JupiterMass
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Georgia State University *
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Course
1010
Subject
Astronomy
Date
Apr 3, 2024
Type
Pages
7
Uploaded by HighnessCloverLapwing39
1 Mass of Jupiter ASTR 1010 Name: Overview In this activity you will calculate the orbital periods of the Galilean moons. Using Isaac Newton’s updated version of Kepler’s Third Law of Planetary Motion for two bodies, y
ou will also calculate the mass of Jupiter. Objectives After completing this activity students will be able to: •
Use Microsoft Excel (or other spreadsheet program) to graph the orbital period of a celestial body. MAKE SURE ALSO YOU SUBMIT YOUR EXCEL SHEET WITH YOUR LAB MANUAL TO THE ICOLLEGE ASSIGNMENT!
•
Compare the orbital periods of other moons in the Solar System to our Moon. •
Calculate the mass of Jupiter using orbital parameters of the Galilean moons. •
Calculate a percent error on measurements. **Note
: If a question is labeled “
THOUGHT QUESTION” we are looking for you to show critical thinking/justification in your answer, not a “correct” answer**
Definitions Here are some terms from lecture that we will be using today in lab:
•
Galilean moons
–
the four largest moons of Jupiter –
Io, Europa, Ganymede, and Callisto. Observed by Galileo in late 1609/early 1610. •
Period –
the length from one peak/trough to the next (or from any point to the next matching point) of a periodic function (see figure in Part 2). •
Amplitude –
The height from the center line to the peak or trough of a periodic function. (see figure in Part 2). •
Kepler’s Third Law of Planetary Motion for two bodies –
P
2
(M
1
+ M
2
) = a
3
, where P is the orbital period in YEARS,
a is the semi-
major axis or radius of the orbit in AUs
, and M
1
and M
2
are the masses of the orbiting bodies in SOLAR MASSES (M
☉
).
Image credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)
2 Part 1. Graphing Orbital periods of Galilean moons In normal lab circumstances, you would get to impersonate Galileo today (minus the whole eye damage from staring at the Sun and the house arrest bits…) and observe Io, Europa, Ganymede, and Callisto move through their orbits around Jupiter. In his work, Sidereus Nuncius
, Galileo included drawings of what he observed between January and March 1610. (Full set of images found here: http://www.astro.umontreal.ca/~paulchar/grps/site/images/galileo.4.html) Using the simulator in the lab, you would do the same, but it would look more like this: There is also a short video in the lab module showing what the simulator looks like as you click through time intervals. Since we are unable to be in the lab right now, you have been provided with an Excel file that has measurements recorded for you. Here is a screen capture of the Excel file, let’s walk through the different parts: •
Red Circle
–
Data columns. This is the data you will use to create your plot. •
Blue Square
–
Once your plot is made, put it here! •
Blue Circle
–
Each moon has its own tab. You must plot and do calculations for each moon! •
Red Square
–
complete your calculations here. This is where your TA will be grading your calculations, so please make sure you place them in the correct spot!
3 Now that you are fa
miliar with the file, let’s make som
e plots! These instructions will also be covered in the pre-lab lecture video. a.
Select the data in columns A (x-axis, time) and B (y-axis, distance from Jupiter. +/- indicate if the moon is being observed on the left or right side of Jupiter) in your Excel file (minus the first row!). b.
On the menu bar, click on ‘Insert’, click on ‘Scatter’, ‘Scatter with Smooth Lines and Markers’
. c.
Place plot in requested area and change the title to the name of the moon you are analyzing. To verify that you are on the right track, your plot for Io should look like this →
If you have a match, make the plots for the other three moons. Your x-axis will be in units of hours and your y-axis will be in units of Jupiter diameters (1 Jupiter diameter is about 140,000 km). Before you start calculating, consider your plots. 1.
THOUGHT QUESTION. Looking at all four of your plots, what do you notice about the differences in values on the x-axis? How do you think difference will influence the orbital periods? (Hint: the x-axis is in units of time!) 2.
THOUGHT QUESTION. Looking again at all four plots, now consider the y-axis. What orbital parameter do you think is influenced by the maximum/minimum y-values? (Hint: the y-axis is in units of distance!) 3.
THOUGHT QUESTION. Based on your answers for Questions 1 and 2, which moon (Io, Europa, Ganymede, Callisto) do you think is closest to Jupiter and which is most distant?
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4 Part 2. Calculating Orbital Parameters! Now that your plots are ready, it is time to find orbital parameters. First, we need the orbital period. The period the length from one peak/trough to the next (or from any point to the next matching point) of a periodic function. This figure shows a period, indicated by the green line, starting at t
1
and ending at t
2
. Look at your plot. Find the x-
values that have matching (or very nearly matching) y-values and record the x-values in t
1
and t
2
in your Excel sheet. Then, next to the cell called ‘in hours’, calculate the period in hours using Equation 1: P = ?
2
− ?
