Fast solar sailing

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Contents 1 Introduction 2 Synopsis 3 Mathematical survey 3.1 The Extended Heliocentric Reference Frame 3.2 A Flight Example 3.2.1 conventional strategy 3.2.2 Optimal Strategy 3.2.3 H-reversal sun flyby trajectory 4 References

[edit] Introduction

This primer to a recent topic of astrodynamics has been arranged into three parts: (1) synopsis, (2) mathematical survey, (3) some references where the interested reader can find the appropriate mathematics. Such topic has not been yet completely investigated in the sense that a full general trajectory optimization is not available hitherto. However, the class of the missions relevant to escaping the solar system have been studied and numerically optimized.

[edit] Synopsis

• A solar sail spacecraft would make plenty of scientific and/or utilitarian space missions possible without consuming any propellant, since sunlight would be utilized directly, namely, without any conversion into heat or electricity. There are several ways for navigating by sail in space; they depend on the mission objectives. In the last two decades, scientists have also proposed a number of missions aimed at exploring the objects and the space beyond Pluto (or, more exactly, the Pluto-Charon system). Scientific probes should go to distances of 50 AU (AU stands for the astronomical unit, that is the mean Sun-Earth distance), at least. More specifically, according to some space mission concepts by NASA, the space-science community is interested in sending some probe(s) to 200 AU and, possibly, to 550-800 AU for exploring the heliosphere boundaries, the interstellar medium and some effects predicted by Einstein's General Relativity.
• For such aims, a high cruise speed of spacecraft is one of the primary goals for reducing flight times and design costs. Before the Nineties, it was a common belief that a space vehicle powered by nuclear-electric engines was required to get very high speed far from the Sun, the only space technology realistically achievable in the near/medium term. However, in the Nineties, it was shown - on a strict mathematical basis - that advanced solar-sail spacecraft could utilize both the solar light and the solar gravity in a special way to get cruise speeds higher or considerably higher than any realistic nuclear-electric engine. This way of conceiving a sail spacecraft and its related trajectory represents the fast solar sailing. By this mode, a space vehicle does not need to pass close to planets to get energy. This is a great advantage in terms of launch flexibility; as a matter of fact, the launch of these sail spacecraft should be no longer tied to rare favourable positions of the planets.

[edit] Mathematical survey

[edit] The Extended Heliocentric Reference Frame

• In the 1991-92 the classical equations of the solar sail motion in the solar gravitational field were written by using a different mathematical formalism, namely, the lightness vector fully characterizing the sailcraft dynamics. In addition, solar-sail spacecraft has been supposed to be able to reverse its motion (in the solar system) provided that its sail were sufficiently light that sailcraft sail loading (σ) is not higher than 2.1 g/m². This value entails a high-performance technology indeed, but much probably within the capabilities of emerging technologies.
• For describing the concept of fast sailing and some related items, we need to define two frames of reference. The first is an inertial Cartesian coordinate system centred on the Sun or a heliocentric inertial frame (HIF, for short). For instance, the plane of reference, or the XY plane, of HIF can be the mean ecliptic at some standard epoch such as J2000. The second Cartesian reference frame is the so-called heliocentric orbital frame (HOF, for short) with the origin in the sailcraft barycenter. The x-axis of HOF is the direction of the Sun-to-sailcraft vector, or position vector, the z-axis is along the sailcraft orbital angular momentum, whereas the y-axis completes the counterclockwise triad. Such definition can be extended to sailcraft trajectories, including both counterclockwise and clockwise arcs of motion, such a way HOF is always a continuous positively-oriented triad. The sail orientation unit vector (defined in sailcraft), say, n can be specified in HOF by a pair of angles, e.g. the azimuth α and the elevation δ. Elevation is the angle that n forms with the xy-plane of HOF (-90° ≤ δ ≤ 90°). Azimuth is the angle that the projection of n onto the HOF xy-plane forms with the HOF x-axis (0 ≤ α < 360 °). In HOF, azimuth and elevation are equivalent to longitude and latitude, respectively.

