We investigate the long-term motion of Saturn's north pole hexagon and the structure of its associated eastward jet, using Cassini imaging science system and ground-based images from 2008 to 2014. We show that both are persistent features that have survived the long polar night, the jet profile remaining essentially unchanged. During those years, the hexagon vertices showed a steady rotation period of 10 h 39 min 23.01 ± 0.01 s. The analysis of Voyager 1 and 2 (1980–1981) and Hubble Space Telescope and ground-based (1990–1991) images shows a period shorter by 3.5 s due to the presence at the time of a large anticyclone. We interpret the hexagon as a manifestation of a vertically trapped Rossby wave on the polar jet and, because of their survival and unchanged properties under the strong seasonal variations in insolation, we propose that both hexagon and jet are deep-rooted atmospheric features that could reveal the true rotation of the planet Saturn.
Images from Voyager 1 and 2 flybys in 1980 and 1981 showed a hexagonal feature at planetocentric latitude 75°N enclosing a strong eastward jet with peak speed of 120 ms−1 [Godfrey, 1988]. The same feature was later reobserved in 1990–1995 with ground-based telescopes and the Hubble Space Telescope [Sánchez-Lavega et al., 1993; Caldwell et al., 1993]. At those times, a large anticyclone ~11,000 km in length (the north polar spot (NPS)) was present in the region outside the hexagon, and it was proposed that the NPS impinging on the jet generated the hexagonal wave [Allison et al., 1990]. Because of this relationship, the NPS was assumed to move with the hexagon, and its motion was put forward as a possible imprint of Saturn's rotation [Godfrey, 1990; Sánchez Lavega et al., 1997]. The north polar region was not reobserved until 2007–2008 when Composite Infrared Spectrometer and visual and infrared mapping spectrometer instruments onboard the Cassini spacecraft revealed the hexagon in Saturn's night both in the temperature field [Fletcher et al., 2008] and in the cloud opacity at 5 µm [Baines et al., 2009]. In this paper we present a study of the region during 5.5 years (2008–2014), determining the precise motion of the hexagon vertices, the absence of the NPS, and the persistence and unchanging properties of the embedded eastward jet stream despite the strong seasonal radiation forcing at Saturn's poles [Pérez-Hoyos and Sánchez-Lavega, 2006]. In addition, we present the first analysis of the motion of the hexagon vertices in 1980–1981 and 1990–1991, extending the studied period to 33 years and allowing us to contextualize the movement of the hexagon in its relationship to the NPS.
2 Image Selection and Measurement Method
Two main data sets have been analyzed along the period 2008–2014: (1) Cassini imaging science system (ISS) images [Porco et al., 2004] obtained from 25 August 2008 to 28 November 2012 with the CB2 filter (wavelength 752 nm) with a typical spatial resolution ranging from 22 to 170 km/pixel. These images were gathered from the NASA Planetary Data System node. (2) Ground-based images captured with telescopes ranging from 0.36 to 2.2 m in aperture operating under the “lucky imaging methodology” [e.g., Law et al., 2006] obtained from December 2012 to January 2014 in the 380 nm–1 µm wavelength range and with a typical spatial resolution of 1900 km. A large number of these images were submitted by observers around the world to the International Outer Planets Watch—Planetary Virtual Observatory and Laboratory (IOPW-PVOL) database [Hueso et al., 2010a]. The data set was complemented with images obtained in different runs along this period with the PlanetCam Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU) [Sánchez-Lavega et al., 2012] and AstraLux [Hormuth et al., 2008] instruments mounted on the 1.23 m and 2.2 m telescopes at Calar Alto Observatory (southern Spain) that cover the visual range (380 nm–1 µm) at specific selected wavelengths. Figure 1 shows a set of images of the polar region. The hexagon stands out as a distinct albedo band outlined by outer and inner edges that also indicate the width of the eastward jet.
