Advanced Topics in Astrodynamics

1
Advanced Topics inAstrodynamics

• Solar Sailing
• Voile Solaire Sonnensegler Vela Solar Vela
  Solare
• Malcolm Macdonald
• Gareth Hughes

Institut d'Estudis Espacials de Catalunya


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University of Glasgow www.gla.ac.uk

• 2nd oldest University in Scotland
• Established 1451 by Pope Nicholas V
• 20 000 full-time students 16 000 as
  undergraduates, studying in 10 faculties
• 50 of students are from Glasgow 26 from
  remainder of Scotland 17 from remainder of EU
• 5750 staff of which approximately 1700 are
  academic
• Oldest engineering faculty in Scotland ( UK),
  now with 5 departments,
• Aerospace Mechanical Electronics Civil and
  Naval
• Home to William Thomson (Lord Kelvin), James
  Watt, John Logie Baird, William Macquorn Rankine
  and many others
• Non-Engineering graduates include Adam Smith,
  Donald Dewar and James Dalrymple

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Department of Aerospace Engineering www.aero.gla.
ac.uk

• 16 Academic Staff, 10 Technical Staff, 40
  research staff
• Only Aerospace Engineering Department in Scotland
• Intake of 100 undergraduates each year
• Courses in spacecraft dynamics and space systems
  engineering, along with a
• new Master of Science in Space Mission Analysis
  and Design in 2004-2005
• Six main research groups
• Air Traffic Management Avionics
• Computational Fluid Dynamics
• Flight Mechanics
• Low Speed Aerodynamics
• Space Systems
• Structures Design

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Space Systems Engineering Research Group

• 13 Members of research group, across 2
  departments
• 3 Members of Academic Staff
• 2 Research Assistants
• 2 Visiting Researchers
• One from Japanese Government and one from Chinese
  Government
• 4 Research Students
• 2 ERASMUS students
• Research activities cover
• Solar sail mission analysis (ESA, NOAA, NASA,
  Lockheed)
• Spacecraft formation-flying (EADS Astrium,
  SciSys Ltd)
• Space robotics (ESA, Astrium GmbH)
• Spacecraft autonomy (BNSC, Astrium UK)

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Space Systems Engineering Research Group

• Co-operating agencies, organisations and
  companies

6
Solar Sailing

• An Introduction and Historical Perspective

7
An Introduction

• Utilise light pressure for propulsion
• Small continuous thrust
• No stowed reaction mass enabling new high
  energy/long duration mission concepts
• Enhance existing mission concepts through
  reduction of launch mass or mission duration
• Numerous technology issues thin films,
  deployable structures, control
• Technology cross-over into other areas
• Large deployable power collectors, antennae,
  optics

8
An Historical Perspective

• 1619 Johannes Kepler using corpuscular theory
  proposes comet tails are pushed outwards from the
  Sun due to sunlight
• 1687 Isaac Newton attempts to explain same
  phenomenon by solely using his theory of
  universal gravitation
• 1744 Euler returns to Keplers original view,
  through adoption of longitudinal wave theory of
  light due to Huygens
• 1754 de Marian and du Fay attempt to measure
  radiation pressure but fail due to residual air
  currents
• 1812 Olbers proposes comet tails are propelled by
  electrostatic forces, however this was eventually
  seen as flawed due to a lack of charging
  mechanisms
• 1873 Maxwell uses his unified theory of
  electromagnetic radiation to provide the correct
  theoretical basis for radiation pressure

9
The Physical Principles Electromagnetic
Description

• Momentum is transported to the solar sail by
  electromagnetic waves
• Electric field component of the wave, E, induces
  current, j, in the sail, the magnetic component
  of induced wave, B, generates a Lorentz force, j
  x B, in direction of propagation of wave and the
  induced current generates another electromagnetic
  wave, observed as the reflection of the incident
  wave
• For a wave along the x axis the force on a
  current element is,
• Where, jz is current density induced in surface
  of reflector
• Resulting pressure is,
• Using Maxwells equations of electrodynamics we
  can replace
• this with field terms, thus the time averaged
  pressure is,
• Term in parenthesis is identified as the energy
  density, U, for the
• electric component, E, and magnetic component,
  B, of the incident wave,
• e0 is permittivity of free space, µ0 the
  permeability of free space

10
The Physical Principles Electromagnetic
Description

• The pressure on surface of thickness ?l is,
• For a perfect reflector, pressure equals total
  energy of the electromagnetic wave
• For 2 plane waves at ?x separation, incident on
  area A, volume between the 2 waves impinging the
  surface is A ?x
• Energy density of electromagnetic wave is,
• Energy flux, W, across surface is,
• Thus,
• Can also use the quantum description of light to
  derive this
• equation using special relativity, thus the
  quantum and
• electromagnetic description of radiation
  pressure is equivalent.
• In electromagnetic description, radiation
  pressure is the energy density of the
  electromagnetic wave
• In quantum description, radiation pressure is the
  conservation of momentum

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An Historical Perspective

• 1889 Faure Graffigny write story mentioning a
  spacecraft propelled by mirrors
• 1900 Peter Lebedew at University of Moscow used
  torsion balance apparatus to validate the theory
  of Maxwell
• 1920s Konstantin Tsiolkovsky writes of the
  potential of a utilising light pressure for space
  navigation
• 1923 H. Oberth proposes the concept of
  reflectors in Earth orbit (Spiegelrakete) to
  illuminate northern regions of Earth, aka Znamya
  experiments in the late 1990s
• 1924 Fredrickh Tsander writes what is today
  considered by many to be the first technical
  report on solar sailing
• 1929 H. Oberth extends his earlier concept for
  several applications of orbit transfer,
  manoeuvring and attitude control using mirrors in
  Earth orbit
• 1951 Carl Wiley (as Russell Sanders) re-invents
  solar sailing concept in America
• 1958 Richard Garwin, in Journal Jet Propulsion,
  coins the phrase Solar Sailing
• 1977 JPL 800 x 800 m 3-axis stabilised comet
  Halley sail mission cancelled

