Research Article

Saturn’s magnetic field revealed by the Cassini Grand Finale

See allHide authors and affiliations

Science  05 Oct 2018:
Vol. 362, Issue 6410, eaat5434
DOI: 10.1126/science.aat5434

Cassini's final phase of exploration

The Cassini spacecraft spent 13 years orbiting Saturn; as it ran low on fuel, the trajectory was changed to sample regions it had not yet visited. A series of orbits close to the rings was followed by a Grand Finale orbit, which took the spacecraft through the gap between Saturn and its rings before the spacecraft was destroyed when it entered the planet's upper atmosphere. Six papers in this issue report results from these final phases of the Cassini mission. Dougherty et al. measured the magnetic field close to Saturn, which implies a complex multilayer dynamo process inside the planet. Roussos et al. detected an additional radiation belt trapped within the rings, sustained by the radioactive decay of free neutrons. Lamy et al. present plasma measurements taken as Cassini flew through regions emitting kilometric radiation, connected to the planet's aurorae. Hsu et al. determined the composition of large, solid dust particles falling from the rings into the planet, whereas Mitchell et al. investigated the smaller dust nanograins and show how they interact with the planet's upper atmosphere. Finally, Waite et al. identified molecules in the infalling material and directly measured the composition of Saturn's atmosphere.

Science, this issue p. eaat5434, p. eaat1962, p. eaat2027, p. eaat3185, p. eaat2236, p. eaat2382

Structured Abstract

INTRODUCTION

Starting on 26 April 2017, the Grand Finale phase of the Cassini mission took the spacecraft through the gap between Saturn’s atmosphere and the inner edge of its innermost ring (the D-ring) 22 times, ending with a final plunge into the atmosphere on 15 September 2017. This phase offered an opportunity to investigate Saturn’s internal magnetic field and the electromagnetic environment between the planet and its rings. The internal magnetic field is a diagnostic of interior structure, dynamics, and evolution of the host planet. Rotating convective motion in the highly electrically conducting layer of the planet is thought to maintain the magnetic field through the magnetohydrodynamic (MHD) dynamo process. Saturn’s internal magnetic field is puzzling because of its high symmetry relative to the spin axis, known since the Pioneer 11 flyby. This symmetry prevents an accurate determination of the rotation rate of Saturn’s deep interior and challenges our understanding of the MHD dynamo process because Cowling’s theorem precludes a perfectly axisymmetric magnetic field being maintained through an active dynamo.

RATIONALE

The Cassini fluxgate magnetometer was capable of measuring the magnetic field with a time resolution of 32 vectors per s and up to 44,000 nT, which is about twice the peak field strength encountered during the Grand Finale orbits. The combination of star cameras and gyroscopes onboard Cassini provided the attitude determination required to infer the vector components of the magnetic field. External fields from currents in the magnetosphere were modeled explicitly, orbit by orbit.

RESULTS

Saturn’s magnetic equator, where the magnetic field becomes parallel to the spin axis, is shifted northward from the planetary equator by 2808.5 ± 12 km, confirming the north-south asymmetric nature of Saturn’s magnetic field. After removing the systematic variation with distance from the spin axis, the peak-to-peak “longitudinal” variation in Saturn’s magnetic equator position is <18 km, indicating that the magnetic axis is aligned with the spin axis to within 0.01°. Although structureless in the longitudinal direction, Saturn’s internal magnetic field features variations in the latitudinal direction across many different characteristic length-scales. When expressed in spherical harmonic space, internal axisymmetric magnetic moments of at least degree 9 are needed to describe the latitudinal structures. Because there was incomplete latitudinal coverage during the Grand Finale orbits, which can lead to nonuniqueness in the solution, regularized inversion techniques were used to construct an internal Saturn magnetic field model up to spherical harmonic degree 11. This model matches Cassini measurements and retains minimal internal magnetic energy. An azimuthal field component two orders of magnitude smaller than the radial and meridional components is measured on all periapses (closest approaches to Saturn). The steep slope in this component and magnetic mapping to the inner edge of the D-ring suggests an external origin of this component.

CONCLUSION

Cassini Grand Finale observations confirm an extreme level of axisymmetry of Saturn’s internal magnetic field. This implies the presence of strong zonal flows (differential rotation) and stable stratification surrounding Saturn’s deep dynamo. The rapid latitudinal variations in the field suggest a second shallow dynamo maintained by the background field from the deep dynamo, small-scale helical motion, and deep zonal flows in the semiconducting region closer to the surface. Some of the high-degree magnetic moments could result from strong high-latitude concentrations of magnetic flux within the planet’s deep dynamo. The periapse azimuthal field originates from a strong interhemispherical electric current system flowing along magnetic field lines between Saturn and the inner edge of the D-ring, with strength comparable to that of the high-latitude field-aligned currents (FACs) associated with Saturn’s aurorae.

A meridional view of the results of the Cassini magnetometer observations during the Grand Finale orbits.

