Magnetopause Compressibility at Saturn with Internal Drivers

We use magnetopause crossings of the Cassini spacecraft to study the response of Saturn's magnetosphere to changes in external and internal drivers. We explain how solar wind pressure can be corrected to account for the local variability in internal plasma particle pressure. The physics‐based method is applied to perform the most robust estimation of magnetopause compressibility at Saturn to date, using 7 years' worth of magnetometer data from the Cassini mission and accounting for variable internal drivers—particle pressure and azimuthal ring current. The concept of magnetopause compressibility is generalized to quantitatively account for its detailed variation with respect to the position of the magnetopause. An analytical fit is provided to map the compressibility index to values of the stand‐off distance. In particular, the procedure shows that the Kronian system appears to behave similarly to that of Jupiter when expanded outwards and more like the Earth's magnetopause when compressed.


Introduction
The boundary separating the internal magnetospheric plasma around a magnetized planet from the external solar wind plasma within the magnetosheath, known as the magnetopause, has been shown to be a highly dynamic system (Escoubet et al., 2013;Kaufmann & Konradi, 1969;Masters et al., 2011). Its shape and position are the results of complex interactions between external influences (e.g., incident solar wind, Interplanetary Magnetic Field) and internal drivers leading to an outward pressure (e.g., magnetospheric magnetic field and plasma population). At the gas giants, the total magnetic field has a "disk-like" structure (Achilleos et al., 2010;Arridge et al., 2008;Connerney et al., 1981) due to the magnetic contribution of an extensive equatorial ring current fed internally by moon ejecta (Bagenal & Delamere, 2011;Dougherty et al., 2006;Jia et al., 2010;Kellett et al., 2010;Khurana et al., 2007;Tokar et al., 2006).

Magnetopause Boundary Position and Pressure Balance Equation
The size and shape of the magnetopause boundary at the gas giants can be estimated, to first order, by solving the pressure balance between external and internal contributions P SW cos 2 + P 0 sin 2 = B 2 2 0 (1 + ) , where P SW is the solar wind pressure, denotes the angle between the local normal to the magnetopause and the solar wind flow direction, is the plasma beta corresponding to the ratio of hot plasma pressure to magnetic pressure, and B is the total magnetic field strength with a magnetodisk structure. In this study, B is modeled using a magnetic dipole-aligned with the planetary rotation axis-and an equatorial CAN (Connerney-Acuña-Ness) disk (Connerney et al., 1981(Connerney et al., , 1983. P 0 denotes the static thermal pressure in the solar wind-assigned a constant value of 10 −4 nPa (Slavin et al., 1985)-and the coefficient sin 2 is introduced to avoid a complex flow velocity in the subsolar region (Petrinec & Russell, 1997). It is worth noting that the CAN disk used here to model the magnetodisk structure of the field was primarily chosen for its simplicity. It assumes a 1∕r radial profile for the ring current density, which has been proven inaccurate by Cassini plasma and field measurements . The current field model was still shown to organize the observed crossings fairly well (Hardy et al., 2019), but a more realistic ring current model may be considered in future work.
The numerical solution of equation (1) can be considered as representing an equilibrium magnetopause boundary with shape and dimensions fixed by two of the three following parameters: the solar wind pressure P SW , the plasma accounting for internal plasma activity, and the magnetopause stand-off distance R MP ; the parameters of the modeled equatorial ring currents, the inner and outer radii, the disk half-thickness, and the current parameter 0 I 0 depend directly on the system's size (Bunce et al., 2007).
The data set used to study the behavior of the magnetosphere at Saturn consists of 1514 magnetopause crossings of the Cassini spacecraft identified using the on-board magnetometer (MAG) and Electron Spectrometer sensor of the Cassini Plasma Spectrometer (CAPS-ELS) instrument, from October 2004 to February 2013 (Pilkington et al., 2015). The trajectory of the spacecraft during this period was shown to adequately sample the mean position of the boundary, with no bias for extreme magnetospheric configurations (Pilkington et al., 2014). Seasonal distortions of the magnetopause are taken into account using the "general deformation method" (Tsyganenko, 1998): the crossing positions are corrected appropriately to model the response of the boundary and current sheet to a dipole tilt with regard to the solar wind flow, observed at Saturn by Arridge et al. (2008).
Local values for the magnetic field strength B and plasma were acquired by the spacecraft at each crossing position. In order to determine the corresponding equilibrium solar wind pressure P SW , it is necessary to have access to the local geometry of the boundary, as it fixes the angle in equation (1). The morphology of the magnetopause is itself dependent on the system size, since the equatorial ring current-and consequently the magnetodisk structure of the field-responds to how close the surface is to the planet. It is thus necessary to estimate the stand-off distance corresponding to each observed crossing, before trying to determine values of the solar wind pressure.