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Recent Activities
ESA/DLR 20 x 20 m ground test (1999)
NASA demo mission studies, ST-5 ST-7 and now
ST-9 2 mm CP-1 film production
New science mission concepts
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ESA / DLR Ground Test 1999
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Future Activities
Cosmos-1 40 kg sail (3rd quarter of 2004)
ESA deployment demo (2006)
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Future Activities Cosmos-1
Antennas
Protective cover
Sun sensor
Solar array
Solar sail blades(stowed position)
Attitude control thrusters
Equipment bay
Apogee solidrocket kick motor
Launch configuration
(L.Friedman)
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Sail Configurations

• Three primary sail configurations can be
  envisaged
• Square sail is the most common current activity
  DLR LGarde AEC-ABLE
• Heliogyro was selected over square sail for Comet
  halley mission before also being dropped, very
  large scale blades
• Disc sail is far-term architecture, provisional
  studies by JPL-NASA have proved promising

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Performance Metrics Key Parameters
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Key Parameters

• Define sail pitch angle as
• Angle from the Sail Sun line to the sail normal
  vector
• Define sail cone angle as
• Angle from the Sail Sun line to the sail thrust
  vector
• Pitch and cone angle are
• 0o to 90o
• Or, -90o to 90o
• Sail clock angle is the projection of the sail
  normal vector into the plane defined by the unit
  vector normal to the orbit plane and a unit
  vector normal to the Sun-line, within the orbit
  plane
• Clock angle is
• 0o to 360o
• Or, 0o to 180o

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Performance Metric

• Most useful metric for astrodynamics is sail
  characteristic acceleration (mm s-2)
• Defined as solar radiation pressure experienced
  by a sail facing the Sun at 1 AU
• Magnitude of the radiation pressure at 1 AU is
  4.56 x 106 Nm-2 varies as (1 / R2)
• Incorporating a sail efficiency to account for
  non-perfect reflection, sail billowing et cetera
  gives sail characteristic acceleration as,
• Current capabilities 0.1 0.25 Requirements are
  0.15 (near) 0.5 (mid) 6 (far)
• Most useful metric for technology development is
  the sail assembly loading (g m-2)
• Defined as ratio of sail mass to sail reflective
  surface area
• Current capabilities 10 20 Requirements are 30
  (near) 7 (mid) lt1.5 (far)

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Perfect Force Model

• The acceleration experienced by the solar sail is
  a
• function of the sail attitude
• Sail of area A with unit vector n normal to
  surface,
• force on the sail due to incident radiation is,
• The reflected radiation exerts a force of equal
  magnitude, in the specular reflected direction
  ur
• Using the identity, ui ur 2(ui.n)2n, the
  total force on the sail is,
• Thus from the quantum description of radiation,
• We is solar energy flux at 1 AU 1368 J s-1m-2
• Thus, sail acceleration is, where, s is
  the sail loading mtotal/A
• Such that the sail normal is, and
  the sail thrust vector is,

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Non-Perfect Force Model

• Traditional non-perfect force model utilises
  standard optics theory, where non-specular
  reflection at gt6-10o is assumed to be of no use
• A solar sail can however utilise ALL reflected
  photons and it can be shown that the traditional
  model is flawed as reflection is highly symmetric
  about the specular line, i.e. no collapse in the
  force vector!
• Rogan et al, 2001, Encounter 2001 Sailing to
  the Stars, SSC01-112, 15th Annual/USU Conf. On
  Small Satellites.
• We will assume a perfect reflector to avoid
  contention between the two theories, unless
  otherwise stated

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Solar Sailing The Basic Idea
UNIVERSITYofGLASGOW

• Very small continuous thrust
• Require large surface area to gain required
  thrust
• Require very light weight structure to gain
  required thrust
• Cannot direct thrust vector towards the Sun
• Direct the thrust vector with the velocity vector
  to gain orbit energy
• Direct the thrust vector against the velocity
  vector to lose orbit energy

Lose Energy
Thrust
v
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Earth Centred Trajectories
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Earth Centred Trajectories

• Discussion of planet-centred applications of
  solar sailing are largely limited to
• Escape manoeuvres
• Lunar fly-by
• Simple orbit transfers, such as inclination
  change, using locally optimal control laws
• Generation of complex orbit transfers is limited
  due to computation difficulties
• Globally optimal solution of multi-revolution
  orbits is difficult
• Planet-centred solar sail manoeuvres tend to be
  locally optimal solutions

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Earth Centred Trajectories

• The general equation of motion for a perfectly
  reflecting solar sail in planet-centred orbit is
• ? is magnitude of sail acceleration, at Earth
  this is the sail characteristic acceleration
• l is the unit vector along the sail Sun line
• n is the unit vector along the sail normal
• Many simplifications can be made, however
  historically the model is over simplified
• We will begin with a simple model and progress
  through to a highly realistic model

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Earth Centred Trajectories

• Earth Escape

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Earth Centred Trajectories Earth Escape

• Minimum time escape trajectories have been
  investigated
• Sackett, L.L., Edelbaum, T.N., 1978, Optimal
  Solar Sail Spiral to Escape, Advances in
  Astronautical Sciences, AAS/AIAA Astrodynamics
  Conference, A78 31-901.
• Using the method of averaging the computational
  effort is significantly reduced
• Only valid when change in orbit elements over the
  averaged interval is small, thus not valid up to
  escape point, instead propagate up to a
  sub-escape point.
• Can be shown that the use of locally optimal
  control laws can generate escape trajectories
  within 1 3.5 of globally optimal for Earth
  escape from high Earth orbit.
• Green, A.J., 1977, Optimal Escape Trajectories
  From a High Earth Orbit by Use of Solar Radiation
  Pressure, T-652, Master of Science Thesis,
  Massachusetts Institute of Technology.
• In 1958 Irving had concluded that in general a
  locally optimal control strategy would be
  sufficiently close to optimal for most
  low-thrust system
• Irving, J.H., 1958, Space Technology, John
  Wiley Sons Inc., New York, 1959
• Also in 1958, Lawden mathematically showed that a
  for low-thrust motor little advantage was to be
  gained by use of a more complex strategy that a
  locally optimal control strategy.
• Lawden, D.F., 1958, Optimal Escape from a
  Circular Orbit, Astronautica Acta, Vol. 4, pp.
  218-234.

