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Keywords:

  • ring current;
  • Saturn's magnetosphere;
  • solar wind interaction

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] We have examined residual magnetic field vectors observed in Saturn's magnetosphere during the first 2 years of the Cassini mission and have fit them to a simple axisymmetric model of the ring current in the middle magnetosphere. We then examine the variations of the ring current parameters with size of the magnetosphere. In addition, we obtain secondary parameters, including the value of the axial field at the center of the ring (equivalently Saturn's Dst) Bz0, the total current IT flowing in the modeled ring current region, and the ratio of the ring current magnetic moment relative to the magnetic moment of Saturn's dipole field, kRC. Results show that the derived parameters increase significantly with system size, due principally to the increasing radius of the outer edge of the ring. We consider the implications of the response of the magnetic moment of the ring current to changing magnetospheric size, by theoretical consideration of the magnetic moment of individual particles in the ring current. The strong positive correlation of the ring current magnetic moment with system size suggests a system in which the ring current is dominated by inertia currents, rather than by thermal effects as in the case of the Earth, with magnetosphere-ionosphere coupling maintaining the angular velocity of the plasma. The variations of Saturn's ring current parameters with system size found in this study are shown to be closely compatible with the size variations in response to the solar wind dynamic pressure recently determined from Cassini data.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Prior to the arrival of Cassini, the magnetic environment of Saturn had been sampled by just three spacecraft, Pioneer-11 (1979), Voyager-1 (1980), and Voyager-2 (1981). An interesting feature of the magnetic field in the central part of the magnetosphere observed in the Voyager data was the signature of a significant “ring current” carried by the charged particles of the magnetospheric plasma. The existence of this current system was first recognized in the Pioneer-11 data by Smith et al. [1980] and in the Voyager data by Ness et al. [1981, 1982] and was subsequently modeled in detail for the Voyager encounters by Connerney et al. [1981b, 1983]. The eastward flowing ring current produces a northward perturbation field close to the planet hence reducing the planetary field, while producing a southward perturbation field further away, thereby enhancing the planetary field in the outer magnetosphere. Increased radial fields also occur, which are in opposite directions on either side of the equatorial plane. Therefore the magnetic effect of the ring current is such that it stretches the planetary field lines away from the planet. The ring current inferred from Voyager data is located at around 8–16 RS in radial range, extends a few RS wide on either side of the equatorial plane, and carries a total current of the order of ∼10 MA. The magnetic perturbation in the poloidal field components are ∼10 nT in magnitude, and are therefore comparable to the magnitude of the planetary magnetic field near to ∼10 RS, which is ∼20 nT.

[3] While the Pioneer-11 flyby data was not initially modeled in this way, the Connerney formulation was eventually employed by Bunce and Cowley [2003] and concurrently by Giampieri and Dougherty [2004]. The result of this modeling work showed that the presence of the ring current was qualitatively similar in both the Pioneer and Voyager encounters. Bunce and Cowley [2003] also showed, however, that the ring current was thinner and located somewhat closer to the planet during the Pioneer-11 flyby than during the Voyager-1 pass, reflecting the more compressed nature of the magnetosphere during the latter flyby. The near-equatorial noon-sector magnetopause crossings occurred at 17 RS and 23–24 RS for Pioneer-11 and Voyager-1, respectively.

[4] The first in situ observations of Saturn's magnetosphere were obtained by Cassini some 23 years later during Saturn Orbit Insertion (SOI) in June 2004. During this first pass of Cassini, Dougherty et al. [2005] showed that the SOI data are not well described by the Voyager-1 model but require a current disk that extends to greater radial distances in the dawn sector in a relatively expanded magnetosphere, a result which was later confirmed by Alexeev et al. [2006]. In addition, Alexeev et al. [2006] have suggested on the basis of these Pioneer, Voyager, and Cassini results that there is evidence for a variation in the ring current parameters as a result of changing solar wind conditions, indicating that the ring current magnetic moment increases with increasing system size. Therefore here we investigate the variation of the ring current modeled parameters from the first 2 years of magnetic field data from the Cassini mission, employing the simple Connerney et al. [1983] model (discussed below), and compare its variability with the location of the subsolar magnetopause. The location of the magnetopause acts as a proxy for the upstream solar wind dynamic pressure, and the nature of this variation will be investigated here in the context of the ring current properties.

2. Magnetic Field Models

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[5] In order to understand the magnetic field vectors from the Cassini spacecraft in terms of the ring current perturbations, we first subtract a model of Saturn's internal magnetic field from the measured values. This yields residual field vectors which are due to external current systems, these are principally due to the ring current in the inner and middle magnetosphere but also have less significant contributions in the outer regions from the magnetopause and tail current systems. Subsequently, we compare a suitably parameterized ring current model (plus a simple correction for the fields due to the magnetopause and tail) with the residual field vectors and vary the parameter values until a good match is achieved. The magnetic field models which we employ are described below.

[6] First, the internal field of the planet is taken to be the “Cassini” model of Dougherty et al. [2005], which was derived specifically from the SOI data set and constitutes the closest measure of the internal planetary field achieved by Cassini thus far. The model is axially symmetric about the planet's spin axis and employs three terms, namely the axial dipole, quadrupole, and octupole terms. The corresponding coefficients are g10 = 21084, g20 = 1544, and g30 = 2150 nT, with reference to a Saturn radius RS (the equatorial 1 bar radius), taken to equal 60,268 km. It is important to note, however, that none of our conclusions would be substantially altered if we had employed other commonly referred to internal field models such as the equally axisymmetric SPV or Z3 models of Davis and Smith [1990] and Connerney et al. [1984], respectively.

[7] To model the ring current, we employ the simple model of Connerney et al. [1981b, 1983], which due to its axisymmetric nature allows such secondary parameters as the ring current magnetic moment and the total current flowing in the system to be calculated in a simple manner. The model was first developed to describe the azimuthal current sheet in Jupiter's magnetosphere [Connerney et al., 1981a] and was later adapted with modified coefficients for Saturn. The basic element in this model is an axially symmetric disc of current of constant half-thickness D, which has an inner radius at a cylindrical radial distance R, and extends radially to infinity. The azimuthal current which flows within the disc varies inversely with the distance from the axis. A current disc of finite radial extent is then simulated by adding another similar disc of larger inner radius, whose current intensity is the same as that of the first but which flows in the opposite sense. The current density distribution of the model is thus given by

  • equation image

for R1ρR2 and −DzD, and is zero otherwise. The Saturn coefficients obtained by Connerney et al. [1983] from fits to the Voyager data were R1 = 8 RS, R2 = 15.5 RS, D = 3 RS, and μoIo = 60.4 nT, while the values obtained for the fit to the Pioneer-11 flyby data by Bunce and Cowley [2003] were R1 = 6.5 RS, R2 = 12.5 RS, D = 2 RS, and μoIo = 76.5 nT (those obtained by Giampieri and Dougherty [2004] being similar). It should be noted that this current system has an essentially “square” cross section, compared with the more disc-like system at Jupiter (i.e., where R1 = 5 RJ, R2 = 50 RJ, and D = 2.5 RJ). Unlike the situation at Jupiter where we may then use analytic approximations to calculate the external field model, here we must obtain model field values obtained numerically from the exact integral formulas as in the original work of Connerney et al. [1981a, 1981b, 1983]. This is a direct result of the “squarer” nature of Saturn's ring current distribution.

[8] As mentioned above, in addition to the planetary field and the field of the ring current, there are two additional external fields that pervade the central magnetosphere. The first is the fringing field of the tail current system, directed northward at Saturn in the equatorial plane, contained within the magnetopause. The second is the interior southward field due to the magnetopause currents that shield both the planetary and the ring current fields. Modeling of the equatorial Bz field due to the magnetopause and tail current systems in the relevant region shows a near-linear dependence on distance along the planet-Sun line (X), varying from small negative values near to the magnetopause through to small positive values at similar distances on the nightside and only a weak dependence in the transverse (Y) direction [Alexeev and Feldstein, 2001; Alexeev and Belenkaya, 2005]. Consequently, we have assumed this basic form in fitting a simple empirical model to the Bz component of the residual field. Specifically, we have employed the following formula

  • equation image

In this formula the combined effects of the magnetopause and tail fields are assumed to vary linearly from a value B1 near to the magnetopause (at X1), to a value B2 at a similar distance in the tail (at X2). Importantly, it will be shown that this effect produces fields which are typically only ∼10% of the main perturbation fields in the inner and middle magnetosphere due to the ring current itself. While the addition of this effect in the model enhances the overall fit to the residual fields, it is worth noting that the Connerney et al. [1983] ring current model parameters would be essentially unchanged if the effect were not included. Details of how we fit the model to the data are given in the following section.

3. Cassini Modeling Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[9] In this study we have investigated the Cassini magnetic field data from the first 2 years of Cassini operations at Saturn, constituting data from 27 orbits. In order to obtain fits to the ring current magnetic field, we first subtract the “Cassini” model of the internal field of Saturn as described above. In general, both the ring current parameters of the Connerney et al. [1983] model and the parameters of the simple magnetopause and tail contribution are then varied until a good fit is found. This is achieved in a straightforward and physical way by examining the features of the residual magnetic field that are good indicators of the ring current perturbation. First, we choose the values of the inner edge R1 (RS) and outer edge R2 (RS) of the current ring by relating the local extrema in the model Bz curve to the features in the observed ΔBz profile. Second, we note that the model Bz field at (and near) the origin is given by

  • equation image

[see Edwards et al., 2001]. The desired amplitude of the ring current ΔBz field near to closest approach of the spacecraft with Saturn therefore defines a relationship between μoIo and D. In general, the iteration of these two parameters together then provides the combination that gives the best fit to ΔB while holding Bzo constant. However, owing to the near-equatorial nature of the Cassini trajectory throughout much of the interval studied here, the half-thickness D of the current sheet is rather difficult to determine with any accuracy, and this difficulty will be addressed below. Finally, for the magnetopause and tail field contribution from equation (2), B1 is the difference between the observed residual field at X1 and the field of the ring current. As such, the ring current field plus B1 represents the field in the initial magnetopause region (i.e., at the maximum X position in the KGS coordinate system (in this system Z points along the rotation axis and the X-Z plane contains the Sun, with X positive toward the solar direction). Similarly, the ring current field plus B2 gives a representation of the field in the tail region (i.e., at the minimum XKGS position). We have then determined the best set of ring current parameters discussed above, plus the B1 and B2 values that describe the overall data.

