Note to readers: This lesson plan, as well as all the other similar articles we've written, may appear, to some folks, like the next thing to Greek language (the math stuff), but bear in mind these S.T.E.M. lessons are really all about the students. Ergo, preparing high school students for the real rocketry world using real math. That being said, we invite you into their world, their minds, and let's support them in this heroic endeavor, because these students are truly going where no high school student has ever gone before.
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5.1 Narrative
In this, the first of a four-part interconnected astronautics-based S.T.E.M. project, students will calculate the change in orbital velocity Delta V (Δv) needed to change the orbital altitude of a spacecraft. Students will also calculate the total round-trip time to transfer between orbits and use that information to determine the duration of their space mission.
Time Frame
About 4 weeks (22 days)
Aerospace Problems
Periapsis Δv
Apoapsis Δv
Δv Budget
Transfer Time
Mission Duration
Mathematics Used
Square Root Equations
Basic Algebra
Material List
A connection to the Internet
Google GMail account
Science Topics
Physics, Astronautics
Activating Previous Learning
Basic Mathematics
Scientific Calculator
Essential Questions
- What is the relationship between the change in velocity and the orbital altitude?
- Why do I need to raise or lower my orbital altitude?
- How long does it take to reach my destination?
- How long can I stay at my destination?
- How does the radius of the earth determine Δv?
- Who are are some of the pioneers in space exploration?
- Wait. I have to do science and technology and engineering and mathematics, all at the same time?
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This lesson is powered by E^8:
1. Engage
Lesson Objectives
Lesson Goals
Lesson Organization
2. Explore
The Circular Orbit
The Elliptical Orbit
Delta V (Δv)
Additional Terms and Definitions
3. Explain
Change in Velocity
4. Elaborate
Other Orbital Examples
5. Exercise
Δv and Transfer Time Parameters
Δv and Transfer Time Scenario
6. Engineer
The Engineering Design Process
SMDA Spacecraft ΔV Plan
Designing a Prototype
SMDA Software
7. Express
Displaying the SMDA
Progress Report
8. Evaluate
Post Engineering Assessment
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Lesson Overview
Students first learn the basics of astronautics involving the Hohmann Transfer Orbit Equations using pencil, paper, and scientific calculator. Students then use what they have learned to create a Space Mission Design App (SMDA), designed according to the Engineering Design Process, that will be used for real-world spacecraft.
Students will learn about circular and elliptical orbits and how orbital altitude can be changed using rocket burns. Students will be using the Periapsis and Apoapsis Δv equations, as well as the Transfer Time equation.
Students will use spreadsheet software to create the app and will use slide-show software for their presentations. They will also create a document of their experiences engineering the SMDA and presenting their findings to the rest of the class.
Constants
Standard Gravitational Parameter μ (m^3/s^2)
Earth Radius (equatorial) (m)
Input
Lower Orbital Altitude (m)
Higher Orbital Altitude (m)
On-Station Time (days)
Output
Periapsis Δv Burn (mps)
Apoapsis Δv Burn (mps)
Δv Budget (mps)
Transfer Time (days)
Round-Trip Time (days)
Mission Duration (days)
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Visual Learning
Here is a video explaining in extruciating detail orbital mechanics and the ideas behind a change in velocity (delta v).
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Continued...
5.2 Vocabulary
Apoapsis Apoapsis Δv Burn Circular Orbit Delta V (Δv)
Δv Budget Elliptical Orbit Gravity Parameter (μ) Hohmann Transfer Orbit
Higher Orbital Altitude Lower Orbital Altitude Mission Duration On-Station Time
Orbital Altitude Periapsis Periapsis Δv Burn Radius of Higher Orbit
Radius of Lower Orbit Round-Trip Time Round-Trip Δv Standard Gravity (g0)
Transfer Orbit #1 Transfer Orbit #2 Transfer Time
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5.3 Analysis
In order to raise or lower a spacecraft that is in orbit around a body, such as the Earth, the Hohmann equations are used. These equations determine the change in orbital velocity (Δv) needed to go to different orbital altitudes.
where
μ = Standard Gravitational Parameter, equal to 398,600.44189 km3/s2
R1 = Radius of the inner (smaller) orbit
R2 = Radius of the outer (larger) orbit
The Δv Budget is simply the two Δv rocket burns added together.
- ΔvBUDGET = ΔvPERIAPSIS + ΔvAPOAPSIS
Using the principle of Reversibility of Orbits, the total Δv needed to get to the destination and back is
- ΔvROUND-TRIP = 2(ΔvBUDGET)
The transfer time is calculated in seconds.
