Positions of Structures of Planetary Magnetospheres as Determined by Eccentric Tilted Dipole Model

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1 WDS'14 Proceedings of Contributed Papers Physics, , ISBN MATFYZPRESS Positions of Structures of Planetary Magnetospheres as Determined by Eccentric Tilted Dipole Model D. Parunakian and I. Alexeev Skobeltsyn Institute of Nuclear Physics of the Moscow State University. Abstract. In this work we determine the geomagnetic eld using an eccentric dipole model instead of spherical harmonics expansions. Among other motivations to do so is that the dipole's contribution is much more pronounced relative to higher-order harmonics at large distances from the Earth, and that the translation of the order of magnitude of about 0.1 Earth radii is signicant enough for many magnetospheric features such as the current sheet and/or the magnetopause, in particular the position of the subsolar point, since due to the similarity of our proposed correction's scale and the thickness of magnetopause region this eect may play an important role in determining if on average a given point is located in the magnetosheath or under the magnetopause. Introduction The de-facto standard approach of modeling the geomagnetic eld, which is utilized in such prominent models as IGRF, CHAOS and MF, is based on spherical harmonic analysis, originally introduced in 1838 by Carl Gauss. In this approach the Earth's magnetic eld can be represented as N max n U(r, θ, λ) = R 0 [gn m cos mλ + h m n sin mλ]( R 0 r )n+1 Pn m (cos θ)+ n=1 m=0 +[G m n cos mλ (t) + H m n sin mλ (t)]( r R 0 ) n P m n (cos θ) Here Pn m (cos Θ) = (sin 2 Θ) m/2 d m P n(cos Θ) d(cos Θ) are associated Legendre functions, where m P n (cos Θ) = 1 d n 2 n n! d(cos Θ) (cos 2 Θ 1) n are Legendre polynomials; R n 0 = km is the mean Earth radius, λ and θ are respectively geocentric longitude and colatitude, N max is the maximum degree and order of the internal expansion, r is the radial distance from the center of the Earth, gn m and h m n are determined by internal currents and terrestrial sources, G m n and Hn m are dened by magnetospheric current systems, and λ = λ ωt; ω = sec 1 is the corrected longitude to take into account the Earth's rotation relative to the Sun-oriented magnetosphere. Note that the original analysis assumes N max to be innite in order to be able to describe the eld with an arbitrary degree of precision. The rst term of this equation describes the contribution of the internal geomagnetic sources, and the second term corresponds to external magnetospheric sources. In Earth-centric Earth-xed (ECEF) coordinate systems the longitude parameters of the second term depend on time, since the Earth's orientation variation is diurnal, while the magnetosphere orientation varies annually the bow shock always being pointed sunwards. Let us rst examine the internal eld. Note that the results obtained by SHA depend solely on the coecients gn m and h m n and are not directly aected by assumed physics of the magnetic eld sources, which is one of the main strengths of this approach. Still, it is evident that the lower-n harmonics are associated with global magnetic eld components (e.g. dipole and quadrupole moments), while the higher-n harmonics correspond to local features that are likely embedded into the lithospheric eld, which can be considered static [Olsen et al., 2014] and into the ocean tidal magnetic signal [Maus et al., 2008]. It should be understood that the higher harmonics of the geomagnetic eld are only relevant at smaller geocentric distances; to illustrate this point, consider Figure 2 which shows the spatial 337

