Post on 25-Aug-2020
Potential Solar Sail Degradation Effects on Trajectoryand Attitude Control
Bernd Dachwald1 and the Solar Sail Degradation Model Working Group2
1German Aerospace Center (DLR), Institute of Space SimulationLinder 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; MarianneGorlich and Franz Lura, DLR, Berlin, Germany; Volodymyr Baturkin, Natl. Tech. Univ. ofUkraine, 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 Conference7–11 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 projectedfor solar sails are assumed to be affected by the erosive effects of the spaceenvironment
Optical solar sail degradation (OSSD) in the real space environment is to aconsiderable degree indefinite (initial ground test results are controversialand relevant in-space tests have not been made so far)
The standard optical solar sail models that are currently used for trajectoryand attitude control design do not take optical degradation into account→ its potential effects on trajectory and attitude control have not beeninvestigated so far
Optical degradation is important for high-fidelity solar sail mission analysis,because it decreases both the magnitude of the solar radiation pressure forceacting on the sail and also the sail control authority
Solar sail mission designers necessitate an OSSD model to estimate thepotential 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 ModelWorking Group” (SSDMWG) with the aim to make the next steptowards a realistic high-fidelity optical solar sail model
We propose a simple parametric OSSD model that describes thevariation of the sail film’s optical coefficients with time, depending onthe sail film’s environmental history, i.e., the radiation dose
The primary intention of our model is not to describe the exactbehavior of specific film-coating combinations in the real spaceenvironment, but to provide a more general parametric framework fordescribing 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 ModelsIdeal ReflectionNon-Perfect ReflectionSimplified Non-Perfect Reflection
2 Degradation ModelData Available From Ground TestingParametric Degradation Model
3 Degradation Effects on Trajectory and Attitude ControlEquations of Motion and Optimal Control LawMars RendezvousMercury RendezvousFast Neptune FlybyFast Transfer to the HeliopauseArtificial 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
OverviewDifferent levels of simplification for the optical characteristics of a solar sail resultin different models for the magnitude and direction of the SRP force acting onthe 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 numbersin total, due to symmetry). Takes the sail shape and local optical variations intoaccount
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42
Solar Sail Force Models
OverviewDifferent levels of simplification for the optical characteristics of a solar sail resultin different models for the magnitude and direction of the SRP force acting onthe 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 numbersin total, due to symmetry). Takes the sail shape and local optical variations intoaccount
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42
Solar Sail Force Models
OverviewDifferent levels of simplification for the optical characteristics of a solar sail resultin different models for the magnitude and direction of the SRP force acting onthe 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 numbersin total, due to symmetry). Takes the sail shape and local optical variations intoaccount
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42
Solar Sail Force Models
OverviewDifferent levels of simplification for the optical characteristics of a solar sail resultin different models for the magnitude and direction of the SRP force acting onthe 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 numbersin total, due to symmetry). Takes the sail shape and local optical variations intoaccount
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 5 / 42
Solar Sail Force Models Ideal Reflection
SRP Forceon an Ideal Solar Sail
The solar radiation pressure (SRP) at a distance r from thesun is
P =S0
c
( r0r
)2
= 4.563µN
m2·( r0
r
)2
FSRP = 2PA cos α cos α n
Nomenclature
S0: solar constant
(1368 W/m2)
c: speed of light invacuum
