Comparing these positions with Halley’s from 1682, Hansteen concluded that all four poles were on the move. Both northern poles and the major southern pole (the Asian, or Australian) were moving eastwards around the geographic poles while the minor southern pole, the South American, was moving westwards. He even estimated how long it would take each magnetic pole to complete a circuit—4320, 1728, 1296 and 864 years, respectively— and suggested that these figures were of astronomical significance. Their lowest common multiple of 25,920 years corresponded, he claimed, to the period of precession of the equinoxes, precession being a variation in Earth’s orbital motion that affects the cycle of seasons.
In the final part of his Magnetismus der Erde, Hansteen made a determined attempt to explain his observations, using the mathematics that Biot and a Swiss mathematician, Leonhard Euler, had recently developed to describe the magnetic field of a dipole. With the discovery of electromagnetism still a year away, permanent magnetism remained the only known source of a magnetic field. A uniformly magnetized terrella and a bar magnet both had two poles, and both produced the same dipolar form of magnetic field. Gilbert had convincingly proposed that Earth’s magnetic field was that of a geocentric axial dipole—that is, it was as though a dipole were located at the center of the Earth and aligned with the Earth’s axis of rotation. Others such as Mercator and de Nautonnier had argued that a dipole tilted at a small angle to the rotation axis better matched the observed pattern of declinations and inclinations.
Top: Chart of geomagnetic declination (then called “variation”) around the world, 1787. This was first published by Christopher Hansteen in Magnetismus der Erde in 1819, and later reproduced by Edward Sabine. Bottom: Chart of geomagnetic inclination (“dip”), around the world, 1780, first published by Christopher Hansteen in Magnetismus der Erde and later reproduced by Edward Sabine.
Gellibrand’s discovery of secular variation, the slow changing of Earth’s magnetic field, had led to the suggestion that the dipole axis was precessing—rotating with time—around the geographic axis. Now Hansteen was suggesting that if the observations were to be satisfactorily explained there had to be not one but two dipoles. As the theory of electromagnetism advanced, models of Earth’s magnetism based on permanent magnets would become obsolete, but Hansteen made one last attempt to support the theory by proposing that these two dipoles were neither axial nor geocentric. By adjusting their positions within the Earth, their orientations and their strengths, he eventually produced a fit with which he was satisfied. It was, he claimed, “as well established, as a means of representing the phenomena, as any hypothesis whatsoever introduced in physical illustration.”
By this time, however, even he was beginning to have doubts. His two-dipole model might fit the observations and be useful in forecasting secular variation, but in his final analysis he carefully avoided implying there actually were two dipoles within the Earth. Indeed, he did not even discuss whether the physical source of Earth’s magnetism was internal or external.
With theories of geomagnetism based on permanent dipoles running into more and more difficulties, Ampère’s discovery that a current loop produced exactly the same form of magnetic field as a dipole offered a new explanation. As early as 1831 Peter Barlow, an English mathematician and engineer, wrote a paper entitled “On the Probable Electric Origin of All the Phenomena of Terrestrial Magnetism.” Barlow had constructed a wooden globe with copper wires wrapped around lines of latitude. On passing a current through the wires he had produced a magnetic field similar in form to the Earth’s, and so demonstrated that Earth’s magnetic field could be electrical in origin.
The question was how might electric currents be generated inside the Earth? Barlow suggested the heat of the sun was somehow transformed into electrical energy, but he did not manage to come up with a credible mechanism.
The Magnetic Crusade
The Earth speaks of its internal workings through the silent voice of the magnetic needle.
—CHRISTOPHER HANSTEEN, 1819
Thanks to Ampère, Biot, Euler and others, mathematics was fast becoming a popular pursuit across Europe. Its language was evolving, and it was increasingly being used to construct accurate descriptions of all sorts of physical phenomena. One distinguished mathematician who would leave his mark on almost every area of physics, including the study of Earth’s magnetism, was Carl Friedrich Gauss. Born into a poor family in Brunswick, Germany in 1777, Gauss had shown astounding mathematical abilities from a young age. At seven, while his classmates fumbled with chalk and slate, he had amazed his teachers by instantly coming up with the sum of the whole numbers from one to a hundred. The answer, he declared, was 5050: 50 times 101. There were 50 pairs of whole numbers, with each pair adding up to 101 (1+100, 2+99 and so on).