The Cassini images were navigated, polar projected, and measured using the software PLIA [Hueso et al., 2010b]. Two different and complementary software packages, LAIA [Cano, 1998] and WinJUPOS [Hahn and Jacquesson, 2011], were used to navigate and measure the ground-based images. The position (longitude in System III and planetocentric latitude) of each hexagon “corner” or vertex in the outer and inner edges were measured with a typical error of 2° in longitude but less than 0.5° in latitude.
The wind profile of the eastward polar jet, where the hexagon resides, was measured on Cassini ISS CB2 images (752 nm) corresponding to three periods: 29 November to 1 December 2008, 2–21 January 2009, and 28 November 2012. Three methods were employed to retrieve the wind profile: cloud tracking on image pairs separated by one planetary rotation (about 10 h), brightness correlation of one-dimensional zonal scans along each hexagon side at different latitudes [García-Melendo et al., 2011], and two-dimensional brightness correlation over polar projected images [Hueso et al., 2009]. Typical velocity errors coming from navigation and tracer misidentifications are ~10–15 ms−1.
In addition, we include new measurements of a set of Voyager 1 and 2 images in 1980 and 1981 and of a few available Hubble Space Telescope and ground-based images in 1990–1991 that display the feature. The details of the method of analysis of these images can be found in the works of Sánchez Lavega et al. [1997, 1993, 2000]. The list of analyzed images are given in the supporting information.
The 2008–2014 analysis shows a mean planetocentric latitude of the six vertices of the hexagon of 74.7° ± 0.2°, with a mean north–south width of 2.8° ± 0.5°. A linear fit to the longitude drift rates for each of the six vertices (Table 1 and Figure 2a) results, when combined, in a mean drift rate for the hexagon of ω = +0.0128 ± 0.0013°/d relative to the System III rotating frame, with angular velocity ΩIII = 810.7939024°/d [Desch and Kaiser, 1981; Seidelmann et al., 2007]. We also used the Linde–Buzo–Gray vector quantization method [Linde et al., 1980; Gray, 1984] to determine the hexagon's rotation period. In our case, vector values are the fixed position in the longitude and the latitude of the hexagon's six vertices. We look for the drift rate that, once subtracted, simultaneously minimizes the dispersion of our position measurements about six optimally chosen vertex positions. To automatically compute vertex positions, we use the Lloyd's iterative technique [Lloyd, 1957]. This method gives ω = +0.0131 ± 0.0010° day−1 (Figure 2b), in excellent agreement with the previous one. The mean drift rate combining both methods is <ω> = +0.0129 ± 0.0020° day−1. This yields an absolute angular velocity for the hexagon <Ω> = 810.7810°/d that translates into a hexagon speed relative to System III of −0.036 ms−1. The corresponding absolute rotation period for the hexagon in 2008–2014 is <τ> = 10 h 39 min 23.01 ± 0.01 s.
Table 1. Individual Drift Rates of the Hexagon Vertices in 2008–2013 and in 1980–1981a
Drift Rate (°/d)
Drift Rate (°/d)
aThe drift rate (ω) is relative to System III longitude and is the slope of the linear fits shown in Figure 2. Latitudes are given in planetocentric degrees. The error of the mean values is calculated as the standard deviation of the average values for individual vertices.
0.0107 ± 0.0005
74.5 ± 1.4
0.0125 ± 0.0003
74.6 ± 1.1
0.0133 ± 0.0003
74.8 ± 1.3
0.0130 ± 0.0005
75.0 ± 1.2
0.0126 ± 0.0004
74.8 ± 1.0
0.0146 ± 0.0005
74.6 ± 1.5
0.0128 ± 0.0013
74.7 ± 0.2
−0.0602 ± 0.0014
A track of the motion of the hexagon vertices in 1980–1981 shows also a small drift but in opposite direction with <ω> = −0.0602 ± 0.014 ° day−1 (<τ> = 10 h 39 min 19.6 ± 0.7 s), close to that of the NPS at the time (<ω> = −0.0444 ± 0.01 ° day−1; <τ> 10 h 39 min 20.3 ± 0.5 s) (Table 1 and Figure 2c) [Godfrey, 1988; Sánchez Lavega et al., 1997, 1993]. The small number of available data for 1990–1991 shows a lower drift rate for the hexagon (Figure 2d; <ω> = −0.0010 ± 0.007 ° day−1), again similar to that of the NPS, indicating that the motion of both features (hexagon and NPS) was coupled. The NPS anticyclone, initially proposed to originate the hexagon [Allison et al., 1990], is no longer present in the images from 2008 to 2014, and it was therefore a transient vortex. We conclude that the NPS is not necessary for the existence of the hexagon, but when present, it could have slightly disturbed the hexagon motion.
