Tag / Stellar Aberration

by David D. Nolte

Maxwellian Steampunk and the Origins of Maxwell’s Equations

Physicists of the nineteenth century were obsessed with mechanical models.  They must have dreamed, in their sleep, of spinning flywheels connected by criss-crossing drive belts turning enmeshed gears.  For them, Newton’s clockwork universe was more than a metaphor—they believed that mechanical description of a phenomenon could unlock further secrets and act as a tool of discovery. 

It is no wonder they thought this way—the mid-eighteenth century was at the peak of the industrial revolution, dominated by the steam engine and the profusion of mechanical power and gears across broad swaths of society. 

Steampunk

The Victorian obsession with steam and power is captured beautifully in the literary and animé genre known as Steampunk.  The genre is alternative historical fiction that portrays steam technology progressing into grand and wild new forms as electrical and gasoline technology fail to develop.  An early classic in the genre is Miyazaki’s 1986 anime´ film Castle in the Sky (1986) by Hayao Miyazaki about a world where all mechanical devices, including airships, are driven by steam.  A later archetype of the genre is the 2004 animé film Steam Boy (2004) by Katsuhiro Otomo about the discovery of superwater that generates unlimited steam power.  As international powers vie to possess it, mad scientists strive to exploit it for society, but they create a terrible weapon instead.   One of the classics that helped launch the genre is the novel The Difference Engine (1990) by William Gibson and Bruce Sterling that envisioned an alternative history of computers developed by Charles Babbage and Ada Lovelace.

Scenes from Miyazaki's Castle in the Sky.

Steampunk is an apt, if excessively exaggerated, caricature of the Victorian mindset and approach to science.  Confidence in microscopic mechanical models among natural philosophers was encouraged by the success of molecular models of ideal gases as the foundation for macroscopic thermodynamics.  Pictures of small perfect spheres colliding with each other in simple billiard-ball-like interactions could be used to build up to overarching concepts like heat and entropy and temperature.  Kinetic theory was proposed in 1857 by the German physicist Rudolph Clausius and was quickly placed on a firm physical foundation using principles of Hamiltonian dynamics by the British physicist James Clerk Maxwell.

DVD cover of Steamboy by Otomo.

James Clerk Maxwell

James Clerk Maxwell (1831 – 1879) was one of three titans out of Cambridge who served as the intellectual leaders in mid-nineteenth-century Britain. The two others were George Stokes and William Thomson (Lord Kelvin).  All three were Wranglers, the top finishers on the Tripos exam at Cambridge, the grueling eight-day examination across all fields of mathematics.  The winner of the Tripos, known as first Wrangler, was announced with great fanfare in the local papers, and the lucky student was acclaimed like a sports hero is today.  Stokes in 1841 was first Wrangler while Thomson (Lord Kelvin) in 1845 and Maxwell in 1854 were each second Wranglers.  They were also each winners of the Smith’s Prize, the top examination at Cambridge for mathematical originality.  When Maxwell sat for the Smith’s Prize in 1854 one of the exam problems was a proof written by Stokes on a suggestion by Thomson.  Maxwell failed to achieve the proof, though he did win the Prize.  The problem became known as Stokes’ Theorem, one of the fundamental theorems of vector calculus, and the proof was eventually provided by Hermann Hankel in 1861.

James Clerk Maxwell.

After graduation from Cambridge, Maxwell took the chair of natural philosophy at Marischal College in the city of Aberdeen in Scotland.  He was only 25 years old when he began, fifteen years younger than any of the other professors.  He split his time between the university and his family home at Glenlair in the south of Scotland, which he inherited from his father the same year he began his chair at Aberdeen.  His research interests spanned from the perception of color to the rings of Saturn.  He improved on Thomas Young’s three-color theory by correctly identifying red, green and blue as the primary receptors of the eye and invented a scheme for adding colors that is close to the HSV (hue-saturation-value) system used today in computer graphics.  In his work on the rings of Saturn, he developed a statistical mechanical approach to explain how the large-scale structure emerged from the interactions among the small grains.  He applied these same techniques several years later to the problem of ideal gases when he derived the speed distribution known today as the Maxwell-Boltzmann distribution.

Maxwell’s career at Aberdeen held great promise until he was suddenly fired from his post in 1860 when Marischal College merged with nearby King’s College to form the University of Aberdeen.  After the merger, the university had the abundance of two professors of Natural Philosophy while needing only one, and Maxwell was the junior.  With his new wife, Maxwell retired to Glenlair and buried himself in writing the first drafts of a paper titled “On Physical Lines of Force” [2].  The paper explored the mathematical and mechanical aspects of the curious lines of magnetic force that Michael Faraday had first proposed in 1831 and which Thomson had developed mathematically around 1845 as the first field theory in physics. 