2 Figure 1. Spatial power spectrum of the geomagnetic eld at Earth's surface (black curve) and at various altitudes of CHAMP (blue). Also shown is the spectrum at a mean altitude of the Ørsted satellite of 750 km (red). Image by Olsen et al. [2014]. power spectra of the geomagnetic eld at various altitudes above the Earth's surface. Note that even at altitudes as small as 750 km the contribution of harmonics of n 20 is negligible. From this follows the reasonable assumption that magnetospheric features located at distances of many R E are mostly sensitive to the dipolar component of the geomagnetic eld and it is fairly safe to ignore the higher harmonics while calculating the positions and dynamics of these features. Another important reason to consider a geomagnetic model built around an eccentric tilted dipole (ETD) for magnetospheric research is that it is dicult to correctly model the location of magnetic dip poles also referred to as true magnetic poles, dened as the points on the Earth's surface where the magnetic eld lines are vertical [Fraser-Smith, 1987] using full-featured global models. This problem has been earlier discussed in multiple papers, such as Olsen and Mandea [2007] and Newitt et al. [2009]. Thus, we believe that locating the ETD directly from satellite observations and, for purposes that require higher-level harmonics, performing SHA in a coordinate system bound to its position and orientation instead of working in a geocentric system is a promising approach. A marked improvement upon IGRF results is the CHAOS model based on CHAMP and Ørsted satellite observations, which has been shown to pinpoint the location of NMP with a 35 km accuracy [Newitt and Chulliat, 2007]. Nevertheless, this approach requires the model to take into account higher harmonics just to predict positions of seemingly dipolar NMP and SNP, since a geocentric dipole cannot correctly reect the positions of the dip poles due to the fact that they are not diametrically opposed (Figure 3) and exhibit drastically dierent secular variations, with NMP travelling now at over 55 km/s in the general direction of Siberia, and SMP moving at about 4 km/s [Mandea and Dormy, 2003]. An ETD-centered and oriented model will therefore be able to produce similar results on large geocentric distances with fewer series members. In reality there is much more confusion surrounding the use of these terms, and the variety of magnetic poles reconstructed from various models is even wider. While further discussion of this topic is outside of the scope of this work, a useful review can be found in Campbell [1996]. Seeing as positions of magnetospheric features respond mainly to the dipolar component of the geomagnetic eld and the dipole is aected by diurnal variations in a more complicated manner than a simple change of orientation due to the Earth's rotation, it is possible to improve our predictions of magnetospheric feature positions by taking this eect into account. To this end, we must rst determine the position of the dipole in a given period of time using magnetic observatory data. 338

3 Figure 2. Locations of geomagnetic poles and magnetic poles based on IGRF-11 from 1900 to 2000 by 10 years and at 2005 and 2010 (red) and 2015 (prediction). Image by WDCG [2014]. Figure 3. North and south magnetic pole locations: from direct measurements (red diamonds); from Jackson et al. model (blue circles); from Langlais et al. (yellow squares). Image by Mandea and Dormy [2003]. Dipole approximation In order to calculate the parameters of the Earth's ETD, we chose to t the dipolar eld to values measured at magnetic observatories or some approximation thereof. To this end, we rst generate a suciently large number of random coordinate pairs in the [ 55, 55 ] geodetic latitude range. We then calculate the value of magnetic eld components in these coordinates using the IGRF model and the final-frontier Python wrapper. The latitude range is limited in order to remove the inuence of inaccurate IGRF predictions in the polar regions. Dipolar magnetic eld calculation in a given point is performed by rst extrinsically rotating and translating the target point relative to the dipole: cos α cos α cos α sin α sin γ sin α cos γ cos α sin α cos γ + sin α sin γ X x y = sin α cos α sin α sin α sin γ cos α cos γ sin α sin α cos γ cos α sin γ sin α cos α sin γ cos α cos γ Y Z z r 1 = x 2 + y 2 + z 2, θ 1 = cos 1 (z/r), φ 1 = tan 1 (y/x) The dipolar eld is then calculated relative to these new r 1, θ 1, φ 1 as usual. Next we perform residue minimization of the candidate dipolar eld in these same coordinates against IGRF-calculated values using NelderMead simplex method [Nelder and Mead, 339