The hexagon encloses an intense and narrow eastward jet. Its velocity profile was measured in 1980–1981 at the altitude level corresponding to a pressure of 0.5–1 bar [Godfrey, 1988] and again in 2008 at level ~2–4 bar [Baines et al., 2009]. Here we present new measurements of the jet in 2008, 2009, and 2012 based on Cassini ISS CB2 images that approximately sense the same altitude level as the Voyager images [García-Melendo et al., 2011]. Figure 3 shows the wind profile in 2008–2009 and 2012 and compares it with that of 1980–1981 from the Voyagers. It is evident that there have been no significant changes on the jet intensity and shape, in spite of the marked seasonal insolation variability at those latitudes [Pérez-Hoyos and Sánchez-Lavega, 2006]. Geostrophic conditions prevail for this jet since the Rossby number Ro = u / f0L ~0.13, with a Coriolis parameter f0 = 2ΩIII sin ϕ0 = 3.17 × 10−4 s−1, a jet peak velocity of u = 120 ms−1 at planetocentric latitude ϕ0 = 75.5°N and L = 2900 km the jet width at half maximum. According to our data, the jet has a peak averaged curvature − < d2u / dy2 > ≈ 6–8 × 10− 11 m− 1s− 1(Figure 3) or − < d2u / dy2 > / β ≈ 20 − 40 violating the Rayleigh–Kuo instability criterion for a purely zonal jet both in a shallow barotropic atmosphere [Sánchez-Lavega, 2011] or in a deep atmosphere [Ingersoll and Pollard, 1982]. Here y is the meridional coordinate, β = (df / dy) = 2ΩIII cos ϕ0 / RS = 1.5 × 10−12 m−1 s−1 is the planetary vorticity gradient, and RS = 55,250 km is the radius of Saturn at the jet latitude.
An early Rossby wave model for the hexagon [Allison et al., 1990] simplified the atmospheric structure and proposed the wave to be forced by the NPS, which is not currently present. Two other studies have been presented to explain the origin of the hexagon: a barotropic linear instability analysis together with laboratory experiments of a fluid in a rotating tank [Barbosa-Aguiar et al., 2010] and a nonlinear numerical model using the Explicit Planetary Isoentropic Code (EPIC) code [Morales-Juberías et al., 2011]. Both studies predict a conspicuous coherent vortex street pattern that is not observed in Cassini images (Figure 1), and the numerical simulations using the EPIC model [Morales-Juberías et al., 2011] predict a nonconsistent phase speed for the hexagon pattern.
