Homework 4 Solution

Applications of the Rocket- Equation

Calculating the Fuel Budget for an Orbital Phasing Maneuver of a GeoStationary Satellite

MAE 6530 - Propulsion Systems TT&C Satellite

• TT&C satellite used to monitor pacific coast battle has failed

• NACSOC has decided to transfer the functions of a spare Atlantic battle group satellite to the pacific until a replacement can be launched

… design an Orbital phasing Maneuver that Allows Transfer of a GEO Synchronous Communication Satellite from 40o 40’ west Longitude To 118o 15’ west longitude

MAE 6530 - Propulsion Systems Phasing Maneuver 40o 40’

118o 15’

MAE 6530 - Propulsion Systems Phasing Maneuver (part 2) 40o 40’

118o 15’

• Design a Reverse Orbital Maneuver that Puts the Satellite Back to the Original Longitude after Mission has been accomplished

MAE 6530 - Propulsion Systems What To Compute

• Compute … Phasing Orbit Parameters … Phasing Orbit Period … Required DV1, DV2

• Assume Rmin > 32,000 km (to stay above Van Allen belts)

• Note: It may take Multiple orbits of Phasing Orbit to accomplish this task

MAE 6530 - Propulsion Systems What To Compute (cont’d)

• Compute … Burn time for Transfer Orbit Insertion … Burn Time for Final Orbit Insertion … Required Fuel Budget for DV1, DV2

MAE 6530 - Propulsion Systems Parameters of the Problem

Original Longitude : 40 deg, 40 min West Destination Longitude: 118 deg, 15 min West

MAE 6530 - Propulsion Systems Parameters of the Problem (cont’d)

MAE 6530 - Propulsion Systems Parameters of the Problem (Concluded)

• Fthruster = 0.500 kNt

• Spacecraft mass

1000 kg “Dry”

MAE 6530 - Propulsion Systems First Compute Radius of Geo Orbit

MAE 6530 - Propulsion Systems Next Compute Angular Velocity of GEO Orbit

MAE 6530 - Propulsion Systems Compute Linear Velocity of GEO Orbit

MAE 6530 - Propulsion Systems Compute Required Time of Flight for Phasing Maneuver

Original Longitude: 40 deg, 40 min West Final Longitude: 118 deg, 15 min West

MAE 6530 - Propulsion Systems Try “Exterior” Phasing Orbit Positive DV 40o 40’

118o 15’

MAE 6530 - Propulsion Systems Required Period for “Exterior” Phasing Orbit

MAE 6530 - Propulsion Systems Compute Eccentricity of the Phasing Orbit

Do we need this?

MAE 6530 - Propulsion Systems Compute Perigee Velocity of Phasing Orbit

MAE 6530 - Propulsion Systems Compute DV Required to Insert Spacecraft into Phasing Orbit

MAE 6530 - Propulsion Systems Compute DV Required to Insert Spacecraft into Final GEO Orbit

Retrograde Burn

MAE 6530 - Propulsion Systems Return Trip 40o 40’

118o 15’

MAE 6530 - Propulsion Systems Look “Interior” Phasing Orbit Retro-grade Burn 40o 40’ (Negative DV) 118o 15’

MAE 6530 - Propulsion Systems Required Period for “Interior” Phasing Orbit

Inside Van Allen Belts!

MAE 6530 - Propulsion Systems Check Value … Compute Semi-Major Axis of Phasing Orbit with Perigee at 32,000 km (as per problem constraints)

MAE 6530 - Propulsion Systems Compute Period of Phasing Orbit with

Perigee at 32,000 km (as per problem constraints)

Check ! • Allowable Minimum size interior orbit is too big For proper phasing

MAE 6530 - Propulsion Systems Must use “Exterior” Phasing Orbit o Positive DV 40 40’

118o 15’

MAE 6530 - Propulsion Systems Required Period for “Exterior” Phasing Orbit

Sec.

