The permanent and induced magnetic dipole moment of the moon

C. T. Russell1, P. J. Coleman, Jr.1, 2, B. R. Lichtenstein1, 2, G. Schubert2

 

1. Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024

2. Department of Planetary and Space Science, University of California, Los Angeles, California 90024

 

  Originally published in: Proceedings of the Fifth Lunar Conference, Vol. 3, pp. 2747-2760, 1974.

 

Abstract

Magnetic field observations with the Apollo 15 subsatellite have been used to deduce the components of both the permanent and induced lunar dipole moments in the orbital plane. The present permanent lunar magnetic dipole moment in the orbital plane is less than 1.3 x 1018 -cm3. Any uniformly magnetized near surface layer is therefore constrained to have a thickness-magnetization product less than 2.5 emu-cm-g-1. The induced moment opposes the external field, implying the existence of a substantial lunar ionosphere with a permeability between 0.63 and 0.85. Combining this with recent measures of the ratio of the relative field strength at the ALSEP and Explorer 35 magnetometers indicates that the global lunar permeability relative to the plasma in the geomagnetic tail lobes is between 1.008 and 1.03.

 

Introduction

While the present-day global lunar magnetic field is weak, the existence of strong regional magnetic fields has been verified by several independent techniques. The returned lunar samples possess substantial natural remanent magnetization (cf. Fuller, 1973 and references therein); strong magnetic fields have often been observed at the Apollo landing sites (Dyal et al., 1970, 1971, 1972a); and orbital studies have shown that there are extensive regions of magnetized material lying at or near the surface (Sharp et al., 1973). We do not know from these measurements whether the fields that magnetized the returned samples, were of lunar or extra-lunar origin. Whatever the source, the magnetizing field must have been ancient and must have been at least several thousand gammas (1 = 10-5 gauss) (Helsley, 1971; Gose et al., 1973), and perhaps even as large as one hundred thousand gammas (Collinson et al., 1973). Furthermore, while the measurements at the landing sites reveal fields at places exceeding 300 (Dyal et al., 1972b), the measured fields at the various sites are apparently oriented at random to one another and show significant variations, even reversals, over distances of the order of kilometers (Dyal et al., 1971, 1972b). Thus, although the remanent magnetization of the returned samples and the surface magnetic measurements imply a strong ancient magnetizing field, perhaps of internal origin, they shed no light on the present-day global field. For this, we must turn to the orbital data.

In addition to the question of the existence of a permanent global magnetic field is the related one of whether the moon contains enough permeable material for a significant dipole moment to be induced when the moon passes through the earth's geomagnetic tail. It has been suggested that such a measurement would provide constraints on the amount of lunar ferromagnetic material at temperatures below the Curie point (Dyal and Parkin, 1973).

We note that if the moon has a significantly large induced dipole moment, then the measurement of the permanent dipole moment may be affected, unless care is taken to give equal weight to data from both lobes of the geomagnetic tail in which the external field is oppositely directed. While data from both lobes have been used in subsatellite studies of the lunar dipole moment (Russell et al., 1973) more weight was implicitly given to the larger data base obtained in the northern lobe. Since Parkin et al. (1973) have reported a relatively large bulk permeability for the moon, = 1.03 + 0.02, it is appropriate to reanalyze the subsatellite data to remove the possibility of contamination by the induced moment. We will also use these data to measure the induced dipole moment, eliminating contributions from the permanent moment. The result of the former analysis is in accord with our previous findings, while that of the latter analysis is somewhat surprising, with important implications for the interpretation of the permeability measurement by Parkin et al. (1973).

 

The Experiment and Data

The Apollo 15 subsatellite was injected into a roughly circular 100 km altitude orbit shortly before the end of the Apollo 15 mission and continued to transmit magnetic field data for over six months. The spin stabilized subsatellite carried a biaxial magnetometer which returned vector samples of the magnetic field every 24 sec, at the slowest data rate, almost continually throughout each 2-hour lunar orbit. A more complete description of the magnetometer has been given by Coleman et al. (1972a,b).

Fig. 1. Magnetic field strength on seven consecutive lunar orbits while the moon was in the geomagnetic tail. (1 = 10-5 gauss).