1
(𝐸??𝑎?𝑖?? 1)
To do your final calculation, you need your period to be in years, so first convert hours to days by dividing your value in hours by 24 and record that next to the ‘in days’ cell, and then divide your value in days by 365 to convert your orbital period into years. Record that value next to the ‘in years’ cell.
Calculate orbital periods for all of the moons. 4.
Which of the Galilean moons has the shortest period and which has the longest? 5.
How do the orbital periods of the Galilean moons compare to the orbital period of the Earth’s Moon? (Note. The orbital period of the Moon is ~27 days)
6.
THOUGHT QUESTION. Are you surprised by the orbital periods of the Galilean moons? If yes, why? And if no, why not?
5 Next, you need to calculate your amplitude. Amplitude is the height from the center line to the peak or trough of a periodic function. This figure shows the amplitude for both the peak and trough in purple. The easiest way to find the amplitude in your data is to look at the ‘Distance from Jupiter’ column in your data and find the largest number (positive or negative). Record that value next to ‘in Jupiter diameters’. NOTE! If your amplitude value was negative, take the absolute value and make it positive!
In order to correctly calculate the mass of Jupiter, the semi-major axis value needs to be in AU instead of Jupiter diameters. MULTIPLY YOUR AMPLITUDE VALUE BY 9.545x10
-4
TO CONVERT FROM JUPITER DIAMETERS TO AU.
Calculate and convert the amplitudes for all of the moons in your Excel sheet. 7.
Which of the Galilean moons is closest to Jupiter and which is the furthest? 8.
THOUGHT QUESTION. The Moon has an average distance from Earth of about 30 Earth diameters (or approximately 3 Jupiter diameters). Why do you think our Moon is closer to Earth than 3 of the 4 Galilean moons are to Jupiter? (Hint. How does gravity’s pull depend on mass and distance?) With orbital periods in YEARS
and semi-major axes in AU
now calculated, you are ready to calculate the mass of Jupiter! Part 3. Calculating the Mass of Jupiter! To calculate Jupiter’s mass, we need to consider Newton’s update of Kepler’s 3
rd
law. In the equation there is a dependence on the total mass of the two orbiting bodies. Under different
6 circumstances, this would complicate things for us but fortunately, we are dealing with a large planet and a tiny moon. In this scenario, we can assume that the moon has almost no relative mass to Jupiter. Looking at the equation, P
2
(M
1
+ M
2
) = a
3
, we can set M
1
equal to Jupiter’s mass and M
2
, which would be the mass of the moon, equal to zero! This simplifies our equation to M
1
*P
2 = a
3
. Rearranging this, you will use Equation 2 to solve for the mass of Jupiter: M
J
=
𝑎
3
𝑃
2
(𝐸??𝑎?𝑖?? 2)
Where a is the semi-major axis in AU and P is the orbital period in YEARS. Go back to your spreadsheet and calculate Jupiter’s mass with each of the moons in solar masses (M
☉
).
Let’s now compare Jupiter’s mass to the mass of the Earth! The Earth has a mass of 3.0x10
-6
M
☉
. To convert your Jupiter masses from solar mass to Earth mass (M
⊕
), divide your mass of Jupiter by the mass of Earth. The accepted mass of Jupiter is about 318 M
⊕
. Calculate the percent error of your measurements versus the accepted value using Equation 3: % error =
|𝑌??? ?𝑎𝑙?𝑒 − 318|
318
∗ 100 (𝐸??𝑎?𝑖?? 3)
Note that the numerator of the percent error equation is an absolute value. This means you are looking at the magnitude of the value, so if your value minus 318 is negative, the absolute value turns it positive. Your percent error should not be negative! Calculate the percent error for your four measurements and let’s analyze your measurements! 9.
Which of your mass measurements had the lowest percent error? 10.
Which one had the highest percent error? 11.
THOUGHT QUESTION. What do you think could have contributed to you error value? 12.
THOUGHT QUESTION. You learned an important lesson today. Scientists don’t base results off of a single data point but gather multiple measurements. Why do you think they do this?
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7 Part 4. Is Jupiter a failed star?!? To finish the lab, let’s put Jupiter in context with other bodies. Jupiter is sometimes referred to (incorrectly) as a failed star. The smallest known stars have a mass of around 0.1 M
☉
(10% the mass of the Sun)
. Using your measurements, let’s find out how many times more massive Jupiter would have be in order to be the smallest star. 13.
Using the moon from Question 9, what is its mass in SOLAR MASSES
? M
J
= 14.
How many times more massive would Jupiter need to be in order to become a small star? (Hint. Think about it like an algebra problem (M
J
* x = 0.1) and solve for x!) To complete this assignment for grading: •
File →
Save As… →
Rename the file ‘YourLastName –
JupiterMass
Lab’
for BOTH
lab manual and Excel file. •
Upload both files to the ‘Mass of Jupiter’
assignment in iCollege (click Add Attachments →
Upload →
upload renamed saved file →
Update). •
Complete the Reflection activity on iCollege •
Consider what the phases of our Moon would look like if its orbital period was as short as Io’s!