Here we interpret the hexagon as a stationary trapped Rossby wave that could be the manifestation of the meanders of a deep unstable polar jet. The three-dimensional Rossby wave is represented by a stream function ψ '(x, y, z, t) = ψ0 exp[(z / 2H) + i(kx + ly + mz − ω0t)] that for quasi-geostrophic conditions in a stratified atmosphere with no vertical shear (∂u / ∂z = 0) obeys the dispersion relationship [Vallis, 2006; Sánchez-Lavega, 2011]
Here ψ0 is the wave amplitude, k = 2π / Lx, l = 2π / Ly, m = 2π / Lz are the zonal, meridional, and vertical wave numbers, respectively, LD = NH/f0~1000 km is the Rossby deformation radius at the troposphere [Read et al., 2009b], ω0 is the wave frequency, H = (Rg*T)/g is the atmospheric scale height ( T is the temperature, g is the gravity acceleration, and Rg* = 3892 Jkg−1 K−1 is the gas constant), and N is the Brunt–Väisälä frequency Lindal et al. . The zonal and meridional wave numbers k = 2π / Lx and l = 2π / Ly are calculated taking Lx = 14,500 km from the hexagon zonal wave number 6 (hexagon side length), and Ly = 5785 km that assumes that the hexagon meridional extent (3° in latitude) corresponds to half the meridional wavelength. We take the hexagon velocity in System III in 2008–2014 as its wave speed cx = −0.036 ms−1. From the jet profile (Figure 3), we take u = 120 ms−1 (cx < <u) yielding− < d2u / dy2 > / u ≈6 × 10−13 m−2. From these numbers, we see in equation ((1)) that and ℓ2 > [< d2u / dy2 > / (−u)]. Therefore, the zonal wave number dominates because of the strong eastward flow velocity. This implies that m2 < 0, and the Rossby wave is vertically trapped within the region of positive static stability in Saturn's atmosphere in agreement with a previous analysis [Allison et al., 1990]. The vertical extent of this trapped region for the wave is unknown and can penetrate well below the lower water clouds at ~10 bar altitude level. To have m2 > 0 (i.e., for vertical propagation with real Lz), the above numbers require Ly > 15,700 km, which is highly unrealistic since the meridional wave oscillation will reach the pole in the north and overcome the neighboring jet in the south.
Because of Saturn's rotation axis tilt of 26.7°, the planet is subject to strong seasonal insolation changes along its year of 29.475 Earth's years [Pérez-Hoyos and Sánchez-Lavega, 2006], with long alternating periods of light and darkness that affect temperatures above the ~2 bar altitude level [Bézard et al., 1984; Sánchez Lavega et al., 1997; Fletcher et al., 2010]. The survival of the hexagon and the stability of the polar jet to the varying radiative forcing suggest that they extend deep in the atmosphere. This is in agreement with what has been deduced for other Saturn jets from studies of a long-lived anticyclone [García-Melendo et al., 2007] and from the giant Great White Spot storm in 2010 [Sánchez-Lavega et al., 2011; García-Melendo et al., 2013]. From those works, the zonal winds were found to extend down to the ~10 bar level or deeper with no vertical shear.
Currently there is an open debate on Saturn's rotation period [Sánchez-Lavega, 2005; Gurnett et al., 2010]. The rotation is determined by the periodic radio emissions with origin in the magnetic field, assumed to be tied to the deep interior. However, Saturn's rotation period is not yet determined since the radio measurements from Saturn's kilometric radiation (SKR) have varied in the last 30 years showing two simultaneous periods that differ by 15 min, one for each hemisphere [Gurnett et al., 2010]. Alternatives based on Saturn's gravitational field [Anderson and Schubert, 2007] and on the stability of the atmospheric zonal jet stream pattern [Read et al., 2009a] also differ from each other and are 5 to 7 min shorter than the mean SKR period. Our long-term study shows that the hexagon has indeed an extremely steady rotation period, with its associated jet remaining unchanged by the strong seasonal cycle, properties that are consistent with deeply rooted features. We suggest that the current steady rotation period of the hexagon of 10 h 39 min 23.01 s ± 0.01 s, free of disturbances from the NPS, could represent the period of Saturn's internal “solid body” rotation.
We gratefully acknowledge the work of the Cassini ISS team that made the data available. Part of the data were based on the observations obtained at Centro Astronómico Hispano Alemán, Observatorio de Calar Alto MPIA-CSIC, Almería, Spain. This work was supported by the Spanish MICIIN projects AYA2009-10701, AYA2012-38897-C02-01, and AYA2012-36666 with FEDER support, PRICIT-S2009/ESP-1496, Grupos Gobierno Vasco IT765-13, and UPV/EHU UFI11/55. A list of the participants of the International Outer Planet Watch Team (IOPW-PVOL) appears in the supporting information.
The Editor thanks two anonymous reviewers for their assistance evaluating this paper.