Towers on the Moon: 1. Concrete Sephora Rupperta,∗, Amia Rossa, Joost Vlassak1, Martin Elvisc aHarvard College, Harvard University, Cambridge, MA, USA bHarvard School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA cHarvard-Smithsonian Center for Astrophysics, Harvard University, Cambridge, MA, USA Abstract The lunar South pole likely contains significant amounts of water in the perma- nently shadowed craters there. Extracting this water for life support at a lunar base or to make rocket fuel would take large amounts of power, of order Gi- gawatts. A natural place to obtain this power are the \Peaks of Eternal Light", that lie a few kilometers away on the crater rims and ridges above the perma- nently shadowed craters. The amount of solar power that could be captured depends on how tall a tower can be built to support the photovoltaic panels. The low gravity, lack of atmosphere, and quiet seismic environment of the Moon suggests that towers could be built much taller than on Earth. Here we look at the limits to building tall concrete towers on the Moon. We choose concrete as the capital cost of transporting large masses of iron or carbon fiber to the Moon is presently so expensive that profitable operation of a power plant is unlikely. Concrete instead can be manufactured in situ from the lunar regolith. We find that, with minimum wall thicknesses (20 cm), towers up to several kilometers tall are stable. The mass of concrete needed, however, grows rapidly with height, from ∼ 760 mt at 1 km to ∼ 4,100 mt at 2 km to ∼ 105 mt at 7 km and ∼ 106 mt at 17 km. arXiv:2103.00612v1 [astro-ph.EP] 28 Feb 2021 Keywords: moon, lunar mining, lunar towers, lunar solar power, concrete structure ∗Corresponding author Email address: [email protected] (Sephora Ruppert) Preprint submitted to Journal of LATEX Templates March 2, 2021 1. Introduction The South pole of the Moon appears to harbor significant resources in the form of water and organic volatiles in the permanently shadowed regions [1]. There is considerable interest in harnessing these resources to support a lunar base [2] or to manufacture rocket fuel to resupply rockets at lower cost than bringing the fuel up from Earth [3]. However, extracting these resources is a power-intensive operation. Kornuta et al. estimate that extraction of 2450 tons/year of water from the permanently dark craters would require power at a level of 0.4 - 1.4 GW (their figure 17) [3]. A promising solution is the nearly continuous energy supply that is poten- tially available a few kilometers away [4] on the \Peaks of Eternal Light" [5]. The \Peaks" are exposed to sunlight for over 90% of the lunar cycle [6]. How- ever, the illuminated area is only a few square kilometers and much of that area would be shadowed by other solar towers [7], limiting the available power. One way of increasing the potential power output is to build higher. The resulting added power is not just due to an increase in the area provided by tall towers; the illumination is also more continuous as the tower rises above local topogra- phy [8]. Ross et al. showed that for towers up to 20 m tall, the maximum power attainable was of order a few megawatts; instead, for towers from 0.5 - 2 km tall several Gigawatts are achievable [7] . Given that Kornuta et al. (Figure 17) estimate that extraction of 2450 tons/year of water from the permanently dark craters would require power at a level of 0.4 - 1.4 GW, a need for towers in the kilometer-high range is indicated [3]. For scale, the Eiffel Tower is 330 m tall [9] and the tallest building on Earth, the Burj Khalifa in Dubai, is 829 m tall [10]. Evidently building comparably tall lunar towers is a challenge. However, the 1=6 gravity on the Moon [11], combined with the lack of an atmosphere and so of winds, and the minimal levels of seismic activity (1010 - 1014 J/yr) [12], suggest that kilometer-scale lunar towers are not ruled out. Here we explore the limits to how tall moon-based solar towers could be using simple modeling. Determining the tallest structure that can be built 2 with a given material is a field with a history stretching back to Greenhill (1881) [13]. General solutions are hard to find, and modeling has to make simplifying assumptions [14]. We considered limits imposed by both compressive strength and buckling. We focused in this first study on towers made of concrete. Transporting materials to the Moon is currently very expensive, of order $0.5 million/kilogram [15]. This makes for an enormous capital cost for a kilometer- scale tower, of order billions At these prices lunar water