4.1 Analytical Solution
In the DTM data published by Giese, Denk et al. (2008), there are 10 profiles numbered r1 to r10, scattering on the Cassini Regio. The sites of the profiles are shown in Fig. 4-1. Profile r1 and r2 only lies on the northern side of the equatorial ridge, and the other profiles are cutting through the ridge. However, r9 is excluded from our modeling since the ridge area in r9 is devastated by cratering. In this study,
Fig. 4-1 The site of DTM profiles. Modified from PIA08406 (courtesy of
NASA) and Giese, Denk et al. (2008).
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the excess load originated from the sunken part of the mountain is ignored in order to simplify condition setting. Thus, the estimated width (W) and height (H) of the ridge are obtained from the DTM profile, and then we use Eq. 3-24 to compute the vertical load (q0). The values of H, W and q0 for every profile are listed in Table 2. In the other hand, we select the basic properties of Iapetus’ material below: density (ρ)
= 1088 kg/m3 (which is the same as the average density of Iapetus); Young’s
modulus = 9×109 Pa, Poisson’s ratio = 0.33 (both based on the study of ice published by Schulson (2001)).
Table 3 Parameters of The Equatorial Ridge for Every Profile
Profile Number Ridge Width (km) Ridge Height (m) Vertical Load (N/m)
r1 74.8* 7032 6.34×1010
r2 119.2* 8983 1.29×1011
r3 98.4 6820 8.09×1010
r4 120.8 7158 1.04×1011
r5 117.6 6818 9.66×1010
r6 157.2 5604 1.06×1011
r7 167.5 6303 1.27×1011
r8 69.0 8398 6.98×1010
r10 63.5 6561 5.02×1010
*: The value is from the doubled width of the north wing because of the lack of DTM data of the ridge’s southern wing.
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Fig. 4-2 to 4-10 show the DTM ridge profile and the deflected surface. In the computation of the deflected surface, the elastic lithospheric thickness is set to 5, 10, 20, 50, 100, 200 km respectively. But these figures will show only up to 100-km thickness since an over 100-km thickness will not cause a significant deflection.
Fig. 4-2 Analytical Flexural Modeling of Profile r1.
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Fig. 4-4 Analytical Flexural Modeling of Profile r3.
Fig. 4-5 Analytical Flexural Modeling of Profile r4.
Fig. 4-3 Analytical Flexural Modeling of Profile r2.
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Fig. 4-7 Analytical Flexural Modeling of Profile r6.
Fig. 4-8 Analytical Flexural Modeling of Profile r7.
Fig. 4-6 Analytical Flexural Modeling of Profile r5.
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These results show at least four noticeable truths:
1) The modeling results of 100-km elastic thickness agree with the previous studies (Dombard et al., 2012; Giese, Denk et al., 2008), proving the validity of the analysis model.
2) Every profile has a bulge at 70-120 km away from the equatorial ridge. The bulge position correlates with the modeling result of 5-km elastic thickness.
3) The height of the bulge ranges from 1 km (r6 & r7) to 7 km (r4). Obviously, Fig. 4-10 Analytical Flexural Modeling of Profile r10.
Fig. 4-9 Analytical Flexural Modeling of Profile r8.
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deflection causes a hundred-meter to 1-km high bulge, but cannot make an ultrahigh uplift. The high bulge is not simply originated from the point load of the ridge.
4) Except the ridge and the bulges, most of the surfaces are affected by the impact craters. For instance, a 3-km deep crater is located in the southern side of profile r8. Large crater directly changed the surface topography a lot, and r4 is the good example since its southern side was devastated by a large impact event. Such a large event would destroy any previous flexure signal caused by the ridge.
4.1.1 Geomorphological Constraints of Elastic Thickness
In fact, we can estimate the elastic lithospheric thickness by the distance between the bulge and the loading point. The peak of the bulge has the distance away from the ridge of 100-200 km from the each profile.
Upon differentiating Eq. 3-31 and setting the result to 0, the x value that yields the maximum w value is obtained:
𝑑𝑤 appropriate variability of Young’s modulus form 1010 to 109 Pa yields the proper range of the elastic thickness with 5-10 km. Thus, if the bulge is made by the ridge load, the elastic thickness should be much thinner than the previous studies. It is significant because the first modeling using DTM data (this study) overrides the
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former elastic lithosphere calculation on Iapetus. We will discuss more, including the outcomes and deduction in the next chapter.
4.2 Numerical Solution
4.2.1 Uniform Thickness
The numerical flexural model of the equatorial ridge has been done in the same material conditions with the analytical model mentioned in section 4.1. The length in our 1-D model is set to 300 km spanning from the both side of the ridge. (That is, the total length is 600 km.) In a numerical analysis with a boundary-conditioned
differential problem, the amount of the map grid enlarges or shrinks the value of the solution. So, we compare the analytical and the numerical solution under the same condition (point load) as a pre-test. After the pre-test, the amount of the map grid is set to 200000 because it correlates well with the analytical model. The width of every gird is 3 m (600 km / 200000 grid points). Fig. 4-11 to 4-19 will show the computed deflected surface of the ridge profile separately. The dashed line indicates the load map (q(x)) the numerical model used. Note that these figures don’t show the deflected curves originating from 50-km, 100-km and 200-km elastic shell since these curves are so flat that the flexure effect can be ignored.
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Fig. 4-12 Numerical Flexural Modeling of Profile r2.
Fig. 4-13 Numerical Flexural Modeling of Profile r3.
Fig. 4-11 Numerical Flexural Modeling of Profile r1.
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Fig. 4-15 Numerical Flexural Modeling of Profile r5.
Fig. 4-16 Numerical Flexural Modeling of Profile r6.
Fig. 4-14 Numerical Flexural Modeling of Profile r4.
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Fig. 4-18 Numerical Flexural Modeling of Profile r8.
Fig. 4-19 Numerical Flexural Modeling of Profile r10.
Fig. 4-17 Numerical Flexural Modeling of Profile r7.
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Because of the bounded edges, the numerical results react smoother than the analytical results. But the bulge area appears in the same position with former results. Moreover, the characteristics of the finite element method is amplifying the highly deflected part and flatting the minor deflected part. So, deflection caused by over 20-km elastic layer is hardly observed.
4.2.2 Variable Thickness
In the cratered ridge scenario, the impact would have heated Iapetus’ surface.
Therefore, the elastic lithospheric thickness would have been thinned, or more critically, decreased to zero. In this paper, we use r4 profile as an example of the scenario. A large crater is located on the south of the profile (e.g. Fig. 4-5). Hence, we set the elastic thickness to 5 km for the northern side and 1 km for the southern side of the ridge. Under the same material properties mentioned before, the modeling result is illustrate in Fig. 4-20.
Fig. 4-20 Modeling of Profile r9 with Various Lithospheric Elastic Thickness.
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In Fig. 4-20, the shape of the ridge is simplified to a symmetrical triangular with a height of 13 km and a width of 120 km (shown as a dashed line on Fig. 4-20). The modeled surface is highly correlated with the ridge area in r4 profile, so it’s plausible that the profile was created by 2 factors: flexure and cratering. Deflecting can form the depression of the northern ridge, and an asteroid heavily bombarded the southern part of the ridge. But the most northern bulge “plateau” area remains unexplainable since it’s too high to be constructed by only flexure. More details of numerical modeling will also be discussed in the next chapter.
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