Numerical studies of charged particles in a magnetic field: Mars application

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1 Cent. Eur. J. Phys. 12(8) DOI: /s Central European Journal of Physics Numerical studies of charged particles in a magnetic field: Mars application Research Article María Ramírez-Nicolás 12, David Usero 3, Luis Vázquez 4 1 Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Avda. Complutense sn, 28040, Spain 2 Instituto de Geociencias (UCM, CSIC), Jose Antonio Novais 2, 28040, Spain 3 Facultad de Ciencias Químicas, Universidad Complutense de Madrid, Avda. Complutense sn, 28040, Spain 4 Facultad de Informática, Universidad Complutense de Madrid, Avda. Complutense sn, 28040, Spain Received 17 December 2013; accepted 18 May 2014 Abstract: An approach to model the Martian magnetization can be done using classical source models. Classical models, such as uniformly magnetized spheres and cylinders, allow for the introduction of additional constraints related to the available information of the magnetic field and its sources. The use of a suitable conservative numerical scheme in Cartesian coordinates was carried out for numerical studies. In this work the motion of different charged particles under the influence of different magnetized sources have been analyzed by using the proposed numerical scheme. For that purpose, electron, proton and alpha particles were used. In addition, the impact of the gravitational effect on the particles motion was also studied. PACS (2008): y, Xx, Tt Keywords: charged particles martian magnetic field trapped orbits numerical schemes Versita sp. z o.o. 1. Introduction The behavior of a charged particle within the Earth s global magnetic field has been widely studied [1]. The motion of a charged particle within the Earth s dipole field is a fundamental tool to understand magnetospheric and ra- marramir@fis.ucm.es diation belt phenomena [2 4]. It is well known that there is no presence of a global magnetic field [5 7] on Mars however some studies have revealed an ancient local magnetic field [8, 9]. This local magnetic field exhibits its greatest values in the Southern Hemisphere [10]. Due to this large difference in the scale of the magnetic fields [11, 12], the behavior of the charged particles under different sources and conditions needs to be analyzed. Some studies [13, 14] have compared physical events, such as, magnetic anomalies, soils characteristics, etc between 521

2 Numerical studies of charged particles in a magnetic field: Mars application Earth and Mars. There are, thus, different theories that explain the magnetic anomalies observed on Mars [15]. The most important theory explains the distribution of the magnetic anomalies as extensive east-west-trending linear features [16]. These anomalies form regions of different polarization, positive and negative, suggesting alternating bands of crustal remanent magnetization [17 19]. This behavior can be also observed on Earth in similar structures associated with the sea floor spreading but on a larger spatial scale [8]. Another theory associates the magnetic anomalies with the magnetic field generated by magnetized sources [17, 20]. The aim of this study is to analyze the behavior of different charged particles under the action of different theoretical magnetic fields. For this purpose, new numerical schemes have been developed to reproduce the trajectories of charged particles under different magnetic and gravitational conditions. In order to explore the numerical solution of the system under study, it is necessary to impose a scheme to satisfy the underlying conservation loss of the continuous system. The conservation law is the responsible for the dynamics of the system and it is necessary to conserve it at a discrete level. For this study, we have assumed that the crustal magnetization of Mars can be modeled using different magnetized sources, since in geophysical exploration, the horizontal gradient of the field is often used to outline separate sources [20]. In this work a selection of some simple sources has been employed to verify the computational strategy suggested here. A comparison between the potential well for the Cartesian scheme with the critical energy value of 1/32 in renormalized units obtained by Störmer for the cylindrical scheme [21] has also been done. In addition, the behavior of alpha particles