MAE 6530 - Propulsion Systems Compute Perigee Velocity of Phasing Orbit

MAE 6530 - Propulsion Systems Compute DV Required to Insert Spacecraft into Phasing Orbit

MAE 6530 - Propulsion Systems Compute DV Required to Insert Spacecraft into Final GEO Orbit

Retrograde Burn

MAE 6530 - Propulsion Systems “Delta-Vee” Summary for Round Trip Maneuver

MAE 6530 - Propulsion Systems Fuel Budgeting

• Spacecraft mass

1000 kg “Dry”

• Start at Burn 4 and Work Backward

Weight of Vehicle at End of Maneuver Series

MAE 6530 - Propulsion Systems Burn 4 Fuel Consumption

MAE 6530 - Propulsion Systems Burn 4 --> Burn Time

MAE 6530 - Propulsion Systems Burn 3 Fuel Consumption and Burn Time

MAE 6530 - Propulsion Systems Burn 2 Fuel Consumption and Burn Time

MAE 6530 - Propulsion Systems Burn 1 Fuel Consumption and Burn Time

MAE 6530 - Propulsion Systems Propellant Burn Summary

• 524.1 kg total propellant consumed

MAE 6530 - Propulsion Systems Burn-Time Summary

• 3187 seconds total burn life of engine expended

MAE 6530 - Propulsion Systems Look at “Double” Interior Orbit .. For Return trip GEO Phasing Orbit Two times around 32,000 km on interior limit

= 39078.1 km MAE 6530 - Propulsion Systems Look at “Double” Interior Orbit ..(2)

• Calculate eccentricity and perigee radius

aphasing = 39078.1 km

= 0.078973

= 35992.0 km check!

MAE 6530 - Propulsion Systems Look at “Double” Interior Orbit ..(3)

• Calculate delta V required to enter phasing orbit

aphasing = 39078.1 km

= - 0.1239 km/sec

Retrograde burn! MAE 6530 - Propulsion Systems Look at “Double” Interior Orbit ..(4)

• Calculate delta V required to insert spacecraft into final orbit

aphasing = 39078.1 km

= 0.1239 km/sec

Retrograde burn MAE 6530 - Propulsion Systems “Delta-Vee” Summary for Round Trip Maneuver, Option 2

retrograde

retrograde 0.1239 0.1239 Exterior Phasing orbit for outward trip “Double” interior orbit for return trip Far more efficient! MAE 6530 - Propulsion Systems Option 2 Fuel Burn Analysis, Burn 4

41.597 kg

MAE 6530 - Propulsion Systems Option 2 Fuel Burn Analysis, Burn 4

41.597 kg

=253.0 sec

MAE 6530 - Propulsion Systems Option 2 Burn 3 Fuel Consumption and Burn Time

43.328 kg

=263.53 sec

MAE 6530 - Propulsion Systems Option 2 Burn 2 Fuel Consumption and Burn Time

67.01 kg

= 407.58 sec

MAE 6530 - Propulsion Systems Option 2 Burn 1 Fuel Consumption and Burn Time

= 71.150 kg

= 432.75 sec

MAE 6530 - Propulsion Systems Option 2 Propellant Burn Summary 41.60 kg 43.33 kg 67.01 kg 71.15 kg • 222.99 kg total propellant consumed

57.5% propellant savings .. Total of 301.11 kg! compared To option 1

MAE 6530 - Propulsion Systems Option 1 Burn-Time Summary 253.0539.0 sec 263.5508.0 sec 407.6 sec 432.8 kg

• 1356.9 seconds total burn life of engine Expended .. 57.4% burn life savings compared to option 1

MAE 6530 - Propulsion Systems General Solution ….

Phasing does not have to be accomplished in a single orbit .. Or within a single target rotation

MAE 6530 - Propulsion Systems General Solution ….

MAE 6530 - Propulsion Systems