3.1. Data Selection and Assumptions

[10] Of the 27 orbits of Cassini which we have examined, we have selected 17 as appropriate to our study, for reasons which we now discuss. We illustrate this discussion by showing an example of the Cassini magnetic field data from Revolution 4. In Figure 1 we show 6 (Earth) days of data, i.e., days 65–70 2005, which span the interval from the inbound magnetopause encounter (leftmost dashed line marked “MP”), through closest approach (central dashed line marked “CA”), to radial distances of ∼25 RS outbound. The local time of the spacecraft inbound is ∼1000 h and similarly 0500 h outbound. With regard to the fitting, the inbound and outbound data are considered equally in the inner region where the field is expected to be near axisymmetric. However, at larger distances we have principally considered the inbound dayside data to which the Connerney et al. [1983] model is most appropriate, rather than the dawn and nightside data where (at least in the midnight sector) the ring current at large distances merges with the tail current sheet [e.g., Alexeev et al., 2006]. Thus the ring current model we derive here is most appropriate to the dayside magnetosphere, though it should apply to significant distances at all local times. Correspondingly, however, we relate the ring current parameters derived here to the magnetopause distances observed on the dayside inbound passes.

image

Figure 1. Plot showing magnetic field and position data for Revolution 4 of the Cassini spacecraft. Data for 6 (Earth) days are shown, corresponding to days 65–70 inclusive of 2005. Shown are the residual field components in cylindrical coordinates referenced to the planet's spin (and magnetic) axis, namely, from the top downward, (a) ΔBρ, (b) ΔBz, and (c) ΔBϕ (in nT). These residuals are the measured field minus the “Cassini” model of the internal planetary field (which is zero in the case of the azimuthal component). The solid line in Figure 1a shows the ρ component of the modeled ring current field, which is essentially zero, while Figure 1b shows the full model of the residual z component, which includes the ring current plus the magnetopause and tail contribution. The dashed line in Figure 1b shows the contribution due to the simple magnetopause and tail field model, taken to vary linearly with distance (X) along the planet-Sun line, which is relatively small compared with the main ring current perturbation. The ring current model parameters in this case are R1 = 6 RS, R2 = 20 RS, D = 2.5 RS, and μoIo ≈ 35.6 nT, and the magnetopause and tail model parameters are B1 = 0.0 nT at X1 ≈ 23 RS, and B2 = 4.0 nT at X2 ≈ −6 RS. Also shown are (d) the position of the spacecraft, namely the cylindrical radial distance from the planet's spin (and magnetic axis), ρ, and (e) the distance along this axis from the equatorial plane, z. Local time information is given at the bottom of the plot. The left-hand vertical dashed line marked “MP” shows the position of the last inbound magnetopause crossing (at a spherical radial distance of 23.7 RS), while the central vertical dashed line marked “CA” shows the closest approach of the spacecraft to the planet. The gray regions indicate the intervals during which the spacecraft lay within the cylindrical radial range of the modeled ring current, that is within 6 ≤ ρ ≤ 20 RS (marked “RC”).

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[11] In Figures 1a–1c we show the residual field components (measured minus “Cassini” internal field model) versus time in cylindrical coordinates referenced to the planet's spin (and magnetic) axis. The azimuthal component is also shown here for completeness, but it is noted that since the “Cassini” model is axisymmetric and hence has zero azimuthal component, the value shown here represents the full azimuthal field measured. Similarly, since the ring current model is also axisymmetric, the model's azimuthal field is also zero. Figures 1d and 1e show the location of the spacecraft in cylindrical (ρ, z) coordinates, while the local time is indicated at the bottom of the figure. The solid line near zero in Figure 1a indicates the ρ field component of the ring current system modeled using the Connerney et al. [1983] formulation as outlined in section 2, while in Figure 1b the solid line indicates the ring current model plus the magnetopause and tail contribution (indicated by the dashed line). Note that the field of the magnetopause and tail model is generally much smaller than the main perturbation field due to the ring current itself, particularly in the region closest to the planet (at CA). The parameters of the Connerney et al. [1983] model in this case are R1 = 6.0 RS, R2 = 20.0 RS, D = 2.5 RS, and μoIo = 35.6 nT, while B1 = 0.0 nT and B2 = 4.0 nT, at X1 ≈ 23 RS and X2 ≈ −6 RS (the latter values having 1 nT resolution). These values have been chosen using the method described above, with the exception of the half-thickness D, the reason for which will now be described.

[12] It can be seen from Figure 1 that the spacecraft is located very close to the equatorial plane during Revolution 4, this also being true of more than 50% of the orbits investigated. As a result of this, we see that the modeled Bρ field is zero, as we would expect for the Connerney et al. [1983] model which assumes that the center of the ring current is located exactly in the equatorial plane at all radial distances. However, the measured residual ΔBρ field is significantly nonzero, which indicates that in reality this simple assumption is not valid. In fact, the residual ΔBρ field in Figure 1 is observed to be negative almost entirely throughout the pass, which indicates that the ring current is displaced northward of the planet's equatorial plane, presumably due to the magnetospheric asymmetry imposed by the flow of the solar wind during northern winter solstice conditions. If the spacecraft passed from north to south through the structure, then it would be possible to characterize both the offset and the thickness. However, owing to the equatorial nature of these orbits this is not possible with these data. Instead, we fix D = 2.5 RS (a value which was chosen to compliment Pioneer-11 and Voyager-1 modeling results) and concentrate on modeling the residual ΔBz. The level of the consequent uncertainties in the other ring current parameters will now be discussed.

[13] Consideration of the Biot-Savart law indicates that the radial distribution of the ring current ΔBz field in the equatorial plane will depend principally on the radial distribution of the total current flowing in the current sheet, integrated in z through its width, and will not depend sensitively on the distribution of that current about the equator, provided it is reasonably compact. Furthermore, the zero divergence of the field also guarantees that ΔBz at a given radius will vary only slowly with distance from the sheet center. In this case, the models fitted to ΔBz using an equatorially centered current sheet with fixed D should provide a good indicator of the total (width integrated) radial current distribution, even if the detailed distribution of that current about the equator remains somewhat uncertain. In terms of the parameters of the Connerney et al. [1983] model, this means that the product μoIoD is well determined from the fit, even if μoIo and D individually are not. This fact was emphasized in the original modeling work by Connerney et al. [1981a, 1981b] and reiterated by Vasyliunas [1983]. To demonstrate that this is the case, in Figure 2a we show the Bz field of the Connerney et al. [1983] ring current versus ρ at z = 0 (i.e., at the center of the model ring current), for three models as follows. The solid line shows the Connerney et al. [1983] Voyager model with R1 = 8.0, R2 = 15.5, D = 3 RS, and μoIo = 60.4 nT. The dot-dashed line then represents the same model parameters except with D = 2 RS and μoIo = (3/2) × 60.4 nT, while the dashed line represents the same model except with D = 4 RS and μoIo = (3/4) × 60.4 nT. All the models thus have the same value of the width-integrated current intensity, proportional to μoIoD. We see that these curves are all close to each other in value, with the most noticeable differences occurring at the edges of the ring current. At the inner edge the maximum difference between the three models is ∼5 nT compared with a total field of ∼17 nT, whereas at the outer edge the effect is smaller at ∼3 nT. Therefore Figure 2a shows that for a fixed value of μoIoD the ring current ΔBz field does not vary greatly at the sheet center as D is varied by a factor of 2. In addition, Figure 2b shows the Bz field of the Connerney et al. [1983] ring current versus ρ for the same Voyager model as in Figure 2a but for three different locations relative to the equator. The solid line shows the model values calculated at z = 0 as in Figure 2a, while the dashed line and the dot-dashed line show the field values at z = 1.5 RS and z = 3.0 RS, respectively. We therefore see that for near-equatorial values of z, the model Bz field values of the Connerney et al. [1983] ring current remain approximately fixed. These results thus justify our assertion above that the fitted models provide a good indication of the total radial current distribution, even if the locations of the current center and its thickness remain somewhat uncertain. As such, the further ring current parameters we derive from our results, such as the ring current magnetic moment and the total current, are also expected to be well estimated.

image

Figure 2. Plots of the Bz field component versus the cylindrical radial distance ρ from the symmetry axis for the Connerney et al. [1983] ring current model. (a) A comparison of the fields for different model parameters at the center of the ring current, z = 0. In each case the inner and outer radii are at R1 = 8 RS, R2 = 15.5 RS, respectively, and the solid line also corresponds to D = 3 RS and μoIo = 60.4 nT (the Voyager Connerney et al. [1983] parameter set). The dot-dashed line is for D = 4 RS, and μoIo = (3/4) × 60.4 nT, and the dashed line is for D = 2 RS, and μoIo = (3/2) × 60.4 nT. Thus the thickness D and current parameter μoIo are changed such that the current intensity integrated through the current sheet is held constant. (b) The parameters are held fixed at the Voyager values throughout, but Bz is shown at differing distances z from the center plane. The solid line is for z = 0 (as in Figure 2a), while the dashed and dot-dashed lines are for z = 1.5 RS and 3 RS, respectively, the latter corresponding to the outer edge of the current system.

