1. AGI’s EAP Curriculum Orbital Mechanics Lesson 3 - Orbital Transfers

2. Introduction This powerpoint was designed to aid professors in teaching several concepts for an introductory orbital mechanics class. The powerpoint incorporates visuals from Systems Tool Kit (STK) that help students better understand important concepts. Any relevant parts of the powerpoint (images, videos, scenarios, etc) can be extracted and used independently. At the end of each lesson there are optional example problems and a tutorial that further expands upon the lesson. Students can complete these as part of a lab or homework assignment. There are multiple supporting scenarios that can be downloaded here. The individual links are also found within the powerpoint.here

3. Lesson 3: Orbital Transfers Lesson Overview Hohmann transfer General coplanar transfers Simple plane changes Time of Flight 3

4. Hohmann Transfer A Hohmann Transfer moves a satellite between two circular, coplanar, and concentric orbits by applying two separate impulsive maneuvers (velocity changes). This type of transfer is the most fuel-efficient. The first impulse is used to bring the satellite out of its original orbit. The satellite then follows a transfer ellipse, known as a Hohmann ellipse, to its apoapsis point located at the radius of the new orbit. A second impulse is then used to return the satellite into a circular orbit at its new radius. 4 Download Scenario: Hohmann transferHohmann transfer Δv2Δv2 Δv1Δv1 Transfer Ellipse Orbit 1 Orbit 2 r1r1 r2r2

5. Hohmann Transfer 5 Δv2Δv2 Δv1Δv1 Transfer Ellipse Orbit 1 Orbit 2 r1r1 r2r2

6. Hohmann Transfer 6 v2v2 v circ1 v1v1 Δv1Δv1 v circ2 Δv2Δv2 2 a t

7. General Coplanar Transfers Although Hohmann transfers use the minimum amount of energy and fuel to reach a new orbit, they also require the most time. For missions with time constraints, a short transfer time can be achieved at the cost of more fuel. Instead of using an elliptical transfer orbit that just reaches the outer orbit, using a transfer ellipse which extends past the outer orbit will result in faster transfer times. 7

8. General Coplanar Transfers 8 v2v2 Δv2Δv2 v circ2 φ2φ2 Download Scenario: Flight Path Angle Flight Path Angle

9. General Coplanar Transfers 9 v2v2 Δv2Δv2 v circ2 φ2φ2 v1v1 v circ1 Δv1Δv1 Download Scenario: Fast vs HohmannFast vs Hohmann

10. Simple Plane Transfer 10 ΔvΔv v v ΔiΔi

11. Time of Flight 11 M Download Scenario: Mean Anomaly Mean Anomaly

12. Time of Flight 12 E Download Scenario: Eccentric Anomaly Eccentric Anomaly

13. Time of Flight 13

14. End Lesson 3 - Tutorial Complete tutorial to further explore lesson: Hohmann Transfer TutorialHohmann Transfer Tutorial 14

15. End Lesson 3 - Exercises True or False: 1. T / F A Hohmann Transfer is the most efficient transfer between two circular orbits. 2. T / F When using a Hohmann transfer to move to a larger circular orbit, the satellite must increase its velocity at the apoapsis of the transfer ellipse to enter the new orbit. 3. T / F If performing a pure inclination change (no other orbital elements change), the old and new orbits will not intersect. Short Answer: 4. We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit using a Hohmann Transfer. What is the total change in velocity required? 5. Given the mass of the satellite to be 500 kg and the gravitational force exerted on the satellite by the Earth (F=2.87×1031 N), what is the specific potential energy of a satellite, u? 6. We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit with a transfer orbit tangent to the LEO orbit and a v1 = 2.75 km/s. What is the total Δv required? 15

