Modeling of Saturn s magnetosphere during Voyager 1 and Voyager 2 encounters

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009ja015124, 2010 Modeling of Saturn s magnetosphere during Voyager 1 and Voyager 2 encounters M. Chou 1 and C. Z. Cheng 1,2 Received 20 November 2009; revised 7 April 2010; accepted 16 April 2010; published 4 August [1] We model Saturn s magnetosphere during the Voyager 1 encounter on days 317 and 318 in 1980 and the Voyager 2 encounter on days 237 and 238 in 1981, respectively. The rotating magnetosphere of Saturn is modeled by two dimensional axisymmetric equilibrium solutions of the Grad Shafranov type equation, which includes effects of the plasma pressure gradient force and the centrifugal force due to plasma toroidal rotation, by prescribing the radial profiles of plasma density and temperature in the equatorial plane. By varying the equatorial plasma profiles the equilibrium solutions are obtained with reasonably good fit to the observed plasma and magnetic field data along the orbits of the Voyager 1 encounter with the Saturn s magnetosphere on days 317 and 318 in 1980 and the Voyager 2 encounter on days 237 and 238 in The numerical equilibrium solutions provide detailed information of the global distribution of plasma environment and magnetic field structure of the Saturn s inner magnetosphere (for L < 24), and the results show that the plasma environment and the magnetic field configuration of the Saturn s magnetosphere are very different between these two spacecraft encounters with the Saturn. In particular, the meridian distributions of heavy ion density, azimuthal current density, and heavy ion beta have thin disk like shapes centered at the equator for the Voyager 1 case, but have fat torus shapes for the Voyager 2 case. The difference results from the difference in the equatorial plasma pressure profile which decreases with R for R >5R S for the Voyager 1 case, but is quite flat between R 5 and 17R S for the Voyager 2 case. Citation: Chou, M., and C. Z. Cheng (2010), Modeling of Saturn s magnetosphere during Voyager 1 and Voyager 2 encounters, J. Geophys. Res., 115,, doi: /2009ja Introduction [2] The measurements of magnetic fields and plasma particles in the Saturn s magnetosphere were carried out early during Pioneer 11, Voyager 1 and 2 flybys and have been followed by Cassini with long term observations. These observations show that positive ions are mainly composed of H + and water group ions W + (i.e., O +,OH +,H 2 O + and H 3 O + ) in the Saturn s magnetosphere [Bridge et al., 1981, 1982; Lazarus and McNutt, 1983; Richardson, 1986; Maurice et al., 1996; Sittler et al., 2006; Schippers et al., 2008] and the plasma environment varies among those encounters. For example, the Voyager 1 data shows that the temperature of heavy ions increases with increasing L shell distance and reaches hundreds of electron volts at L 10, beyond which the temperature decreases with increasing L. Nevertheless, the heavy ion temperature becomes much higher in the mantle region during Voyager 2 flyby with 1 Plasma and Space Science Center, National Cheng Kung University, Tainan, Taiwan. 2 Institute of Space, Astrophysical, and Plasma Sciences, National Cheng Kung University, Tainan, Taiwan. Copyright 2010 by the American Geophysical Union /10/2009JA T max ( 1 kev) at L 15 [Richardson, 1986]. By assuming a dipole field configuration and taking average value of plasma bulk quantities such as density and temperature between Voyager 1 and 2 data [Richardson and Sittler, 1990] solved the force balance equation along magnetic field lines and constructed the meridian plasma density contour inside L = 12. Similar works were performed by solving the magnetic field aligned force balance equation including the ambipolar electric field term based on the Cassini data to show the 2D meridian distribution of plasma in the Saturn s inner magnetosphere of L 10 [Sittler et al., 2008; Persoon et al., 2009]. [3] The previous modelings of Saturn s magnetosphere are mostly based on 2D axisymmetric assumption and provides the ring current distribution based on input from the magnetic field data observed not too far (L ] 25) from the planet [Connerney et al., 1981, 1983; Bunce and Cowley, 2003; Giampieri and Dougherty, 2004; Bunce et al., 2007, 2008]. The ring current plays an important role in determining the magnetic field configuration in the mid magnetosphere of Saturn. By introducing a model ring current with its current density inversely proportional to the radial distance from 8.5R S to 15.5R S with a half thickness of 2.5R S, the Voyager 1 magnetic field data were fitted through the 1of17