As Maxwell explored the interrelationships among electric and magnetic phenomena, he derived a wave equation for the electric and magnetic fields and was astounded to find that the speed of electromagnetic waves was essentially the same as the speed of light.  The importance of this coincidence did not escape him, and he concluded that light—that rarified, enigmatic and quintessential fifth element—must be electromagnetic in origin. Ever since Francois Arago and Agustin Fresnel had shown that light was a wave phenomenon, scientists had been searching for other physical signs of the medium that supported the waves—a medium known as the luminiferous aether (or ether). With Maxwell’s new finding, it meant that the luminiferous ether must be related to electric and magnetic fields.  In the Steampunk tradition of his day, Maxwell began a search for a mechanical model.  He did not need to look far, because his friend Thomson had already built a theory on a foundation provided by the Irish mathematician James MacCullagh (1809 – 1847)

The Luminiferous Ether

The late 1830’s was a busy time for the luminiferous ether.  Agustin-Louis Cauchy published his extensive theory of the ether in 1836, and the self-taught George Green published his highly influential mathematical theory in 1838 which contained many new ideas, such as the emphasis on potentials and his derivation of what came to be called Green’s theorem. 

In 1839 MacCullagh took an approach that established a core property of the ether that later inspired both Thomson and Maxwell in their development of electromagnetic field theory.  What McCullagh realized was that the energy of the ether could be considered as if it had both kinetic energy and potential energy (ideas and nomenclature that would come several decades later).  Most insightful was the fact that the potential energy of the field depended on pure rotation like a vortex.  This rotationally elastic ether was a mathematical invention without any mechanical analog, but it successfully described reflection and refraction as well as polarization of light in crystalline optics. 

In 1856 Thomson put Faraday’s famous magneto-optic rotation of light (the Faraday Effect discovered by Faraday in 1845) into mathematical form and began putting Faraday’s initially abstract ideas of the theory of fields into concrete equations.  He drew from MacCullagh’s rotational ether as well as an idea from William Rankine about the molecular vortex model of atoms to develop a mechanical vortex model of the ether.  Thomson explained how the magnetic field rotated the linear polarization of light through the action of a multiplicity of molecular vortices.  Inspired by Thomson, Maxwell took up the idea of molecular vortices as well as Faraday’s magnetic induction in free space and transferred the vortices from being a property exclusively of matter to being a property of the luminiferous ether that supported the electric and magnetic fields. 

Maxwellian Cogwheels

Maxwell’s model of the electromagnetic fields in the ether is the apex of Victorian mechanistic philosophy—too explicit to be a true model of reality—yet it was amazingly fruitful as a tool of discovery, helping Maxwell develop his theory of electrodynamics. The model consisted of an array of elastic vortex cells separated by layers of small particles that acted as “idle wheels” to transfer spin from one vortex to another .  The magnetic field was represented by the rotation of the vortices, and the electric current was represented by the displacement of the idle wheels. 

Maxwell's vortex model
Fig. 1 Maxwell’s vortex model of the electromagnetic ether.  The molecular vortices rotate according to the direction of the magnetic field, supported by idle wheels.  The physical displacement of the idle wheels became an analogy for Maxwell’s displacement current [2].

Two predictions by this outrightly mechanical model were to change the physics of electromagnetism forever:  First, any change in strain in the electric field would cause the idle wheels to shift, creating a transient current that was called a “displacement current”.  This displacement current was one of the last pieces in the electromagnetic puzzle that became Maxwell’s equations. 

Maxwell's discovery of the displacement current
Fig. 2 In “Physical Lines of Force” in 1861, Maxwell introduces the idea of a displacement current [RefLink].

In this description, E is not the electric field, but is related to the dielectric permativity through the relation

Maxwell went further to prove his Proposition XIV on the contribution of the displacement current to conventional electric currents.

Maxwell completing the laws of electromagnetics
Fig. 3 Maxwell’s Proposition XIV on adding the displacement current to the conventional electric current [RefLink].

Second, Maxwell calculated that this elastic vortex ether propagated waves at a speed that was close to the known speed of light measured a decade previously by the French physicist Hippolyte Fizeau.  He remarked, “we can scarcely avoid the inference that light consists of the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” [1]  This was the first direct prediction that light, previously viewed as a physical process separate from electric and magnetic fields, was an electromagnetic phenomenon.

