>>  Non-Fiction – General Science Writing

Robert A. Metzger¹ and Geoffrey Landis²

¹Quad W. Associates, 6605 Williamson Dr. NE, Atlanta, GA 30328
²Ohio Aerospace Institute, NASA Research Center, 21000 Brookpark Road, MS: 302-1, Cleveland, OH 44135 (404) 705-8475, rametzger@aol.com

Abstract. A laser-based lightsail craft is proposed in which the laser beam is operated in a multi-bounce mode, such that after first striking the lightsail, the beam is reflected back to a source reflector where it is again directed to the lightsail. Recent developments in nearly ideal reflectors permit more than 1000 bounces, which reduce the laser power requirments by a factor of 1000 as compared to conventional laser-based lightsail proposals. Using the multi-bounce lightsail approach coupled with lasers operating in the power range of 100 MW to 1 GW, the details of a mission to Mars requiring only a sub-100 day transit, and an orbital transfer vehicle designed to transport cargo between Phobos and Deimos are examined.

INTRODUCTION

The concept of using the momentum transfer of a photon to propel a spacecraft dates back to the beginning of the 20th century (Tsander 1924), while detailed analysis using a solar-based lightsail concept was later proposed by Garwin (1958) and by Tsu (1959). These initial studies concluded that using the sun as a source of photons would be suitable for propulsion systems operating in the inner solar system, but would quickly lose efficiency further away from the sun, since the solar flux falls off as a square of the distance from the sun. With the invention of the laser (Maiman 1960), it quickly became obvious that lasers would not suffer from the same decrease in strength as a function of distance as photons from the sun do, and that lasers in fact could be used not only to propel lightsails anywhere within the solar system, but even over interstellar distances (Forward 1962). It was later shown that multistage lightsail craft could be designed which not only could be sent to nearby stars, but actually deaccelerated, thereby allowing rendevous with nearby stars (Forward 1984). The major challenge faced in the actual implementation of these proposed laser-based sailcraft, both interstellar and even within the solar system, is that extremely large laser power levels are required, even for small payloads. It has been proposed that extremely small payloads (10 kg) could be delivered to Mars in only 10 days of travel time using laser-based lightsail caft (Meyer, 1984), but in order to do so, would require a 47 GW laser system. Those missions envisioned for interstellar distances require lasers with powers ranging from 65 GW to 7.2 TW, power levels approaching the total power generating capabilities of Earth (Hoffert 1998). Table 1 shows typical output power levels of several optical sources, including the power generating ability of a typical commercial nuclear (or natural gas/oil/coal) power plant. Currently, even large military lasers are 4 orders of magnitude too small to even operate the envisioned small package missions to Mars.

The reason why such large laser power levels are required for lightsail craft is that the momentum transfer of a photon being reflected from a lightsail is extremely inefficient. The basic equation which governs the motion of a lightsail being accelerated by photons is (Forward 1962):

where a is the acceleration of the sail, R is the sail reflectivity (ranges from 0 for a transparent material, to 1.0 for a perfectly reflecting material), P is the incident photon power striking the sail, M is the total mass of the sail (and any payload which it may be carrying) and c is the speed of light. Under constant acceleration, the distance the sail travels, §, and its velocity, v, at any time, t, can be described by:

As an example to illustrate the basic laser-based lightsail concept, we will consider a mission to send a 20,000 kg craft + payload to Mars using a lasersail craft. While most proposed missions have power requirements which could only be supplied by a large numbers of commercial power generation facilities, for purposes of this example, we will limit ourselves to the power available from a single facility, that of 1 GW. A typical transit distance for an Earth-Mars mission for purposes of this example will be taken as 100 million km. Using equations 1 and 3, the time to make the Earth-Mars transit can be expressed by:

For a high reflectivity sail, in which R can be approximated by a value of 1.0, the time for the Earth-Mars transit as determined by Equation 4 is 283 days. However, this assumes that all the power of the laser beam is able to strike the lasersail craft. Due to laser beam diffraction, there exists a diffraction limited distance, S§, at which point the diameter of the diffracted laser beam has the same diameter as that of the sailcraft. At a distance greater than Sd, the edge of the expanded, diffracted beam begins to spill past the edge of the sail, no longer contributing to accelerating the craft. Typically, lightsail missions are envisioned in which laser power is applied to the sail until it reaches a distance of, S§, at which point the laser is turned off and the craft allowed to continue its trip at the velocity it had attained at the time the laser was turned off. If the laser is operated past the point where the craft has exceeded Sd, the power efficiency of the propulsion system rapidly decreases as the craft travels further away. The diffraction limited distance Sd, can be expressed by (Forward 1962):

Where D is the diameter of the lightsail craft's reflector, d is the initial, undiffracted diameter of the laser beam, and l is the laser wavelength. For purposes of this example we will use a lightsail reflector diameter, D, of 1000m, an initial laser beam diameter, d, of 10 m, and a laser wavelength of 1x10-6 m. For these values, Sd has a value of 4.2 million km, or 4.2% of the travel distance for the Earth-Mars transit. Once the lightsail craft has reached a distance of 4.2 million km, the laser is turned off and the craft continues onto Mars at the velocity it had attained when the laser was turned off. Substituting Sd for d in equation 4 results in a time for the craft to reach the diffraction limit distance of 4.2 million km in 1.55x106 sec, or 18 days. A that point, the laser is turned off and the lightsail continues its Earth-Mars transit under a constant velocity which can be determined by equations 1 and 2 to be 530 m/sec. Under that velocity, the total Earth-Mars transit time is 6 years. However, in reality, the craft would never even reach Mars, since the DV requirement between Earth and Mars orbit is approximately 3 km/sec, a velocity far in excess of what the lightsail craft of this mass and size could attain with a 1 GW laser.

The major limitation to the lasersail approach as the above example illustrates, is that despite the large power levels being used (1 GW laser), relying on the momentum transfer during a single laser beam reflection is insufficient to move the lightsail craft to other planets. For the lightsail craft idea to be feasible, there needs to be a way to greatly enhance the photon momentum transfer to the lightsail craft. Such a method was first proposed by Meyer (1998). If the reflected photon could be recycled, to once again be reflected against the sail, another momentum transfer would take place, and the efficiency doubled. We will examine the possibility of recycling the laser beam, allowing it to make multiple lightsail reflections.

MULTI-BOUNCE LASER-BASED SAIL APPLICATIONS

Conceptually, the process for recycling the laser beam is quite easy. Once the beam strikes the lightsail, the beam can be reflected back to its point of origin, and then reflected back again to the lightsail craft for another reflection. The limitations on the number of bounces the laser beam can make are dictated by the ability to reaim the reflected beam, the ability of the both the sailcraft and the reflector at the laser to efficiently reflect the beam, and the heating of the lightsail. For purposes of this discussion, we will assume that reflectivity and heating issues represent the rate limiting constraints, and that beam positioning can be adequately maintained through implementation of adaptive optics and steering lasers (steering lasers not used for power, but used to determine the position and angle of both the sailcraft and reflector mirror).

Very high reflectivity thin film technologies now exist which can produce reflectivities, R, for a specific wavelenth in excess of 99.9%. High reflectivity Bragg Reflectors constructed of alternating layers of thin materials, including both semiconductors and insulators (1-100 nm) with different dielectric constants can be tailored to produce nearly ideal reflectors. Such composite thin films with reflectivities of 99.95% would enable at least 1000 bounces before the power in the beam was greatly reduced due to absorption of the laser beam in the reflector or lightsail. Under these conditions, the effective power which the lightsail experiences is NxP where N is the number of bounces the laser makes from the sail. For purposes of this example, we will assume that N can have a value of 1000. This increases the effective power that the lightsail experiences by a factor of 1000.

