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Potential Solar Sail Degradation Effects on Trajectory and Attitude Control

Bernd Dachwald1 and the Solar Sail Degradation Model Working Group2

1German Aerospace Center (DLR), Institute of Space Simulation

Linder Hoehe, 51170 Cologne, Germany, bernd.dachwald@dlr.de

2Malcolm Macdonald, Univ. of Glasgow, Scotland; Giovanni Mengali and Alessandro A.

Quarta, Univ. of Pisa, Italy; Colin R. McInnes, Univ. of Strathclyde, Glasgow, Scotland; Leonel Rios-Reyes and Daniel J. Scheeres, Univ. of Michigan, Ann Arbor, USA; Marianne G¨

• rlich and Franz Lura, DLR, Berlin, Germany; Volodymyr Baturkin, Natl. Tech. Univ. of

Ukraine, Kiev, Ukraine; Victoria L. Coverstone, Univ. of Illinois, Urbana-Champaign, USA; Benjamin Diedrich, NOAA, Silver Spring, USA; Gregory P. Garbe, NASA MSFC, Huntsville, USA; Manfred Leipold, Kayser-Threde GmbH, Munich, Germany; Wolfgang Seboldt, DLR, Cologne, Germany; Bong Wie, Arizona State Univ., Tempe, USA

AAS/AIAA Astrodynamics Specialists Conference 7–11 August 2005, Lake Tahoe, CA

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 1 / 42

Outline

The Problem

The optical properties of the thin metalized polymer films that are projected for solar sails are assumed to be affected by the erosive effects of the space environment Optical solar sail degradation (OSSD) in the real space environment is to a considerable degree indefinite (initial ground test results are controversial and relevant in-space tests have not been made so far) The standard optical solar sail models that are currently used for trajectory and attitude control design do not take optical degradation into account → its potential effects on trajectory and attitude control have not been investigated so far Optical degradation is important for high-fidelity solar sail mission analysis, because it decreases both the magnitude of the solar radiation pressure force acting on the sail and also the sail control authority Solar sail mission designers necessitate an OSSD model to estimate the potential effects of OSSD on their missions

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 2 / 42

Outline

Our Approach

We established in November 2004 a ”Solar Sail Degradation Model Working Group” (SSDMWG) with the aim to make the next step towards a realistic high-fidelity optical solar sail model We propose a simple parametric OSSD model that describes the variation of the sail film’s optical coefficients with time, depending on the sail film’s environmental history, i.e., the radiation dose The primary intention of our model is not to describe the exact behavior of specific film-coating combinations in the real space environment, but to provide a more general parametric framework for describing the general optical degradation behavior of solar sails

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 3 / 42

Outline

1

Solar Sail Force Models Ideal Reflection Non-Perfect Reflection Simplified Non-Perfect Reflection

2

Degradation Model Data Available From Ground Testing Parametric Degradation Model

3

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law Mars Rendezvous Mercury Rendezvous Fast Neptune Flyby Fast Transfer to the Heliopause Artificial Lagrange-Point Missions

4

Summary and Conclusions

5

Outlook

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 4 / 42

Solar Sail Force Models

Overview

Different levels of simplification for the optical characteristics of a solar sail result in different models for the magnitude and direction of the SRP force acting on the sail: Model IR (Ideal Reflection) Most simple model Model SNPR (Simplified Non-Perfect Reflection) Optical properties of the solar sail are described by a single coefficient Model NPR (Non-Perfect Reflection) Optical properties of the solar sail are described by 3 coefficients Generalized Model by Rios-Reyes and Scheeres Optical properties are described by three tensors of rank 1, 2, and 3 (19 numbers in total, due to symmetry). Takes the sail shape and local optical variations into account

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42

Solar Sail Force Models

Overview

Different levels of simplification for the optical characteristics of a solar sail result in different models for the magnitude and direction of the SRP force acting on the sail: Model IR (Ideal Reflection) Most simple model Model SNPR (Simplified Non-Perfect Reflection) Optical properties of the solar sail are described by a single coefficient Model NPR (Non-Perfect Reflection) Optical properties of the solar sail are described by 3 coefficients Generalized Model by Rios-Reyes and Scheeres Optical properties are described by three tensors of rank 1, 2, and 3 (19 numbers in total, due to symmetry). Takes the sail shape and local optical variations into account

