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Solar sail mission applications for non-Keplerian orbits
PII:
Acta Astronautica Vol. 45, Nos. 4-9, pp. 567-575, 1999 0 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765199 $ - see front matter SOO94-5765(99)00177-O
Solar Sail Mission Applications
for Non-Keplerian
Orbits
Colin R McInnes Department of Aerospace Engineering, University of Glasgow, Glasgow, G12 8QQ, Scotland, UK
Abstract Solar sails have long been considered for a diverse range of mission applications. h4ost of these applications have been conventional missions where the solar sail is utilised primarily as a means of low thrust propulsion for orbit transfer. For this reason solar sailing has historically been neglected, with more conventional and lower risk forms of propulsion utilised. In this paper new families of non-Keplerian orbits are described which are quite unique to solar sailing. These orbits have novel mission applications, several of which require the use high performance solar sails with small science payloads. Since the orbits and applications are unique to solar sails, they are mission enabling rather than mission enhancing, compelling mission planner; to develop and utilise solar sail technology. 0 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Solar sails have been considered for a range of high energy applications such as comet rendezvous [ 11, solar polar [2] and Mercury orbiter missions [3]. These missions envisage the use of deployable solar sails with a characteristic acceleration of order 1 mms-2. This paper will explore the rich variety of orbits and mission applications which are enabled using high performance solar saiIs with a characteristic acceleration of order 6 rnms-2. This is a significant design parameter as it corresponds to the gravitational acceleration of the Sun at 1 AU. Therefore, with a solar sail which can generate an acceleration of the same order as the local gravitational acceleration, large families of exotic, non-Keplerian orbits become possible. Firstly, the various families of non-Keplerian orbits enabled by high performance solar sails will be surveyed. Displaced orbits high above the ecliptic plane [4] and polar orbits displaced behind solar system bodies will be discussed [5]. In addition, it will be demonstrated that a continuum of new Lagrange points can also be generated [6]. Mission applications will then be investigated for each of these orbit families. Such applications include mapping the 3D structure of the heliosphere and planetary magnetic tails from displaced orbits. In addition, by patching Keplerian and non-Keplerian orbits together, elaborate new trajectories may be generated which have their own potential applications. Lastly, it will be shown that small payloads can be rapidly delivered to the outer solar system using high performance solar sails. After deployment of the sail on a C3=0 Earth escape trajectory, a close solar approach may be used to generate asymptotic speeds of order 15-20 AU per year allowing 100 AU class missions to be executed in only 5 years.
2. High Performance
Solar Sails
The performance of a solar sail can be characterised in a number of ways, the most convenient of which is the lightness number p. This is defined as the ratio of solar radiation pressure force to the solar gravitational force exerted on the sail. Since both of these forces 567
Low-Cost Planetary Missions
568
have an inverse square variation, the lightness number is constant for a given sail design. For a perfectly reflecting solar sail it can be shown that the lightness number is related to the sail loading (mass per unit area) CTby
p=iT3 pi2
1
In addition, the characteristic
acceleration a, is defined as the sail acceleration at 1AU and is
related to the lightness number by @O. 168 a, (mms-2). Therefore, for a lightness number of unity, a high performance solar sail is required with a sail loading of approximately 1.5 gm-* and a characteristic acceleration of order 6 mms- 2. Current solar sail designs [2,3] envisage a characteristic acceleration of order lrnrns-2. Most conventional solar sail designs use some from of boom structure to hold the sail flat after deployment. Such structures are always in compression and hence contribute significantly to the total solar sail mass. The most efficient means of fabricating a high performance solar sail is to use spin induced tension to delete most of the sail structural mass [7]. A schematic high performance sail is shown in Fig. la with Kevlar tapes used to provide load paths. In addition to deleting a significant amount of structural mass, a high performance solar sail must also have a greatly reduced sail film mass. Again, for a conventional solar sail design, a plastic substrate is required to allow handling, packing and deployment of the sail, as shown in Fig. lb. Once deployed however, the substrate is no longer required and only lowers the sail performance. One means of increasing the solar sail performance, while allowing safe packing and deployment, is to use a substrate which vaporises in vacuum or under the action of solar UV radiation. Kapton strips may be left to provide rip-stops and secondary load paths in the sail film. Small scale production of 0.05 micron all metal films has already been demonstrated. Aluminium 0.1 micron Kapton Plastic 7,/,//f///////,
2 micron 0.025 micron
Chromium Aluminium 0.05 micron 2 micron UV Sensitive Plastic
Figure 1.100 m radius solar sail (p=l) (a) configuration
(b) sail films
An approximate mass breakdown for a 100 m radius high performance sail would include a 4 kg metallic film, 4 kg of ripstops, 10 kg of Kevlar tapes, a 25 kg bus and 5 kg of contingency mass. Such a sail would have a total mass of 48 kg and a lightness number of approximately 1. If a 2 micron UV sensitive plastic substrate is included this adds 90 kg (assuming similar mass properties to Kapton) giving a launch mass of 138 kg. This is well within the C3=0 capacity of a small launcher such as Taurus. Concepts exist to further increase the sail performance by perforating the sail film with holes smaller than the mean wavelength of visible sunlight [7]. Such perforation is possible using techniques well established in the
Low-Cost Planetary Missions
semiconductor industry. The perforations significantly reflectivity largely unchanged. Using such techniques appear possible.
