lunar transfers and the circular restricted three-body...

Lunar Transfers and the CircularRestricted Three-Body Problem

Bennington College Science Workshop

Michael S. Reardon

Visiting Assistant Professor of MathematicsBennington College

[email protected]

November 30th, 2012

Introduction: What is a CubeSat?

CubeSats belong to a class of satellites called nanosatellites:I dimensions: 10cm10cm10cmI mass: < 1.33kg

Double 20cm10cm10cm and Triple 30cm10cm10cmCubeSats are also becoming common.

CubeSats are usually constructed using basic kits and are thenfitted with mission specific technology including:

I CommunicationsI Remote SensingI Propulsion and Navigation

CubeSats piggy-back on larger missions

The Project

The Vermont Lunar Lander CubeSat project is a collaborativeeffort between students and faculty from:

I Vermont Technical CollegeI Saint Michaels CollegeI Norwich UniversityI University of Vermont

Project Director: Carl Brandon, VTC Featured in an emmy nominated VT PBS episode of EmergingScience:Out of This World

Goal: Develop a triple CubeSat capable of reaching a 100 kmlunar orbit and possibly conducting a lunar landing.

2013 test launch to test communications and a guidance system

The Project

Project Director: Carl Brandon, VTC

Featured in an emmy nominated VT PBS episode of EmergingScience:Out of This World

The Project

Low Thrust vs High Thrust

High thrust chemical propellants: Fuel+Oxidizer highpressure byproduct gases Exhaust Nozzle Thrust

I Hydroxyl ammonium nitrate and methanol monopropellantI 4N of thrustI Isp = 270sI v budget of 2250 m/s

Low thrust SEP propulsion: Solar Power Strong ElectricFields Ion Acceleration High Exhaust Velocity

I Xenon Ion SEP systemI 1mN of thrustI Isp 3000sI v budget of 4000 m/s

Figure: JPL Miniature Xenon Ion Thruster

Low Thrust vs High Thrust High thrust chemical propellants: Fuel+Oxidizer high

pressure byproduct gases Exhaust Nozzle ThrustI Hydroxyl ammonium nitrate and methanol monopropellantI 4N of thrustI Isp = 270sI v budget of 2250 m/s

Low Thrust vs High Thrust High thrust chemical propellants: Fuel+Oxidizer high

pressure byproduct gases Exhaust Nozzle ThrustI Hydroxyl ammonium nitrate and methanol monopropellantI 4N of thrustI Isp = 270sI v budget of 2250 m/s

Figure: Ion Thruster (Image courtesy of Wikipedia Commons)

Ion Propulsion Advantages vs Chemical Propulsion:1. Lower fuel mass requirements2. Larger range3. Considered safer for launch with other satellites

Ion Propulsion Disadvantages vs Chemical Propulsion:1. Longer transfer times2. Require larger battery/solar panels3. Radiation damage4. Subject to thruster shutdown due to eclipsing5. Unable to perform large, nearly instantaneous velocity

corrections

Ion Propulsion Advantages vs Chemical Propulsion:1. Lower fuel mass requirements2. Larger range3. Considered safer for launch with other satellites

Ion Propulsion Disadvantages vs Chemical Propulsion:1. Longer transfer times2. Require larger battery/solar panels3. Radiation damage4. Subject to thruster shutdown due to eclipsing5. Unable to perform large, nearly instantaneous velocity

corrections

High Thrust: Direct Transfer (STK)

2 Impulse Transfer based on Hoffman transfer1. 1st impulse takes the CubeSat through the L1 gateway to a

100 km lunar periapsis2. 2nd impulse at lunar periapsis to circularize the lunar orbit

Segment v (m/s) Time (days)1 1039 4.82 738 -

Total 1777 m/s 4.8

(a) (b)

Figure: Direct transfer to lunar orbit

Low Thrust Transfer (STK)

Low thrust transfer similar to the ESA SMART-1 mission1. 1st series of thrust arcs near perigee to increase the radius of

apogee2. 2nd series of thrust arcs near apogee to raise the radius of

perigee and ensure temporary lunar capture3. 3rd series of thrust arcs and spirals to stabilize the lunar orbit4. 4th series of thrust arcs and spirals to circularize and decrease

the orbit radius

Segment v (m/s) Time (day)1 1157 1832 150 83 450 194 910 155

Total 2667 365

Low Thrust Transfer (STK) Low thrust transfer similar to the ESA SMART-1 mission

1. 1st series of thrust arcs near perigee to increase the radius ofapogee

2. 2nd series of thrust arcs near apogee to raise the radius ofperigee and ensure temporary lunar capture

3. 3rd series of thrust arcs and spirals to stabilize the lunar orbit4. 4th series of thrust arcs and spirals to circularize and decrease

the orbit radius

Total 2667 365

Low Thrust Transfer (STK) Low thrust transfer similar to the ESA SMART-1 mission

1. 1st series of thrust arcs near perigee to increase the radius ofapogee

2. 2nd series of thrust arcs near apogee to raise the radius ofperigee and ensure temporary lunar capture

3. 3rd series of thrust arcs and spirals to stabilize the lunar orbit4. 4th series of thrust arcs and spirals to circularize and decrease

the orbit radius

Total 2667 365

Low Thrust Transfer (STK)

(a) Lunar transfer (b) Lunar spiral-in

The CRTBP

Computing lunar transfer trajectories requires a goodunderstanding of the Circular Restricted Three Body Problem(CRTBP).