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 Forceon an Ideal Solar Sail
The solar radiation pressure (SRP) at a distance r from thesun is
P =S0
c
( r0r
)2
= 4.563µN
m2·( r0
r
)2
incoming radiation
reflected radiation
sail
sun-line
α α
α
α
αSRPFn
t
FSRP = 2PA cos α cos α n
Nomenclature
S0: solar constant
(1368 W/m2)
c: speed of light invacuum
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 modelparameterizes the optical behavior of the sail film by theoptical coefficient set
P = {ρ, s, εf , εb,Bf ,Bb}
The optical coefficients for a solar sail with a highlyreflective aluminum-coated front side and with a highlyemissive 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 reflectionfactor
εf and εb: emissioncoefficients of the frontand back side,respectively
Bf and Bb:non-Lambertiancoefficients of the frontand 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 SailSRP Force in a sail-fixed coordinate frame S = {n, t}
incoming radiation
reflected radiation
sail
sun-line
αα
α
SRPF
n
t
⊥F
||F θφm
FSRP = 2PA cos α [(a1 cos α + a2)n− a3 sinα 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 forcecomponent along n
F||: SRP force
component along t
θ: thrust cone angle
φ: centerline angle
P: solar radiationpressure (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 SailSRP Force on along the radial and sail normal direction
FSRP can also be decomposed along the radial directioner 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 radiationpressure (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 ModelSRP 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) sinα t]
Typically, however, the reflection coefficient ρ is denoted as ηwithin this model
Nomenclature
FSRP: SRP force
P: solar radiationpressure (SRP)
A: sail area
α: sail pitch angle
n: sail normal vector
t: sail tangential vector
ρ: reflection coefficient
s: specular reflectionfactor
εf and εb: emissioncoefficients of the frontand back side,respectively
Bf and Bb:non-Lambertiancoefficients of the frontand 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 numbersin total, due to symmetry). Takes the sail shape and local optical variations intoaccount
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 opticaldegradation of metalized polymer films as second surface mirrors (metalizedon the back side)
No systematic testing to measure the optical degradation of candidate solarsail films (metalized on the front side) has been reported so far andpreliminary test results are controversial
I Lura et. al. measured considerable OSSD after combined irradiationwith VUV, electrons, and protons
I Edwards et. al. measured no change of the solar absorption andemission 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 environmentis 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 (averagesun 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 becomestime-dependent, p(t). With the simplifications stated before, p(t) is a function ofthe 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 ′ = S0r20
∫ t
t0
cos α
r2dt ′
SRD per year on a surface perpendicular to the sun at 1 AU
Σ0 = S0 · 1 yr = 1368W/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
=r20
T
∫ t
t0
cos α
r2dt ′ 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 becomestime-dependent, p(t). With the simplifications stated before, p(t) is a function ofthe 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 ′ = S0r20
∫ t
t0
cos α
r2dt ′
SRD per year on a surface perpendicular to the sun at 1 AU
Σ0 = S0 · 1 yr = 1368W/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
=r20
T
∫ t
t0
cos α
r2dt ′ 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 becomestime-dependent, p(t). With the simplifications stated before, p(t) is a function ofthe 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 ′ = S0r20
∫ t
t0
cos α
r2dt ′
SRD per year on a surface perpendicular to the sun at 1 AU
Σ0 = S0 · 1 yr = 1368W/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
=r20
T
∫ t
t0
cos α
r2dt ′ 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
=r20
T
∫ t
t0
cos α
r2dt ′
Σ(t) depends on the solar distance history and the attitude history z[t] = (r , α)[t]of 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 =r20
T
cos α
r2with 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