569
reduce the sail mass while leaving the sail lightness numbers of order 10
3. Displaced Solar Orbits
Now that the design of high performance solar sails has been considered, various families of non-Keplerian orbits will be investigated. Firstly, it will be shown that circular Sun-centred orbits can be displaced above the ecliptic plane by directing a component of the solar radiation pressure force normal to the ecliptic. An idealised, perfectly reflecting solar sail will be considered at position r in a frame of reference rotating with angular velocity o relative to an inertial frame I, as shown in Fig. 2. The sail orientation is defined by the unit vector n, fixed in the rotating frame of reference, and the solar sail performance is characterised by the sail lightness number p_ Since the solar sail orientation is fixed in the rotating frame, the sail must rotate once per orbit with respect to an inertial frame. A detailed analysis of the dynamics, stability and control of such orbits can be found elsewhere [4]. Z Disolaced orbit
t
Figure 2. Displaced By requiring
an equilibrium
Sun-centred
number /I for a displaced
z and orbital angular velocity
(2 / p)(
0 f
q2 ’
ta~~=(Z,p~2+[l_-(0,iU)2]
01
solar sail orbit
solution in the rotating frame of reference
that the sail pitch angle a and sail lightness radius p, displacement
n
ti2=s
it can be shown [4] Sun-centred
orbit of
w is given by
(24
2 “z[(z/p)2+(*-(w/iu)2)2p’2
l+
p=
[
;
(2b) [(zIp)2+(1-(wIo)2)]2
570
Low-Cost Planetary Missions
where CT,is the orbital angular velocity of a circular Keplerian orbit of radius r. Various families of orbits can now be obtained by a suitable choice of co. For example, o can be considered as a free parameter to minimise the required sail lightness number for a given orbit radius and displacement. Alternatively, @can be fixed to generate, for example, 1 year orbits synchronous with the Earth, as she un in Fig. 3. // ‘//I ‘i \’ 3 0.95 1
0.5
4
0.99
5
1.0
6
1.3
20 N
-0 5
-1
L
-1
-0 5
1
Figure 3. Sail lightness number contours (boundary surface St )
The contours shown in.Fig. 3 represent surfaces where the solar sail may be located in the p-z plane. The orientation of the sail normal is also shown. It can be seen that 1 year orbits in the ecliptic plane can be obtained closer to the Sun than 1AU. Similarly, for a solar sail with unit lightness number the sail may continuously levitate over the poles of the Sun. Between these two limits there is a continuum of 1 year orbits displaced above the ecliptic plane. The outer surface SI represents a boundary outside which orbits are no longer possible. In addition to individual displaced orbits, orbits may be patched together or patched to Keplerian orbits by turning the sail edgewise to the Sun, as shown in Fig. 4. It can be shown [4] that the patched, displaced orbits must be perpendicular to each other.