The CRTBP models the trajectory of a massless satellite subjectto the gravity of two bodies in circular orbits about their CM

The coordinate system is co-rotating with the massive bodiesabout their CM

The origin is fixed at their CM The x-y plane is aligned to their plane of rotation. m1 and m2 are fixed at (x , y) = (, 0) and (1 , 0)

respectively where = m2/(m1 + m2), m1 > m2.

The CRTBP

The origin is fixed at their CM

The x-y plane is aligned to their plane of rotation. m1 and m2 are fixed at (x , y) = (, 0) and (1 , 0)

The CRTBP

The origin is fixed at their CM The x-y plane is aligned to their plane of rotation.

m1 and m2 are fixed at (x , y) = (, 0) and (1 , 0)respectively where = m2/(m1 + m2), m1 > m2.

The CRTBP

The equations of motion are given by:

x =d

dt

xyzuvw

=

uvw

2v + xU

2u + yU

zU

= F(x)

U =1

2(x2 + y 2) +

1 r1

+

r2

r1 =

(x + )2 + y 2 + z2

r2 =

(x 1 + )2 + y 2 + z2

The CRTBP

The equations of motion are given by:

x =d

dt

xyzuvw

=

uvw

2v + xU

2u + yU

zU

= F(x)

U =1

2(x2 + y 2) +

1 r1

+

r2

r1 =

(x + )2 + y 2 + z2

r2 =

(x 1 + )2 + y 2 + z2

The Jacobi Energy The Jacobi Energy is given by:

C (x) = x2 + y 2 + 21 r1

+ 2

r2 (x2 + y 2 + z2)

C is constant on all trajectories By setting x = y = z = 0 we obtain the zero velocity surfaceC (x , y , z) = C

(c) top view (d) side view

Figure: A zero velocity surface

C (x) = x2 + y 2 + 21 r1

+ 2

r2 (x2 + y 2 + z2)

C is constant on all trajectories

By setting x = y = z = 0 we obtain the zero velocity surfaceC (x , y , z) = C

(a) top view (b) side view

C (x) = x2 + y 2 + 21 r1

+ 2

r2 (x2 + y 2 + z2)

C (x) = x2 + y 2 + 21 r1

+ 2

r2 (x2 + y 2 + z2)

The Jacobi Energy

0.5 0 0.5 1 1.5

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

z

(a)

0.2 0 0.2 0.4 0.6 0.8 1 1.20.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

x

(b)

1 0.5 0 0.5 1 1.5

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

z

(c)

1 0.5 0 0.5 1 1.5 2

1

0.5

0

0.5

1

xz

(d)

1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

x

(e)

2 1.5 1 0.5 0 0.5 1 1.5 22

1.5

1

0.5

0

0.5

1

1.5

2

x

y

(f)

Figure: Planar projection of C and sample orbits for decreasing C

Lagrange Points

5 Lagrange points where forces balance (in the rotatingcoordinate system)

I Solutions to x = F(x) = 0I L1L3 are on located on the x-axisI L4 and L5 are located at the tips of equilateral triangles

Ideal for observation, communication, and (eventually) supply

Figure: The E-M Lagrange points (courtesy of Wikipedia Commons)

Lagrange Points 5 Lagrange points where forces balance (in the rotating

coordinate system)I Solutions to x = F(x) = 0I L1L3 are on located on the x-axisI L4 and L5 are located at the tips of equilateral triangles

Lagrange Points 5 Lagrange points where forces balance (in the rotating

coordinate system)I Solutions to x = F(x) = 0I L1L3 are on located on the x-axisI L4 and L5 are located at the tips of equilateral triangles

Stabilty

Small displacements x = x xi about the ith Lagrange pointevolve locally according to the equation:

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

J(xi) is the (constant) Jacobian matrix of F at xi Solutions are of the form vet where (, v) is an

eigenvalue/eigenvector pair of J(xi)

6 eigenvalues/eigenvectors describe how trajectories approach,leave, or orbit the fixed point

If Re() > 0, perturbations from Li may grow and the Lagrangepoint is unstable

L1L3 are unstable, L4L5 are stable

Stabilty

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

J(xi) is the (constant) Jacobian matrix of F at xi

Solutions are of the form vet where (, v) is aneigenvalue/eigenvector pair of J(xi)

