© ESO 2020

1 Introduction

The Cassini spacecraft has measured Saturn’s gravitational field to high precision using five of the 22 Grand Finale orbits (Iess et al. 2019). The accuracy of the even harmonics J₂, J₄, and J₆ has been significantly improved compared with the previous values detected by the flyby missions and by the earlier Cassini orbits (Jacobson et al. 2006). The higher-order harmonics J₈–J₁₂ were measured through precise Doppler tracking of the Cassini spacecraft. These new gravity data provide excellent observational constraints on Saturn’s internal structure. The Cassini Grand Finale gravity measurements reveal larger values for the gravitational harmonics J₆–J₁₀ with respect to the predictions by typical rigid-body interior models (Iess et al. 2019; Galanti et al. 2019). In order to reconcile the calculated gravitational harmonics with these large values of J₆–J₁₀, Iess et al. (2019) proposed two different approaches to introduce differential rotation into Saturn’s interior models. The first approach is to approximate the wind profile as differential rotation on cylinders and simultaneously optimize interior parameters as well as differential rotation profiles, for which the gravity measurements J₂–J₁₀ are well reproduced within the error bars. Militzer et al. (2019) further generalized this approach with an accelerated concentric maclaurin spheroid (CMS) method to explore Saturn’s interior and rotation. The second approach is to assume the interior shows rigid-body rotation and consider the zonal-flow effect using thermal wind equations, for which the higher-order gravity harmonics J₈ and J₁₀ are well reproduced. In order to interpret all the even gravity harmonics measured by Cassini, Galanti et al. (2019) constructed a wide range of rigid-body interior models and reanalyzed Saturn’s flow-induced gravity signal using thermal wind balance. Additionally, Kong et al. (2019) introduced a dynamo region into Saturn’s interior and estimated the flow-induced gravity corrections using perturbation approaches. The model interprets all the even zonal gravity harmonics measured by the Cassini Grand Finale mission.

Based on the first approach, the works of Iess et al. (2019) and Militzer et al. (2019) not only interpreted the even gravitational harmonics measured by Cassini but also discussed the interior composition of Saturn in detail. By contrast, the studies using the second approach paid great attention to deducing the depth of zonal flows from the even gravitational harmonics (Galanti et al. 2019; Kong et al. 2019), but still little information is available about Saturn’s interior composition. In this study, we aim to understand the internal structure and interior composition of Saturn using a similar model for Jupiter, in order to judge whether a consistent picture works for both planets. The interiors of Jupiter and Saturn were compared by Guillot (1999) within the framework of the three-layer assumption, and the effects of shape and rotation on Saturn’s interior were also explored using three-layer interior models (Helled & Guillot 2013). Recently, we constructed a wide range of four-layer interior models to investigate Jupiter’s internal structure and composition from Juno gravity measurements (Ni 2019). This motivates consideration of how the four-layer model works for the understanding of Saturn’s interior. Furthermore, it is of great interest to discern whether there are common features with respect to the predictions of Militzer et al. (2019). In Sect. 2, we describe four-layer structure models specified for Saturn. We calculate the gravitational harmonics in terms of the fifth-order theory of figures. We use the newly detected data from the Cassini Grand Finale, corrected by the dynamical contributions of Galanti et al. (2019) and Kong et al. (2019), to constrain the internal structure of Saturn. In Sect. 3, the practical aspects of optimization calculations are presented and three cases of four-layerinterior models are specified in terms of Saturn’s rotation period and flow-induced gravity signals. Section 4 presents the interior model predictions of gravitational harmonics, helium mass fractions, and heavy element amounts. The effects of rotation periods and atmospheric helium mass fractions on the internal structure of Saturn are discussed in detail, together with the sensitivity of the predictions to the equation of state (EOS) adopted in the interior models. Finally, the main results of this work are summarized in Sect. 5.