has been studied due to their important role on Mars. The Radiation Assessment Detector (RAD) is an instrument carried by the Curiosity rover whose preliminary role was to characterize the broad spectrum of radiation environments found inside the spacecraft during the cruise phase. The main objectives of RAD are to characterize the energetic particle spectrum on the surface of Mars and thus determine the radiation dose received on the Martian surface ( Motivated by the work mentioned above, Störmer theory has been analyzed in Section 2. Relationship between the dimensionless potential in Cylindrical [22] and in Cartesian coordinates is thus obtained. Furthermore, a re-analysis of the methodology presented in [23] for different particles is shown in Section 3.1. The results have been used in Section 3.2 to generate a suitable numerical scheme in Cartesian coordinates. The main results and discussion of this scheme are provided in subsection 3.3. Finally, a summary of the main conclusions is given in Section Analytical approach In this section the expressions for the magnetic fields used in this study are provided. Using Störmer s theory, the threshold of the potential well for a proton has been calculated. To remove the constraints, a new expression for the potential has been developed in Cartesian Coordinates, and the results have been verified Störmer s theory revisited It is assumed that the particles under consideration are non-relativistic. To simplify the interpretation for nonrelativistic equations, the integral of the energy in cylindrical coordinates (ρ, z, φ) can be expressed as ρ 2 + ż 2 + ρ 2 φ 2 = Constant. (1) Due to axial symmetry B φ =0, so that conservation of angular momentum leads to a constant value of p φ = Constant. In [21, 22], the motion of a charge in a pure magnetic dipole field (Störmer model) is considered. Results and explanations for the dynamics of light particles, which are present in the radiation belts surrounding magnetized planets, are given by this model [1, 24]. The dimensionless energy used by Störmer [21] is given by the Hamiltonian as follows H s = 1 2 ( ρs 2 + ż s 2 ) + V s (ρ s, z s ), (2) where the potential field V s can be written as: V s (ρ, z) = 1 2 ( 1 ρ s ρ s (ρ 2 s + z 2 s )3/2 ) 2. (3) The potential of an electron and a proton were calculated and plotted versus the ρ coordinate in Figure 1. This figure allows identification of the trapped region, that region where the particle does not have enough energy to escape from the potential field, and the untrapped region, where the energy of the particle is enough to escape from the potential field. From Figure 1 it can be concluded that the boundary energy for an electron to become untrapped is 1/32 for a dimensionless system. To calculate the real value of this potential barrier, which the particles must overcome to avoid being trapped under the action of a magnetic field, the methodology developed by Störmer using cylindrical coordinates and Gaussian 522

3 María Ramírez-Nicolás, David Usero, Luis Vázquez Figure 1. Potential well calculated from Störmer theory, using dimensionless equation (3), for an electron (left) and an proton (right). units has been used. According to these conditions, the potential well can be written as: V (ρ, z) = 1 2 ( pφ ρ e ) 2 C m m z ρ, (4) c (ρ 2 + z 2 ) 3/2 where p φ is the azimuthal component of angular momentum, which has a constant value and only depends on the initial conditions of the problem, e is the electron charge, C m is a constant value, m z is the z component of the dipolar momentum and c is the speed of light (table 3). In this case, the initial conditions are the values corresponding to a charged particle which describes a confined trajectory. To characterize the potential well, and calculate the maximum and the minimum of the potential wells for the trapped particles, equation (4) has been evaluated at z=0 and calculated for a variation of V relative to ρ. The obtained equations for ρ max and ρ min are described in (5) and (6). ρ min = ec mm z cp φ, and (5) ρ max = 2eC mm z cp φ. (6) These values have been assessed for a charged particle in the expression (4). The behavior of the potential considering z=0 for an electron (left) and a proton (right) are plotted in Figure 2. Table 1 shows the maximum and the minimum values of the potential wells defined in expressions (5) and (6) for an