While the flow of plasma past the moon distorts the lunar field in the solar wind and magnetosheath, the lobes of the geomagnetic tail are relatively free of such effects and evidence of the lunar field becomes readily apparent. Figure 1 shows the magnetic field strength measured on seven consecutive orbits in the north lobe of the geomagnetic tail. The portion gf the orbit labeled shadow occurs over the farside of the moon. The most striking feature of these data is the similarity of magnetic field from pass to pass, which changes only slowly as the moon rotates under the orbit plane of the subsatellite. Transient features do occur, as on orbits 1080 and 1083, and such periods are eliminated from the analysis to follow. The large feature that occurs near 1150, 1350, etc., arises from an extensive magnetized region centered between the craters Van de Graaff and Aitken at approximately 170o E and 20o S selenographic coordinates. A more complete discussion of the mapping of such features has been given by Sharp et al. (1973) and Russell et al. (1973; 1974).

 

The Analysis Technique

The magnetic field measured at the subsatellite consists of contributions from distant sources, from the interaction of the plasma environment with the moon, and from permanent and induced fields in the moon itself. In the lobes of the geomagnetic tail, the field due to external sources, i.e. the geomagnetic field, is steadier than in any other region encountered by the moon and the energy density of the plasma is least. Thus, we expect fields due to the interaction of the moon with its plasma environment, such as limb compressions (Russell et al., 1973), and temporally varying external and induced fields, to be minimal.

To separate the lunar field which is fixed in selenographic coordinates from the geomagnetic field which is ordered in solar oriented coordinates, we have removed the average and the linear trend for each orbit of data, assuming these to be due to the geomagnetic field and then rotated the residual fields into a local lunar coordinate system, with one direction radially outward, one tangential to the orbit plane (eastward), and one perpendicular to the orbit plane (northward). These residual fields are assumed to be due to lunar sources. This method does not permit measurement of the dipole moment perpendicular to the orbit plane, since any such moment will give a constant field perpendicular to the orbit plane everywhere along the circular subsatellite orbit. On the other hand, we would not expect the lunar dipole moment either induced or permanent to lie in this direction (116o E, 62o N). In analogy with the earth and Jupiter (Smith et al., 1974), a permanent moment would be expected to be nearly aligned with the rotation axis. An induced moment should be approximately in the equatorial plane, i.e. parallel to the tail field through the center of the moon. We note that this technique reduces the contribution of the dipole field, since a sinusoidally varying field in the radial component, say, consists of a steady field and a component at twice the orbital period in the inertial frame. We have compensated for this effect in the analyses to follow.

In addition to separating geomagnetic fields from lunar fields, we wish to separate the permanent moment from the induced moment. During periods in which the moon is in the south lobe of the geomagnetic tail, any induced dipole moment should be in the antisolar direction, while in the northern lobe the opposite situation should prevail. On the other hand, any permanent dipole moment would be fixed in the moon, and independent of the lobe in which the moon was situated. Thus, given equal amounts of data in both lobes of the tail, the induced dipole moment would not contribute to the measured permanent dipole moment and vice versa. However, the amount of data obtained with the Apollo 15 subsatellite in each lobe is not equal. Thus, we have divided the data into two groups according to the lobe of the geomagnetic tail in which the data were obtained, and analyzed each group separately, later averaging the results from both lobes.

While the data shown in Fig. 1 are fairly representative of the data obtained in the lobes of the geomagnetic tail, there are also many times when there is only partial coverage or only a portion of the orbit is suitable for mapping due to the presence of telemetry errors, or transient events. In this study, only orbits with at least 75% coverage were used. There were a total of 29 such orbits, 10 in the south lobe and 19 in the north lobe. Of these orbits, we eliminated the 4 noisiest orbits (2 from each lobe) on the basis of the r.m.s. amplitude of the residual fields.

 

The Permanent Dipole Moment

Fig. 2. The average radial, tangential and perpendicular residual field as a function of azimuthal angle, along the orbit plane measured eastward from the 0o selenographic meridian for data obtained in the north lobe.

Since there were at least short gaps in the data coverage for every orbit, the residual fields in local lunar coordinates were first averaged by azimuthal angle from the 0o selenographic meridian in 1o bins, creating a time series without gaps. Figure 2 shows the average residuals as a function of orbit azimuth for the radial, tangential, and perpendicular components in the north lobe of the tail.

Fig. 3. Fourier amplitudes of the first 20 harmonics of the data shown in Fig. 2.