4 Figure 4. Stand-o distance of the subsolar point per Shue et al. [1998] model for the geocentric and eccentric dipoles. 1965] as implemented in the lmfit module, while varying up to six parameters: dx, dy, dz displacements of the dipole from the Earth's center and α, β, γ Euler angles (later used to determine the latitudinal and longitudinal components of the dipole's tilt. B x sin λ m cos λ m sin φ m cos λ m cos φ m B y = cos λ m sin λ m sin φ m sin λ m cos φ m B z 0 cos φ m sin φ m In order to counter overtting eects, which would cause the dipole to be located and oriented in a way that best corresponds to the eld in a random set of coordinates but not necessarily anywhere else, we then calculate reduced chi-squared χ 2 = 1 Bed B igrf ν, where the σ 2 B ed and B igrf vectors are comprised of magnetic eld components for 181 magnetic observatory locations calculated by the both models, and use this value to measure the accuracy of the candidate dipole produced as a result of the tting procedure. Since the NelderMead method has a tendency to complete in local minima, this procedure is repeated multiple times (N runs = 50 in our work). As the nal result, we choose the dipole that has the lowest χ 2. The best displacement estimate calculated so far for the 2000 epoch is x = km, y = km, z = km in geographic ECEF coordinates while the tilt angles are θ = 9.38, φ = The total dipole oset is about 414 km. These values roughly correspond to the 1968 epoch eccentricity and tilt reported by Hilton and Schulz [1973], although the eccentricity was determined to be about 450 km at the time (which is not surprising if we take into account the speed at which the NMP position evolves). Finally, let us turn to determining positions of magnetospheric features in GSM while taking into account that the dipole's position and tilt changes over time in this system (Figure 4). While it may seem that the contribution on the scale of several hundred kilometers is not much, we must keep in mind that the thickness of the magnetopause itself is estimated to be of this order of magnitude, with dierent estimates ranging from 200 km [Le and Russell, 1994] to 800 km [Panov et al., 2008]. Therefore, a correction of the magnetopause position on the scale of its thickness may be critically important in determining if a given point in space is on average located below the magnetopause or in the magnetosheath. Conclusion A logical continuation of our work would be a synthesis of the both approaches. While the SHA-wielding models known to the authors are geocentric, we propose analysing the geomag- B e B n B u 340

5 netic eld using high-precision SWARM data in a coordinate system bound to the empirically determined geomagnetic dipole. We envision that this will bring a marked improvement of magnetic eld values estimated in circumpolar regions and also will allow better positioning of magnetospheric features. Acknowledgments. This work is supported by the joint grant of RFBR and the London Royal Society KO_A, RFBR grant A and FP7EC grant IMPEx. Our work utilizes SWARM mission data provided by the European Space Agency, and Ørsted space mission data by the National Space Institute of the Technical University of Denmark. We cordially thank Jana afránková and the Charles University in Prague sta for the invitation and the possibility to participate in the WDS2014 conference. References Campbell, W., Magnetic pole locations on global charts are incorrect, Eos, Transactions American Geophysical Union, 77, Fraser-Smith, A. C., Centered and eccentric geomagnetic dipoles and their poles, , Reviews of Geophysics, 25, 1, Hilton, H. H. and Schulz, M., Geomagnetic potential in oset dipole coordinates, Journal of Geophysical Research, 78, , Le, G. and Russell, C., The thickness and structure of high beta magnetopause current layer, Geophysical research letters, 21, , Mandea, M. and Dormy, E., Asymmetric behavior of magnetic dip poles, Earth, planets and space, 55, , Maus, S., Yin, F., Lühr, H., Manoj, C., Rother, M., Rauberg, J., Michaelis, I., Stolle, C., and Müller, R., Resolution of direction of oceanic magnetic lineations by the sixth-generation lithospheric magnetic eld model from champ satellite magnetic measurements, Geochemistry, Geophysics, Geosystems, 9, Nelder, J. A. and Mead, R., A simplex method for function minimization, Computer journal, 7, , Newitt, L., Chulliat, A., and Orgeval, J., Location of the north magnetic pole in april 2007, Earth Planets and Space (EPS), 61, 703, Newitt, L. R. and Chulliat, A., Comment on will the magnetic north pole move to siberia?, Eos, Transactions American Geophysical Union, 88, , Olsen, N. and Mandea, M., Will the magnetic north pole move to siberia?, Eos, Transactions American Geophysical Union, 88, , Olsen, N., Luhr, H., Finlay, C. C., Sabaka, T. J., Michaelis, I., Rauberg, J., and Toner-Clausen, L., The CHAOS-4 geomagnetic eld model, Geophysical Journal International, 197, , Panov, E., Büchner, J., Fränz, M., Korth, A., Savin, S., Reme, H., and Fornacon, K.-H., High-latitude earth's magnetopause outside the cusp: Cluster observations, Journal of Geophysical Research: Space Physics, 113, Shue, J.-H., Song, P., Russell, C. T., Steinberg, J. T., Chao, J. K., Zastenker, G., Vaisberg, O. L., Kokubun, S., Singer, H. J., Detman, T. R., and Kawano, H., Magnetopause location under extreme solar wind conditions, WDCG web-page, World Data Center for Geomagnetism, Kyoto, Japan, URL online