electron and a proton. It can be verified, as expected from the results of the Störmer potential, that the relationship between the maximum and the minimum of the well potential for all the studied cases reproduces ρ max =2ρ min. Table 1. Maximum and minimum values of the potential wells corresponding to Figure 2 for an electron an a proton (left and right). These values have been obtained from equations (3), (4). All the results are expressed in Gaussian units. Particle Störmer Proposed Electron ρ min Electron V min Electron ρ max Electron V max Proton ρ min Proton V min Proton ρ max Proton V max Connection between both approaches In this section we analyze the correlation between the dimensionless equation from Störmer (3) and the expression in cylindrical coordinates and Gaussian units shown in (4). For this purpose, both equations were assesed at z = 0. To establish this relationship we have considered ρ = aρ s. After comparing both equations under the metioned considerations, it can be concluded that, for z=0 (in both cases), the correlation between ρ and ρ s coordinates is a = ecmmz p φ. c In order to move from one potential to another by a simple relationship, we have suggested a relationship between both which is described by V = AV s. From this equation and taking into account the calculated a factor, the relationship beetween both potentials is given by the parameter A: 523

4 Numerical studies of charged particles in a magnetic field: Mars application Figure 2. Potential well calculated from equation (4) in Gaussian unit system for an electron (left) and a proton (right). V = AV s = A 1 ( ) 2 ( pφ 1 1 ) 2. (7) 2m a ρ s ρ 2 s From (7) it was obtained that A = ma2 p φ 2. With this simple calculation it has been possible to establish a conection between both potentials facilitating the change of coordinates between the dimensionless potential and the potential in Gaussian units. 3. Numerical schemes In order to obtain and analyze the trajectories described by charged particles under the action of a magnetic field [25, 26], this section provides a scheme in Cartesian coordinates that allows solving the equations of motion in a simplified manner Numerical scheme in cylindrical coordinates Introducing dimensionless variables, the equations of motion can be written as (8), (9). These equations have been simplified into a two dimensional system using a magnetic field with azimuthal symmetry [17]. d 2 ρ dt 2 d 2 z dt 2 = U, and (8) ρ = U z. (9) where ρ = x 2 + y 2. As proposed in [23], a numerical scheme with a conserved discrete energy and a time-inversion symmetry preserving can be used to integrate these equations of motion (10), (11). These two properties are also fulfilled by real paths. On the other hand, the original continuous problem (8), (9) implies the solution of a system of coupled non-linear equations. However, a numerical approach reduces the complexity: we have to solve a system of two non-linear equations sequentially; we first solve equation (10) and then it becomes possible to solve equation (11). The great advantage of this numerical scheme is that it is not necessary to solve the system of equations simultaneously. ρ n+2 2ρ n+1 + ρ n t 2 z n+2 2z n+1 z n = U t 2 = U ( ρ n+2,z n) U (ρ n,z n ), and ρ n+2 ρ n (10) ( ρ n+2,z n+2) U ( ρ n+2,z n) z n+2 z n. (11) With these equations, a family of trajectories in the plane (ρ, z) can be obtained. One of these paths, which corresponds to the trajectories described by a charged particle, is shown in Figure 3. The initial conditions used for each particle in the scheme (10), (11) are shown in Table 2, where h is the discretization parameter, N is the number of points (iteration step) and (ρ 0,z 0 ) and (v(ρ 0 ),v(z 0 )) are the initial position and velocity of the charged particle, respectively Numerical scheme in cartesian coordinates In a nonrelativistic system, the motion equation of a charged particle in a generic magnetic field B is: m dv dt = e (v B), (12) c 524