Low-Cost Planetary Missions
571
orbit
Figure 4. Patched orbits Clearly, these displaced non-Keplerian orbits have a diverse range of applications for solar physics and space weather missions. ,The Ulysses mission demonstrated the value of field and particle observations from out of the ecliptic plane. However, while Ulysses provided relatively short 4 months passes of the solar poles, displaced solar sail orbits provide a continuous out-of-plane view. For example, a 1 year Earth-synchronous orbit with p=O.5 AU and z=OS AU requires a lightness number of 0.88, similar to the design discussed in section 2. From this vantage point high above the ecliptic plane coronal mass ejections moving towards Earth can be clearly imaged. Such imaging is of importance for space weather studies [2] where CMEs propagating directly to Earth cannot be imaged from Earth against the solar disk. In addition, if the sail is rotated edgewise to the Sun a patch to a Kepleri an orbit is obtained with a perihelion of 0.387 AU and an inclination of 45O, allowing further direct investigation of the heliosphere. On return to the aphelion of the ellipse the sail can be patched back to a displaced orbit. Alternatively, the sail could patch directly to another displaced orbit normal to the first, allowing observations from above and below the echptic.
4. Displaced Planetary Orbits Following the discussion of displaced Sun-centred orbits, displaced planet-centred orbits will now be investigated. These orbits are again generated by orienting the solar sail such that a component of the solar radiation pressure force is directed out of the orbit pj!ane. Therefore, planet-centred displaced orbits are again circular orbits, but are displaced behind the planet in the anti-Sun direction_ As before, the orbital angular velocity w, orbit radius p and out-of-plane displacement distance z may be chosen independently with a suitable choice of sail pitch angle 01and sail acceleration a, as shown in Fig. 5. It will be assumed that the radiation field is uniform over the scale of the problem (tens of planetary radii) so that the ratio of the solar radiation pressure force to the local gravitational force increases with increasing distance from the planet. It is this relation that leads to interesting new orbits. For example, Mercury provides large displaced orbits due to its small mass and close proximity to the Sun. However, for Mercury the rapid rotation of the Sun-line and solar gravitational perturbations are important. It can be shown that control laws exist for stabilising any
Low-Cost Planetary Missions
572
unstable orbits against Derturbations. such as the Sun-line rotation and solar grav
Figure 5. Displaced planet-centred orbit For displaced planet-centred orbits it is found [5] that the required sail pitch angle a and sail acceleration a are given by
[II 2
1- w
tana=g Z
[
i;,
&2&. ’
r3
(34
(3b) where the required sail acceleration can be transformed to a lightness number by appropriate re-scaling. Again, various families of displaced orbits can be obtained by a suitable choice of orbital angular velocity CO.From Eq. (3b) it can be seen that an optimal family of displaced orbits which minimises the required sail acceleration may be generated through the choice of w = ti . A section of the resulting sail lightness number surfaces is shown in Fig. 6.
Low-Cost Planetary Missions
573
30 --
--
20
----
--IO
-
30
Planet ---4
I
”
f
I-
..l.
10
_)
20
30 z (Planetary
Figure
40
50
60
71
radii)
6. Sail lightness number contours
It can be seen that the requirements for Contour displaced orbits about Mercury are P Earth P Mere ury significantly less demanding than for those 1 0.49 0.029 about the Earth. For Earth centred orbits applications include investigation of the full 3D 2 0.66 0.039 structure of the geomagnetic tail. For 3 0.99 0.057 conventional missions, long elliptical orbits are required to gather data along the tail. However, 4 1.64 0.094 it then becomes difficult to deconvolve spatial variations and temporal variations due to the 5 3.30 0.189 spacecraft motion. Using displaced orbits, _ instruments are maintained at a fixed distance along the tail. For example, a 17.5 day orbit with ~20 Earth radii and z=40 Earth radii requires a lightness number of 0.74. By orienting the sail edgewise to the Sun a 12 day Keplerian orbit with a perigee of 25 Earth radii and inclination of 270 is achieved, allowing motion along the geomagnetic tail, if required. In addition, patching the initial displaced orbit to other displaced orbits is possible as detailed elsewhere [5]. 5. New Lagrange Points The circular restricted three-body problem has five well known Lagrange point equilibrium solutions Lj cj= l-5) where an infinitesimal mass will remain at rest with respect to two primary masses in orbit about their common centre of mass. For the planet-Sun-sail three-body problem rich new possibilities arise for artificial equilibrium points. In fact, it can be demonstrated that there is a continuum of new equilibrium solutions parameterised by the sail attitude and lightness number [6]. A section of sail lightness number surfaces about the Earth-Sun L1 and L2 points are shown in Fig. 7.
