The Moon has been regarded as a natural Weber bar capable of amplifying gravitational waves (GWs) for detecting events across a wide range of frequencies. However, accurately determining the amplification effects remains challenging due to the absence of 3D numerical simulation methods. In this study, we develop a high-order 3D finite element method (spectral-element method, SEM) to numerically simulate the lunar response to GWs below 20 mHz. We verify the accuracy of our method by comparing the resonant peaks of our results with those from semi-analytical solutions and find that the frequency deviation is less than 3% for the first peak at about 1 mHz and less than 0.8% for the subsequent peaks up to 10 mHz. Using this method, we evaluate the amplification of GW signals due to 3D topographic effects of the Moon, and we find enhancements at a series of specific frequency components. These results highlight the non-negligible effect of surface topography on the lunar response to GWs, as a fundamental factor that holds significant implications across both global and regional analyses. Our work paves the way for a comprehensive evaluation of the Moon's resonant response to GWs, helpful for the strategic planning of lunar GW detections.
The Moon is thought to have greater potential for amplifying the gravitational-wave (GW) signals in decihertz frequency band, compared with ground-and space-based laser interferometry [1][2][3][4][5], and pulsar timing array [6][7][8][9], especially for detecting astrophysical sources caused by supernovae, compact binaries, intermediate-mass black holes, intermediate mass-ratio inspirals, and stochastic GW backgrounds [10][11][12]. As the ideas of detecting GWs on the Moon become increasingly popular [13][14][15], strategic premission planning-encompassing site selection, instrument configuration, parameter optimization, and landing zone assessment-has become imperative and must be tightly coupled with specific scientific objectives to ensure mission success. However, it remains unclear where the ideal landing zone is for deploying GW detectors, due to the difficulty in evaluating the actual response to GWs on the Moon.
Previous studies have considered the layered interior and/or subsurface structures of the Moon [16][17][18][19][20], and showed that the Moon has great potential for amplifying the GWs at multiple resonant frequencies [18,20]. However, their approaches are based on ideally spherical models thus cannot include the effect caused by fluctuating topography and crustal thickness of the Moon. To solve this issue, Zhang et al. [21] employed a two-dimensional (2D) FEM to numerically simulate the Moon’s response to GWs. Their work points to a new way of accurately simulating the realistic lunar response to GWs in the future. However, since their model is 2D, it could not account for real three-dimensional (3D) amplification effects of both topographic fluctuations and interior lateral heterogeneity of the Moon. Consequently, the resonant peaks from their 2D simulations apparently deviate from the 3D semi-analytical results [21]. Therefore, it is necessary to develop 3D FEM to clarify the actual lunar response to GWs.
Here, we construct a 3D lunar FEM model and develop a high-order FEM (spectral-element method, SEM) to numerically simulate the Moon’s response to GWs. We also compare our numerical results with semi-analytical solutions (Section II) to verify the accuracy of the proposed method. Then, we evaluate 3D topographic effects by comparative analyses of the lunar response to GWs with and without the incorporation of realistic lunar topography (Fig. 1, Section III). In addition, we discuss the limitations of our current method and point out potential directions for future development (Section IV). Finally, we draw a conclusion to this work (Section V).
The theory of calculating the force density imposed by GW on an elastic body was first laid down by Dyson [22], who introduced a coupling term between GW and elastic body and derived an external-force density, 𝑓 ⃗ ∇ ⋅ 𝜇𝐡 , 1 where 𝜇 is the shear modulus and h refers to 3D spatial components of the GW tensor. Based on the previous studies on analytical and semi-analytical solutions of planet’s response to GWs [18,23,24], the GW-induced elastic force density acting on the Moon can be expressed as are the orthogonal unit base vectors for GW propagating along 𝒆 .
Based on a spherically layered model [18,21], we add the force densities (i.e., surface force density and body force density) at the interface where the shear modulus and S-wave velocity vary significantly. We also add force vectors at the lunar surface in accordance with the discontinuous boundary conditions.
Here we describe the 3D SEM which we use to simulate the lunar response to GWs. Fig. 2 presents the spectral-element model constructed for the entire Moon and illustrates the key parameters of the GW simulation setup. Our numerical model and simulation are based on ABAQUS [25] and SPECFEM3D codes [26][27][28] and we incorporate GW-generated forces into the simulation. A global 3D lunar model with 44,512 fourth-order spectral elements has been constructed [Fig. 2(a)]. This model is adapted from references [18,21]. Up to 800 receivers are uniformly set on the lunar surface for recording the lunar response to GWs. 3). To cover a broad frequency range [21], the Gaussian wavelet STF with a halfduration of 50 s is adopted [21]. The total simulation time duration is up to 120,000 s (Fig. 3), enabling seismic waves to propagate multiple times through the entire Moon volume. From the perspective of numerical stability, the maximum allowable time step is approximately 0.2 s. The 3D numerical simulation was conducted on 256 cores in parallel mode at the National Supercomputing Center in Wuxi, China, whose wall-clock time is about 1.5 hours.
Fig. 4 shows the displacement amplitudes of the lunar surface in response to GWs at several representative time steps, in both 3D and 2D visualizations. The seismic waves induced by GWs of a given polarization can propagate throughout the entire Moon, resulting in global-scale oscillations. This process excites the Moon’s normal modes, which can be obse
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