Magnetic field observations from the Cassini spacecraft are averaged into 1 × 1 RS (1 RS = 60238 km) meridional bins at Saturn. A Runge-Kutta procedure uses the bin-averaged field components to estimate the global configuration of Saturn's magnetic field and map field lines from the equator to Saturn's ionosphere. Within ∼18 RS of Saturn, the mapping gives a good representation of the meridional shape of Saturn's magnetic field and confirms that beyond L ∼ 6, the field departs from a dipole and becomes a smoothly-warped magnetodisk. The disk warps upwards by as much as ∼1 RS at radial distances of ∼15 RS, with the warping increasing with radial distance. When traced back to Saturn, the bin average field lines intersect the ionosphere at latitudes similar to those expected on the basis of recent magnetic field models; the ionospheric pierce latitudes can differ by as much as 6° from those expected from a simple offset dipole.
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 Empirical models of planetary magnetic fields can be divided into two types. In the first type (the “fit-model”), presumed functional forms are fitted to observations having analytic dependences on parameters such as distance and latitude. In the second type (the “bin-model”), the observed fields are averaged into relevant spatial bins without reference to prescribed functions. Models of the first type can provide an analytic formulation of the magnetic field, ensuring ∇ · B = 0 and continuity. However, the fit-models may not offer unique solutions. Models of the second type use no pre-conceived functional constraints and can describe the field in a unique fashion. Both types of models rely on suitably large numbers of observations of the field.
 Both types of models have a long history in describing the Earth's magnetosphere. The well-known Mead-Fairfield model represents a particular fit-model that has been utilized for decades in estimating the external terrestrial field [Mead and Fairfield, 1975]. The fitting process has been greatly extended in the Tsyganenko models, which embrace a number of satellite data sets and various currents and include effects of disturbance indices such as Dst or Kp [e.g., Tsyganenko, 2002]. On the other hand, bin-models have also served to define the topology of the terrestrial field. The fields of the magnetotail and that of the inner magnetosphere have been characterized using bin-averaging techniques [e.g., Fairfield, 1979; Nakabe et al., 1997], and the technique has been applied to derive a very sensitive model of both the symmetric and partial ring currents [Le et al., 2004].
 So far, all the magnetic field models at Saturn have been of the fit-model variety. Aligned with the spin axis, the internal field can be represented by only a few terms in a multipole expansion resulting from fits to the data [e.g., Davis and Smith, 1990; Dougherty et al., 2005; Burton et al., 2009]. Beyond L ∼ 7 RS, Saturn's field requires a correction to this dipole. From the Voyager observations, this correction was modeled using a “slab” ring current between 8 and 15 RS [Connerney et al., 1983]. The slab model has been employed by a number of Cassini-era studies of Saturn's magnetosphere [e.g., Sittler et al., 2006; Bunce et al., 2007], although its efficacy is questionable. More sophisticated models have appeared based on the Cassini observations themselves. A Tsyganenko-type model has been derived from Cassini magnetometer data; this model adds to the internal field the perturbations caused by magnetopause currents as well as a current sheet [Khurana et al., 2006]. By balancing pressure gradient, centrifugal and J × B forces, a ring current model has produced an extended magnetodisc based on Cassini field observations [Achilleos et al., 2010].
 This investigation uses the bin-average approach to find the magnetic field. Cassini has accumulated a sufficiently comprehensive sampling of Saturn's magnetosphere to make this approach reasonable. In cylindrical coordinates, the Br and Bz (meridional) components are averaged into 1 × 1 RS bins for the year 2007. Azimuthal or local time dependences are ignored, although the reader is cautioned that local time dependence becomes likely beyond radial distances of ∼15 RS. Examination of the meridional fields gives a good picture of Saturn's magnetodisk, ring current structure, and any warping that may be present.
2. Data Set and Data Processing
 Since Saturn Orbit Insertion (SOI) in July 2004, the Cassini magnetometer has measured magnetic fields throughout the magnetosphere of Saturn [Dougherty et al., 2004, 2005]. The instrument measures three components of the magnetic field at time resolution of seconds. This report considers magnetic field components in cylindrical coordinates in which the z axis is aligned with the spin axis of Saturn, r is the radial distance perpendicular to z, and ϕ represents an azimuthal phase angle. Because only meridional components Br and Bz are analyzed, this phase remains arbitrary and is ignored. To make the computations more tractable, the Br and Bz components are time-averaged into half hour bins for an entire year. Figure 1 shows the totality of these half-hour samples in rz-space. Of course, some regions were under-sampled or not sampled at all, and that is a drawback of this approach. The particular year of 2007 was chosen because Cassini sampled both high and low latitudes at a wide range of radial distances that year. A time span of one year was used to limit effects of possible seasonal changes at Saturn.
