	(5) Europa	(6) Ganymede	(7) Callisto
7/24/93	00:00:00	1.0	+2.95	+2.75	–7.43	+13.15
7/24/93	12:00:00	1.5	–0.86	+4.70	–6.30	+13.15
Your	Data:
(1) Date	(2) Time	(3) Day	(4) Io	(5) Europa	(6) Ganymede	(7) Callisto

** Keep Collecting Data Until You Have 18 non-cloudy Observations Above**

p (period) = _________ days a (semi-major axis) = ________ J.D.

p (period) = _________ years a (semi-major axis) = ________ A.U.

p (period) = _________ days a (semi-major axis) = ________ J.D.

p (period) = _________ years a (semi-major axis) = ________ A.U.

p (period) = _________ days a (semi-major axis) = ________ J.D.

p (period) = _________ years a (semi-major axis) = ________ A.U.

p (period) = _________ days a (semi-major axis) = ________ J.D.

p (period) = _________ years a (semi-major axis) = ________ A.U.

Use the space below to show your four separate calculations of Jupiter's mass based on your observations of each Galilean moon. Don't forget to include the appropriate units for the mass of Jupiter.

Summarize your calculations for the mass for Jupiter in each of the four cases below:

From moon I, Io: m_J = ______________________ solar masses

From moon II, Europa: m_J = ______________________ solar masses

From moon III, Ganymede: m_J = ______________________ solar masses

From moon IV, Callisto: m_J = ______________________ solar masses

Hint: If one of the values is very different from the other three, look for a source of error. Perhaps the data are not adequate for a better result, in which case leave the value as you obtained it.

Average m_J = ______________________ solar masses.

Additional Questions and Discussion:

1. Convert your average value for m_J into kilograms by multiplying your previous result (in solar masses) times the mass of the Sun in kilograms (this can be found in your textbook). What is the percent difference between your average value for m_J and the accepted value (also given in your textbook) for the mass of Jupiter in kilograms ?

2. To express the mass of Jupiter in units of the mass of the Earth, divide your result from Question 1 above by 5.97 ´ 10²⁷ grams, which is the mass of the Earth. SHOW YOUR WORK BELOW.

m_J = ______________________ Earth masses.

3. Which of the four Galilean moons was the most difficult to fit a smooth sine curve to? Why?

4. How would you change the time interval between observations for this "difficult" moon? (Remember that since you are observing from the surface of the Earth, you are still stuck with a certain amount of time each day when Jupiter is not visible!)

5. When a moon was not visible because it was behind Jupiter, you entered 0 J.D. for its apparent distance from Jupiter. However, all you really know from your measurement is that it's apparent distance was somewhere between the edges of Jupiter at +0.5 J.D. and at –0.5 J.D. How would you account for this uncertainty when drawing your smooth sine curve through all of your "0 J.D." points?

6. Suppose that you were only able to get telescope time to observe Jupiter's moons once every 4 days instead of once every 12 hours as in this lab. Which of Jupiter's four moons would you still be able to fit accurately to a sine curve and which ones would be very difficult to fit? Why?

Conclusions and Comments

Name: __________________

Lab Partner: __________________

Pre-lab Exercises: The Moons of Jupiter

1. The orbit of the Earth's Moon has a period of 27.3 days (7.47 ´ 10^-2 years) and a radius (semi-major axis) of 3.84 ´ 10⁵ km (2.56 ´ 10^-3 A.U.). What is the mass of the Earth? In what units? Show all of your work below using Kepler's Third Law as shown in the lab write-up.

2. There are actually many smaller moons beyond the orbit of Callisto. Will they have longer or shorter periods than Callisto? Why?

3. Use the tables in your astronomy text to find the diameter of Jupiter in kilometers.

1 J.D. = km

Also use the tables in your textbook to find the value of 1 Astronomical Unit in kilometers.

1 A.U. = km

(You may have to convert from centimeters or meters into kilometers in order to do this.)

Now, divide the length of an A.U. (in km) by the diameter of Jupiter (in km) to find out how many J.D. are in 1 A.U. SHOW ALL OF YOUR WORK BELOW.

1 A.U. = J.D.

4. Which do you think would cause the larger error in determining the mass m when using Kepler's Third Law:

(a) A ten percent error in p? You can evaluate this error by using a = 1 and p = 1 + 10% = 1.1 in Kepler's Third Law above to calculate:

=

= %

(b) A ten percent error in a? You can evaluate this error by using a = 1 + 10% = 1.1 and p = 1 in Kepler's Third Law above to calculate:

=

= %

5. Based on your answers in Exercise 4 above, which measurement should therefore be done the most carefully for each of Jupiter's moons: the period p or the semi-major axis a? Why?