4.1. Physical Origins of the Variation in Ring Current Magnetic Moment
 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
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
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.
 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.
 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
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
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
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
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. . 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/r′3), 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.
 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
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
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
 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 R′M, 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
which is readily solved numerically for (r′/R′M) given the initial value (r/RM) and the expansion/compression factor (R′M/RM). It may be noted that if (r/RM) = 1, then (r′/R′M) = 1 for all (R′M/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′/R′M) versus (r/RM) for various (R′M/RM) values. The dotted line shows the initial position, i.e., (r′/R′M) = (r/RM) for (R′M/RM) = 1, while the dashed and solid lines show results for (R′M/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′/R′M) < (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′/R′M) > (r/RM) in this case, except at the origin and at the magnetopause.
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 R′M. 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′/R′M = r/RM for no change in the size of the system.
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 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 r′2Ω′ = 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.
 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
With the limiting assumption that Ω′ = Ω (which we term “inertia case a”) we then have
Thus for example, at the magnetopause where (r′/R′M) = (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 (μ′i/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
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.”
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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 Assuming instead that the thermal terms are dominant in equation (18), applicable to electrons and hot ions, the ratio instead becomes
In this case the magnetic moment does not vary at all at the magnetopause, i.e., (μ′th/μth) = 1 for all (R′M/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.
 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.
 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  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.  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
 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 PMS ≈ kSWpSW, 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
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 BM2 ∝ pSW. 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.
 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 BM ≈ kM(1 + kRC)BP(RP/RM)3, and the pressure balance condition equation (24) gives
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.
 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. , 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
where RS is again Saturn's radius. Arridge et al.  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. , 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  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.
 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. , 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., kRC ≈ a(RM/RS) + b, where a ≈ 0.044 and b ≈ −0.49 (see Table 2). We then obtain from equation (25)
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
Results are shown in Figure 8, where we plot (RM/RS) versus p*SW (nPa). The Arridge et al.  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.
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.  (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).
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 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  (and also neglecting dipole tilt effects), this field may be approximated along the planet-Sun line by
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 BzM ≈ kM(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.  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.  is entirely consistent with expectations based on the variations in ring current parameters found here.
 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. , 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.