5 María Ramírez-Nicolás, David Usero, Luis Vázquez Figure 3. Trajectories for trapped electron (left) and proton (right) in the z ρ plane described by equations (10), (11). Table 2. Summary of the dimensionless initial conditions for the ρ z scheme (10), (11) for an electron and a proton. Parameter Electron Proton N h ρ v(ρ 0 ) 0 0 z v(z 0 ) where m, e and v are the mass, the charge and the velocity of the particle, respectively, c is the speed of the light and B is the magnetic field. In order to analyze the behavior of a charged particle in different magnetic fields, a numerical scheme in Cartesian coordinates based on the discretization of the motion equation (12) is proposed. The analysis of the long-time behavior of the particles and the described trajectories must be made by using numerical schemes that preserve the discrete energy [24]. Using these numerical schemes, the cumulative error for long integrations is negligible. The proposed numerical scheme can be expressed as: v n+1 v n = e ( ( v n+1 + v n) B n ), (13) h mc 2 where h = t. Multiplying equation (13) in a scalar way by (v n+1 v n ) we obtain the discrete kinetic energy conservation: 1 ( ) v n = 2 2 (vn ) 2. (14) On the other hand, rearranging the terms of the system of equations (13), and writing in a matrix form to simplify the calculations, we obtain the following expression: 1 eh 2mc Bn z eh 2mc Bn y eh 2mc Bn z 1 eh eh 2mc Bn y 2mc Bn x eh 2mc Bn x 1 eh 1 2mc Bn z eh 2mc Bn y eh 2mc Bn eh z 1 2mc Bn x eh 2mc Bn y eh 2mc Bn x 1 vx n+1 vy n+1 vz n+1 vx n vy n vz n =. (15) The velocity components can be obtained from this scheme and using (16) the position of the particle can be also calculated. r n+1 = r n + v n t. (16) It is important to emphasize that scheme (13) in Cartesian coordinates, beside being conservative (it preserves kinetic energy), allows for suitable numerical simulations with any magnetic field. The magnetic field for the three sources proposed in this study, the sphere, the cylinder and a linear combination of them (showed in following sections) present a behavior similar to a dipole [27, 28]. It should be noted that the potential vector associated with each magnetic field does not satisfy the same assumptions. It is assumed that magnetic fields are related with a scalar potential, U, by B = U. The scalar potential for a sphere and a cylinder is described by the following expressions respectively: C m m z z U sphere =, and (17) (x 2 + y 2 + z 2 3/2 ) U cylinder = 2C mm z z (x 2 + y 2 + z 2 ). (18) 525

6 Numerical studies of charged particles in a magnetic field: Mars application The analysis in this study has been done using three different configurations of sources of the local magnetic field: a magnetized sphere, a magnetized infinite cylinder and the linear combination of both. The expressions of the magnetic field used are given next. Sphere: The magnetic field generated by an uniformly magnetized sphere can be described as B sphere = C ( me 3 (m r) r r 2 m ), (19) r 5 where C me is a constant equal to 10 7 in the international system units (SI), m is the dipolar momentum and r is the position vector [28, 29]. Horizontal cylinder: the equation of the magnetic field B due to a uniformly magnetized cylinder of infinite length is: B cylinder = 2C ( mc 2 (m r) r r 2 m ), (20) r 4 where C mc is a constant equal to 10 5 in SI [27 29]. Linear combination of a Sphere and a Cylinder: it is assumed that the magnetic field can be generated by a linear combination of the expressions provided above multiplied by a constant factor, (α and β), which indicates the contribution of each magnetic field. B ec = α C ( me 3 (m r) r r 2 m ) +β 2C ( mc 2 (m r) r r 2 m ), r 5 r 4 (21) It should be noted that the magnetic field generated by the cylinder depends on the inverse square of distance while the sphere decreases by the cube of the distance Numerical results The aim of this study is to analyze the exact solution of (12) and compare it with the numerical one. To obtain the analytical solution of motion equation (12), the system of equations has been solved using a wave-like solution. To solve the system, it has been considered a time independent and uniform magnetic field in the z direction B = (0, 0, B). The equations for the three components of the particle velocity can be obtained by integrating (12). Using the same methodology, the particle position with respect to time has been obtained by integrating the rate equations. Figure 4 shows the expected behavior of the trajectories of an electron in a constant magnetic field, B. The picture on the left shows the numerical and analytical trajectories. The picture on the right shows the difference between both trajectories with the largest difference being 2.5%. These results verify the viability of