2 combination of the fields obtained from the postulated ring current together with a central dipolar field [Connerney et al., 1981]. Using the above method, the data of Voyager 1 and Voyager 2 were combined to model the ring current density distribution, which becomes thicker with a half thickness of 3R S with the corresponding total current of about A[Connerney et al., 1983]. Recent studies concerning the asymmetry of magnetosphere [Alexeev et al., 2006] and the current sheet warping [Arridge et al., 2008] are important in the outer part of the Saturn s magnetosphere. [4] In this paper we perform the modeling of 2D axisymmetric equilibrium configurations of the Saturn s inner magnetosphere for L < 24 by solving the Grad Shafranov type equation by adopting the previously developed magnetospheric equilibrium code [Cheng, 1992] with the plasma density and temperature profiles in the equatorial plane as input. The Grad Shfranov type equation takes into account the force balance in both parallel and perpendicular directions with respect to the magnetic field lines. The 2D equilibrium solutions of plasma density, temperature, magnetic field, current density and plasma beta are solved numerically and are shown in the meridian plane. By choosing proper input plasma and rotation profiles in the equatorial plane we obtain global solutions of magnetic field and plasma density and temperature of heavy ions that fit very well with the observed data along the orbits of Voyager 1 encounter with the Saturn s magnetosphere on days 317 and 318 in 1980 and Voyager 2 encounter on days 237 and 238 in 1981, respectively. Then, we discuss the reconstructed global plasma and magnetic field environments for both Voyager 1 and 2 encounters within the computation domains. 2. Basic Formulation [5] Since Saturn has a fast rotating magnetosphere with multiple ion species, we consider the steady state, MHD equations including the continuity and force balance equations as well as the Ampère s law and Gauss s law for magnetism: rðvþ ¼ 0 ð1þ V rv ¼ J B rp rb ¼ 0 J rb ¼ 0; where the mass density is given by r = S a m a n a, the single fluid velocity is given by rv = S a m a n a V a, and the total plasma pressure is given by P = S a P a, and m a, n a, V a and P a stand for mass, density, velocity and pressure for particle species a, respectively. Note that we have assumed that the plasma particles have isotropic pressure. The electrostatic force term is ignored in equation (2) by assuming that the charge quasi neutrality condition is a good approximation for large scale phenomenon. In addition, the gravity term is ignored because the gravity is important only for radial distance R ] 1.6R S in the Saturn s magnetosphere [Ip, 1983]. With the help of equation (1) and by assuming that ð2þ ð3þ ð4þ the plasma has the toroidal velocity V = wr^e, where w = w(r, Z) is the angular frequency of plasma flow for axisymmetric magnetospheres and ^e is the unit vector in the azimuthal direction, the force balance equation becomes J B rp ¼! 2 R 2^e r^e ¼! 2 R^e R ¼ 1 2!2 rr 2 where ^e R is the unit vector in the radial direction, and the right hand side of equation (5) represents the centrifugal force. [6] We will further simplify the force balance equation, equation (2), by considering the Saturn observation data. From observations, the density ratio of water group ions to proton n W /n H is significantly greater than unity for latitude within ±0.4 as shown by Cassini [Wilson et al., 2008]. In the off equatorial region the values of n W /n H is in general 3 to 10 from the Voyager 1 data (when the latitude ] 15 ) and is 1 in the region outside L 6.5 from the Voyager 2 data (in which the latitude 30 ). However, the proton density becomes significantly dominant near the closest approach of Voyagers (where L 4 to 7, latitude 30 ) but then the heavy ions dominate again when the two spacecrafts are near the equator [Lazarus and McNutt, 1983; Richardson, 1986]. Because the centrifugal force is proportional to m a and n a and the water group ion mass is at least 16 times larger than the proton mass, the centrifugal force is mainly contributed by the water group ions along the most portion of Voyager flyby trajectories. Thus, we can ignore the contribution of centrifugal force from protons in the force balance equation. The temperature ratio of oxygen ions to protons T O /T H is 6 to 8 from Voyager 1 observations and is 3 to 6 from Voyager 2 observations, so the heavy ion pressure dominates over the proton pressure. Electrons in the Saturn s magnetosphere can be divided into thermal and suprathermal components with temperature in order of tens of ev and kev, respectively [Sittler et al., 1983; Maurice et al., 1996; Schippers et al., 2008]. The electron pressure is mainly contributed by suprathermal electrons and the contribution from thermal electrons can be ignored. Thus, the total plasma pressure is mainly contributed by heavy ions and suprathermal electrons and is given by P ffi n W T W + n h T h, in which the subscript h stands for (hot) suprathermal electrons. [7] The divergence free condition of magnetic field from equation (4) implies that the magnetic field lines can be represented as the intersection of two flux surfaces [Stern, 1970, 1976; Cheng, 1992]. For an axisymmetric system the magnetic field can be expressed as B ¼rY r where Y = Y (R, Z) is a single valued flux function and is the azimuthal angle in the cylindrical coordinates (R,, Z). From the above consideration of the total plasma pressure and the centrifugal force and with the help of equations (6) and (3), the force balance equation, equation (5), in the perpendicular direction leads to the Grad Shafranov type equation [Cheng, 1992] r ry 1 2 m W n ð5þ ð6þ ð7þ 2of17