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Earth Centred Trajectories Earth Escape

• Simplest Earth escape strategy is the On-Off
  law
• Sail faces the sail for half an orbit as it
  travels away from the Sun and is turned off for
  half an orbit as it travels towards the Sun
• Requires two rapid 90o slew manoeuvres per. orbit
• Using,
• For small eccentricity this becomes,
• Neglecting periods of eclipse the sail transverse
• acceleration is,
• Change in semi-major axis over one revolution is,

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Earth Centred Trajectories Earth Escape

• On-Off steering law model, using polar equation
  of motion
• From GEO radius in a simple 2D model,
• with a fixed Sun-line
• Fix sail characteristic acceleration
• at 1 mm s-2
• Integrate for 3 days
• Semi-Major axis always increases
• Eccentricity also always increases
• thus ?a derived previously only valid
• for short time interval from start

30
Earth Centred Trajectories Earth Escape

• A more sophisticated scheme is called Orbit Rate
  Steering
• Sands, N., 1961, Escape from Planetary
  Gravitational Fields by Using Solar Sails
  American Rocket Society Journal, Vol. 31, pp.
  527-531.
• Solar sail is rotated at one half of the orbit
  rate
• Rate of rotation varies with semi-major axis and
• eccentricity, hence is not constant
• Sail pitch rate is,
• Ensuring correcting phasing, the sail pitch is,
• Transverse sail acceleration is,
• Repeating analysis performed for On-Off
  steering,

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Earth Centred Trajectories Earth Escape

• Orbit Rate steering law model, from same orbit
  with same sail acceleration
• Semi-major axis is seen to increase more
• Semi-major axis is seen to dip at end of
• each orbit
• Again, eccentricity increases
• Requires one rapid 90o slew
• manoeuvres per. orbit

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Earth Centred Trajectories Earth Escape

• Locally optimal energy gain control law
• Fimple, W.R., 1962, Generalized
  Three-Dimensional Trajectory Analysis of
  Planetary Escape by Solar Sail, American Rocket
  Society Journal, Vol. 32, pp. 883-887.
• Maximise instantaneous energy gain by aligning
  the sail pitch to maximise the solar radiation
  pressure force along the velocity vector
• Scalar product of equation of motion with
  velocity vector gives,
• Left side can be written as,
• which represents the total orbit energy rate of
  change
• Hence, instantaneous rate of energy change is,

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Earth Centred Trajectories Earth Escape

• Defining the angle between the velocity vector
  and Sun-line as ? we see that,
• The instantaneous rate of energy change is found
  from turning point of this equation
• Must find the sail pitch angle which maximises
  the orbit energy gain for a given value of ?
• Turning point is found from,
• Giving optimal sail pitch as,
• Transverse sail acceleration is,
• For a quasi-circular orbit ? f p/2, so
  transverse
• acceleration is function of true anomaly only
• Repeating prior analysis,

34
Earth Centred Trajectories Earth Escape

• Locally Optimal steering law model, from same
  orbit with same sail acceleration
• Semi-major axis is seen to increase more
• Semi-major axis rate of change is always
• positive
• Again, eccentricity increases
• Requires one rapid 90o slew
• manoeuvres per. orbit
• A distinct difference is now clear from
• SEP type trajectories where eccentricity
• remains low throughout early stage of
• escape spiral

35
Earth Centred Trajectories Earth Escape

• Locally Optimal steering law model, from same
  orbit with same sail acceleration
• Run model for 60 days until escape
• Semi-major axis and eccentricity
• continue to increase
• Sail pitch corrects for varying orbit size
• Escape asymptote is away from the Sun

36
Earth Centred Trajectories Earth Escape

• Avoid rapid sail rotations and ensure energy gain
  at all points
• Consider a circular polar orbit can write
  transverse acceleration as,
• For a near-circular orbit,
• From this we can note that ?a is maximised if,
  , the optimal fixed pitch
  sail angle
• Change in semi-major axis is thus,
• From,
  can obtain change in eccentricity per.
  orbit as,
• Giving ?e 0, since transverse force is constant
• Although sail pitch is constant the sail must
  roll through 360o once per. orbit to keep
  transverse acceleration tangent to the trajectory

37
Earth Centred Trajectories Earth Escape

• From a polar orbit at GEO radius, with sail
  characteristic acceleration 1.0 mm s-2
• Semi-major axis monotonically increases
• Eccentricity remains low but not zero
• Sail clock angle rotates once per. orbit
• Since orbit is quasi-circular can use
• variational equation of semi-major axis
• to obtain closed loop estimate of outward
• spiral solution
• Write variational equation as,
• Thus,

38
Earth Centred Trajectories Earth Escape

• Using the estimate of outward spiral for an orbit
  from GEO radius with sail characteristic
  acceleration 1.0 mm s-2
• Estimate number of orbits until escape as,
• Escape is found to occur after 23.62 orbits
• Note the rate of spin varies during escape,
• but since e lt 0.5 is not overly demanding

39
Solar Sail Trajectories

• Realistic Model

40
Solar Sail Trajectories Realistic Model

• In order to accurately model a trajectory about a
  planet the use of a more advanced model is
  required
• The use of modified equinoctial elements is
  increasingly popular (and my preferred choice)
• Giacaglia, G.E.O., The Equations of Motion of an
  Artificial Satellite in Nonsingular Variables,
  Celestial Mechanics, Vol. 15, pp. 191-215, 1977.
• Walker, M.J.H., Ireland, B., Owens, J., A Set of
  Modified Equinoctial Elements, Celestial
  Mechanics, Vol. 36, pp. 191-215, 1985.
• These employ a fast-variable (phase angle) as the
  sixth element, allowing a regular perturbation
  technique to be used with the fast variable as
  independent variable
• True longitude semi-latus rectum, in place of
  mean longitude semi-major axis gives a set of
  non-singular equations of motion, excluding the
  case of 180o inclination, the modified
  equinoctial elements
• An important feature of this set of equations is
  the ability to model eccentricities of zero,
  equal to one and greater than one, i.e. can model
  a trajectory accurately through the point of
  escape

Also, define the auxiliary (positive) variables
41
Solar Sail Trajectories Realistic Model

• Lagrange first introduced this element set in
  1774 for his study of secular variation
• His notation was h, l, p, q rather than f, g, h,
  k
• Lagrange used i instead of (i/2), the use of the
  half-angle simplifies the resulting Lagrange
  planetary equations in non-singular elements and
  allows the use of Allans expansion of the
  geopotential
• Additionally, the use of tangent in place of sine
  and cosine allows for the further simplification
  of Lagranges equations, while still allowing the
  use of Allans expansion of the geopotential
• Having defined the modified equinoctial elements,
  the equations of motion of the modified
  equinoctial elements in terms of the auxiliary
  (positive) variables, in Gaussian form, reduce to,