The two sets of 360 average residual fields, one for each lobe, were then Fourier analyzed to determine the amplitude and phase at the orbital period and its harmonics. Figure 3 shows the amplitudes of the first 20 Fourier coefficients corresponding to the vector time series given in Fig. 2. We note that the dipolar component of the lunar field is not as dominant in the lunar field as the earth's dipolar component. The terrestrial dipolar field strength is 7 times the quadrupolar field at the surface of the earth (Mead, 1970). Table 1 gives the amplitude and phase of the first harmonic of the radial, tangential, and perpendicular field components for both lobes of the tail, as measured at the subsatellite.

Combining north lobe and south lobe data, we obtain two independent estimates of the lunar dipole moment, one from the radial component and one from the tangential component. The first harmonic of the perpendicular component can also be converted to an equivalent dipole moment and used as an error estimate. These three moments, the mean of the tangential and radial estimates, and the difference between the radial estimate and the mean, are listed in Table 2.

We have three estimates of the error in an individual component: each of the components of the difference vector and the mean of the two perpendicular components. (The mean is taken to obtain the same statistical accuracy as in the average and difference vectors.) The resulting root mean square error is 0.94 x 1018 -cm3. Thus, our estimate of the dipole moment, i.e. [(1.19 + 0.94)i + (0.54 + 0.94)j] x 1018 -cm3, where i is in the direction of the intersection of the subsatellite orbit and the 0o selenographic meridian and j is in the orbit plane 90o to the east. While the ith component excludes zero up to the 80% confidence level, the j-component excludes zero only up to the 40% confidence level. Further, the magnitude of the observed moment, 1.31 x 1018 -cm3, is less than the magnitude of the expected error vector 1.33 x 1018 -cm3 (0.94 x 21/2 x 1018 -cm3). Thus, we conclude that the lunar dipole moment is most likely below our present limit of detectability, i.e. less than 1.33 x 1018 -cm3, and, assuming that the permanent dipole moment should be along the lunar rotation axis, this places an upper limit of 2.8 x 1018 -cm3 on the permanent moment, a value less than 3.5 x 10-8 that of the earth. We note that this value is 40% smaller than the upper limit obtained when no explicit steps were taken to exclude the contribution of the induced moments (Russell et al., 1973). This, plus the fact that the north and south lobe estimates of both the radial and tangential fields in Table 1 are nearly 180o out of phase, suggests that there is a significant induced component. However, before pursuing this, we will discuss the implications of such a small lunar dipole field.

 

Discussion

Even if we could overlook the ancient lunar field determination of Collinson et al. (1973), fields of over 2000 are required to cause the remanent magnetization observed in the returned lunar samples (Gose et al., 1973). Thus, an ancient lunar dipole moment, if the cause of these fields, must have had a strength of at least 1.5 x 1023 -cm3. The present-day dipole moment is less than 2 x 10-5 of this value. If the ancient field were due to a lunar dynamo, this dynamo has effectively stopped. Since dynamo action is associated with mass motion, the absence of evidence for a present-day dynamo is in accord with the extremely weak seismological activity (cf. Latham et al., 1973).

On the other hand, if the moon were once uniformly magnetized, a magnetization of approximately 2 x 10-3 emu g-1 would be required. To be consistent with the present-day moment, all but the outer 11 meters of this material must have been demagnetized. It is obvious then, that if an ancient magnetic moment ever existed it has effectively completely disappeared. We note that if the moon were originally uniformally magnetized with an intensity of 10-4 emu-g-1, a layer of 0.3 km thickness might still remain. However, such an ancient lunar field would fail by a factor of 20 to account for ancient field strength deduced by Gose et al. (1973) and 1000 to account for the ancient field strength deduced by Collinson et al. (1973).

It has been postulated that the small scale magnetic fields may be attributed to holes punched into otherwise uniformly magnetized layers (Sonett and Runcorn, 1973). Assuming a uniform layer of magnetization of 10-4 emu-gm-1, which has been often observed in the returned lunar samples (cf. Pearce et al., 1973), the measured dipole moment constrains this layer to be 0.3 km thick, while the amount of magnetized material of such magnetization required to create the Van de Graaff magnetic feature would have a thickness of 20 km (Sharp et al., 1973). This suggests that the Van de Graaff region is not a hole in a uniformly magnetized shell. Alternatively, the limit on the remaining surface area of such a 20 km deep layer, if it once existed, is 1.2% of the lunar surface. If, however, the moments of the various magnetized regions were randomly oriented, 10,000 regions with the magnetic moment of the principal Van de Graaff feature, could exist without the resultant moment exceeding our observed upper limit. Since 96% of the lunar surface surveyed by the subsatellite is devoid of magnetic features with fields 0.4 , this does not appear to be the case.