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[14] The final data selection criterion relates to the timing of the last observed inbound magnetopause crossing with respect to the spacecraft encounter with the outer edge of the ring current, since it is a prime aim of our study to relate the ring current parameters to the size of the magnetosphere. In nine out of the 27 orbits studied here, the model outer edge of the ring current was encountered several days after the observation of the last magnetopause encounter inbound. This occurred when the spacecraft trajectory moved almost parallel to the magnetopause boundary on approach to Saturn but did not cross it. In some cases the magnetopause was last crossed up to 7 days before closest approach, and as such we are unable to be confident that the boundary location accurately reflects the conditions when the middle/inner magnetosphere was traversed several days later. The approach we have used therefore is to exclude any orbits where there is a significant interval (i.e., >4 days) between the last inbound magnetopause crossing and closest approach. We note that there is then no additional requirement to select passes on the basis of local time, as all of the inbound magnetopause observations that we finally select (shown in Table 1) lie within the relatively narrow window between ∼0800 and ∼1015 UT. In conclusion, 17 orbits of Cassini remain for study, along with the two previous flyby models which are used for comparison.

Table 1. Results of the Ring Current Modeling for Both the Cassini Data Used in This Study, and the Previous Results for the Pioneer-11 and Voyager-1 Flybysa
Year/doyPassR1/RSR2/RSμ0I0/nTBz0/nTIT/MAkRCMP/RsdoyRM/RspSW/nPa
  • a

    The first two columns show the year and day numbers and orbit name, respectively, followed in the third, fourth, and fifth columns by the modeled ring current parameters R1, R2, and μoIo. For the Cassini data the latter parameter is determined assuming D = 2.5 RS, while Pioneer-11 and Voyager-1 models are for D = 2 and 3 RS, respectively. The sixth, seventh, and eighth columns indicate the derived parameters Bz0, IT, and kRC as discussed in section 3. For each pass we also show in the ninth and tenth columns the spherical radial distance of the observed magnetopause and the day on which it was observed. Finally, in the 11th and 12th columns we show the calculated subsolar magnetopause location and the corresponding solar wind dynamic pressure derived using the Arridge et al. [2006] model.

79/243–245P116.512.576.511.09.60.2117.024317.00.096
80/317–318VG18.015.560.410.511.50.3823.531723.50.025
04/181–184SOI6.520.063.716.017.20.6830.618124.80.018
04/300–305TA7.518.063.512.013.30.5021.530020.20.045
04/348–353TB7.019.050.311.012.00.4721.934820.60.042
05/045–050T37.020.053.312.013.40.5528.34525.50.016
05/065–070Rev 46.020.035.610.010.30.3823.76621.90.032
05/085–090Rev 56.018.037.510.09.90.3219.88718.70.065
05/102–107Rev 66.016.044.011.010.30.2919.010317.40.091
05/121–126Rev 76.516.045.410.09.80.2918.512117.90.079
05/138–143Rev 87.022.050.712.013.90.6529.013726.10.014
05/157–162Rev 96.516.054.512.011.80.3520.415719.40.054
05/175–180Rev 105.015.031.710.08.40.1919.717518.80.064
05/193–198Rev 117.016.056.611.011.20.3521.519320.30.045
05/212–217Rev 127.020.048.811.012.30.5125.821123.40.023
05/246–251Rev 146.015.041.710.09.20.2318.124616.70.099
05/264–269Rev 156.518.042.110.010.30.3522.626420.10.045
05/283–288Rev 167.017.039.38.08.40.2818.828317.30.086
06/54–59Rev 218.015.056.58.08.50.2720.15417.10.089

3.2. Comparison of the Ring Current Modeled and Derived Parameters With Subsolar Magnetopause Location

[15] In Figure 3, we now show three examples from the Cassini modeling which illustrate the variability of the ring current distribution on the size of the system. We show only the residual ΔBz field, since as we explained above, we are no longer fitting to the residual ΔBρ field. The top panel shows the residual ΔBz field with the solid line representing the chosen model field and the dashed line indicates the proportion which is due to the magnetopause and tail fields, while the bottom two panels indicate the location of the spacecraft in cylindrical (ρ, z) coordinates. The local time is given at the bottom of the plot and ranges between ∼0900 and ∼0500 UT from inbound to outbound. Each of the examples in Figure 3 shows an interval of data which includes the last inbound magnetopause crossing (indicated by a dashed line marked “MP”), through closest approach (shown by the dashed line marked “CA”), extending outbound to where the spacecraft is at radial distances of ∼25–35 RS. Where the outbound MP is included in the plot, it is also indicated by a dashed line. The grey region marked “RC” encompasses the ring current region as modeled individually for each case. In Figure 3a we show data from Revolution 16 (days 283–288 of 2005), which is an example of a compressed magnetosphere during an interval of high solar wind dynamic pressure. The magnetopause crossing marked on the left of the plot occurs on day 283 at ∼1135 UT at a spherical radial distance of 18.8 RS, and at a local time of ∼0930 LT. The outer edge of the inbound ring current is rather difficult to determine due to the disturbed outer magnetospheric conditions but is associated with a negative minimum in the residual ΔBz field at ∼17 RS, which is reproduced by the model. Following the local minimum, the residual field increases near-linearly with time before peaking at a positive value of 13 nT, associated with the inner ring current radius. The two model edge values indicated by these data are thus R1 = 7.0 and R2 = 17.0 RS, along with a best fit value of μoIo = 39.3 nT (and D = 2.5 RS, as assumed throughout). The chosen magnetopause and tail field model parameters are B1 = −1.0 nT and B2 = 3.0 nT, at X1 ≈ 16 RS and X2 ≈ −7 RS. It can be seen that the chosen model is a good representation of the data during the inbound portion of the plot but fits less well to the outbound data. During the outbound pass, we see that the large-scale planetary-modulated wave field which is found to be present throughout much of the Cassini data [e.g., Cowley et al., 2006] has a particularly large-amplitude in the z component, which makes the fitting of the ring current parameters more problematic.

image

Figure 3. Examples of Cassini magnetic field data and the fitted model external field values for various states of the magnetosphere, shown in a similar format to Figure 1, showing (a) a fairly compressed magnetosphere, (b) “typical” conditions, and (c) an example of an expanded magnetosphere. Figure 3a shows in the top panel the z component of the residual field ΔBz (nT), plotted over 6 (Earth) days during Revolution 16 of the Cassini spacecraft, corresponding to days 283–288 of 2005. The residual field is the measured field minus the “Cassini” model of the internal planetary field. The solid line shows the total modeled ring current field plus contribution from the magnetopause and tail field model. The dashed line indicates the modeled magnetopause and tail field model alone, taken to vary linearly with distance (X) along the planet-Sun line. The ring current parameters are R1 = 7 RS, R2 = 17 RS, D = 2.5 RS, and μoIo ≈ 39.3 nT, while the magnetopause and tail field model are B1 = −1.0 nT at X1 ≈ 16 RS, and B2 = 3.0 nT at X2 ≈ −7 RS. The two lower panels show the position of the spacecraft, namely the cylindrical radial distance from the planet's spin (and magnetic axis) ρ, and the distance along this axis from the equatorial plane, z, both in RS, while local time (in hours and minutes) is indicated at the bottom of the figure. The left-hand vertical dashed line marked “MP” shows the position of the last inbound magnetopause crossing (at a spherical radial distance of 18.8 RS), while the central vertical dashed line marked “CA” shows the closest approach of the spacecraft to the planet. The first outbound magnetopause crossing is also marked in this case. The gray regions indicate the intervals during which the spacecraft lay within the cylindrical radial range of the modeled ring current (marked “RC”). Also shown are (b) the data from Revolution 11, corresponding to days 193–198 of 2005 in the same format as Figure 3a. Here the model parameters are R1 = 7 RS, R2 = 16 RS, D = 2.5 RS, μoIo ≈ 56.6 nT, B1 = −3.0 nT at X1 ≈ 18 RS, and B2 = 4.0 nT at X2 ≈ −7 RS, and the last inbound magnetopause (shown on the left) is located at a spherical radial distance of 21.5 RS. Finally shown are (c) data for 7 days of the Titan 3 orbit, corresponding to days 44–50 of 2005. The chosen model parameters for this example are R1 = 7 RS, R2 = 20 RS, D = 2.5 RS, μoIo ≈ 53.3 nT, B1 = 0.0 nT at X1 ≈ 25 RS, and B2 = 2.0 nT at X2 ≈ −6 RS. The last inbound magnetopause is located at a spherical radial distance of 28.3 RS.

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image

Figure 3. (continued)

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image

Figure 3. (continued)

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[16] In Figure 3b, we show 6 days of data from Revolution 11 (days 193–198 of 2005) in the same format as discussed above. This is an example of a “typical” size magnetosphere, where the last inbound magnetopause position marked on the left of the plot is observed on day 193 at ∼2125 UT at a spherical radial distance of 21.5 RS, and at a local time of ∼1035 LT. Again, the residual ΔBz field increased near-linearly with time during the inbound pass, plateauing at a positive value of ∼12 nT near to closest approach, associated with the ring current inner edge. In this example, the inner and outer edge values are chosen to be R1 = 7.0, and R2 = 16.0 RS, with a best fit value of μoIo = 56.6 nT. In this case, the magnetopause and tail field model parameters are B1 = −3.0 nT and B2 = 4.0 nT, at X1 ≈ 18 RS and X2 ≈ −7 RS. As in Figure 3a, we can see the large-scale planetary-modulated wave superposed on the residual fields of the middle magnetosphere. We also note the difference between the residual ΔBz field inbound and outbound, which is accounted for quite well by the asymmetry associated with the magnetopause and tail field model. Finally, in Figure 3c we show 7 days of data from the Titan 3 orbit (days 44–50 of 2005), again in the same format as discussed above. This orbit represents an example of an expanded magnetosphere during more rarefied solar wind conditions. This is evidenced by the location of the last inbound magnetopause crossing on day 45 at ∼0140 UT, which lies at a radial range of 28.3 RS at ∼0930 LT. Here we see a “quieter” magnetosphere during the inbound pass than for the previous two examples in the sense that the planetary period oscillations are less evident. Once more, the outer and inner edge of the ring current is chosen from the location of the (slightly) negative minimum and positive maximum in the residual ΔBz on the inbound pass. The values for the fitted model for this example are R1 = 7.0, and R2 = 20.0 RS, along with a best fit value of μoIo = 53.3 nT. The magnetopause and tail field model parameters are B1 = 0.0 nT and B2 = 2.0 nT, at X1 ≈ 25 RS and X2 ∼ −6 RS.