2 Four-layer interior structure models for Saturn

We consider Saturn’s interior structure to be similar to the four-layer structure of Jupiter. Following the previous work of Ni (2019), the radial structure of Saturn is assumed to consist of four layers: (1) a homogeneous molecular hydrogen atmosphere in which helium is depleted, (2) a homogeneous metallic hydrogen envelope in which helium is enriched, (3) a dilute core region with enriched heavy elements, and (4) an isothermal central compact core of rock. The outer two layers should be divided by a nonadiabatic region of hydrogen–helium immiscibility, but there are still uncertainties in the H/He phase separation (Lorenzen et al. 2011; Morales et al. 2013; Schöttler & Redmer 2018), and such a nonadiabatic behavior has rather little influence on the gravitational harmonics with respect to the uncertainties discussed in this work (Galanti et al. 2019; Nettelmann et al. 2015). Therefore, the alternative procedure is to ignore the nonadiabatic region in Saturn’s internal structure and to consider the nonadiabatic effect on Saturn’s thermodynamical properties. In this case, a temperature jump ΔT₁₋₂ is introduced across the interface between these two layers, since the nonadiabatic region reduces the heat transport from Saturn’s deeper interior (Hubbard & Militzer 2016; Nettelmann et al. 2015). The mass fraction of helium and heavy elements in the outer three layers are denoted by (Y₁, Z₁), (Y₂, Z₂), and (Y₃, Z₃), respectively. Based on current knowledge of gas giant planets, there are three requirements on the element abundances in Saturn’s interior. First, the upper molecular atmosphere has a smaller helium mass fraction than the lower metallic envelope (Y₁ < Y₂) owing to helium rainout. Second, the dilute core region is a diffusive region in which the heavy elements are enriched compared to the exterior envelope region. This is attributed to the heavy elements dissolved from the central rocky core into the metallic hydrogen and expanded outward through a portion of the planet (Wahl et al. 2017). Therefore, there is an increase in the heavy element abundance for the dilute core region, ΔZ₂₋₃ = Z₃ − Z₂ > 0. Given the enriched heavy elements of ices and rocks, we introduced one free parameter f_rock for the dilute core region, which is defined as the rock fraction in heavy elements f_rock = Z_rock∕Z₃. Third, the mean helium mass fraction in Saturn Y∕(1 − Z) should be equal to its value in the protosolar nebula Y_proto∕(1 − Z_proto) = 0.277 ± 0.006 (Serenelli & Basu 2010; Guillot et al. 2018; Ni 2019). The correlation between Y₁, Y₂, and Y₃ can be found in the appendix of Ni (2019). Figure 1 illustrates a schematic diagram of Saturn’s four-layer structure. The interface between layers 1 and 2 is characterized by the pressure P₁₋₂ and the interface between layers 2 and 3 is described with the pressure P₂₋₃ (P₂₋₃ > P₁₋₂).

The CMS method and theory of figures (ToF) have been used to calculate the self-consistent shape and gravity field of rotating liquid planets (Hubbard 2013; Hubbard & Militzer 2016; Debras & Chabrier 2018; Nettelmann 2017; Ni 2018; Guillot et al. 2018). The CMS confers an advantage in that it is mathematically simple and shows good accuracy. Its disadvantage is that it requires quite a large amount of computation despite the accelerated CMS method (Militzer et al. 2019). By contrast, the ToF is famous for its high numerical efficiency (Zharkov & Trubitsyn 1978), but a high-order approximation is required for the ToF in order to match the error bars of the Cassini Grand Finale gravity measurements. In view of a large ensemble of possible four-layer configurations and a number of iterations necessary to fit the available measurements, we employ the ToF to fifth order (ToF5) to calculate Saturn’s level surface (Zharkov & Trubitsyn 1975; Ni 2018): $$r(s,\theta )=s[1+{\sum }_{i=0}^5{s}_{2i}(s)P_{2i}(\cos \theta )].$$(1)

The ToF5 accuracy is examined in Appendix A by comparison with the accelerated CMS results. It is found that its accuracy is comparable with the accelerated CMS method and it is able to interpret the Cassini Grand Finale gravity measurements. Using the computed figure functions s_2i (s) and abbreviating the level surface as r = s(1 + ∑), one can evaluate the even zonal harmonics as (Guillot et al. 2018; Ni 2019) $$\begin{array}{ll}{J}_{2i}=\hfill & -\frac{3}{2}{\left(\frac{R_{\text{p}}}{R_{\text{eq}}}\right)}^{2i}{\displaystyle {\int }_{-1}^1\text{d}}\cos \theta P_{2i}(\cos \theta ){\displaystyle {\int }_0^1\text{d}}x \hfill \\ \hfill & \times x^{2i+2}{\left(1+\sum \right)}^{2i+2}\eta (x)\left(\frac{\text{d}r}{\text{d}s}\right),\hfill \end{array}$$(2)

where R_eq is the equatorial radius of the planet, R_p is the mean radius of the planet, x is the scaled mean radius, x = s∕R_p, η(x) is the normalized density distribution, and (dr∕ds) is the derivative of r with respectto s obtained from Eq. (1).