• The sailcraft lightness vector L = [λ_r , λ_t , λ_n] depends on α and δ (non-linearly) and the thermo-optical parameters of the sail materials (linearly). Neglecting a small contribution coming from the aberration of light, one has the following particular cases (irrespective of the sail material):

1. α = 0 , δ = 0 ⇔ [λ_r , 0 , 0] ⇔ λ=|L|=λ_r
2. α ≠ 0 , δ = 0 ⇔ [λ_r , λ_t , 0]
3. α = 0 , δ ≠ 0 ⇔ [λ_r , 0 , λ_n]

[edit] A Flight Example

[edit] conventional strategy

• Now suppose we have built a sailcraft with an all-metal sail of Aluminium and Chromium such that σ = 2 g/m². A launcher delivers the (packed) sailcraft at some million kilometers from the Earth. There, the whole sailcraft is deployed and begins its flight in the solar system (here, for the sake of simplicity, we neglect any gravitational perturbation from planets). A conventional spacecraft would move approximately in a circular orbit at about 1 AU from the Sun. In contrast, a sailcraft like this one is sufficiently light to be able to escape the solar system or to point to some distant object in the heliosphere. If n is parallel to the local sun-light direction, then λ_r = λ = 0.725 (i.e. 1/2 < λ < 1); as a result, this sailcraft moves on a hyperbolic orbit. Its speed at infinity is equal to 20 km/s. Strictly speaking, this potential solar sail mission would be faster than the current record speed for missions beyond the planetary range, namely, the Voyager-1 speed, which amounts to 17 km/s or about 3.6 AU/yr (1 AU/yr = 4.7404 km/s). However, three kilometers per second are not meaningful in the context of very deep space missions.
• As a consequence, one has to resort to some L having more than one component different from zero. The classical way to gain speed is to tilt the sail at some suitable positive α. If α= +21°, then the sailcraft begins by accelerating; after about two months, it achieves 32 km/s. However, this is a speed peak inasmuch as its subsequent motion is characterized by a monotonic speed decrease towards an asymptotic value, or the cruise speed, of 26 km/s. After 18 years, the sailcraft is 100 AU away from the Sun. This would mean a pretty fast mission. However, considering that a sailcraft with 2 g/m² is technologically advanced, is there any other way to increase its speed significantly? Yes, there is. Let us try to explain this effect of non-linear dynamics.

[edit] Optimal Strategy

• The above figures show that spiralling out from a circular orbit is not a convenient mode for a sailcraft to be sent away from the Sun since it would not have a high enough excess speed. On the other hand, it is known from astrodynamics that a conventional Earth satellite has to perform a rocket maneuver at/around its perigee for maximizing its speed at "infinity". Similarly, one can think of delivering a sailcraft close to the Sun to get much more energy from the solar photon pressure (that scales as 1/R²). For instance, suppose one starts from a point at 1 AU on the ecliptic and achieves a perihelion distance of 0.2 AU in the same plane by a two-dimensional trajectory. In general, there are three ways to deliver a sailcraft, initially at R₀ from the Sun, to some distance R < R₀:
  • using an additional propulsion system to send the folded-sail sailcraft to the perihelion of an elliptical orbit; there, the sail is deployed with its axis parallel to the sun-light for getting the maximum solar flux at the chosen distance;
  • spiralling in by α slightly negative, namely, via a slow deceleration;
  • strongly decelerating by a "sufficiently large" sail-axis angle negative in HOF.

The first way - although usable as a good reference mode - requires another high-performance propulsion system.

The second way is ruled out in the present case of σ = 2 g/m²; as a matter of fact, a small α < 0 entails a λ_r too high and a negative λ_t too low in absolute value: the sailcraft would go far from the Sun with a decreasing speed (as discussed above).

In the third way, there is a critical negative sail-axis angle in HOF, say, α_cr such that for sail orientation angles α < α_cr the sailcraft trajectory is characterized as follows:

1. the distance (from the Sun) first increases, achieves a local maximum at some point M, then decreases. The orbital angular momentum (per unit mass), say, H of the sailcraft decreases in magnitude. It is suitable to define the scalar H = H•k, where k is the unit vector of the HIF Z-axis;
2. after a short time (few weeks or less, in general), the sailcraft speed V = |V| achieves a local minimum at a point P. H continues to decrease;
3. past P, the sailcraft speed increases because the total vector acceleration, say, A begins by forming an acute angle with the vector velocity V; in mathematical terms, dV / dt = A • V / V > 0. This is the first key-point to realize;
4. eventually, the sailcraft achieves a point Q where H = 0; here, the sailcraft's total energy (per unit mass), say, E (including the contribution of the solar pressure on the sail) shows a (negative) local minimum. This is the second key-point;
5. past Q, the sailcraft - keeping the negative value of the sail orientation - regains angular momentum by reversing its motion (that is H is oriented down and H < 0). R keeps on decreasing while dV/dt augments. This is the third key-point;
6. the sailcraft energy continues to increase and a point S is reached where E=0, namely, the escape condition is satisfied; the sailcraft keeps on accelerating. S is located before the perihelion. The (negative) H continues to decrease;
7. if the sail attitude α has been chosen appropriately (about -25.9 deg in this example), the sailcraft flies-by the Sun at the desired (0.2 AU) perihelion, say, U; however, differently from a Keplerian orbit (for which the perihelion is the point of maximum speed), past the perihelion, V increases further while the sailcraft recedes from the Sun.
8. past U, the sailcraft is very fast and pass through a point, say, W of local maximum for the speed, since λ < 1. Thus, speed decreases but, at a few AU from the Sun (about 2.7 AU in this example), both the (positive) E and the (negative) H begin a plateau or cruise phase; V becomes practically constant and, the most important thing, takes on a cruise value considerably higher than the speed of the circular orbit of the departure planet (the Earth, in this case). This example shows a cruise speed of 14.75 AU/yr or 69.9 km/s. At 100 AU, the sailcraft speed is 69.6 km/s.