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42

Solar Sail Force Models

Overview

Different levels of simplification for the optical characteristics of a solar sail result in different models for the magnitude and direction of the SRP force acting on the sail: Model IR (Ideal Reflection) Most simple model Model SNPR (Simplified Non-Perfect Reflection) Optical properties of the solar sail are described by a single coefficient Model NPR (Non-Perfect Reflection) Optical properties of the solar sail are described by 3 coefficients Generalized Model by Rios-Reyes and Scheeres Optical properties are described by three tensors of rank 1, 2, and 3 (19 numbers in total, due to symmetry). Takes the sail shape and local optical variations into account

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42

Solar Sail Force Models

Overview

Different levels of simplification for the optical characteristics of a solar sail result in different models for the magnitude and direction of the SRP force acting on the sail: Model IR (Ideal Reflection) Most simple model Model SNPR (Simplified Non-Perfect Reflection) Optical properties of the solar sail are described by a single coefficient Model NPR (Non-Perfect Reflection) Optical properties of the solar sail are described by 3 coefficients Generalized Model by Rios-Reyes and Scheeres Optical properties are described by three tensors of rank 1, 2, and 3 (19 numbers in total, due to symmetry). Takes the sail shape and local optical variations into account

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42

Solar Sail Force Models Ideal Reflection

SRP Force

• n an Ideal Solar Sail

The solar radiation pressure (SRP) at a distance r from the sun is P = S0 c r0 r 2 = 4.563 µN m2 · r0 r 2 FSRP = 2PA cos α cos α n

Nomenclature S0: solar constant (1368 W/m2) c: speed of light in vacuum r0: 1 astronomical unit (1 AU) α: sail pitch angle n: sail normal vector t: sail tangential vector FSRP: SRP force A: sail area Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 6 / 42

Solar Sail Force Models Ideal Reflection

SRP Force

• n an Ideal Solar Sail

The solar radiation pressure (SRP) at a distance r from the sun is P = S0 c r0 r 2 = 4.563 µN m2 · r0 r 2

i n c

• m

i n g ฀ r a d i a t i

• n

r e f l e c t e d ฀ r a d i a t i

• n

sail sun-line

α α α α α

SRP

F n t

FSRP = 2PA cos α cos α n

Nomenclature S0: solar constant (1368 W/m2) c: speed of light in vacuum r0: 1 astronomical unit (1 AU) α: sail pitch angle n: sail normal vector t: sail tangential vector FSRP: SRP force A: sail area Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 6 / 42

Solar Sail Force Models Non-Perfect Reflection

The Non-Perfectly Reflecting Solar Sail

The non-perfectly reflecting solar sail model parameterizes the optical behavior of the sail film by the

• ptical coefficient set

P = {ρ, s, εf, εb, Bf, Bb} The optical coefficients for a solar sail with a highly reflective aluminum-coated front side and with a highly emissive chromium-coated back side are: PAl|Cr = {ρ = 0.88, s = 0.94, εf = 0.05, εb = 0.55, Bf = 0.79, Bb = 0.55}

Nomenclature ρ: reflection coefficient s: specular reflection factor εf and εb: emission coefficients of the front and back side, respectively Bf and Bb: non-Lambertian coefficients of the front and back side, respectively Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 7 / 42

Solar Sail Force Models Non-Perfect Reflection

The Non-Perfectly Reflecting Solar Sail

SRP Force in a sail-fixed coordinate frame S = {n, t}

i n c

• m

i n g ฀ r a d i a t i

• n

reflected฀radiation sail sun-line

α α α

SRP

F n t

⊥

F

||

F θ φ m

FSRP = 2PA cos α [(a1 cos α + a2) n − a3 sin α t] with the derived optical coefficients

a1 1 2 (1 + sρ) a2 1 2 ✧ Bf(1 − s)ρ + (1 − ρ) εfBf − εbBb εf + εb ★ a3 1 2 (1 − sρ) Nomenclature α: sail pitch angle n: sail normal vector m: thrust unit vector t: sail tangential vector FSRP: SRP force F⊥: SRP force component along n F||: SRP force component along t θ: thrust cone angle φ: centerline angle P: solar radiation pressure (SRP) A: sail area Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 8 / 42

Solar Sail Force Models Non-Perfect Reflection

The Non-Perfectly Reflecting Solar Sail

SRP Force on along the radial and sail normal direction

FSRP can also be decomposed along the radial direction er and the sail normal direction n:

FSRP = 2PA cos α [b1er + (b2 cos α + b3)n] with the derived optical coefficients

b1 1 2 (1 − sρ) b2 sρ b3 1 2 ✧ Bf(1 − s)ρ + (1 − ρ) εfBf − εbBb εf + εb ★ Nomenclature FSRP: SRP force P: solar radiation pressure (SRP) A: sail area α: sail pitch angle er : radial unit vector n: sail normal vector Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 9 / 42

