are the caverns in mined asteroids suitable for space ...in particular, in situ asteroid mining is...

Are the caverns in mined asteroids suitable for space colonies with artificial gravity?

Grant No: MBR036

Final summary T. I. Maindl, B. Loibnegger, R. Miksch

Department of Astrophysics, University of Vienna, Austria January 15, 2019

1. Executive summary Several suggestions regarding the design and implementation of colonies inside of mined asteroids and comets already exist. The “shell” of such a space station would shield cosmic radiation. A necessity for long-term colonies is (artificial) gravity. Present studies include habitats in rotating wheels or tori that create gravity (Grandl & Bazsó 2013). Other studies somewhat vaguely mention ‘augmenting the natural rotation with additional artificial rotation’ (Taylor et al. 2008). We elaborate on the latter and explore the feasibility of creating artificial gravity for a habitat by putting the entire asteroid to rotation. The novelty in our approach is to investigate whether the asteroidal hull – once set to rotation as a whole – can sustain the material loads resulting from a sufficiently high rotation rate. While constructing a spinning wheel/torus in a cavern inside an asteroid is a demanding engineering task involving to construct bearings which sustain the rotation rate over the life-time of the station, setting the whole parent body to rotation will avoid many problems. Gravity would then act directly on the interior surface of the cavern. This approach involves challenges of its own, mostly connected with the rotation rate required for sustaining adequate artificial gravity. We investigate the stability of a rotating asteroid with a substantial cavern. For estimating the material loads we built two analytical models based on a simplified configuration of a spheroidal asteroid with a cylindrical cavern inside. The models differ in the assumed surfaces the centrifugal forces act upon – model 1 considers tensile stress only, model 2 both tensile and shear stress. The material loads acting on the hull of the cavern inside a mined asteroid are checked numerically by simulating a typical scenario with an adapted version of our own 3D smooth particle hydrodynamics (SPH) hypervelocity impact code. We find that the material loads resulting from centrifugal forces are of the order of magnitude of material strength of rock or even less. This renders a space station in the cavern of a mined rocky asteroid feasible. However, dimensions need to be chosen right and the material composition and material strength of the asteroid have to be known to sufficient accuracy. From comparing our analytical models with numerical simulations, we conclude the need for thorough numerical studies based on accurate material composition, mass distribution, and porosity data before a specific asteroid can be considered as a rotating parent body for a habitat with artificial gravity inside. As a next step, a study focusing on possible spin-up processes will be beneficial and will constitute a project comparable in effort and cost to the presented one. Questions such a study will answer are “How much fuel will it take to spin up?”, “What is a feasible time frame for spinning up?”, and

Grant No: MBR036, Final summary, T. I. Maindl, B. Loibnegger, R. Miksch

2

“Can transport starts and landings during the mining operations contribute to building up the body’s angular momentum contributing to the spin-up?”.

2. Main results It is well established that space settlements will depend on resources that are produced in space. In particular, in situ asteroid mining is key for space habitation. Due to the very low gravity on asteroids of a few hundred meters in diameter the perhaps most promising mining concept is based on underground mining. After completing such a mining operation, a ‘hollow’ asteroid is left. This project investigates the suitability of such a large cavern as a habitat or space colony. One of the biggest challenges is establishing gravity. Rather than focusing on building a rotating torus-shaped ‘space station’ inside the cavern we investigate the stability of the hollow asteroid when set to rotation at a rate that provides a sufficient fraction of Earth’s gravity for the inhabitants. We start with estimates from an analytical study and continue with numerical simulations with our own SPH (smooth particle hydrodynamics) code that allows modeling different ductile and brittle materials with varying porosity. The latter will be necessary due to the limited amount of information we currently have on asteroid composition and exact material properties.