Maxwell's estimate of the speed of light
Fig. 4 Maxwell’s calculation of the speed of light in his mechanical ether. It matched closely the measured speed of light [RefLink].

These two predictions—of the displacement current and the electromagnetic origin of light—have stood the test of time and are center pieces of Maxwells’s legacy.  How strange that they arose from a mechanical model of vortices and idle wheels like so many cogs and gears in the machinery powering the Victorian age, yet such is the power of physical visualization.

[1] pg. 12, The Maxwellians, Bruce Hunt (Cornell University Press, 1991)

[2] Maxwell, J. C. (1861). “On physical lines of force”. Philosophical Magazine. 90: 11–23.

Books by David Nolte at Oxford University Press

by David D. Nolte

The Aberration of Starlight: Relativity’s Crucible

The Earth races around the sun with remarkable speed—at over one hundred thousand kilometers per hour on its yearly track.  This is about 0.01% of the speed of light—a small but non-negligible amount for which careful measurement might show the very first evidence of relativistic effects.  How big is this effect and how do you measure it?  One answer is the aberration of starlight, which is the slight deviation in the apparent position of stars caused by the linear speed of the Earth around the sun.

This is not parallax, which is caused the the changing position of the Earth around the sun. Ever since Copernicus, astronomers had been searching for parallax, which would give some indication how far away stars were. It was an important question, because the answer would say something about how big the universe was. But in the process of looking for parallax, astronomers found something else, something about 50 times bigger—aberration.

Aberration is the effect of the transverse speed of the Earth added to the speed of light coming from a star. For instance, this effect on the apparent location of stars in the sky is a simple calculation of the arctangent of 0.01%, which is an angle of about 20 seconds of arc, or about 40 seconds when comparing two angles 6 months apart.  This was a bit bigger than the accuracy of astronomical measurements at the time when Jean Picard travelled from Paris to Denmark in 1671 to visit the ruins of the old observatory of Tycho Brahe at Uranibourg.

Fig. 1 Stellar parallax is the change in apparent positions of a star caused by the change in the Earth’s position as it orbits the sun. If the change in angle (θ) could be measured, then based on Newton’s theory of gravitation that gives the radius of the Earth’s orbit (R), the distance to the star (L) could be found.

Jean Picard at Uranibourg

Fig. 2 A view of Tycho Brahe’s Uranibourg astronomical observatory in Hven, Denmark. Tycho had to abandon it near the end of his life when a new king thought he was performing witchcraft.

Jean Picard went to Uranibourg originally in 1671, and during subsequent years, to measure the eclipses of the moons of Jupiter to determine longitude at sea—an idea first proposed by Galileo.  When visiting Copenhagen, before heading out to the old observatory, Picard secured the services of an as yet unknown astronomer by the name of Ole Rømer.  While at Uranibourg, Picard and Rømer made their required measurements of the eclipses of the moons of Jupiter, but with extra observation hours, Picard also made measurements of the positions of selected stars, such as Polaris, the North Star.  His very precise measurements allowed him to track a tiny yearly shift, an aberration, in position by about 40 seconds of arc.  At the time (before Rømer’s great insight about the finite speed of light—see Chapter 1 of Interference (Oxford, 2023)), the speed of light was thought to be either infinite or unmeasurably fast, so Picard thought that this shift was the long-sought effect of stellar parallax that would serve as a way to measure the distance to the stars.  However, the direction of the shift of Polaris was completely wrong if it were caused by parallax, and Picard’s stellar aberration remained a mystery.

Fig. 3 Jean Picard (left) and his modern name-sake (right).

Samuel Molyneux and Murder in Kew

In 1725, the amateur Irish astronomer Samuel Molyneux (1689 – 1828) decided that the tools of astronomy had improved to the point that the question of parallax could be answered.  He enlisted the help of an instrument maker outside London to install a 24-foot zenith sector (a telescope that points vertically upwards) at his home in Kew.  Molyneux was an independently wealthy politician (he had married the first daughter of the second Earl of Essex) who sat in the British House of Commons, and he was also secretary to the Prince of Wales (the future George II).  Because his political activities made demands on his time, he looked for assistance with his observations and invited James Bradley (1693 – 1762), the newly installed Savilian Professor of Astronomy at Oxford University, to join him in his search.

Fig. 4 James Bradley.

