AAE450 Spring 2009

Analysis of Trans-Lunar Spiral Trajectory

[Levi Brown] [Mission Ops]

February 12, 2009

1

AAE450 Spring 2009 [Levi Brown] [Mission Ops]

Spiral Transfer

2

Equations of Motion (EOM)s : r: distance from central body to

spacecraft (s/c)μ: Gravitational parameter (G*M)

θ: Angle of rotationT: Thrust from engine

m: Mass of spacecraftAssumes: • Two body problem

• Point mass central body/spacecraft

• Coplanar Transfer to Moon

• Thrust is in

• Force of gravity in the

• Circular lunar orbit

• Constant thrust

Created Code:• Input s/c mass, thrust levels, time of flight (TOF)

• Integrate EOMs for spiral outward to

moon’s sphere of influence (r ≈ 3.2 x 105 km)

• “Turn off” Earth’s gravity and “turn on” Moon

• Determine velocity and position relative to

moon (integrate with a “braking” thrust)

• Output history of s/c position and velocity

AAE450 Spring 2009 [Levi Brown] [Mission Ops] 3

ResultsLimiting Case: 1 Year TOFInputs:mo = 595.4 kg (Latest OTV Mass)TOFout = 356 daysTOFin ≈7 days (Keep low spiral time)

Results:Required Tout = 109.5 mNRequired Tin = 93.5 mNPerilune Altitude ≈ 8.0 kmPerilune Velocity = 2.3 km/sNote:• Code uses physics to determine s/c motion and is independent of engine properties

• Can we produce this much thrust for this amount of time with a s/c of this mass?

• ΔV is dependent on engine characteristics because of low thrust

• Will use this code iteratively with propulsion to determine if mission can be completed

for a given system

-4 -3 -2 -1 0 1 2 3 4

x 105

-3

-2

-1

0

1

2

3

x 105

X Distance (km)

Y D

ista

nce

(km

)

EarthLunar OrbitSpiral OutMoon PositionSpiral In

AAE450 Spring 2009

Back-up Slides

[Levi Brown] [Mission Ops] 4

AAE450 Spring 2009 [Levi Brown] [Mission Ops]

Case 1 Spiral Out

5

-4 -3 -2 -1 0 1 2 3 4

x 105

-3

-2

-1

0

1

2

3

x 105

X Distance (km)

Y D

ista

nce

(km

)

EarthLunar OrbitSpiral Trajectory to L1

AAE450 Spring 2009 [Levi Brown] [Mission Ops]

Case 1 Spiral In

6

-5 -4 -3 -2 -1 0 1 2

x 104

-5

-4

-3

-2

-1

0

x 104

X Distance (km)

Y D

ista

nce

(km

)

MoonSpiral In

AAE450 Spring 2009 [Levi Brown] [Mission Ops] 7

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7x 10

4

Time (days)

Rad

ius

from

Moo

n (k

m)

Spacecraft Radius for Lunar Spiral

AAE450 Spring 2009 [Levi Brown] [Mission Ops] 4

Other Future Work:

• Work with descent group

• Many possibilities for altitude and velocity for descent

• What works best?

• Should we try to circularize orbit more first?

• Look at effects of varying departure orbit (higher LEO, GTO)

• Add in 3rd body effect

• Have both Earth and Moon gravity in both integrations

• More efficient thrusting

• Is it more efficient to thrust in velocity direction?

AAE450 Spring 2009 [Levi Brown] [Mission Ops] 4

Notes on iteration with propulsion:• DeltaV is caused by 2 factors (gravity and thrust)• For impulse type maneuvers, it is assumed that the thrusting is instantaneous• This means that the deltaV between two points on a trajectory is only caused by gravity• Because code indicates an initial mass, a constant thrust, and a TOF, it was determined that the best method of calculating deltaV was through the equations for the engine• The current mass was based off of our initial deltaV assumption of 8 km/s• The sizing equations take a deltaV and various specifications for a particular engine• Using the deltaV and specifications, parameters such as TOF, propellant mass, mass flow rate, initial mass, and thrust are produced.• If we fix a TOF, we can vary deltaV until the TOF match and then find the mass and thrust required for that engine• Then use the trajectory code to determine if we can reach the moon• If not, vary the deltaV and do it again