A coherent model of the crustal magnetic field of Mars

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2004je002265, 2004 A coherent model of the crustal magnetic field of Mars Jafar Arkani-Hamed Earth and Planetary Sciences, McGill University, Montreal, Quebec, Canada Received 12 March 2004; revised 12 July 2004; accepted 27 July 2004; published 15 September [1] The immense amount of the mapping-phase Mars Global Surveyor (MGS) magnetic data allows us to derive a highly accurate magnetic map of Mars. The data acquired at nighttimes within the first 3 years of the mapping phase are divided into two almost equal parts, and each part is expressed in spherical harmonics of degree up to 90. The two models are almost identical over harmonics of degree up to 62 but show appreciable differences over higher-degree harmonics, indicating that the higher-degree harmonic coefficients have appreciable noncrustal contribution. The most repeatable components of the two models are combined to produce an accurate 62-degree harmonic model of the magnetic field of the Martian crust. It is further demonstrated that the altitude of MGS is the major limiting factor of the resolution of the mapping-phase data. The small-scale features of the data, with wavelengths shorter than 400 km, have significant contribution from noncrustal sources. They are not useful for delineating the details of the magnetic source bodies in the crust. INDEX TERMS: 5440 Planetology: Solid Surface Planets: Magnetic fields and magnetism; 5475 Planetology: Solid Surface Planets: Tectonics (8149); 5464 Planetology: Solid Surface Planets: Remote sensing; 5499 Planetology: Solid Surface Planets: General or miscellaneous; KEYWORDS: Magnetic field of Mars, magnetic anomaly resolution, spherical harmonic model of the magnetic field of Mars, Mars, terrestrial planets Citation: Arkani-Hamed, J. (2004), A coherent model of the crustal magnetic field of Mars, J. Geophys. Res., 109,, doi: /2004je Introduction [2] Several magnetic anomaly maps of Mars have been published in the last 5 years on the basis of the Mars Global surveyor (MGS) magnetic data [Acuna et al., 1999, 2001; Purucker et al., 2000; Arkani-Hamed, 2001, 2002; Cain et al., 2003; Langlais et al., 2004]. There is good agreement among the maps over the strong magnetic anomalies in the south hemisphere. But the maps show appreciable differences over small and weak anomalies. In the early maps no conclusive correlation was found between the anomalies and the surface features, except over some large areas such as the lack of appreciable magnetic anomalies over the northern lowlands and inside the giant impact basins, and the absence of magnetic signature associated with large shield volcanoes. However, as more data became available and were incorporated in the magnetic maps some correlations started to emerge [e.g., Langlais and Purucker, 2003; Raymond et al., 2004; Arkani-Hamed, 2004]. After the first phase, i.e., the derivation of the magnetic anomaly maps of Mars that illustrated the major characteristics of the magnetic field, it is now the time to resume the second phase and identify the correlation between the magnetic anomalies and tectonic features. [3] With the available high-altitude magnetic data it is possible to derive an accurate magnetic anomaly map of Mars, though with a somewhat lower resolution. I use the Copyright 2004 by the American Geophysical Union /04/2004JE radial component data acquired at nighttime to derive a 62-degree spherical harmonic model of the magnetic potential of the Martian crust. It is shown in this paper that the nighttime tangential component data are relatively more contaminated than the radial component data. The tangential components of the Martian crustal field determined from this potential model are then compared with the corresponding measured data to assess the external field contribution to the measured tangential components of the magnetic field. I also discuss the resolution of the final magnetic anomaly model. The geological interpretation of the magnetic anomalies is well beyond the scope of this paper, and is not considered. 2. A Coherent Magnetic Field Model of the Crust [4] A highly repeatable and reliable magnetic anomaly map of Mars is essential for the investigation of the relationship between the tectonic features and the magnetization of the Martian crust. The crustal magnetization has likely been acquired in the early, 500 Myrs, history of the planet, when a strong core dynamo was active. The secondary magnetization of the deeper parts of the crust induced by the magnetic field of the upper parts of the crust, and in the absence of the core dynamo, is likely negligible [Arkani-Hamed, 2003]. The original magnetization of the crust has been reduced by later processes associated with meteorite impacts and tectonic activities [e.g., Hood and Richmond, 2002; Hood et al., 2003; Arkani- Hamed, 2004; Mohit and Arkani-Hamed, 2004]. The result- 1of8