Solar Sail Force Models Simplified Non-Perfect Reflection

The Simplified Model

SRP Force in a sail-fixed coordinate frame S = {n, t}

Recall that FSRP = 2PA cos α [(a1 cos α + a2) n − a3 sin α t] with

a1 1 2 (1 + sρ) a2 1 2 ✧ Bf(1 − s)ρ + (1 − ρ) εfBf − εbBb εf + εb ★ a3 1 2 (1 − sρ)

Assumptions: s = 1, εfBf = εbBb

FSRP = PA cos α [(1 + ρ) cos α n − (1 − ρ) sin α t] Typically, however, the reflection coefficient ρ is denoted as η within this model

Nomenclature FSRP: SRP force P: solar radiation pressure (SRP) A: sail area α: sail pitch angle n: sail normal vector t: sail tangential vector ρ: reflection coefficient s: specular reflection factor εf and εb: emission coefficients of the front and back side, respectively Bf and Bb: non-Lambertian coefficients of the front and back side, respectively Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 10 / 42

Degradation Model

Overview (Reprise)

Model IR (Ideal Reflection) Most simple model Model SNPR (Simplified Non-Perfect Reflection) Optical properties of the solar sail are described by a single coefficient Model NPR (Non-Perfect Reflection) Optical properties of the solar sail are described by 3 coefficients Generalized Model by Rios-Reyes and Scheeres Optical properties are described by three tensors of rank 1, 2, and 3 (19 numbers in total, due to symmetry). Takes the sail shape and local optical variations into account

Those models do not include optical solar sail degradation (OSSD)

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 11 / 42

Degradation Model Data Available From Ground Testing

Data Available From Ground Testing

Much ground and space testing has been done to measure the optical degradation of metalized polymer films as second surface mirrors (metalized

• n the back side)

No systematic testing to measure the optical degradation of candidate solar sail films (metalized on the front side) has been reported so far and preliminary test results are controversial

◮ Lura et. al. measured considerable OSSD after combined irradiation

with VUV, electrons, and protons

◮ Edwards et. al. measured no change of the solar absorption and

emission coefficients after irradiation with electrons alone Respective in-space tests have not been made so far The optical degradation behavior of solar sails in the real space environment is to a considerable degree indefinite

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 12 / 42

Degradation Model Parametric Degradation Model

Simplifying Assumptions

For a first OSSD model, we have made the following simplifications:

1 The only source of degradation are the solar photons and particles 2 The solar photon and particle fluxes do not depend on time (average

sun without solar events)

3 The optical coefficients do not depend on the sail temperature 4 The optical coefficients do not depend on the light incidence angle 5 No self-healing effects occur in the sail film Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 13 / 42

Degradation Model Parametric Degradation Model

Solar radiation dose (SRD) Let p be an arbitrary optical coefficient from the set P. With OSSD, p becomes time-dependent, p(t). With the simplifications stated before, p(t) is a function of the solar radiation dose ˜ Σ (dimension

• J/m2

) accepted by the solar sail within the time interval t − t0: ˜ Σ(t) t

t0

S cos α dt′ = S0r 2 t

t0

cos α r 2 dt′ SRD per year on a surface perpendicular to the sun at 1 AU ˜ Σ0 = S0 · 1 yr = 1368 W/m2 · 1 yr = 15.768 TJ/m2 Dimensionless SRD Using ˜ Σ0 as a reference value, the SRD can be defined in dimensionless form: Σ(t) = ˜ Σ(t) ˜ Σ0 = r 2 T t

t0

cos α r 2 dt′ where T 1 yr

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 14 / 42

Degradation Model Parametric Degradation Model

Solar radiation dose (SRD) Let p be an arbitrary optical coefficient from the set P. With OSSD, p becomes time-dependent, p(t). With the simplifications stated before, p(t) is a function of the solar radiation dose ˜ Σ (dimension

• J/m2

) accepted by the solar sail within the time interval t − t0: ˜ Σ(t) t

t0

S cos α dt′ = S0r 2 t

t0

cos α r 2 dt′ SRD per year on a surface perpendicular to the sun at 1 AU ˜ Σ0 = S0 · 1 yr = 1368 W/m2 · 1 yr = 15.768 TJ/m2 Dimensionless SRD Using ˜ Σ0 as a reference value, the SRD can be defined in dimensionless form: Σ(t) = ˜ Σ(t) ˜ Σ0 = r 2 T t

t0

cos α r 2 dt′ where T 1 yr

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 14 / 42

Degradation Model Parametric Degradation Model

Solar radiation dose (SRD) Let p be an arbitrary optical coefficient from the set P. With OSSD, p becomes time-dependent, p(t). With the simplifications stated before, p(t) is a function of the solar radiation dose ˜ Σ (dimension