[edit] H-reversal sun flyby trajectory

The Figure below shows the mentioned sailcraft trajectory. Only the initial arc around the Sun has been plotted. The remaining part is rectilinear, in practice, and represents the cruise phase of the spacecraft. The sail is represented by a short segment with a central arrow that indicates its orientation. Note that the complicate change of sail direction in HIF is very simply achieved by a constant attitude in HOF. That brings about a net non-Keplerian feature to the whole trajectory.

Some remarks are in order.

• As mentioned in point-3, the strong sailcraft speed increase is due to both the solar-light thrust and gravity acceleration vectors. In particular, dV / dt, or the along-track component of the total acceleration, is positive and particularly high from the point-Q to the point-U. This suggests that if a quick sail attitude maneuver is performed just before H vanishes, α → -α, the sailcraft motion continues to be a direct motion with a final cruise velocity equal in magnitude to the reversal one (because the above maneuver keeps the perihelion value unchanged). The basic principle both sailing modes share may be summarised as follows: a sufficiently light sailcraft needs to lose most of its initial energy for subsequently achieving the absolute maximum of energy compliant with its given technology.

• The above 2D class of new trajectories represents an ideal case. The realistic 3D fast sailcraft trajectories are considerably more complicate than the 2D cases. However, the general feature of producing a fast cruise speed can be further enhanced. Some of the enclosed references contain strict mathematical algorithms for dealing with this topic. Recently (July 2005), in an international symposium an evolution of the above concept of fast solar sailing has been discussed. A sailcraft with σ = 1 g/m² could achieve over 30 AU/yr in cruise (by keeping the perihelion at 0.2 AU), namely, well beyond the cruise speed of any nuclear-electric spacecraft (at least as conceived today). Such paper has been published on the Journal of the British Interplanetary Society (JBIS) in 2006.

[edit] References

• http://interstellar.jpl.nasa.gov/
• G. Vulpetti, The Sailcraft Splitting Concept, JBIS, Vol.59, pp. 48-53, February 2006
• G. L. Matloff, Deep-Space Probes: to the Outer Solar System and Beyond, 2nd ed., Springer-Chichester, UK, 2005
• T. Taylor, D. Robinson, T. Moton, T. C. Powell, G. Matloff, and J. Hall, Solar Sail Propulsion Systems Integration and Analysis (for Option Period), Final Report for NASA/MSFC, Contract No. H-35191D Option Period, Teledyne Brown Engineering Inc., Huntsville, AL, May 11, 2004
• G. Vulpetti, Sailcraft Trajectory Options for the Interstellar Probe: Mathematical Theory and Numerical Results, the Chapter IV of NASA/CR-2002-211730, “The Interstellar Probe (ISP): Pre-Perihelion Trajectories and Application of Holography”, June 2002
• G. Vulpetti, Sailcraft-Based Mission to The Solar Gravitational Lens, STAIF-2000, Albuquerque (New Mexico, USA), 30 Jan - 3 Feb, 2000
• G. Vulpetti, General 3D H-Reversal Trajectories for High-Speed Sailcraft, Acta Astronautica, Vol. 44, No. 1, pp. 67-73, 1999
• C. R. McInnes, Solar Sailing: Technology, Dynamics, and Mission Applications, Springer-Praxis Publishing Ltd, Chichester, UK, 1999
• Genta, G., and Brusa, E., The AURORA Project: a New Sail Layout, Acta Astronautica, 44, No. 2-4, pp. 141-146 (1999)
• S. Scaglione and G. Vulpetti, The Aurora Project: Removal of Plastic Substrate to Obtain an All-Metal Solar Sail, special issue of Acta Astronautica, vol. 44, No. 2-4, pp. 147-150, 1999
• J. L. Wright, Space Sailing, Gordon and Breach Science Publishers, Amsterdam, 1993