Overlain on the spacecraft trajectory is the measured azimuthal field from the first Grand Finale orbit, revealing high-latitude auroral FACs and a low-latitude interhemispherical FAC system. Consistent small-scale axisymmetric internal magnetic field structures originating in the shallow interior are shown as field lines within the planet. A tentative deep stable layer and a deeper dynamo layer, overlying a central core, are shown as dashed semicircles. The A-, B-, C-, and D-rings are labeled, and the magnetic field lines are shown as solid lines. RS is Saturn’s radius, Z is the distance from the planetary equator, ρ is the perpendicular distance from the spin axis, and Bϕ is the azimuthal component of the magnetic field.

Abstract

During 2017, the Cassini fluxgate magnetometer made in situ measurements of Saturn’s magnetic field at distances ~2550 ± 1290 kilometers above the 1-bar surface during 22 highly inclined Grand Finale orbits. These observations refine the extreme axisymmetry of Saturn’s internal magnetic field and show displacement of the magnetic equator northward from the planet’s physical equator. Persistent small-scale magnetic structures, corresponding to high-degree (>3) axisymmetric magnetic moments, were observed. This suggests secondary shallow dynamo action in the semiconducting region of Saturn’s interior. Some high-degree magnetic moments could arise from strong high-latitude concentrations of magnetic flux within the planet’s deep dynamo. A strong field-aligned current (FAC) system is located between Saturn and the inner edge of its D-ring, with strength comparable to the high-latitude auroral FACs.

The internal magnetic field of a planet provides a window into its interior (1). Saturn’s magnetic field was first measured by the Pioneer 11 magnetometers (2, 3) and shortly afterward by Voyagers 1 and 2 (4, 5). Those observations (27) revealed that Saturn’s magnetic axis is tilted with respect to the planetary spin axis by less than 1°. The axisymmetric part of Saturn’s internal field is more complex than a dipole (commonly denoted with the Gauss coefficient g10), with the axisymmetric quadrupole and octupole magnetic moment contributions (coefficients g20 and g30, respectively) amounting to ~10% of the dipole moment contribution when evaluated at the 1-bar surface of the planet. Three decades later, Cassini magnetometer (MAG) measurements (812) made before the Grand Finale phase refined Saturn’s dipole tilt to <0.06° (11). Secular variation is at least an order of magnitude slower than at Earth (11), and magnetic flux is expelled from the equatorial region toward mid-to-high latitude (12). Magnetic moments beyond spherical harmonic degree 3, with Gauss coefficients on the order of 100 nT, were suggested by the MAG measurements during Cassini Saturn Orbit Insertion (SOI) in 2004 (12) but were not well resolved because of the limited spatial coverage of SOI, with periapsis (closest approach) at 1.33 RS (where 1 RS = 60,268 km is the 1-bar equatorial radius of Saturn) and latitudinal coverage within ±20° (12).

Saturn’s main magnetic field is believed to be generated by rotating convective motion in the metallic hydrogen (13, 14) layer of Saturn through the magnetohydrodynamic (MHD) dynamo process (1, 1517). The highly symmetric nature of Saturn’s internal magnetic field with respect to the spin axis defies an accurate determination of the rotation rate of Saturn’s deep interior and challenges our understanding of the MHD dynamo process because Cowling’s theorem (18) precludes the maintenance of a perfectly axisymmetric magnetic field through an active MHD dynamo. The possibility that we are observing Saturn’s magnetic field at a time with no active dynamo action can be ruled out given the sizable quadrupole and octupole magnetic moments because these would decay with time at a much faster rate than that of the dipole [(19), chapter 2]. In addition to the main dynamo action in the metallic hydrogen layer, deep zonal flow (differential rotation) and small-scale convective motion in the semiconducting region of Saturn could lead to a secondary dynamo action (20).

We analyzed magnetometer measurements made during the last phase of the Cassini mission. The Cassini Grand Finale phase began on 26 April 2017, with the spacecraft diving through the gap between Saturn’s atmosphere and the inner edge of its D-ring 22 times, making a final dive into the atmosphere on 15 September 2017. Periapses were ~2550 ± 1290 km above the atmosphere’s 1-bar level, equivalent to ~1.04 ± 0.02 RS from the center of Saturn (fig. S1). The low periapses, high inclination (~62°), and proximity to the noon-midnight meridian in local time provided an opportunity to investigate Saturn’s internal magnetic field and the magnetic environment between Saturn and its rings (figs. S1 and S2). The fluxgate magnetometer (8) made continuous (32 measurements per second) three-component magnetic field measurements until the final second of spacecraft transmission. For analysis, we averaged the data to 1-s resolution. During ~±50 min around periapsis, the instrument used its highest dynamic range mode, which can measure magnetic field components up to 44,000 nT, with digitization noise of 5.4 nT (8). This mode had not been used since an Earth flyby in August 1999 (21) and therefore required recalibration during the Grand Finale orbits. Four spacecraft roll campaigns and numerous spacecraft turns enabled this recalibration. The spacecraft attitude needed to be known to a high accuracy of 0.25 millirads, but intermittent suspension of the spacecraft star identification system during phases of instrument targeting resulted in gaps in the required information. These gaps were filled by reconstructing spacecraft attitude using information from the onboard gyroscopes. Attitude reconstruction is currently available for nine of the first 10 orbits, so we restricted our analysis to those orbits. Individual orbits of Cassini are commonly referred to with a rev number; these nine orbits are rev 271 to 280, excluding rev 277.