Low-Cost Planetary Missions
574 0.015
0.01
0.005
s so N
-0.005
-0.01
-0.015
Figure 7. Sail lightness number contours (boundary surfaces St and S2) It can be seen that near the classical Lagrange points only low values of sail lightness number are required. For example, the solutions sunwards of the Ll point provide an ideal location for providing advance warning of CMEs incident on the Earth [8]. Other applications include stationing instruments Earthwards of the L2 point to provide monitoring of the geomagnetic tail. In addition, equilibrium points high above the Earth have also been proposed for high latitude communications and observation [9].
6. 100 AU Missions A solar sail with unit lightness number provides a unique opportunity to efficiently extract energy from the gravitational well of the Sun. If the sail is deployed on a Cj=O Earth escape trajectory with the sail oriented to brake the spacecraft, it will slowly reverse its direction of motion [lo], as shown in Fig. 8. As the solar sail then falls sunwards it rapidly accelerates as gravitational potential energy is converted to kinetic energy. After the solar pass this energy would be converted back to potential energy and lost. However, using a solar sail of unit lightness number solar gravity can be effectively switched off, thus allowing the spacecraft to escape without loosing the energy gained during the infall to the Sun.
Low-Cost Planetary Missions
575
Deployment
-15
-1
-0.5
cl X NJ)
05
1
1.5
1 2
0
50
100
150
200
250
300
35lJ
t (dvs)
Figure 8. Solar fly-past (a) trajectory (b) heliocentric speed The trajectory shown in Fig. 8 is obtained with a solar sail of unit lightness number and a pitch angle of -400 relative to the Sun-line to limit the solar pass to 0.2 AU. This attitude is fixed and allows the sail to both brake, and escape after the solar pass. Such a trajectory allows 100 AU class missions to be executed in approximately 6 years.
7. Conclusions It has been demonstrated that high performance solar sails enable unique mission applications using families of non-Kepler&n orbits. For the most demanding orbits a lightness number of order unity is required. However, for less demanding orbits, many of which have exciting mission applications, only a low performance sail is required. Effort is now required to investigate UV or vacuum sensitive sail substrates which will allow safe manufacture and packing of the sail film, but will also allow high performance once deployed on-orbit. 8. References [l] Friedman L., et. al.: ‘Solar Sailing - The Concept Made Realistic’, Paper 78-82, AIAA 16th Aerospace Sciences Meeting, Huntsville, Alabama, January 1978. [2] Neugebauer, M., et. al.: ‘Solar Polar Sail Mission’, NASA/JPL NRA Response, 1996. [3] Leipold, M., et. al.: ‘Mercury Orbiter with a Solar Sail Spacecraft’, Acta Astronautica, Vol. 35, pp. 635-644, 1995. [43 McInnes, C.R., and Simmons, J.F.L.: ‘Halo Orbits for Solar Sails I - Heliocentric C&e’, Journal of Spacecraft and Rockets, Vol. 29, No.4, pp. 466-47 1, 1992. [5] McInnes, CR., and Simmons, J.F.L.: ‘Halo Orbits for Solar Sails II - Geocentric Case’, Journal of Spacecraft and Rockets, Vol. 29, No. 4, pp. 472-479, 1992. [6] McInnes, CR., et. al.: ‘Solar Sail Parking in Restricted Three-Body Systems’, Journal of Guidance, Dynamics and Control, Vol. 17, No. 2, pp. 399-406, 1994. [7] Uphoff, C.: ‘Very Fast Solar Sails’, Space Missions and Astrodynamics III, Politecnico di Torino, Italy, June 1994. [8] West, J.L.: ‘NOAA/DOD/NASA Geostorm Warning Mission’, JPL Report D-13986, 1996. [9] Forward, R.L.: ‘The Statie: A Spacecraft That Does Not Orbit’, Journal of Spacecraft and Rockets, Vol. 29, pp. 466-471, 1992. [ lO]Vulpetti, G.: ‘3D High-Speed Escape Heliocentric Trajectories by All-Metallic-Sail LowMass Sailcraft’, Acta Astronautica, Vol. 39, pp. 161-170, 1996.