Magnetopause Crossings and Magnetospheric Scales
We start by solving the pressure balance equation at Saturn in order to determine a set of equilibrium magnetopause models (Hardy et al., 2019) with integer stand-off distances ranging from 15 to 40 Saturn radii (R S ≈ 60 268 km), each with consistent plasma disk parameters according to the results of Bunce et al. (2007).
The method used to determine the system size is illustrated in Figure 1, in the special case of a crossing M observed in the noon-midnight meridional plane. In the general case, we determine the intersections-shown in purple-between the crossing direction OM and the reference surfaces, shown in blue. A spline function is defined to map these intersections with the matching values for the stand-off distance, shown in green along the Sun-planet line. This function is then used to estimate the system size corresponding to the observed crossing position, and the procedure is repeated throughout the entire data set.

From System Size to Solar Wind Pressure
Now that the equilibrium system size has been determined for each crossing, local values of the solar wind pressure can be estimated. The main difficulty of this step lies in the magnetopause geometry depending on the system's size and us having access to a finite number-rather than a continuum-of equilibrium surfaces. This was addressed through the following procedure: at each crossing M, • consider the two equilibrium boundaries whose scales are the closest to the stand-off distance estimate; • these surfaces are used alongside the spacecraft measurements to solve equation (1) at M, resulting in two values for P SW ; • the relative position of the subsolar nose with respect to the scales of each reference surface is used to estimate the solar wind pressure at M as a weighted average.
For example, if the stand-off distance corresponding to a crossing M was found to be 22.3 R S , the reference surfaces of scale 22 R S and 23 R S would be used to determine two values for the solar wind pressure, noted P * SW, 22 and P * SW, 23 , respectively. The solar wind pressure at M would then be estimated as P SW = 0.7 P * SW, 22 + 0.3 P * SW, 23 . In order to assess the magnetopause compressibility and study the response of the magnetosphere to changes in solar wind pressure, it becomes necessary to account for the variability in local plasma . Pilkington et al. (2015) used a K-clustering algorithm to group the crossings into three clusters depending on the values of , with a surface model that includes a 19 % polar flattening (Pilkington et al., 2014). Though this method was able to quantify the impact of internal plasma activity on the stand-off distance, it reduced the number of crossings available to study the boundary compressibility within each cluster. In particular, the uncertainty in for the high-cluster-that is, describing a plasma-loaded magnetosphere-was too high to illustrate any definite impact of plasma activity on magnetopause compressibility. We describe in the following section a method that reduces the number of parameters impacting system size, while accounting for the variability in internal plasma activity over the entire data set.

Magnetopause Compressibility and Impact of Internal Particle Pressure
Let R MP and P SW denote the stand-off distance and effective solar wind pressure of a magnetospheric state perturbed by a small change in pressure dP SW . Assuming a regime devoid of magnetospheric plasma, the consequent displacement of the subsolar nose dR MP is assumed to satisfy, to first order, where is the compressibility parameter of the boundary; the larger the value of , the smaller the impact of a change in pressure on system size, and the more "rigid" the magnetopause boundary, and vice versa.
Considering infinitesimal changes in pressure, integrating equation (2) leads to the linear relationship or the power law This expression has been shown to be valid over a wide range of stand-off distance Bunce et al., 2007), though it is affected by the magnetospheric plasma content. Given a list of crossings with consistent values for the stand-off distance R MP and solar wind pressure P SW , the compressibility parameter could then be inferred semi-empirically from a linear fit of equation (3). The relationship found between the magnetospheric scales R MP and the solar wind pressure estimates P SW is shown in Figure 2a. The long "trailing off" of the crossings towards the top right of the plane illustrates the broad range in both solar wind pressure and plasma , which prevents a direct determination of magnetopause compressibility over the entire data set. This trend also shows the large impact of internal plasma activity over magnetospheric scales, consistent with previous observations showing that hot plasma dynamics are competitive with solar wind conditions in determining the system's size (Pilkington et al., 2015). This factor needs to be addressed before performing any fit to the data for determining the value of .