using the proposed scheme. Table 3. Summary of constants used by the numerical scheme in Gaussian Units. (NOTE: 1esu = 1statC = 1g 1/2 cm 3/2 s 1 ). Parameter Electron charge (esu) Proton charge (esu) Value Alpha particle charge (esu) Electron mass (g) Proton mass (g) Alpha particle mass (g) Speed of light (cms 1 ) Sphere C m (gcm/esu 2 ) Cylinder C m (gcm/esu 2 ) Dipolar momentum (erg/g) Preliminary parameters assessment An assessment of the parameters of the numerical scheme is required in order to determine and verify the trajectory described by different particles when they are moving in each of the magnetic fields proposed in section 2.2. It has been assumed that the particles are moving in the vacuum and there is no influence from other external forces (like electric fields, atmospheric wind or the interaction with the solar wind). According to previous studies [2, 23], the constants of the numerical scheme used hereafter are summarized in the Table 3. Figure 5 depicts the trajectory followed by a trapped (top) and untrapped (bottom) electron under the effect of a magnetic field due to an uniformly magnetized sphere (left), a cylinder (middle) and a lineal combination of them (right). The results are shown in cylindrical coordinates for convenience. The initial velocity used in the simulation was in the range of cm/s and the range of the number of steps configured varied from 200 to 400. Figures 6, 7 show the same results as Figure 5 but for a proton and an alpha particle, respectively. For both cases, the range of the initial velocity has been taken between cm/s and the number of steps ranged between 300 to The values of the initial positions and the discretization parameter for all the cases studied are shown in Table 4. As expected from the theoretical analysis, values of the parameter of discretization are lower for untrapped particles. The stability of the scheme was confirmed by the conservation of the kinetic energy (for long time simulations and all speed values used, energy values remained unchanged). Table 5 shows the kinetic energy obtained for the electron, proton and alpha particle under the effect of a magnetic field generated by a magnetized sphere. 526

7 María Ramírez-Nicolás, David Usero, Luis Vázquez Figure 4. Left, comparison between analytical solution (red) and numerical (blue) in three dimensions (x,y,z). Right, representation of the errors between both solutions, the first panel is the real solution module (MSR), the second panel is the numerical solution module (MSN) and the third one is the percentage error defined as ( MSN MSR )100 MSR Figure 5. Trapped (top) and untrapped (bottom) orbits described by an electron under the effect of a magnetic field due to an uniformly magnetized sphere (left), a cylinder (middle) and a lineal combination of them (right). From left to right and top to bottom, each row of figures corresponds to trapped and untrapped: electron-sphere (TES-UES), electron-cylinder (TEC-UEC) and electron-sphere+cylinder (TESCUESC) Real initial conditions In previous section, the different trajectories of an electron, a proton and an alpha particle under different magnetic sources have been described and analyzed. These results have shown confined orbits for all cases and particles studied. To validate the proposed numerical scheme, more realistic initial velocities were used. For that purpose, velocities close to 102 km/s, 107 cm/s in Gaussian units, similar to those associated with the solar winds were considered. In this section only the magnetic field generated by a magnetized sphere has been assessed. Figure 8 shows the trapped trajectories of an electron under the conditions described above. As in section 3.3.1, trapped trajectories are obtained under specific initial conditions for the velocity and position vector. In this case, the initial conditions configured were (in Cartesian Coordinates): ( 9 107, 5 107, ) for velocity, (0.02, 0.009, 0.03) for position, N=200 and h = (in Gaussian units). The kinetic energy obtained for the case represented in Figure 8 was erg. Moreover, obtained magnetic field components ranged between 102 nt and