3 The parallel component of equation (5) is obtained by the inner product with the magnetic field ¼ 1 2 m W n where s is the arc length along the B field line. Note that in the right hand side of equation (8) we have ignored the contribution from electrons due to small mass and from protons due to small number density in comparison with the water group ion number density from observations. In solving equation (8) we assume the plasma temperature T W and T h to be constant along magnetic field lines, which is reasonable because the thermal conduction along magnetic field lines is very fast. Also, the suprathermal electron density n h is roughly constant along field lines because the centrifugal force is quite small for electrons due to small electron mass and the electric field is perpendicular to the magnetic field in the MHD model so that B rn h 0 from the electron momentum equation. This is also supported by the observation that no latitudinal effect is found for n h between the Voyager 1 and Voyager 2 data [Richardson and Sittler, 1990]. Then, from equation (8) an analytical form of the heavy ion density n W along the field lines can be obtained and is given by " n W ¼ n W eq exp m W! 2 ðr 2 R 2 eq Þ # 2T W where the subscript eq denotes quantities at the equator. The mass of water group ions m W is taken to be 16 proton mass in this paper. Note that in deriving equation (9) the angular frequency w is assumed to be constant along the field lines, i.e., only a function of Y, or L shell, to allow for differential toroidal rotation. [8] Equation (7) can be solved numerically by using the iterative metric method to obtain the inverse functions R(Y, c) and Z(Y, c) [Cheng, 1992; DeLucia et al., 1980] in which c is the coordinate along flux surface in the poloidal plane. By prescribing pressure at equator P eq = P(R, Z =0) (i.e., given n eq and T eq ), the pressure along field line will be determined by equation (9) times T = T eq. In the beginning the discrete points R(Y i, c i ) and Z(Y i, c i ) are guessed before iteration, then the coordinates [R(Y i, c i ), Z(Y i, c i )] are adjusted to satisfy equation (7) in each iterative step in the iterative metric method. When the converged solution is obtained within an accepted error, the field lines and hence the plasma pressure distribution are determined everywhere in the computational domain. The shapes of the inner and outer boundaries are chosen as dipole L shells, which are determined by the field line equation r/r S = Lsin 2 in the spherical coordinate system, and the dipole flux on the boundaries is related to the L shell value by Y D = B D R S 2 /L [Cheng, 1992]. B D is chosen to be 0.21G as implied from Voyager observations [Connerney et al., 1982]. After we have chosen the L shells for the inner and outer boundaries, the coordinates of the inner and outer boundaries are held fixed during numerical iterations. In the numerical solutions presented in this paper the computational domain is between the L min = 1.4 and L max = 24 dipole flux surfaces for the ð8þ ð9þ Voyager 1 case and is between the L min = 1.2 and L max =21 dipole flux surfaces for the Voyager 2 case. 3. Density and Temperature Distributions in the Equatorial Plane [9] To solve the Grad Shafranov type equation one needs to specify the source terms in the right hand side of equation (7) by prescribing the radial profiles of n W, T W, n h, T h and w in the equatorial plane. We model the radial profile of the water group ion density in the equatorial plane n Weq as n Weq ¼ n W max 4R min R 4R2 min R 2 ð10þ where n Wmax is the maximum value of density and R min represents the radial position of the inner boundary of the computational domain in the equatorial plane. Equation (10) is formulated such that the maximum density n Wmax is located at R =2R min and the density n Weq scales as R when R R min. Observations show that the suprathermal electron density increases rapidly in the inner part and decreases in a relative slow manner with increasing R in the outer part of Saturn s magnetosphere [Schippers et al., 2008], so the suprathermal electron density radial profile in the equatorial plane n heq is modeled by n heq ¼ n hmax 4R2 h 3 3=4 R 6 R 8 þ R 8 h ð11þ in which the maximum value of suprathermal electron density n hmax is located at R 1.14R h. The suprathermal electron temperature T heq is also modeled according to Voyager observations [Richardson and Sittler, 1990; Sittler et al., 1983] in the form T heq ¼ T hmax 5R3 Th R 2 ð Þ 1=5 R 5 þ R 5 Th ð12þ with the suprathermal electron maximum temperature T hmax located at R 1.13R Th. The temperature profiles of heavy ions for Voyager 1 and 2 data are similar inside L 10 but different in the outer region. Therefore, we use different models for the heavy ion temperature with T W eq ¼ T W max 5 2 3=5 R 2 T W R R 5 þ R 5 ð13þ T W for the Voyager 1 flyby data and T W eq ¼ T W max 7 3 4=7 R 3 T W R R 7 þ R 7 T W ð14þ for the Voyager 2 flyby data, respectively. The maximum temperature is located at R 1.08R TW from equation (13) and at R 1.04R TW from equation (14). Also, the radial profile of the angular frequency in the equatorial plane w eq is modeled in a simple form! eq ¼! s exp R ; ð15þ ar max 3of17