Dimensionality Reduction and Plasma Scaling
Let us start by noticing that the term accounting for the static thermal pressure P 0 sin 2 in equation (1) only plays an important role at high-latitude positions, close to the cusp on the dayside at a latitude of ≈ 71 • (Hardy et al., 2019). Since most of our observed crossings of the Cassini spacecraft occurred at low latitudes around Saturn (with a maximum observed latitude of around 62 • and a median latitude of ≈ 6 • ), it is relevant to consider the approximate pressure balance equation

10.1029/2019GL086438
This is equivalent to with ref denoting a prescribed reference value of plasma beta and P SW, eff = an effective, scaled solar wind pressure. Thus, considering the effective pressure P SW, eff in place of the pressure estimates P SW allows us to artificially scale all the crossings to a common reference value of plasma beta ref . In other words, under the assumption of pressure balance, P SW, eff are the values of the external solar wind pressure that we would expect had all crossings been acquired with the same plasma .
For ref = 3.58, for example-the mean value of plasma over the data set-the relationship between the stand-off distance and the effective solar wind pressure is shown in Figure 2b. The color bar seems to indicate that the trend shown in Figure 2a vanishes, and the crossings appear to cluster much closer to each other, as expected. Choosing any other value for ref would only displace the cluster horizontally, without disrupting the distribution shown in Figure 2b.

Revisiting the Impact of Internal Plasma Pressure on System Size
Another consequence of scaling the solar wind pressure by considering P SW, eff can be seen in equation (2); it may be expanded as In the context of the Earth's magnetosphere-which is relatively devoid of plasma at the magnetopause boundary (Shue et al., 1997)-only the first term of the right-hand side of equation (7) contributes to a displacement of the subsolar nose. In this case, the magnetic field can be well approximated by a vacuum dipole and the compressibility index is found to be = 6.
The additional term necessary for Saturn and Jupiter shows that an enhancement in internal plasma activity acts towards inflating the magnetosphere (note the plus sign in front) in such a way that a relative change in has the same impact as a relative change in P SW if ≫ 1. This is consistent with the large impact of plasma on system size illustrated in Figure 2. Additionally, the compressibility parameter is expected to be smaller at the gas giants due to the "disk-like" structure of their magnetic fields: ionized moon ejecta are accelerated towards corotation with the rapidly rotating magnetospheres, harboring an azimuthal ring current that acts towards stretching the field lines radially outwards along the equatorial plane. This would lead to the magnetopause being more compressible when it is expanded (i.e., in a plasma-loaded state) and more similar to the dipole case as it is compressed (i.e., in a plasma-depleted state). This variability of the compressibility with regard to the system size is studied in the following section.

Filtering Crossings Far From Pressure Balance
In order to estimate how the magnetopause compressibility at Saturn varies depending on the system size, it is necessary to filter out the crossings that were observed while the magnetopause boundary was not close to equilibrium, but strongly accelerating instead.
To do so, at each crossing, the solar wind pressure estimates-derived from the data and the reference surfaces-can be compared with the weighted average of the values corresponding to the equilibrium surfaces of similar scales. Figure 3 shows the crossings that remained after eliminating those for which the aforementioned difference in pressure was larger than 40 % of the corresponding averaged equilibrium values.
Two observations can be made from Figure 3: the crossings appear not to be distributed along a line, but rather along a slightly convex curve instead; this illustrates the impact of system size on magnetopause compressibility. This feature was previously hidden by the variability in plasma in Figure 2a and drowned by the scatter in Figure 2b; it is studied further in the next subsection. Second, there is an apparent "flaring" in the crossing distribution when moving towards the top left. This could be due to the magnetosphere being less rigid when subjected to changes in solar wind pressure, as the system is expanded: the boundary is then Figure 3. Relationship between the stand-off distance R MP and the effective solar wind pressure estimates P SW, eff introduced in equation (6). The color bar indicates the difference in solar wind pressure ΔP SW between the estimated value P SW, eff and the reference value from pressure equilibrium P SW, ref ; crossings with a difference smaller than 40 % were kept. The dashed horizontal line indicates R MP = 24 R S , and the green and orange lines are linear fits of equation (3)  more easily pushed away from pressure balance, and a larger number of observed crossing is thus likely to correspond to an accelerating magnetopause.