16. End Lesson 3 - Answers True or False: 1. T / F A Hohmann Transfer is the most efficient transfer between two circular orbits. 2. T / F When using a Hohmann transfer to move to a larger circular orbit, the satellite must increase its velocity at the apoapsis of the transfer ellipse to enter the new orbit. 3. T / F If performing a pure inclination change (no other orbital elements change), the old and new orbits will not intersect. Short Answer: 4. Given the mass of the satellite to be 500 kg and the gravitational force exerted on the satellite by the Earth (F=2.87×1031 N), what is the specific potential energy of a satellite, u? u= 3.3216×107 m 2 /s 2 5. We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit using a Hohmann Transfer. What is the total change in velocity required? See Example on slide 14 6. We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit with a transfer orbit tangent to the LEO orbit and a v1 = 2.75 km/s. What is the total Δv required? See Example on slide 15 16

17. End Lesson 3 - Answers 5. Hohmann Transfer Example: We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit. What is the total change in velocity required? Step 1 - Solve for semi-major axis of the Hohmann transfer. a t = (r LEO + r GEO ) / 2 = (7,000 + 42,164) / 2 = 24,582 km Step 2 - Solve for circular orbit velocities. v circ1 = v LEO = √(/r LEO ) = √(3.986x10 5 /7,000) = 7.546 km/s v circ2 = v GEO = √(/r GEO ) = √(3.986x10 5 /42,164) = 3.075 km/s Step 3 - Determine the energy of the Hohmann transfer. t = -/2a t = -3.986x10 5 /2*24,582 = -8.108 Step 4 - Solve for periapsis and apoapsis velocities of the transfer. v 1 = √(2( t + /r 1 )) = √(2(-8.108 + 3.986x10 5 /7,000)) = 9.883 km/s v 2 = √(2( t + /r 2 )) = √(2(-8.108 + 3.986x10 5 /42,164)) = 1.640 km/s Step 5 - Determine the change in velocities for each maneuver. Then find the total change, Δv Tot. Δv 1 = v 1 - v LEO = 2.337 km/s Δv 2 = v GEO – v 2 = 1.435 km/s Δv Tot = 2.337 + 1.435 = 3.772 km/s 17 v2v2 v circ1 v1v1 Δv1Δv1 v circ2 Δv2Δv2 2a t

18. End Lesson 3 - Answers 6. Fast Transfer Example: We would like to transfer our satellite from a LEO orbit (r LEO = r 1 = 7,000 km) to a GEO (r GEO = r 2 = 42,164 km) orbit with a transfer orbit tangent to the LEO orbit and a v 1 = 2.75 km/s. What is the total Δv required? Step 1 - Solve for circular orbit velocities. v circ1 = v LEO = √(/r LEO ) = 7.546 km/s v circ2 = v GEO = √(/r GEO ) = 3.075 km/s Step 2 - Determine v 1 and v 2 energy of the transfer orbit. Δv 1 = 3 km/s = v 1 -v circ1 → v 1 = 10.296 km/s t = v 1 2 /2- /r LEO = -3.939 km 2 /s 2 t = v 2 2 /2- /r GEO → v 2 = 3.321 km/s Step 3 - Determine the flight path angle at location of Δv 2. h t = r 1 v 1 cos(φ 1 ) = 72072 km 2 /s h t = r 2 v 2 cos(φ 2 ) → cos(φ 2 ) = 0.5147 Step 4 - Determine Δv 2 and Δv Tot. Δv 2 2 = v 2 2 + v GEO 2 – 2 v 2 v GEO cos(φ 2 ) → Δv 2 = 3.312 km/s Δv Tot = 2.75 + 3.158 = 5.908 km/s 18 Download Scenario: Fast vs Hohmann to compareFast vs Hohmann Δv Tot =5.91 km/s vs Δv TotHohm = 3.77 km/s ToF = 2.65 hours vs ToF Hohm = 4.97 hours (ToF=time of flight) v2v2 Δv2Δv2 v circ2 φ2φ2 v1v1 v circ1 Δv1Δv1

19. Additional Resources Visit agi.com/resources/educational-alliance-program/ to learn more about the available curriculum and the Educational Alliance Program (EAP).agi.com/resources/educational-alliance-program/ To learn more about STK: Beginner-level video tutorials Complete STK Certification Questions / Comments: Education@agi.com