• J/m2

) accepted by the solar sail within the time interval t − t0: ˜ Σ(t) t

t0

S cos α dt′ = S0r 2 t

t0

cos α r 2 dt′ SRD per year on a surface perpendicular to the sun at 1 AU ˜ Σ0 = S0 · 1 yr = 1368 W/m2 · 1 yr = 15.768 TJ/m2 Dimensionless SRD Using ˜ Σ0 as a reference value, the SRD can be defined in dimensionless form: Σ(t) = ˜ Σ(t) ˜ Σ0 = r 2 T t

t0

cos α r 2 dt′ where T 1 yr

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 14 / 42

Degradation Model Parametric Degradation Model

Dimensionless SRD Using ˜ Σ0 as a reference value, the SRD can be defined in dimensionless form: Σ(t) = ˜ Σ(t) ˜ Σ0 = r 2 T t

t0

cos α r 2 dt′ Σ(t) depends on the solar distance history and the attitude history z[t] = (r, α)[t]

• f the solar sail, Σ(t) = Σ(z[t])

Differential form for the SRD The equation for the SRD can also be written in differential form: ˙ Σ = r 2 T cos α r 2 with Σ(t0) = 0

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 15 / 42

Degradation Model Parametric Degradation Model

Dimensionless SRD Using ˜ Σ0 as a reference value, the SRD can be defined in dimensionless form: Σ(t) = ˜ Σ(t) ˜ Σ0 = r 2 T t

t0

cos α r 2 dt′ Σ(t) depends on the solar distance history and the attitude history z[t] = (r, α)[t]

• f the solar sail, Σ(t) = Σ(z[t])

Differential form for the SRD The equation for the SRD can also be written in differential form: ˙ Σ = r 2 T cos α r 2 with Σ(t0) = 0

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 15 / 42

Degradation Model Parametric Degradation Model

Assumption that each p varies exponentially with Σ(t) Assume that p(t) varies exponentially between p(t0) = p0 and lim

t→∞ p(t) = p∞

p(t) = p∞ + (p0 − p∞) · e−λΣ(t) The degradation constant λ is related to the ”half life solar radiation dose” ˆ Σ (Σ = ˆ Σ ⇒ p = p0+p∞

2

) via λ = ln 2 ˆ Σ Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6 half life SRDs ˆ Σp (too much for a simple parametric OSSD analysis) Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 16 / 42

Degradation Model Parametric Degradation Model

Assumption that each p varies exponentially with Σ(t) Assume that p(t) varies exponentially between p(t0) = p0 and lim

t→∞ p(t) = p∞

p(t) = p∞ + (p0 − p∞) · e−λΣ(t) The degradation constant λ is related to the ”half life solar radiation dose” ˆ Σ (Σ = ˆ Σ ⇒ p = p0+p∞

2

) via λ = ln 2 ˆ Σ Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6 half life SRDs ˆ Σp (too much for a simple parametric OSSD analysis) Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 16 / 42

Degradation Model Parametric Degradation Model

Assumption that each p varies exponentially with Σ(t) Assume that p(t) varies exponentially between p(t0) = p0 and lim

t→∞ p(t) = p∞

p(t) = p∞ + (p0 − p∞) · e−λΣ(t) The degradation constant λ is related to the ”half life solar radiation dose” ˆ Σ (Σ = ˆ Σ ⇒ p = p0+p∞

2

) via λ = ln 2 ˆ Σ Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6 half life SRDs ˆ Σp (too much for a simple parametric OSSD analysis) Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 16 / 42

Degradation Model Parametric Degradation Model

Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P EOL optical coefficients Because the reflectivity of the sail decreases with time, the sail becomes more matt with time, and the emissivity increases with time, we use: ρ∞ = ρ0 1 + d s∞ = s0 1 + d εf∞ = (1 + d)εf0 εb∞ = εb0 Bf∞ = Bf0 Bb∞ = Bb0 Degradation of the optical parameters in dimensionless form p(t) p0 =     

• 1 + de−λΣ(t)

/ (1 + d) for p ∈ {ρ, s} 1 + d

• 1 − e−λΣ(t)

for p = εf 1 for p ∈ {εb, Bf, Bb}

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 17 / 42

Degradation Model Parametric Degradation Model

Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P EOL optical coefficients Because the reflectivity of the sail decreases with time, the sail becomes more matt with time, and the emissivity increases with time, we use: ρ∞ = ρ0 1 + d s∞ = s0 1 + d εf∞ = (1 + d)εf0 εb∞ = εb0 Bf∞ = Bf0 Bb∞ = Bb0 Degradation of the optical parameters in dimensionless form p(t) p0 =     