Under these conditions, a major concern becomes the heating of the lightsail. For a reflector of 99.95%, a single pass would result in a power absorption by the lightsail of only 0.05% of the incoming beam, resulting in minimal heating. However, for 1000 bounces, an appreciable fraction of the laser beam power will be absorbed by the sail. We will examine what power limitations may exist under the condition, that all of the laser power becomes absorbed by the lightsail. Under this condition, the lasersail will act as a blackbody radiator, where its equilibrium temperature T, is defined by:

Where a is the absorption/bounce, s is the Stefan-Boltzmann constant, and e is the emissivity of the lightsail. We will examine a worst case example where we assume that after 1000 bounces that all the power has been absorbed by the lightsail, resulting in an a of 1x10-3, where in addition, an e of 0.5 will be used (surface engineering can typically increase this value of e nearer to one which makes the material a better emitter of heat). For a maximum allowable temperature, the minimum sail area, A, can be determined from equation 6. Assuming that it would be prudent to maintain sail temperatures at least 300 °C below the melting point of the materials used, this would imply that for semiconductor Bragg reflectors that the sail temperature should not exceed 500 °C, while insulator-based Bragg Reflectors should not exceed a temperature of 700 °C. For an effective power (NxP) of 1x1012 W, and a maximum allowable sail temperature of 500 °C, the minimum lightsail area is 4x104 m2. Therefore, when relying soley on radiative cooling, an effective power level (NxP) of 1x1012 W requires a sail with a diameter greater than 200 m.

MULTI-BOUNCE MISSIONS

We will first examine the case of sending the 20 tonne lightsail craft to Mars using a 1GW laser operating under the conditions of 1000 bounces. In order to insure that the lightsail stays safely within an allowable temperature range we will use a sail with a diameter, D, of 1000 m. One final consideration for this analysis is to consider for the reflecting materials used, if a sail of the desired dimensions can be built within the weight limit of 20 tonnes. Forward (1984) proposed an extremely thin (16 nm) Al sail with a resulting areal density of 0.1 gm/m2. Although extremely light weight, the reflectivity is only 0.82, making it sufficient for a single bounce application, but unsuitable for a multi-bounce approach. In order to maintain a high degree of reflectivity, multiple semiconductor or dielectric layers will be required, resulting in a sail thickness which could be estimated to be two orders of magnitude thicker than the thin Al lightsail. Such a thickness would result in an areal density of 10 gm/m2. For a sail with a 1000 m diameter the resulting total weight would be 7850 kg, within the weight budget of the 20 tonne lightsail craft. Therefore, for purposes of this example we will consider a lightsail with a mass of 10 tonne, carrying 10 tonne of cargo.

Power for the mission would be supplied by two, 1 GW laser installations. One installation would be Earth-based for launching, and a second installation would be placed on Mars (or in orbit) and would be used to capture the lightsail by deacclerating it. In addition, we will utilize an Earth-based reflector to return the laser beam to the sailcraft with a diameter, d, of 1000 m. A similar reflector would be incorporated into the Mars-base deaccelerator. Laser beam diffraction still limits the distance at which the laser can be usfully used. Using equation 5, Sd can be determined to be 4.2x1011 m. However, since the laser beam makes 1000 bounces, if the diffraction distance limit of the laser beam is reached after it has traveled 4.2x1011 m, the actual distance the sailcraft will have traveled is only 4.2x1011m/2000, which is 2.1x108 m, or roughly half the distance from the Earth to the moon. Using equation 1, the acceleration the lightsail experiences while under laser propulsion is 0.33 m/sec2 (or 3.4% of 1 standard Earth-g). Under this acceleration for a distance of 2.1x108 m, Equation 4 can be used to determine that the time required for the laser to be on is 3.6x104 sec, or approximately 10 hours. This represents a significant reduction in time as compared to a single bounce mission. When the laser is turned off the velocity of the lightsail can be determined from equation 2 to be 1.2x104 m/sec, or 12 km/sec (a value far greater than that required for the Earth-Mars DV). Under this velocity the lightsail will transit the 100 million km to Mars in 96 days at which time a 10-hour laser deaccleration will be executed from the Mars-based laser to capture the lightsail. This transit time is significantly less than the time required using conventional rockets.