2 ing magnetic anomalies, if determined accurately, provide good information about the magnetic properties of the crust. [5] The magnetic anomaly maps derived from the sciencephase and aerobraking-phase low-altitude data acquired at km altitudes [Acuna et al., 1999; Connerney et al., 1999; Purucker et al., 2000; Arkani-Hamed, 2001] are not useful for the investigation of small-scale weak magnetic anomalies. This is partly because of the acrossorbit wide gaps in the data that severely reduce the resolution of the maps despite the high resolution of the data along the orbit, and partly because the data are acquired in daytimes and have contribution from the external field. At these low elevations, the crustal field over the strong anomalies in the south is much greater than the external field, but it may not be the case over the weak anomalies. Some of the small and weak anomalies seen in the low-altitude maps may have appreciable contribution from the external field that obscures the relationship between the anomalies and their crustal magnetic source bodies. [6] The magnetic maps derived by combining the lowaltitude data with the high-altitude data acquired in the early stages of the mapping-phase of MGS [Arkani-Hamed, 2002; Cain et al., 2003; Langlais et al., 2004] used data from all three components of the magnetic measurements. However, the east-west component, and to a lesser extent the northsouth component, of the measured magnetic data have contribution from the external magnetic field, because of the proximity of the satellite orbits to the Martian ionosphere [Mitchell et al., 2001]. A time-varying external magnetic field of period t, measured by a satellite moving at a velocity n, is translated to a special variation of wavelength l = nt. This noncrustal component gets incorporated in the crustal magnetic models and again distorts the shape and amplitude of the weak anomalies, making it difficult to accurately determine their crustal magnetic sources. [7] To derive an accurate map I use the radial component of the mapping phase data alone and select their most common features on the basis of covariance analysis. A map is considered accurate if the amplitudes of all of the wavelengths contained are repeatable. Mars Global Surveyor has provided a huge amount of magnetic data acquired within km altitudes since it has been put in the mapping-phase orbit. The altitude referred to in this paper is relative to a mean spherical surface of radius 3390 km. Although the high altitude of the spacecraft limits the resolution of the map [Acuna et al., 2001; Connerney et al., 2001; Arkani-Hamed, 2002] the huge amount of the data that densely covers the entire globe, except for the small polar region, yields highly repeatable and accurate magnetic anomalies. The radial component data are least contaminated by the external magnetic field compared to the tangential components (see below). The mapping phase data are acquired primarily at two local times, 0200 and Only the nighttime data are used in order to further minimize the effects of external magnetic field. The vast amount of the high-altitude data allows us to divide the entire data into two almost equal sets. Each data set almost uniformly covers the entire surface (except for the small areas over the poles) with a mean track spacing less than 15 km. Each set of data is binned over grid cells of degree and the mean data value of a given grid cell is calculated. Grid cells with only one original data point are not used. Those with 2 or 3 original data points are simply averaged. The data in a cell with more than 3 original data points are averaged after removing the outliers. A data value is assumed an outlier if it differs from the mean value by more than two standard deviation of the values in the cell. Figure 1 shows the histogram of the number of data points remained after removing the outliers. Majority of the cells, about 90%, have more than 15 data points acquired at different times. The averaging of all data within a given grid cell further suppresses the remaining time-varying external field contribution. [8] Figure 2 shows the variations of the orbital radius of MGS versus the colatitude during the nighttime when data are selected. The altitude variations within a cell, <±5 km, are 80 times shorter than the altitude of the data points in the cell. The MGS mean altitude changes by ±30 km over the entire globe, which is 13 times shorter than the mean altitude of the spacecraft. However, as seen in Figure 2, the trends of the altitude variations are similar for both data sets, i.e., higher over the northern lowlands and lower over the southern highlands. Also, the difference