External magnetic field from sources in the magnetosphere and ionosphere of Saturn

In deriving the internal planetary magnetic field from in situ MAG measurements, it is necessary to separate the internal planetary field from exterior sources of field (currents in the magnetosphere and ionosphere). In the saturnian system, the simplest external source to correct is the magnetodisk current, which flows in the equatorial region far out in the magnetosphere (2226). Like the terrestrial ring current, the magnetodisk current produces a fairly uniform depression in the field strength close to the planet and is normally treated as axisymmetric (fig. S3). At Saturn, the magnetodisk current contributes ~15 nT of magnetic field parallel to the spin axis of Saturn (BZ) in the inner magnetosphere (2226). Magnetopause and magnetotail currents, which determine the properties of the outer magnetosphere, contribute ~2 nT BZ field in the inner magnetosphere of Saturn (22, 26). More complex are the ubiquitous planetary period oscillations (PPOs), global 10.6- to 10.8-hour oscillations, at close to the expected planetary rotation period that are present in all Saturn magnetospheric parameters [reviewed in (27, 28)]. The PPOs arise from a rotating electrical current systems with sources and sinks in Saturn’s ionosphere and magnetosphere and closure currents that flow along magnetic field lines (commonly referred to as FACs) and map to the auroral regions. Normally, two periods are present at any one time, with one dominant in the northern hemisphere and the other in the southern (2933), which during the interval studied here were 10.79 and 10.68 hours, respectively (33). The rotation periods, amplitudes, and relative amplitudes of the two PPO systems vary slowly over Saturn’s seasons (3436). These seasonal changes indicate that the source of the periods is most likely to be of atmospheric origin rather than deep within the planet (37, 38). Moreover, the PPO periods are a few percent longer compared with any likely rotation period of the deep interior of the planet (3941). There is also a nonoscillatory system of strong FACs in the auroral region with current closure through the planetary ionosphere, and there are distributed FACs at other latitudes (4244). This nonoscillatory current system is associated with magnetospheric plasma that rotates more slowly than the planet (42). During the Grand Finale phase, the Cassini spacecraft was close enough to the planet during periapse to directly detect currents in the ionosphere. We describe results on these external current systems first because they must be considered during analysis of the internal field.

Shown in Fig. 1 is the vector magnetic field measurements in International Astronomical Union (IAU) Saturn System III right-hand spherical polar coordinates, with r, θ, and ϕ denoting radial, meridional, and azimuthal directions, respectively (45) from the nine Cassini Grand Finale orbits for which accurate spacecraft attitude information is available. This coordinate system has its origin at the center of mass of Saturn, and the spin axis of Saturn is the polar axis (zenith reference). The peak magnetic field strength encountered during these Grand Finale orbits is ~18,000 nT, which is well within the range of the fluxgate magnetometer (8). The peak field was not encountered during periapsis but at mid-latitude in the southern hemisphere. The dominant radial and meridional magnetic field components, Br and Bθ, which exhibit similar behavior on each orbit, are shown in Fig. 1, A and B. This repeatability, combined with the fixed local time of the orbits and the slight differences in orbital period, demonstrate that the planetary magnetic field is close to axisymmetric.

Fig. 1 Vector magnetic field measurements from nine Cassini Grand Finale orbits.

(A to C) The IAU System III Saturn-centered spherical polar coordinates are adopted here. The peak measured magnetic field strength is ~18,000 nT, whereas the azimuthal component of the field (plotted on a different scale) is ~1/1000 of the total field outside the high-latitude auroral FACs region, which are labeled. Vertical dashed lines indicate the MAG range 3 (Embedded Image nT) time period. (C) The vertical dotted lines mark the inner edge of the D-ring mapped magnetically, and the solid vertical solid lines mark the outer edge of the D-ring mapped magnetically, as described in the text.

Shown in Fig. 1C is the azimuthal component of the field, Bϕ, which is almost three orders of magnitude smaller than the radial and meridional components outside the auroral FACs region. If it originates from the interior of the planet, Bϕ would be expected to decay with radial distance and also be part of a nonaxisymmetric field pattern (Eq. 5). However, much of the observed Bϕ signal seems to be of external rather than internal origin. The magnetic perturbations from the northern and southern auroral currents through which the spacecraft flew are marked in Figs. 1 and 2. In Fig. 2, the cylindrical coordinate system is adopted, where ρ is the perpendicular distance to the spin axis of Saturn and Z is the distance from the planetary equator of Saturn, which is defined by its center of mass. These current sheets have been crossed many times during the Cassini mission, but previous crossings occurred much further from the planet (43, 44, 46). The sharp field gradients observed are due to local FACs, and the short scale of the variation in the field indicates local current sources rather than relatively distant currents in the planet’s interior. The scatter in the auroral FACs from orbit to orbit is partly due to the variability of the aurora over time and partly due to the large contribution from the rotating PPO signals, which are encountered at different phases on each orbit.