42
Solar Sail Trajectories Realistic Model

• Spacecraft are acted on by many forces other than
  just point-mass gravity due to the planet
• Planetary oblateness and 3rd body gravity terms
  must be considered for all spacecraft
• Of special importance for solar sailing are
• The non-constant rate of rotation of the Sun
  about the Earth, i.e. Earths eccentric orbit
• Earths atmosphere, drag and lift
• Planetary albedo
• The limb-darkened, finite solar disk variation of
  light pressure from a point-source
• Shadow cone
• Solar wind

43
Solar Sail Trajectories Realistic Model

• Earths eccentric orbit means that the level of
  sail acceleration varies throughout the year from
  the nominal characteristic acceleration
  variation amplitude is 3.5
• If model includes Earths eccentricity then it
  must also include a correction of sail
  acceleration
• Planetary albedo can have an effect for low
  altitude orbits at Mercury, but typically can be
  neglected, at Earth is typically over 3 orders of
  magnitude smaller than light pressure
• Solar wind exerts a small force on the sail
• At solar maxima the mean solar proton number
  density is 4x10-6 m-3
• With mean solar wind speed of order 700 km s-1
• Solar wind pressure can be estimated transported
  momentum as,
• Giving pressure 3x10-9 N m-2
• Solar wind pressure is of scale 4 orders of
  magnitude less than light pressure at 1 AU
• Calculation of when sail is in shadow is a simple
  but critical element, as zero thrust is available
• Also, numerous shadow events can have notable
  thermal effect on all spacecraft systems

44
Solar Sail Trajectories Realistic Model

• Earths residual atmosphere effects all
  satellites up to 500 1000 km depending on solar
  activity
• Perturbation is a strong function of solar cycle
• At periods of low solar activity atmospheric
• drag sail acceleration balance at 430 km
• At periods of mean solar activity atmospheric
• drag sail acceleration balance at 560 km
• At periods of high solar activity atmospheric
• drag sail acceleration balance at 940 km
• Safe mean minimum altitude bound is gt 800 km,
  although at times of solar maxima this may double
• At Mars the minimum solar sail altitude is 300
  km
• At Venus the minimum solar sail altitude is 900
  km

45
Solar Sail Trajectories Realistic Model

• The varying direction of incidence of solar
  radiation from different parts of the solar disc
  means that the (1/R2) variation breaks down at
  low solar distances
• Initially assume the solar disc is uniformly
  bright i.e. time independent and isotropic
  across the disc
• Solar radiation pressure on a perfectly
  reflecting sail at radius r may be written
• where ?0, the angular radius of the solar disc,
  is given by sin-1(Rs/r)
• From diagram, note azimuthal symmetry of geometry
  and that
• specific intensity is independent of r, thus,
• where I0 is frequency integrated specific
  intensity
• Performing integration and substituting xo it is
  found that,

46
Solar Sail Trajectories Realistic Model

• Expanding this equation, in powers of (Rs/r)2 and
  for r gtgtRs, to the first order we get,
• However, at large r this must match
  asymptotically with the point-source expression,
  viz,
• Hence, by comparison the frequency integrated
  specific intensity I0 can be identified as,
• Thus, from last equation on prior slide, we
  obtain
• an expression for the solar radiation pressure
  on a
• radially orientated sail from a uniformly
  bright, finite
• angular sized solar disc as,
• Or,

47
Solar Sail Trajectories Realistic Model

• A more realistic solar model is gained through
  consideration of solar limb darkening in the
  functional form of the specific intensity
• Limb darkening is due to the specific intensity
  of solar radiation having a directional
  dependence, thus when viewing from an oblique
  angle the associated specific intensity falls
• Thus, the limb appears darker than the centre
• Empirically, solar limb darkening has a complex
  functional form.
• Can use approximate model of solar atmosphere to
  gain analytical expression
• The grey solar atmosphere model, which assumes
  the atmosphere is in both radiative and local
  thermodynamic equilibrium
• Specific intensity is thus, , where ? is
  the aspect angle
• Limb is darker by a factor of 0.4
• For limb darkened specific intensity maximum
  deviation rises to 0.708
• Limb-darkened solar radiation pressure deviates
  less from inverse square law

48
Optimal Inclination for Planet-Centred Solar
Sailing
49
Optimal Inclination for Planet-Centred Solar
Sailing

• Previously, a slightly variation, up to 7 , has
  been noted for Earth escape times depending on
  the launch date through the year.
• The reason for this has only just been explained
  mathematically, with previous studies suggesting
  contradictory explanations and predictions
• Macdonald, M., McInnes, C.R., Realistic Earth
  Escape Strategies for Solar Sailing, J.
  Guidance, Control Navigation, In Press.
• Using a derivation of the optimal energy gain
  control law we can derive an optimal inclination
• Define the Sun Vector coordinate system, Xsun,
  Ysun, Zsun
• Zsun velocity component is zero
• N.B. If velocity and Sun vectors are parallel
  system is
• undefined, thus use for theory development only
  and
• not orbit propagation!
• Sail normal vector is,

50
Optimal Inclination for Planet-Centred Solar
Sailing

• Deriving optimal energy gain control law
  function to be maximised is
• Sail acceleration vector is defined as,
  , thus,
• Since Zsun velocity is zero,
• Forming the 1st derivative with respect to a and
  d gives conditions for turning points as,
• Re-arrange a equation as,
• Can thus solve for a and d,
• Clock angle solution states sail normal, velocity
  vector Sun vector must be coplanar for optimal
  energy gain, however only the sail normal vector
  can be controlled, thus clock angle must be fixed
  at 0o or 180o

51
Optimal Inclination for Planet-Centred Solar
Sailing

• BUT, optimal energy gain control law can also be
  called the optimal semi-major axis controller and
  derived from the variational equation of
  semi-major axis,
• We note that the variation of semi-major axis
  depends on only the in-plane perturbations
• Thus, to maximise energy gain we must direct the
  sail thrust within the orbit plane
• The plane defined by the velocity vector and Sun
  vector are only coincident with the orbit plane
  when orbit is in the Ecliptic plane
• When orbit is outside this plane the sail force
  cannot be directed within all of the required
  planes and an out-of-orbit-plane force is
  generated, essentially wasting sail thrust
• The optimal plane for planet-centred energy gain
  is thus the Ecliptic plane, at Earth i
  23.439o
• Optimal plane holds for variation of all elements
  with variation dependent on only the in-plane
  forces
• A 7o orbit at European winter solstice has
  inclination 16.4o to ecliptic, but in summer this
  becomes 30.4o, thus can expect worse performance
  for same orbit at summer escape spirals.

