3. Enceladus
3.3 Shell Thickness
Eq. (2-65) shows how to calculate the shell libration amplitude for a decoupled system, where the coefficients 𝜔𝜔𝑖𝑖2 and 𝛤𝛤�𝑧𝑧 depend on the mass distribution of the two parts of the system. In models for Enceladus harboring a global subsurface ocean, both the inner and outer solid regions include part of the fluid ocean. In this section, we demonstrate all the components needed for obtaining the theoretical libration amplitude in terms of the model parameters.
We divide the ocean into three parts with two spheres of radius 𝑟𝑟1 and 𝑟𝑟2 (see Figure 3-5 and 3-6) so that the spherical region in between exerts no torque on the shell or the core. The value of 𝑟𝑟1 (𝑟𝑟2) is the maximum (minimum) radius of the quadratic surface of the core (inner shell). The maximum is at the tip of the largest-elongation axis 𝑎𝑎 (i.e., 𝜃𝜃 = 90° and 𝜙𝜙 = 0°) and the minimum is at the north/south pole (i.e., 𝜃𝜃 = 0°).
According to Eq. (2-27), the radii can be written as 𝑟𝑟1 = 𝑅𝑅𝑐𝑐�1 +𝛼𝛼𝑐𝑐
3 + 𝛽𝛽𝑐𝑐
2 � (3-4)
and
𝑟𝑟2= 𝑅𝑅𝑜𝑜�1 −2
3 𝛼𝛼𝑜𝑜�. (3-5)
Volume 1 contains the ocean interior to 𝑟𝑟1 and volume 2 contains the ocean exterior to 𝑟𝑟2. The liquid in volume 2 is co-rotating with the shell, relative to the rest parts including the core and volume 1. Analogous to the two-concentric-objects-model in Section 2.4.2.2,
Figure 3-5. Interior structure for studying Enceladus’ physical libration (not to scale).
There are three constant density layers: rocky core (brown), water ocean (deep blue and white), and ice shell (light blue). The ocean is further divided into three regions by two sphere 𝑟𝑟1 and 𝑟𝑟2, which are the smallest and the largest sphere in the ocean, respectively. Volume 1 (2) is the ocean interior to 𝑟𝑟1 (exterior to 𝑟𝑟2). Region 1 (2) is the combination of the core (shell) and volume 1 (2). This figure is adopted from Van Hoolst et al. (2008).
Figure 3-6. Interior structure for studying Enceladus’ physical libration (to the scale).
The mean values of the core radius is 𝑅𝑅𝑐𝑐 = 192.88 (𝑘𝑘𝑚𝑚) , the ocean depth is 42.91 (𝑘𝑘𝑚𝑚) , and the shell thickness is ℎ = 16.31 (𝑘𝑘𝑚𝑚) . This is the equatorial profile (i.e., 𝜃𝜃 = 90° ). The a-axis of the core and the shell are aligned, lying horizontally. The volume 1 and volume 2 are actually very small. The maximum thickness of volume 1 and 2 on the equatorial plane is 3.2 and 5.6 (𝑘𝑘𝑚𝑚).
region 1 – including the core and volume 1 – is the “core,” and region 2 – including the shell and volume 2 – is the “shell.” Even though the dynamics of the liquid in the librating object could be complicated (Lemasquerier et al., 2017), we assume a rather simple and idealized model to test our equations.