Stabilty

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

Stabilty

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

Stabilty

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

Stabilty

x = F(xi) + J(xi)x + O(||x||2) J(xi)x (1)

Chaos in the CRTBP

The CRTBP is a dynamical system where knowledge of thepresent state can be used to uniquely determine any future state

Dynamical systems may exhibit chaotic behavior. Calling cardsof chaos include:

I Sensitivity to small changes of initial conditionsI Topological mixingI Periodic orbits are dense in the phase space

Large C solutions are tightly bound to the earth and moon Small C solutions, especially those passing near Lagrange points

typically more chaotic

Chaos in the CRTBP

Large C solutions are tightly bound to the earth and moon

Small C solutions, especially those passing near Lagrange pointstypically more chaotic

Chaos in the CRTBP

1.5 1 0.5 0 0.5 1 1.51.5

1

0.5

0

0.5

1

1.5

xzz

y

(a) (x , y , z) trajectory

0 10 20 30 40 50 600.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

x(t)

(b) x vs. t

Figure: Sensitivity to initial conditions

Chaos in the CRTBP

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

(a) t = 2.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

(b) t = 5

0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

(c) t = 7.5

0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

(d) t = 10

0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

(e) t = 12.5

0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

(f) t = 15

Figure: Topological mixing

Chaos in the CRTBP

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.24

3

2

1

0

1

2

3

4

x

Figure: A Poincare map

Chaos in the CRTBP

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.051

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

x

0.4 0.45 0.5 0.55 0.60.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

xx

0.4 0.45 0.5 0.55 0.60.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

x

Figure: Periodic orbits and the Poincare map

Chaos in the CRTBP

z

Figure: Periodic and quasiperiodic orbits

Periodic orbits are fixed points? Stability is determined by the eigenvalues of the MonodromyMatrix

The eigenvalues/eigenvectors determine how trajectoriesapproach, leave, or orbit the periodic orbit

Chaos in the CRTBP

z

Periodic orbits are fixed points?

Stability is determined by the eigenvalues of the MonodromyMatrix

Chaos in the CRTBP

z

Chaos in the CRTBP

z

Periodic Orbits

The Lagrange points are also surrounded by periodic orbitsincluding:

I Lyopunov (planar)I Halo (non-planar)I Lissajous (quasi-periodic)

0.7 0.75 0.8 0.85 0.9 0.95 1

0.1

0.05

0

0.05

0.1

(a) Lyapunov family (b) Halo families

Figure: Periodic orbit families about L1

Periodic Orbits The Lagrange points are also surrounded by periodic orbits

including:I Lyopunov (planar)I Halo (non-planar)I Lissajous (quasi-periodic)

0.7 0.75 0.8 0.85 0.9 0.95 1

0.1

0.05

0

0.05

0.1

Periodic Orbits The Lagrange points are also surrounded by periodic orbits

including:I Lyopunov (planar)I Halo (non-planar)I Lissajous (quasi-periodic)

0.7 0.75 0.8 0.85 0.9 0.95 1

0.1

0.05

0

0.05

0.1

Transfer via Lyopunov Manifolds

Information on how trajectories leave fixed points is useful forefficient station keeping

It can also be used for transport to/from fixed points andperiodic orbits along invariant manifolds

0.8 0.9 1 1.1 1.2

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

(a) Unstable Manifold

0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

x

(b) Stable Manifold

Figure: Lyopunov orbit manifolds

Transfer via Lyopunov Manifolds Information on how trajectories leave fixed points is useful for

efficient station keeping

It can also be used for transport to/from fixed points andperiodic orbits along invariant manifolds

0.8 0.9 1 1.1 1.2

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

x

(b) Stable Manifold

efficient station keeping It can also be used for transport to/from fixed points and

periodic orbits along invariant manifolds

0.8 0.9 1 1.1 1.2

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

x

(b) Stable Manifold

efficient station keeping It can also be used for transport to/from fixed points and

periodic orbits along invariant manifolds

0.8 0.9 1 1.1 1.2

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

x

(b) Stable Manifold

Transfer via Lyopunov Manifolds Example: A Impulse Transfer via Lyopunov manifolds

1. 1st impulse takes the CubeSat to stable manifold of aLyopunov orbit

2. 2nd impulse adjusts the velocity to that of the manifold3. 3rd impulse to leave orbit to ensure lunar capture and 100 km

lunar periapsis upon leaving the L1 orbit.4. 4th impulse at lunar periapsis to circularize the orbit about the

moon

Segment v (m/s) Time (day)1 677 1.62 851 28.33 3 9.44 635 -

Total 2165 39.3

Transfer via Lyopunov Manifolds

(a) CRTBP frame (Matlab) (b) Earth frame (STK)

Summary

Questions?

Acknowledgements:

lunar transfers and the circular restricted three-body...

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