• 1 + de−λΣ(t)

/ (1 + d) for p ∈ {ρ, s} 1 + d

• 1 − e−λΣ(t)

for p = εf 1 for p ∈ {εb, Bf, Bb}

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 17 / 42

Degradation Model Parametric Degradation Model

Reduction of the number of model parameters We use a degradation factor d and a single half life SRD for all p, ˆ Σp = ˆ Σ ∀p ∈ P EOL optical coefficients Because the reflectivity of the sail decreases with time, the sail becomes more matt with time, and the emissivity increases with time, we use: ρ∞ = ρ0 1 + d s∞ = s0 1 + d εf∞ = (1 + d)εf0 εb∞ = εb0 Bf∞ = Bf0 Bb∞ = Bb0 Degradation of the optical parameters in dimensionless form p(t) p0 =     

• 1 + de−λΣ(t)

/ (1 + d) for p ∈ {ρ, s} 1 + d

• 1 − e−λΣ(t)

for p = εf 1 for p ∈ {εb, Bf, Bb}

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 17 / 42

Degradation Model Parametric Degradation Model

OSSD Effects

• n the optical coefficients and the maximum sail temperature

0.02 . 2 0.04 0.04 0.06 0.06 0.08 . 8 0.1 . 1 0.12 0.12 0.14 0.14 0.16 . 1 6 0.18 . 1 8 . 2 . 2

Σ ρ/ρ0 , s/s0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.8 0.85 0.9 0.95 1

0.02 0.02 0.04 . 4 0.06 0.06 0.08 0.08 0.1 0.1 0.12 . 1 2 . 1 4 . 1 4 . 1 6 0.16 0.18 . 1 8 0.2 . 2

Σ εf /εf0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.05 1.1 1.15 1.2

d d

ρ/ρ0, s/s0, and εf/εf 0 Σ Tmax (α=0) [°C]

1 2 3 4 100 200 300 400

r = 0.2 AU r = 1.0 AU r = 0.8 AU r = 0.6 AU r = 0.4 AU

Tmax for different solar distances (d = 0.2) Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 18 / 42

Degradation Model Parametric Degradation Model

OSSD Effects

• n the SRP force bubble and the control angles

. = Σ 5 . = Σ . 1 = Σ . 5 = Σ r

e

t

e

FSRP-bubble

10 20 30 40 50 60 70 80 90 −10 10 20 30 40 50 60 Pitch angle [deg] Cone angle [deg] Σ = 5.0 Σ = 1.0 Σ = 0.5 Σ = 0.0

θ(α)

0.5 1 1.5 2 2.5 3 3.5 4 35.2 35.3 35.4 α* [deg] 0.5 1 1.5 2 2.5 3 3.5 4 24 26 28 30 32 θ* [deg] 0.5 1 1.5 2 2.5 3 3.5 4 −30 −20 −10 Σ ∆Ft

* [%]

α∗, θ∗, and ∆F ∗

t Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 19 / 42

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Simulation Model

Considerations for trajectory control

Gravitational forces of all celestial bodies Solar wind Finiteness of the solar disk Reflected light from close celestial bodies Aberration of solar radiation (Poynting-Robertson effect) The solar sail bends and wrinkles, depending on the actual solar sail design Finite attitude control maneuvers

Simplifications for mission feasibility analysis and to isolate the effects of OSSD

The solar sail is a flat plate The solar sail is moving under the sole influence of solar gravitation and radiation The sun is a point mass and a point light source The solar sail attitude can be changed instantaneously

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 20 / 42

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Simulation Model

Considerations for trajectory control

Gravitational forces of all celestial bodies Solar wind Finiteness of the solar disk Reflected light from close celestial bodies Aberration of solar radiation (Poynting-Robertson effect) The solar sail bends and wrinkles, depending on the actual solar sail design Finite attitude control maneuvers

Simplifications for mission feasibility analysis and to isolate the effects of OSSD

The solar sail is a flat plate The solar sail is moving under the sole influence of solar gravitation and radiation The sun is a point mass and a point light source The solar sail attitude can be changed instantaneously

Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 20 / 42

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Problem Statement

Equations of motion For a solar sail in heliocentric cartesian reference frame: ˙ r = v ˙ v = − µ r 3 r + a where a = a(r, n, b1(t), b2(t), b3(t)) is the SRP acceleration acting on the solar sail Problem Minimize the time tf necessary to transfer the sail from x0 = (r0, v0) to xf = (rf , vf ) by maximizing the performance index J = −tf