 

The Induced Dipole Moment

Fig. 4. Induced field ratio for the radial component in both the north and south lobes of the geomagnetic tail as a function of azimuthal angle measured eastward from the projection of the external field on the orbit plane.

The technique for the measurement of the induced dipole moment is identical to that for the permanent moment with two modifications. First, the azimuthal angle along the orbit track is measured from the projection of the orbital average field into the orbit plane. The orbital average field is considered to be the external field. Second, since this analysis is insensitive to induced moments perpendicular to the orbit plane, we use the field projected on the orbit plane to normalize the induced field. Figure 4 shows the normalized radial component as a function of azimuthal angle for the north and south lobes. The large fields near 160o in the north lobe and near 340o in the south lobe are associated with the permanent fields in the Van de Graaff region. Table 3 shows the amplitude and phase of the normalized induced field for the radial, tangential, and perpendicular field components for both lobes of the tail.

Combining north and south lobe data, we obtain two independent estimates of the induced lunar dipole moment, and an equivalent induced moment from the perpendicular component. These moments, the mean of the tangential and radial estimates and the difference between the radial estimate and the mean are listed in Table 4. We have the same three estimates of the error as we had before, and under the assumption that the lunar permeability is isotropic, the magnitude of the induced j-component provides a fourth estimate.

The resulting root mean square error is 2.4 x 1022 -cm3 / . Thus, our estimate of the induced dipole moment is (-6.25 + 2.4) x 1022 -cm3 / . Somewhat surprisingly the induced moment opposes the external field. In fact, the probability that the induced moment is positive is 0.4%, assuming random errors with a gaussian distribution. In support of this we note that each of the four measurements in Table 3, north and south, radial and tangential, give a negative induced moment. Thus, the region below the subsatellite orbit must be diagmagnetic with respect to the region above it. Table 5 lists the ratio of the equatorial induced field to the external field at the altitude of the subsatellite, the equivalent permeability, and the probability that our measurements are consistent with these values.

 

Discussion

The principal source of error in this analysis is the change in the external field during the course of the measurement. This sets the standard deviation, which is small compared to the induced field. As a further check, we processed all 29 orbits, and then only the 8 most quiet ones. Both these cases returned results similar to those above, except that the standard deviations were greater; in the former because of the addition of noisy orbits and in the latter because of the large reduction in the number of independent samples. Finally, we processed test signals and assured ourselves that there were no phase changes introduced by our computer processing of the data. We note that since the same magnetometer is used for measuring both the external field and the induced field, the moments obtained by this technique do not contain systematic errors introduced by possible drifts in instrument calibration and the like. Thus, there is little doubt that the region below the subsatellite orbit is diamagnetic with respect to the region above it.

Parkin et al. (1974) have reported a paramagnetic or ferromagnetic whole body permeability, 1.006 1.018. While their technique is dependent on the relative calibration of the ALSEP and Explorer 35 magnetometers and though the surface magnetometers may be responding to some local property rather than the global permeability, we would have expected a permeability similar to the one they reported. The only conclusion consistent with both observations is that a diamagnetic layer must lie between the subsatellite and the moon. A substantial lunar ionosphere would provide such a layer.

The calculation of the lunar permeability by Parkin et al. (1974) did not include the effect of a lunar ionosphere. Such a layer would tend to shield the interior from the external field and hence the ratio of the field strength at the ALSEP, to that at Explorer 35, would be less than would be observed in the absence of an ionosphere. Thus, the permeabilities deduced by Parkin et al. (1974) should be revised upward. Similarly, our estimates of the global permeability of the region below the subsatellite given in Table 5 do not include the effect of a core of high permeability, and hence the permeability of the ionosphere is much less than that given in Table 5.

In order to obtain a reasonable estimate of the permeabilities of both the ionosphere and the moon from the induced field at the subsatellite and the ALSEP/Explorer 35 field ratio, we have assumed that in the tail lobe (permeability o = 1), an ionosphere with permeability 1, fills the space between the subsatellite and the moon, which has a uniform permeability 2. Following Jackson (1962), we obtain the ratio, R, of the induced field at the subsatellite to the external field

and the ratio of the field at the ALSEP magnetometer to the external field

We have used a = 1738 km and b = 1838 km in our calculations.