1.1. Analytical study In the following we briefly describe the developed models and refer to our submitted manuscript for the detailed calculations (Maindl et al. 2018). In that manuscript we present how to estimate the necessary spin rate and discuss the implications arising from substantial material stress given the required rotation rate. We develop two analytical models for estimating the resulting material stress and apply them to fictitious, yet realistic rocky near-Earth asteroids that might be the target for asteroid mining operations. The analytical models are based on an idealized setup as follows: a spheroidal asteroid with semi-axes a and b has a cavern that resembles a cylinder with radius rc and height hc, respectively (see Fig. 1). The asteroidal material is assumed to be homogeneous and the cylinder’s symmetry axis and the semi-major axis a of the spheroid coincide with the y-axis so that artificial gravity acts on the cavern’s inside lateral surface if the asteroid is rotating about the y-axis. Depending on the spin rate, a volume element of the asteroid will be subject to a centrifugal force denoted by F/2, acting in the xz-plane (only x is shown in Fig. 1). The lateral distance between the asteroid’s surface and the space station is denoted by d.

Figure 1. Sketch of an asteroid with a cylindrical cavern, picture: Asteroid Vesta,

NASA/JPL-Caltech/UCAL/MPS/DLR/IDA (Maindl et al. 2018)


3

2.1.1. Rotation rate For a certain artificial gravity level gc on the lateral surface of the space station, the body has to rotate at a rate of ω = (gc/rc)½. Recent studies on the amount of gravity necessary for a person’s orientation judgement, astronauts’ ability to orient themselves, and for maintaining balance suggest that Martian gravity (38% of Earth’s surface value gE) should be enough (Harris et al. 2014). For space stations in the size range rc = 50...250 m, the required rotation rates for 38% of Earth’s gravity would be between 1.17 and 2.6 revolutions per minute. Figure 2 plots required rotation rates for a range of radii and artificial gravity levels.

2.1.2. Material stress We developed two analytical models that determine the stresses acting inside the rotating asteroid. For the analytic estimates we assume the asteroid to be composed of homogeneous material with density ρ. Model 1 focuses on tensile loads, model 2 considers tensile and shear stress based on different assumptions. Both result in a stress pattern that depends on the dimensions of the asteroid and the space station inside as well as on the rotation rate. Given a certain material strength and provided that the asteroid mining operations can be adjusted accordingly, the task is then to find the best size of the cavern that provides the desired artificial gravity while maximizing the usable lateral surface of the cylindrical space station. This lateral surface S is the usable area in the space station and is given by S = 2π rc hc.

2.1.2.1. Model 1 Model 1 estimates centrifugal forces acting on the asteroid material and exerting a load on an arbitrary symmetry plane. A certain tensile stress results from this load. The two halves of the asteroid are pulled apart “as a whole” by the centrifugal force F1. This force acts on the cross-

Figure 2. Rotation rates ω in rads per second and revolutions per minute (rpm) resulting in artificial gravity gc (expressed in units of the Earth’s surface gravity gE) on the lateral

surface of a cylinder of radius rc (Maindl et al. 2018).


4

section A being the interior of an ellipse with a carved out rectangle (the cylindrical cavern), the tensile stress acting on A is given by σ1 = F1/A. It turns out that σ1 can be parametrized such that it scales linearly with the material density ρ, the artificial gravity gc, and the asteroid’s semi-minor axis b if hc and rc are replaced by their dimensionless counterparts hc’ = hc /a and rc’ = rc /b, respectively:

Figure 3. Model 1 material stress for different space station sizes (dimensionless radius rc’ and height hc’). The blue contour lines give the ratio σ1/(ρ gc b) for ρ given in g/cm3, gc

measured in units of gE, and b measured in meters, respectively. The dotted red lines give contours of the ratio S/(ab), the upper x-axis gives the scaled rotation rate (Maindl et al.

2018).


5

Figure 3 plots this function f (the scaled tensile stress) for all possible hc and rc (shades of blue and blue contour lines) along with the scaled usable surface of the cylindrical space station S/(ab). An application of this diagram might be as follows: the material strength of a specific asteroid corresponds to a maximum tensile stress and – along with its material density ρ, semi-minor axis b, and desired artificial gravity gc – corresponds to a specific shade of blue. For that value, the cavern with the maximum usable area (red contours) is sought. Note however, that in this model the cavern is allowed to extend all the way to the asteroid’s surface (corresponding to d = 0 in Fig. 1) so that some safety margin needs to be allowed for.