Acta Astronautica Vol. 45, Nos. 4-9, pp. 567-575, 1999 0 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765199 $ - see front matter SOO94-5765(99)00177-O
Solar Sail Mission Applications
for Non-Keplerian
Orbits
Colin R McInnes Department of Aerospace Engineering, University of Glasgow, Glasgow, G12 8QQ, Scotland, UK
Abstract Solar sails have long been considered for a diverse range of mission applications. h4ost of these applications have been conventional missions where the solar sail is utilised primarily as a means of low thrust propulsion for orbit transfer. For this reason solar sailing has historically been neglected, with more conventional and lower risk forms of propulsion utilised. In this paper new families of non-Keplerian orbits are described which are quite unique to solar sailing. These orbits have novel mission applications, several of which require the use high performance solar sails with small science payloads. Since the orbits and applications are unique to solar sails, they are mission enabling rather than mission enhancing, compelling mission planner; to develop and utilise solar sail technology. 0 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Solar sails have been considered for a range of high energy applications such as comet rendezvous [ 11, solar polar [2] and Mercury orbiter missions [3]. These missions envisage the use of deployable solar sails with a characteristic acceleration of order 1 mms-2. This paper will explore the rich variety of orbits and mission applications which are enabled using high performance solar saiIs with a characteristic acceleration of order 6 rnms-2. This is a significant design parameter as it corresponds to the gravitational acceleration of the Sun at 1 AU. Therefore, with a solar sail which can generate an acceleration of the same order as the local gravitational acceleration, large families of exotic, non-Keplerian orbits become possible. Firstly, the various families of non-Keplerian orbits enabled by high performance solar sails will be surveyed. Displaced orbits high above the ecliptic plane [4] and polar orbits displaced behind solar system bodies will be discussed [5]. In addition, it will be demonstrated that a continuum of new Lagrange points can also be generated [6]. Mission applications will then be investigated for each of these orbit families. Such applications include mapping the 3D structure of the heliosphere and planetary magnetic tails from displaced orbits. In addition, by patching Keplerian and non-Keplerian orbits together, elaborate new trajectories may be generated which have their own potential applications. Lastly, it will be shown that small payloads can be rapidly delivered to the outer solar system using high performance solar sails. After deployment of the sail on a C3=0 Earth escape trajectory, a close solar approach may be used to generate asymptotic speeds of order 15-20 AU per year allowing 100 AU class missions to be executed in only 5 years.
2. High Performance
Solar Sails
The performance of a solar sail can be characterised in a number of ways, the most convenient of which is the lightness number p. This is defined as the ratio of solar radiation pressure force to the solar gravitational force exerted on the sail. Since both of these forces 567
Low-Cost Planetary Missions
568
have an inverse square variation, the lightness number is constant for a given sail design. For a perfectly reflecting solar sail it can be shown that the lightness number is related to the sail loading (mass per unit area) CTby
p=iT3 pi2
1
In addition, the characteristic
acceleration a, is defined as the sail acceleration at 1AU and is
related to the lightness number by @O. 168 a, (mms-2). Therefore, for a lightness number of unity, a high performance solar sail is required with a sail loading of approximately 1.5 gm-* and a characteristic acceleration of order 6 mms- 2. Current solar sail designs [2,3] envisage a characteristic acceleration of order lrnrns-2. Most conventional solar sail designs use some from of boom structure to hold the sail flat after deployment. Such structures are always in compression and hence contribute significantly to the total solar sail mass. The most efficient means of fabricating a high performance solar sail is to use spin induced tension to delete most of the sail structural mass [7]. A schematic high performance sail is shown in Fig. la with Kevlar tapes used to provide load paths. In addition to deleting a significant amount of structural mass, a high performance solar sail must also have a greatly reduced sail film mass. Again, for a conventional solar sail design, a plastic substrate is required to allow handling, packing and deployment of the sail, as shown in Fig. lb. Once deployed however, the substrate is no longer required and only lowers the sail performance. One means of increasing the solar sail performance, while allowing safe packing and deployment, is to use a substrate which vaporises in vacuum or under the action of solar UV radiation. Kapton strips may be left to provide rip-stops and secondary load paths in the sail film. Small scale production of 0.05 micron all metal films has already been demonstrated. Aluminium 0.1 micron Kapton Plastic 7,/,//f///////,
2 micron 0.025 micron
Chromium Aluminium 0.05 micron 2 micron UV Sensitive Plastic
Figure 1.100 m radius solar sail (p=l) (a) configuration
(b) sail films
An approximate mass breakdown for a 100 m radius high performance sail would include a 4 kg metallic film, 4 kg of ripstops, 10 kg of Kevlar tapes, a 25 kg bus and 5 kg of contingency mass. Such a sail would have a total mass of 48 kg and a lightness number of approximately 1. If a 2 micron UV sensitive plastic substrate is included this adds 90 kg (assuming similar mass properties to Kapton) giving a launch mass of 138 kg. This is well within the C3=0 capacity of a small launcher such as Taurus. Concepts exist to further increase the sail performance by perforating the sail film with holes smaller than the mean wavelength of visible sunlight [7]. Such perforation is possible using techniques well established in the
Low-Cost Planetary Missions
semiconductor industry. The perforations significantly reflectivity largely unchanged. Using such techniques appear possible.