Fig. 2 Cassini’s Grand Finale trajectory shown in the meridional plane.

The trajectory of Cassini Grand Finale orbit rev 275 (black/blue trace with arrows) in the ρ-Z cylindrical coordinate system—where ρ is the perpendicular distance from the spin axis, and Z is distance from Saturn’s planetary equator—is overlain on the average positions of Saturn’s auroral FACs region (gray region) and the newly discovered low-latitude FACs (red region). Thin gray traces with arrows are magnetic field lines. The interval along the trajectory during which the measured magnetic field was >10,000 nT is highlighted in blue. Small circles are at 3-hour intervals, and the beginning of day 142 of year 2017 is shown.

The narrow central peak in Bϕ ranging between 5 and 30 nT observed along these Grand Finale orbits was unexpected. It can be seen from Fig. 1C that the large change in field slope takes place where the spacecraft crosses the magnetic field lines that map to the inner edge of the D-ring. We adopted 1.11 RS from the center of Saturn in the equatorial plane as the D-ring inner edge. If the azimuthal field gradient is due to a local FAC, the current within the D-ring is interhemispheric and always flows from the northern to the southern hemisphere along these nine Grand Finale orbits. There must be return currents either at lower altitude than the spacecraft penetrates in the ionosphere or at other local times. Assuming approximate axisymmetry (for azimuthal spatial scales much larger than latitudinal scales), Ampère’s law combined with current continuity leads to a total interhemispheric current flow in the range of 0.25 million to 1.5 million ampere per radian of azimuth. Thus, this low-latitude FAC is comparable in strength with the currents observed on field lines connected to the high-latitude auroral region (43, 44, 46). It thus may be part of a global magnetospheric–ionospheric coupling system. Alternatively, the system might be purely of ionospheric origin, in which case it could be due to skewed flow between northern and southern ends of the field line (47). All the orbit periapses are near local noon, so the repeatability from periapsis to periapsis does not rule out a dependency on local time.

There is a clear difference between the signatures of northern and southern auroral FACs (Fig. 1C), and again, magnetic mapping reveals the source. The locations of the auroral current sheets are well known magnetically from previous Cassini measurements (Fig. 2) (43, 44, 46). During Cassini’s inbound trajectory in the north, the spacecraft is located on the field lines just poleward of the main FAC region, as shown by the decreasing Bϕ, which is indicative of the distributed downward current on polar field lines, as previously shown (43, 44, 46). The less structured Bϕ during the outbound southern trajectories (Fig. 1C) indicates that the southern hemisphere auroral FACs are not being fully crossed by the spacecraft, and instead we observed the center of FAC activity moving dynamically back and forth over the spacecraft.

Saturn’s internal magnetic field

Next we analyzed the radial and meridional components of the measured magnetic field, which are dominated by the axisymmetric internal magnetic field. The highly inclined nature of the Grand Finale orbits allows direct determination close to the planet of Saturn’s magnetic equator, which is defined as where the cylindrical radial component of the magnetic field (Bρ) vanishes. The measured Bρ as a function of the spacecraft’s distance from the planetary equator of Saturn is shown in Fig. 3A. The measured value along these orbits shows that Saturn’s magnetic equator is 0.0466 ± 0.0002 (1 SD) RS northward of the planetary equator, measured at distance ~1.05 RS from the spin axis of Saturn. This indicates a small yet non-negligible north-south asymmetry in Saturn’s magnetic field. Shown in Fig. 3B are the distribution of the directly measured magnetic equator position of Saturn in the ρ-Z plane and predictions from various preexisting degree-3 models (4, 7, 10, 11). Several degree-3 internal field models have been derived in (10) from different combinations of datasets; the two shown in Fig. 3 were (i) derived from data between Cassini SOI to July 2007 and (ii) data from Cassini SOI to July 2007 combined with Pioneer 11 and Voyager 1 and 2 data. A nominal magnetodisk field (table S2) was added to the internal field when predicting the magnetic equator positions from existing models. A systematic decrease of the magnetic equator position with distance from the spin axis is evident. This is a consequence of the axisymmetric part of the magnetic field, as illustrated by the predictions from existing axisymmetric models. In addition to this systematic trend, variations of measured magnetic equator positions at similar ρ were also observed. These additional variations of the magnetic equator positions serve as a direct bound on possible nonaxisymmetry of Saturn’s internal magnetic field because any internal nonaxisymmetry would cause longitudinal variations of the magnetic equator positions. The measured peak-to-peak variation at similar cylindrical radial distances, ~0.00030 RS (18 km), translates to a dipole tilt of 0.0095°, a guide value for possible internal nonaxisymmetry. This value is about one order of magnitude smaller than the upper limit on dipole tilt placed before the Grand Finale (10, 11, 48). The corresponding field components at spacecraft altitude are only ~3 nT (in a background field of 18,000 nT), which is too small to be discerned from external fields in the measured Bϕ. Such an extreme axisymmetric planetary magnetic field is difficult to explain with a dynamo model (18, 4952).