Gauss’s father, Gerhard, worked variously as a gardener, bricklayer and canal tender, and assumed his son would also earn his living as a laborer. However, Gauss’s mother Dorothea and his uncle Friedrich encouraged and fostered the child’s interest in mathematics, and in 1792 the Duke of Brunswick granted the fifteen year old a stipend to attend the local Collegium Carolinum, today the Technische Universität Carolo-Wilhelmina zu Braunschweig (Technical University of Braunschweig). The young man was soon deep into the study of number theory; he derived the binomial theorem, a central result of number theory, and independently discovered what is now known as Bode’s Law, a numerical relationship that relates the sizes of the orbits of successive planets.
In 1795 Gauss moved from Brunswick to the University of Göttingen, where his career and reputation flourished. In 1807, at the age of twenty-nine, he was appointed director of the astronomical observatory, and for the next ten years he worked mainly on mathematical problems in astronomy, using the now popular methods of calculus. He also perfected and published his “least squares” method for finding the best mathematical fit for a set of observational data marred by random errors, or “noise.” The method proved itself in verifying the orbit of the asteroid Ceres, until then something of an enigma because of the difficulty of making accurate measurements of its position.
Before long, Gauss’s interests had shifted to geodesy, the study of Earth’s shape and dimensions, and potential theory, an emerging mathematical method that was found to be particularly useful for analyzing Earth’s gravitational field. These would set the scene for his future work in geomagnetism.
Gauss was encouraged by von Humboldt. The two men first met in 1828, and it is likely that they discussed the problems involved in obtaining absolute measurements of magnetic intensity and mathematical representations of Earth’s magnetic field. Also on the agenda would have been von Humboldt’s vision of a worldwide network of geomagnetic observatories. Within the next few years Gauss had solved the absolute intensity problem, produced a mathematical model of Earth’s magnetic field, built his own geomagnetic observatory, and thrown his weight into the geomagnetic observatory movement by setting up Göttingen Magnetische Verein (Göttingen Magnetic Union).
Carl Friedrich Gauss, born 1777. A mathematical genius from childhood, Gauss became fascinated by geomagnetism, producing the first mathematical model of Earth’s magnetic field, building his own geomagnetic observatory, and setting up the Göttingen Magnetic Union, which aimed to establish a worldwide network of observatories.
Gauss’s answer to the intensity problem was as simple as it was ingenious. The obstacle to calibrating the period of oscillation of a dip needle or compass had been the need to know what amounted to the strength of the magnet used for the needle—in technical terms, its “dipole moment.” The period of oscillation depended on the product of this dipole moment and Earth’s magnetic field intensity—that is, the one multiplied by the other.
Gauss figured that by making a second measurement he could obtain the ratio of the two—the one divided by the other. He took another compass, and while letting Earth’s magnetism pull its tip towards the north he arranged the original magnet to pull it at right angles
—towards the east or west. Like a rotational tug-of-war, the angle at which the compass needle ended up depended on how strong the magnet was compared with Earth’s magnetic intensity. Once both the product and the ratio of two things are known it is a straightforward matter of arithmetic to untangle them: Gauss was able to calculate not just the intensity of Earth’s magnetism, but also the dipole moment of the magnet.
For obvious reasons this became known as the oscillation-deflection method. After Gauss published the details in 1832 it was rapidly adopted, not just in permanent observatories as he had intended, but also in some temporary observatories and in survey work. It also got around the problem of compass needles losing their magnetization during long sea voyages, something that had dogged explorers and surveyors for centuries and was the main reason ships routinely carried several compasses. Von Humboldt and others must have rued not having thought of such a simple solution.