Nomenclature a: propulsive (+ disturbing) acceleration

• n the sail

r: sail position r: radius, |r| v: sail velocity µ: gravitational parameter

• f the sun

b1(t), b2(t), b3(t): functions of the sail’s

• ptical parameters

n: sail normal vector x: sail state 0: initial value of f : final value of Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 21 / 42

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Problem Statement

Equations of motion For a solar sail in heliocentric cartesian reference frame: ˙ r = v ˙ v = − µ r 3 r + a where a = a(r, n, b1(t), b2(t), b3(t)) is the SRP acceleration acting on the solar sail Problem Minimize the time tf necessary to transfer the sail from x0 = (r0, v0) to xf = (rf , vf ) by maximizing the performance index J = −tf

Nomenclature a: propulsive (+ disturbing) acceleration

• n the sail

r: sail position r: radius, |r| v: sail velocity µ: gravitational parameter

• f the sun

b1(t), b2(t), b3(t): functions of the sail’s

• ptical parameters

n: sail normal vector x: sail state 0: initial value of f : final value of Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 21 / 42

Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Variational Problem

Using standard COV, the optimal direction of n is found by maximizing the Hamiltonian H H = λ λ λr · v − µ r 3λ λ λv · r + λ λ λv · a + λΣ r 2 r 2T er · n where λ λ λr, λ λ λv are the vectors adjoint to the position, and λΣ is the radiation dose costate The result is n =    sin (αλ − α) sin αλ er + sin α sin αλ eλ

λ λv

for αλ = 0 er for αλ = 0 where er = r |r| eλ

λ λv = λ

λ λv |λ λ λv| cos αλ = er · eλ

λ λv

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Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Remarks about the Optimal Solution

The optimal control law requires the thrust vector to lie in the plane defined by the position vector r and the primer vector λ λ λv. This generalizes a similar conclusion obtained for model IR by C. Sauer and for model NPR without degradation by G. Mengali and A. Quarta The equation giving the optimal cone angle as a function of αλ can be written analytically and solved numerically The next slide shows, how the optimal solutions typically vary with the solar radiation dose

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Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law

Typical Optimal Solutions

• α

αλ

Σ

• α

αλ

Σ

• α

αλ

Σ

• α

αλ

Σ Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 24 / 42

Degradation Effects on Trajectory and Attitude Control Mars Rendezvous

Mars Rendezvous

Solar sail with 0.1 mm/s2 ≤ ac < 6 mm/s2 C3 = 0 km2/s2 2D-transfer from circular orbit to circular orbit Trajectories calculated by G. Mengali and A. Quarta using a classical indirect method with an hybrid technique (genetic + gradient-based algorithm) to solve the associated boundary value problem Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit) Half life SRD: ˆ Σ = 0.5 (S0·yr) Three models: ⊲ Model (a): Instantaneous degradation ⊲ Model (b): Control neglects degradation (”ideal” control law) ⊲ Model (c): Control considers degradation

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Degradation Effects on Trajectory and Attitude Control Mars Rendezvous

Mars Rendezvous

Trip times for 5% and 20% degradation limit

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 500 1000 1500 2000 2500 3000

ac [mm/s2] tf [days] d=0.05

no degradation model a model b model c 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 200 300 400 500 600 700

ac [mm/s2] tf [days]

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 500 1000 1500 2000 2500 3000 3500 4000

ac [mm/s2] tf [days] d=0.2

no degradation model a model b model c 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 200 300 400 500 600 700 800 900 1000

ac [mm/s2] tf [days]

OSSD has considerable effect on trip times The results for model (b) and (c) are indistinguishable close

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Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous

Mercury Rendezvous

Solar sail with ac = 1.0 mm/s2 C3 = 0 km2/s2 Trajectories calculated by B. Dachwald with the trajectory optimizer GESOP with SNOPT Arbitrarily selected launch window MJD 57000 ≤ t0 ≤ MJD 57130 (09 Dec 2014 – 18 Apr 2015) Final accuracy limit was set to ∆rf ,max = 80 000 km (inside Mercury’s sphere of influence at perihelion) and ∆vf ,max = 50 m/s Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit) Half life SRD: ˆ Σ = 0.5 (S0·yr)

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Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous

Mercury Rendezvous

Launch window for different d

Launch date (MJD) Trip time [days] 57000 57050 57100 57150 350 400 450 Degradation limit: 0% 5% 10% 20% Launch window for Mercury rendezvous Solar sail with ac=1.0mm/s