The even zonal gravitational harmonics measured by the Cassini Grand Finale are contributed from a rigidly rotating planet $J_{2i}^{\text{rigid}}$ and differentially rotating zonal flows ΔJ_2i. The rigid-rotation contribution is obtained by subtracting the flow-induced contribution from the Cassini Grand Finale gravity measurements, $J_{2i}^{\text{rigid}}=J_{2i}^{\text{Cassini}}-\Delta J_{2i}$. Very recently, Galanti et al. (2019) explored Saturn’s flow-induced gravity signals using thermal wind balance and Kong et al. (2019) estimated Saturn’s flow-induced gravity corrections using perturbation approaches. The flow-induced gravitational harmonics ΔJ_2i and the resulting rigid-rotation contribution $J_{2i}^{\text{rigid}}$ are shown in Table 1. We note that the rigid-rotation gravity harmonics are used for reference in the following optimization calculations.

Fig. 1 Schematic four-layer structure of Saturn. There is a temperature jump ΔT1−2 between layers 1 and 2 owing to a thin helium rain layer, and the interface between layers 2 and 3 is described with the pressure P2−3.

3 Practical aspects

One important input quantity in the interior models of giant planets is the EOS (Miguel et al. 2016; Saumon & Guillot 2004), that is, the density as a function of temperature, pressure, and composition. For hydrogen and helium, we use two ab initio EOSs in this work: the REOS3b for which the zero point of the specific internal energy in the REOS.3 tables of Becker et al. (2014) is changed to coincide in the ideal gas regime with the SCvH EOS, and the MH13+SCvH which is extracted from the tables of Militzer & Hubbard (2013, MH13) for hydrogen–helium mixtures using the SCvH EOS. For a minor amount of heavy elements, we use the EOS of silicates dry sand in SESAME for rocks and the SESAME EOS 7154 of water for ices (Lyon & Johnson 1992). For mixtures of hydrogen, helium, and heavy elements, the EOS is constructed using the separate EOSs for pure species in terms of the additive volume rule.

The numerical model is established based on the CEPAM package (Guillot & Morel 1995). The outer boundary condition at 1 bar is set as T = 135 ± 5 K based on Voyager measurements (Lindal 1992). The heavy element abundances Z₁ and Z₂ as well as the compact core mass M_core are adjusted to reproduce Saturn’s equatorial radius R_eq and even low-order gravitational harmonics J₂, J₄, J₆. The other parameters are poorly constrained at present, such as the pressure P₁₋₂ for the H/He phase transition, the temperature jump ΔT₁₋₂ accounting for the nonadiabatic effect of helium rainout, and the size and composition of dilute cores P₂₋₃, Y₃, Z₃, and f_rock. Their effects on interior models have been investigated in previous works (Miguel et al. 2016; Ni 2019) and are beyond the scope of this work; in our calculations, they are adopted beforehand based on the current knowledge of Saturn’s interior. The pressure for the H/He phase transition is fixed at P₁₋₂ = 1.0 Mbar (Morales et al. 2013; Lorenzen et al. 2011) and the temperature jump ΔT₁₋₂ is uniformly varied from 0 to 1000 K. We note that increasing the P₁₋₂ value causes fewer heavy elements in Saturn’s core and hotter deep interiors allow a larger amount of heavy elements in Saturn’s interior (Miguel et al. 2016; Ni 2019). The pressure P₂₋₃ of the dilute core region is uniformly varied from 2 to 8 Mbar and some interior models without dilute cores are considered as well to cover all possible configurations in Saturn’s interior. As demonstrated in Ni (2019), the compositions in the deeper layer, Y₃ and Z₃, are chosen randomly within a broad range of ΔZ₂₋₃ = 0−0.2 and ΔY₂₋₃ ≤ 0.2 under the requirement that the heavy elements become enriched in the dilute core region and the density steadily increases from the metallic envelope to the dilute core region. The rock fraction in the dilute core region is fixed at a median value of f_rock = 0.50. We note that increasing the rock fraction f_rock tends to lower the amount of heavy elements in Saturn’s interior (Ni 2019).