2.1.2.2. Model 2 Rather than modeling two entire halves of the hollowed-out asteroid that are pulled apart by the centrifugal forces, this model studies the “mantle” outside the space station. This mantle is the solid torus created by sweeping the right part of the hashed surface between the red lines at y = ±hc/2 in Fig. 4 around the y-axis. The mantle is subject to centrifugal forces that exert tensile stress on the asteroid material along the symmetry plane (the hashed area in Fig. 5). Additional shear stress acts on the two annuli resulting from rotating the red lines in Fig. 4 about the y-axis. Calculating the total stress is more lengthy than in case of model 1 and results in a combined tensile and shear stress σ2 of

where F2 is the total centrifugal force acting on the mantle and At and As are the areas subject to tensile and shear loads, respectively. In contrast to the model 1 approximation, this model does not easily scale with the asteroid size, density, and desired artificial gravity. Therefore, a diagram will always be specific to the asteroid under consideration. An application example is given further below.

1.2. Numerical study In order to validate the predictions of the simplified models 1 and 2, we performed a series of smooth particle hydrodynamics (SPH) simulations of a typical configuration. We are primarily

Figure 4. Simplified sketch of cavern for clarification (Maindl et al. 2018)


6

interested in the load distribution inside the asteroidal body and use an adapted version of our own CUDA-parallel 3D SPH hypervelocity impact code (Maindl et al. 2013, Schäfer et al. 2016). It implements full elasto-plastic continuum mechanics, the p-α porosity model (Jutzi et al. 2008, Haghighipour et al. 2018), and a tensorial correction for first-order consistency (Schäfer et al. 2007). The material model used for this study is based on the Tillotson equation of state with parameters for basaltic rock as given in Haghighipour et al. (2018) and porosities varying between 0% (solid) and 75%. Each scenario is resolved in about 500k SPH particles. In each simulation run, the asteroid is set to rotation at the required rate and subsequently allowed to equilibrate during a 200 second time horizon with output frames every 0.5 seconds. As indicator for material load we use the von Mises stress which is defined as σe = (3J2)½ with J2 denoting the second invariant of the deviatoric stress tensor. For a more exhaustive description please refer to Maindl et al. (2019).

1.3. Application to a realistic asteroid We apply the analytic models 1 and 2 as well as the SPH-based numerical framework to a rocky asteroid with dimensions 500 m by 390 m. These values are inspired by a number of similar-sized rocky near-Earth asteroids such as 3757 Anagolay, 99942 Apophis, 3361 Orpheus, 308635 (2005 YU55), and 419624 (SO16) (cf. JPL 2018) that might serve as destinations of future mining missions. While little is known about the material properties and composition of these objects, we assume they are composed of basaltic rock with a non-porous bulk density of ρ = 2.7 g/cm3. Tensile strength values for basalt are in the range of approx. 12...14 MPa (Stowe 1969), shear strengths are approx. 8...36 MPa (Karaman et al. 2015), which provides an order-of-magnitude framework of the expected material strength data. Finally, we will assume a desired artificial gravity level of gc = 0.38 gE as discussed above. The diagram in Fig. 5 shows the resulting tensile stress along with the space station surface area and required rotation rates predicted by model 1. The stress levels are mostly of the same order of magnitude as the assumed material strength (about 10 MPa) or even smaller. However, the solution resulting in the maximum area S ~ 0.3 km2 would have the cylindrical cavern extend to the asteroid’s surface, which seems unrealistic and will lead to the asteroid becoming unstable. Also, for realistic scenarios a material stress (about 4 MPa in this case) very close to the – poorly constrained – material strength will be unacceptable so that a cavern size resulting in less tensile stress will be desirable (more towards the lower-right of the diagram). Investigating the combined tensile and shear stresses according to our model 2 results in a different stress-pattern, depicted in Fig. 6. While material loads are systematically lower for space stations deeper inside the asteroid in their entirety, stresses for “thinner” tori (i.e., larger rc) are of the same order of magnitude as predicted by model 1. At the same time, the stresses tend to increase in value if rc gets close to touching the asteroid’s surface, which discourages stations that extend up to near the parent body’s surface.