=r20
T
∫ t
t0
cos α
r2dt ′
Σ(t) depends on the solar distance history and the attitude history z[t] = (r , α)[t]of 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 =r20
T
cos α
r2with 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 limt→∞
p(t) = p∞
p(t) = p∞ + (p0 − p∞) · e−λΣ(t)
The degradation constant λ is related to the ”half life solar radiation dose” Σ(Σ = Σ ⇒ p = p0+p∞
2 ) via
λ =ln 2
Σ
Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6half life SRDs 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, Σp = 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 limt→∞
p(t) = p∞
p(t) = p∞ + (p0 − p∞) · e−λΣ(t)
The degradation constant λ is related to the ”half life solar radiation dose” Σ(Σ = Σ ⇒ p = p0+p∞
2 ) via
λ =ln 2
Σ
Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6half life SRDs 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, Σp = 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 limt→∞
p(t) = p∞
p(t) = p∞ + (p0 − p∞) · e−λΣ(t)
The degradation constant λ is related to the ”half life solar radiation dose” Σ(Σ = Σ ⇒ p = p0+p∞
2 ) via
λ =ln 2
Σ
Note that this model has 12 free parameters additional to the 6 p0, 6 p∞ and 6half life SRDs 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, Σp = 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, Σp = Σ ∀p ∈ P
EOL optical coefficients
Because the reflectivity of the sail decreases with time, the sail becomes morematt with time, and the emissivity increases with time, we use:
ρ∞ =ρ0
1 + ds∞ =
s01 + d
εf∞ = (1 + d)εf 0
εb∞ = εb0 Bf∞ = Bf 0 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, Σp = Σ ∀p ∈ P
EOL optical coefficients
Because the reflectivity of the sail decreases with time, the sail becomes morematt with time, and the emissivity increases with time, we use:
ρ∞ =ρ0
1 + ds∞ =
s01 + d
εf∞ = (1 + d)εf 0
εb∞ = εb0 Bf∞ = Bf 0 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, Σp = Σ ∀p ∈ P
EOL optical coefficients
Because the reflectivity of the sail decreases with time, the sail becomes morematt with time, and the emissivity increases with time, we use:
ρ∞ =ρ0
1 + ds∞ =
s01 + d
εf∞ = (1 + d)εf 0
εb∞ = εb0 Bf∞ = Bf 0 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 Effectson the optical coefficients and the maximum sail temperature
0.020.02
0.04
0.04
0.060.06
0.080.08
0.1
0.1
0.12
0.12
0.14
0.14
0.16
0.16
0.18
0.18
0.2
0.2
Σ
ρ/ρ
0 , s/
s 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.8
0.85
0.9
0.95
1
0.020.02
0.040.04
0.060.06
0.080.08
0.10.1
0.12
0.12
0.14
0.14
0.16
0.16
0.18
0.18
0.20.2
Σ
εf /ε
f0
0 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
Tm
ax(α
=0)
[°C
]
0 1 2 3 4
0
100
200
300
400
r = 0.2 AU
r = 1.0 AU
r = 0.8 AU
r = 0.6 AU
r = 0.4 AU
Frame 001 ⏐ 23 Jun 2005 ⏐
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 Effectson the SRP force bubble and the control angles
0.0=Σ
5.0=Σ0.1=Σ0.5=Σ
re
te
FSRP-bubble
0 10 20 30 40 50 60 70 80 90−10
0
10
20
30
40
50
60
Pitch angle [deg]
Con
e an
gle
[deg
] Σ = 5.0
Σ = 1.0
Σ = 0.5
Σ = 0.0
θ(α)
0 0.5 1 1.5 2 2.5 3 3.5 435.2
35.3
35.4
α* [deg
]
0 0.5 1 1.5 2 2.5 3 3.5 424
26
28
30
32
θ* [deg
]
0 0.5 1 1.5 2 2.5 3 3.5 4−30
−20
−10
0
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 trajectorycontrol
Gravitational forces of all celestialbodies
Solar wind
Finiteness of the solar disk
Reflected light from close celestialbodies
Aberration of solar radiation(Poynting-Robertson effect)
The solar sail bends and wrinkles,depending on the actual solar saildesign
Finite attitude control maneuvers
Simplifications for missionfeasibility analysis and to isolatethe effects of OSSD
The solar sail is a flat plate
The solar sail is moving under thesole influence of solar gravitationand radiation
The sun is a point mass and apoint light source
The solar sail attitude can bechanged 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 trajectorycontrol
Gravitational forces of all celestialbodies
Solar wind
Finiteness of the solar disk
Reflected light from close celestialbodies
Aberration of solar radiation(Poynting-Robertson effect)
The solar sail bends and wrinkles,depending on the actual solar saildesign
Finite attitude control maneuvers
Simplifications for missionfeasibility analysis and to isolatethe effects of OSSD