In contrast to Jupiter’s interior models, there are two additional issues in modeling Saturn’s interior. One is the uncertainty on the planet’s deep rotation rate due to the near-perfect alignment of the magnetic field dipole with the rotation axis (Anderson & Schubert 2007). Galanti et al. (2019) explored Saturn’s deep atmospheric flows with two rotation periods 10 h 32 min 45 s (Helled et al. 2015, fast) and 10 h 39 min 22 s (Smith et al. 1982, slow) for deep rigid-body rotation. Kong et al. (2019) investigated Saturn’s zonal circulation of the molecular layer with the bulk rotation period of 10 h 33 min 38 s (Mankovich et al. 2019, medium). Combining the flow-induced gravity corrections with different rotation periods, we consider three cases in this work. Cases (I) and (II) correspond to the flow-induced gravity corrections of Galanti et al. (2019) with the fast and slow rotation rates, respectively, and Case (III) corresponds to the flow-induced gravity corrections of Kong et al. (2019) with the medium rotation rate. The other issue in modeling Saturn’s interior is the uncertainty of the atmospheric helium mass fraction, because no entry probe has been achieved for Saturn’s atmosphere.The reexamination of Voyager measurements suggested a helium mass fraction in Saturn’s atmosphere of 0.18−0.25 relative to total hydrogen and helium (Conrath & Gautier 2000). The evolutionary models of Hubbard et al. (1999) constrained the atmospheric helium massfraction between 0.11 and 0.21 relative to total helium and hydrogen. In this work, we uniformly vary the atmospheric helium mass fraction Y₁∕(X₁ + Y₁) from 0.130 to 0.230 in order to discern its effect on Saturn’s interior. Given Y₁ and Y₃, the helium mass fraction in the middle envelope Y₂ is adjusted to make sure that the total helium to hydrogen ratio in Saturn Y∕(1 − Z) is equal to its value in the protosolar nebula Y_proto∕(1 − Z_proto) = 0.277 ± 0.006 (Serenelli & Basu 2010). Here, great attention is paid to the effect of rotation periods and atmospheric helium abundances on Saturn’s interior models.

Fig. 2 Gravity harmonics for Saturn obtained with various models in the Ji − Ji+2 planes. The model solutions are shown for Cases (I-III) with the different EOSs used in the interior models: REOS3b (solid points) and MH13+SCvH (open points). The rigid-rotation gravity harmonics $J_{2i}^{\text{rigid}}$ corrected by the dynamical contributions of Galanti et al. (2019) and Kong et al. (2019) are displayed as green and yellow solid squares, respectively, with 1σ error bars.

4 Model results

4.1 Saturn’s even zonal gravity harmonics

In view of the high precision of Cassini Grand Finale gravity measurements and the dynamical uncertainties of differentially rotating zonal flows, we allow the model solutions to deviate from the measured data within ~30σ to cover a wide range of interior configurations. Figure 2 shows the even gravity harmonics obtained with various models compared with the rigid-rotation gravity harmonics $J_{2i}^{\text{rigid}}$. Green and yellow solid squares stand for the rigid-rotation gravity harmonics $J_{2i}^{\text{rigid}}$ corrected bythe dynamical contributions of Galanti et al. (2019) and Kong et al. (2019), respectively. One can see that the $J_{2i}^{\text{rigid}}$ values corrected by Galanti et al. (2019) exhibit lower absolute values for the high-order harmonics J₆ to J₁₀ than those corrected by Kong et al. (2019). Also shown is the uncertainty σ associated with Cassini Grand Finale measurements. The dynamical contributions to Saturn’s zonal gravity harmonics are larger than the very small measurement uncertainty σ, as shown in Table 1. In this case, a wide range of possible internal flows and dynamical considerations significantly increase the effective uncertainty of the rigid-rotation harmonics $J_{2i}^{\text{rigid}}$. Kaspi et al. (2017) explored theeffective uncertainty of Jupiter’s static gravity harmonics by investigating the effect of differential rotation. The effective uncertainty on the staticJ₂ − J₈ is indeed greater than the Juno measurement uncertainty by orders of magnitude. The effect of differential rotation is expected to be greater for Saturn than for Jupiter. Future work should consider the effective uncertainty on Saturn’s rigid-rotation gravity harmonics in the fitting procedure.

