2.1 Narrative
In this, the second of four aerospace-based S.T.E.M. projects, students will calculate the payload capacity of the R.E.L. Skylon spaceplane. Students will use the launch site latitude to determine the weight of an orbiting payload.
Time Frame
About 4.5 weeks (22 days)
Aerospace Problems
Launch Latitude
Orbital Inclination
Orbital Altitude
Payload Weight
Mathematics Used
Quadratic Equations
Linear Equations
Material List
A connection to the Internet
Google GMail account
Science Topics
Physics, Aerospace
Activating Previous Learning
Basic Algebra
Scientific Calculator
Essential Questions
- Who are the pioneers of spaceplane technology?
- What is the Orbital Inclination of a spacecraft?
- Where is the payload bay of the R.E.L. Skylon located?
- When will be the first flight of the Skylon spaceplane?
- Why do people want to fly payload into orbit on a spaceplane?
- How does the latitude of the Launch Site effect an orbital payload?
- How does the desired orbital altitude effect an orbital payload?
- Wait. I have to do science and technology and engineering and mathematics, all at the same time?
::
This lesson is powered by E^8:
1. Engage
Lesson Objectives
Lesson Goals
Lesson Organization
2. Explore
The Quadratic Equation
The Linear Equation
The Altitude-Payload Line and its Components and Definitions
Additional Terms and Definitions
3. Explain
Orbital Inclination
Orbital Altitude vs. Payload Weight
Payload Weight vs. Orbital Altitude
4. Elaborate
Other Orbital Spacecraft Examples
5. Exercise
Orbital Space Mission Parameters
Orbital Space Mission Design Scenario
6. Engineer
The Engineering Design Process
SMDA Spaceflight Plan
Designing a Prototype
SMDA Software
7. Express
Displaying the SMDC
Progress Report
8. Evaluate
Post Engineering Assessment
::
Lesson Overview
Students first learn the basics of spaceflight launch payload using pencil, paper, and scientific calculator. They 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 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
(none)
Input
Launch Site Latitude (deg)
Payload Weight (lbs)
Orbital Altitude (mi)
Output
Payload Weight (kg)
At Latitude (km)
To I.S.S. (km)
To Polar Orbit (km)
Orbital Altitude (km)
At Latitude (kg)
To I.S.S. (kg)
To Polar Orbit (kg)
::
Visual Learning
Here is a short (7 min) video of the R.E.L. Skylon in operational mode. The video explains the entire flight sequence, from payload installation to propellant loading to takeoff and landing; and shows different commercial spaceflight profiles.
The company claims a 48 hour turnaround time between spaceflights.
::
Continued...
2.2 Vocabulary
International Space Station Latitude Launch Site Launch Site Latitude
Orbital Altitude Orbital Inclination Payload Payload Weight
Polar Orbit
::
2.3 Analysis
To determine the weight and orbital altitude of a spacecraft climbing into earth orbit, we need information on the space plane's capabilities at various launch latitudes. Fortunately, R.E.L. provides us with exactly what we need.
Skylon Payload Capability verses Orbital Altitude for a particular Orbital Inclination and Launch Site Latitude
Using these graphs, we can determine the general formula for each of these lines. So, let's make a couple of tables, shall we?
We will concentrate on the Launch Site Latitude equal to the Orbital Inclination. For the Launch Site Latitude of 0 degrees, we will look for the graph at 250 km for the Orbital Inclination of 0 degrees. For the 15 degree graph, we will look at the 250 km point on the 15 degree Orbital Inclination line. This process continues for each graph:
Skylon Payload Capability Table
Now we can analyse the tables to see what kind of equation that we have. We can use the old trick of subtracting the dependent variables (absolute value) to determine the degree of the polynomial equation. If all the subtractions keep coming up with the same number it is a linear (degree 1) polynomial. If after 2 subtractions we get a constant, then it is a quadratic (degree 2) polynomial. If 3, then a cubic (degree 3), etc. Let's see what we get for the first table:
15,500
15,250 >> |15,500 - 15,250| = 250
14,500 >> |15,250 - 14,500| = 750 >> |250 - 750| = 500
13,250 >> |14,500 - 13,250| = 1,250 >> |750 - 1,250| = 500
11,750 >> |13,250 - 11,750| = 1,750 >> |1,250 - 1,750| = 500
So we get a constant after two iterations. Therefore, we are dealing with a second degree polynomial, or a quadratic equation.