 Each half hour sample is associated with a specific (r,z) coordinate at which the measurement was made. The samples are averaged within 1 × 1 RS bins on a regular grid within the ranges 1 ≤ r ≤ 24 RS and −11 ≤ z ≤ 11 RS (1 RS = 60268 km). The isolated square in Figure 1 indicates one of these bins. Data gaps and extraneous measurements (field strength exceeding 104 nT) are omitted in these accumulations. A total of 4637 half hour samples (2318.5 hours) contribute to the bin averages. Of these, 2356 (1178.0 hours) are north of the equator, and 2281 (1140.5 hours) are south of the equator. An average bin holds 14.4 half hour samples (7.2 hours). Even though this number seems statistically sparse, the resulting bin averages exhibit a surprisingly consistent picture of the magnetic field topology.
3. Bin Average Magnetic Field
Figure 2 displays the bin averages on the regular (r,z) grid in a “whisker” format. The length of each whisker represents logarithm of the magnetic field strength, while the orientation represents the direction of the field. Bins with open circles represent those bins for which fewer than 2 samples are available. Open circles with whiskers in the outer magnetosphere indicate those bins for which a two-dimensional interpolation or extrapolation could be used to estimate the field; open circles with whiskers in the inner magnetosphere indicate blank bins that were filled using an offset dipole [Dougherty et al., 2005; Burton et al., 2009].
 The general features of Saturn's magnetic field are apparent in Figure 2. The equatorial field has a southward direction from the inner magnetosphere (r ∼ 2 RS) to beyond the orbit of Titan (r ∼ 20 RS). North and south of the equator at distances of ∣z∣ ∼ 6 RS, the field tends to be strongly radial (Bz ≈ 0) throughout approximately the same radial range. At greater distances from the equator (z ∼ 10 RS), the field becomes more dipole-like with significant Bz components. There is no indication of an X-configuration that might be associated with merging, nor is a true neutral sheet present in which Bz = 0 and Br abruptly changes sign at the equator. Because merging is a transient event, the long-term averaging of the data may mitigate against its appearance in the mean field configuration. Furthermore, reconnection X lines have been observed only on the nightside beyond ∼25 RS [Jackman et al., 2007; Hill et al., 2008], and this study does not extend to such distances.
4. Field Line Traces
 Given the mean field on a regular grid, the global configuration of the field can be estimated by simultaneously integrating the field line equations:
where dr and dz represent increments corresponding to distance ds along the field lines and B = (Br2 + Bz2)1/2 is the field magnitude. Equations (1) are solved numerically using a fourth-order Runge-Kutta procedure [e.g., Press et al., 1992]. The dr and dz steps are computed simultaneously using an algorithm adapted from an Excel spreadsheet procedure [Bilbao y Léon et al., 1996]. Field lines were traced from the equator beginning at r = 4, 6, 8, … 18 RS using a constant step size of ds = 0.1 RS. The Runge-Kutta process requires Br and Bz at non-grid points, and the non-grid values were computed using 2D LaGrange interpolation. The Runge-Kutta algorithm was tested using a dipole field, which reproduced the field lines to within ∼0.1 RS. When the field lines reach the edge of the bin area at r = 1.5 RS, the field traces are extrapolated to Saturn's ionosphere using an offset dipole [Dougherty et al., 2005; Burton et al., 2009].
 The consistency of the bin model field and the Khurana field at the bin centers was checked by examining (∇ · B)/B ≈ 0 where B is the magnitude of the field. The derivatives were estimated using five-point differencing for both fields in r and z. The left-hand side of the equation averaged 5.9 × 10−3 RS−1 for the bin field and 1.1 × 10−3 RS−1 for the Khurana field, the analytic version of which exactly satisfies the Maxwell equation. Both numerical averages are very small, although the bin value is six times larger than the Khurana value. Some errors are expected for not including azimuthal derivatives of B, for differencing errors, and for “noise” in the measured bin averages. Nevertheless, the bin field satisfies the ∇ · B = 0 condition, on the average, to better than 1% within the region considered by the binning.