8 Numerical studies of charged particles in a magnetic field: Mars application Figure 6. Same as Figure 5 considering a proton. From left to right and top to bottom, each row of figures corresponds to trapped and untrapped: proton-sphere (TPS-UPS), proton-cylinder (TPC-UPC) and proton-sphere+cylinder (TPSC-UPSC). Figure 7. Same as Figure 5 considering an alpha particle. From left to right and top to bottom, each row of figures corresponds to trapped and untrapped: alpha-sphere (TAS-UAS), alpha-cylinder (TAC-UAC) and alpha-sphere+cylinder (TASC-UASC). nt, similar to those found on the surface of Mars to modify the trajectories of the particles. Now the motion equation (12) becomes Gravity effect m Finally, to complete the model we include, in the numerical scheme, the additional forces due to gravity. According to the low values of the magnetic field of Mars. This implies that particles affected by this field are close to the surface. Gravity is the main force that can be considered dv e = (v B) + mg, dt c (22) where g = 0, 0, GM is the gravity value along the zr2 axis, G is universal gravitational constant, M = kg is the mass of Mars and r = R + hp, where hp is 528

9 María Ramírez-Nicolás, David Usero, Luis Vázquez Table 4. Initial conditions for the positions, r o, ((x,y,z) coordinates)) and the parameter of discretization, h, in gaussian units for an electron, a proton and an alpha particle. Parameter TES UES TPS UPS TAS UAS h r , 1.10, , 0.40, , 0.008, , 0.05, , 0.009, , 0.009, 0.01 T EC UEC T PC UPC TAC UAC h r 0 60, 200, 60 60, 200, , 0.1, , 0.1, , 0.02, , 0.02, 0.01 TESC UESC TPSC UPESC TASC UASC h r 0 70, 7000, 70 70, 7000, , 0.05, , 0.05, , 0.07, , 0.07, 0.01 Table 5. Kinetic energy values obtained for a magnetic field generated by a magnetized sphere. Particle Electron Proton Alpha Kinetic energy (erg) difference between magnetic field and gravitational magnetic field trajectories, presents greatest values as the initial velocity increases. It is noted that configuration 1 and configuration 3 have the lowest and the greatest initial velocities values respectively (see Table 6). It should be noted that N and h are configuration parameters of the representation of the schemes and do not have any impact on the chosen values. 4. Conclusions Figure 8. Representation of the trapped orbit described by an electron when it moves at velocities of hundreds km/s using the numerical scheme proposed in Cartesian coordinates. the height of the particle and R = km is the radius of Mars. Table 6 shows the initial conditions selected for each of the cases plotted in Figure 9. This figure depicts the trapped trajectories for an electron moving in a magnetic field generated by a magnetized sphere (blue) and considering the gravitational effect (red). The bottom pictures show the differences between the trajectories of the models considered. Preliminary results show a correlation between the initial velocity configured and the differences of the trajectories due to the gravitational effect described by the particle. This correlation shows that Error, the The study of the behavior of a charged particles can be done in different ways. Based on the results discussed by other authors [29], the present paper proposes a scheme, in Cartesian coordinates, that can reproduce these results in a much more simplified way. In fact, it has been demonstrated that, in the case of a dipolar magnetic field, the cartesian scheme reproduces Störmer s results. It has also been tested to show that instead of using a scheme of second order partial derivatives [29], a more simplified scheme in first-order derivatives can be used to solve the motion equations. Therefore, all computational loads and approaches of the schemes can be strongly simplified. This scheme has been obtained by using a matrix formulation. Another important improvement of this scheme is that it makes possible to use magnetic field expressions without restrictions. Comparing our Cartesian approach to Störmer theory [21], it has been possible to reproduce the potential wells where we obtain the energy values for bounded particles. 529