52
Optimal Inclination for Planet-Centred Solar
Sailing

• Can test this through use of realistic orbit
  model
• Initially assume no perturbations, perfect sail,
  point-source Sun and no shadow
• Optimal inclination is clearly visible for this
  scan of escape times from GEO radius, right-hand
  graph is section through surface plot for sail
  characteristic acceleration 0.75 mm s-2
• Introduction of shadow could be expected to
  effect this due to lack of thrust in this plane

53
Optimal Inclination for Planet-Centred Solar
Sailing

• Re-run analysis with Earth shadow effects
  included
• Optimal inclination remains, however surface is
  much more irregular
• We note that a change in orbit inclination can be
  as influential on escape time as a change in sail
  characteristic acceleration of 0.25 mm s-2
• Difference in maximum and minimum escape time
  falls as sail acceleration is increased

54
Locally Optimal Control Laws
55
Locally Optimal Control Laws

• The rate of change of any orbital element can be
  calculated, hence a locally optimal control law
  generated
• Such control laws maximise the instantaneous rate
  of change of the element and provide the sail
  control angles in closed analytical form
• Local optimality does not guarantee global
  optimality
• The optimal energy gain control is derived from
  the rate of change of semi-major axis
• Variational equation of element is written as,
• Unit vector of ?k gives direction along which
  sail thrust should be maximised, to maximise
  dk/dt
• ?k must be transposed from satellite RTN
  reference frame into the Sun sail line
  reference frame, within which sail control angles
  are defined

56
Locally Optimal Control Laws

• Note slight change of emphasis in control angle
  coordinate system definition
• Conversion of ?k is a simple matter of coordinate
  conversion, however we must be clear as to the
  precise coordinate systems being used
• With ?k in the Sun sail line reference frame we
  define,
• This is the ideal force control angles
• A standard optimisation derivative is used to get
  sail pitch
• which maximises sail thrust vector along ideal
  vector, ?k
• Locally optimal clock angle is not optimised as
  sail force
• does not depend on the clock angle

57
Locally Optimal Control Laws

• The semi-major axis variational equation is,
• Thus,
• Using,
• We can get the locally optimal sail pitch angle
  as,
• And the locally optimal clock angle as,
• Can repeat for any orbital element, such as
  radius of pericentre or argument of pericentre,

58
Locally Optimal Control Laws

• Variational equation of inclination and right
  ascension are dependent on only the out-of-plane
  term
• However, taking unit vector of ?i allows locally
  optimal control law to be generated
• Define a switching term using the signum
  function of the cosine term, such that
• Hence, conversion into Sun sail line reference
  frame allows pitch and clock angles to be found
• For inclination and d switches 180o at (??)
  90o and 270o, thus total slew 70o

59
Locally Optimal Control Laws

• Can blend each individual control law to generate
  orbit transfers, escape manoeuvres or
  station-keeping algorithms, blending locally
  optimal control laws has been discussed widely
  for low-thrust propulsion
• The basics of the blending process is very simple
• The blended vector is found using, ,then we can
  find , a and d as before
• The complex/difficult part is defining the weight
  function of each control law
• Previously, SEP trajectories used an optimiser to
  set the weights as a function of time from the
  start epoch, this re-introduces a time dependence
  which we wish to avoid
• The approach detailed for solar sail uses the
  osculating orbit elements to set the weight
  function, thus maintaining independence from time
• Macdonald, M., McInnes, C.R., Analytic Control
  Laws for Near-Optimal Geocentric Solar Sail
  Transfers (AAS 01-473), Advances in the
  Astronautical Sciences, Vol. 109, No. 3, pp.
  2393-2413, 2001.
• Macdonald, M., McInnes, C.R., Realistic Earth
  Escape Strategies for Solar Sailing, J.
  Guidance, Control Navigation, In Press.
• A similar approach has now been detailed for SEP
  propulsion
• Petropoulos, A. E., Simple Control Laws for
  Low-Thrust Orbit Transfers, AAS/AIAA
  Astrodynamics Specialists Conference, Big Sky,
  Montana, 3-7 August 2003.

60
Realistic Earth Escape
61
Realistic Earth Escape

• Due to the high number of orbit revolutions
  typical of low-thrust planet-centred trajectories
  accurate analysis is hindered by computational
  difficulties
• We require to generate planet centred
  trajectories which are computationally simple and
  near-optimal
• Hence the popularity of locally optimal control
  solutions, such as the energy gain control law
• Locally optimal control has the advantage that
  sail control angles can be calculated independent
  of time, making the system suitable for on-board
  autonomous sail control
• Can utilise locally optimal control laws to
  examine the feasibility of a range of potential
  initial orbits
• GTO GEO 1000 km Polar
• Will use a high-fidelity model including orbit
  perturbations discussed earlier
• Earth oblateness to the 18th order Lunar Solar
  gravity as point masses Earth and Lunar shadow
  Sun is modelled as a uniformly bright finite disc
  and the sail using the optical model

62
Realistic Earth Escape

• Exclusive use of the locally optimal energy gain
  control law results in very low final perigee
  passage prior to escape
• Recall prior 2D escape from GEO, note the final
  passage of much closer to Earth than GEO radius
• In this 2D escape minimum radii is above
• Earths atmosphere, however many authors
• have noted that other scenarios can result
• in negative altitudes prior to escape
• This is a unique feature of solar sailing
• over other low-thrust systems, where the
• thrust vector is unconstrained
• Can avoid planetary collision through
• blended locally optimal control laws

63
Realistic Earth Escape

• We will compare both exclusive use of locally
  optimal energy gain control law and blended with
  radius of pericentre control law
• The blended system should only use the pericentre
  controller when required, i.e. at low altitudes
• Through application of engineering judgment we
  define the weights as,
• Rapid change over at low perigee altitudes
• Allows us to realistically consider Earth
• escape trajectories