7

As the analytical models differ to quite some extent, we test them with numerical simulations. As mentioned above, we use our own 3D SPH code to investigate the behavior of a rotating spheroidal asteroid with a cylindrical cavern. We chose one scenario where the cavern is well inside the parent body (see Table 1). In addition, we vary the rocky material’s porosity between 0% (solid) and 75% porous. The resulting bulk densities along with the material loads predicted by models 1 and 2 are given in Table 2.

Figure 5. Model 1 results for gravity gc = 0.38 gE in a space station of radius rc and height hc. The color code and the blue contour lines give the tensile stress σ1 resulting from the required

rotation rate (a function of gc and rc). The red dotted contours state the surface area of the space station S, the upper x-axis denotes the required rotation rate (Maindl et al. 2018).


8

Table 1. Scenario parameters

a b rc hc gc ω ω

[m] [m] [m] [m] [gE] [rad s-1] [rpm]

250 195 125 300 0.38 0.173 1.65

Figure 6. Model 2 results for gravity gc = 0.38 gE in a space station of radius rc and height hc. The color code and the blue contour lines give the combined tensile and shear stress σ2

resulting from the required rotation rate (a function of gc and rc). The red dotted contours state the surface area of the space station S, the upper x-axis denotes the required rotation rate

(Maindl et al. 2018).


9

Table 2. Material load predictions from models 1 and 2

Matrix density Porosity Bulk density Model 1 load Model 2 load

ρm ρ σ1 σ2

[kg m-3] [kg m-3] [MPa] [MPa]

2700 0% 2700 3.4 2.2

2700 20% 2160 2.8 1.8

2700 50% 1350 1.7 1.1

2700 75% 675 0.86 0.56 After the SPH scenarios equilibrate they result in a stress pattern inside the rotating asteroid. Figure 7 shows snapshots of the σe stress pattern in the xy symmetry plane of the asteroid for different material porosities. Qualitatively, the patterns look similar to each other, the biggest stresses occur at the corners of the cylinder and extend to the surface. Compared to the analytical model, with increasing porosity the stress drops way steeper in magnitude (cf. the scale of the color bars). While the solid material results approximately resemble the analytical values in Table 2, porous material yields significantly lower σe figures with the deviations reaching one order of magnitude for 75% porosity (cf. Table 2). The latter clearly indicates that the lower stresses are not a bulk density-effect alone but are also driven by porosity. In all cases, the simulations show that the stress patterns inside the asteroids are much more complex than what is assumed in the analytical models. Simple analytical models can still provide reasonable order-of-magnitude estimates for a limited number of cases. However, thorough numerical studies based on accurate material composition data are essential before a specific asteroid can be considered as a rotating parent body for a habitat with artificial gravity inside. In summary, our results show that both the analytical models and the numerical simulations predict material loads in the order of or less than the assumed material strength of rocky bodies. This holds especially for bodies made of competent rock. Thus, hollowed-out asteroids as parent bodies for potential space stations inside may be able to sustain the loads exerted by spin rates necessary for artificial gravity.


10

Figure 7. Color-coded pattern of σe in the xy symmetry plane of the rotating asteroid (note the

varying scales). The assumed material porosities are indicated in the center (Maindl et al. 2019).

