### Data analysis, calculations, and report

#### Directions

**1. Plot your data**

In Lab 1, Part 1, you should have taken data using the Moons of Jupiter simulator for 20 days for the orbit of the moon Ganymede. Now, I would like you to create a plot of this data.

For information on plotting and tools you can use to create your plot, see "Plotting, Part I" in EARTH 501.

Using a program like Microsoft Excel, one of the tools listed on the EARTH 501 lesson, or plain old paper and pencil, please produce a 2 dimensional plot with the Julian Date plotted on the x-axis and the amplitude of the separation between Ganymede and Jupiter (in Jovian diameters) as the y-axis. When plotting your data, be careful to use the + / - signs the simulator reports for the separation between Jupiter and Ganymede.

You should see, as mentioned on the content background page, that your data describes a fair approximation of a sine curve.

**NOTE:** You will be submitting this lab as a single document that is in either Microsoft Word (.doc) or PDF (.pdf) format so I can open it. Your document will need to include the plot you just created, as well as a written report you will complete below.

Since there are a variety of ways you may create your plot, you will need to consider how you will incorporate it into your submitted document. I suggest you either save it first as a graphics files (.jpg, .pdf, or .tiff) which you then paste into your document, or if you use a web plotting program that allows you to save your plot as a hyperlink, simply include the URL in your document.

**2. Analyze your data**

When scientists are faced with a plot like the one that you just created, they wish to extract useful information from it. Your ultimate goal is to calculate the mass of Jupiter. In order to do so, you need to extract from your plot the period of Ganymede's orbit and its semi-major axis.

I think your data quality should be high enough that you can simply eyeball these two quantities. To do this, estimate how many days there are between two successive peaks or two successive troughs on the plot. If you do not have two peaks or two troughs worth of data, the time from peak to trough is half of the period, so you can also measure that quantity and then double it.

To get the semi-major axis, you should determine the height of a peak or the depth of a trough.

If you have experience doing data analysis like this, you may be familiar with the technique of calculating a "line of best fit" or a "curve of best fit." If you have this experience, please feel free to use any tools you are familiar with to do the fitting of a sine curve to your data, which should then provide you with a more precise estimate of the period and semi-major axis for Ganymede's orbit. This is * not* a requirement for completing this lab—it is an option only for those wishing to approach the problem in this way.

**3. Calculate**

At this point, you should have a determination of the period of Ganymede's orbit in days and its semi-major axis in units of Jovian diameters.

For the first calculation, please convert these quantities to years and astronomical units (AU) using the following relationships:

1 year = 365.25 days

1 AU = 1,050 Jovian diameters

Next, we want to calculate the mass of Jupiter. Recall the formula for Kepler's Third Law from the content background:

P^{2} = (4π ^{2} x a^{3}) / G(m_{Jupiter} + m_{moon})

If you use the units of P in years, a in AU, and the masses in solar masses, the constants drop out, and the simplified equation is:

P^{2} = (a^{3}) / (m_{Jupiter} + m_{moon})

Calculate the sum of Jupiter's and its moon's mass in solar masses.

Although this equation gives you the sum of Jupiter's and Ganymede's mass, you can assume that Ganymede's mass is small enough that it can be ignored, so the value you calculated for the mass previously is an estimate of Jupiter's mass alone. Finally, let's convert that to a more useful unit:

1 solar mass = 1.99 x 10^{30} kilograms

Convert the mass of Jupiter from solar masses to kilograms.

**4. Report your results**

You should now have successfully determined the mass of Jupiter by making simple observations of the position of its moons over the course of about a month! Please write a brief summary of your work. In your summary, include the following:

- your table of data from the simulator;
- your estimates for the period and semi-major axis of Ganymede's orbit in both sets of units (days & Jovian diameters, as well as years and AU);
- your calculated mass of Jupiter in solar masses and in kilograms;
- look up the mass of Jupiter in kg, and note how well your calculated value compares to the accepted value;
- a brief (paragraph or so) discussion of the lab and how well you think it illustrates some of the concepts we studied in Lessons 2 and 3.

**5. Save your work **

Save your work AS A SINGLE DOCUMENT in either a Microsoft Word or PDF file in the following format:

Lab1_AccessAccountID_LastName.doc (or .pdf).

For example, student Elvis Aaron Presley's file would be named "Lab1 _eap1_presley.doc" - This naming convention is important, as it will help me make sure I match up each submission with the right student!

#### Submit your work

Please submit your work to Lab 1 in Canvas by the due date indicated on our Canvas calendar.

#### Grading criteria

See the grading rubric for specifics on how this assignment will be graded.