in the mean altitudes of the two data sets over a given cell is less than 10 km. [9] Each of the two binned radial component data sets thus obtained is used to determine the spherical harmonic coefficients of a magnetic potential model, specified by harmonics of degree up to 90. The magnetic potential model V at a given observation point (r, q, j) is expressed as Vr; ð q; jþ ¼a X N n¼1 ða=r Þ ðnþ1þ X n ½ g m¼0 nm cos mj þ h nm sin mjšp m n ; where a is the radius of a reference sphere that contains all of the magnetic source bodies. r is the distance from the origin of the coordinate system (the center of the sphere) to the observation point (r > a), P n m is the Schmidt normalize associated Legendre function of degree n and order m, N (=90) is the highest harmonic degree retained in the expansion, and g nm and h nm are the spherical harmonic coefficients. The radial component of the magnetic field, B r, is related to the magnetic potential through B r [10] The spherical harmonic coefficients of the potential model are determined through least squares fitting of the radial component data (hereafter by data I mean the binned data) to the radial component of the model field. Figure 3 shows the scatter diagram of the fitted models to the two data sets, demonstrating the goodness of the fit. Figure 4 shows the models and the difference between the models and the data. Note the different color bars used for the model and the difference maps. The difference maps are dominated by very small amplitude noncrustal noise. Major features of the data are well represented by the models. [11] The two potential models thus obtained provide good opportunity to further remove the possible time-varying, 2of8

3 Figure 1. Histogram of the number of cells containing more than a given number of data points. HA1 and HA2 stand for the two high-altitude data sets. The vertical axis shows the percentage of the grid cells that contain data points larger than that indicated by the corresponding number on the horizontal axis. noncrustal parts of the models introduced by the remaining external magnetic field and by data processing. The power spectra of the radial components of the models and their degree correlation coefficients (Figure 5) show that the two models are almost identical over harmonics of degree up to 60. The models are well correlated over these harmonics, with correlation coefficients higher than The power spectra, however, display distinct characteristics over harmonics of degree higher than 65. Not only the two spectra appreciably differ from each other but also the general trends of the spectra change. Also, the correlation between the two models deteriorates, indicating that the higher-degree harmonic coefficients do not have appreciable repeatability. They have significant contributions from noncrustal sources. [12] Included in Figure 5 are the power spectra and degree correlation coefficients of the covarying harmonics (harmonics with positive degree correlation coefficients (see Arkani-Hamed et al. [1994] for the details of covariance analysis), which are least contaminated by noncrustal sources. The degree correlation coefficients are improved over the higher-degree harmonics. They are now greater than 0.85 over the entire harmonics of degree lower than 62, and greater than 0.95 over harmonics of degree lower than 50. These harmonic coefficients are highly repeatable, and thus reliable. It is worth mentioning, that the procedure adopted in this paper selects only the part of the magnetic data that had minor changes during 3 years of data acquisition. The averaging of many data values in a given grid cell that are measured at different times and the subsequent covariance analysis suppress the time varying noncrustal contribution. However, the data processing is not capable of removing the possible quasi-steady components of the external magnetic field (see below). Nevertheless, I believe that these harmonic coefficients present the least contaminated magnetic potential of the Martian crust that is derived from the mapping-phase data. This is because the original radial magnetic data are least contaminated, and the stringent procedures adopted in the data selection to avoid near dusk data (suggested by J. E. P. Connerney, Goddard Space Flight Center, personal communication, 2003) and the covariance analysis have further suppressed the noncrustal effects. [13] Figure 6 shows the crustal magnetic field components at 370 km altitude, the mean altitude of the mapping phase data, derived by averaging the covarying harmonic coefficients of the two potential models and retaining harmonics of degree up to 62. The figure presents the coherent model of the magnetic field of Mars, because the tangential components shown in the figure are calculated from the magnetic potential that in turn is determined from the least contaminated radial component magnetic data. The strong magnetic anomalies seen in the figure are very similar, although not identical, to those in the magnetic maps derived by many investigators [e.g., Acuna et al., 1999; Purucker et al., 2000; Arkani-Hamed, 2001, 2002; Cain et al., 2003; Langlais et al., 2004]. There are differences over the small-scale and weak anomalies. 