Estimation of the Needed Regolith for Covering Lunar Habitat by Protective Layer

The type of a structure chosen for the analysis is based on a thin concrete shell supported from the inside by air pressure. To counterbalance the air (which wants to fly out the “balloon” of the created habitat), the weight of the structure as well as the protective overlay of lunar soil should be used. Regolith will be used to create a concrete-like composite for the creation of a thin-wall structure of the habitat and for the protective overlay. The overlay will play multiple roles: protection against space radiation, thermal insulation, and a layer absorbing small meteorites (and preventing ricochets). Such a strategy based on the structure made from concrete-like material and regolith overlay was discussed preliminarily in a previous publication (Konecny and Katzer, 2021). The structural performance is related to the dimensions that depend on the size of the habitat. Using the recommendations of Ruess, Schaenzlin and Benaroya (2006), the effective height of the storey of 3.5 m was chosen for further analyses. To limit the height of the habitat, it was decided to adapt the design based on a half sphere (as the ending) and a half cylinder (as the central part). In this way, the size of the habitat can be easily extended (by creating a longer central half cylinder part) without changing its height. In case of traditional dome, the number of people would influence its overall dimensions (including height). From civil engineering point of view, the larger the height, the more complicated the construction process will be.

The conducted computations were divided into two stages: a) half sphere and b) half cylinder. The cylindrical part is a straight part with the cross section defined by a half circle forming a horizontal half cylinder with the radius r and length d. The half sphere is represented by two quarters of a sphere that form the ending segments of the habitat on both ends of the straight part. The sphere has also the radius r. The scheme of the habitat is presented in Fig. 1, while the cross section of the proposed habitat is presented in Fig. 2.

Figure 1.

Figure 2.

The effective floor area is computed first. It is based on the known radius of the spherical part r_Str and considered effective height of the habitat storey h. The effective usable area of the spherical cupola is computed based on the formula of the y coordinate (center line of the structure, see Fig. 1): (1) y=2rStrx−x2, y = \sqrt {2{r_{Str}}x - {x^2}} , which shall be larger than the limiting value of the storey height h. This is fulfilled for x > x_h as follows: (2) xh=2rStr−4rStr2−4h22, {x_h} = {{2{r_{Str}} - \sqrt {4r_{Str}^2 - 4{h^2}} } \over 2}, which is the second solution of quadratic formula.

The floor area is computed for the cylindrical part and the spherical parts separately. Therefore, the sum of floor areas is computed as follows: (3) SE=SE,d+SE,c, {S_E} = {S_{E,d}} + {S_{E,c}}, where the area of the rectangular floor projection of the cylinder is S_E,d: (4) SE,d=2drStr−xh {S_{E,d}} = 2d\left( {{r_{Str}} - {x_h}} \right) and the area under the circular floor projection of the cupola is computed from the projections of two quarters of the sphere S_E,c yielding: (5) SE,c=πrStr−xh2. {S_{E,c}} = \pi {\left( {{r_{Str}} - {x_h}} \right)^2}.

The volume of the habitat's living space V_Str,E can be computed from the floor area S_E by multiplication by height h as follows: (6) VStr,E=SE⋅h. {V_{Str,E}} = {S_E} \cdot h.

The suitable number of inhabitants np is derived based on eq. (3) from the necessary floor area for one inhabitant S_n: (7) np=sEsn. {n_p} = {{{s_E}} \over {{s_n}}}.

If the desired number of inhabitants np and the radius of the structure r are given, then the above-mentioned computations shall be repeated iteratively with a variable length of the straight part d until satisfactory result is achieved.

The regolith volume V_Rl is obtained as a computation of the total volume of the habitat, including the regolith cover, V_Tot, without the volume of the structure itself V_Str: (8) VRl=VTot−VStr. {V_{Rl}} = {V_{Tot}} - {V_{Str}}.

The cylindrical part and the spherical parts are computed separately in a manner similar to that of floor area. Therefore, the sum of volumes, either for the structure or the total volume, is computed as follows: (9) VRl=Vd+Vc, {V_{Rl}} = {V_d} + {V_c}, where the volume of the half cylinder is V_d: (10) Vd=πr22⋅d {V_d} = {{\pi {r^2}} \over 2} \cdot d and the volume of the cupola formed from two-quarters of the sphere V_c is (11) Vc=2πr33. {V_c} = {{2\pi {r^3}} \over 3}.

It is worth mentioning that the radius of the cupola or the cylinder stands for r_Str in case of V_Str computation or for r_Tot in case of V_Tot computation, as shown in in equations (10) and (11).

The volume of regolith is computed as follows: (12) VRl=π2rhTh+hTh22⋅d+2π3r2hTh+3rhTh2+hTh33. {V_{Rl}} = {{\pi \left( {2r{h_{Th}} + h_{Th}^2} \right)} \over 2} \cdot d + {{2\pi \left( {3{r^2}{h_{Th}} + 3rh_{Th}^2 + h_{Th}^3} \right)} \over 3}.

The last parameter to be computed is the volume of the supporting systems, storage, and equipment V_Supp, that is: (13) VSupp=VStr−VStr,E. {V_{{\rm{Supp}}}} = {V_{Str}} - {V_{Str,E}}.

Computations were based on the input parameters given in Table 1. Output parameters of the calculated model of habitat are given in Table 2. The necessary volume of the regolith depending on the thickness of the cover to be excavated for the lunar habitat V_Rl occupied by four inhabitants is presented in Table 3.