Lab-4-Keplers-Laws
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Louisiana State University *
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1108
Subject
Astronomy
Date
Apr 3, 2024
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8
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1 Name: Astronomy 1108 Lab 4: Kepler’s Laws Part 1: Introduction In this lab, you will use the NAAP Planetary Orbit Simulator to explore various aspects of how planets move. You will be able to explore how the distance from a planet to its star affects its orbital period, how eccentricity influences orbital speed, and the patterns in orbital speed in general. Part 2: Background Overview Newton’s 3 laws:
Newton’s First Law: An object in motion or at rest stays in motion or at rest unless acted on by another object.
Newton’s Second Law: Force is equal to the mass of an object multiplied by its acceleration.
Newton’s Third Law: Every action has an equal and opposite reaction. Newton’s Law of Gravitation: 𝐹 =
𝐺𝑚
ଵ
𝑚
ଶ
𝑟
ଶ
Kepler’s 3 Laws:
Kepler’s First Law: All planets move in elliptical orbits, with the sun as one focus.
Kepler’s Second Law: A line that connects a plant to the sun sweeps out equal areas in equal times.
Kepler’s Third Law: The period of a planet’s orbit squared is proportional to its average distance from the sun cubed.
2 1.
For each of the statements below, indicate which laws they apply to (i.e. which of Kepler’s Laws or Newton’s Laws) a.
Only a force acting on an object can change its motion. b.
Planets move faster when close to their host star. c.
Planets orbit the sun in an elliptical path. d.
Planets with larger orbits take a longer time to complete a single orbit. 2.
Kepler’s 3
rd
can be expressed as: 𝑃
ଶ
(𝑦𝑒𝑎𝑟𝑠) = 𝑎
ଷ
(𝐴𝑈)
Which scenario does this apply to? a.
A planet orbiting the Sun. b.
Anything orbiting our Sun. c.
A planet orbiting a star. d.
Any object orbiting another object. 3.
What is the approximate eccentricity of the ellipse below? Hint: Think about what values are possible, and what values such as 0 and 1 mean. a.
e = 0 b.
e = 0.5 c.
e = 1 d.
e = 1.2 4.
If a planet is twice as far away at aphelion, dap, as when it is at perihelion, d
pe
, then the ratio of the gravitational forces ி
ೌ
ி
=
? (Hint: Use Newton’s Law of Gravitation.) a.
1 b.
¼ c.
½ d.
2 e.
4
3 Part 3: Kepler’s First Law Open the Nebraska Astronomy Applet Project’s Labs software (NAAP Labs) on the desktop. Click on Planetary Orbits, then Planetary Orbit Simulator. You will need to perform the following setup:
Open the Kepler’s 1
st
Law tab and enable all 5 check boxes.
The white dot is the simulated planet. This can be dragged.
The orbit size is changed with the semi-major axis slider. There are some limitations in this simulation for simplicity. The semi-major axis cannot be larger than 50 AU, and the eccentricity must be between 0 and 0.7. Adjust the various parameters of the simulation and familiarize yourself with the effects of each parameter. The simulation can also be animated. 5.
Write a short description of Kepler’s First Law. 6.
Create an orbit with a semi-major axis of 20 AU and an eccentricity of 0. Drag the planet to the far left and far right and record r
1
and r
2
values. r
1 r
2 Far left Far right 7.
Change the eccentricity of the previous orbit to 0.5 and record the new r values. r
1 r
2 Far left Far right
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4 8.
Is there a point at which the r values are equivalent (regardless of the eccentricity)? Are there multiple? Mark the below ellipse appropriately. 9.
Is there a relationship between the sum of the r values and the properties of the ellipse? Does this hold for other ellipses (of different semimajor axes)? Part 4: Kepler’s Second Law Within the same simulator, follow these steps to reset and initialize for the following experiment:
Select “Clear Optional Features.”
Open the Kepler’s 2
nd
Law tab and select “Start Sweeping.”
Adjust the semi-major axis and the animation rate so that everything is moving at a reasonable pace.
Explore the other buttons to add, remove, and size various sweeps. 10.
Erase all current sweeps and create an orbit with a semi-major axis of 1 AU and an eccentricity of 0. Set the fraction sweep to 1/12
of the period. How does the shape of the sweep change at various points throughout the orbit? 11.
Change the eccentricity of the previous orbit to 0.5. Familiarize yourself with the effects this change has. Where is the sweep thinnest and widest? What are the names of these positions? (Hint: These names are used in question 4.)
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5 12.
Vary the eccentricity freely and find the value at which the variation of sweep segment are is greatest. What is this eccentricity? 13.
The below ellipse matches the parameters of Halley’s comet’s orbit. This orbit has a semi-
major axis of 18.5 AU, a period of 76 years, and an eccentricity of 0.97. Explain why the comet is only observable for 6 months every orbit.
6 Part 5: Kepler’s Third Law Within the same simulator, follow these steps to reset and initialize for the following experiment:
Select “Clear Optional Features”
Open the Kepler’s 3
rd
Law tab 14.
Use the simulator to complete the following table: Object Period (yr) a (AU) e 𝑷
𝟐
𝒂
𝟑
Earth 1 0.017 Jupiter 5.2 0.049 Hale-Bopp 354 0.700 15.
As a planet’s semi-major axis increases, how does the orbital period change? 16.
Create the Earth’s orbit with an eccentricity of 0.6. How does this affect the orbital period?