The solar sail is a flat plate
The solar sail is moving under thesole influence of solar gravitationand radiation
The sun is a point mass and apoint light source
The solar sail attitude can bechanged 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
r3r + 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 fromx0 = (r0, v0) to xf = (rf , vf ) by maximizing theperformance index J = −tf
Nomenclature
a: propulsive(+ disturbing) accelerationon the sail
r: sail position
r : radius, |r|
v: sail velocity
µ: gravitational parameterof the sun
b1(t), b2(t), b3(t):functions of the sail’soptical 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
r3r + 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 fromx0 = (r0, v0) to xf = (rf , vf ) by maximizing theperformance index J = −tf
Nomenclature
a: propulsive(+ disturbing) accelerationon the sail
r: sail position
r : radius, |r|
v: sail velocity
µ: gravitational parameterof the sun
b1(t), b2(t), b3(t):functions of the sail’soptical 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 theHamiltonian H
H = λλλr · v −µ
r3λλλv · r + λλλv · a + λΣ
r20
r2Ter · n
where λλλr , λλλv are the vectors adjoint to the position, and λΣ is the radiation dosecostate
The result is
n =
sin (αλ − α)
sin αλer +
sin α
sin αλeλλλv
for αλ 6= 0
er for αλ = 0
where
er =r
|r|eλλλv
=λλλv
|λλλv |cos αλ = er · eλλλv
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 22 / 42
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 planedefined by the position vector r and the primer vector λλλv . Thisgeneralizes a similar conclusion obtained for model IR by C. Sauer andfor model NPR without degradation by G. Mengali and A. Quarta
The equation giving the optimal cone angle as a function of αλ canbe written analytically and solved numerically
The next slide shows, how the optimal solutions typically vary withthe solar radiation dose
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 23 / 42
Degradation Effects on Trajectory and Attitude Control Equations of Motion and Optimal Control Law
Typical Optimal Solutions
0 15 30 45 60 75 900
30
60
90
120
150
180
α [deg]
αλ [deg]
Σ =0, d =0.2
0 15 30 45 60 75 900
30
60
90
120
150
180
α [deg]
αλ [deg]
Σ =0.5, d =0.2
0 15 30 45 60 75 900
30
60
90
120
150
180
α [deg]
αλ [deg]
Σ =1, d =0.2
0 15 30 45 60 75 900
30
60
90
120
150
180
α [deg]
αλ [deg]
Σ =10, d =0.2
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 = 0km2/s2
2D-transfer from circular orbit to circular orbit
Trajectories calculated by G. Mengali and A. Quarta using a classicalindirect method with an hybrid technique (genetic + gradient-basedalgorithm) to solve the associated boundary value problem
Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit)
Half life SRD: Σ = 0.5 (S0 ·yr)Three models:
B Model (a): Instantaneous degradationB Model (b): Control neglects degradation (”ideal” control law)B Model (c): Control considers degradation
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 25 / 42
Degradation Effects on Trajectory and Attitude Control Mars Rendezvous
Mars RendezvousTrip times for 5% and 20% degradation limit
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5500
1000
1500
2000
2500
3000
ac [mm/s2]
tf [days]
d=0.05
no degradationmodel amodel bmodel c
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5200
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 0.5500
1000
1500
2000
2500
3000
3500
4000
ac [mm/s2]
tf [days]
d=0.2
no degradationmodel amodel bmodel c
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5200
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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 26 / 42
Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous
Mercury Rendezvous
Solar sail with ac = 1.0 mm/s2
C3 = 0km2/s2
Trajectories calculated by B. Dachwald with the trajectory optimizerGESOP 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’ssphere of influence at perihelion) and ∆vf ,max = 50m/s
Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit)
Half life SRD: Σ = 0.5 (S0 ·yr)
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 27 / 42
Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous
Mercury RendezvousLaunch window for different d
Launch date (MJD)
Trip
time
[day
s]
57000 57050 57100 57150
350
400
450
Degradation limit:0%