3. Quasi-Steady External Magnetic Field Contribution [14] It is important to assess the external magnetic field contribution to the tangential components of the MGS data, because many investigators have used all three components of the data to derive the crustal magnetic models [Arkani- Hamed, 2001, 2002; Cain et al., 2003; Langlais et al., Figure 2. The orbital radius of MGS when the HA1 and HA2 data are selected. HA1 and HA2 stand for the two high-altitude data sets. 3of8

4 Figure 3. The scatter diagrams of the two spherical harmonic models. The strong linearity of the distribution emphasizes the goodness of the model fitting. HA1 and HA2 stand for the two high-altitude data sets. 2004]. For this purpose, I extracted the tangential components of the magnetic field directly from the two mappingphase data sets and binned them as done above for the radial component data. The resulting binned data sets were expanded in the spherical harmonics using harmonics of degree up to 90. Included in Figure 5 are the power spectra of the two models for each of the components, and their degree correlation coefficients. The higher-degree harmonics of the tangential components are more contaminated compared to those of the radial component, as manifested in their relatively smaller correlation coefficients. The power spectra of the tangential components start deviating from each other at harmonics of degree 55 for the north-south component and of degree 50 for the east-west component, compared to degree 62 for the radial component. The degree correlation coefficients of the tangential components also deteriorate at lower-degree harmonics than those of the radial component. [15] The first-degree harmonics have remarkably different power, especially those of the east-west component. The power spectra of the magnetic anomaly models previously derived using all three components of the MGS data also show appreciable differences over these harmonics [see Arkani-Hamed, 2002, Figure 2; Langlais et al., 2004, Figure 5]. The difference is more evident in their degree correlation coefficients, indicating appreciable contribution from noncrustal sources, such as the possible quasi-steady part of the magnetic field associated with the magnetic pileup boundary and draping of the interplanetary magnetic field around the planet [Brain et al., 2003; Bertucci et al., 2003; Vennerstrom et al., 2003]. For detailed investigation of this phenomenon, the first degree covarying harmonic coefficients of the east-west component of the two data sets were averaged. A similar averaging is done for the first degree harmonic coefficients of the north-south components. A horizontal magnetic field intensity map is then derived using the averaged first-degree harmonic coefficients of the east-west and north-south components. The resulting map has the longest wavelength, 20,000 km, and very small amplitude, less than 3 nt, emphasizing that these harmonics have negligible contribution to the crustal magnetic field of Mars shown in Figure 6. [16] The nighttime mapping-phase data selected for this study were acquired at solar zenith angles greater than 120 degree, where the amplitude of the induced magnetic field was less than 20 nt and the induced field had some radial component [Crider et al., 2003]. Detailed study of the morphology of the external magnetic field of Mars [Brain et al., 2003] revealed a radial component of no more than 5 nt at the mapping-phase altitudes. To avoid the effects of the crustal field, Brain et al. [2003] used only those measurements with amplitudes smaller than 10 nt. Although such a selection avoided strong crustal magnetic field, it was not capable of removing the crustal field entirely. Inspection of Figure 8 of Brain et al. [2003] shows that the mean value of the radial component at solar zenith angle greater than 150 degrees increases as the satellite s altitude decreases below 500 km, implying that part of the sources of the anomalies are below the observation points. Moreover, the variability of the radial component shows appreciable increase at these altitudes. According to the authors, that the variability of the data was a reflection of the crustal field effects, the appreciable increase of the variability indicates that their selected data still contained crustal field at the mapping-phase altitude, km, and the actual radial component of the external field was likely smaller