As canbe seen in Fig. 2, the models yield J₄ closely around the fitted $J_4^{\text{rigid}}$ values and the calculated harmonics J₆ to J₁₀ fall in between the $J_{2i}^{\text{rigid}}$ values corrected by Galanti et al. (2019) and by Kong et al. (2019). These harmonics show good agreement with the rigid-rotation gravity harmonics within the effective uncertainties. For Cases (I) and (II) respectively with the fast and slow rotation rates, the former seems to yield larger absolute values for the high-order gravity harmonics J₆ to J₁₀. This is consistent with the results of Galanti et al. (2019). Moreover, MH13+SCvH also yields larger absolute values for J₆ to J₁₀ than REOS3b. This can be explained as MH13+SCvH exhibiting higher densities for all the temperatures than REOS3b (Miguel et al. 2016). The high-order gravity harmonics are sensitive to the density in the outer part of a planet (Guillot 2005) and their absolute values generally increase with the local density (Ni 2019).

Fig. 3 Heavy element abundances in the molecular and metallic envelopes obtained with REOS3b for Cases (I) (open circles), Case (II) (solid circles), and Case (III) (solid stars). The optimization solutions are displayed in different colors according to different helium depletions Y1 ∕(X1 + Y1).

4.2 Saturn’s element abundance in the envelopes

With the Cassini Grand Finale gravity measurements, it would be interesting to explore Saturn’s element abundances in the outer envelopes. In the four-layer structure model, the heavy element abundances in the outer two envelopes (Z₁ and Z₂) and the mass of the central compact core (M_core) are adjustedto match the observational constraints. In this section, we take the optimization slolutions using REOS3b for example. The resultingheavy element abundances in the outer two envelopes are shown in Fig. 3 for Cases (I-III), which are derived with different Y₁∕(X₁ + Y₁) values. With less helium depletion in the molecular envelope corresponding to an increase of Y₁ ∕(X₁ + Y₁), the Z₁ value is decreased but the Z₂ value is allowed to vary in a wider range. Taking Case (I) for example, the Z₁ value is decreased from 5 to 0.5 times solar fraction while the Z₂ variation is changed from 0−5 to 0−7 times solar fraction. The increased flexibility of Z₂ can be understood because the dilute core region is allowed to vary in both size (P₂₋₃) and composition (Z₃ and Y₃) while the mean helium-to-hydrogen ratio is fixed for the molecular envelope. In addition, comparing Case (I) with Case (II), we find that the solutions with the slow rotation rate are confined to a narrow area in the figure, suggesting more heavy-element enrichment in the molecular envelope. By contrast, the solutions of Case (III) tend to the area of Z₂ > Z₁ in particular for less helium depletion in the molecular envelope.

The correlations between the abundances of helium and heavy elements are illustrated in Fig. 4 for the outer two envelopes. Figure 4a shows the heavy-element abundance Z₁ as a function of the helium mass fraction Y₁ in the molecular envelope. In practical calculations, the Y₁∕(X₁ + Y₁) value is uniformly adopted beforehand and the Z₁ value is determined from the observational constraints. As can been seen, there exists a good correlation between Y₁ and Z₁ for all three cases and for various amounts of helium depletion. The amount of heavy elements in the molecular envelope decreases from Case (II) with the slow rotation rate, to Case (I) with the fast rotation rate, and then again to Case (III) with the flow-induced gravity correction of Kong et al. (2019). In Fig. 4b, we plot the heavy element abundance Z₂ versus the helium mass fraction Y₂ in the metallic envelope. The Y₂ value is determined by the requirement that the total helium-to-hydrogen ratio in Saturn be equal to its value in the protosolar nebula, and the Z₂ value is adjusted in terms of the observational constraints. We note that only the solutions with Y₁ ∕(X₁ + Y₁) = 0.180 are displayed for the sake of clarity and the solutions with the other Y₁∕(X₁ + Y₁) values show very similar behaviors. One can identify a tendency toward the smaller Z₂ value with increasing Y₂ value, although the solutions have a large spread of available space. The solutions of Case (II), with the slow rotation rate, tend to smaller Z₂ values in the metallic envelope, while most solutions of Case (III) yield larger Z₂ values. This is in stark contrast to the situation shown in Fig. 4a. Such contradicting behavior brings in small differences of the heavy element mass in the envelopes for all three cases; this is further discussed below.

