Hohmann Transfer Orbit Equation

S.T.E.M. Project #1 Pulp Science Fiction Cover
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) 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.

Start Date
Beginning of Fall Semester

Due Date
Midterm Fall Semester

Time Frame
About 6 weeks

Mathematics Used
Square Root Equations

Activating Previous Learning
Basic Mathematics
Scientific Calculator

::

Constants
Gravity Parameter (km^3/s^2)
Earth Radius (equatorial) (km)

Input
Lower Orbital Altitude (km)
Higher Orbital Altitude (km)
On-Station Time (days)

Output
Periapsis delta V (kps)
Apoapsis delta V (kps)
Delta V Budget (kps)
Transfer Time (days)
Mission Duration (days)

::

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 (Delta V) needed to go to different orbital altitudes.
Hohmann Transfer Orbit Equations
where r1 and r2 are the distances of the spacecraft from the center of the orbiting body.

The time (in seconds) it takes to change the orbital altitude of a spacecraft is:
Hohmann Transfer Time Equation
Therefore, orbits are circular and the transfer orbit is an ellipse:
Hohmann Transfer Orbit 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 delta V 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.

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 their destination. The Delta V Budget is found by adding the 2 delta v numbers together. The Mission Duration is the On-Station Time added to twice the Transfer Time.

For example:

INPUT
Lower Orbital Altitude = 200 km
Higher Orbital Altitude = 400,000 km
On-Station Time = 10 days

OUTPUT
Periapsis Delta V
Apoapsis Delta V
Delta V Budget
Transfer Time
Round Trip Transfer Time
Mission Duration

We must first determine r1 and r2.

r1 = Lower Orbital Altitude + Radius of Earth = 200 + 6,371 = 6,571 km
r2 = Lower Orbital Altitude + Radius of Earth = 400,000 + 6,371 = 406,371 km
r1 + r2 = 412,942 km
2r1 = 13,142 km
2r2 = 812,742 km

Plugging everything into the Hohmann equations, we get:

Periapsis Delta V = 3.296 kps
Apoapsis Delta V = 0.855 kps
Delta V Budget = 4.150 kps
Transfer Time = 5 days
Round Trip Transfer Time = 10 days
Mission Duration = 21 Days

The app is initially constructed with the Input/Output sheet and the Constants sheet in place and the numbers entered.




The project info can be entered after the app has been built.


https://docs.google.com/spreadsheet...

Working Prototype of the Space Mission Design App


Note: The sample spreadsheet has the Project 2, 3 and 4 information that is not needed for this project grayed-out. It is recommended that space is allowed for these future calculations.

Students that know how to use spreadsheet software should be encouraged to create their own app (remember, Google Apps are free!)

::

Teacher Lesson Plan
Use the slide-show presentation below to give a lesson about basic orbital mechanics and calculating Delta V and Orbital Transfer Time.

In this lesson, students will identify the various aspects of an orbital transfer diagram, matching them with the terms and definitions.

Students then practice the calculations using pencil, paper, and scientific calculator.

Students then learn about the Engineering Design Process, and begin the process of laying the ground work for the app built with a spreadsheet. Sample Open Source computer code is provided to aid students with their spreadsheet formulas.


Some screenshots of the Teacher Presentation:

Vocabulary Fill In of the Hohmann Transfer Orbit Diagram
Other properties that describe an orbit
Delta V and Mission Duration Student Worksheet
Open Source Cloud Computer Code
  1. Engage
    • Lesson Objectives
    • Lesson Goals
    • Lesson Organization
  2. Explore
    • The Rocket Equation
    • Delta V Components and Definitions
    • Additional Terms and Definitions
  3. Explain
    • Basic Orbital Mechanics
    • Hohmann Transfer Orbit Equations
    • Hohmann Transfer Time Equation
    • Mission Duration Equation
  4. Elaborate
    • Advanced Orbital Mechanics
  5. Exercise
    • Space Mission Parameters
    • Space Mission Design Scenario 1
    • Space Mission Design Scenario 2

  6. Engineer
    • The Engineering Design Process
    • SMDA Spaceflight Plan
    • Designing a Prototype
    • SMDA Software
  7. Express
    • Displaying the SMDA
    • Progress Report
  8. Evaluate
    • Post Engineering Assessment
This lesson can be delivered in one or two class periods, with students working on the project after school. It is recommended that a few minutes of a few class periods be set aside for student help.

The Student Workbook (below) accompanies the Teacher Lesson Plan.

::

Missions
Spaceships are useless unless they have a place to go. These missions will add a sense of realism to the student project. Students will be placed into groups and asked to determine the total Crew Size and the total Spacecraft Weight for a given space mission:


Other missions can be created and modified to suit the interest of the students. For example, a group that is interested in dinosaurs could be given a mission to find fossils on Mars. Or a group that wants to start their own business one day could get a mission to make a profit on placing a satellite in orbit.

Encourage students to design their own missions. This project is very flexible in that regard.

::

Presentation
Students will be asked to present their findings to the rest of the class. Parents are, of course, encouraged to attend (it is suggested that a pot luck would make things more interesting).

Each presentation will have slides that introduces the group, describes the mission, and displays the calculations. A short biography of the person named after the spacecraft should also be included.

Students that know how to use presentation software should be encouraged to create their own presentations (remember, Google Apps are free!)

::

Rubric
Students will also create a website and embed their slide presentation and their S.T.E.M. app in a webpage. Their journal will be kept on the webpage as well. If the class presentation is recorded to video, it can be uploaded to Youtube, then embedded in the webpage.

Therefore, each webpage (one for each project) should have the following items:
  1. Embedded Slide Presentation
  2. Embedded view of S.T.E.M. app
  3. Link to S.T.E.M. app
  4. Link to working prototype of S.T.E.M. app
  5. Journal Entries
  6. Embedded Youtube video of the presentation

This is not a scoring rubric; rather it is a guide of what is expected for the project.

The presentation should take between 5 and 10 minutes, unless there are a lot of questions from the audience. For a class with 6 groups, this comes out to between 30 and 60 minutes.

Students should be encouraged to dress professionally, and to practice their presentations beforehand.

Scoring and grading these projects is left up to the professionalism of the teacher.

::

Conclusion
The linear equations used in this project should be easy enough for the average high school Pre-Calculus student. The teacher may need to guide students through the setup of the equations and the calculations. As the semester progress, the concepts and the mathematics will become more challenging.

Using technology to do a high school math project should be easy and fun.

But above all, it should be free.

Both the spreadsheet and the presentation were built using Google Docs, a free application when you sign up for gmail through Google. Therefore, any student with an Internet access can use these tools for free, whether at the school, or at the library, or at the coffee shop, or at home, etc.

Students that do well on this project will learn many important skills that will help to succeed in whatever field they desire to choose.

<<  |  TOP OF PAGE  |  >>