Qualitatively Illustrating the Impact of System Size on Magnetopause Compressibility
The impact of system size on magnetopause compressibility can be illustrated by performing two separate linear fits of equation (3) to the crossing distribution shown in Figure 3.
The magnetopause crossings are chosen to be separated into two subsets: one corresponding to an expanded state (R MP ≥ 24 R S ) and one corresponding to a compressed state of the magnetosphere (R MP ≤ 24 R S ). In the first case, in a plasma-loaded regime (and/or low dynamic pressure regime), the compressibility is found to be = 4.51, with a 95 % confidence interval CI 95 = [4.31, 4.72]; in a plasma-depleted regime (and/or high dynamic pressure regime), as the boundary is pushed closer the planet, = 5.71, with a 95 % confidence interval CI 95 = [5.25, 6.25] (the statistical interval is given as is, though the compressibility index has a physical upper bound of 6, corresponding to a vacuum dipole case). Such a "bimodal" behavior of the magnetopause is consistent with placing Saturn's magnetosphere in between the Earth's, where ≈ 6, and Jupiter's, where ≈ 4 (Bagenal & Delamere, 2011). The cut-off value of 24 R S also echoes with previous observation and modeling studies (Arridge et al., 2011;Sorba et al., 2017) in which a shift in behavior related to magnetic field structure was found around ≈ 22 − 25 R S .
It is worth noting at this stage that the precise position of this "bend" in the crossing distribution-arbitrarily identified here at R MP = 24 R S -is of little significance. However, it does qualitatively illustrate how the magnetopause compressibility varies with system size and thus motivates further study in its response to changes in the position of the magnetopause.

Generalizing Magnetopause Compressibility to Account for the Impact of System Size
In the most general case, the response of the system's size R MP to changes in effective solar wind pressure can be described by an equation of the form where R MP denotes the magnetopause stand-off distance, P SW, eff the effective solar wind pressure introduced in equation (6), and a real function monotonically increasing on the domain considered.
The differentiation of equation (8) If the function is chosen to be a first-degree polynomial, the relationship described by equation (8) is equivalent to the case of equation (3) with a constant compressibility . Because we expect to vary with system size (as shown in Figure 3), it seems necessary to introduce nonlinear terms in the expression of .
In the case where is defined as a second-degree polynomial which hints at a hyperbolic expression for .
Let us then generalize this idea one step further by considering the parametric expression where c 0 , c 1 and c 2 are real numbers. The additional parameter c 0 introduces a new degree of freedom in allowing a vertical translation of the hyperbola.
The following procedure can now be performed: • Using the expression of the compressibility from equation (13), the integration of equation (10) leads to a functional form for . • A fit of equation (8) to the crossing distribution shown in Figure 3 provides the coefficients in the expression for and thus . • Using the relationship between the system size R MP and P SW, eff shown in Figure 3, can be plotted with respect to R MP ; this is shown in Figure  4.
This final relationship between and R MP is further described by fitting a hyperbolic expression of the form to the crossing distribution in Figure 4. The coefficients are found to be a = 3.  Figure 4. In particular, it is found that (R MP = 15 R S ) = 5.97 and (R MP = 35 R S ) = 4.00, which is consistent with the discussion concluding section 4.2. It is worth noting, however, that the uncertainties become relatively large as the system approaches either very compressed or very expanded states; this is mainly due to these extreme states being represented by a relatively small number of observed crossings. In the case of Kanani et al. (2010) and Pilkington et al. (2015), we find that previous considerations of the classic linear relationship of equation (3) may have led to a slight overestimation of the mean magnetopause compressibility. The value and uncertainty for determined by Arridge et al. (2006) seems to be, on average, in good agreement with our findings. In particular, Sorba et al. (2017) identified a shift in behavior around 25 R S , with two distinct values for the compressibility depending on whether the system is more compressed or expanded: interestingly, the value for that we find at 25 R S is very close to the average of the two values determined by the authors. This seems to show that their bimodal modeling approach was able to capture the mean response of the system, though the finer behavioral structure evidenced in Figure 4 was lost. For reference, a comparison of the magnetopause profiles-both in the equatorial and noon-midnight meridional planes-from the aforementioned models is shown in the supporting information.

Conclusion
An extensive set of observed magnetopause crossings at Saturn was used to study the response of the planetary magnetosphere to changes in solar wind pressure. Our physics-based three-dimensional magnetopause model that includes an equatorial ring current (dependent on system size) and internal hot plasma particle pressure (with constant plasma ) was used to estimate magnetospheric scales and local values of solar wind pressure, incorporating magnetic and plasma observations from the Cassini spacecraft. FH was supported by a studentship jointly funded by UCL's Perren and Impact schemes. NA and PG were supported by the UK STFC Consolidated Grant (UCL/MSSL Solar and Planetary Physics, ST/N000722/1). The magnetopause crossings of the Cassini spacecraft mentioned in this study were identified by Pilkington et al. (2015) using the Cassini MAG, CAPS-ELS, and MIMI data available from the Planetary Data System (http://pds.nasa.gov/).