569
reduce the sail mass while leaving the sail lightness numbers of order 10
3. Displaced Solar Orbits
Now that the design of high performance solar sails has been considered, various families of non-Keplerian orbits will be investigated. Firstly, it will be shown that circular Sun-centred orbits can be displaced above the ecliptic plane by directing a component of the solar radiation pressure force normal to the ecliptic. An idealised, perfectly reflecting solar sail will be considered at position r in a frame of reference rotating with angular velocity o relative to an inertial frame I, as shown in Fig. 2. The sail orientation is defined by the unit vector n, fixed in the rotating frame of reference, and the solar sail performance is characterised by the sail lightness number p_ Since the solar sail orientation is fixed in the rotating frame, the sail must rotate once per orbit with respect to an inertial frame. A detailed analysis of the dynamics, stability and control of such orbits can be found elsewhere [4]. Z Disolaced orbit
t
Figure 2. Displaced By requiring
an equilibrium
Sun-centred
number /I for a displaced
z and orbital angular velocity
(2 / p)(
0 f
q2 ’
ta~~=(Z,p~2+[l_-(0,iU)2]
01
solar sail orbit
solution in the rotating frame of reference
that the sail pitch angle a and sail lightness radius p, displacement
n
ti2=s
it can be shown [4] Sun-centred
orbit of
w is given by
(24
2 “z[(z/p)2+(*-(w/iu)2)2p’2
l+
p=
[
;
(2b) [(zIp)2+(1-(wIo)2)]2
570
Low-Cost Planetary Missions
where CT,is the orbital angular velocity of a circular Keplerian orbit of radius r. Various families of orbits can now be obtained by a suitable choice of co. For example, o can be considered as a free parameter to minimise the required sail lightness number for a given orbit radius and displacement. Alternatively, @can be fixed to generate, for example, 1 year orbits synchronous with the Earth, as she un in Fig. 3. // ‘//I ‘i \’ 3 0.95 1
0.5
4
0.99
5
1.0
6
1.3
20 N
-0 5
-1
L
-1
-0 5
1
Figure 3. Sail lightness number contours (boundary surface St )
The contours shown in.Fig. 3 represent surfaces where the solar sail may be located in the p-z plane. The orientation of the sail normal is also shown. It can be seen that 1 year orbits in the ecliptic plane can be obtained closer to the Sun than 1AU. Similarly, for a solar sail with unit lightness number the sail may continuously levitate over the poles of the Sun. Between these two limits there is a continuum of 1 year orbits displaced above the ecliptic plane. The outer surface SI represents a boundary outside which orbits are no longer possible. In addition to individual displaced orbits, orbits may be patched together or patched to Keplerian orbits by turning the sail edgewise to the Sun, as shown in Fig. 4. It can be shown [4] that the patched, displaced orbits must be perpendicular to each other.