4 Figure 1. Equatorial plasma profiles used for modeling Voyager 1 observation. (top) The radial density profiles of the water group ions (solid line) and suprathermal electrons (dashed line) in the equatorial plane with the input parameters n Wmax = 110 cm 3, = 4.6, n hmax = 0.2 cm 3, and R h =8R S. (middle) The radial temperature profiles of water group ions (solid line) and suprathermal electrons (dashed line) in the equatorial plane by choosing T Wmax =150eV,R TW =14R S, T hmax =700eV,andR Th =18R S. (bottom) The pressure profiles of water group ions, suprathermal electrons, and the sum of these two contributions. Note that the total plasma pressure is mainly contributed by heavy ions inside 14R S. where w s denotes the angular frequency of Saturn rotation and R max is the maximum radial distance in the computational domain. Note that the Saturn angular frequency w s = /s is used in this paper. In summary, the input parameters introduced in the equatorial radial profile models include n Wmax, n hmax, T Wmax, T hmax,, R min, R max, R h, R TW, R Th, a, w s and the mass of water group ions m W. 4. Results [10] In the following we present the global 2D axisymmetric equilibrium solutions that fit reasonably well with the observed plasma and magnetic field data along the orbits of Voyager 1 encounter of the Saturn s magnetosphere on days 317 and 318 in 1980 and the Voyager 2 encounter on days 237 and 238 in We then discuss the differences in the plasma and magnetic field distributions of the Saturn s magnetosphere for these two spacecraft observations Voyager 1 Case [11] Because the measured electron number density is in general 2 3 times less than the ion number density due to the spacecraft charging in the Voyager 1 and Voyager 2 measurements [Richardson, 1986; Richardson and Sittler, 1990], and because the thermal electrons as well as protons give only a minor contribution to the total plasma pressure and the centrifugal force in the modeling domain, the equilibrium solutions are obtained in this model by choosing appropriate equatorial profile input parameters to give reasonably good fit to the observed heavy ion density and magnetic field along the spacecraft trajectories. For the Voyager 1 case the radial profile parameters are chosen to be n Wmax = 110 cm 3, n hmax = 0.2 cm 3, T Wmax = 150 ev, T hmax = 700 ev, a = 3, and = 4.6, respectively. The other radial distance parameters are set as R min = 1.4R S, R max = 24R S, R h =8R S, R TW =14R S, and R Th =18R S, respectively. With the above input parameters the radial density (Figure 1, top), temperature (Figure 1, middle) and pressure (Figure 1, bottom) profiles of the water group ions and the suprathermal electrons in the equatorial plane are shown. Note that, the heavy ion density is much larger than the suprathermal electron density and the suprathermal electron temperature is larger than the heavy ion temperature in the whole region, so that the heavy ion pressure is larger (smaller) than the suprathermal electron pressure inside (outside) 17.5R S. However, the total plasma pressure is mainly contributed by heavy ions inside 14R S. 4of17

5 Figure 2. The observed heavy ion density (dotted points) and the model equilibrium solution (solid line) along the Voyager 1 orbit during days 317 and 318 in The left (right) curve represents the inbound (outbound) orbit. The heavy ion density data are not all available throughout the Voyager 1 encounter, and the model solution agrees reasonably well with the observation data. [12] Figure 2 shows the data of heavy ion density (dotted points) from Voyager 1 flyby observations during days 317 and 318 in 1980 and the model solution (solid line) along the spacecraft trajectory obtained from the equilibrium model. Note that the observation data is not all available through the encounter orbit, and the model solution agrees reasonably well with the observation data. The contours (solid lines) of heavy ion density n W in the meridian plane is shown in Figure 3. The dash dotted lines show the computed magnetic field lines and the field lines deviate significantly from the dipole field in between R 10R S and R 20R S. The thick solid line represents the Voyager 1 trajectory, and in the inbound journey of Voyager 1 the heavy ion density increases with decreasing R to a local maximum of about 2 cm 3 at (R, Z) (7.5, 2) R S, where the corresponding L shell is L 8.5. For the outbound orbit the maximum heavy ion density is located at (R, Z) (5.5, 0.5) R S with n W 30 cm 3. The n W contours in Figure 3 show a disk like structure in the equatorial plane of the Saturn s magnetosphere during Voyager 1 visiting and the disk half thickness is about 1 R S inside 5R S and is about 1.5 2R S from 6R S out to the boundary. Figure 3 also shows that n W is quite small in for high latitude inside R =3R S and Voyager 1 passed through the southern low density region near its closest approach to Saturn. [13] Figure 4 shows contours (solid lines) of the azimuthal current density J and the magnetic field lines (dasheddotted lines) in the meridian plane for modeling the Voyager 1 flyby observation of the Saturn s magnetosphere. The current density J is concentrated in the equatorial plane with major concentration in between R =8R S and 18R S with the maximum current density of about 0.09 na/m 2 at R 11R S and a half thickness of about 1.5R S. Overall, the current density extends from R 4R S to 20R S and the contours display a thin torus structure. The calculated current density distribution is different from the center dipole + ring current model [Connerney et al., 1981] with a thinner and more extended structure. In addition, our equilibrium model results in a total current of Ampère for the Saturn s magnetosphere during Voyager 1 flyby, which is larger than that obtained by Connerney et al. [1981] by about 40%. [14] Figure 5 (top) shows the B field intensity obtained from our equilibrium solutions (dashed line) and the measured B field intensity data (solid line) along the Voyager 1 trajectory. The dotted line represents the dipole field intensity and the difference between the dashed and dotted lines is the contribution from the azimuthal (or ring) current. Note that the portions (8R S < R <12R S for the inbound orbit and 6R S < R <10R S for the outbound orbit) of the Voyager 1 orbit with magnetic field intensity less than the dipole field 5of17