2

Half life SRD = 0.5 (S0*yr) Launch date (MJD) Trip time increase [%] 57000 57050 57100 57150 5 10 15 20 25 30 35 Half life SRD=0.5 (S0*yr) Degradation limit: 5% 10% 20% Launch window for Mercury rendezvous Solar sail with ac=1.0mm/s

2

Sensitivity of the trip time with respect to OSSD depends considerably on the launch date Some launch dates considered previously as optimal become very unsuitable when OSSD is taken into account For many launch dates OSSD does not seriously affect the mission

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Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous

Mercury Rendezvous

Optimal α-variation for different d

Time (MJD) Pitch angle [rad]

57000 57100 57200 57300 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Degradation limit: 0% 5% 10% 20%

Mercury rendezvous (solar sail with ac=1.0mm/s

2)

Launch date = MJD 57000.0

Half life SRD = 0.5 (S0*yr)

Launch at MJD 57000.0

Time (MJD) Pitch angle [rad]

57100 57200 57300 57400 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Half life SRD = 0.5 (S0*yr) Degradation limit: 0% 5% 10% 20%

Mercury rendezvous (solar sail with ac=1.0mm/s

2)

Launch date = MJD 57030.0

Launch at MJD 57030.0 OSSD can also have remarkable consequences on the optimal control angles Given an indefinite OSSD behavior at launch, MJD 57000.0 would be a very robust launch date

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Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby

Fast Neptune Flyby

Solar sail with ac = 1.0 mm/s2 C3 = 0 km2/s2 Trajectories calculated by B. Dachwald with the trajectory optimizer InTrance To find the absolute trip time minima, independent of the actual constellation of Earth and Neptune, no flyby at Neptune itself, but

• nly a crossing of its orbit within a distance ∆rf ,max < 106 km was

required, and the optimizer was allowed to vary the launch date within a one year interval Sail film temperature was limited to 240◦C by limiting the sail pitch angle Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit) Half life SRD: 0 ≤ ˆ Σ ≤ 2 (S0·yr)

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Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby

Fast Neptune Flyby

Topology of optimal trajectories for different d

x [AU] y [AU]

• 1
• 1

1 1 2 2

• 1
• 1
• 0.5
• 0.5

0.5 0.5 1 1 1.5 1.5 2 2

520 510 500 400 300 200 100

Trip time = 5.80 years

rmin Tlim Sail Temp. [K]

Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s

2)

= 0.204 AU = 513.15 K = 240°C

without degradation

C3=0km

2/s 2 at Earth

Flyby at Neptune orbit within ∆r<1.0E6km

d = 0

x [AU] y [AU]

• 1
• 1

1 1 2 2

• 1.5
• 1.5
• 1
• 1
• 0.5
• 0.5

0.5 0.5 1 1 1.5 1.5

520 510 500 400 300 200 100

Trip time = 7.87 years

rmin Tlim Sail Temp. [K]

Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s

2)

= 0.294 AU = 513.15 K = 240°C

20% degradation limit, half life SRD = 0.5 (S0*yr)

C3=0km

2/s 2 at Earth

Flyby at Neptune orbit within ∆r<1.0E6km

d = 0.2 With increasing degradation: Increasing solar distance during final close solar pass Increasing solar distance before final close solar pass Longer trip time

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Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby

Fast Neptune Flyby

Trip time and trip time increase for different d and ˆ Σ

Sail degradation limit [%] ___ Trip time [yrs] _ _ Trip time increase [%] 5 10 15 20 1 2 3 4 5 6 7 8 10 20 30 40 Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s

2)

Half life SRD = 0.5 (S0*yr) C3=0km

2/s 2 at Earth

Flyby at Neptune orbit within ∆r<1.0E6km

Different degradation factors d (ˆ Σ = 0.5 (S0 ·yr))

1 / Half life SRD [1/(S0*yr)] ___ Trip time [yrs] _ _ Trip time increase [%] 0.5 1 1.5 2 1 2 3 4 5 6 7 8 5 10 15 20 25 Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s

2)

Degradation limit = 10% C3=0km

2/s 2 at Earth

Flyby at Neptune orbit within ∆r<1.0E6km faster degradation without degradation

Different half life SRDs ˆ Σ (d = 0.1) Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 32 / 42

Degradation Effects on Trajectory and Attitude Control Fast Transfer to the Heliopause

Fast Transfer to the Heliopause

Solar sail with ac = 1.75 mm/s2 C3 = 0 km2/s2 Trajectories calculated by M. Macdonald with AnD-blending (blending

• f locally optimal control laws)

Transfer to the nose of the heliosphere at a latitude of 7.5 deg and a longitude of 254.5 deg at 200 AU from the sun Sail jettison at 5 AU to eliminate any potential interference with the interplanetary/interstellar medium Solar distance limited to 0.25 AU Degradation factor: 0 ≤ d ≤ 0.3 (0–30% degradation limit) Half life SRD: ˆ Σ = 0.5 (S0·yr)