Regarding the coefficients 𝜔𝜔𝑖𝑖2 (Eq. 2-64) and 𝛤𝛤�𝑧𝑧 (Eq. 2-55), there are three factors to be elaborated with respect to their integration volume. First, the equatorial moment of inertia difference, which is related to 𝑄𝑄22 (Section 2.1.1), can be written as (Eq. D-3)
(𝐵𝐵 − 𝐴𝐴)𝑟𝑟𝑐𝑐𝑟𝑟𝑖𝑖𝑜𝑜𝑐𝑐1= 8𝜋𝜋
15 � 𝜌𝜌(𝑟𝑟)𝜕𝜕𝑟𝑟5𝛽𝛽(𝑟𝑟)
𝜕𝜕𝑟𝑟 𝑑𝑑𝑟𝑟
𝑟𝑟𝑐𝑐𝑟𝑟𝑖𝑖𝑜𝑜𝑐𝑐1
=8𝜋𝜋
15 � 𝜌𝜌(𝑟𝑟)𝜕𝜕𝑟𝑟5𝛽𝛽(𝑟𝑟)
𝜕𝜕𝑟𝑟 𝑑𝑑𝑟𝑟
𝜕𝜕𝑐𝑐
0 +8𝜋𝜋
15 � 𝜌𝜌(𝑟𝑟)𝜕𝜕𝑟𝑟5𝛽𝛽(𝑟𝑟)
𝜕𝜕𝑟𝑟 𝑑𝑑𝑟𝑟
𝑟𝑟1
𝜕𝜕𝑐𝑐
= 8𝜋𝜋
15(𝜌𝜌𝑐𝑐𝑅𝑅𝑐𝑐5𝛽𝛽𝑐𝑐 − 𝜌𝜌𝑜𝑜𝑅𝑅𝑐𝑐5𝛽𝛽𝑐𝑐) (3-6) and
(𝐵𝐵 − 𝐴𝐴)𝑟𝑟𝑐𝑐𝑟𝑟𝑖𝑖𝑜𝑜𝑐𝑐2= 8𝜋𝜋
15 � 𝜌𝜌(𝑟𝑟)𝜕𝜕𝑟𝑟5𝛽𝛽(𝑟𝑟)
𝜕𝜕𝑟𝑟 𝑑𝑑𝑟𝑟
𝑟𝑟𝑐𝑐𝑟𝑟𝑖𝑖𝑜𝑜𝑐𝑐2
=8𝜋𝜋 The second term in Eq. 6) is due to the ocean in volume 1, and the first term in Eq. (3-7) is due to the ocean in volume 2. Second, the polar moments of inertia for region 1 and 2 are given by (e.g., Eq. 36 in Van Hoolst et al., 2009) Substituting Eq. (3-6) and (3-7) into (2-51), we arrive at the external gravitational torques exerted by Saturn on region 1 (core + volume 1) and region 2 (shell + volume 2) Substituting Eq. (3-6) and (3-10) into (2-55), the torsion constant (i.e., the internal torque) can be written as
𝛤𝛤�𝑧𝑧 =8𝜋𝜋𝐺𝐺
5 [𝜌𝜌𝑜𝑜𝛽𝛽𝑜𝑜+ 𝜌𝜌𝑠𝑠(𝛽𝛽𝑠𝑠− 𝛽𝛽𝑜𝑜)] �8𝜋𝜋
15 𝑅𝑅𝑐𝑐5𝛽𝛽𝑐𝑐(𝜌𝜌𝑐𝑐− 𝜌𝜌𝑜𝑜)�. (3-13)
If we consider Saturn’s oblateness, the conventional gravitational torque Eq. (2-51) need to be modified by a factor (see Eq. 2-50)
Δ ≡ �1 +5 2 𝐽𝐽2�𝑅𝑅
𝑑𝑑�
2
� (3-14)
which was neglected in Section 2.4.2 about the physical libration. Given the reference radius for Saturn’s gravity field 𝑅𝑅 = 60,330 𝑘𝑘𝑚𝑚 , Saturn’s gravitational coefficients 𝐽𝐽2 = 1.6291 × 10−2 (Anderson and Schubert, 2007), and the orbital semi-major axis of Enceladus 𝑎𝑎 = 2.38 × 105 𝑘𝑘𝑚𝑚 (Table 1.4 in de Pater and Lissauer, 2015) in place of 𝑑𝑑 , we get the value Δ ≈ 1.0026 . We treat it as a constant and multiply it with the aforementioned components related to the external gravitational torque (e.g., Eq. 3-6 and 3-7) in order to introduce the effect due to Saturn’s oblateness.
For each interior model, we apply Eq. (2-65) to solve for the shell libration amplitude, which is used to compared with the observed value 0.120° ± 0.014° (Thomas et al., 2016) in order to estimate the shell thickness (Figure 3-7). Given 𝜌𝜌𝑠𝑠 = 920 𝑘𝑘𝑘𝑘/𝑚𝑚3, the estimated shell thickness is 16.24 𝑘𝑘𝑚𝑚 when Δ = 1 and 16.35 𝑘𝑘𝑚𝑚 when Δ = 1.0026.
Given 𝜌𝜌𝑠𝑠 = 950 𝑘𝑘𝑘𝑘/𝑚𝑚3, the estimated shell thickness is 15.88 𝑘𝑘𝑚𝑚 when Δ = 1 and 15.99 𝑘𝑘𝑚𝑚 when Δ = 1.0026. The estimated thickness is increased by less than 1% after considering Saturn’s oblateness. The discrepancy is almost negligible, so we take 𝜌𝜌𝑠𝑠 = 950 𝑘𝑘𝑘𝑘/𝑚𝑚3 and Δ = 1 to discuss the feature in the librating system.
Since the shape of the inner and outer surface of the shell is different (see Figure 3-4), the shell thickness depends on the co-latitude 𝜃𝜃 and longitude 𝜙𝜙. Given the mean shell thickness of 16.31 𝑘𝑘𝑚𝑚, the variation in the shell thickness is plotted in Figure 3-8.
The thickness is the smallest at the polar region and increases toward the equatorial region.