6 Figure 3. The density contours of heavy ions n W (solid lines) in the meridian plane obtained from the equilibrium solution for modeling the Voyager 1 flyby observations of the Saturn s magnetosphere. The dash dotted lines show the computed magnetic field lines, and the spacecraft trajectory is plotted as thick solid line. The heavy ion density is fitted along the trajectory, and the half thickness of plasma disk is about 1R S inside R =5R S and is about 1.5 2R S in the radial distance beyond 6R S. 6of17

7 Figure 4. The contours of the azimuthal current density J in the meridian plane for modeling the Voyager 1 flyby observations of the Saturn s magnetosphere. The contours (solid lines) show a thin torus of ring current with half thickness 1.5R S. The dash dotted lines show the computed magnetic field lines. 7of17

8 Figure 5. (top) The Voyager 1 magnetic fields data (solid line) during the Saturn encounter on days 317 and 318 in 1980 are compared with the modeling solutions. The dashed line represents the equilibrium magnetic field strength, and the dotted line shows the Saturn dipole magnetic field strength. (middle and bottom) The polar and radial components of the magnetic field in the spherical coordinates for observed data (solid line) and modeling solution (dashed line). 8of17

9 Figure 6. The solid line contours show the distribution of the computed heavy ion beta b W in the meridian plane for modeling the Voyager 1 flyby observations of the Saturn s magnetosphere. The dash dotted lines show the computed magnetic field lines. The high beta region (b W ^ 0.6) is confined in Z ] 1R S between 9 and 18R S radial distance. The maximum b W value of about 1.3 is located at 13R S. Note that b W value decreases rapidly away from this region, which indicates significant latitudinal variation of b W. intensity are located on the inner side of the peak azimuthal (or ring)current density, which is located at R 11R S as shown in Figure 4. This indicates that the ring current produces a magnetic field with opposite direction to the dipole magnetic field and thus depresses the magnetic field intensity in these orbit portions. The polar and radial components of the magnetic field in the spherical coordinates for observed data (solid line) and modeling solution (dashed line) are shown in Figures 5 (middle) and 5 (bottom), respectively. The deviation of field components between modeling solution and observed data is ]10%. [15] The contours (solid lines) of the computed heavy ion beta b W distribution in the meridian plane for modeling the Voyager 1 flyby observation of the Saturn s magnetosphere are shown in Figure 6. The heavy ion beta b W distribution shows a thin torus structure. The high beta region (b W ^ 0.6) is confined in Z ] 1R S between 9 and 18R S radial distance. The maximum b W value of about 1.3 is located at (R, Z) (13, 0) R S. Note that the b W value decreases rapidly away from this region, which indicates significant latitudinal variation of b W Voyager 2 Case [16] Voyager 2 entered the Saturn s magnetosphere during days 237 and 238 in 1981, and its orbit is different from the Voyager 1 with higher latitude except near the closest encounter location. We model the Saturn s magnetosphere for the Voyager 2 case with the equatorial profile parameters n Wmax = 140 cm 3, n hmax = 0.2 cm 3, T Wmax = 900 ev, T hmax = 800 ev, a = 1.5, and = 5, respectively. The other radial distance parameters are set as R min = 1.2R S, R max = 21R S, R h =8R S, R TW =16R S and R Th =15R S, respectively. With the above input parameters the radial profiles of density, temperature and pressure of the water group ions and the suprathermal electrons in the equatorial plane are plotted in Figures 7 (top), 7 (middle), and 7 (bottom), respectively. Although the suprathermal electron temperature is higher than the heavy ion temperature for R <14R S, the heavy ion pressure is higher than the suprathermal electron pressure due to much higher heavy ion density. The plasma pressure is thus mainly contributed by heavy ions in the whole computational domain. [17] As shown in Figure 8, the heavy ion density data (dots) agree reasonably well with the model equilibrium solution (solid line) along the Voyager 2 inbound trajectory. For the outbound journey the available data of heavy ions is more limited and is close to the equator with n W reaching a maximum value of 100 cm 3 at the equator [Lazarus and McNutt, 1983]. The solid line contours in Figure 9 show the heavy ion density distribution in the meridian plane. Note that the heavy ion density is concentrated in the equatorial plane at about (R, Z) = (2.6, 0) R S and forms a torus around 9of17