Fig. 4 Saturn’s element abundances in the outer envelopes obtained with REOS3b for Case (I) (red open circles), Case (II) (blue solid circles), and Case (III) (black solid stars). Panel a: trade-off between heavy element and helium mass fractions in the molecular envelope. Panel b: trade-off between heavy element and helium mass fractions in the metallic envelope. We note that panel b applies to Y1 ∕(X1 + Y1) = 0.180, but would remain very similar for the other Y1∕(X1 + Y1) values.

Fig. 5 Mass of heavy elements in the core and in the envelopes predicted with MH13+SCvH. The heavy element abundances in the envelopes are indicated by dashed lines in units of the solar abundance of heavy elements. (a) The results of Cases (I) and (II) are displayed in red circles and in blue stars, respectively, reflecting the effect of the different rotation period. (b) The results of Case (III) are displayed according to different helium depletions Y1 ∕(X1 + Y1) as specified in the legend.

Fig. 6 Mass of the heavy elements in the core and the total heavy element mass in Saturn predicted with different EOSs for hydrogen and helium: MH13+SCvH (blue stars) and REOS3b (red circles). Horizontal lines denote the global mass fraction of heavy elements in units of the solar abundance of heavy elements.

4.3 Saturn’s core and heavy element mass

The interior models of Saturn are affected by the uncertainties in the zonal flows, rotation periods, atmospheric helium mass fractions, and EOSs. Figure 5 shows the comparison of the heavy-element masses in the core and in the envelopes derived using MH13+SCvH for Cases (I-III). In Fig. 5a, we plot the solutions of Cases (I) and (II) to discern the effect of rotation periods on the model predictions. As can be seen, decreasing the rotation rate causes the solutions to be confined to a narrow area of larger $M_Z^{\text{core}}$ and smaller $M_Z^{\text{env}}$. In Fig. 5b, we plot the solutions of Case (III) according to different Y₁ ∕(X₁ + Y₁) values to discern the effect of atmospheric helium mass fractions. It seems that less helium depletion in the molecular envelope tends to allow a larger core mass $M_Z^{\text{core}}$ and a smaller amount of heavy elements in the envelopes $M_Z^{\text{env}}$. Militzer et al. (2019) investigated the effects of rotation periods and core radii using seven sets of interior models with five different rotation periods and three different core radii. As the rotation period is increased, the model solutions move toward an area of larger $M_Z^{\text{core}}$ and smaller $M_Z^{\text{env}}$ and meanwhile the helium mass fraction in the molecular envelope becomes enhanced with a smaller amount of helium rainout (Militzer et al. 2019). Our results presented here are consistent with the predictions of these latter authors in spite of the different four-layer structure assumptions regarding Saturn’s interior. In addition, the heavy-element masses in the core and in the envelopes have some dependence on the flow-induced gravity corrections used to deduce the rigid-rotation gravity harmonics $J_{2i}^{\text{rigid}}$. As can be seen from panels a and b, the $M_Z^{\text{core}}$ values derived in Case (III) reach core masses 1 M_⊕ smaller than the results of Case (I) and the largest $M_Z^{\text{env}}$ value of Case (III) is 1 M_⊕ larger than the value of Case (I). On the whole, results derived with MH13+SCvH reveal a heavy-element mass in the core of 12−18 M_⊕ and a heavy-element mass in the envelopes of up to five times the solar fraction. These results agree with the MH13 predictions of 15−18 M_⊕ and 1.2−4.0 times the solar fraction suggested by Militzer et al. (2019). We note that the predictions of Militzer et al. (2019) are dependent on core size. If smaller fractional core radii were taken into account, smaller core masses would be achieved together with a higher heavy-element abundance in the envelopes, leading to better agreement between the two.

Figure 6 shows the sensitivity of Saturn’s heavy elements to the EOSs for Cases (I–III). As can be seen, the interior model predictions exhibit some dependence on the adopted EOSs. However, this dependence is less considerable than the corresponding dependence found for Jupiter (Ni 2019). The REOS3b solutions seem to allow a larger amount of heavy elements in Saturn’s core and in Saturn as a whole with respect to the MH13+SCvH solutions. This feature is also found in the analogous interior models for Jupiter (Ni 2019). For MH13+SCvH, the heavy element mass in the core $M_Z^{\text{core}}$ spans over the range of 12−18 M_⊕ and the global heavy element abundance Z_global is estimated as 11.5−13.0 times the solar fraction. For REOS3b, $M_Z^{\text{core}}$ spans over a range of 13−19 M_⊕ and Z_global is as high as 12.0−13.5 times the solar fraction. In contrast with the interior model predictions for Jupiter (MH13+SCvH: 6.5−27.0 M_⊕ and five to six times solar fraction, REOS3b: 7−32 M_⊕ and roughly seven times solar fraction), the four-layer structure models of Saturn yield a narrower range for the core mass but a higher value for the global heavy element abundance. One can also see that, out of all three cases, the solutions of Case (III) show the greatest range for the core mass and the largest amount of heavy elements in Saturn’s interior. The effect of the flow-induced gravity signals is even comparable with that of the rotation period. Furthermore, it should be pointed out that the interior temperature in Saturn could be hotter than what we consider here owing to the nonadiabatic structure with compositional gradients in the deep interior (Vazan et al. 2016). This increased temperature would lead to a larger amount of heavy elements in Saturn’s interior (Ni 2019).