Fig. 5. Joint permeabilities ratios of ionosphere, 1 and moon, 2 consistent with ALSEP/Explorer 35 field ratio and the induced field measured at the subsatellite. Shaded region indicates values within one standard deviation of the measured value for both measurements.

Figure 5 is a plot of the permeabilities pairs consistent with the observations. The possible permeability pairs range from the case of a strongly diamagnetic ionosphere 1 = 0.63 and a strongly ferromagnetic moon 2 = 1.03 to that of a much weaker ionosphere ,u 1 = 0.85 and a slightly ferromagnetic moon 2 = 1.008. Such a permeability for the ionospheric region implies a substantial energy density, of the order of 100 eV-cm-3 for the ionospheric plasma. This could be provided, for example, by a density of 500 cm-3 at 2000oK, a temperature similar to that in the terrestrial ionosphere, or by a density of 10 cm-3 with an average energy of 10 eV. The former model is consistent with the ionization of from 0.2% to 5% of the lunar atmosphere (Hodges et al., 1973; Hoffman et al., 1973); the latter model is consistent with the observations of Lindeman et al. (1973) who report Suprathermal Ion Detector Experiment observations of a density of 20 cm -3 in the energy range from 10 to 500 eV in the geomagnetic tail. While our data in themselves are consistent with either source, the ionospheric scale height must be comparable to or less than the subsatellite altitude. We note that this would be true in the former case in which the scale height would be similar to the neutral scale height. The lunar surface would have to have a potential of several hundred volts to maintain such a small scale height for several hundred volt ions. Finally, the observed photoelectron layer would only make a minor contribution to the diamagnetism of this region (Reasoner and Burke, 1972).

Returning to the question of the lunar permeability, the primary effect of the ionosphere is to widen the range of possible lunar permeabilities, and to increase slightly the minimum probable permeability. Since the permeabilities derived by Parkin et al. (1974) were used to model the lunar interior, these models should be revised. The primary source of the lunar permeability is free iron. Thus, raising the permeability estimate requires increasing the estimate of the iron content. However, since the lunar moment of inertia is close to that of a uniform sphere, we cannot distribute the relatively heavy free iron at will. We may add free iron to the outer cool layer, (cool relative to the Curie point) keeping its thickness constant or we may increase their thickness keeping the specific iron content constant. The latter implies a cooler moon. Finally, we note that the permeability ratios calculated are measured relative to the diamagnetic tail lobes. While the energy density of the plasma in the tail lobes is small compared to other regions of space, it may be large enough to contribute significantly to the apparent lunar permeability. CONCLUSIONS The present permanent lunar magnetic dipole moment is less than 1.3 x 1018 -cm3, in the orbit plane of the Apollo 15 subsatellite and less than 3 x 1018 -cm3 if the moment is aligned with the lunar spin axis. There is no evidence in these data for an ancient lunar dipole field. If there exists a uniformly magnetized near surface layer, the subsatellite measurements constrain the product of the magnetization of this layer and its thickness to be less than 2.5 emu-cm-g-1. Thus, a layer of magnetization of 10-4 emu-g-1 would be constrained to a thickness of less than 0.3 km. Since modeling the Van de Graaff magnetic anomaly requires a 20 km thickness of material of this magnetization, magnetic features such as Van de Graaff should be considered magnetized patches rather than holes in a magnetized layer.

There is a measurable induced field at the orbit of the subsatellite, opposing the external field. This diamagnetic behavior indicates the presence of a substantial lunar ionosphere, with a permeability in the range from 0.63 to 0.85. Previous estimates of the global lunar permeabilities should be raised to account for this ionosphere. The ratios of the field at ALSEP 15 to that at Explorer 35 imply a lunar permeability of between 1.008 and 1.03. Compositional and thermal models should be revised by increasing the total free iron content and possibly decreasing the interior temperature, if it can be shown that the permeability ratio observed is not principally due to the diamagnetism of the tail lobes.

 

Acknowledgments

We wish to thank C. W. Parkin, P. Dyal, and W. D. Daily for discussions of their latest analyses before publication, and C. P. Sonett for several helpful suggestions. This work was supported by the National Aeronautics and Space Administration under contract NAS 9-12236 and by NASA grant NGR 05-007-351.

 

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