10 Figure 7. Equatorial plasma profiles used for modeling Voyager 2 observation. (top) The radial density profiles of the water group ions (solid line) and suprathermal electrons (dashed line) in the equatorial plane with the input parameters n Wmax = 140 cm 3, =5,n hmax =0.2cm 3,andR h =8R S.(middle) The radial temperature profiles of water group ions (solid line) and suprathermal electrons (dashed line) in the equatorial plane by choosing T Wmax = 900 ev, R TW =16R S, T hmax = 800 ev, and R Th =15R S. (bottom) The pressure profiles of water group ions, suprathermal electrons, and the summation of these two contributions. Note that the plasma pressure is mainly contributed by the heavy ions. 10 of 17

11 Figure 8. Heavy ion densities from Voyager 2 inbound observation. The solid line represents the fitted densities along the spacecraft trajectory in the equilibrium model. 11 of 17

12 Figure 9. The density contours of heavy ions n W (solid lines) in the meridian plane obtained from the equilibrium solution for modeling the Voyager 2 flyby observations of the Saturn s magnetosphere. The dash dotted lines show the computed magnetic field lines and the spacecraft trajectory is plotted as thick solid line. The heavy ion density is fitted along the spacecraft trajectory. The heavy ion density is concentrated in the equatorial plane in the radial region from 2 to12r S with the half thickness of about 3R S. On the inner edge the thickness of the high density region decreases rapidly to about 0.45R S at 3R S. 12 of 17

13 Figure 10. The contours of the azimuthal current density J in the meridian plane for modeling the Voyager 2 flyby observations of the Saturn s magnetosphere. The contours (solid lines) show a torus of ring current with half thickness 5R S. The dash dotted lines show the computed magnetic field lines. Saturn with a half thickness of 3R S in the north south direction and extending from R 2to 12R S. On the inner edge of the torus the heavy ion density decreases rapidly inward and the density scale length reduces to 0.45R S near the spacecraft equator crossing location. During the inbound journey of Voyager 2 the heavy ion density increases with decreasing R to a local maximum of n W 1.5 cm 3 at (R, Z)= (5, 2.5) R S, where the corresponding L shell is L 7. For the outbound journey the maximum of n W is about 100 cm 3 at (R, Z) = (2.9, 0) R S, which is properly modeled in our equilibrium solution shown in Figure 9. The density scale length near the equator crossing obtained in this model is also in agreement with the observation data [Lazarus and McNutt, 1983]. As indicated in Figure 9, Voyager 2 passed by the northern part of the low density region near the closest approach at about (R, Z) = (2.6, 1.2) R S. In the outer part of the density torus (beyond R 12R S ) the density decreases in a relatively moderate manner and has a larger scale length of 5R S. [18] The contours (solid lines) of the azimuthal current density J distribution for the Voyager 2 case are shown in Figure 10. J has a maximum of 0.04 na/m 2 at (R, Z) (13, 0) R S and has a half thickness of about 5R S in the northsouth direction and extends from 8R S to 20R S in the radial direction. Inside R 8R S the current extends inward to 3R S with decreasing thickness. The strength of J at the equator for the Voyager 2 encounter is in general 2 times smaller than that of the Voyager 1 case (see Figure 13, top). However, since the current density is distributed in a broader region for the Voyager 2 case, the total current integrated in the computation domain is Ampère which approximately equals to the value obtained for the Voyager 1 case. [19] Figure 11 (top) shows the B field intensity obtained from the equilibrium solution (dashed line) and the measured B field intensity data (solid line) along the Voyager 2 trajectory during days 237 and 238 in The dotted line represents the Saturn dipole field intensity and the difference between the dashed and dotted lines is that contributed by the ring current. As Voyager 2 entered and left the Saturn s magnetosphere, the observed field strength deviates significantly from the dipole value smoothly along the Voyager 2 path except near the equator at the closest approach. Note that the magnetic field intensity observed by Voyager 2 along its orbit is larger than the dipole field intensity. This is because magnetic field produced by the azimuthal current shown in Figure 10 along the Voyager 2 orbit is not opposite to the dipole field and thus the observed magnetic field intensity is larger than the dipole values. The polar and radial components of the magnetic field in the spherical coordinates for observed data (solid line) and modeling solution (dashed line) are shown in Figures 11 (middle) and 11 (bottom), respectively. The deviation of field components between modeling solution and observed data is ]10%. [20] The solid line contours of the computed heavy ion beta b W distribution in the meridian plane for modeling the 13 of 17