Fig. 3 Saturn’s magnetic equator position as directly measured along the Cassini Grand Finale orbits.

(A) The measured cylindrical radial component of the magnetic field versus distance from Saturn’s planetary equator from nine Cassini Grand Finale orbits. (B) The distribution of the magnetic equator northward displacements (triangles) versus cylindrical radial distance compared with predictions from existing degree-3 internal field models (lines) (4, 7, 10, 11). The Z3 model is from (4), the Cassini 3 model is from (11), the SPV model is from (7), the Cassini (SOI, June 2007) model, and the Cassini, P11, V1, V2 model are both from (10). The Cassini (SOI, June 2007) model were derived from Cassini SOI data to July 2007 data, and the Cassini, P11, V1, V2 model is derived from Cassini SOI to July 2007 data combined with Pioneer 11 and Voyager 1 and 2 data (10).

Although the internal magnetic field of Saturn is highly axisymmetric (structureless in the longitudinal direction), it exhibits latitudinal structures across many different length scales. This is revealed by our retrieval of the axisymmetric Gauss coefficients of Saturn’s internal magnetic field from these nine Grand Finale orbits. In this analysis, the magnetodisk field was explicitly included, adopting an analytical formula (25) with initial parameters taken from (22). The axisymmetric internal Gauss coefficients and the magnetodisk current field were then retrieved via an iterative process. To determine the internal Gauss coefficients, only measurements from range 3 of the fluxgate magnetometer were adopted (Embedded Image nT, r < 1.58 RS), and measurements from all nine orbits were treated as a single dataset. These measurements were outside the auroral FACs region, as shown in Figs. 1 and 2. For retrieval of the magnetodisk current field, measurements from both range 3 and range 2 of the fluxgate magnetometer were used (Embedded Image nT, r < 4.50 RS), and each orbit was treated separately (table S2). Along the trajectory of the close-in portion of the Cassini Grand Finale orbits, the contributions from the magnetopause and magnetotail currents cannot be practically separated from the quasi-uniform magnetodisk current contribution and thus are subsumed into our magnetodisk current model. The time-varying external (magnetodisk + magnetopause + magnetotail) BZ field could create a time-varying component of the internal dipole g10 through electromagnetic induction. The maximum magnetodisk field variation we have observed along the nine orbits is ~10 nT, which for an induction depth of 0.87 RS (Methods and fig. S4) would induce a time variation in the internal dipole coefficient of ~3.3 nT. This is less than the digitization level of the highest range of the fluxgate magnetometer onboard Cassini (5.4 nT); thus, the detection of which is not straightforward.

We first investigated the minimum parameter set (in terms of internal Gauss coefficients) needed to adequately describe the observations, while applying no regularization to the parameters (Methods) (53, 54). This experiment revealed that axisymmetric internal Gauss coefficients up to degree 9 are required to bring the root-mean-square (RMS) residual in range 3 below 10 nT (fig. S5). The residual (Br, Bθ) after removal of the unregularized degree 3 model (coefficients provided in table S1) are on the order of 100 nT but have larger amplitude and larger spatial scales in the northern hemisphere as compared with those in the southern hemisphere (Fig. 4). Shown in Fig. 5 is the residual (Br, Bθ) as a function of latitude after removal of the unregularized degree 6 model and the best-fitting magnetodisk model for each orbit (coefficients provided in tables S1 and S2), which exhibits consistent small-scale features with amplitude ~25 nT and typical spatial scale ~25° in latitude. The highly consistent nature (from orbit to orbit) of these small-scale magnetic perturbations in (Br, Bθ), in contrast to the orbit-to-orbit varying Bϕ signature, suggests a source below the highly time-variable currents in the ionosphere of Saturn.

Fig. 4 Saturn’s magnetic field beyond spherical harmonic degree 3 as directly measured during the Grand Finale orbits.

(A and B) Residuals of the radial and meridional magnetic fields from the unregularized degree-3 internal field model (parameters are provided in table S1). It can be seen that the residuals are on the order of 100 nT, which is 10 times larger than the magnetodisk field.

Fig. 5 Saturn’s magnetic field beyond spherical harmonic degree 6 as directly measured during the Grand Finale.

(A and B) Residuals of the radial and meridional magnetic fields from the unregularized degree-6 internal field model (parameters are provided in table S1). The field corresponding to the magnetodisk current has also been removed (parameters of the magnetodisk current model are provided in table S2). Latitudinally banded magnetic structures on the order of 25 nT are evident. Orbit-to-orbit deviations in Bθ between –60° and –40° latitudes are mostly due to the influences of southern hemisphere high-latitude FACs.