64
Realistic Earth Escape GTO

• GTO has been identified by many studies as a
  potential starting orbit for Earth escape
  missions, particularly the DLR ODISSEE study
• GTO delivery would be by Ariane 5 Structure for
  Auxiliary Payloads (ASAP), giving maximum perigee
  560 km, future Ariane 5 GTO may be 250 km
• At this altitude atmospheric drag and aerodynamic
  torque will act on sail
• As such when below 1000 km we turn the sail
  edge-on to the atmosphere, to maintain a
  minimum drag profile
• This has additional effect of reducing the steep
  gravity gradient across the sail and easies
  attitude control system design
• Further, lack of atmospheric drag within model
  can be justified
• This addition to the blended escape strategy
  allows GTO to be considered as a viable option
  for a highly agile sail

65
Realistic Earth Escape GTO

• GTO can have a midday or midnight launch, giving
  a sun-pointing perigee or apogee
• A midnight launch is consider the norm, with a
  sun-pointing apogee
• Semi-major axis controller strikes planet for
  acceleration gt 0.3 mms-2 in midnight launch and
  for escapes trajectories of only a few
  revolutions in midday launch
• Blended controller avoids planet and maintains
• similar escape durations, thus the low-drag
• profile addition to sail control has minimal
  effect
• on performance
• Escape time varies exponentially with
  acceleration
• Short orbit period means we require a rapid slew
• manoeuvre capability

66
Realistic Earth Escape GTO

• Examine specific case of midnight GTO with sail
  characteristic
• acceleration of 2.0 mm s-2
• Single controller collides with Earth at day
  35, while blended
• controller avoids this

Semi-major axis controller
Pericentre controller
67
Realistic Earth Escape GEO

• GEO is probably the most attractive of the
  potential starting orbits
• Large orbit radius reduces rapid slew capability
  requirement slightly
• Outside atmospheric drag and step gravity well of
  Earth
• Find for escape with only a few revolutions
  semi-major axis controller collides with Earth
  just prior to reduction in number of revolutions
• Again, see escape time varies exponentially with
  sail acceleration
• Also, note exponential curve is mixed with short
  period
• oscillations, seen as maxima minima in curve
• This is a unique characteristic of solar sail
  propulsion
• caused by an inability to gain substantial
  amounts of
• energy as we travel towards the Sun
• True location of these peaks would be very
  difficult to
• determine, thus utilise or avoid

68
Realistic Earth Escape GEO

• Can demonstrate the benefit of defining weight
  functions independent of time by removing all the
  orbit perturbations, modelling the Sun as a point
  source and the sail as a perfectly specular
  reflector
• The blended sail control system alters the sail
  control angles to adjust for the new scenario,
  while maintaining a safe orbit altitude
• This self-correcting feature of the control
  system offers the potential to significantly
  increase sail autonomy

69
Realistic Earth Escape 1000 km Polar Orbit

• A high polar orbit can be achieved at low cost,
  for example with a Dnepr or other ICBM
• Several potential advantages and perhaps even
  more problems as a starting orbit
• Short orbit period requiring rapid slews,
  atmosphere, steep gravity well
• Potential parallel applications as part of a
  Mercury sample return mission
• Use semi-major axis exclusively
• if e lt 0.07 rp gt 500 km
• Blended controller maintains a safe
• minimum altitude
• Without blended control we would conclude
• midnight GTO was not a good option and that
• either a midday GTO or GEO orbit were
• preferable
• With blended we can reduce this problem to that

?????
70
Earth Escape Without Shadow
71
Earth Escape Without Shadow

• Shadow passage results in loss of thrust and
  prolonged thermal cycling can cause thermal
  damage to sail and spacecraft
• Shadow passage can impart severe thermal loads on
  to the spacecraft, dynamically exciting the
  structure and potentially over stressing the sail
  film, thus requiring heavier booms and/or thicker
  film coatings
• Furthermore, eclipse can cause large charging
  swings in spacecraft
• Loss of thrust means we require a secondary
  propulsion system to maintain sail control
• This would mean additional mass and loss of
  performance
• Using the blended control can examine escape from
  a range of polar orbits to determine the required
  sail acceleration to eliminate shadow events
• Sail model initially considers only the idealised
  scenario

72
Earth Escape Without Shadow

• As initial altitude is decreased the required
  sail acceleration increases exponentially
• The minimum time, shadow free Earth escape
  duration is 141 days and is independent of
  initial altitude

73
Earth Escape Without Shadow

• Can examine single case from 20 000km polar
  orbit, with all perturbations re-introduced
• Un-perturbed case suggest sail acceleration of
  0.8 mm s-2 perturbations and non-ideal sail mean
  we should increase sail acceleration to 0.85 mm
  s-2
• Trajectory is confirmed shadow free through
  examination of position vectors, with escape on
  day 141

RAAN
Inclination
74
Sail to the Moon
75
Solar Sailing to the Moon

• In 1981 the Union pour la Promotion de la
  Propulsion Photonique, U3P, and the World Space
  Foundation, WSF, proposed a Moon race
• WSF, was formed in 1979 principally by JPL
  engineer Robert Staehle and others after
  termination of the JPL sail work, attempting to
  raise private funds to fly a small-scale demo
  flight
• U3P, was formed in 1981 and rapidly proposed the
  idea of a race to advance the technology
• The Solar Sail Union of Japan, SSUJ, was formed
  in 1982 to compete in the Moon race
• Officially the race is still on, but in reality
  there has been little recent activity
• U3P are however talking of re-launching the
  race
• From the U3P website, The race is still going
  ... this page will be soon updated !
• In 1992 the US Columbus Quincentennial Jubilee
  commission attempted to stimulate interest in a
  Mars race
• Both race proposals generated significant
  interest and design concepts, however both also
  foundered, perhaps the length of race was to long
  for TV, thus limiting sponsorship
• We note however that solar sails are not well
  suited to operations near Earth, thus perhaps the
  choice of target was the problem!