14 Figure 11. (top) The Voyager 2 magnetic fields data (solid line) during the Saturn encounter on days 237 and 238 in 1981 are compared with the modeling solutions. The dashed line represents the equilibrium magnetic field strength, and the dotted line shows the Saturn dipole magnetic field strength. (middle and bottom) The polar and radial components of the magnetic field in the spherical coordinates for observed data (solid line) and modeling solution (dashed line). Voyager 2 flyby observation of the Saturn s magnetosphere are shown in Figure 12. The high beta region (b W ^ 0.6) is confined between R 10 and 20R S radial distance with a thickness of Z ] 3R S, which is three times thicker than that for the Voyager 1 case. The maximum b W value of about 2.8 is located at R 13R S which roughly coincides with the maximum J location. Note that the b W value decreases smoothly away from this region to latitude ^±30 with b W ] Summary and Discussion [21] In this paper we have obtained two two dimensional axisymmetric equilibrium solutions that properly model the rotating Saturn s magnetosphere with reasonably good fit to the observed plasma and magnetic field data along the orbits of Voyager 1 on days 317 and 318 in 1980 and Voyager 2 on days 237 and 238 in 1981, respectively, during their brief Saturn encounters. These two model solutions are obtained by solving the Grad Shafranov type equation, which includes effects of the pressure gradient force and the centrifugal force due to plasma rotation, by prescribing the radial profiles of plasma density and temperature in the equatorial plane as shown in Figure 1 for the Voyager 1 case and Figure 7 for the Voyager 2 case. For R >2R S in the equatorial plane the heavy ion density profile scales as R 4.6 for the Voyager 1 case and R 5 for the Voyager 2 case, so the density decreases faster for the Voyager 2 case and has smaller values beyond 3R S in the equatorial plane. However, the heavy ion temperature is about six times smaller for the Voyager 1 case than for the Voyager 2 case. Thus, the heavy ion pressure profile still decreases with R for the Voyager 1 case as shown by the solid curve in Figure 13 (middle), and is quite flat between R 5 and 17R S for the Voyager 2 case as shown by the dashed curve in Figure 13 (middle). Also, the heavy ion pressure, which dominates over the suprathermal electron pressure inside 17.5R S, for the Voyager 1 case (solid curve) is smaller than the Voyager 2 case (dash curve) in the outer region with R > 5.5R S. The difference in the plasma pressure profile is the main cause of the difference in the plasma and magnetic field distribution (as discussed below) between the Voyager 1 and Voyager 2 cases. [22] The modeling results show that the Saturn s magnetosphere and its plasma environment are considerably different for the Voyager 1 and Voyager 2 encounters with the Saturn. For the Voyager 1 case as shown in Figure 3 the heavy ion density distribution in the meridian plane is concentrated around (R, Z) (3, 0) R S and distributes as a disk like structure with a half thickness of 1R S in the Z direction and 6R S in the R direction. For the Voyager 2 case, although the peak heavy ion density is also concen- 14 of 17