A new internal magnetic field model for Saturn, which we refer to as the Cassini 11 model, is presented in Table 1 and Fig. 6, with 11 denoting the highest spherical harmonic degree Gauss coefficient above the uncertainties (Methods). Because there was incomplete latitudinal coverage during the Grand Finale orbits, which can lead to nonuniqueness in the solution, a regularized inversion technique (53, 54) has been used to construct the Cassini 11 model. With the regularized inversion technique (53, 54), we explicitly sought internal field solutions that not only match the observations but also contain minimum magnetic power beyond degree 3, when evaluated at a reference radius (Methods). Presented in Table 1 are the axisymmetric Gauss coefficients, five times the formal uncertainty (5σ) (definition is provided in Methods), and the RMS residual of the Cassini 11 model. The resulting Mauersberger-Lowes (M-L) magnetic power spectrum (5557) of the Cassini 11 model evaluated at the surface of Saturn is shown in Fig. 6. In the Cassini 11 model, the axisymmetric Gauss coefficients between degree 4 and degree 11 are on the order of 10 to 100 nT, whereas those above degree 11 are below the 5σ uncertainty.

Table 1 Gauss coefficients of a new model for Saturn’s internal magnetic field, which we refer to as the Cassini 11 model, constructed from nine orbits of Cassini Grand Finale MAG data with regularized inversion.

The reported uncertainty is five times the formal uncertainties associated with the chosen regularization (Methods).

View this table:
Fig. 6 Magnetic power spectrum of the Cassini 11 model.

This new internal field model for Saturn is constructed from nine Cassini Grand Finale orbits with regularized inversion. Central values and five times the formal uncertainties derived from the regularized inversion are shown (Methods). It can be seen that the high-degree moments between degree 4 and degree 11 are on the order of 10 to 100 nT, whereas those above degree 11 are below the derived uncertainty.

We have also investigated whether it is possible to obtain a simpler description of Saturn’s internal magnetic field (in terms of Gauss coefficients) in a shifted coordinate system. We shifted the coordinate system northward from Saturn’s center of mass along the direction of the spin axis, in the range of 0.03 to 0.05 RS denoted by zS, and solved for the internal Gauss coefficients. We found that no simpler description of the field in terms of Gauss coefficients can be obtained in the Z-shifted coordinates because of the complex nature of Saturn’s internal magnetic field in the latitudinal direction. Only the quadrupole moment g20 can be reduced to zero in the Z-shifted coordinates with zS ~ 0.0337 RS; all other even-degree moments (such as g40, g60, and g80) remain nonzero (fig. S6). In addition, g40 becomes more than an order of magnitude larger in these shifted coordinates compared with that in the Saturn-centered coordinates. The RMS residual remains unchanged as we shifted the coordinate system. We therefore did not consider models in shifted coordinates any further.

Implications for Saturn’s dynamo and interior

Mathematically, downward continuation of the magnetic field measured outside of a planet toward its interior is strictly valid through regions with no electrical currents J. From the perspective of dynamo action, the downward continuation of the magnetic field is possible through regions with no substantial dynamo action. For Saturn, becasue of the smooth increase of electrical conductivity as a function of depth (13, 14, 58), the vigor of dynamo action is expected to rise smoothly with depth (20) and can be quantified by the magnetic Reynolds number (Rm), which measures the ratio of magnetic field production/modification by flow in an electrically conducting fluid against ohmic diffusion (Methods) (59). Assuming 1 cm s−1 flow and an electrical conductivity model from (58, 60), the local Rm reaches 10 at 0.77 RS, 50 at 0.725 RS, and 100 at 0.70 RS (fig. S9). For dynamo action in the shallow semiconducting layer, Rm = 20 is likely to be sufficient given the existence of the main magnetic field originating from the deep dynamo. Thus, the top of the shallow dynamo is likely to fall between 0.77 and 0.725 RS. We chose 0.75 RS as an approximation for the surface of the shallow dynamo; the properties of the downward continued magnetic field are broadly similar in this range of depths.

The small-scale Br beyond degree 3 within ±60° latitude at 0.75 RS based on the Cassini 11 model is shown in Fig. 7. The small-scale magnetic structures measured along the spacecraft trajectory map to latitudinally banded magnetic structures with amplitude ~5000 nT and typical spatial scale ~15° in latitude at 0.75 RS, which amount to ~5 to 10% of the local background field (fig. S7). The inferred small-scale magnetic field pattern is broadly similar for either a spherical surface with radius 0.75 RS or a dynamically flattened elliptical surface with equatorial radius 0.75 RS and polar radius 0.6998 RS (fig. S8). The magnetic power in the low-degree (degrees 1 to 3) part of the M-L spectrum still greatly exceeds that in the higher degree part when evaluated at 0.75 RS; the dipole power is one order of magnitude larger than the octupole power, whereas the octupole power is almost two orders of magnitude larger than those of degrees 4 to 11. The orders-of-magnitude difference between the low-degree power and high-degree power at 0.75 RS suggests that the latitudinally banded magnetic perturbations could originate from a secondary dynamo action in the semiconducting region of Saturn (20). This secondary dynamo action likely only produces perturbations on top of the primary deep dynamo, as suggested by a kinematic mean-field dynamo model (20). Fully three-dimensional (3D) self-consistent giant planet dynamo models with deep differential rotation (zonal flows) also produced latitudinally banded magnetic fields (61, 62).