[17] This procedure has then been repeated for the remaining orbits with the results shown in Table 1. The interval and pass number are shown in the first two columns of the table, where the first two entries refer to Pioneer-11, and Voyager-1, the third and fourth columns then show the inner and outer edges of the ring current in RS. The fifth column indicates the value of μoIo (nT) obtained for the fixed value of D = 2.5 RS (except for Pioneer-11 and Voyager-1, which correspond to D = 2 and 3 RS, respectively). The sixth column shows the corresponding value of Bz0 (nT) obtained from equation (2). The seventh and eighth columns indicate the total current flowing in the ring, and the ratio of the total magnetic moment of the model ring current relative to Saturn's internal dipole moment. These integral values will be discussed further below. Finally, in the ninth, tenth, 11th, and 12th columns we indicate the radial position of the last inbound magnetopause crossing observed by Cassini, the day on which it was observed, the inferred subsolar magnetopause location, and the corresponding solar wind dynamic pressure, respectively. The last two parameters were both inferred from the Arridge et al. [2006] model, as will now be discussed.

[18] Arridge et al. [2006] have recently employed Cassini data to derive a new model of the size and shape of Saturn's magnetopause with variable solar wind dynamic pressure, based in functional form on the terrestrial model of Shue et al. [1997]. In this model the distance to the magnetopause is given by

  • equation image

where r is the observed radial distance of the magnetopause, RM is the subsolar distance to the magnetopause, K is a parameter describing the shape of the magnetopause, and θ is the polar angle measured from the x-axis (in KSM coordinates). The variables RM and K are assumed to be functions of the solar wind dynamic pressure such that

  • equation image

and

  • equation image

where a1 = 9.7 RS, a2 = 0.24, a3 = 0.77, a4 = −1.5, and pSW (nPa) is the dynamic pressure of the upstream solar wind estimated from pressure balance with the magnetic pressure in the magnetosphere just inside the boundary (see Arridge et al. [2006] and section 4.2 for details). From the observed positions (r, θ values) of the magnetopause on the 17 Cassini orbits included in this study, we can thus use the Arridge et al. [2006] model to provide an estimate of both the magnetopause distance at the subsolar point, along with the corresponding solar wind dynamic pressure. These are the values shown in columns 11 and 12 of Table 1. As can be seen from Figures 1 and 3, it is the field perturbation due to the ring current which is by far the most important of the residual fields, and therefore we will now focus on comparing the ring current parameters with magnetospheric size. Using the Arridge et al. [2006] model described above, we are therefore now able to compare the values of the subsolar magnetopause location with the parameters of the ring current models which we have derived, to investigate their variation with system size.

[19] In Figure 4 we thus show scatter plots of the Connerney et al. [1983] model parameters versus subsolar magnetopause distance RM, together with least squares fits to these data (solid lines). The gray bands in each case correspond to plus and minus one standard deviation of the points from the best fit line. For clarity, the results from the linear least squares fitting procedure are summarized in Table 2. The first column indicates the fitted parameter as a function of subsolar magnetopause distance, RM, the second and third columns show the gradient and intercept of best fit, respectively. The final column indicates the standard deviation of the points about the fitted value, σ. In Figure 4 the data from the 17 orbits of Cassini are indicated by the crosses, while the values derived from the Voyager-1 data by Connerney et al. [1983] are shown by the square symbol, and the Pioneer-11 values obtained by Bunce and Cowley [2003] are indicated by the diamond symbol. These are shown for completeness but not included in the fitting procedures that follow, due to the different epoch of the flybys and the modestly different approach that was undertaken to obtain the ring current parameters for these studies. Figure 4a shows the inner edge of the ring current model R1 (RS), Figure 4b shows the outer edge of the ring current R2 (RS), and Figure 4c shows the value μoIo (nT). In the latter case the Voyager-1 and Pioneer-11 values have been normalized to the equivalent model with D = 2.5 RS, such that μoIoD = constant as discussed above. Taking each figure in turn, in Figure 4a we find that the best fit straight line of the form R1 = mRM + c, has a gradient of value m = 0.030, and intercept c = 6.1 RS. The standard deviation of the points from the best fit line is ±0.67 RS (indicated by the gray band), and the variability relative to the mean is ∼10%. Therefore we can see that while there is a small increase of the distance of the inner edge of the ring current with increasing subsolar magnetopause location, the effect is very small, as may be expected. Over the total variation of the subsolar magnetopause of ∼10 RS, we see an increase in the radius of the inner edge of the modeled ring current by only 0.5 RS. In Figure 4b we show the best fit to the variation of the radius of the outer edge of the ring current, R2, as a function of subsolar magnetopause distance. Here we find the best fit gradient m = 0.64, with intercept c = 4.8 RS. The standard deviation of the points from the best fit line is ±0.97 RS, and the variability relative to the mean is ∼6%. The variation of the distance of the outer edge of the ring current with system size is therefore much more apparent, increasing from ∼15 to ∼21 RS over the range shown. Finally, in Figure 4c we show the fit of the μoIo parameter of the model, and derive the best fit gradient m = 1.12 nT RS−1, and intercept c = 26.7 nT. The standard deviation of the points from the best fit line is ±8.6 nT, and the relative variability is ∼18%. These data indicate a modest increase in the current density parameter with increasing system size, from ∼45 to ∼55 nT over the range shown. We can therefore infer that the plasma content of the ring current must be reasonably steady according to these results. There are, however, specific examples shown in Figure 4c where the current is anomalously low (and a few higher), and these may represent variations resulting from production-loss dynamics.

image

Figure 4. Scatterplots and linear least squares fits to the ring current model parameters as functions of the subsolar magnetopause distance, RM, showing (a) the modeled inner radius R1 (RS) of the ring current for the Cassini data used in this study (crosses), along with the Pioneer-11 (diamond) and Voyager-1 (square) values from previous studies. The solid line indicates the best linear fit to these data (Pioneer and Voyager points excluded), with gradient m = 0.030 (with RM in RS) and intercept c = 6.1 RS. Also shown is (b) the modeled outer radius R2 (RS) of the ring current, with the solid line again indicating the best linear fit to the data. Here, the best fit gradient is m = 0.63, and the intercept is c = 4.8 RS. Finally shown is (c) the μoIo parameter of the model, where the gradient of the best linear fit is given by m = 1.12 nT RS−1, while the intercept is c = 26.7 nT. The gray bands indicate one standard deviation of the points from the best fit line.

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Table 2. Results of the Linear Least Squares Fitting Procedure Described in Section 3.2 and Relating to Figures 4 and 5a
Fitted ParameterGradient, mIntercept, cSD, σ% dev.
  • a

    The first column indicates the parameter which was fitted as a function of subsolar magnetopause distance, RM. The second and third columns show the results of the fit, namely the best fit gradient and intercept values. Finally, the fourth and fifth columns indicate the standard deviation of the points from the fitted value, σ, and the percentage difference of the standard deviation from the mean value, respectively.

R1 Rs0.0306.1 RS0.67 RS10.1
R2 Rs0.634.8 RS0.97 RS5.5
μ0I0 nT1.12 nT Rs−126.7 nT8.6 nT17.9
Bz0 nT0.42 nT Rs−12.35 nT1.3 nT12.0
IT MA0.65 MA Rs−1−1.92 MA1.3 MA12.0
kRC0.044 Rs−1−0.490.06015.3

[20] Beyond the initial fits to the Connerney et al. [1983] model parameters discussed above, we can also derive secondary parameters of interest which result from the modeling work. The first of these is the Bz0 parameter given by equation (3). This is the north-south component of the magnetic field due to the ring current at the center of the ring, i.e., essentially at the center of the planet, which is thus equivalent to a kronian DST index. The second parameter is the total current flowing in the ring current based on our derived parameters. Given the current density variation in the Connerney et al. [1983] model given by equation (1), we can simply obtain an expression for the total current

  • equation image

and hence by integration across the radial extent of the ring current region we find

  • equation image

where the parameters are the same as in section 2 above. Finally, we can estimate the ratio of the magnetic moment of the ring current to the magnetic moment of the planetary dipole. The magnetic moment of a ring of current of radius r carrying current I is πr2I, i.e., the area of the loop times the current. Integrating over the entire Connerney et al. [1983] model current system using equation (1) then yields

  • equation image

The planetary dipole moment is given by

  • equation image

where μ0 is the permeability of free space, and BP is the planetary dipole field on the magnetic equator, at radial distance RP. As above, we take a value of RP = RS = 60268 km and BP = 21084 nT, appropriate to the “Cassini” internal field model [Dougherty et al., 2005]. Dividing equation (7) by equation (8) then yields

  • equation image

Note that both IT and mRC contain only the product μoIoD as expected, such that they are insensitive to the specific choice of D as discussed above.