There are 86,400 seconds per day, so the number of days it takes to change the orbital altitude of a spacecraft becomes:
- Transfer-TimeDAYS = Transfer-TimeSECONDS / 86400
And the Round-Trip Time is double the Transfer Time:
- Round-TripDAYS = 2(Transfer-TimeDAYS)
The Mission Duration is the Round-Trip Time added to the On-Station Time.
- MissionDuration = On-StationDAYS + Round-TripDAYS
Orbits are circular and the transfer orbit is an ellipse:
Hohmann Transfer diagram
In the diagram above, green represents the lower circular orbital altitude and red represents the higher circular orbital altitude. Yellow represents the elliptical transfer orbit from green to red (or dashed yellow representing from red to green).
The first Δv rocket engine firing is done at the lowest point in the elliptical transfer orbit (periapsis). This puts the spacecraft on the path in yellow. At the highest point in the elliptical orbit (apoapsis), another rocket engine firing occurs, this time circularizing the orbit (in red). To go home, simply reverse the procedure, using the principle of reversibility of orbits.
We will use as inputs the lower and higher orbital altitudes, as well as the On-Station Time, which is the number of days the astronauts spend at the mission destination. To review, the Δv Budget is found by adding the two Δv numbers together. The Round-Trip Time is twice the Transfer time, and the Mission Duration is the On-Station Time plus the Round-Trip Time.
Orbital Mechanics!
Example
You are the Captain of a spacecraft that is currently in a Low Earth Orbit (LEO) at an orbital altitude of 200 km. You need to increase your orbital altitude to 8,500 km so that you can deposit a new satellite and bring back the old one. Your On-Station Time is 5 days. Find the change in velocity, transfer time, Round-Trip Time, and the Mission Duration for this space mission.
AltitudeLOWER = 200 km
AltitudeHIGHER = 8,500 km
First, we need to calculate R1 and R2.
R1 = AltitudeLOWER + RadiusEARTH
= 200 + 6378 = 6,578 m
R2 = AltitudeHIGHER + RadiusEARTH
= 8500 + 6378 = 14,878 m
Therefore,
ΔvPERIAPSIS = SQRT(398600/6578) * SQRT(2(14878)/21456 - 1)
= 1,383 mps
ΔvAPOAPSIS = SQRT(398600/14878) * SQRT(1 - 2(6578)/21456 )
= 1,123 mps
ΔvBUDGET = 1383 + 1123
= 2,506 mps
ΔvROUND-TRIP = 2(2506)
= 5,011 mps
Transfer-TimeSECONDS = (6578 + 14878)38(398600)
= 5,529 s
Transfer TimeDAYS = Transfer TimeSECONDS86400
= 552986400 = 0.064 days
Round-TripDAYS = 2(Transfer TimeDAYS)
= 2(0.064) = 0.128 days
Mission-Duration = On-StationDAYS + Round-TripDAYS
= 5 + 0.128
= 5.13 days
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R.A.F.T. Writing
Role: Teacher
Audience: Middle School students
Format: Five paragraph essay
Topic: The Gemini spacecraft. Who were the astronauts that flew the mission? What spacecraft was used to boost the Gemini to a higher orbit? What was unique about the missions? What was in common with all the missions? How does a Gemini change in orbital altitude differ from the one presented in this textbook? How are they the same? Why even bother to change an orbit anyway?
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5.4 Space Mission Design App
Given the above information, we can use a spreadsheet to enter equations and data to create a Space Mission Design App (SMDA).
The S.T.E.M. for the Classroom/Google App is broken down into four (4) parts:
1. Input/Output Interface
2. Graph
3. Constants
4. Calculations
The App can now be developed.
Sample Open Source Code
Once the cells have been named referencing cells is easy.
CALCULATIONS
(Coming Soon)
Hohmann Transfer Orbit Space Mission App
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5.5 Chapter Test
I. VOCABULARY
Match the astronautics term with its definition.
1. Apoapsis
2. Δv Budget
3. Mission Duration
4. Periapsis Δv Burn
5. Radius of Lower Orbit
A. The lower circular orbital altitude of a spacecraft as measured from the center of an orbiting body.
B. The rocket firing at the lowest point of a Transfer Orbit.
C. The highest point in an elliptical orbit.
D. The total time necessary to accomplish a space mission.
E. The total amount of Delta V needed to accomplish a space mission.
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II. MULTIPLE CHOICE
Circle the correct answer.