Fig. 7 Small-scale axisymmetric magnetic field of Saturn at 0.75 RS.

The values of ΔBr were computed by using the central values of the degree 4 to degree 11 Gauss coefficients (Table 1). These small-scale magnetic perturbations at 0.75 RS are on the order of 5000 nT, which amount to ~5 to 10% of the local background field (fig. S7). The plotted latitudinal range corresponds to the region of Cassini MAG range 3 measurements along the Grand Finale orbits.

The dynamo α-effect, which describes the generation of a poloidal magnetic field from a toroidal magnetic field [poloidal and toroidal magnetic field are defined in (63)] by rotating convective motion, is likely to be of the traditional mean-field type in the shallow dynamo (16, 17, 20). The validity of the traditional α-effect has been extensively discussed in (20). In this scenario, no substantial low-degree nonaxisymmetric magnetic field is expected from the shallow dynamo.

From an empirical perspective, some but not all of the high-degree magnetic moments could still have a deep interior origin. One extreme interpretation of the Cassini 11 model is to regard the magnetic moments up to degree 9 as entirely being of deep interior origin because the magnetic power of degrees 7 and 9 is within a factor of three of that of degree 3 when evaluated at 0.50 RS. When discussing internal magnetic fields well below 0.75 RS, we have (i) treated the shallow dynamo as a small perturbation and (ii) implicitly assumed stable stratification between the surface of the deep dynamo and the shallow dynamo. Under this assumption, there exists no local dynamo α-effect between the surface of the deep dynamo and the shallow dynamo. Although the ω-effect, which describes the generation of the toroidal magnetic field from differential rotation acting on the poloidal magnetic field, likely operates in the stably stratified layer, the ω-effect will not affect the poloidal component of the magnetic field. Downward continuation of the magnetic field well below 0.75 RS can only be justified if there are no effective electrical currents generating the poloidal magnetic field in the layer between the surface of the deep dynamo and the shallow dynamo.

The radial magnetic field is shown in fig. S10 as a function of latitude at 0.50 RS computed from the Gauss coefficients of the Cassini 11 model up to degree 9. It can be seen that the radial component of Saturn’s magnetic field at 0.50 RS is strongly concentrated toward high latitude, leaving minimum magnetic flux at latitudes below ±45°, a weaker version of which is evident from the magnetic moments up to degree 3 (fig. S10). The possibility of weak magnetic field at the poles of Saturn near 0.50 RS is suggested in fig. S10. In this alternative interpretation, magnetic moments beyond degree 9 would be of shallower origin. This highlights the ambiguity in the separation of magnetic moments of shallow origin from those of deep origin, which cannot be resolved with MAG data alone.

The main dynamo of Saturn does appear to be deeply buried, originating from a depth deeper than the expected full metallization of hydrogen at ~0.64 RS (13). The surface strength of Saturn’s internal magnetic field is more than one order of magnitude weaker than that of Jupiter (6467). From dynamo scaling analysis, Saturn’s magnetic field appears to be weaker than expected from both force-balance considerations and energy-balance considerations if the outer boundary of the deep dynamo is much shallower than 0.40 RS (60). The most widely accepted theoretical explanation for Saturn’s highly axisymmetric magnetic field (49, 50) suggests that differential rotation in a stably stratified and electrically conducting layer above the deep dynamo region electromagnetically filters out nonaxisymmetric magnetic moments. Although 3D numerical dynamo simulations (51, 52) that implement stable stratification and differential rotation have only produced an average dipole tilt of 0.6°, about two orders of magnitude larger than the newly derived guide value 0.0095°, the Rm in these 3D numerical dynamo simulations (51, 52) are only in the range of 100 to 500, which is orders of magnitude smaller than possible values (~30,000) inside the metallic hydrogen layer of Saturn (68). At present, we regard the extreme axisymmetry of Saturn’s magnetic field as a continuing enigma of the inner workings of Saturn.

Methods

Un-regularized and regularized inversion of Saturn’s internal magnetic field

The internal magnetic field of Saturn outside the dynamo region is described using Gauss coefficients (gnm , hnm) where n and m are the spherical harmonic degree and order, respectively:Embedded Image(1)Embedded Image(2)Embedded Image(3)Embedded Image(4)Embedded Image(5)It can be seen from Eq. 5 that the azimuthal magnetic field of internal origin, Embedded Image, must be part of a non-axisymmetric (m ≠ 0) field pattern.

The forward model can be represented asdata = G model(6)For this particular problem, data represents the magnetic field measurements (Br, Bθ, Bϕ), model represents the Gauss coefficients (gnm, hnm), and G represents the matrix expression of Eqs. 3 to 5.

In un-regularized inversion, one seeks to minimize the difference between the data and the model field only|dataG model|2(7)whereas in regularized inversion, additional constraints are placed on the model parameters. Instead, one seeks to minimize|dataG model |2 + γ 2|L model|2(8)in which L represents a particular form of regularization and γ is a tunable damping parameter adjusting the relative importance of model constraints.