Low-Cost Planetary Missions
571
orbit
Figure 4. Patched orbits Clearly, these displaced non-Keplerian orbits have a diverse range of applications for solar physics and space weather missions. ,The Ulysses mission demonstrated the value of field and particle observations from out of the ecliptic plane. However, while Ulysses provided relatively short 4 months passes of the solar poles, displaced solar sail orbits provide a continuous out-of-plane view. For example, a 1 year Earth-synchronous orbit with p=O.5 AU and z=OS AU requires a lightness number of 0.88, similar to the design discussed in section 2. From this vantage point high above the ecliptic plane coronal mass ejections moving towards Earth can be clearly imaged. Such imaging is of importance for space weather studies [2] where CMEs propagating directly to Earth cannot be imaged from Earth against the solar disk. In addition, if the sail is rotated edgewise to the Sun a patch to a Kepleri an orbit is obtained with a perihelion of 0.387 AU and an inclination of 45O, allowing further direct investigation of the heliosphere. On return to the aphelion of the ellipse the sail can be patched back to a displaced orbit. Alternatively, the sail could patch directly to another displaced orbit normal to the first, allowing observations from above and below the echptic.
4. Displaced Planetary Orbits Following the discussion of displaced Sun-centred orbits, displaced planet-centred orbits will now be investigated. These orbits are again generated by orienting the solar sail such that a component of the solar radiation pressure force is directed out of the orbit pj!ane. Therefore, planet-centred displaced orbits are again circular orbits, but are displaced behind the planet in the anti-Sun direction_ As before, the orbital angular velocity w, orbit radius p and out-of-plane displacement distance z may be chosen independently with a suitable choice of sail pitch angle 01and sail acceleration a, as shown in Fig. 5. It will be assumed that the radiation field is uniform over the scale of the problem (tens of planetary radii) so that the ratio of the solar radiation pressure force to the local gravitational force increases with increasing distance from the planet. It is this relation that leads to interesting new orbits. For example, Mercury provides large displaced orbits due to its small mass and close proximity to the Sun. However, for Mercury the rapid rotation of the Sun-line and solar gravitational perturbations are important. It can be shown that control laws exist for stabilising any
Low-Cost Planetary Missions
572
unstable orbits against Derturbations. such as the Sun-line rotation and solar grav
Figure 5. Displaced planet-centred orbit For displaced planet-centred orbits it is found [5] that the required sail pitch angle a and sail acceleration a are given by
[II 2
1- w
tana=g Z
[
i;,
&2&. ’
r3
(34
(3b) where the required sail acceleration can be transformed to a lightness number by appropriate re-scaling. Again, various families of displaced orbits can be obtained by a suitable choice of orbital angular velocity CO.From Eq. (3b) it can be seen that an optimal family of displaced orbits which minimises the required sail acceleration may be generated through the choice of w = ti . A section of the resulting sail lightness number surfaces is shown in Fig. 6.
Low-Cost Planetary Missions
573
30 --
--
20
----
--IO
-
30
Planet ---4
I
”
f
I-
..l.
10
_)
20
30 z (Planetary
Figure
40
50
60
71
radii)
6. Sail lightness number contours
It can be seen that the requirements for Contour displaced orbits about Mercury are P Earth P Mere ury significantly less demanding than for those 1 0.49 0.029 about the Earth. For Earth centred orbits applications include investigation of the full 3D 2 0.66 0.039 structure of the geomagnetic tail. For 3 0.99 0.057 conventional missions, long elliptical orbits are required to gather data along the tail. However, 4 1.64 0.094 it then becomes difficult to deconvolve spatial variations and temporal variations due to the 5 3.30 0.189 spacecraft motion. Using displaced orbits, _ instruments are maintained at a fixed distance along the tail. For example, a 17.5 day orbit with ~20 Earth radii and z=40 Earth radii requires a lightness number of 0.74. By orienting the sail edgewise to the Sun a 12 day Keplerian orbit with a perigee of 25 Earth radii and inclination of 270 is achieved, allowing motion along the geomagnetic tail, if required. In addition, patching the initial displaced orbit to other displaced orbits is possible as detailed elsewhere [5]. 5. New Lagrange Points The circular restricted three-body problem has five well known Lagrange point equilibrium solutions Lj cj= l-5) where an infinitesimal mass will remain at rest with respect to two primary masses in orbit about their common centre of mass. For the planet-Sun-sail three-body problem rich new possibilities arise for artificial equilibrium points. In fact, it can be demonstrated that there is a continuum of new equilibrium solutions parameterised by the sail attitude and lightness number [6]. A section of sail lightness number surfaces about the Earth-Sun L1 and L2 points are shown in Fig. 7.