5 Summary

In order to discern whether Jupiter and Saturn share similar internal structure, the four-layer interior model, which was developed for Jupiter to interpret Juno gravity measurements, has been used to understand the internal structure and compositionof Saturn from the Cassini Grand Finale gravity measurements. Particular attention is paid to the additional uncertainties for Saturn, such as the deep rotation periods, the atmospheric helium mass fraction, and the flow-induced gravity corrections. Based on current knowledge of Saturn, three rotation periods of Saturn’s deep interior are considered and two sources of the flow-induced gravity corrections are adopted. Various helium depletions in the molecular envelope are assumed in the four-layer interior models. Also, two ab initio EOSs for hydrogen and helium (REOS3b and MH13+SCvH) are used in interior calculations based on the work of Miguel et al. (2016).

The rigid-rotation gravity harmonics are calculated using two-dimensional integrals with the level surface derived from the ToF5 method. Considering two possible dynamical contributions of Galanti et al. (2019) and Kong et al. (2019), the effective uncertainty of the rigid-rotation gravity harmonics is remarkably increased with respect to the error bar of the Cassini Grand Finale measurements. The four-layer models accurately reproduce the even rigid rotation J₂ –J₁₀ within the effective uncertainty. The high-order gravity harmonics J₆ –J₁₀ are found to have some dependence on the rotation period and the EOS. As shown in Fig. 2, the short-period model yields the larger absolute values of J₆ –J₁₀, and MH13+SCvH yields larger absolute values than REOS3b. Saturn’s deep rotation period and atmospheric helium mass fraction affect the distribution of helium and heavy elements in the outer two envelopes. Increasing the deep rotation rate or reducing helium depletion in the molecular envelope lead to a smaller amount of heavy elements in the molecular envelope and allow a larger heavy element abundance in the metallic envelope. Also, there are correlations between themass fractions of helium and heavy elements in the molecular and metallic envelopes. The heavy element abundances tend to decrease with increasing helium mass fraction, as shown in Fig. 4.

The core mass and heavy element abundance in Saturn’s interior depend on the deep rotation rate, the atmospheric helium mass fraction, the flow-induced gravity corrections, and the EOSs for hydrogen and helium. In our baseline simulations, decreasing the rotation rate causes the solutions of $M_Z^{\text{core}}$ and $M_Z^{\text{env}}$ to be confined to a narrow area, and decreasing helium depletion in the molecular envelope leads to larger $M_Z^{\text{core}}$ and smaller $M_Z^{\text{env}}$. The flow-induced gravity corrections of Galanti et al. (2019) and Kong et al. (2019) also result in different solutions of $M_Z^{\text{core}}$ and $M_Z^{\text{env}}$, with a difference of up to 1 M_⊕ in $M_Z^{\text{core}}$ and 1 M_⊕ in $M_Z^{\text{env}}$. The models using MH13+SCvH yield $M_Z^{\text{core}}$ of 12−18 M_⊕ and a global heavy element abundance Z_global of 11.5−13.0 times the solar fraction. The models using REOS3b yield $M_Z^{\text{core}}$ of 13−19 M_⊕ and Z_global of 12.0−13.5 times the solar fraction. By contrast, the analogous interior models for Jupiter predict a wider range of $M_Z^{\text{core}}$ (6.5−32 M_⊕) but a lower value of Z_global (5−7 times the solar fraction), together with a stronger dependence on the ab initio EOSs for hydrogen and helium (Ni 2019). New information on Saturn’s deep rotation period, atmospheric helium mass fraction, and flow-induced gravity signals will contribute to improving the uncertainty in Saturn’s four-layer interior models and to a better understanding of the internal structure and interior composition of Saturn.