10 Numerical studies of charged particles in a magnetic field: Mars application Table 6. Initial conditions values for an electron under the effect of a magnetic field generated by a sphere and with gravity force which varies with altitude. Configuration 1 Configuration 2 Configuration 3 h N 200 r , , , , , , v , 7 106, , 5 107, , 7 107, Figure 9. Top, trajectories for an electron for different initial configurations. In blue the trajectories without considering the effect of the gravity field and in red including the effect of this field. Bottom, differences between both trajectories, Error. Working always in Gaussian units, it has been possible to check that for potential values, V, higher than , an electron can escape from the potential well, and for lower values it gets trapped. Protons and alpha particles have a similar behavior but in this case the threshold is 1021 (in Gaussian units). From these calculations, it is possible to obtain the trajectories for different particles under the effect of different magnetic field sources. Depending on the initial conditions, the above results show that an electrically charged particle (electron, proton and alpha particle) can describe trapped or untrapped trajectories. Also, it has been confirmed that the kinetic energy remains constant in all computed intervals, which indicates the stability of the proposed scheme. This value is directly proportional to the mass of the particle we are working with. Taking this into account, the numerical values obtained for the proton and alpha particles kinetic energy is three orders of magnitude higher than electron s energy. These energy values have been obtained for the initial conditions corresponding to confined particle trajectories. The addition of a constant gravity field, considering its value as 1/3 of the value of gravity on Earth, has an impact on the trajectories of the particles in a magnetic field. This impact is shown in the values observed in Figure 9 on the bottom, where we have represented the difference between trajectories described with and without the action of the gravitational field. Tests carried out with protons show smaller deviations than in the case of electrons. This is directly related to the difference in mass between the two particles. Acknowledgments The authors wish to thank the MetNet project (AYA C05-02, AYA C05-02) for facilitating all the means to carry out this work. Ramı rez-nicola s wants also to thank to the Secretary of State for Research, Development and Innovation of the Ministry of 530

11 María Ramírez-Nicolás, David Usero, Luis Vázquez Economy and Competitiveness for the FPI grant (BES ) associated to this project. References [1] A. J. Dragt, Rev. Geophys. 3, 255 (1965) [2] J. E. Howard, M. Horányi, G. R. Stewart, Phys. Rev. Lett (1999) [3] M. Iñarrea, V. Lanchares, J. Palacián, A. I. Pascual, J. P. Salas, P. Yanguas, Els. Sci. 197, 242 (2004) [4] L. Dorman, Cosmic rays in magnetospheres of the Earth and other planets, Astrophys. Space Sc. L. (Springer, Virginia and University of Leiden, 2009) [5] B. Langlais, M. E. Purucker, M. Mandea, J. Geophys. Res. 109 (2004) [6] B. Langlais, V. Lesur, M. E. Purucker, J. E. P. Connerney, M. Mandea, Space Sci. Rev. 152, 223 (2010) [7] C. C. Resse, V. S. Solomatov, Icarus. 207, 82 (2010) [8] M. H. Acuña et al., Science. 279, 1676 (1998) [9] M. H. Acuña et al., J. Geophys. Res. 106, (2001) [10] M. H. Acuña et al., Science 284, 790 (1999) [11] H. H. Kieffer, B. M. Jakosky, C. W. Snyder, M. S. Matthews, Mars (Arizona Press, Arizona, 1992) 1498 [12] G. Kletetschka, R. J. Lillis, N. F. Ness, M. H. Acuña, J. P. Connerney, P. J. Wasilewski, Meteorit Planet Sci. 44, 131 (2009) [13] S. A. McEnroe, L. L. Brown, P. Robinson, J. Appl. Geophys. 56, 195 (2004) [14] J. A. Bowles, J. E. Hammer, S. A.Brachfeld, J. Geophys. Res. 114, E10003 (2009) [15] E. Kallio, R. A. Frahm, Y. Futaana, A. Fedorov, P. Janhunen, Planet Space Sci. 56, 852 (2008) [16] J. E. P. Connerney et al., Science 284, 794 (1999) [17] J. Arkani-Hamed, J. Geophys. Res. 110, E08005 (2005) [18] J. Arkani-Hamed, J. Geophys. Res. 107, 5032 (2002) [19] B. Langlais, Y. Quesnel, CR Geosci. 340, 791 (2008) [20] J. E. P. Connerney, M. H. Acuña, N. H. Ness, T. Spohn, G. Schubert, Space Sci. Rev. 102, (2004) [21] C. Störmer, Archives des Sciences Physiques el Naturalles, 24 (Anonymous, 1907) [22] C. Störmer, The Polar Aurora (Claredon Press, Oxford, 1955) [23] L. Vázquez, S. Jiménez, Appl. Math. Comput. 25, 207 (1988) [24] A. Dragt, J. Finn, J. Geophys. Eng. 81, 2327 (1976) [25] E. W. Hones, J. Geophys. Res. 68, 1209 (1963) [26] E. C. Ray, E. C. Ray, Ann. Phys. N Y. 24, 1 (1963) [27] R. J. Blakely, Potential theory in gravity and magnetic applications (Cambridge University Press, Cambridge, 1995) [28] P. Gay, Geophysics 30, 818 (1965) [29] Y. Quesnel, B. Langlais, C. Sotin, A. Galdéano, J. Geophys. Eng. 5, 387 (2008) 531