Meanwhile, Gauss’s keen mathematical brain had been racing ahead. If he could devise a simple mathematical description of Earth’s magnetic field at any epoch of time—a standard set of equations from which, by inserting a few appropriate numbers or “coefficients,” the declination, inclination and intensity at any location on Earth could be calculated—everyone everywhere would be able to compute whatever feature of the field they happened to need.
A group of French mathematicians, including Pierre Simon Laplace, Adrien-Marie Legendre and Siméon-Denis Poisson, had already made significant progress in developing such a method for describing Earth’s field of gravity. This used the idea of gravitational potential (a simple intermediary quantity from which the strength and direction of the field could be calculated) and a technique they called “spherical harmonic analysis.”
This technique capitalized on Earth’s near spherical shape. The gravitational potential and force of gravity vary so little over Earth’s surface that when you move from one place to another you cannot feel any difference. However, the Earth is not perfectly spherical. At the equator, for example, it bulges slightly and so locations on the equator are further from the center of the Earth than are the poles. This—and other, more complex, irregularities in the shape and the distribution of mass in the Earth—mean the gravitational potential and field are quite complicated. The essence of spherical harmonic analysis was the idea that this complex detail could be pictured as the combination of a series of smooth waves around lines of both latitude and longitude. These waves, or “spherical harmonic functions,” involved a series of mathematical expressions (called Legendre polynomials), and the relative contribution of each was given by its own numerical coefficient. Adding together all the spherical harmonic functions, each multiplied by its coefficient, produced the mathematical model of the gravitational potential. From this the gravitational field could be calculated.
One of the most appealing features of spherical harmonic analysis was, and is, that each of the harmonic functions corresponds to a conceptual physical source. The first and simplest does not depend on latitude or longitude at all, and so represents a symmetrical, spherical mass. This makes by far the biggest contribution, and so dominates the mathematical model of Earth’s gravitational field.
The next function represents a dipole, the next a quadrupole and describes the “fat-in-the-middle, squashed-at-the-poles” effect of the equatorial bulge—and so on. You can choose to fit as many or as few spherical harmonic functions as you want, giving as simple or as detailed a picture as you want, or as the data will allow. Adding more “higher degree” harmonics simply adds more detail; it does not alter the contributions already calculated for the lower degree ones.
Gauss reasoned that with enough observational data it should be possible to carry out a similar spherical harmonic analysis of the Earth’s magnetic field. Both the gravitational and geomagnetic fields involved action-at-a-distance forces that originated within the Earth. Both could be described via the intermediary idea of a potential. A mathematical model based on spherical harmonic functions would be more widely applicable, and might eventually prove more durable, than the physically appealing but ultimately unrealistic models based on permanent magnetic dipoles.
By 1838 Gauss was impatient to try out this idea. First, however, he needed up-to-date measurements of all three elements of the geomagnetic field—declination, inclination and intensity— from as many locations as possible and preferably spread evenly around the globe. By now declination and inclination charts were being produced and updated regularly. He chose Barlow’s 1833 chart of declination, and an 1836 chart of inclination attributed to a cartographer by the name of Horner and published in the German Physikalishe Wörterbuch.
Most importantly, in 1837 the Irish-born English mariner, geodesist and geomagnetist Edward Sabine had collated and rationalized all available geomagnetic intensity observations and published the first global intensity chart. As he had explained in his report to the British Association for the Advancement of Science, the chart incorporated “753 distinct determinations at 670 stations widely distributed over the Earth’s surface.” But he freely admitted that even this wealth of data was less than ideal, “leaving, it is true, much still to be desired.”
Sabine gave a critical evaluation of each of the 753 determinations, which dated back to the explorations and publications of von Humboldt and Hansteen. The data, all given in von Humboldt’s relative intensity units, included measurements made in North America by a Scottish botanist, David Douglas (after whom the Douglas fir is named), and many made by Sabine himself on his own numerous voyages. He credited d’Entrecasteaux and de Rossel with having made the very first intensity measurements, but rather than use measurements from the early 1800s he chose instead those made in the 1830s by Captain Robert Fitzroy on his epic five-year voyage of the Beagle with Charles Darwin.