[21] In Figure 5 we thus show scatterplots versus subsolar distance of the three derived parameters given by equations (2), (6), and (9), respectively, together with least squares fits indicated by the solid lines. The results from the fitting procedures are summarized in Table 2. As in Figure 4, gray bands indicate plus and minus one standard deviation of the points from the best fit line. Figure 5a shows the variation of Bz0 (nT) as a function of the subsolar magnetopause position. In this case, we find that the best fit gradient is m = 0.42 nT RS−1, while the best fit intercept value is c = 2.4 nT. The standard deviation of the points from the best fit line is ±1.3 nT, and the variability relative to the mean is ∼12%. We can therefore see that there is a significant increase in the kronian DST with increasing subsolar magnetopause location. Over the maximum variation of the subsolar magnetopause of ∼10 RS, we see that the “best fit” values increase from ∼9 to ∼13 nT, while the individual data points range over a factor of two, between ∼8 and ∼16 nT. As is the case for Figure 4c, we note that this scatter may be an indicator of the level of variation in the plasma content of the magnetosphere resulting from dynamic effects. In particular it may possibly indicate occasions where the magnetosphere has recently undergone significant restructuring following (for example) compression-related dynamics [e.g., Cowley et al., 2005; Bunce et al., 2005], and perhaps a recent change in the plasma content of the ring current region. In Figure 5b we show the variation of the total current with the subsolar magnetopause distance. The least squares fit gives a best fit value of the gradient in this case of m = 0.65 MA RS−1, while the best fit intercept value is c = −1.92 MA. The standard deviation of the points from the best fit line is ±1.3 MA, and the variability relative to the mean is ∼12%. We note that the total current increases significantly by a factor of ∼1.6 across the range of subsolar magnetopause locations observed, from ∼9 to ∼15 MA. We again note some significant scatter in the data points, clearly reflecting the scatter in Figure 4 discussed above. Finally, in Figure 5c we show the variation of the ratio of the magnetic moment of the ring current to the planetary dipole moment, kRC. Here, we find that the best fit gradient is m = 0.044 RS−1, while the best fit intercept is c = −0.49. The standard deviation of the points from the best fit line is ±0.06, and the variability relative to the mean is ∼15%. We observe that the value of kRC, and hence the magnetic moment of the ring current, increases significantly with the system size, and in particular that across the range of magnetopause distances kRC increases by a factor of ∼3, from ∼0.21 to ∼0.65. The physical origin of this effect will be discussed in section 4 below, together with its consequences for the dependence of the magnetopause position on solar wind dynamic pressure.

image

Figure 5. Scatterplots and linear least squares fits to the derived parameters as functions of the subsolar magnetopause location, showing (a) the Bz0 (nT) parameter (given by equation (3)) for the Cassini data used in this study (crosses), along with the Pioneer-11 (diamond) and Voyager-1 (square) values from previous studies. The solid line indicates the best linear fit to the data (both Pioneer and Voyager points excluded), with gradient m = 0.42 nT RS−1 and the intercept c = 2.35 nT. Also shown is (b) the total current IT (MA) derived from model values, where the gradient is m = 0.65 MA RS−1, and the intercept is c = −1.92 MA and (c) the ratio of the ring current magnetic moment to the planetary dipole moment kRC, with the solid line again indicating the best linear fit to the data. Here, the best fit gradient is m = 0.044 RS−1, while the intercept is c = −0.49. The gray bands indicate one standard deviation of the points from the best fit line.

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4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

4.1. Physical Origins of the Variation in Ring Current Magnetic Moment

[22] We now discuss some implications of the above results, specifically that the magnetic moment of Saturn's ring current increases strongly with the size of the magnetosphere. We begin by considering the physical origin of this effect. The magnetic moment of the ring current as a whole, enclosed within volume V, is given by

  • equation image

where j is the current density in the ring current plasma, and r is the position vector within it (the result being independent of the choice of origin). For example, for current I flowing in a closed loop this formula (with jdV being replaced by Idl) yields the well-known result m = IA, as indicated in section 3.2 above. Here A is the vector area of the loop, whose component in a particular direction is given by the area of the loop projected onto a perpendicular plane, with a sense relative to the current given by the right hand rule. The current density j in the plasma consists of the sum of two components. The first is the magnetization current due to the gyration of the particles about their guiding centers, given by jM = curl M, where magnetization M is the gyration magnetic dipole moment per unit volume. Substituting this into equation (10) gives

  • equation image

where the final form is obtained by explicit term-by-term integration of the components of the initial integrand. Thus the total ring current magnetic moment due to the magnetization current is equal to the vector sum of the individual particle gyration magnetic moments, as may have been anticipated. These magnetic moments are directed opposite to the direction of the local magnetic field for both positive and negative particles, thus producing an overall ring current magnetization magnetic moment in a quasi-dipolar planetary field that is in the same sense as that of the planet.

[23] The second current component is that due to the cross-field guiding center drifts of the particles relative to the E × B drift, the latter being common to all the charged particles in the plasma. The drifts that are important in the present context are the grad-B and curvature magnetic inhomogeneity drifts, together with the inertia drift associated with plasma acceleration, specifically that due to plasma corotation with the planet. Again, the total ring current magnetic moment associated with these drifts is given by the sum of the guiding center drift “current loop” contributions provided by each of the ring current particles. In a quasi-dipolar field the magnetic moments associated with these drifts (grad-B, curvature, and inertia) are all in the same sense as that of the planetary dipole for both positive and negative particles and thus are also all in the same sense as the overall magnetic moment produced by the magnetization current. Here we will thus examine how the total magnetic moment of individual ring current particles due to gyration plus guiding center drift varies under magnetospheric expansion and compression due to changes in the solar wind dynamic pressure, the variation in magnetic moment of the ring current as a whole then in principle being obtained by summation over all the particles. Of course, the magnetic moment of the ring current as a whole can also change due to variations in its plasma particle content, through injections from plasma sources and ejections to sinks. However, for magnetospheric compressions and expansions on short time scales the plasma particle content should remain essentially fixed, such that the origin of the effect found in section 3 should be found in the behavior of individual particles under expansion and compression, as we now examine.

[24] We first consider the magnetic moment associated with particle guiding center drift, and begin by noting (as above) that in a quasi-dipolar field the grad-B and curvature drifts are in the same sense as each other, with comparable magnitudes for comparable perpendicular and parallel particle thermal energies. Thus in such a field the bounce-averaged magnetic inhomogeneity drift is only modestly dependent on equatorial pitch angle for particles of given thermal energy W. This being the case, here we only consider particles that are confined to the equatorial plane with zero parallel velocity, thus retaining the essential physics while introducing considerable simplicity of analysis. We also note that this approximation may indeed be reasonable for much of Saturn's equatorially confined plasma of internal origin [e.g., Richardson, 1995]. The particle guiding-center drift velocity is then given by

  • equation image

where m, q, and W are the mass, charge, and perpendicular thermal energy of the particle, respectively, V is the plasma flow velocity (the E × B drift), and B the equatorial magnetic field. We also consider for simplicity an axisymmetric closed field line system in which the equatorial magnetic field points southward (as at Jupiter and Saturn) with a strength B which therefore varies only with radial distance r, and with a plasma flow which is purely azimuthal at angular velocity Ω which also varies only with r. The particle drift velocity relative to the E × B flow is then purely azimuthal and given by

  • equation image

Since dB/dr < 0, we note that both terms on the right-hand side are such that ions drift to the east relative to the flow, and electrons to the west, both resulting in magnetic moments that are in the same sense as the planetary dipole moment (northward in this case) as indicated above. From the discussion related to equation (10), the magnitude of the associated magnetic moment is μ = πr2iϕ, where the iϕ is the azimuthal particle current (charge per time), such that the magnetic moment associated with the guiding center drift is

  • equation image

This expression gives the total magnetic moment associated with the drift current when summed over all the particles in the (equatorially confined) ring current plasma. To this we must then add the magnetic moment associated with the particle gyration about the field lines, which as shown above similarly sums to give the total magnetic moment associated with the magnetization current. This is given by μg = W/B directed antiparallel to the magnetic field lines for both positive and negative particles, thus for a quasi-dipolar field having the same sense and comparable magnitude to the magnetic moment associated with the magnetic inhomogeneity guiding center drift. Summing the drift and gyration contributions thus yields the total particle magnetic moment as

  • equation image

We note that the first term, associated with the rotation of the plasma, is the same as that given in a related context by Alexeev et al. [2006]. The magnetic perturbation produced by the particle at radial distance r′ large compared with r is then the dipole field given in the equatorial plane by ΔB ≈ (μo/4π)(μtot/r3), in the same sense as the planetary equatorial field. We note in passing that the magnetic perturbation field produced by the particle at the center of the planet, a location that is now “inside” the drift current ring but again “outside” the gyration current loop, is in the opposite direction, and assuming a planetary dipole field is given by the Dessler-Parker-Sckopke relation appropriate to these circumstances, i.e., by twice the particle kinetic energy mr2Ω2/2 + W divided by the planetary magnetic dipole moment [Dessler and Parker, 1959; Sckopke, 1966]. Thus the field due to the particles' motion strengthens the planetary field in the outer part of the magnetosphere and weakens it in the inner, in conformity with the discussion in section 3 and the results shown in Figures 1 and 3.

