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)

Description: Possible configurations of the magnetic field in the outer magnetosphere during geomagnetic polarity reversals are investigated by considering the idealized problem of a magnetic multipole of order m and degree n located at the centre of a spherical cavity surrounded by a boundless perfect diamagnetic medium. In this illustrative idealization, the fixed spherical (magnetopause) boundary layer behaves as a perfectly conducting surface that shields the external diamagnetic medium from the compressed multipole magnetic field, which is therefore confined within the spherical cavity. For a general magnetic multipole of degree n, the non-radial components of magnetic induction just inside the magnetopause are increased by the factor {1 + [(n + 1)/n]} relative to their corresponding values in the absence of the perfectly conducting spherical magnetopause. An exact equation is derived for the magnetic field lines of an individual zonal (m = 0), or axisymmetric, magnetic multipole of arbitrary degree n located at the centre of the magnetospheric cavity. For such a zonal magnetic multipole, there are always two neutral points and n-1 neutral rings on the spherical magnetopause surface. The two neutral points are located at the poles of the spherical magnetopause. If n is even, one of the neutral rings is coincident with the equator; otherwise, the neutral rings are located symmetrically with respect to the equator. The actual existence of idealized higher-degree (n〉1) axisymmetric magnetospheres would necessarily imply multiple (n + 1) magnetospheric cusps and multiple (n) ring currents. Exact equations are also derived for the magnetic field lines of an individual non-axisymmetric magnetic multipole, confined by a perfectly conducting spherical magnetopause, in two special cases; namely, a symmetric sectorial multipole (m = n) and an antisymmetric sectorial multipole (m = n-1). For both these non-axisymmetric magnetic multipoles, there exists on the spherical magnetopause surface a set of neutral points linked by a network of magnetic field lines. Novel magnetospheric processes are likely to arise from the existence of magnetic neutral lines that extend from the magnetopause to the surface of the Earth. Finally, magnetic field lines that are confined to, or perpendicular to, either special meridional planes or the equatorial plane, when the multipole is in free space, continue to be confined to, or perpendicular to, these same planes when the perfectly conducting magnetopause is present.Key words. Geomagnetism and paleomagnetism (reversals-process, time scale, magnetostratigraphy) · Magnetospheric physics (magnetopause, cusp, and boundary layers; magnetospheric configuration and dynamics)

Description: The earliest known drawing of sunspots appears in The Chronicle of John of Worcester, which was compiled in the first half of the twelfth century. In this medieval chronicle, the Latin text describing the sunspots is accompanied by a colourful drawing, albeit idealised, which shows the apparent positions and sizes of two sunspots on the solar disk. The date of this observation of sunspots from Worcester, England is firmly established as AD 1128 December 8. Assuming that the drawing was prepared fairly carefully, the angular diameters of the two sunspots are at least about 3 arcmin and 2 arcmin in the northern and southern hemispheres, respectively. Similarly, the heliographic latitudes of both sunspots are within the approximate range of 25°–35°. About five days after this observation of sunspots on the solar disk, on the night of AD 1128 December 13, a red auroral display was observed from Songdo, Korea (the modern city of Kaesong). This auroral observation was recorded in the Koryo-sa, the official Korean chronicle of the period. In addition, five Chinese and five Korean descriptions of auroral displays were recorded in various East-Asian histories between the middle of AD 1127 and the middle of AD 1129. The ten oriental auroral records in this particular interval correspond to six distinct auroral events, which provide evidence for recurrent, though possibly intermittent, auroral activity on a timescale almost exactly equal to the synodic-solar-rotation period (approximately 27 days). The six distinct auroral events were apparently associated with two series of recurrent geomagnetic storms, both of which were sufficiently intense to produce mid-latitude auroral displays in East Asia. These ancient solar and auroral observations are interpreted in terms of present-day understanding of solar-terrestrial physics. Con-temporary ground-based and satellite measurements during the last few decades have indicated that recurrent geomagnetic storms are usually a feature of the declining phase of the solar cycle. In addition, the strength of such recurrent geomagnetic storms has been classified as moderate rather than intense. The recurrent geomagnetic storms occurring during the interval AD 1127–1129 must have been sufficiently intense to produce mid-latitude auroral displays over China and Korea, some of which appeared or extended south of the observing site. This last statement remains true even after proper allowance is made for the fact that during the twelfth century, the north geomagnetic pole was probably situated at the usual high geographic latitude, but in the same geographic longitude range as East Asia. Therefore, it may be inferred that the two series of intense recurrent geomagnetic storms occurred near a medieval maximum in the "eleven-year" solar cycle. Moreover, the overall level of solar activity appears to have been especially high at the end of the second decade of the twelfth century.b〉Key words. Magnetospheric physics (auroral phenomena; storms and substorms) – Solar physics, astrophysics and astronomy (photosphere and chromosphere)

Description: All the accessible auroral observations recorded in Chinese and Japanese histories during the interval AD 1840–1911 are investigated in detail. Most of these auroral records have never been translated into a Western language before. The East Asian auroral reports provide information on the date and approximate location of each auroral observation, together with limited scientific information on the characteristics of the auroral luminosity such as colour, duration, extent, position in the sky and approximate time of occurrence. The full translations of the original Chinese and Japanese auroral records are presented in an appendix, which contains bibliographic details of the various historical sources. (There are no known reliable Korean observations during this interval.) A second appendix discusses a few implausible "auroral" records, which have been rejected. The salient scientific properties of all exactly dated and reliable East Asian auroral observations in the interval AD 1840–1911 are summarised succinctly. By comparing the relevant scientific information on exactly dated auroral observations with the lists of great geomagnetic storms compiled by the Royal Greenwich Observatory, and also the tabulated values of the Ak (Helsinki) and aa (Greenwich and Melbourne) magnetic indices, it is found that 5 of the great geomagnetic storms (aa〉150 or Ak〉50) during either the second half of the nineteenth century or the first decade of the twentieth century are clearly identified by extensive auroral displays observed in China or Japan. Indeed, two of these great storms produced auroral displays observed in both countries on the same night. Conversely, at least 29 (69%) of the 42 Chinese and Japanese auroral observations occurred at times of weak-to-moderate geomagnetic activity (aa or Ak≤50). It is shown that these latter auroral displays are very similar to the more numerous (about 50) examples of sporadic aurorae observed in the United States during the interval AD 1880–1940. The localised nature and spatial structure of some sporadic aurorae observed in East Asia is indicated by the use of descriptive terms such as "lightning", "rainbow", "streak" and "grid".

Description: A further study is made of 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. Previously, the validity of this technique was corroborated using scientific observations of aurorae in Japan during the interval 1957–2004 and contemporaneous 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. The present investigation utilises a list of major geomagnetic storms in the interval 1868–2008, which is based on the magnitude of the AA* magnetic index, and reconstructed solar images based on the sunspot observations acquired by the Royal Greenwich Observatory during the shorter interval 1874–1976. 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 for almost 90% of these major geomagnetic storms. Even an "average" observer would easily achieve a corresponding success rate of 70% and this success rate increases to about 80% if a minority of ambiguous situations are interpreted favourably. The use of information on major geomagnetic storms, rather than modern auroral observations from Japan, provides a less direct corroboration of the technique for identifying historical occurrences of intense geomagnetic storms, if only because major geomagnetic storms do not necessarily produce auroral displays over East Asia. Nevertheless, the present study provides further corroboration of the validity of the original technique for identifying intense geomagnetic storms. This additional corroboration of the original technique is important because early unaided-eye observations of sunspots and aurorae provide the only possible means of identifying individual geomagnetic storms during the greater part of the past two millennia.