For such a mission, the cost of delivering 1 kg of cargo to Mars from Earth orbit can be approximated. The total energy expended by the lasers (both the Earth- and Mars-based lasers) can be calculated from the expression:

Where P is the laser power of 1x109 W, the total time the lasers run, t, is 20 hours, and h is the wall-plug efficiency of the laser, which for purposes of this example will be assumed to be 25%. Under these conditions the total energy requirement becomes 8x107 kW-hr. The cost of producing electricity in the US is currently on the order of $0.03/kW-hr, which would result in a total power cost of $2.4 million, or only $240/kg for the delivered cargo. Using a single bounce laser system, (assuming that it could transport the craft to Mars - which it could not) the cost would be approximately 1000 times higher, for a total cost of $2.4 billion or $240,000 /kg.

As a second example, a less aggressive mission will be examined, in which 10 tonnes of cargo is transferred between the Martian moons of Phobos to Deimos. Phobos has an orbital height 5900 km above the surface of Mars and an orbital velocity of 2.11 km/sec, while that of Deimos is 20,000 km above the Martian surface and has an orbital velocity of 1.35 km/sec. In order to use a lightsail craft as an orbital transfer vehicle (OTV), a DV of 0.76 km/sec will be required. If once again a system is used which can operate for 1000 bounces, the total distance which a Mars-surface based laser must travel would be on the order of 20,000 km x 2000 x 1/2, or 2x1010 m (where the ½ value reflects the fact that the first bounces will be occurring when the OTV is at an orbital height of only 5900 km, and it will only be the final bounces which occur for an orbital height of 20,000 km). For a value of Sd of 2x1010 m (the total distance the laser beam will need to travel), equation 5 can be used to show that a mirror and sail of 250 m in diameter will be of sufficient size. With an areal density of 10 gm/m2 the resulting OTV would weigh approximately 1000 kg. If we allow the laser to fire for 20 hours as was the case of the Earth-Mars transit, the resulting acceleration, a, during the transit can be calculated from DV/Dt to be 0.011 m/sec2. For a transfer of 10 tonnes of cargo, equation 1 can be used to calculate a required laser power, P of 1.75x107 W, which for a wallplug efficiency, h, of 25% would require a power source of approximately 1x108 W, or 1/10 of what was required for the Earth-Mars transit system. Equation 7 can again be used to calculate the total energy requirement for the OTV, which becomes 1.4x106 kW-hr and represents a total cost of $42,000 for 10 tonnes of cargo transfer, or $4.20/kg. Costs for a single bounce laser system would be approximately 1000 times higher

We have shown that the advent of nearly ideal reflectors allows for the potential of a multi-bounce sailcraft. Such systems can efficiently and economically be used for both deep space missions as well as for smaller orbital transfer missions. The use of a multi-bounce approach radically reduces the power requirement of the laser system (by a factor of 1000) as compared to conventional single bounce laser sail schemes, making the possibilities of such multi-bounce lightsail craft feasible within the coming decades.

a = acceleration (m/sec2)
R = sail reflectivity
P = incident photon power (W)
M = mass of sail and payload (kg)
c = speed of light (3.0x108 m/sec)
v = velocity of sail (m/sec)
d = distance sail has traveled
t = time (sec)
Sd = diffraction limited distance (m)
l = wavelength of laser (m)
D = laser sail diameter (m)
d = initial laser beam diameter (m)
T = Temperature (K)
N = number of laser bounces
s = Stefan-Boltzmann Consant (5.67x10-8 W/m2-°K4)
e = emissivity of back of lasersail
A = area of sail (m2)
DV - change in orbital velocity (m/sec)
a - laser absorption/bounce
E - energy (joule)
h - laser wallplug efficiency