6: Kepler Models
- Page ID
- 162998
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Lab Overview
Lab Goals
The goal of this lab is to develop the scientific practices of taking measurements, identifying patterns and relationships between variables, and making scientific claims based on evidence with clear reasoning. You will develop these lab skills while reinforcing/furthering your understanding of Kepler’s Laws for gravitational orbits. As a result of this lab, you should be able to:
- Explain your methodology for performing measurements with explanations for the choice of precision in measurement and analysis of how finer/coarser measurements might affect results.
- Make scientific claims about orbital characteristics/values that are based on evidence with clear reasoning/analysis of that evidence to support those claims.
Equipment and Setup
This lab requires the following equipment on each table.
- Orbit Templates (3)
- 3-5 different types (sizes) of beans/legumes
- Centimeter rulers
- Graph paper
- Calculator
Submission Instructions
Submit a brief lab report that includes the following sections:
Methodology
Write a Methodology section that details the steps you took to create your Kepler Orbital Model. This section should include your responses to the Methodology Reflection Questions as well as a narrative of how you used the beans to create the model. Your audience is someone who knows a little about Kepler’s laws but was not present in the lab and does not have this lab activity handout.
Results
Organize your data tables, orbital models, and any other measurements or calculations. Include a brief introduction/explanation before the tables/models/measurements. Include captions that have a number and description. Consider any annotations or color coding that will be helpful to reference in the discussion.
Discussion
Write/type your answers to the Discussion Questions using the Claim-Evidence-Reasoning format with a separate (3-5 sentence) paragraph for each question.
Kepler Orbits
Background Information
Gravitational orbits of objects, such as a planet around a star, follow Kepler’s Three Laws of Planetary Motion:
- Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
- The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
- The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.
In this lab we will explore those elliptical orbits by primarily exploring the second law, which relates to area and time. We will create a diagram similar to the one on the right but with more detailed measurements.
Lab Instructions
Measuring the area of the wedges (A or B) on an ellipse can be challenging. In this lab, we will use beans as a way to approximate areas and build the wedges from these beans. Beans should be in a single layer.
Measurement Calibration
- Identify the area of a 4x4 grid and 16x16 grid: 4x4 grid: _____ in2 16x16 grid: _____in2
- Identify the number of beans that fit in a 2x2 grid and a 10x10 grid for each bean type.
- Record those values below, then identify the average area of a bean is in in2 for each bean type. Area of 1 bean is Area of Grid divided by # of Beans.
|
Bean Type |
# Beans in 4x4 |
Area of 1 Bean |
# Beans in 16x16 |
Area of 1 Bean |
- Are your results different for the 4x4 grid and the 16x16 grid? If so, decide as a group which you think is more accurate and circle those measurements on the table. Briefly explain your reasoning below.
Make an Kepler’s 2nd Law Orbital Model
Goal: Use beans to identify equal area wedges along the orbit to show the position of the orbiting planet at equal time-intervals. For an accurate model, you should have enough beans to make a clear wedge throughout the orbit.
- Decide as a group which type of bean to use and how many beans you will use (it doesn’t need to be the whole bag). Consider the following to make the best selection for this lab:
- You want to use enough beans that they form a wedge even at the largest distances between the planet and star
- Using fewer beans will give you more positions of the planet (ie, more data) and more data makes patterns easier to find and minimizes error.
- Use an amount of beans that is less than ¼ of the overall area to create more than 4 total wedges
- Align the beans along the line between the planet and the star so that they form a wedge. Try to make both sides of the wedge as straight of a line as possible.
- Draw a line along the edge of the beans. Use a ruler and try to be as exact as possible. Put a big dot at the edge of the line where it hits the elliptical orbit.
- Shift the beans to the other side of the line you drew, then continue to repeat steps 2-3 until you have gone around the entire orbit. Make sure to always use the same beans for each wedge.
Methodology Reflection Questions
As a team, discuss and record notes for the following questions:
- How did the different bean types compare in area? What bean type did you use and what were the benefits/drawbacks of that selection?
- How many beans did you use for each area wedge? Why did you choose that number? What were the benefits/drawbacks of that choice?
- What might be the benefits/drawbacks of using a much larger object (like an orange) instead of a bean?
- What might be the benefits/drawbacks of using a much smaller object (like sand) instead of a bean?
- What actions did your group take to make measurements as accurate as possible?
- Were there any challenges to taking accurate measurements? If so, what could you do (possibly with different materials) to be more accurate?
Apply Kepler’s 3rd Law to a Larger/Smaller Orbit
Kepler’s 3rd law relates the period of orbit to the distance of the semi-major axis:
P2 ∝ a3
And for our solar system using years and AU as units:
P2 = a3
Using this relationship, we can identify that for the same amount of time, the area swept out (ie, number of beans) for planets is proportional to the square root of the semimajor axis:
Area ∝ √a
We can then identify the number of beans to use in the larger/smaller orbit to make a Kepler’s Second Law Orbital Model that is consistent with the first. For this process, Orbit 1 is the orbit you previously modeled. Use the table below to make this conversion:
- Measure the semimajor axis of each orbit
- Divide the result for Orbit 2 by Orbit 1
- Take the square root of that number
- Fill in the number of beans used in Orbit 1, then multiply that number by the previous column (√a2 / a1). Round to the nearest whole bean and put that result in the last column.
- Repeat the process you used in part 1 of the lab to create a Kepler Orbital Model for this second planet.
|
Semimajor Axis (a) |
Ratios |
Number of Beans |
|||
|
Orbit 1 |
Orbit 2 |
a2 / a1 |
√(a2 / a1) |
Orbit 1 |
Orbit 2 |
|
Semimajor Axis (a) |
Ratios |
Number of Beans |
|||
|
Orbit 1 |
Orbit 3 |
a3 / a1 |
√(a3 / a1) |
Orbit 1 |
Orbit 23 |
Lab Extension Options
- Repeat the experiment with a different number of beans (same bean type) and compare the differences as part of your methodology section.
- Repeat the experiment with a different bean (or other object) choice and compare the two bean choices as part of your methodology section.
- Something else! Make a proposal and check in with your instructor. Be sure to identify if the extension will relate to your methodology section, discussion section, or both!
Lab Analysis
Use your orbits to answer the discussion questions below as a group. Write your responses as a separate paragraph for each question. Each paragraph should include:
- Claim: A succinct one-sentence answer to the question.
- Evidence: A sentence or two that selects and summarizes measurements, descriptions, or patterns from your lab. Be as clear as possible. You might want to add labels or color-coding to your models to make it easy to refer to specific parts or components.
- Reasoning: Explain how the specific evidence you selected supports/justifies your claim. Imagine you are explaining it to someone who is not in this lab so they do not have the background information that you have.
Discussion Questions
- Is the speed of the planet always the same? If not, make a claim about any patterns related to the speed of the planet at different parts of the orbit.
- Is the period (how much time it takes for one complete orbit) of two planets orbiting the same star the same or different? If not, make a claim about any patterns related to the period of planets with different orbits
- Does the size of an orbit seem to affect the speed of a planet? If so, make a claim about any patterns related to the speeds of planets with different orbits.