Acknowledgements

This work is supported by the Science and Technology Development Fund, Macau SAR (File No. 0005/2019/A1), the Pre-Research Projects on Civil Aerospace Technologies of China National Space Administration (Grant Nos. D020308 and D020303), and the National Natural Science Foundation of China (Grant No. 11761161001).

Appendix A ToF to fifth order compared with the accelerated CMS method

Here, we examine the accuracy of the ToF to fifth order (ToF5). The comparison with the accelerated CMS method is performed with a similar procedure for Jupiter (Ni 2019). First, we converted the ToF-based density profile as a function of the mean radius s to a density profile as a function of the equatorial radius r using the figure functions s_2i(s). The gravitational harmonics are then recalculated using the accelerated CMS method and the differences between the ToF5-fit and CMS gravitational harmonics are evaluated. Second, the accelerated CMS method is used to fit the rigid-rotation gravitational harmonics instead of ToF5. The resulting gravitational harmonics and model predictions are compared with the ToF5 results. Table A.1 shows the comparison of the ToF5 and accelerated CMS results for two representative models using REOS3b and MH13+SCvH. Their offsets are compared with the error bar of the Cassini Grand Finale gravity measurements (Iess et al. 2019) and the flow-induced gravity corrections of Galanti et al. (2019) and Kong et al. (2019). While the ToF5-fit and CMS-fit results show approximately the same J₂ values, the offsets in J₄ –J₁₀ are smaller than the error bars of Iess et al. (2019) in particular for the higher-order harmonics. Given the uncertainties in the flow-induced gravity corrections (Galanti et al. 2019; Kong et al. 2019), the offsets in J₂ –J₁₀ become more negligible. Besides, one can see that the effect on the heavy element abundances in Saturn’s interior is negligible with respectto the uncertainties discussed in this work. With these in mind, the accuracy of the ToF5 method is sufficient to infer Saturn’s interior from the Cassini Grand Finale gravity measurements.

References

All Tables

All Figures

	Fig. 1 Schematic four-layer structure of Saturn. There is a temperature jump ΔT1−2 between layers 1 and 2 owing to a thin helium rain layer, and the interface between layers 2 and 3 is described with the pressure P2−3.
In the text

Fig. 2 Gravity harmonics for Saturn obtained with various models in the Ji − Ji+2 planes. The model solutions are shown for Cases (I-III) with the different EOSs used in the interior models: REOS3b (solid points) and MH13+SCvH (open points). The rigid-rotation gravity harmonics $J_{2i}^{\text{rigid}}$ corrected by the dynamical contributions of Galanti et al. (2019) and Kong et al. (2019) are displayed as green and yellow solid squares, respectively, with 1σ error bars.

In the text

	Fig. 3 Heavy element abundances in the molecular and metallic envelopes obtained with REOS3b for Cases (I) (open circles), Case (II) (solid circles), and Case (III) (solid stars). The optimization solutions are displayed in different colors according to different helium depletions Y1 ∕(X1 + Y1).
In the text

Fig. 4 Saturn’s element abundances in the outer envelopes obtained with REOS3b for Case (I) (red open circles), Case (II) (blue solid circles), and Case (III) (black solid stars). Panel a: trade-off between heavy element and helium mass fractions in the molecular envelope. Panel b: trade-off between heavy element and helium mass fractions in the metallic envelope. We note that panel b applies to Y1 ∕(X1 + Y1) = 0.180, but would remain very similar for the other Y1∕(X1 + Y1) values.

In the text

Fig. 5 Mass of heavy elements in the core and in the envelopes predicted with MH13+SCvH. The heavy element abundances in the envelopes are indicated by dashed lines in units of the solar abundance of heavy elements. (a) The results of Cases (I) and (II) are displayed in red circles and in blue stars, respectively, reflecting the effect of the different rotation period. (b) The results of Case (III) are displayed according to different helium depletions Y1 ∕(X1 + Y1) as specified in the legend.

In the text

	Fig. 6 Mass of the heavy elements in the core and the total heavy element mass in Saturn predicted with different EOSs for hydrogen and helium: MH13+SCvH (blue stars) and REOS3b (red circles). Horizontal lines denote the global mass fraction of heavy elements in units of the solar abundance of heavy elements.
In the text