Writing the quadratic in aerospace form, the general equation becomes:
- AtLatitudePayloadALT = a*DEG^2 + b*DEG + Payload0
where
AtLAtitudePayloadALT = Weight of the orbital cargo headed to a certain altitude in space
a = Constant
DEG = Orbital Inclination of Payload
b = Constant
Payload0 = Initial Payload
Since the initial weight of the payload is irrelevant, we can zero out b:
- AtLatitudePayloadALT = a*DEG^2 + Payload0
Using the table of data for a 250 km orbital altitude, we can calculate Payload0 by plugging in DEG = 0 and Payload250 = 15,500:
- AtLatitudePayload250 = a*DEG^2 + 15,500
We can then easily calculate a by using the table (again) and plugging DEG = 15 and Payload0 = 15,250:
- AtLatitudePayload250 = -0.0011*DEG^2 + 15,500
We now have the equation that we need to determine the payload capability (PayloadALT) depending on the latitude of the launch site (DEG) for a 250 km Orbital Altitude.
The other polynomial equation can be determined using the same technique on the 800 km table:
- AtLatitudePayload800 = -0.0011*DEG^2 + 11,000
We now have the equation that we need to determine the payload capability (PayloadALT) depending on the latitude of the launch site (DEG), this time for an 800 km Orbital Altitude.
We can now determine the two points needed to draw the linear equation for the payload.
::
Example
An R.E.L. Skylon is conducting spaceflight operations from Spaceport America. A customer has a satellite that needs to be placed in an orbital altitude of 490 miles. The Orbital Inclination is irrelevant. What is the maximum weight that the satellite can be?
Spaceport America has a location in New Mexico of 32.9980 North Latitude, so we will set our independent variable DEG to 33.
- AtLatitudePayload250 = -0.0011*(33)^2 + 15,500 = 14,290 kg
and
- AtLatitudePayload800 = -0.0011*(33)^2 + 11,000 = 9,790 kg
So, the endpoints to to our linear equation are (250, 14,290) and (800, 9,790). We can finally write the linear equation in slope-intercept (y=mx+b) form by finding the slope (m) and the y-intercept (b). The slope is the change in y divided by the change in x. Plugging one of the points back into the equation yields the y-intercept.
- Slope = m = (9,790 - 14,290) / (800 - 250) = -8.18
and
- y-int = b = y1 - m*x1 = 14,250 - (-8.18)(250) = 16,335 km
Therefore, the linear equation for the Skylon operating out of Spaceport America given a desired orbital altitude of between 250 km and 800 km is:
- Spaceport-to-AtLatitudeALT = -8.18*ALT + 16,335
Converting 490 miles to 789 kilometers, and plugging that into our formula, we get:
Spaceport-to-AtLatitude789 = -8.18(789) + 16,335
= 9,883 kg
Therefore, the satellite can have a maximum weight of almost ten thousand kilograms.
The same technique described above can be used to determine the equations to reach the NASA’s International Space Station (I.S.S.) and for a polar orbit from Spaceport America. These will be exercises left up to the student.
Spaceport-to-ISSALT = -7.73*ALT + 13,982
Spaceport-to-PolarALT = -7.27*ALT + 8,118
::
R.A.F.T. Writing
Role: Teacher
Audience: Middle School students
Format: Five paragraph essay
Topic: The Space Transportation System (Space Shuttle). Who were some of the astronauts that flew the missions? What payload did they deposit in orbit? What was unique about their missions? What was in common with all the missions? How does the Space Shuttle differ from the spaceplane presented in this textbook? How are they the same? Why even bother to build a spaceplane anyway?
