Project1-Instructions-FALL2023
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School
Georgia Southern University *
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Course
1000
Subject
Astronomy
Date
Jan 9, 2024
Type
Pages
3
Uploaded by ElderIbex1351
1
ASTR1000A
Fall 2023
Project 1 -
Kepler’s Laws
–
Instructions
–
To better understand this project, you may need to review the folio Module “
Week 2
Orbits and Gravity”,
section
“Kepler’s Laws of Planetary Motion”.
Follow the instructions below to complete the project worksheet.
Part I. Elliptical Orbits
For this part you will need to run a computer simulation that visualizes the motion of planets around a star. The
simulation is available online. Go to:
https://phet.colorado.edu/sims/html/gravity-and-orbits/latest/gravity-and-orbits_en.html
When the initial menu appears, click on option
“
To Scale
”
.
STEP 1.
Once the screen of the simulation appears, check the options “Velocity”, “Path” and “Grid”
in the boxes at
the right side of the screen. This simulation calculates the orbital motion of a planet about its star. Initially, all
the parameters of the simulation are set to correspond to the motion of the Earth around the Sun, but the
parameters can be adjusted to simulate other planets and even planets around stars other than the Sun.
STEP 2.
To start the simulation, c
lick the “play”
button:
You will see the planet Earth moving around the Sun, and the count of the days running at the bottom right.
The green arrow indicates the direction o
f the planet’s velocity, its length shows how fast the planet is moving.
Wait until the Earth completes one orbit. Play with the simulation a little bit and become familiar with the
parameters that you can vary. Try moving the planet to a different position and drag the tip of the arrow to
decrease or increase a bit the velocity. Observe how the initial position and velocity of the planet affect the
shape and size of its orbit.
STEP 3.
Reset the simulation clicking on the button:
and then
.
The simulation will be ready to reproduce the Sun-Earth system.
STEP 4.
Run the simulation, and when the planet has completed one revolution take a screenshot of the
simulation
. Make sure the “Path” and “Grid”
options are checked. To take a screenshot of the simulation left
click on the PhET logo, located at the bottom-right of the screen, and insert the saved image in the designated
blank space in the Worksheet document.
STEP 5.
Let’s explore
what happens to the size of the orbit when the initial velocity of the planet increases. Reset
the simulation (STEP 3), and then increase the length of the velocity arrow.
Don’t
overdo it, vary the velocity
by a small amount. To increase the velocity, make
sure the “Velocity” option is checked. Then grab the head of
the velocity arrow and pull it to make it about 5% longer. The velocity arrow must be vertical on the screen
before the simulation starts, correct it if it looks inclined. Run the simulation, you may need to zoom-in or
zoom-out using the slider located at the top left of the window. Take a screenshot after the planet completes
one orbit; paste it on the designated space in the Worksheet document.
STEP 6.
Now
let’s analyze
what happens to the size and shape of the orbit when the initial velocity is decreased a
little bit. Repeat the procedure in STEP 5, but now decrease the length of the velocity arrow. Once the planet
completes its orbit, stope the simulation, take a screenshot, and paste it on the designated space in the
Worksheet document.
2
Part II.
Kepler’s 3rd La
w
Kepler’s third law
of planetary motion states that the cube of the semimajor axis
𝑎
of the orbit of a planet equals
the square of the orbital period
𝑝
,
𝑝
2
= 𝑎
3
(equation 1)
The semimajor axis
𝑎
is half the longest diameter of the elliptical orbit, measured in astronomical units (the
average distance between the Sun and the Earth). The orbital period
𝑝
is the time the planet takes to complete
exactly one orbit, measured in Earth years.
In this section, you will use the PhET
simulation to verify Kepler’s third law
. You will measure the semimajor axis
and period of three different orbits and confirm equation 1 is satisfied. Complete the following steps.
STEP 1.
Reset the simulation so that it is set to reproduce the Sun-Earth system (see Part I, STEP 3).
STEP 2.
Shorten a bit the length of the velocity arrow to decrease the initial speed. The velocity arrow must be
vertical on the screen before the simulation starts, correct it if it looks inclined. Keep the original star mass, so
that Kepler’s 3
rd
law has the form shown on of equation 1. Run the simulation and pause it when the planet
completes
exactly one orbit
.
