ALBERT

Description: The discussion in the preceding paper is restricted to the uncertainties in magnetic-field-line tracing in the magnetosphere resulting from published standard errors in the spherical harmonic coefficients that define the axisymmetric part of the internal geomagnetic field (i.e. gn0 ± δgn0). Numerical estimates of these uncertainties based on an analytic equation for axisymmetric field lines are in excellent agreement with independent computational estimates based on stepwise numerical integration along magnetic field lines. This comparison confirms the accuracy of the computer program used in the present paper to estimate the uncertainties in magnetic-field-line tracing that arise from published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) internal geomagnetic field (i.e. gnm ± δgnm and hnm ± δhnm). An algorithm is formulated that greatly reduces the computing time required to estimate these uncertainties in magnetic-field-line tracing. The validity of this algorithm is checked numerically for both the axisymmetric part of the internal geomagnetic field in the general case (1 ≤ n ≤ 10) and the complete internal geomagnetic field in a restrictive case (0 ≤ m ≤ n, 1 ≤ n ≤ 3). On this basis it is assumed that the algorithm can be used with confidence in those cases for which the computing time would otherwise be prohibitively long. For the complete internal geomagnetic field, the maximum characteristic uncertainty in the geocentric distance of a field line that crosses the geomagnetic equator at a nominal dipolar distance of 2 RE is typically 100 km. The corresponding characteristic uncertainty for a field line that crosses the geomagnetic equator at a nominal dipolar distance of 6 RE is typically 500 km. Histograms and scatter plots showing the characteristic uncertainties associated with magnetic-field-line tracing in the magnetosphere are presented for a range of illustrative examples. Finally, estimates are given for the maximum uncertainties in the locations of the conjugate points of selected geophysical observatories. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, including the associated uncertainties in the locations of the conjugate points of geophysical observatories, should be regarded as "first approximations'' in the sense that these estimates are only as accurate as the published standard errors in the full set of spherical harmonic coefficients. As in the preceding paper, however, all computational techniques developed in this paper can be used to derive more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Description: The validity of a technique developed by the authors to identify historical occurrences of intense geomagnetic storms, which is based on finding approximately coincident observations of sunspots and aurorae recorded in East Asian histories, is corroborated using more modern sunspot and auroral observations. Scientific observations of aurorae in Japan during the interval 1957–2004 are used to identify geomagnetic storms that are sufficiently intense to produce auroral displays at low geomagnetic latitudes. By examining white-light images of the Sun obtained by the Royal Greenwich Observatory, the Big Bear Solar Observatory, the Debrecen Heliophysical Observatory and the Solar and Heliospheric Observatory spacecraft, it is found that a sunspot large enough to be seen with the unaided eye by an "experienced" observer was located reasonably close to the central solar meridian immediately before all but one of the 30 distinct Japanese auroral events, which represents a 97% success rate. Even an "average" observer would probably have been able to see a sunspot with the unaided eye before 24 of these 30 events, which represents an 80% success rate. This corroboration of the validity of the technique used to identify historical occurences of intense geomagnetic storms is important because early unaided-eye observations of sunspots and aurorae provide the only possible means of identifying individual historical geomagnetic storms during the greater part of the past two millennia.

Description: The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a "magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. gn0 ± δgn0). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as "first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Description: A previous application of extreme-value statistics to the first, second and third largest geomagnetic storms per solar cycle for nine solar cycles is extended to fourteen solar cycles (1844–1993). The intensity of a geomagnetic storm is measured by the magnitude of the daily aa index, rather than the half-daily aa index used previously. Values of the conventional aa index (1868–1993), supplemented by the Helsinki Ak index (1844–1880), provide an almost continuous, and largely homogeneous, daily measure of geomagnetic activity over an interval of 150 years. As in the earlier investigation, analytic expressions giving the probabilities of the three greatest storms (extreme values) per solar cycle, as continuous functions of storm magnitude (aa), are obtained by least-squares fitting of the observations to the appropriate theoretical extreme-value probability functions. These expressions are used to obtain the statistical characteristics of the extreme values; namely, the mode, median, mean, standard deviation and relative dispersion. Since the Ak index may not provide an entirely homogeneous extension of the aa index, the statistical analysis is performed separately for twelve solar cycles (1868–1993), as well as nine solar cycles (1868–1967). The results are utilized to determine the expected ranges of the extreme values as a function of the number of solar cycles. For fourteen solar cycles, the expected ranges of the daily aa index for the first, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude, contrary to the situation for the half-daily aa index over nine solar cycles. The observed range of the first extreme daily aa index for fourteen solar cycles is 159–352 nT and for twelve solar cycles is 215–352 nT. In a group of 100 solar cycles the expected ranges are expanded to 137–539 and 177–511 nT, which represent increases of 108% and 144% in the respective ranges. Thus there is at least a 99% probability that the daily aa index will satisfy the condition aa 〈 550 for the largest geomagnetic storm in the next 100 solar cycles. The statistical analysis is used to infer that remarkable conjugate auroral observations on the night of 16 September 1770, which were recorded during the first voyage of Captain Cook to Australia, occurred during an intense geomagnetic storm.

Description: This paper provides a complete generalization of the classic result that the radius of curvature (ρ) of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particle's equatorial distance (ϖ) from the dipole (i.e. ρ ∝ ϖ3). Comparable results are derived for the radii of curvature of all possible planar charged-particle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ϖn+2. This result reduces to the classic one in the case of a zonal magnetic dipole (n =1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 〈 m 〈 n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m(Φ – Φnm)] = 0, where Φnm = (1/m) arctan (hnm/gnm); gnm and hnm denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis (θ = 0,π) of a tesseral magnetic multipole is a magnetic neutral line if m 〉 1. A further 2m(n – m) neutral lines exist at the intersections of the 2m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation Pnm(cos θ) = 0 in the range 0 〈 θ 〈 π, where Pnm(cos θ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ϖn+2 and inversely proportional to cos[m(Φ – Φnm)]. Since this last expression vanishes at the 2m equatorial neutral lines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rn+2, where r denotes radial distance from the multipole, and inversely proportional to Pnm(cos θ)/sin θ. Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral line (m 〉 1) or any one of the 2m(n – m) neutral lines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are presented for low-degree (n ≤ 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the "non-adiabatic'' scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.

Description: Early auroral observations recorded in various oriental histories are examined in order to search for examples of strictly simultaneous and indisputably independent observations of the aurora borealis from spatially separated sites in East Asia. In the period up to ad 1700, only five examples have been found of two or more oriental auroral observations from separate sites on the same night. These occurred during the nights of ad 1101 January 31, ad 1138 October 6, ad 1363 July 30, ad 1582 March 8 and ad 1653 March 2. The independent historical evidence describing observations of mid-latitude auroral displays at more than one site in East Asia on the same night provides virtually incontrovertible proof that auroral displays actually occurred on these five special occasions. This conclusion is corroborated by the good level of agreement between the detailed auroral descriptions recorded in the different oriental histories, which furnish essentially compatible information on both the colour (or colours) of each auroral display and its approximate position in the sky. In addition, the occurrence of auroral displays in Europe within two days of auroral displays in East Asia, on two (possibly three) out of these five special occasions, suggests that a substantial number of the mid-latitude auroral displays recorded in the oriental histories are associated with intense geomagnetic storms.Key words. Magnetospheric physics (auroral phenomena; storms and substorms)