15 Figure 12. The solid line contours show the distribution of the computed heavy ion beta b W in the meridian plane for modeling the Voyager 2 flyby observations of the Saturn s magnetosphere. The dash dotted lines show the computed magnetic field lines. The high beta region (b W ^ 0.6) is confined in Z ] 3 R S between 10 and 20 R S radial distance. The maximum b W value of about 2.8 is located at 13 R S which roughly coincides with the maximum J location. Note that b W value decreases away from this region, with the low beta region (b W ] 0.1) distributed outside latitude ±30. trated around (R, Z) (3, 0) R S, its distribution is much more spread out in the Z direction with a fat torus like structure as shown in Figure 9. [23] The azimuthal current density distribution is concentrated around (R, Z) (11, 0) R S with a peak value of 0.09 na/m 2 and spreads as a thin disk like shape with a half thickness of 1.5R S for the Voyager 1 case as shown in Figure 4. But, for the Voyager 2 case the azimuthal current density distribution is concentrated around (R, Z) (13, 0) R S with a peak value of 0.04 na/m 2 and spreads as a fat bullet torus like shape with a half thickness of 5R S as shown in Figure 10. The total integrated current is approximately the same at A for these two cases. Figure 13 (top) shows the current density radial profile in the equatorial plane for the Voyager 1 case (solid curve) and the Voyager 2 case (dashed curve) with the current density for the Voyager 1 case larger than the Voyager 2 case for R >3R S. [24] The heavy ion b W distribution is peaked around (R, Z) (13, 0) R S with a peak value of 1.3 and spreads as thin torus like shapes with a half thickness of 1.5R S and the low beta region (b W ] 0.1) distributed outside Z ^ 2R S for the Voyager 1 case as shown in Figure 6. But for the Voyager 2 case the heavy ion b W distribution is peaked around (R, Z) (13, 0) R S with a peak value of 2.8 and spreads like fat torus like shapes with a half thickness of 2R S as shown in Figure 12. The heavy ion b W radial profile in the equatorial plane is similar for these two cases, but the b W value is larger for the Voyager 2 case for R >9R S than for the Voyager 1 case with the Voyager 2 maximum value (at R 13R S ) about 2 times the Voyager 1 maximum value as shown in Figure 13 (bottom). Sergis et al. [2007] performed statistical study of the Cassini data near the equator and proposed that the quiet time ring current and disturbed time ring current can be classified by using the heavy ion beta. However, from our model results the equatorial plasma pressure profile may be a more important factor to classify the phase of ring currents as shown in Figure 13 (middle). [25] In testing the force equilibrium model we found that by fixing all other parameters the distribution of ring current can extends outward as the outer boundary extends. This property is based on force equilibrium and it is in agreement with the result from the simple ring current model [Bunce et al., 2007, 2008] in which the position of outer edge of ring current are sensitive to the sub solar magnetopause radius R SSMP due to solar wind conditions. However, we found that the thickness of ring current is determined by the pressure of heavy ion in Saturn s magnetosphere as discussed in the 15 of 17

16 Figure 13. (top) The radial profiles of the azimuthal current density, (middle) the heavy ion pressure, and (bottom) the heavy ion beta in the equatorial plane obtained by the equilibrium model. The solid lines and dashed lines refer to the Voyager 1 case and the Voyager 2 case, respectively. pervious paragraph and this is closely related to the phenomena of energetic particle injections [Mauk et al., 2005; Mitchell et al., 2005]. [26] Previous studies on the spherical harmonic expansion of Saturn magnetic field data [Acuna et al., 1981; Connerney et al., 1981, 1982, 1983] often suggested that there is a current free region inside 8R S for which the magnetic field can be represented as r B = r 2 V = 0 such that the spherical harmonic expansion can be applied to solve the magnetic field. The current free solution seems to be not very practical as implied by the considerable ring current density inside 8R S for the Voyager 1 and Voyager 2 encounters as shown in Figures 4 and 10. Nevertheless, because the fitting on B field is the first priority in the Z 3 +ring current model, Connerney et al. [1983] obtains a fit on B field with the deviation of ]1% on field components to the observed data, which is better than ours (]10%), but the estimated ion density in their model by stress balancing deviates from Voyagers data by several times. In general, our model provides good fit for both B field and ion density simultaneously. [27] The equilibrium solutions obtained in this paper provide detailed global structures of plasma density, temperature and beta, ring current, and magnetic field line configuration in the Saturn s magnetosphere, which helps understand the plasma environment of the planet. The analysis shows that the physical states of the Saturn s magnetosphere are different between these two Voyager flybys with considerably different distributions of n W, J and b W due to different equatorial plasma pressure profiles during these two flybys. Finally, the equilibrium model employed in this paper assumes isotropic temperature which may not reflect the true plasma state in the Saturn s magnetosphere. A more satisfactory model should incorporate the anisotropic temperature effect which will be considered in future works. [28] Acknowledgments. This work is supported by the National Science Council under project NSC M and the National Cheng Kung University s Top University Project. The traveling expense was supported by the Plasma Science Focus Group of the National Center for Theoretical Sciences. [29] Masaki Fujimoto thanks Vytenis Vasyliunas and another reviewer for their assistance in evaluating this paper. References Acuna, M. H., J. E. P. Connerney, and N. F. Ness (1981), Topology of Saturn s main magnetic field, Nature, 292, Alexeev, I. I., V. V. Kalegaev, E. S. Belenkaya, S. Y. Bobrovnikov, E. J. Bunce, S. W. H. Cowley, and J. D. Nichols (2006), A global magnetic model of Saturn s magnetosphere and a comparison with Cassini SOI data, Geophys. Res. Lett., 33, L08101, doi: /2006gl Arridge, C. S., K. K. Khurana, C. T. Russell, D. J. Southwood, N. Achilleos, M. K. Dougherty, A. J. Coates, and H. K. Leinweber (2008), Warping of Saturn s magnetospheric and magnetotail current sheets, J. Geophys. Res., 113, A08217, doi: /2007ja Bridge, H. S., et al. (1981), Plasma observations near Saturn Initial results from Voyager 1, Science, 212, of 17

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