We seek to minimize the power in the surface integral of the radial component of the magnetic field beyond spherical harmonic degree 3Embedded Image(9)at a reference radius rref. Thus,Embedded Image(10)for n > 3 and L = 0 for n ≤ 3.

After experimenting with different rref and γ, we choose rref = 1.0 and γ = 0.6 which yield smooth magnetic field perturbations when viewed between 1.0 RS and 0.75 RS.

In this framework, the model solution can be computed viamodel = (GTG + γ2 LTL)−1GT data = GN data(11)in which superscript T represents matrix transpose.

The model covariance matrix can be computed frommodcov = GN datacov (GN)T(12)where datacov is the data covariance matrix, taken to be σdata2I, in which σdata is the RMS of the data model misfit and I is the identity matrix (54). The formal uncertainty of the model solution is the square root of the diagonal term in the model covariance matrix.

Electromagnetic induction inside Saturn

Due to the rapidly radial varying nature of electrical conductivity inside Saturn, the three factors controlling the electromagnetic (EM) induction at Saturn are: the frequency of the inducing field ωind, the electrical conductivity σ(r) and the electrical conductivity scale height Embedded Image. The EM induction inside Saturn would occur at a depth where the frequency dependent local EM skin depth Embedded Image becomes similar to or less than the local electrical conductivity scaleheight Hσ(r), where μ0 is the magnetic permeability.

Figure S4 shows the frequency dependent EM skin depth compared to the electrical conductivity scaleheight inside Saturn, adopting the electrical conductivity profile (58). Two frequencies have been chosen, the of Saturn Embedded Image hours ∼1.7 × 10–4 rad s–1 and the orbital period of Cassini Grand Finale orbits Embedded Image Earth days ∼1.0 × 10–5 rad s–1. It can be seen from fig. S4 that EM induction inside Saturn would occur at depth 0.87 RS for ωind equal to the rotational frequency of Saturn and at depth 0.86 RS for ωind equal to the orbital frequency of Cassini Grand Finale orbits. Thus, the induction depth inside Saturn would likely be 0.87 RS or less.

A 10 nT inducting BZ field would create a time-varying internal dipole field with equatorial strength of 5 nT and polar strength of 10 nT when evaluated at this depth, which corresponds to a time varying dipole coefficient g10 of 3.3 nT when defined with respect to the equatorial radius of Saturn’s 1 bar surface.

Supplementary Materials

References and Notes

  1. The local magnetic Reynolds number is defined as Rm = UconvHσ/λ, where Uconv is the typical convective velocity; λ is the magnetic diffusivity, defined as the inverse of the product of electrical conductivity and magnetic permeability; and Hσ is the conductivity scale-height defined as Embedded Image.
  2. Due the divergence free nature of magnetic field, a poloidal-toroidal decomposition of the magnetic field, Embedded Image, is commonly adopted in the dynamo region. The toroidal magnetic field, Embedded Image has no radial component, and is associated with local electrical currents. The poloidal magnetic field Embedded Image has no azimuthal component if it is axisymmetric.
  3. Assuming 1 cm s–1 flow and an electrical conductivity of 2 × 105 S/m, Rm associated with a layer with a thickness of 0.2 RS would be ~30,000.
Acknowledgments: All authors acknowledge support from the Cassini Project. H.C. acknowledges Royal Society Grant RP\EA\180014 to enable an academic visit to Imperial College London, during which some of the work has been carried out. Funding: Work at Imperial College London was funded by Science and Technology Facilities Council (STFC) consolidated grant ST/N000692/1. M.K.D. is funded by Royal Society Research Professorship RP140004. H.C. is funded by NASA’s CDAPS program NNX15AL11G and NASA Jet Propulsion Laboratory (JPL) contract 1579625. Work at the University of Leicester was supported by STFC grant ST/N000749/1. E.J.B. is supported by a Royal Society Wolfson Research Merit Award. Work at the University of California, Los Angeles is funded by NASA JPL contract 1409809. K.K.K. is funded by NASA JPL contract 1409806:033. M.G.K. is funded by JPL under contract 1416974 at the University of Michigan. M.E.B. and T.A.B. are supported by the Cassini Project. Author contributions: M.K.D. led the instrument team and supervised the data analysis. H.C., K.K.K., and S.K. carried out the magnetometer calibration analysis. T.A.B. carried out the spacecraft attitude reconstruction. H.C., G.J.H., and G.P. carried out the magnetic field data analysis. M.E.B., E.J.B., S.W.H.C., M.G.K., C.T.R., and D.J.S. provided theoretical support and advised on the data analysis. All authors contributed to the writing of the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: The derived model parameters are given in Table 1 and tables S1 and S2. Fully calibrated Cassini magnetometer data are released on a schedule agreed with NASA via the Planetary Data System at https://pds-ppi.igpp.ucla.edu/mission/Cassini-Huygens/CO/MAG; we used data from rev 271 through to rev 280, excluding rev 277.
View Abstract

Navigate This Article