1.4. More accurate analytical model The following analytical model is currently in preparation and will be published in Miksch et al. (2019). Assuming an asteroid as a symmetric rotating ellipsoid, a simple approximation is an egg-shaped object. Here, it is quite easy to compute the volume by integration in one dimension and letting the calculated area rotate around the axis of symmetry. The inserted volume of the cylindrical space station is then subtracted from the volume of the asteroid and the result is a simple approach of the volume of the remaining asteroid. In this case, the thickest location of the mantle is at half of the height of the cylinder (location of dmax in Fig. 8). Hence, the strongest force is acting here. In a real scenario, the stability of the cavern and therefore the asteroid itself depends on the density and mass of each location on the space-stations “floor“ K (see Fig. 8). The centrifugal force depends on the mass in motion and a possible solution to minimize the force acting is to simply shift the “floor” towards the surface of the asteroid while having the same


11

angular velocity. The inner surface now is not flat and therefore, the created g-force has not the same value along the height of the space station h. A range of g-force needs to be defined, which determines the degree of curvature (see Fig. 8). A bonus in shaping the cavern this way is the flexibility against the real shape of the asteroid. Since the curve can be divided in several small intervals, one can design the “floor” in a way such that the g-force has a constant value (resulting in kind of a step function). Another achievement of this method is the independence of the asteroids shape itself. The asteroid just has to have a stable rotation axis.

Figure 8. Schematic view on the asteroid. K is the “floor” of the space-station. By knowing the density of the asteroid, K can follow a law that the g-force acting can be held constant

in a defined range.


12

3. Obstacles or changes of direction during the project Our SPH code needed to be adapted to accurately simulate small, relatively rapidly rotating bodies with high precision. Background: the code was designed as a hypervelocity impact code and successfully tested and applied to collisions of bodies ranging in size from cm-sized objects, small to large asteroids, moons and protoplanets up to giant planetary collisions. Highly precise simulations of rotating bodies kept being a challenge. Resolution: the implemented tensorial correction dealing with rotating bodies was refined in the CUDA code-version and tested in a development/test cycle of about 1.5 months involving several dozen submits to the code repository. Consequence for this project: the code enhancement enabled us to numerically check the analytic results. By far most of the work was done by our partners in Tübingen, but we still needed to reduce the scope of this project and cut investigating how to set a mined asteroid to rotation.

4. Potential impact and opportunities for implementation of the results

We established that in principle a rocky asteroid in the considered size-range should be able to sustain rotation rates that provide sufficient artificial gravity for a space station built in its inside. Even though both the analytic and numerical results are rough estimates at best – partly because of the simplifying assumptions, partly because of the unknown material composition and density distribution inside the asteroid – the potential use of a mined asteroid as parent body to a space station should be considered early in the mining process. In that way, the shape of the cavern can be optimized for its later use as a habitat and to balance out density and/or shape effects to avoid tumbling of the asteroid. Our results – while not substituting detailed in-situ studies – provide a viable basis for deciding whether a potential candidate for asteroid mining may be suitable for housing a space habitat at an early stage of the selection process and the involved missions.

5. Conclusion and next steps We developed two analytical models for estimating the load acting on the material of a hollowed-out asteroid when the whole body rotates at a rate necessary for providing sufficient gravity. Additionally, we started to formulate a more accurate model geared towards minimizing the total material load from rotation which will be published in the near future. We applied the first two models to a realistic 500 m x 390 m sized asteroid with a substantial cavern and validated our results by executing a suite of numerical simulations of the same rotating rocky object where we also vary the porosity of the asteroidal material. The material stresses determined by the analytical models and the SPH-based numerical simulations are comparable to or less than assumed material strengths, which renders an asteroid-hosted space station in