5%
10%
20%
Launch window for Mercury rendezvousSolar sail with ac=1.0mm/s2
Half life SRD = 0.5 (S0*yr)
Frame 001 ⏐ 05 Jul 2005 ⏐ Mercury Launch Window with OSSD
Launch date (MJD)
Trip
time
incr
ease
[%]
57000 57050 57100 571500
5
10
15
20
25
30
35
Half life SRD=0.5 (S0*yr)
Degradation limit:
5%
10%
20%
Launch window for Mercury rendezvousSolar sail with ac=1.0mm/s2
Frame 001 ⏐ 27 Jun 2005 ⏐ Mercury Launch Window with OSSD
Sensitivity of the trip time with respect to OSSD depends considerably on thelaunch date
Some launch dates considered previously as optimal become very unsuitable whenOSSD is taken into account
For many launch dates OSSD does not seriously affect the mission
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 28 / 42
Degradation Effects on Trajectory and Attitude Control Mercury Rendezvous
Mercury RendezvousOptimal α-variation for different d
Time (MJD)
Pitc
han
gle
[rad
]
57000 57100 57200 573000.3
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/s2)Launch date = MJD 57000.0
Half life SRD = 0.5 (S0*yr)
Frame 001 ⏐ 14 Jul 2005 ⏐
Launch at MJD 57000.0
Time (MJD)P
itch
angl
e[r
ad]
57100 57200 57300 574000.3
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/s2)Launch date = MJD 57030.0
Frame 001 ⏐ 14 Jul 2005 ⏐
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 veryrobust launch date
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 29 / 42
Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby
Fast Neptune Flyby
Solar sail with ac = 1.0 mm/s2
C3 = 0km2/s2
Trajectories calculated by B. Dachwald with the trajectory optimizerInTrance
To find the absolute trip time minima, independent of the actualconstellation of Earth and Neptune, no flyby at Neptune itself, butonly a crossing of its orbit within a distance ∆rf ,max < 106 km wasrequired, and the optimizer was allowed to vary the launch datewithin a one year interval
Sail film temperature was limited to 240◦C by limiting the sail pitchangle
Degradation factor: 0 ≤ d ≤ 0.2 (0–20% degradation limit)
Half life SRD: 0 ≤ Σ ≤ 2 (S0 ·yr)
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 30 / 42
Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby
Fast Neptune FlybyTopology of optimal trajectories for different d
x [AU]
y[A
U]
-1
-1
0
0
1
1
2
2
-1 -1
-0.5 -0.5
0 0
0.5 0.5
1 1
1.5 1.5
2 2
520510500400300200100
Trip time = 5.80 years
rmin
Tlim
Sail Temp. [K]
Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s2)
= 0.204 AU
= 513.15 K
= 240°C
without degradation
C3=0km2/s2 at EarthFlyby at Neptune orbit within Δr<1.0E6km
d = 0
x [AU]
y[A
U]
-1
-1
0
0
1
1
2
2
-1.5 -1.5
-1 -1
-0.5 -0.5
0 0
0.5 0.5
1 1
1.5 1.5
520510500400300200100
Trip time = 7.87 years
rmin
Tlim
Sail Temp. [K]
Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s2)
= 0.294 AU
= 513.15 K
= 240°C
20% degradation limit, half life SRD = 0.5 (S0*yr)
C3=0km2/s2 at EarthFlyby at Neptune orbit within Δr<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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 31 / 42
Degradation Effects on Trajectory and Attitude Control Fast Neptune Flyby
Fast Neptune FlybyTrip time and trip time increase for different d and Σ
Sail degradation limit [%]
___
Trip
time
[yrs
]
__
Trip
time
incr
ease
[%]
0 5 10 15 200
1
2
3
4
5
6
7
8
0
10
20
30
40Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s2)Half life SRD = 0.5 (S0*yr)
C3=0km2/s2 at EarthFlyby at Neptune orbit within Δr<1.0E6km
Frame 001 ⏐ 27 Jun 2005 ⏐ Flight Time
Different degradation factors d (Σ = 0.5 (S0 ·yr))
1 / Half life SRD [1/(S0*yr)]
___
Trip
time
[yrs
]
__
Trip
time
incr
ease
[%]
0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
0
5
10
15
20
25Fast Earth-Neptune transfer with solar sail (ac=1.0mm/s2)Degradation limit = 10%
C3=0km2/s2 at EarthFlyby at Neptune orbit within Δr<1.0E6km
faster degradationwithoutdegradation
Frame 001 ⏐ 27 Jun 2005 ⏐ Flight Time
Different half life SRDs 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 = 0km2/s2
Trajectories calculated by M. Macdonald with AnD-blending (blendingof locally optimal control laws)
Transfer to the nose of the heliosphere at a latitude of 7.5 deg and alongitude of 254.5 deg at 200 AU from the sun
Sail jettison at 5 AU to eliminate any potential interference with theinterplanetary/interstellar medium