[25] We now wish to examine how the total particle magnetic moment given by equation (15) varies as the magnetosphere expands and contracts under the action of the solar wind dynamic pressure. To do this, we employ a specific simple axisymmetric field model formed by the vector addition of the planetary magnetic dipole field with a spatially uniform field which is directed everywhere parallel to the planetary equatorial field, i.e., southward at Jupiter and Saturn. The combined field produces magnetic nulls on the planetary magnetic axis on both sides of the planet at a radial distance RM given by

  • equation image

where Bu is the strength of the uniform field, and the planetary field is of strength BP on the planet's equator at radial distance RP. The field line surface connecting the nulls then forms an exactly spherical “magnetopause” of radius RM centered on the planet, which completely separates all the planetary field lines (lying inside the spherical boundary) from unconnected field lines on the outside. There are thus no “open” field lines in this simple magnetic model. The uniform field Bu inside the “magnetosphere” may thus be viewed as being due to the magnetopause currents that confine the planetary field inside the sphere of radius RM. Adding the uniform field given by equation (16) to the planetary dipole field then gives the equatorial field strength versus r for magnetospheric radius RM as

  • equation image

Note that in this model the total equatorial field strength at the spherical magnetopause at r = RM is exactly three times the planetary field alone. We recall that the plane magnetopause boundary approximation employed originally by Chapman and Ferraro gives a factor of exactly two in the subsolar region, such that for a boundary of more realistic intermediate shape the subsolar multiplication factor will lie somewhere between, as discussed further in section 4.2 below. Substituting equation (17) into equation (15) then yields

  • equation image

[26] We now consider how the particle magnetic moment changes as the size of the system changes. The initial system is taken to be of radius RM, with the particle drifting at radius r within it, having angular velocity Ω and perpendicular thermal energy W. The boundary then moves to radius RM, such that the particle orbit moves to radius r′, with angular velocity Ω′ and perpendicular thermal energy W. We now discuss in turn how each of these new parameters is determined. First, the new radius of the orbit is determined from the fact that the particles are fixed to given flux shells as the boundary moves, in accordance with Alfvén's theorem. The equatorial radius of corresponding shells in the initial and final states is readily determined, e.g., by considering the amount of magnetic flux between the shell and the magnetopause (determined by suitable integration of equation (17)), and requiring that this remains constant for a given shell. The required condition is

  • equation image

which is readily solved numerically for (r′/RM) given the initial value (r/RM) and the expansion/compression factor (RM/RM). It may be noted that if (r/RM) = 1, then (r′/RM) = 1 for all (RM/RM), i.e., particles on the magnetopause remain so for any motions of the boundary. Solutions of equation (19) are shown in Figure 6, where we plot (r′/RM) versus (r/RM) for various (RM/RM) values. The dotted line shows the initial position, i.e., (r′/RM) = (r/RM) for (RM/RM) = 1, while the dashed and solid lines show results for (RM/RM) = 0.5 and 2, respectively, i.e., for a compression and an expansion by a factor of two. While on expansion all particles of course move outward in absolute terms, it can be seen from the figure that they are then located at a smaller distance than initially relative to the magnetopause, i.e., (r′/RM) < (r/RM), except at the origin and at the magnetopause. Similarly, while on compression all particles move inward in absolute terms, they are then located at a larger distance relative to the magnetopause, i.e., (r′/RM) > (r/RM) in this case, except at the origin and at the magnetopause.

image

Figure 6. Plot showing how the equatorial radius of circular particle drift motion changes in response to expansions and contractions of the magnetosphere, using the simple “spherical” magnetic field model given by equation (17). The initial radial position r is shown on the horizontal axis, normalized to the initial radius of the magnetopause RM, while the final position r′ is plotted on the vertical axis, normalised to the final radius of the magnetopause RM. The solid line corresponds to an expansion of the magnetosphere by a factor of two, while the dashed line corresponds to a contraction of the magnetosphere by a factor of two, as determined from equation (19) based on Alfvén's theorem. The dotted line corresponds simply to r′/RM = r/RM for no change in the size of the system.

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[27] Second, with regard to the angular velocity of the plasma, we employ two extreme assumptions. The first is that the plasma angular momentum is preserved in the displacement, such that r2Ω′ = r2Ω. The angular velocity of the plasma is thus decreased on expansion and increased on compression. This case ignores the effect of magnetosphere-ionosphere coupling, however, which tends to bring the plasma back toward rigid corotation with the planet. The other extreme assumption therefore is that the angular velocity is essentially constant during such changes, at least on timescales sufficient for magnetosphere-ionosphere coupling to act. We then have Ω′ = Ω, equal, for example, to some significant fraction of the planetary angular velocity. Third, the change in W is given simply by conservation of the first (magnetic moment) adiabatic invariant, i.e., W/B′ = W/B, where B′ is the equatorial field strength at the new radius r′. We also note that the second “bounce” invariant of the particles is also preserved in our calculation, being zero throughout due to the zero parallel velocity simplifying assumption.

[28] Rather than consider all of these effects acting together in equation (18), we here examine the behavior in two limits, depending on whether the particle magnetic moment is dominated by the inertia term in the equation (the first term on the right-hand side) or the thermal term (the second term proportional to W). It can be seen from the equation that the inertia term dominates the thermal term if the kinetic energy associated with the rotation mr2Ω2/2 dominates the thermal energy W, and vice versa. If we consider particles near-rigidly corotating in the central ring current region at a radial distance of ∼10 RS, for example, then the rotational kinetic energy is ∼0.03 eV for electrons, ∼50 eV for protons, and ∼800 eV for oxygen ions. While the thermal energy will thus generally dominate the rotational energy for electrons, the opposite may be true for ions, particularly for heavy ions that may dominate the mass density. Initially assuming the dominance of inertia applicable to heavy ions, therefore the ratio of the initial and final moments is given by

  • equation image

With the limiting assumption that Ω′ = Ω (which we term “inertia case a”) we then have

  • equation image

Thus for example, at the magnetopause where (r′/RM) = (r/RM) = 1, we find that the magnetic moment associated with the plasma rotation increases very rapidly as the fifth power of the size of the magnetosphere. Results computed by numerical solution of equation (19) are shown by the solid lines in Figure 7, where we plot the ratio of the moments (μ′/μ) versus initial position (r/RM), and we note the logarithmic scale on the vertical axis. The upper solid line shows (μi/μi)a for a magnetospheric expansion by a factor of two, while the lower solid line is similarly for a compression by a factor of two. The change in magnetic moment for particles at the magnetopause is thus by a factor of 25 = 32 in either direction, with significant but lesser effects at smaller distances within the magnetosphere. In the other limit in which the particle angular momentum is conserved during these changes (which we term “inertia case b”) we have instead (Ω′/Ω) = (r/r′)2, so that equation (20) becomes

  • equation image

At the magnetopause the ratio now varies only linearly with the size of the system. Overall results for an expansion and compression by factors of two are shown in Figure 7 by the upper and lower dashed lines, respectively. Significant changes in magnetic moment again occur, but much less than for “case a.”

image

Figure 7. Plot showing the ratio of the final over the initial magnetic moment per particle for charged particle motion in a spherical magnetosphere of variable radius, plotted versus the initial particle orbit radius r normalised to the initial magnetopause radius RM. Note the logarithmic scale of the vertical axis. The upper set of three lines with (μ′/μ) ≥ 1 correspond to a magnetospheric expansion by a factor of two, while the lower set of three lines with (μ′/μ) ≤ 1 correspond to a magnetospheric compression by a factor of two. The solid and dashed lines correspond to particles whose magnetic moment is dominated by the inertia term associated with plasma rotation, i.e., for particles whose rotational kinetic energy is greater than their thermal energy. The solid lines correspond to the case in which the plasma angular velocity is assumed to be constant as the magnetosphere expands or contracts (case a, equation (21)), while the dashed line corresponds to the case in which the plasma angular momentum is assumed to be constant (case b, equation (22)). The dot-dashed lines then correspond to particles whose magnetic moment is dominated by the thermal term, i.e., particles whose thermal energy is greater than their rotational kinetic energy (equation (23)). The horizontal dotted line shows (μ′/μ) = 1 corresponding to no change in the radius of the magnetosphere.

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[29] Assuming instead that the thermal terms are dominant in equation (18), applicable to electrons and hot ions, the ratio instead becomes

  • equation image

In this case the magnetic moment does not vary at all at the magnetopause, i.e., (μth/μth) = 1 for all (RM/RM), and changes by only very modest factors inside the system. Results for factor of two changes in system size are shown in Figure 7 by the dot-dashed lines, where the upper line again corresponds to expansion and the lower to compression. The variation in the central region of the magnetosphere peaks at factors of only ∼20% in either direction, from which we thus conclude that the “thermal” magnetic moment varies with system size by significantly smaller factors than for either of the inertia-dominated cases considered here. Of course, the details of this behavior may be expected to be specific to the particular case considered, in which the parallel particle velocity has been neglected, and its variation during system expansion and compression through conservation of the second (bounce) adiabatic invariant. An alternative interesting case that does not neglect the parallel motion is that of a particle population that remains quasi-isotropic during adiabatic transport. An approximate treatment, not presented here for brevity, shows that in this case the “thermal” magnetic moment per particle, equivalent to the second term on the right-hand side of equation (18), increases as the cube root of the system size in the outer part of the magnetosphere adjacent to the magnetopause (and by lesser factors at smaller distances as in Figure 7). Thus for changes in system size by factors of ∼2 in either direction, the particle “thermal” magnetic moments change by factors of ∼30% (or less) in this case. Such factors are thus similar in magnitude to those shown in Figure 7, from which we conclude more generally that the “thermal” magnetic moment varies with system size by modest factors that are significantly smaller than for either of the inertia-dominated cases considered.