13

principle feasible. We find an order-of-magnitude agreement between the analytical models and the numerical simulation for competent rock and significant deviations for porous material. The latter is a clear indication that bulk density alone is not enough to describe the behavior of porous material. This underlines the need for thorough numerical studies based on accurate material composition, mass distribution, and porosity data before a specific asteroid is considered as a rotating parent body of a habitat. Missions to candidate objects seem inevitable in order to determine these parameters. While the exact design of the cavern’s size and shape that will (a) cause the hull material to be strong enough to endure the material stress during rotation and (b) avoid tumbling of the spinning body, will have to be determined specifically for the individual asteroid. Hence, a study focusing on possible spin-up processes will be beneficial and will constitute a project comparable in effort and cost to the present one. Such a study will investigate methods of actually initiating the rotation at the required rate and will answer questions such as “How much fuel will it take to spin up?”, “What is a feasible time frame for spinning up?”, and “Can transport starts and landings during the mining operations contribute to building up the body’s angular momentum contributing to the spin-up?” Questions that may remain open until specific asteroids are visited include “How much of the regolith and/or loose gravel-like material on the surface will be lost due to the rotation?” and “How does that material loss influence the asteroid’s stability?”.

References This research received seed funding from the Dubai Future Foundation through the Guaana.com open research platform. References marked with an asterisk (*) were published under this grant. Grandl, W., Bazsó, Á, 2013, Near Earth Asteroids - Prospection, Orbit Modification, Mining and Habitation, in: Badescu, V. (ed.), Asteroids: Prospective Energy and Material Resources, 415-438, Springer, Berlin Haghighipour N., Maindl, T. I., Schäfer, C. M., Wandel, O. J., 2018, Triggering the Activation of Main-belt Comets: The Effect of Porosity, The Astrophysical Journal, 855, 60 Harris, L. R., Herpers, R., Hofhammer, T., Jenkin, M., 2014, How much gravity is needed to establish the perceptual upright?, PLOS One 9 JPL, 2018, Small Body Database (2018-11-10) Jutzi M., Benz, W., Michel, P., 2008, Numerical simulations of impacts involving porous bodies. I. Implementing sub-resolution porosity in a 3D SPH hydrocode, Icarus, 198, 242 Karaman, K., Cihangir, F., Ercikdi, B., Kesimal, A., Demirel, S., 2015, Utilization of the Brazilian test for estimating the uniaxial compressive strength and shear strength parameters, Journal of the Southern African Institute of Mining and Metallurgy 115, 185 – 192


14

(*) Maindl, T. I., Miksch, R., Loibnegger, B., 2018, Stability of a rotating asteroid housing a space station, arXiv:1812.10436 (https://arxiv.org/abs/1812.10436), submitted to Frontiers in Astronomy and Space Sciences (open access) (*) Maindl, T. I., Schäfer, C. M., Loibnegger, B., Miksch, R., 2019, Tensile loads in porous rotating asteroids with artificial caverns, submitted to 50th Lunar and Planetary Science Conference, The Woodlands, Texas, March 18–22, 2019 (online available Feb 1, 2019 on https://www.hou.usra.edu/meetings/lpsc2019/program/) Maindl T. I., Schäfer, C., Speith, R., Süli, Á., Forgács-Dajka, E., Dvorak, R., 2013, SPH-based simulation of multi-material asteroid collisions, Astronomical Notes, 334, 996–999 (*) Miksch, R., Loibnegger, B., Maindl, T. I., 2019, in prep. Schäfer, C., Riecker, S., Maindl, T. I., Speith, R., Scherrer, S., Kley, W., 2016, A smooth particle hydrodynamics code to model collisions between solid, self-gravitating objects, Astronomy & Astrophysics, 590, A19 Schäfer C., Speith, R., Kley, W., 2007, Collisions between equal-sized ice grain agglomerates, Astronomy & Astrophysics, 470, 733 Stowe, R. L., 1969, Strength and Deformation Properties of Granite, Basalt, Limestone, and Tuff at Various Loading Rates. Misc. paper C-69-1, Army Engineer Waterways Experiment Station Vicksburg, MS Taylor, T. C., Grandl, W., Pinni, M., Benaroya, H., 2008, Space Colony from a Commercial Asteroid Mining Company Town, AIP Conference Proceedings, 969, 934

are the caverns in mined asteroids suitable for space ...in particular, in situ asteroid mining is...

Documents