Figure 3 shows the field line traces (red) resulting from the 2007 bin averages. For comparison, dipole field lines appear in green, field lines from the fitted field model in blue [Khurana et al., 2006], and those from the force-balance model are dotted lines [Achilleos et al., 2010]. The traces of the model fields employ the same Runge-Kutta algorithm used for the bin average traces, while the dipole model uses r = Lcos2λ.
Figure 3 reveals interesting details about the global topology of the field. First, the magnetic field lines depart considerably from those of a dipole, even at radial distances as small as ∼6 RS. This departure from a dipole has important consequences for where equatorial field lines map to the ionosphere. Second, the inflection points of the field generally lie above the equator, indicating a northward warping of the field similar to that found by examining current sheet crossings in the outer magnetosphere [Arridge et al., 2008]. At radial distances of ∼15 RS, this warping amounts to ∼1 RS and confirms a general “bowl” shape of the field. A similar warping of the nightside plasma sheet has been reported in ENA observations [Carbary et al., 2008]. Because the Sun and solar wind flow were south of Saturn's equator in 2007, northward displacement of the current sheet might be expected for a field averaged in local time. Third, the general shapes of the bin average traces agree fairly well with those derived from the Khurana and Achilleos fields, which has general implications for understanding force balance within the magnetosphere. However, the bin average lines tend to lie more southward than their model counter-parts. Fourth, the magnetic field lines are not symmetric about the equator, suggesting the southern aurora is not at the same latitudes at the northern aurora.
5. Mapping to the Ionosphere
 The magnetic field lines are traced to their ionospheric pierce points in Figure 4, which displays the plantocentric latitude of these points vs. equatorial crossing radius. The pierce latitudes of the bin average field (red) are compared with those of the fit model (blue), the force-balance model (dotted), the offset dipole (green), and a slab-current field (black). The ionosphere is assumed to lie 1000 km above Saturn's ellipse at the equator and at the poles (polar radius/equatorial radius = 0.902). Using an offset dipole, the field lines are traced from the inner edge of the bin average box (r = 1.5 RS) to the ellipsoidal ionosphere. Simulations of this tracing method using a centered dipole reveal errors in the pierce point latitude of ∼1% at 4 RS dropping to ∼0.1% at 20 RS.
 The 0.037 RS offset of Saturn's dipole generally produces different pierce latitudes in the north and the south, and this feature is common to all the models (except the Achilleos model because it assumes north-south symmetry). The pierce latitudes of the bin average field generally tend to lie somewhat more poleward than those of the three fit models. At large crossing distances (r > 10 RS), the models and the bin average field tend to map to pierce latitudes lower than those of the offset dipole. Conversely, at smaller crossing distances, the fit models and the bin average field map to pierce latitudes higher than those of the offset dipole. This mapping can be used to connect magnetospheric phenomena to ionospheric features. One application is to trace satellite L-shells to ionospheric pierce points. Table 1 lists these pierce latitudes for Saturn satellites lying within the bin region. Should the satellites produce ionospheric signatures, they should appear close to these latitudes.
Table 1. Latitudes of Pierce Points for Field Lines Tracing From Saturn Moons
2007 Observed Bin
 Averaging one year's worth of magnetic field observations into 1 × 1 RS bins exposes a warped magnetodisk at Saturn. The magnetodisk configuration begins at radial distances of ∼6 RS and continues until at least 20 RS. The shape of the field differs considerably from that of a dipole, and generally agrees with models that balance the magnetospheric forces and also with models derived from fitting magnetic field data to currents. The disk is warped northward of the equator; the warping increases with radial distance, amounting to ∼1 RS in z at r ≈ 15 RS. The ionospheric pierce latitudes of the bin-average field are similar to those predicted from tracing the model field lines. Depending on the L shell, these pierce points can differ from those of an offset dipole by up to ∼6° in planetocentric latitude.
 This research was supported in part by the NASA Office of Space Science under Task Order 003 of contract NAS5-97271 between NASA Goddard Space flight Center and the Johns Hopkins University.