Thermal Limitations of Starwisp-Type Interstellar Probes

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1 Thermal Limitations of Starwisp-Type Interstellar Probes By Gregory L. Matloff 1) 1) Physics Department, New York City College of Technology, CUNY, Brooklyn, NY, USA (Received 1st Dec, 2016) Starwisp, an interstellar beam-accelerated nano-probe, is considered from a thermal-limitation point of view. Aluminum, beryllium, graphene and reflective monolayer sails are considered for a 100-m radius sail, constant beam power, and a million-kilometer acceleration distance. A 30-nm Al sail can achieve 102 km/s, an advanced monolayer sail with a reflectivity of 0.6 and an absorption of can achieve 0.03c. In all cases, beam power is < W and beam energy is < J. Adaptive optics must be used to insure that the beam is always larger than the sail to reduce the risk of the sail melting Key Words: Beamed Sails, Nanoprobes, Interstellar Spacecraft Nomenclature V: velocity a: acceleration s: acceleration distance REF: sail reflectivity P: beam power M: spacecraft mass c: speed of light in vacuum R: disc-sail radius ρ : sail density t: sail thickness σ: Stefan-Boltzmann-Constant ε : sail emissivity ABS: sail absorption λ : beam wavelength Dl : transmitting optics aperture S: transmitter-sail separation Δ Q/ Δ T: heat flow per unit time K: thermal conductivity Acon: heat transfer area Δ T/Δ x: temperature gradient Subscripts tr: partially transmissive op: operational collimation length. The solar-pumped laser or maser power station would be constructed in space and might also serve as a source of solar energy to be beamed down to Earth s surface when not engaged in space-probe acceleration. Because of recent space flight experience with small photon sails and advances in nanotechnology, interest in this concept has revived. Most theoretical treatments of the Starwisp concept consider acceleration of the sail/payload and design elements for the power station. Here, thermal limitations of Starwisp are examined. Namely, what is the maximum terminal velocity that can be obtained by a highly reflective thin-film probe, as determined from consideration of the thermodynamics of beam/sail interaction? Figure 1 presents the Starwisp concept. The laser power station, with an aperture diameter Dl broadcasts a collimated beam with constant power P towards the spacecraft over a distance S. At distance S from the power station, the beam completely fills the sail. The disc-shaped sail has a mass M and a radius R. The sail is assumed to be at rest relative to the power station at the start of power beaming. The velocity increment during the power-beaming process is V. The sail acceleration, a, is assumed to be constant during the power-beaming process. The sail is assumed to be oriented normal to the power beam during acceleration. From the time-independent equation of Newtonian kinematics, sail acceleration can be related to sail final velocity V and acceleration distance S: 1.Introduction:A Low-Mass Interstellar Probe Concept In 1985, Robert Forward presented Starwisp [1]. This very-low mass interstellar probe concept consists of a thinfilm payload deposited on a small photon sail that is accelerated by a laser or maser over a comparatively short a = 2S Sail/payload acceleration can be related to sail reflectivity (REF), sail/payload mass, beam power and the speed of light in vacuum(c) using the standard relation [2], for an opaque sail: (1) 1

2 (2) For the case of a partially transmissive sail [2], the factor (1+REF) in Eq. (2) should be replaced by (ABS + 2REF), where ABS is fractional absorption of incident beamed energy by the sail. Next, Eqs. (1) and (2) are equated. A disc sail is assumed with the mass (M) equal to π R 2 ρ t, where ρ is sail density and t is sail thickness. The numerical value of the speed of light in vacuum, 3 X 10 8 m/s, is substituted: 2. Thermal Considerations of Opaque and Partially Transmissive Sails Equation (3) is a significant result in considering the thermal limitations of Starwisp. The power absorbed by an opaque sail is (1-REF) P. Since both faces of the disc sail can re-radiate absorbed photons, the power radiated per unit sail area is: where σ is the