Solar distance limited to 0.25 AU
Degradation factor: 0 ≤ d ≤ 0.3 (0–30% degradation limit)
Half life SRD: Σ = 0.5 (S0 ·yr)
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 33 / 42
Degradation Effects on Trajectory and Attitude Control Fast Transfer to the Heliopause
Fast Transfer to the HeliopauseTrajectories for different d (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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 34 / 42
Degradation Effects on Trajectory and Attitude Control Fast Transfer to the Heliopause
Fast Transfer to the HeliopauseTrajectories for different d (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
0 5 10 15 20 25 30
Degradation limit [%]
Trip
tim
e ––
[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
Aph
elio
n ra
dius
– –
[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
0 5 10 15 20 25 30
Degradation limit [%]
Velo
city
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
tim
e to
5 A
U –
– [y
rs]
Trip time and velocity at 5 AU (sail jettison point)
With increasing degradation:
Increasing radius of aphelion passage
Longer trip time
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 35 / 42
Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions
Artificial Lagrange-Point Missions
Sun-Earth restricted circular three-body problem with non-perfectlysolar sailSRP acceleration allows to hover along artificial equilibrium surfaces(manifold of artificial Lagrange-points)Solutions calculated by C. McInnes
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 36 / 42
Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions
Artificial Lagrange-Point MissionsContours 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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 37 / 42
Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions
Artificial Lagrange-Point MissionsContours 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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 38 / 42
Degradation Effects on Trajectory and Attitude Control Artificial Lagrange-Point Missions
Artificial Lagrange-Point MissionsContours 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
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 39 / 42
Summary and Conclusions
Summary and Conclusions
Based on the current standard model for non-perfectly reflecting solarsails, we have developed a parametric model that includes the opticaldegradation of the sail film due to the erosive effects of the spaceenvironment
Using this model, we have investigated the effect of differentpotential degradation behaviors on trajectory and attitude control forvarious exemplary missions
All our results show that, in general, optical solar sail degradation hasa considerable effect on trip times and on the optimal steering profile.For specific launch dates, especially those that are optimal withoutdegradation, this effect can be tremendous
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 40 / 42
Outlook
Outlook
Having demonstrated the potential effects of optical solar saildegradation on future missions, more research on the real degradationbehavior has to be done because the degradation behavior of solarsails in the real space environment is to a considerable degreeindefinite
To narrow down the ranges of the parameters of our model, furtherlaboratory tests have to be performed
Additionally, before a mission that relies on solar sail propulsion isflown, the candidate solar sail films have to be tested in the relevantspace environment
Some near-term missions currently studied in the US and Europewould be an ideal opportunity for testing and refining our degradationmodel
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 41 / 42
Outlook
Potential Solar Sail Degradation Effects on Trajectoryand Attitude Control
Bernd Dachwald1 and the Solar Sail Degradation Model Working Group2
1German Aerospace Center (DLR), Institute of Space SimulationLinder 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; MarianneGorlich and Franz Lura, DLR, Berlin, Germany; Volodymyr Baturkin, Natl. Tech. Univ. ofUkraine, 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 Conference7–11 August 2005, Lake Tahoe, CA
Dachwald & SSDMWG (DLR & . . . ) Solar Sail Degradation AAS/AIAA ASC 2005 42 / 42