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Degradation Effects on Trajectory and Attitude Control Fast Transfer to the Heliopause

Fast Transfer to the Heliopause

Trajectories for different d (ˆ Σ = 0.5 (S0·yr))

Inner solar system trajectories Variation of solar distance and inclination

With increasing degradation: Constant solar distance during final close solar pass Increasing radius of aphelion passage

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Degradation Effects on Trajectory and Attitude Control Fast Transfer to the Heliopause

Fast Transfer to the Heliopause

Trajectories for different d (ˆ Σ = 0.5 (S0·yr))

21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 5 10 15 20 25 30 Degradation limit [%] Trip time –– [yrs] 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Aphelion radius – – [AU]

Trip time to 200 AU and radius of aphelion passage

9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 5 10 15 20 25 30 Degradation limit [%] Velocity at 5 AU –– [AU/yr] 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 Trip time to 5 AU – – [yrs]

Trip time and velocity at 5 AU (sail jettison point)

With increasing degradation: Increasing radius of aphelion passage Longer trip time

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Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions

Artificial Lagrange-Point Missions

Sun-Earth restricted circular three-body problem with non-perfectly solar sail SRP acceleration allows to hover along artificial equilibrium surfaces (manifold of artificial Lagrange-points) Solutions calculated by C. McInnes

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Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions

Artificial Lagrange-Point Missions

Contours of sail loading in the x-z-plane ρ = 1 ρ = 0.9

[1] 30 g/m2 [2] 15 g/m2 [3] 10 g/m2 [4] 5 g/m2

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Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions

Artificial Lagrange-Point Missions

Contours of sail loading in the x-z-plane ρ = 1 ρ = 0.8

[1] 30 g/m2 [2] 15 g/m2 [3] 10 g/m2 [4] 5 g/m2

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Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions

Artificial Lagrange-Point Missions

Contours of sail loading in the x-z-plane ρ = 1 ρ = 0.7

[1] 30 g/m2 [2] 15 g/m2 [3] 10 g/m2 [4] 5 g/m2

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Summary and Conclusions

Summary and Conclusions

Based on the current standard model for non-perfectly reflecting solar sails, we have developed a parametric model that includes the optical degradation of the sail film due to the erosive effects of the space environment Using this model, we have investigated the effect of different potential degradation behaviors on trajectory and attitude control for various exemplary missions All our results show that, in general, optical solar sail degradation has a considerable effect on trip times and on the optimal steering profile. For specific launch dates, especially those that are optimal without degradation, this effect can be tremendous

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Outlook

Outlook

Having demonstrated the potential effects of optical solar sail degradation on future missions, more research on the real degradation behavior has to be done because the degradation behavior of solar sails in the real space environment is to a considerable degree indefinite To narrow down the ranges of the parameters of our model, further laboratory tests have to be performed Additionally, before a mission that relies on solar sail propulsion is flown, the candidate solar sail films have to be tested in the relevant space environment Some near-term missions currently studied in the US and Europe would be an ideal opportunity for testing and refining our degradation model

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Outlook

Potential Solar Sail Degradation Effects on Trajectory and Attitude Control

Bernd Dachwald1 and the Solar Sail Degradation Model Working Group2

1German Aerospace Center (DLR), Institute of Space Simulation

Linder Hoehe, 51170 Cologne, Germany, bernd.dachwald@dlr.de

2Malcolm Macdonald, Univ. of Glasgow, Scotland; Giovanni Mengali and Alessandro A.

Quarta, Univ. of Pisa, Italy; Colin R. McInnes, Univ. of Strathclyde, Glasgow, Scotland; Leonel Rios-Reyes and Daniel J. Scheeres, Univ. of Michigan, Ann Arbor, USA; Marianne G¨

• rlich and Franz Lura, DLR, Berlin, Germany; Volodymyr Baturkin, Natl. Tech. Univ. of

Ukraine, Kiev, Ukraine; Victoria L. Coverstone, Univ. of Illinois, Urbana-Champaign, USA; Benjamin Diedrich, NOAA, Silver Spring, USA; Gregory P. Garbe, NASA MSFC, Huntsville, USA; Manfred Leipold, Kayser-Threde GmbH, Munich, Germany; Wolfgang Seboldt, DLR, Cologne, Germany; Bong Wie, Arizona State Univ., Tempe, USA

AAS/AIAA Astrodynamics Specialists Conference 7–11 August 2005, Lake Tahoe, CA

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