mining would be hard to make into a profitable industry. Instead it has been shown that concrete can be made out of the loose lunar surface material (\regolith") [16]. Doing so would greatly reduce the up-front capital cost as only the relatively lightweight photovoltaic panels would need to be supplied from Earth. Hence, we explore the possibilities for concrete towers on the Moon in this paper. We used an analytic approach to estimating the stresses in the modeled towers. In this way we could expose the scalings of maximum tower height to the model parameters. We first describe the model in section 2. In section 3 we then describe the results after optimizing tower geometry and imposing a minimum wall thickness. We discuss the limitations of these calculations, and so the need for further work, in section 4. We present a summary and our conclusions in section 5. 2. Theory To explore the structural limitations of a concrete tower, we modeled a cir- cular structure that gets exponentially thinner with height. The cross-sectional area at a given height x above the base is described by −kx A(x) = A0e ; (1) where A0 is the cross-sectional area at the base of the tower, k is the exponent by which the tower cross-section shrinks (k ≥ 0), and x is the height above the base. 3 The thickness of the tower's walls also decrease with the same exponent, k. The cross-section of the concrete walls by height is given by −kx Ac(x) = Ac;0e ; (2) where Ac;0 is the cross-sectional area of the walls at the base of the tower. Furthermore, Ac;0 = A0(1 − b); (3) where b is unitless and determines the fraction of the tower that is hollow. 2.1. Stress At any point, the tower's walls are under compressive stress by the weight of the concrete above the point. Because of the circular symmetry of the model, any point of equal height, i.e. any point of a given cross-section, essentially experiences the same amount of stress. As a function of height, the stress is therefore F (x) σ(x) = ; (4) Ac(x) where F (x) is the weight of the tower section above acting on the cross- section. Z L F (x) = ma = ρg Ac(x)dx; (5) x where ρ is the density of concrete, g describes the gravity on the surface of the moon, and L is the total height of the tower. Applying a safety factor fs to the load, the resulting stress in the tower is f ρg σ(x) = s 1 − ek(x−L) (6) k For an infinitely tall tower, this reduces to σ(x) = fsρg=k, which makes the compressive stress independent of height x, i.e. constant throughout the tower. The parameter k can be picked to fix the compressive stress in the structure and optimize the tower's dimensions. It is also worth mentioning that the stress is independent of the base area. 4 Figure 1: Tower specifications: L is the tower's total height, x the height of a considered cross section, and Ac(x) the cross-sectional area of the concrete at height x. F (x) is the total force applied on a given cross section of height x by the weight of the above concrete. 2.2. Buckling Buckling is the sudden change in shape of a structure under load. For columns that means bending or bowing under a compressive load. If an applied load reaches the so called critical load, the column comes to be in a state of un- stable equilibrium - the slightest lateral force will cause the column to suddenly bend, which decreases the carrying capacity significantly and likely causes the the column to collapse. The tower in our model is essentially a column and as such, buckles under its own weight at a certain height, also known as self-buckling. To find this critical height, we need to derive the stability conditions for the tower's specific geometry. For towers with uniform cross section, that is towers that do not get thinner 5 with height (k = 0), Greenhill [13] found that the critical self-buckling height is EI 1=3 Lc ≈ 7:8373 (7) ρgAc where E is the elastic modulus, I is the second moment of area of the beam cross section, ρ is the density of the material, g is the acceleration due to gravity and Ac is the cross-sectional area of the body [13]. Self-Buckling of a column of non-uniform cross-section (k>0) The Euler{Bernoulli theory, also known as the classical beam theory, pro- vides means of calculating the deflection behaviour of beams. The theory entails the bending equation, which relates the bending moment of a beam or column to its deflection: d2y M(x) = −EI (8) dx2 where M(x) is the bending moment at some position x, E is the elastic modulus, I(x) is the second moment of area of the beam's cross-section at x, y(x) describes the deflection of the beam in the y-direction at x.