Low-Cost Planetary Missions
574 0.015
0.01
0.005
s so N
-0.005
-0.01
-0.015
Figure 7. Sail lightness number contours (boundary surfaces St and S2) It can be seen that near the classical Lagrange points only low values of sail lightness number are required. For example, the solutions sunwards of the Ll point provide an ideal location for providing advance warning of CMEs incident on the Earth [8]. Other applications include stationing instruments Earthwards of the L2 point to provide monitoring of the geomagnetic tail. In addition, equilibrium points high above the Earth have also been proposed for high latitude communications and observation [9].
6. 100 AU Missions A solar sail with unit lightness number provides a unique opportunity to efficiently extract energy from the gravitational well of the Sun. If the sail is deployed on a Cj=O Earth escape trajectory with the sail oriented to brake the spacecraft, it will slowly reverse its direction of motion [lo], as shown in Fig. 8. As the solar sail then falls sunwards it rapidly accelerates as gravitational potential energy is converted to kinetic energy. After the solar pass this energy would be converted back to potential energy and lost. However, using a solar sail of unit lightness number solar gravity can be effectively switched off, thus allowing the spacecraft to escape without loosing the energy gained during the infall to the Sun.
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Figure 8. Solar fly-past (a) trajectory (b) heliocentric speed The trajectory shown in Fig. 8 is obtained with a solar sail of unit lightness number and a pitch angle of -400 relative to the Sun-line to limit the solar pass to 0.2 AU. This attitude is fixed and allows the sail to both brake, and escape after the solar pass. Such a trajectory allows 100 AU class missions to be executed in approximately 6 years.
7. Conclusions It has been demonstrated that high performance solar sails enable unique mission applications using families of non-Kepler&n orbits. For the most demanding orbits a lightness number of order unity is required. However, for less demanding orbits, many of which have exciting mission applications, only a low performance sail is required. Effort is now required to investigate UV or vacuum sensitive sail substrates which will allow safe manufacture and packing of the sail film, but will also allow high performance once deployed on-orbit. 8. References [l] Friedman L., et. al.: ‘Solar Sailing - The Concept Made Realistic’, Paper 78-82, AIAA 16th Aerospace Sciences Meeting, Huntsville, Alabama, January 1978. [2] Neugebauer, M., et. al.: ‘Solar Polar Sail Mission’, NASA/JPL NRA Response, 1996. [3] Leipold, M., et. al.: ‘Mercury Orbiter with a Solar Sail Spacecraft’, Acta Astronautica, Vol. 35, pp. 635-644, 1995. [43 McInnes, C.R., and Simmons, J.F.L.: ‘Halo Orbits for Solar Sails I - Heliocentric C&e’, Journal of Spacecraft and Rockets, Vol. 29, No.4, pp. 466-47 1, 1992. [5] McInnes, CR., and Simmons, J.F.L.: ‘Halo Orbits for Solar Sails II - Geocentric Case’, Journal of Spacecraft and Rockets, Vol. 29, No. 4, pp. 472-479, 1992. [6] McInnes, CR., et. al.: ‘Solar Sail Parking in Restricted Three-Body Systems’, Journal of Guidance, Dynamics and Control, Vol. 17, No. 2, pp. 399-406, 1994. [7] Uphoff, C.: ‘Very Fast Solar Sails’, Space Missions and Astrodynamics III, Politecnico di Torino, Italy, June 1994. [8] West, J.L.: ‘NOAA/DOD/NASA Geostorm Warning Mission’, JPL Report D-13986, 1996. [9] Forward, R.L.: ‘The Statie: A Spacecraft That Does Not Orbit’, Journal of Spacecraft and Rockets, Vol. 29, pp. 466-471, 1992. [ lO]Vulpetti, G.: ‘3D High-Speed Escape Heliocentric Trajectories by All-Metallic-Sail LowMass Sailcraft’, Acta Astronautica, Vol. 39, pp. 161-170, 1996.