Stefan-Boltzmann Constant (5.67 X 10-8 W m -2 K -4 ), T is maximum sail operational temperature and ε is sail emissivity. Rearranging Eq. (3), and substituting this result and the numerical value of the Stefan-Boltzmann Constant into Eq. (4), /S becomes: sails: 1+ REF a = ( )P Mc 1+ REF S = 2.1X10 9 P R 2 W = S S 1 REF ( ) 2π R 2 P = εσt 4 op op εt 4 = 7.48X REF 1 REF Equation (3) becomes, for partially transmissive εt 4 = 7.48X10 16 ABS + 2REF 1 REF The emissivity of a partially transmissive sail can be written [2,3] ( ) ( 1 REF ) REF + ABS ε tr = 1 ( 1 REF ABS)REF (3) (4) (5) (6) (7) It is now possible using Eqs. (5-7) to compare the performance of various sail materials. First, /S can be calculated for various materials with known or calculated emissivity, and known reflectance, maximum operational temperature, thickness and density. Next, a value of beam collimation distance S is selected, which allows for the calculation of sail velocity V at the end of the acceleration process. Acceleration time is determined by dividing distance S by average velocity (V/2). Selection of an arbitrary sail radius allows for the calculation of sail mass M. Substitution of M and (assumed constant) acceleration in Eq. (2) determines the required value of (constant) beam power P striking the sail during acceleration. A beam collimation distance of 10 6 km, about three times the Earth-Moon separation, is selected. An arbitrary sail radius (R= 100 m) is selected, which is intermediate between existing solar sail radii and those projected for the NASA Heliopause Sail [4], 3. Sail Materials Considered Three basic types of sail materials are considered in this analysis (Table 1). The first is 30-nm aluminum. This pure metal sail is the thinnest entirely opaque aluminum sail [5]. The maximum operational temperature of this sail is taken as 900 K, below the melting point of aluminum but well above the maximum operational temperature of existing aluminum-coated solar sails. Drexler proposed that such a sail would likely be manufactured in space [5]. But as described by Scaglione and Vulpetti, it is possible to construct an aluminum-plastic sail similar to present-day designs using a plastic substrate that would rapidly degrade in the space environment when exposed to solar UV radiation [6]. Note that the value of density for this sail in presented in Table 1 is somewhat larger than the the density of solid aluminum. This is to account for the payload, which likely will be deposited as a thin-film on the side of the sail facing away from the energy beam. The second sail considered is a partially transmissive10-nm beryllium sail. Analysis indicated that space-manufactured solar-photon sails constructed using beryllium film offer superior performance compared with sails constructed using other metals examined. Once again, the maximum operational temperature (Tmax) listed in Table 1 is less than the melting point of solid beryllium and the density of the sail is slightly greater than the density of pure beryllium. Other parameters are from a published analysis of the performance of beryllium hollow-body solar sails [7]. Although sail survival in the near-sun environment is not a primary issue in the consideration of beam-accelerated sails, an exhaustive examination of space radiation effects indicates that beryllium sails could function within ~0.07 AU from the Sun [8]. Next, three graphene-sail varieties are considered. As before, the values listed in Table 1 for sail density and maximum operational temperature are conservative. The first graphene sail considered (graphene A) uses values of thickness, reflectivity, fractional absorption and emissivity from an examination of this monolayer as a solar-photon sail [9]. Graphene B and graphene C have increased values of reflectivity to beamed radiation and appropriately modified values of emissivity. 2