[30] We now consider the implication of these findings for the interpretation of the results derived in section 3. These show (Figure 5c) that the dipole moment of the ring current increases from ∼0.2 of the planetary dipole moment for a compressed magnetosphere with a subsolar magnetopause radius of ∼16 RS, to ∼0.6 of the planetary dipole moment for an expanded magnetosphere with a subsolar magnetopause radius of ∼26 RS (these being the limiting subsolar magnetopause radii in our data set). These values thus imply that over this radial range the total ring current magnetic moment increases approximately as the square of the subsolar magnetopause distance. This effect is thus much larger than can be accounted for by a ring current whose magnetic moment is dominated by either hot plasma currents or by inertia currents in which plasma angular momentum is preserved during magnetospheric expansions and contractions. Rather, it is indicative of a ring current whose total magnetic moment is dominated by inertia currents in which the plasma angular velocity is at least partly maintained by magnetosphere-ionosphere coupling. Generally, we may suppose that the observed ring current moment shown in Figure 5c consists of the sum of a thermal component that is essentially unvarying with system size (variations of a few tens of percent over the range being negligible in this context), and an inertia current component that increases strongly with system size. These two components cannot be uniquely separated in our results, but we know at least that the thermal component cannot exceed some fraction of the minimum magnetic moment observed for the most compressed magnetosphere, i.e., some fraction of the ∼0.2 planetary dipoles that is found in this case. Further, the larger the assumed contribution from the thermal component in this case, the faster must be the subsequent growth of the inertia current contribution with system size. For example, if we assume that the thermal component and the inertia current component contribute equally to the magnetic moment for the most compressed magnetosphere (each providing ∼0.1 planetary dipoles), then the moment of the inertia current must increase as the cube of the system size to account for the growth in the total magnetic moment shown in Figure 5c. In this case the moment of the inertia current would then be three times the thermal component for a subsolar magnetopause radius in the midpoint of our range (∼21 RS) and five times the thermal component for the most expanded system considered. Thus the magnetic moment of the inertia current must dominate for noncompressed systems, even if not over the whole range of system size. We note that this conclusion is opposite to the case of the Earth, where pressure effects in the ring current exceed rotational effects by many orders of magnitude.

[31] With regard to previous results at Saturn, we note that stress-balance calculations based on Voyager particle and field data confirm that inertia currents indeed dominate in the outer part of the ring current, at equatorial distances beyond ∼14 RS [McNutt, 1983, 1984; Mauk et al., 1985], in agreement with our conclusions. At smaller distances, however, the Voyager results are less conclusive. Inside ∼14 RSMcNutt [1984] found that the inertia stress inferred from in situ plasma data falls to increasingly negligible values but suggested that at least part of this effect could be due to the increasing distance of the spacecraft from the magnetic equatorial plane in the inner region, thus reducing the plasma density estimates. On the other hand, Mauk et al. [1985] presented some circumstantial evidence for the importance of plasma pressure gradient effects in this inner region. With these results in mind, we have examined how the ring current magnetic moment varies with system size in various radial ranges within the empirical current disk model derived in section 3, as given by the linear fits to the current parameters shown in Figure 4. Taking approximate account of the changing equatorial radial extension of the field lines as the current increases, we find that the magnetic moment in given flux shells increases significantly with system size throughout the model ring current but less rapidly in the inner region than in the outer. Specifically, we find that inside ∼10 RS the magnetic moment increases approximately as the ∼1.5 power of the subsolar magnetopause radius, while outside that distance it increases as the ∼2.5 power. Overall, the increase is approximately as the square of the radius, as indicated above. Taken at face value these results indicate the significance of inertia currents throughout the ring current disk, in potential (but not conclusive) conflict with the above Voyager results. Of course it is possible that the properties of Saturn's plasma environment have changed between the Voyager and Cassini epochs. Principally, however, in our view this potential discrepancy provides motivation for more refined analysis in future.

4.2. Consequences for System Response to Solar Wind Pressure Changes

[32] We now consider some consequences of the variation of the ring current magnetic moment, specifically how it affects the size of the magnetospheric cavity and its response to changes in the solar wind dynamic pressure. The size of the cavity is, of course, set by the balance between the field and plasma pressures on either side of the boundary. Here for simplicity we consider specifically the subsolar region, which sets the spatial scale for the size of the magnetosphere as a whole. The pressure of the shocked solar wind in the magnetosheath adjacent to the magnetopause, PMS, is given by PMSkSWpSW, where pSW = ρSWVSW2 is the dominant upstream dynamic pressure of the solar wind, ρSW being the mass density and VSW the flow speed, and kSW ≈ 0.88 for the high Mach number regime appropriate to Saturn [Spreiter and Alksne, 1970]. The subsolar magnetopause pressure balance condition is then

  • equation image

where BM is the magnetospheric field strength inside the boundary, and βM is the ratio of the plasma pressure at the boundary divided by the field pressure. Away from the subsolar region the right-hand side of equation (24) becomes modified by the factor cos2 Ψ, where Ψ is the angle between the solar wind flow direction and the boundary normal, such that the solar wind pressure is reduced and the boundary moves to larger radial distances from the planet. We will suppose for the purposes of this discussion that βM is small or can be treated as an approximate constant as the boundary moves under the action of the solar wind pressure, such that BM2pSW. We also first suppose that the magnetospheric field in question is the planetary dipole field only, enhanced by magnetopause currents by a factor kM. We then have BM = kMBP(RP/RM)3, where BP is the field at the planet's equator at radial distance RP, and RM is the magnetopause distance, as in the discussion above. We also take kM ≈ 2.44 for a subsolar boundary of realistic shape, thus lying between a value of two appropriate to a plane boundary and three appropriate to a sphere (as noted in section 4.1 above) [e.g., Mead and Beard, 1964; Alexeev, 2005]. Substitution of BM into equation (24) then produces the usual result that the boundary distance RM varies as the one sixth root of pSW.

[33] If we now also include a ring current contribution, modeled simply as a magnetic moment at the origin which is a factor kRC times the planetary moment, a valid approximation provided the main part of the ring current does not approach RM too closely, then we simply have BMkM(1 + kRC)BP(RP/RM)3, and the pressure balance condition equation (24) gives

  • equation image

Thus if the ring current moment is a constant, the size of the magnetosphere will be increased by the factor (1 + kRC)1/3, but it will still vary inversely with the sixth root of the solar wind dynamic pressure. According to the above discussion of particle magnetic moment variations, this is then the behavior expected for systems whose ring currents are dominated by hot plasma currents like that of the Earth, whose ring current magnetic moment is typically a few tens of percent of that of the planet, similar to that of Saturn as shown above. In fact empirical models of the Earth's magnetosphere appear to favor an even slightly “stiffer” response to solar wind dynamic pressure, varying as pSW−1/6.6 [e.g., Shue et al., 2000]. The magnetic moment of the Earth's ring current (as indicated, e.g., by the DST index) does of course vary considerably with time, but this is primarily due to variable input of hot plasma into the inner quasi-dipolar magnetosphere from the tail plasma sheet, i.e., to changing plasma content of the ring current, contrary to the initial assumption made in the analysis above. These variations have no direct physical connection with the size of the magnetosphere but may generally be largest under disturbed conditions when the magnetosphere may be more compressed by the solar wind than is usual. If so, this would lead to an anticorrelation of the ring current moment with system size, opposite to the effect found for Saturn above, such that the overall magnetopause response is then somewhat “stiffer” than for a fixed dipole moment alone, as observed.

[34] Now suppose instead that the ring current magnetic moment increases with system size, as expected for a rotation inertia-dominated system, and as found empirically for Saturn in section 3. Then, as the solar wind pressure falls and the boundary moves to larger distances, so the ring current moment will grow, causing the boundary to move out to a larger distance than for the case where the magnetic moment is fixed. Similarly, as the solar wind pressure increases and the boundary moves to smaller distances, so the ring current magnetic moment will be reduced, causing the boundary to move in to a smaller distance than for the case of a fixed magnetic moment. In other words, in this case the size of the magnetosphere will be more responsive to the solar wind dynamic pressure than for the case of a fixed dipole. As discussed above in section 3.2, the issue of the responsiveness of Saturn's magnetopause boundary position has recently been investigated empirically by Arridge et al. [2006], using Cassini observations of magnetopause crossings in the dayside and dawn sector. However, since simultaneous measurements of the upstream solar wind dynamic pressure were unavailable, this study employed the value of the magnetic pressure BM2/2μo just inside the boundary to estimate the solar wind dynamic pressure from equation (24) (suitably modified, of course, for the boundary angle effects mentioned above away from the subsolar region). In addition, since routine magnetospheric plasma parameters were also unavailable, it was further assumed that βM could be neglected in this equation. In other words, the size and shape of the boundary were effectively parameterized as functions of p*SW = BM2/2μokSW = pSW/(1 + βM). The empirical fit obtained to the variation of the boundary position in the subsolar region (see equations (3) and (4)) was then

  • equation image

where RS is again Saturn's radius. Arridge et al. [2006] thus indeed found that the boundary position is more responsive to the solar wind dynamic pressure than for a fixed dipole moment. A similar result was found earlier for Jupiter by Huddleston et al. [1998], and, as here, was attributed by them to the extension of the magnetosphere by the middle magnetosphere ring current field. Similarly, the modeling work by Alexeev and Belenkaya [2005] at Jupiter found that the magnetopause boundary followed a similar relationship with the solar wind dynamic pressure and suggested that the solar wind dynamic pressure and the size of the magnetosphere controls the outer boundary of the jovian plasma disc.