76
Solar Sailing to the Moon

• The Moon race stimulated much hardware and
  astrodynamics work
• Generating many journal and conference papers on
  sail strategies to target the Moon
• Most lunar transfer strategies follow the same
  basic structure
• Phase 1 is to maximise the sails orbit energy,
  using the locally optimal control law
• Typically this constitutes the vast majority of
  the trajectory
• The winner in the Moon race was the first to pass
  behind the Moon, such as the sail was not visible
  from Earth
• Phase 2 of the Moon race was thus to target the
  final few revolutions such that the sail / moon
  phasing was correct
• This can be done by variable thrust levels, SSUJ
  approach, or targeted rendezvous
• Some limited work has also been conducted on
  lunar rendezvous

77
Solar Sailing to the Moon

• 593 day solar sail transfer from GTO Polar
  Lunar orbit
• Phase 1 (blue) maximise the local energy gain
  until a semi-major axis of 47000 km
• Phase 2 (yellow) is a time
• optimal transfer
• Phase 3 is a 7 day free flying
• phase
• Phase4 (red) is a rendezvous
• orbit to the Moon

78
GeoSail

• A Solar Sailing SMART Mission

79
GeoSail Introduction

• Motivated by the desire to achieve long residence
  in Earths magnetosphere
• Enables high-resolution statistical
  characterisation of plasma environment
• Solar sailing enables long residence (multi-year)
  in Earths magnetosphere by precessing the orbit
  major axis at 1 deg/day to continually track the
  Sun-Earth line
• Without a propulsion system, residence in
  magnetosphere is significantly curtailed
• Reaction propulsion methods critically limit
  mission duration (3.5 km s-1 per yr)
• Prior work for Lockheed-Martin become focused on
  NASA technology and funding
• Revised analysis was performed in the context of
  an ESA SMART mission

SMART-2
SMART-3 (?)
SMART-1
SMART-x ?
80
GeoSail Science Objectives

• Long duration residence enables high resolution
  temporal analysis of processes within
  magnetosphere
• Providing breakthrough in understanding the
  physical processes in magnetosphere
• The primary science goals are
• Understand how spontaneous magnetic reconnection
  occurs in a magnetic current sheet (near tail
  phenomena 22-30 Earth radii).
• Understand the mechanisms behind reconnection
  mode destabilisation and saturation in the
  magnetotail.
• Analyse the plasma structure at the sub-second
  resolution.
• Understand reconnection and particle dynamics at
  the day/dawn side low-latitude boundary layer
  along the Earths magnetopause.

81
GeoSail Instruments

• Instrument suite based on heritage and current
  developments
• Incorporation of instruments onto solar sail may
  require new innovations

82
GeoSail Instruments

• Payload integration is a key issue
• Current instrumentation is located using best
  engineering practice
• A technology goal of GeoSail is understanding the
  interaction between sail and space environment
• Further study is required to better predict this
  interaction, allowing instruments to be best
  located to enable science goal attainment
• The DLR-ESA sail concept utilises a deployable
  central boom for yaw and pitch control
• The SailBus and magnetometers are therefore 10
  m from the sail film
• Mounting on the boom may provide acceptable
  isolation

83
GeoSail Trajectory Analysis

• The GeoSail orbit is designed to achieve the
  science goals
• Perigee 11 Earth radii - Located on planetary
  day-side
• Apogee 23 Earth radii - Located on planetary
  night-side
• Perigee aligned with magnetopause
• Apogee aligned with tail reconnection
• region, between 22 30 Earth radii
• Forced orbit precession at 1 deg/day
• artificial non-Keplerian orbit
• Solar sail enables extended study of this
• key region, within the lunar orbit

Solar Sail
Earth
Sun-Earth line
84
GeoSail Trajectory Analysis

• Aim is to rotate argument of perigee at
  360/365.25 deg per. day
• Natural precession of apse-line due to Earth
  oblateness is of order 10-3 deg day-1
• Argument of perigee depends on all three orbital
  elements
• By placing orbit in ecliptic plane and setting
  sail pitch at zero we can remove out-of-plane
  terms
• Directing sail thrust along major axis, the
  components of the
• solar radiation pressure experienced by the sail
  are,
• For rotation of apse line to be synchronous with
  annual rotation
• of Sun-line then must always hold
• Using a fixed sail pitch of zero degrees is a
  simple strategy to
• implement in reality

85
GeoSail Trajectory Analysis

• Change in elements over single orbit can be
  obtained by considering variational equations of
  motion,
• Thus,
• Integrating rate of change of argument of perigee
  over a single orbit gives,
• Thus, the apse-line will rotate due to the solar
  radiation pressure
• Mean rate of precession of the apse line can be
  determined dividing through by orbit period,
• Thus, for the Sun-synchronous condition,
  the required solar sail characteristic
  acceleration is,
• where, deg day-1

86
GeoSail Trajectory Analysis

• Preceding analysis used a constant apse-line
  precession rate, however due to Earths
  eccentricity the Sun line does not rotate at a
  constant rate
• From conservation of angular momentum, can be
  shown that the Sun-line rotation rate will vary
  as the inverse square of planets heliocentric
  distance,
• Solar radiation pressure also has an
  inverse-squared variation with heliocentric
  distance, thus the forced precession of the apse
  line has the same functional relationship
• The modified mean precession rate is,
• Thus, the simple steering law will maintain the
  apse line as Sun-synchronous even if the planets
  orbit is non-circular
• The required sail characteristic acceleration is
  0.096 mm s-2

87
GeoSail Trajectory Analysis

• Additionally, we should correct sail
  acceleration/ rate of change of perigee for
  shadow
• Since orbit is Sun-synchronous shadow will be of
  same duration every orbit
• For eclipse of span (2?f) centred on the apogee,
  the change in argument of perigee during an orbit
  is now,
• Thus, integration yields
• The required sail characteristic acceleration is
  0.100 mm s-2
• Thus, shadow has only small effect on required
  sail performance
• Furthermore, we correct the sail for 85
  Reflective efficiency 94 Specular reflective
  efficiency to get the required sail
  characteristic acceleration of 0.113 mm s-2
• Can be shown that the use of a locally optimal
  argument of perigee control requires sail
  characteristic acceleration of 0.096 mm s-2

88
GeoSail Trajectory Analysis

• Simulation of orbit evolution shows perigee
  apogee vary by up to 1 Earth radii
• Variation is dependent on start epoch
• Start epoch shown is 03 January 2010
• Delay to June equinox reverse variation
• Angle between orbit major-axis Sun Earth line
  lt 8 deg over 2 years, mainly due to Lunar gravity