6. The Apoapsis Δv rocket burn occurs at the highest point of the Hohmann transfer orbit.
A. TRUE B. FALSE
7. The Δv Budget is the total change in velocity needed to conduct a round-trip space mission.
A. TRUE B. FALSE
8. A spacecraft is orbiting the Earth at an orbital altitude of 1,000 km. What is the orbital radius of the spacecraft?
A. 5,371 km B. 6,371 km C. 7,371 km D. Cannot be determined
9. What is the Hohmann transfer time of a spacecraft headed for the apoapsis ΔV rocket burn if the Round-Trip Transfer Time is 7 hrs?
A. 3.5 hrs B. 7.0 hrs C. 10.5 hrs D. Cannot be determined
10. There are always two Δv rocket burns whenever a spacecraft needs to raise or lower its orbital altitude. The second Δv rocket burn is used to change the shape of an orbiting spacecraft into a ____.
A. Ellipse B. Circle C. Parabola D. Cannot be determined
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III. CALCULATIONS
A wayward satellite is need of repairs and to have some electronic parts replaced. The satellite is in a stable orbit 1,250 km above the Earth. A crew inside a repair vehicle is also in a stable orbit, but at an orbital altitude of 400 km below the satellite. It is estimated that the crew will need 3 days to conduct all the necessary repairs.
11. What is the Lower Orbital Radius?
12. What is the Higher Orbital Radius?
13. What is the On-Station Time?
14. What is the Periapsis Δv Rocket Burn?
15. What is the Apoapsis Δv Rocket Burn?
16. What is the Δv Budget?
17. What is the Round Trip Δv Budget?
18. What is The Transfer Time?
19. What is Round Trip Transfer Time?
20. What is the Mission Duration?
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IV. WRITING
Write a one paragraph essay on the topics below.
21. Explain why a spacecraft must first push off and then stop when it wants to go from one point in space to another, such as a higher (or lower) orbital altitude.
22. Explain how the On-Station Time effects the Mission Duration.
23. Explain how a spacecraft would return back to its original orbital altitude if the Apoapsis ΔV rocket burn was not performed.
24. Explain why a spacecraft must have a larger Round Trip Δv Budget if it needs to go to a higher orbital altitude.
25. Write a short story about what it would feel like to float weightlessly in space, while gazing at the curvature of the Earth as it transfers from a lower orbital altitude to a higher orbital altitude.
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CLICK HERE TO OPERATE THE DELTA V AND TRANSFER TIME APP
CLICK HERE FOR THE TEACHER SLIDE SHOW
CLICK HERE FOR THE STUDENT HANDOUT
CLICK HERE FOR THE SPACE MISSION DESIGN PARAMETERS HANDOUT
CLICK HERE TO GO TO THE EXAMPLE RUBRIC STUDENT WEBSITE
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END OF DIARY
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A (partial) list of future topics in the series:
- S.T.E.M. Education For the 21st Century and Beyond
An Introduction to S.T.E.M. For the Classroom
- Go Where No Student Has Gone Before
A more indepth discussion of what we’re trying to accomplish.
- Suborbital Spaceflight - Quadratic Equations
Students calculate the height that SpaceShipTwo reaches space.
- Orbital Payload - Quadratic and Linear Equations
Students calculate the payload that the R.E.L. Skylon can place into Low Earth Orbit (LEO).
- A City in the Sky - Matrices
Students design a space station, and find the cost to place it into orbit. They also find the total volume and the number of crew that can safely occupy the station.
- Landing is the Hardest Thing to Do - Trigonometry
Students calculate the ground speed and altitude of a spacecraft returning from space.
- Delta V and Transfer Time - Square Root Equations
Students calculate the change in orbital velocity needed to go from a lower orbital altitude to a higher orbital altitude and find the time it takes for the maneuver.
- Spacecraft Weight Analysis - Linear Equations
Students find the weight of a real crew capsule that was designed in 1971 and determine the mission duration and the number of crew that can fly the mission.
- The Rocket Equation - Exponential Equations
Students determine the amount of cryogenic propellant needed to fly a space mission using an engine module designed in 1971.
- Fly Me to the Moon - Finance
Students calculate the amount of cryogenic propellant needed to land on the Moon and find the amount of profit you can make by selling moon rocks.
- Delta V and the Gravity of the Situation - Square Root Equations
Where we ask the question: does the mathematics add up to what the astronauts are depicted doing?
- The Thrill(e) in the Rille - Trigonometry
Students calculate the amount of rope needed for Apollo astronauts to safely descend into a lunar canyon.
- The Bone of Contention - Proportions
Students determine the identities of fictitious astronauts who have perished on a lunar landing mission using their recovered femur bones.
- TBA - Mathematics Topic is also TBA
Lesson plans that are still in the works...
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