[35] It is thus of interest to compare the variation in the position of the subsolar magnetopause expected from the ring current results derived above with the empirical result of Arridge et al. [2006], to examine their mutual consistency. As a first approximation we employ the simplest centered dipole model of the ring current field as in equation (25) above, together with the linear fit to the ring current results shown in Figure 5c above, which relates the magnetic moment of the ring current to the radius of the subsolar magnetopause, i.e., kRCa(RM/RS) + b, where a ≈ 0.044 and b ≈ −0.49 (see Table 2). We then obtain from equation (25)

  • equation image

where, as in section 3.2, BP ≈ 21084 nT for a planetary radius of RP = RS = 60268 km. Equation (27) can be readily solved numerically for (RM/RS) for given p*SW and compared with equation (26). For the planetary dipole field alone we obtain from equation (25) with kRC = 0

  • equation image

Results are shown in Figure 8, where we plot (RM/RS) versus p*SW (nPa). The Arridge et al. [2006] empirical formula (equation (26)) is shown by the solid line, together with the Cassini boundary data to which it is fitted (solid circles). The short-dashed line then shows the magnetopause position expected for the planetary dipole alone, given by equation (28). The resulting values are clearly too small, and vary too slowly with the dynamic pressure, to explain the observed positions. The long-dashed line, however, shows the result obtained from equation (27) which includes the effect of the varying ring current moment, which we see is in much better agreement with the observations. The absolute values are larger as a consequence of the inflating effect of the ring current moment, and the slope is also steeper, more comparable with the observations and empirical model, due to the effect of the ring current moment variation with system size as outlined above. However, the calculated distances still fall a little short of those observed, by ∼5% in the middle of the range.

image

Figure 8. Plot in log-log format showing the variation of RM (in planetary radii RS), the radial distance of Saturn's subsolar magnetopause, with p*SW (in nPa), the solar wind dynamic pressure divided by (1 + βM), where βM is the plasma “beta” inside the subsolar magnetopause (see text). The solid line is the empirical model of Arridge et al. [2006] (equation (26)), as fitted to the Cassini data shown by the solid circles. The short-dashed line then shows the expected variation for the planetary dipole field alone (equation (28)), while the long-dashed line also includes the effect of the varying Saturn ring current determined empirically in this study, whose external field is approximated by that of a variable centered dipole (equation (27)). The dot-dashed line then shows results in which the full numerically computed field of the ring current at the magnetopause position is employed, together with the planetary field and a correction for the effect of the fringing field of the tail current system given by equation (29), as described in the text. The gray band about the dot-dashed line indicates the variation in magnetopause radius expected to be associated with ring current variability, specifically with the scatter in the μoIo current density parameter shown in Figure 4c (see text for further details).

Download figure to PowerPoint

[36] One reason why the subsolar magnetopause distances given by equation (27) are too small is that the centered dipole approximation to the ring current field will generally somewhat underestimate its value at the magnetopause. The centered dipole approximation to the field of a current ring becomes a close approximation only for radial distances in excess of about twice the radius of the ring, which is clearly not satisfied at the position of the magnetopause for the outer part of Saturn's ring current (see Figure 4b, relating the radius of the outer boundary of the ring current to the radius of the subsolar magnetopause). On approaching the current ring from the outside, the field strength becomes higher than for the equivalent centered dipole. To take account of this effect we have therefore calculated the dependence of the subsolar boundary position on p*SW using the full ring current results found above. That is to say, for a given subsolar magnetopause radius RM, we have used the appropriate ring current model given by the linear fits shown in Figure 4 to numerically compute the full ring current field at the distance of the subsolar magnetopause, BzRC(RM), a negative quantity (the dipole tilt being neglected as a first approximation). To this has been added the planetary dipole field BzP(RS) = −BP(RS/RM)3, also negative, and the magnetopause compression factor kM = 2.44 has been applied to both. We have finally also taken account of the fringing field of the tail current system confined inside the dayside magnetopause, which provides a small but significant contribution in the positive z direction. According to the model of Alexeev and Belenkaya [2005] (and also neglecting dipole tilt effects), this field may be approximated along the planet-Sun line by

  • equation image

where X is the distance from the planet toward the Sun on the planet-Sun line, RT is the radial distance of the inner edge of the cross-tail current on the midnight meridian (thus located on the meridian at X = −RT), and ΦT is the total open flux in each tail lobe. Here we have taken typical values of RT ≈ 0.7 RM [e.g., Alexeev et al., 2006], and ΦT ≈ 35 GWb [Badman et al., 2005]. The total field at the magnetopause has thus been taken as BzMkM(BzP(RM) + BzRC(RM)) + BzT(RM), and the associated solar wind pressure has then been determined from equation (24), i.e., p*SW = BzM2/2μokSW. The resulting variation of RM with p*SW is shown by the dot-dashed line in Figure 8. It can be seen that this is in excellent agreement with the data and the empirical formula of Arridge et al. [2006] over the full range of p*SW for which data exists. We thus conclude that the variation of the magnetopause position determined by Arridge et al. [2006] is entirely consistent with expectations based on the variations in ring current parameters found here.

[37] Concerning the variability in the magnetopause position, the gray band shown about the dot-dashed line in Figure 8 indicates the range of magnetopause positions that would be expected from the scatter in the ring current parameters shown in Figure 4. Specifically, given that the current density parameter μoIo shown in Figure 4c is by far the most scattered parameter, the top and bottom edges of the gray band in Figure 8 have been determined in the same manner as the dot-dashed line but using a μoIo variation corresponding to the top and bottom edges of the gray band in Figure 4c, respectively. Since the variability of μoIo for a given magnetopause radius gives a measure of the temporal variability in the mass content of the ring current, the gray band in Figure 8 shows the expected corresponding level of variability in the subsolar magnetopause radius for given solar wind pressure. The width of this band can be seen to be comparable to the scatter in magnetopause positions found by Arridge et al. [2006], such that variability in the ring current mass content is likely to be at least a significant contributor to magnetopause variability at the level shown. However, since the scatter in the magnetopause radius at a given p*SW remains small compared with the overall variation, as can be seen in Figure 8, the dominant variations in magnetopause radius are still those associated with variations in the solar wind dynamic pressure.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[38] We have studied the magnetic field data from the first 2 years of the Cassini mission and have fit models of the ring current magnetic perturbation using the axisymmetric model of Connerney et al. [1981a, 1983], plus a modest correction due to the magnetopause and tail fields. In order to fit to the data, we first subtract the internal planetary field model which yields residual fields which are due to three effects: the ring current in the inner/middle magnetosphere, the magnetopause currents near to the dayside boundary, and the tail current systems in the nightside magnetosphere. It is the field perturbation due to the ring current which is by far the most important in the middle magnetosphere, and as such the focus of this work is to compare the ring current parameters with magnetospheric size. In order to compare the fitted model parameters to the size of the magnetosphere, we also need to measure the location of the last inbound magnetopause crossing on each orbit. We find that the 2 years of data provide 17 orbits with good data coverage together with a recent measurement of the spherical radial position of the magnetopause. Finally, we map the measured magnetopause location to the subsolar point using the Arridge et al. [2006] model, which then gives a consistent indication of the state of the magnetosphere from pass to pass.

[39] The fitted parameters of the Connerney et al. [1981a, 1983] model are the inner and outer edges of the ring current region, R1 and R2, respectively, and the μ0I0 parameter which describes the magnitude of the current density within it. We fix the value of the half-thickness of the ring current, D = 2.5 RS, for reasons outlined in section 3.1. Each model parameter has been plotted as a function of the subsolar magnetopause distance, and a least squares fit has been used to reveal the dependence of each value on the size of the magnetosphere. We find that the only parameter which increases significantly with the system size is the radius of the outer edge of the ring current. The inner edge remains essentially fixed with increasing magnetospheric size, whilst the μ0I0 parameter increases by ∼20% over the range observed, though within considerable (∼20%) scatter. Within this scatter therefore we conclude that the ring current is reasonably steady (and hence the mass content is also steady). However, there are specific examples where the current appears to be anomalously low (or high), and these may be cases relating from production-loss dynamics. We have then gone on to derive secondary parameters, the north-south field at the center of the ring Bz0 (or “kronian DST”), the total current flowing within the model region IT, and the ratio of the ring current magnetic moment to the planetary dipole moment, kRC. Our results indicate that each of these parameters increase significantly with increasing system size, due mainly to the increasing outer radius of the ring current and also the modestly increasing current parameter.

[40] We then investigated the physical origin of the increase in the ring current magnetic moment with system size and some of the consequences that result. Our analysis shows that the strong positive correlation of the ring current magnetic moment with system size is indicative of a system in which inertia currents provide the dominant contribution to the ring current magnetic moment at least for noncompressed systems, if not over the whole range of system sizes. The magnitude of the effect also suggests that magnetosphere-ionosphere coupling must be effective in maintaining the angular velocity of the plasma as the system expands and contracts, presumably at some reasonable fraction of the planetary angular velocity. The magnetic moments associated with ring currents that are instead dominated by hot plasma currents, such as that of the Earth, are not expected to vary significantly during expansions and compressions, other than by variations in the hot plasma content due to production, radial transport, and loss.

[41] The behavior of the ring current magnetic moment is reflected in the response of the size of the magnetosphere to variations in the dynamic pressure of the solar wind. For systems dominated by hot plasma currents, the magnetosphere will be inflated by the magnetic effect of the ring current, but its size will still vary approximately as the sixth root of the dynamic pressure (as for the planetary dipole field alone), if the ring current magnetic moment is not strongly influenced by correlated variations in the hot plasma content. Empirically, this is the situation that applies in the case of the Earth. For inertia-dominated ring currents, however, the magnetic moment increases strongly with system size, particularly if the angular velocity is maintained by magnetosphere-ionosphere coupling, thus causing the size of the magnetosphere to be more responsive to the dynamic pressure of the solar wind than for a fixed dipole. A recent study of Cassini magnetopause data by Arridge et al. [2006] has indeed demonstrated the occurrence of such responsive behavior at Saturn. We have shown that the magnetopause response found by these authors is entirely consistent with expectations based on the variations in ring current parameters found here.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[42] EJB was supported during the course of this study by PPARC Postdoctoral Fellowship PPA/P/S/2002/00168. SWHC was supported by PPARC/STFC grants PPA/G/O/2003/00013 and PPA/E000983/1 and by a Royal Society Leverhulme Trust Senior Research Fellowship. Work at Moscow and Leicester was also supported by INTAS grant 03-51-3922. MKD was supported by PPARC Senior Fellowship PPA/Y/S/2003/00244.

[43] Zuyin Pu thanks Donald Mitchell and another reviewer for their assistance in evaluating this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Magnetic Field Models
  5. 3. Cassini Modeling Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
jgra18796-sup-0001-t01.txtplain text document2KTab-delimited Table 1.
jgra18796-sup-0002-t02.txtplain text document1KTab-delimited Table 2.

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