3 Performance of the five sail materials considered here is presented in Table 2. Note that Graphene C can accelerate to about 0,03c. This sail can reach Alpha Centauri in ~130 years. 4. Rayleigh s Criterion and the Power Station According to Rayleigh s Criterion [2], the power beam will diverge after transmission. Equation (7.3) of Ref. 2 relates the laser/maser wavelength (λ), the diameter of the transmitting optics exit aperture (Dl), the sail radius (R) and the separation between the transmitter and sail (S) for the case where the beam completely fills the sail at distance S: 2.44λ D l = 2R S Assuming a yellow-light wavelength of 0.5 microns, a 100-m sail radius and a 10 6 km sail acceleration distance, the diameter of the beam-transmitting station is about 6 meters. This is about equivalent in size to the largest single terrestrial telescope mirrors. To assure that the beam always impacts the sail, the pointing accuracy must exceed 2R/S radians or 2 X 10-7 radians. But even if this can be achieved and maintained during the 6 hour acceleration period, the concept is in deep trouble. Early in the acceleration period, the beam size will be smaller. When the beam radius is one-tenth the sail radius, the radiant flux per unit area striking the sail will be 100X greater than at the termination of sail acceleration. Unless the thin-film sail can rapidly reduce the flux level by conduction to the rest of the sail, the sail must radiate at a temperature approaching 3,000 K. Aluminum, of course, will melt far below this temperature. To investigate whether such a level of heat conduction is possible, we first apply the standard thermal conductivity relationship for bulk aluminum [10]. This equation can be written : ΔQ Δt = KA con ΔT Δx where ΔQ / Δ t is the heat flow per unit time, K is the thermal conductivity of aluminum, Acon is the area into which heat is transferred, and ΔT/Δx is the temperature gradient. We assume here that the inner 1% of the sail is irradiated. The maximum temperature at the center of the sail is 933 K. To continue our optimistic stance, it is assumed very unrealistically that the temperature at the sail rim is 0 K. The temperature gradient is therefore about 10 K/m. The area into which heat is transferred by conduction is the circumference (8) (9) of the spot size (20π m) multiplied by the sail thickness (30 nm). From Ref. 4, K = 205 W/m-K for bulk aluminum. Substituting in Eq. (10), we find that conduction can account for only a tiny fraction of the heat input to the sail. But in reality, the situation is even worse. Rigorous experiments have demonstrated that the thermal conductivity of thin-film aluminum is considerably less than that of bulk aluminum [11,12]. 5. Conclusions The aluminum sail considered is the easiest to construct and unfurl. But it is totally unsuited for extrasolar missions beyond the heliopause. A beryllium sail would likely require in-space construction. With a solar-system exit velocity of 215 km/s, it could reach the Sun s inner gravity focus in about 10 years. Although some science could be done at and beyond this location [13], it is very unlikely that a Starwisp type probe without cross-range capability could fully exploit the observational astrophysical advantages of this location [14]. To perform interstellar missions with Starwisp, a graphene-like monolayer seems to be essential. But graphene fractional reflectivity to beamed energy must exceed 0.3 without any mass penalty to equal the performance of a highabsorption graphene sail with a 5% sunlight reflectivity unfurled close to the Sun [15]. If one wishes to reach Alpha Centauri within a human lifetime, it is necessary to increase the monolayer sail beam reflectivity above 60% without increasing sail mass. In all monolayer-sail cases, absorption of beamed energy must be low. The technology of monolayers in is its infancy. It is not impossible that a substance more capable for this application, such as an alloy or compound of monolayer hafnium, might be found. Another possibility, discussed by Landis, is to construct the sail as a high-refractive-index, lowrefractive-index dielectric sandwich, which might increase fractional reflectivity nearly to unity, at the price of a mass increase [16]. In many considerations of beamed-energy interstellar propulsion, it is assumed using Rayleigh s Criterion that the hot spot is initially much smaller than the sail size, expanding to fill the sail near the end of the acceleration process. Because of the huge energy density in required to accelerate Starwisp over a small beam-collimation