The variation is the largest along the arc that pass through the a-axis (𝜙𝜙 = 0°), while smallest through b-axis (𝜙𝜙 = 90°).
The trend that the libration amplitude increases with the decreasing shell thickness is the result of resonance (Figure 3-9). Resonance occurs when one of the eigenperiods approaches the forcing period. Eigenperiods depend on the polar moment of inertia 𝐶𝐶, the equatorial moment of inertia difference (𝐵𝐵 − 𝐴𝐴) , and the coupling factor 𝛤𝛤�𝑧𝑧 . Assuming the coupling is negligible, we revert back to a simple case where the eigenperiod is given in Eq. (2-61), which is proportional to the squared root of the ratio
𝐶𝐶/(𝐵𝐵 − 𝐴𝐴) . If there were no subsurface ocean, the eigenperiod would not have been decreasing so rapidly. 𝐶𝐶𝑠𝑠 and (𝐵𝐵 − 𝐴𝐴)𝑠𝑠 would decrease simultaneously as the icy shell becomes thinner (Figure 3-10 and 3-11). If the ocean is present, the upper-most ocean (i.e., volume 2) serving as part of region 2 would slow down the decreasing trend of (𝐵𝐵 − 𝐴𝐴)2 as well as 𝐶𝐶2. Yet, the extent to which 𝐶𝐶2 decreases is still much larger than (𝐵𝐵 − 𝐴𝐴)2, so the eigenperiod would decrease with the thinner icy shell.
Figure 3-7. Libration amplitude and shell thickness.
The libration amplitude is calculated based on the conventional torque (Δ = 1; solid line) and the torque including Saturn’s oblateness (Δ = 1.0026; dashed line). Two sets of density models are used, where the shell densities are 920 𝑘𝑘𝑘𝑘/𝑚𝑚3 (blue) and 950 𝑘𝑘𝑘𝑘/𝑚𝑚3 (red). The observed shell libration amplitude is 0.120° ± 0.014°
(Thomas et al., 2016; black line). Given 𝜌𝜌𝑠𝑠 = 920 𝑘𝑘𝑘𝑘/𝑚𝑚3 , the estimated shell thickness is 16.24 𝑘𝑘𝑚𝑚 when Δ = 1 and 16.35 𝑘𝑘𝑚𝑚 when Δ = 1.0026. Given 𝜌𝜌𝑠𝑠 = 950 𝑘𝑘𝑘𝑘/𝑚𝑚3 , the estimated shell thickness is 15.88 𝑘𝑘𝑚𝑚 when Δ = 1 and 15.99 𝑘𝑘𝑚𝑚 when Δ = 1.0026 . The estimated thickness is increased by less than 1%
after considering Saturn’s oblateness.
Figure 3-8. The variation in shell thickness.
The mean shell thickness 16.31 (𝑘𝑘𝑚𝑚) is chosen to demonstrate the variation. The variation is the largest along the arc that pass through the a-axis (𝜙𝜙 = 0° ), while smallest through b-axis (𝜙𝜙 = 90°).
Figure 3-9. Eigenperiods and shell thickness.
The forcing period 𝑇𝑇 (black line) equals to the orbital period. The eigenperiod 𝑇𝑇2
(blue line) is sensitive to the shell thickness and decreases as the shell becomes thinner. The crossing of 𝑇𝑇2 and forcing period indicates resonance amplification of the eigenmotion (i.e., the shell libration).
Figure 3-10. Equatorial moment of inertia difference and shell thickness.
Since moment of inertia is directly proportional to the mass, the equatorial moment of inertia difference of the shell (𝐵𝐵 − 𝐴𝐴)𝑠𝑠 (dashed blue line) decreases as the shell becomes thinner. If the ocean in volume 2 is accounted, it will fill in the void due to the thinner shell and sustain the high values of (𝐵𝐵 − 𝐴𝐴)2 (solid blue line). The ocean in volume 1 acts in the opposite way on the (𝐵𝐵 − 𝐴𝐴)𝑐𝑐 (dashed red line), so (𝐵𝐵 − 𝐴𝐴)1
(solid red line) is lower.
Figure 3-11. Polar moment of inertia and shell thickness.
The polar moment of inertia of the core 𝐶𝐶𝑐𝑐 (solid red line) is bigger than the shell 𝐶𝐶𝑠𝑠
(solid blue line) due to its large size. Since moment of inertia is directly proportional to the mass, 𝐶𝐶𝑠𝑠 decreases as the shell becomes thinner. When considering the ocean in volume 2, the decreasing trend of 𝐶𝐶2 (dashed blue line) is slow but still significant.
Compared to 𝐶𝐶𝑐𝑐, the effect of volume 1 on the 𝐶𝐶1 (dashed red line) is small.