89
GeoSail Trajectory Analysis
Inertial frame
Rotating frame
Sun Earth line
90
GeoSail Spacecraft Design

• Design methodology assumes SMART framework, hence
  multiple new technologies
• Design requirement that if sail deployment fails
  mission can continue to demonstrate new
  technologies.
• Spacecraft is 3-axis stabilised driven by
  current sail designs
• Sail boom for yaw and pitch with reaction wheels
  and cold gas system for roll
• SailBus is highly autonomous fully integrated
  with sail systems.
• AOCS system includes low mass power Sun and
  star trackers technology goal
• SailBus has five primary modes of operation

91
GeoSail Spacecraft Design

• OBDH and TTC system is a single integrated
  avionics bundle technology goal
• Provide a standard, low mass unit for future
  European near-Earth missions
• Contains standard bus components common across
  all near-Earth mission
• Power regulation distribution
• Communications
• Command Telemetry handling
• Data processing and storage
• Sub-systems such as power generation attitude
  determination are separate
• Power supply by a single body mounted silicon
  solar array (15 AM0)
• No deployment or pointing requirements
• SilverZinc primary battery and three LiIon
  secondary batteries
• 174 charge discharge cycles, at 250 minutes
  shadow
• Shadow duration is 0.4 of perigee perigee
  orbit period

92
GeoSail Spacecraft Design

• Mean sail film temperature, in sunlight, varies
  between 271.5 K and 266.8 K
• Due to long Earth shadow periods spacecraft
  thermal environment is severe
• Sub-system includes 12 layers of MLI
• Heater Thermostat
• 15 Temperature sensors
• Radiation environment is suitable for soft COTS
  technology
• TID at 4 mm shielding is 3.5 krad (Si) over
• two-year mission (SPENVIS)
• Meteoroid analysis (Grun Model) suggests no
• impacts with objects greater than 0.16 mm
• diameter over two-year mission

93
GeoSail Solar Sail Requirements

• Sail design based on DLR-ESA sail concept
• 3-axis stabilised with deployable central mast
  for pitch and yaw control
• SailBus mounted on end of deployable central mast
• A more future-proof design would perhaps employ
  sail-tip vanes as well
• Sensor and antennae mounting is a significant
  design issue for all sail missions
• Providing a 4p steridian view requires sensors on
  both sides of the sail
• If a deployed boom is used design complexity and
  risk is increased
• Implementation of a 30 mass margin
• has minimal impact over sub-system level
• margins on sail size

94
GeoSail Solar Sail Requirements

• Required sail size is 41.2 m for core payload
  42.8 m for enhanced payload
• Using DLR designed CFRP booms
• Fixed 25 kg mass of non-jettisoned mechanisms
• 7.5 µm Kapton film substrate with aluminum and
  chromium coatings
• 10 of coated film mass for bonding
• Decreased film thickness and boom density has
  only limited effect on sail size
• 2 µm Kapton film with 50 gm-1
• booms reduces sail size by
• 4-metres launch mass by
• 6.8 kg (12 reduction)

95
GeoSail Mission Mass Budget
UNIVERSITYofGLASGOW

• Orbit insertion is by bi-prop kick-motor active
  sail maneuvering from LEO/GTO prohibited by
  duration and complexity
• Vega launch assumed no trade performed, is most
  expensive option
• For launch into a 1500 km circular 23.4o
  inclination orbit, in the ecliptic plane, launch
  margin is 42 for core mission and 40 for
  enhanced payload

96
GeoSail Spacecraft Configuration and Cost

• Current high-end ROM cost estimate is
  approximately 104 M (FY 2003)
• ROM Cost includes all costs
• SMART-1 cost 110 M (FY 2003)
• GeoSail fits well into the SMART mission cost cap

Vega Launch Fairing
SailBus Solar Array
Gimballed Boom
Boom deployment
Stowed Sail Booms
Solar Sail Deployment Module
Sail Film Storage
Bi-prop Motor
97
GeoSail Conclusions

• GeoSail is a unique opportunity to demonstrate
  sail technology under the SMART framework
• Allows technology demonstration with unique
  science from a non-Keplerian orbit
• Integration and design issues require further
  study, however it is clear the concept is strong
• Technology requirements are within European
  capabilities and awaiting flight opportunities
  and funding
• GeoSail is the logical choice for a first
  operational European solar sail mission is
  currently the only concept that is truly enabled
  by solar sail propulsion

SMART-2
SMART-3 (?)
SMART-1
SMART-x ?
98
Geostorm Polesitter Missions
99
Solar Sailing At Mercury
100
Solar Sailing At Mercury

• Mercurys orbit is highly eccentric, e 0.2056.
• Perihelion 0.31 AU
• Aphelion 0.46 AU
• Solar radiation varies according to,
• Variation in SRP over the 88 day orbit period is
• shown
• SRP varies from 4 10 times the Earth value
• Mercury is an attractive environment for solar
  sailing

101
Solar Sailing At Mercury

• Mercury sample return mission is significantly
  enhanced by solar sail propulsion
• Due to the significant variations in SRP
  trajectory optimisation is hindered
• Initial assumption is that escape would be of
  minimum duration at Mercury perihelion

102
Solar Sailing At Mercury - Escape

• Mercury escape times found using MEE
• Sail assumed a perfect reflector no orbit
  perturbations
• Shadow is not included.
• Sun assumed a parallel point source
• With Mercury radius taken as sail/sun distance.
• Initial orbit is 500km altitude, in orbit plane
  of Mercury
• (i.e. i ioptimal).
• Start epoch is 01 January 2015, incremented in 1
  day intervals
• Perihelion passage at day 20.86
• Aphelion passage at day 64.84
• Perihelion re-passage at day 108.83
• Time until escape varies in sinusoidal fashion.
• Corresponds to SRP distribution through Hermian
  year.
• For low acceleration sail maxima is just after
• perihelion passage.

103
Solar Sailing At Mercury - Escape

• Migration of maxima minima escape times for
  selected initial altitudes.
• Launch date of maximum duration escape tends
  towards Mercury aphelion passage as sail
  acceleration is increased.
• And similarly for minimum towards perihelion.
• The difference between minimum maximum escape
• times decreases as sail acceleration is
  increased.
• A