distance, the limits of thermal conduction rule this approach out, even for a metallic sail. One alternative is to use adaptive optics to vary the beam size at the sail during the acceleration process. Beam power requirement is enormous in all cases, increasing as a function of performance. It is interesting that total beam energy is constant, varying within a factor of ~2. The goals of the Breakthrough Initiatives Starshot project ( are very challenging accelerating a nano-probe directed towards Alpha Centauri at 0.2c using projected beamed energy within the next two decades. But the possible accomplishments of the development are well worth the effort. A high-reflectivity nano layer may result, which will have applications to interplanetary and interstellar travel. Another possibility is 3

4 development of a terrestrial or near-earth beamed energy system capable of diverting Earth-threatening asteroids [17]. Acknowledgments The author appreciates the assistance of Les Johnson and members of his group at the NASA Marshall Space Flight Center. Although the work discussed in this paper was completed before the author joined the Breakthrough Initiative Project Starshot Advisors Board, the author greatly appreciates comments from other team members. The scenarios described here are not necessarily representative of those considered in Project Starshot. References 1) Forward, R. L.: Starwisp: An Ultra-Light Interstellar Probe, Journal of Spacecraft and Rockets, 22, (1985), pp ) Matloff, G. L.: Deep-Space Probes, 2nd ed., Springer- Praxis, Chichester, UK,2005, Chap. 7. 3) Wolfe, W. L., ed,: Handbook of Military Infrared Technology, Office of Naval Research, Dept. of the Navy, Washington, D.C.1965, p Microelectrothermal Test Structures and Finite-Element- Model-Based Data Analysis, Journal of Microelectromechanical Systems, 16, (2007), pp ) Maccone, C.: Deep Space Flight and Communications: Exploiting the Sun as a Gravitational Lens, Springer- Praxis, Chichester, UK (2010). 14) Landis, G. A. : Mission to the Gravitational Focus of the Sun: A Critical Analysis, arxiv.1604.o6351v1 (2016). 15) Matloff, G. L.: The Speed Limit for Graphene Interstellar Solar Photon Sails, JBIS, 66, (2013), pp ) Landis, G. A.:Advanced Solar- and Laser-Pushed Lightsail Concepts, final report to NASA Institute of Advanced Concepts (May 31, 1999). 17) Lubin, P. and Hughes,G. B.: Directed Energy for Planetary Defense, Handbook of Cosmic Hazards and Planetary Defense, Pelton, J. N. and Allahdadi, F. (eds.), Springer- Verlag, NY, 2015, pp Earth Power Station Beam Sail 4) Johnson, L. and S. Leifer, S.: Propulsion Options for Interstellar Exploration, AIAA Paper , ) Drexler, K. E. : High Performance Solar Sails and Related Reflecting Devices, AIAA Paper , Presented at 4th Princeton AIAA Symposium on Space Manufacturing Facilities, Princeton, NJ, May, ) S. Scaglione, S. and G. Vulpetti, G.: The Aurora Project: Removal of Plastic Substrate to Obtain an All-Metal Solar Sail, Acta Astronautica, 44, (1999), pp ) Matloff, G. L. :The Beryllium Hollow-Body Solar Sail and Interstellar Travel, JBIS, 59, (2006), pp ) Kezerashvili, R. Ya : Solar Sail Interstellar Travel: (1) Thickness of Solar Sail Films, JBIS, 61, (2008), pp ,. Also see references cited in this paper. Fig. 1. The Starwisp Concept 9) Matloff, G. L. : Graphene: The Ultimate Solar Sail Material?, JBIS, 65, (2012), pp ) Ohanian, H. C. : Physics, 2nd ed., Norton, NY, 1989, pp ) Bai, S.-Y. Tang, Z. A., Huang, Z.-X., Yu, J., and Wang, J.-Q.: Thermal Conductivity Measurement of Submicron- Thick Aluminum Oxide Thin Films by a Transient Thermo- Reflectance Technique, Chinese Physics Letters, 25, (2008), pp ) Stonjanovic, N., Yun, J., Washington, E. B. K., Berg, J. M., M. W. Holtz, M. W., and H. Temkin, H.: Thin-Film Thermal Conductivity Measurement Using 4

5 Table 1. Input data for five photon sails. Material ρ, kg/m 3 t,nm Tmax, K REF ABS ε Aluminum K Beryllium Graphene A Graphene B Graphene C Table 2. Output data for photon sails in Table 1. (Al: aluminum, Be: beryllium, G: graphene. Beam collimation distance = 10 6 km, sail radius = 100 m. Material Final Acceleration Sail a, m/s 2 P,W Energy, J Velocity,km/s Time,s Mass,kg Al X X Be X X G A X X GB X X GC X X