than 5 nt. 4. Resolution of the Magnetic Maps of Mars [17] There is some consensus about the resolution limit of the crustal magnetic anomaly maps of Mars derived by several authors. Extensive analyses of the aeromagnetic and satellite magnetic anomalies of Earth show that the altitude of the observation imposes a strong constraint on the resolution of the anomalies. It is conventionally agreed that magnetic measurements do not resolve crustal magnetic anomalies with wavelengths shorter than the altitude of the observation. A similar suggestion was made about the magnetic anomalies of Mars derived from the high-altitude MGS data [Connerney et al., 2001; Arkani-Hamed, 2002]. Figure 7 compares the power spectra of the spherical harmonic models of the magnetic potential of Mars derived by many investigators [Arkani-Hamed, 2002; Cain et al., 2003; Langlais et al., 2004]. The curve CONN is the power spectrum of the spherical harmonic model I derived using Connerney et al. s [2001] radial map. The models are very similar over harmonics of degree less than 50 corresponding to wavelengths longer than 430 km, except for first and second degree harmonics which contain some quasi-steady external field component as mentioned above. 4of8

5 Figure 4. The radial component of the magnetic field models and the difference between the models and the binned data. Note the large difference in the scale of the color bars for the models and the difference maps. HA1 and HA2 stand for the two high-altitude data sets. The present model agrees well with both Connerney et al. s [2001] and Cain et al. s [2003] models up to degree 62, corresponding to wavelengths of about 350 km. The power spectra of the models significantly deviate over higherdegree harmonics, reflecting the effects of different data and data processing techniques used by the authors. The models derived using the mapping phase data alone [Connerney et al., 2001; Arkani-Hamed, 2002; and the present model] show higher power than those derived either from the low-altitude science phase and aerobraking phase data [Arkani-Hamed, 2002], or from a combination of the low-altitude and high-altitude data [Cain et al., 2003; Langlais et al., 2004]. Due to the proximity of the satellite to the ionosphere during the mapping phase, the highaltitude data are noisier than the low-altitude data. The close proximity of Cain et al. s model to the low-altitude model of Arkani-Hamed [2002], as compared to the highaltitude model of Arkani-Hamed [2002], is largely due to the fact that the low-altitude data have much greater amplitudes than the high-altitude data. The fitting of the model to the combined low- and high-altitude data simultaneously tends to favor the low-altitude data. The lower power of Langlais et al. s model at higher-degree harmonics is likely because the dipole source model used by the authors did not sample the high degree harmonics at sufficient resolution and resulted in aliasing of the shorter wavelengths toward the longer ones. The appreciable differences of the models over higher-degree harmonics emphasize that these harmonic coefficients are not reliable. They are dominated by noncrustal sources. Cain et al. [2003] showed that their model fits the data better than the models proposed by Purucker et al. [2000] and Arkani-Hamed [2001], claiming that their model better reflects the crustal magnetic field. It is worth mentioning that it is always 5of8

6 Figure 5. (a) The power spectra of the magnetic field components of the two models. The power n 2 spectrum of degree n is calculated from R n = S m=0 (n + 1) [C nm +S 2 nm ], where C nm and S nm stand for the spherical harmonic coefficients of a given magnetic field component of degree n and order m. Thin solid and thin dotted curves are HA1 and HA2 models. Thick solid and thick dashed curves are the covarying harmonics of the HA1 and HA2 models, respectively. (b) The degree correlation of HA1 and HA2 models (thin solid), the covarying HA1 and HA2 models (thick solid), and the averaged covarying HA1 and HA2 model and the 62-degree models directly derived from the original binned data (thick dashed). possible to better fit the observed magnetic data by including higher-degree harmonics. A one-dimensional example is a polynomial of degree n that perfectly fits n + 1 data points. But the details of the data, and thus the higher orders of the polynomial, may not reflect the basic physics presented by the data. Cain et al. s [2003] model better fits the data, but the data have appreciable noncrustal components. Cain et al. [2003] combined the low- and high-altitude data and included some daytime data that are affected by external field, the low-altitude data were acquired during day times. Also, during the mapping phase MGS is partly inside the ionosphere and thus the short wavelength