James Bradley was a rising star in the scientific circles of England.  He came from a modest background but had the good fortune that his mother’s brother, James Pound, was a noted amateur astronomer who had set up a small observatory at his rectory in Wanstead.  Bradley showed an early interest in astronomy, and Pound encouraged him, helping with the finances of his education that took him to degrees at Baliol College at Oxford.  Even more fortunate was the fact that Pound’s close friend was the Astronomer Royal Edmund Halley, who also took a special interest in Bradley.  With Halley’s encouragement, Bradley made important measurements of Mars and several nebulae, demonstrating an ability to work with great accuracy.  Halley was impressed and nominated Bradley to the Royal Society in 1718, telling everyone that Bradley was destined to be one of the great astronomers of his time. 

Molyneux must have sensed immediately that he had chosen wisely by selecting Bradley to help him with the parallax measurements.  Bradley was capable of exceedingly precise work and was fluent mathematically with the geometric complexities of celestial orbits.  Fastening the large zenith sector to the chimney of the house gave the apparatus great stability, and in December of 1725 they commenced observations of Gamma Draconis as it passed directly overhead.  Because of the accuracy of the sector, they quickly observed a deviation in the star’s position, but the deviation was in the wrong direction, just as Picard had observed.  They continued to make observations over two years, obtaining a detailed map of a yearly wobble in the star’s position as it changed angle by 40 seconds of arc (about one percent of a degree) over six months. 

When Molyneux was appointed Lord of the Admiralty in 1727, as well as becoming a member of the Irish Parliament (representing Dublin University), he had little time to continue with the observations of Gamma Draconis.  He helped Bradley set up a Zenith sector telescope at Bradley’s uncle’s observatory in Wanstead that had a wider field of view to observe more stars, and then he left the project to his friend.  A few months later, before either he or Bradley had understood the cause of the stellar aberration, Molyneux collapsed while in the House of Commons and was carried back to his house.  One of Molyneux’s many friends was the court anatomist Nathaniel St. André who attended to him over the next several days as he declined and died.  St. André was already notorious for roles he had played in several public hoaxes, and on the night of his friend’s death, before the body had grown cold, he eloped with Molyneux’s wife, raising accusations of murder (that could never be proven). 

James Bradley and the Light Wind

Over the following year, Bradley observed aberrations in several stars, all of them displaying the same yearly wobble of about 40 seconds of arc.  This common behavior of numerous stars demanded a common explanation, something they all shared.  It is said that the answer came to Bradley while he was boating on the Thames.  The story may be apocryphal, but he apparently noticed the banner fluttering downwind at the top of the mast, and after the boat came about, the banner pointed in a new direction.  The wind direction itself had not altered, but the motion of the boat relative to the wind had changed.  Light at that time was considered to be made of a flux of corpuscles, like a gentle wind of particles.  As the Earth orbited the Sun, its motion relative to this wind would change periodically with the seasons, and the apparent direction of the star would shift a little as a result.

Fig. 5 Principle of stellar aberration.  On the left is the rest frame of the star positioned directly overhead as a moving telescope tube must be slightly tilted at an angle (equal to the arctangent of the ratio of the Earth’s speed to the speed of light–greatly exaggerated in the figure) to allow the light to pass through it.  On the right is the rest frame of the telescope in which the angular position of the star appears shifted.

Bradley shared his observations and his explanation in a letter to Halley that was read before the Royal Society in January of 1729.  Based on his observations, he calculated the speed of light to be about ten thousand times faster than the speed of the Earth in its orbit around the Sun.  At that speed, it should take light eight minutes and twelve seconds to travel from the Sun to the Earth (the actual number is eight minutes and 19 seconds).  This number was accurate to within a percent of the true value compared with the estimates made by Huygens from the eclipses of the moons of Jupiter that were in error by 27 percent.  In addition, because he was unable to discern any effect of parallax in the stellar motions, Bradley was able to place a limit on how far the distant stars must be, more than 100,000 times farther the distance of the Earth from the Sun, which was much farther away than any had previously expected.  In January of 1729 the size of the universe suddenly jumped to an incomprehensibly large scale.

Bradley’s explanation of the aberration of starlight was simple and matched observations with good quantitative accuracy.  The particle nature of light made it like a wind, or a current, and the motion of the Earth was just a case of Galilean relativity that any freshman physics student can calculate.  At first there seemed to be no controversy or difficulties with this interpretation.  However, an obscure paper published in 1784 by an obscure English natural philosopher named John Michell (the first person to conceive of a “dark star”) opened a Pandora’s box that launched the crisis of the luminiferous ether and the eventual triumph of Einstein’s theory of Relativity (see Chapter 3 of Interference (Oxford, 2023)), .

By David D. Nolte, Sept. 27, 2023

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