STEP 3.
Use
the “Measuring Tape” tool t
o determine the length of the longer axis of the orbit, in kilometers. If you
placed the velocity vector vertical at the start of the simulation, the longer axis of the elliptical orbit must be
oriented horizontally. If that is not the case, you will need to figure out the orientation of the longer axis of the
orbit. However, the longer axis always will pass through the position of the star. The figure below shows how
to measure the length of the major axis. Note that the measuring tape extends from side to side across the
oval enclosed by the orbit. Divide the width of the longer axis by 2 to get the length of the semimajor axis (half
the width) “
𝑎
” in units of kilome
ters. Record this value on the
worksheet, Table 1,
column 2.
STEP 4.
Record the orbital period
“
𝑝
”
of the planet, in units of Earth days, in the
worksheet, Table 1,
column 4.
This information is displayed at the bottom right
of the simulation, just above the “Clear” button.
STEP 5.
Take a screenshot of the simulation making certain that it shows the orbit, the Measuring Tape tool
reading the length of the longer axis, and the orbital period. Paste the image on the designated location of the
worksheet
.
STEP 6.
Repeat steps 2-5 to get measurements for a total of three different orbits.
STEP 7.
Now we need to convert the widths to astronomical units (AU) and the periods to years. To convert the
widths to AU, divide the width in kilometers by 149 600 000, and record that value in the worksheet Table 1,
column 3. To convert the period to years, divide the period in days, by 365.25 . Record the results in the
worksheet Table 1
,
column 5.
STEP 8.
Next, calculate a
3
and p
2
, and record the results in the worksheet
, Table 1
, columns 6 and 7.
3
Part III
. Kepler’s 3rd
Law and the moons of Jupiter
In this part of the project, you will use Kepler’s 3
rd
law to calculate the mass of Jupiter, from the orbital
characteristics of its four largest moons. Since you will not need the PhET simulation anymore, you can close it.
In this case, Jupiter is the central object and the moons orbit around it. When the central object is not the Sun, we
need to use Kepler’s 3
rd
law in its Newtonian form,
𝑝
2
=
4𝜋
2
𝐺(𝑀
1
+𝑀
2
)
𝑎
3
(equation 2)
For the conditions of our particular problem,
?
1
: mass of Jupiter
?
2
: mass of the
satellite. Can be taken as zero since they are much smaller than the planet’s
mass.
𝑝
: orbital period of the satellite
𝑎
: semi-major axis of the orbit of the satellite
𝐺
: Gravitational constant (
𝐺 = 6.674 × 10
−11
? ∙ 𝑚
2
/𝑘𝑔
2
)
When the mass
?
2
of the satellite is much smaller than the mass
?
1
of the planet, we can isolate
?
1
from
equation 2,
?
1
=
4𝜋
2
𝐺𝑝
2
𝑎
3
(equation 3)
You will use equation 3 to calculate the mass of planet Jupiter.
STEP 1.
Using the textbook or internet resources search the orbit size
𝑎
(semi-major axis or radius) and orbital
period
𝑝
of the four bigger satellites of Jupiter, listed in Table 2, column 1 of the worksheet. Record those
values in seconds and meters as indicated in the table.
STEP 2.
Calculate
𝑝
2
and
𝑎
3
and record them in the appropriate columns of Table 3. You may need to use
scientific notation to write very large values involved (check Appendix C in the textbook if you need to review
scientific notation). Round your results to four significant figures.
STEP 3.
Using equation 3, calculate the mass of Jupiter. Apply that formula to the orbital parameters of each
satellite. Record the results in Table 3 of the worksheet. You will get very large numbers, write the results
using scientific notation.
STEP 4.
Calculate the average of the four values obtained in step 4.
How to Submit the Project
To submit your project
1.
Rename the completed worksheet file using the following format:
Project1-WorkSheet_
LastNameFirstName
.docx
where
LastName
and
FirstName
must be replaced by your last and first names, as they appear in folio.
2.
Export the file to pdf format. In Word, go to “File >> Save As”. In the
drop-down
menu, choose “PDF” and
save.
3.
Upload your pdf file
in the corresponding folder in the “Assignments” tool of
folio.
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