components of the data, especially the daytime data, are contaminated by noncrustal sources. Figure 7 emphasizes that the degree 62 presents likely an optimum upper limit of the harmonic degrees of the crustal magnetic field that can be resolved by the available high-altitude, mapping-phase magnetic data. Two different philosophies have been adopted in the spherical harmonic modeling of the magnetic field of Mars. The first philosophy, to derive a model that better fits the observation, has resulted in the high-degree model of Cain et al. [2003], which better represents the actual magnetic field (crustal plus external) at satellite altitude and is useful for studying the dynamics of charged particles at satellite altitudes. The second philosophy, to derive the most reliable magnetic field of the crust, relies on the fact that the higherdegree harmonic coefficients are dominated by time varying, noncrustal components. This modeling philosophy resulted in the low-degree models of Arkani-Hamed [2002] and the present paper, which are useful for determining the magnetization of the crust and geological interpretation of the magnetic anomalies. 5. Conclusions [18] A highly repeatable and coherent model of the magnetic field of the Martian crust is derived using the nighttime mapping-phase MGS magnetic data. The data are first divided into two almost identical parts, and each part is expressed in terms of the spherical harmonics of degree up to 90. The power spectra and degree correlation coefficients of these two models show that the models are almost identical over harmonics of degree up to 62, with correlation coefficients larger than The higher-degree harmonics show distinct differences, Their power spectra differ appreciably and their degree correlation coefficients deteriorate, emphasizing that they have appreciable contributions from noncrustal sources. The general characteristics of the power spectra and degree correlation of the two models suggest that the resolution of the mapping-phase data is limited by the altitude of the measurements. The final 62-degree model of the Martian crustal magnetic field is derived using the 6of8

7 Figure 7. The power spectra of the magnetic potential models. The power spectrum of degree n is calculated from n R n = S m=0 (n + 1) [g 2 nm +h 2 nm ], where g nm and h nm stand for the spherical harmonic coefficients of a given magnetic potential model. The curve CONN stands for Connerney et al. [2001], JAH-HA and JAH-LA are the high-altitude and the low-altitude models of Arkani-Hamed [2002], CAIN stands for Cain et al. [2003], and LANG stands for Langlais et al. [2004]. JAH is the present model. covarying harmonics of degree up to 62, which have degree correlations higher than 0.85 over the entire harmonics retained, and higher than 0.95 over the harmonics of degree lower than 50. [19] Acknowledgments. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. I would like to thank Norman F. Ness for his constructive and useful comments. Figure 6. The 62-degree harmonic model of the magnetic field of Mars at 370 km altitude. References Acuna, M. H., et al. (1999), Global distribution of crustal magnetization discovered by the Mars Global Surveyor MAG/ER experiment, Science, 284, Acuna, M. H., et al. (2001), Magnetic field of Mars: Summary of results from aerobreaking and mapping orbits, J. Geophys. Res., 106, 23,403 23,417. Arkani-Hamed, J. (2001), A 50 degree spherical harmonic model of the magnetic field of Mars, J. Geophys. Res., 106, 23,197 23,208. Arkani-Hamed, J. (2002), An improved 50-degree spherical harmonic model of the magnetic field of Mars derived from both high-altitude and low-altitude data, J. Geophys. Res., 107(E10), 5083, doi: / 2001JE Arkani-Hamed, J. (2003), Thermoremanent magnetization of the Martian lithosphere, J. Geophys. Res., 108(E10), 5114, doi: / 2003JE Arkani-Hamed, J. (2004), Timing of the Martian core dynamo, J. Geophys. Res., 109, E03006, doi: /2003je Arkani-Hamed, J., R. Langel, and M. Purucker (1994), A magnetic anomaly map of the Earth derived from POGO and Magsat data, J. Geophys. Res., 99, 24,075 24,090. Bertucci, C., et al. (2003), Magnetic field draping enhancement at the Martian magnetic pileup boundary from Mars global surveyor observations, Geophys. Res. Lett., 30(2), 1099, doi: /2002gl Brain, D. A., F. Bagenal, M. H. Acuña, and J. E. P. Connerney (2003), Martian magnetic morphology: Contributions from the solar wind and crust, J. Geophys. Res., 108(A12), 1424, doi: /2002ja Cain, J. C., B. B. Ferguson, and D. Mozzoni (2003), An n = 90 internal potential function of the Martian crustal magnetic field, J. Geophys. Res., 108(E2), 5008, doi: /2000je Connerney, J. E. P., et al. (1999), Magnetic lineations in the ancient crust of mars, Science, 284, of8

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