::
2.4 Orbital 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
AtLatitudePayload250=-0.0011* DEG^2+15500
AtLatitudePayload800=-0.0011* DEG^2+11000
GRAPHING
m=(AtLatitudePayload800-AtLatitudePayload250)/550
b=AtLatitudePayload250 -m*250
AtLatitudePayload=m*ALT+b
Orbital Payload Space Mission App
::
2.5 Chapter Test
I. VOCABULARY
Match the aerospace term with its definition.
1. Launch Site Latitude
2. Orbital Altitude
3. Orbital Inclination
4. Payload Weight
5. Polar Orbit
A. The height above Mean Sea Level (MSL) of a spacecraft.
B. The mass of a payload that is effected by Earth’s gravity.
C. An orbit that flies above the North and South poles; it has an Orbital Inclination of 98 degrees.
D. The latitude (measured in degrees) of the launch site.
E. The number of degrees that an orbit subtends relative to the equator.
::
II. MULTIPLE CHOICE
Circle the correct answer.
6. The R.E.L. Skylon Payload Equation when graphed forms a parabola which can be describe using a quadratic equation.
A. True B. False
7. The R.E.L. Skylon Payload Equation when graphed forms a straight line which can be describe using a linear equation.
A. True B. False
8. The further north the R.E.L. Skylon launches from, the ____ payload weight it can carry into Low Earth Orbit.
A. More B. Less C. Neither D. Cannot be determined
9. The higher the orbital altitude of the R.E.L. Skylon, the ____ payload weight it can carry into Low Earth Orbit.
A. More B. Less C. Neither D. Cannot be determined
10. The more the R.E.L. Skylon carries into Low Earth Orbit, the ____ the final orbital altitude of the spaceplane.
A. More B. Less C. Neither D. Cannot be determined
::
III. CALCULATIONS
An orbital spacecraft launches from Baikonur Cosmodrome to the International Space Station (I.S.S.).
11. What is the Launch Site Latitude of the cosmodrome?
12. What is the orbital inclination that the spacecraft needs to attain?
13. What is the Orbital Altitude of the International Space Station?
14. What is the Baikonur Cosmodrome-I.S.S. general Quadratic Equation?
15. What is the Baikonur Cosmodrome-I.S.S. 250 km general Quadratic Equation?
16. What is the Baikonur Cosmodrome-I.S.S. 800 km general Quadratic Equation?
17. What is the slope of the Baikonur Cosmodrome-I.S.S. general Linear Equation?
18. What is the y-intercept of the Baikonur Cosmodrome-I.S.S. general Linear Equation?
19. What is the Baikonur Cosmodrome-I.S.S. general Linear Equation?
20. What is the maximum weight that the R.E.L. Skylon can lift to the International Space Station from the Baikonur Cosmodrome?
::
IV. WRITING
Write a one paragraph essay on the topics below.
21. Explain how to find the leading coefficient of the R.E.L. Skylon payload Quadratic Equation.
22. Explain why the further north a launch site is located, the less payload that can be carried into space.
23. Explain why the higher the orbital altitude needed, the less the amount of payload that can be carried.
24. Explain why the more the amount of payload is needed to be carried into space, the less the orbital altitude that the R.E.L. Skylon can attain.
25. Write a short story about what it would feel like to float weightlessly inside of an R.E.L. Skylon spaceplane while gazing at the curvature of the Earth as it flies an orbital spaceflight profile.
::
CLICK HERE TO OPERATE THE ORBITAL PAYLOAD SPACEFLIGHT APP
CLICK HERE FOR THE TEACHER SLIDE SHOW
(coming soon)
CLICK HERE FOR THE STUDENT HANDOUT
(coming soon)
CLICK HERE FOR THE ORBITAL PAYLOAD SPACE MISSION DESIGN PARAMETERS HANDOUT
(coming soon)
CLICK HERE TO GO TO THE EXAMPLE RUBRIC STUDENT WEBSITE
(coming soon)
::
END OF DIARY
::
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...
::