Gauss interpolated the data from these charts to estimate values of declination, inclination and intensity at twelve evenly spaced locations around each of seven lines of latitude—a grid of eighty-four locations. He then worked out a series of equations, 168 in all, from which he could calculate the numerical coefficient of each spherical harmonic function in his model. Solving these equations to obtain the twenty-four coefficients was a formidable task; with only pencil and paper it must have taken his team of “calculators” many days to complete.
In the case of the gravitational potential the first term in the spherical harmonic analysis, the one representing a uniform spherical mass, was easily the most important. However, since magnetic monopoles (magnets with only one pole) do not exist, in the geomagnetic potential this term was zero.
By far the biggest contribution in Gauss’s analysis came from the second term, which represented a geocentric axial dipole. This term was five times bigger than any other, and vindicated Gilbert’s original hypothesis based on his observations of his uniformly magnetized terrella.
The principle of spherical harmonic analysis is to describe the potential of the magnetic (or gravitational) field at any particular time as a sum of terms of decreasing size and importance. Shown here are pictorial representations of some of the terms that are symmetric about the rotation axis: (a) the biggest term—the geocentric axial dipole; (b) the second degree, or quadrupole, which is much smaller; (c) the third degree, or octupole, which is smaller again; (d) the even smaller fourth degree term. (Not drawn to scale.)
The next biggest terms were the ones that, when added to this, represented a geocentric but tilted dipole. Taken together, they represented a dipole equivalent in strength to about seventy thousand million billion of the bar magnets typically found in school physics laboratories, with poles at 77°84′ N, 296°30′ E in northwestern Greenland and 77°8′ S, 116°30′ E in Antarctica. Since Gauss’s time, this part of the field has been known as the dipole field.
All the other terms in Gauss’s analysis, the so-called “higher degree” spherical harmonic terms which represented more complicated, “non-dipole” features of the magnetic field, very rapidly
became smaller and smaller.
One of the best ways to test a mathematical model is to see how well it can reproduce actual measurements. Hansteen had done this with his four-pole model and now Gauss did the same. Using the twenty-four numerical coefficients of his model, he calculated values of declination, inclination and intensity at the locations of the original observations. The average difference between intensities calculated from Gauss’s model and the observed intensity measurements was 0.046 (von Humboldt’s relative intensity units), or less than five percent. The average differences in declination and inclination were 1.5° and 1.1°, respectively. Given the small amount of data, the poor distribution of the observation sites around the globe, and the fact that all the calculations had been carried out manually, this was astonishingly accurate.
Gauss’s spherical harmonic analysis provided a neat and elegant mathematical way of describing Earth’s magnetic field. (It would evolve into the present-day International Geomagnetic Reference Field, in which the numerical coefficients are known as “Gauss coefficients.”) However, it brought scientists no closer to understanding what actually caused this magnetism.
Gauss himself seemed curiously uninterested in this. Being at heart a mathematician, he was motivated more by the analytical description offered by the new mathematical methods than by the need for a physical explanation of the phenomenon. He did, however, point out that when better data were available it should be possible to use spherical harmonic analysis to distinguish between internal sources and any possible external ones.
Gauss was sure that the primary source of Earth’s magnetism would turn out to be internal, but compared with the elegance of his mathematics his physical reasoning was rather unsophisticated. Curiously, nearly twenty years after Ørsted’s discovery that a magnetic field could be the result of an electric current, Gauss stuck steadfastly to the idea of permanent magnetism. However, he did have difficulty accepting the notion of great permanent dipoles deep within the Earth, revolving or precessing fast enough to produce the observed rates of change of declination and inclination—and now intensity—of the magnetic field.
North Pole, South Pole: The Epic Quest to Solve the Great Mystery of Earth's Magnetism Page 11
