Archive for Johannes Kepler

A Piece on a Paradox

Posted in The Universe and Stuff with tags , , , , , , , , , on March 7, 2012 by telescoper

Not long ago I posted a short piece about the history of cosmology which got some interesting comments, so I thought I’d try again with a little article I wrote a while ago on the subject of Olbers’ Paradox. This is discussed in almost every astronomy or cosmology textbook, but the resolution isn’t always made as clear as it might be. The wikipedia page on this topic is unusually poor by the standards of wikipedia, and appears to have suffered a severe attack of the fractals.

I’d be interested in any comments on the following attempt.

One of the most basic astronomical observations one can make, without even requiring a telescope, is that the night sky is dark. This fact is so familiar to us that we don’t imagine that it is difficult to explain, or that anything important can be deduced from it. But quite the reverse is true. The observed darkness of the sky at night was regarded for centuries by many outstanding intellects as a paradox that defied explanation: the so-called Olbers’ Paradox.

The starting point from which this paradox is developed is the assumption that the Universe is static, infinite, homogeneous, and Euclidean. Prior to twentieth century developments in observation (Hubble’s Law) and theory  (Cosmological Models based on General Relativity), all these assumptions would have appeared quite reasonable to most scientists. In such a Universe, the intensity of light received by an observer from a source falls off as the inverse square of the distance between the two. Consequently, more distant stars or galaxies appear fainter than nearby ones. A star infinitely far away would appear infinitely faint, which suggests that Olbers’ Paradox is avoided by the fact that distant stars (or galaxies) are simply too faint to be seen. But one has to be more careful than this.

Imagine, for simplicity, that all stars shine with the same brightness. Now divide the Universe into a series of narrow concentric spherical shells, in the manner of an onion. The light from each source within a shell of radius r  falls off as r^{-2}, but the number of sources increases in the same manner. Each shell therefore produces the same amount of light at the observer, regardless of the value of r.  Adding up the total light received from all the shells, therefore, produces an infinite answer.

In mathematical form, this is

I = \int_{0}^{\infty} I(r) n dV =  \int_{0}^{\infty} \frac{L}{4\pi r^2} 4\pi r^{2} n dr \rightarrow \infty

where L is the luminosity of a source, n is the number density of sources and I(r) is the intensity of radiation received from a source at distance r.

In fact the answer is not going to be infinite in practice because nearby stars will block out some of the light from stars behind them. But in any case the sky should be as bright as the surface of a star like the Sun, as each line of sight will eventually end on a star. This is emphatically not what is observed.

It might help to think of this in another way, by imagining yourself in a very large forest. You may be able to see some way through the gaps in the nearby trees, but if the forest is infinite every possible line of sight will end with a tree.

As is the case with many other famous names, this puzzle was not actually first discussed by Olbers. His discussion was published relatively recently, in 1826. In fact, Thomas Digges struggled with this problem as early as 1576. At that time, however, the mathematical technique of adding up the light from an infinite set of narrow shells, which relies on the differential calculus, was not known. Digges therefore simply concluded that distant sources must just be too faint to be seen and did not worry about the problem of the number of sources. Johannes Kepler was also interested in this problem, and in 1610 he suggested that the Universe must be finite in spatial extent. Edmund Halley (of cometary fame) also addressed the  issue about a century later, in 1720, but did not make significant progress. The first discussion which would nowadays be regarded as a  correct formulation of the problem was published in 1744, by Loys de Chéseaux. Unfortunately, his resolution was not correct either: he imagined that intervening space somehow absorbed the energy carried by light on its path from source to observer. Olbers himself came to a similar conclusion in the piece that forever associated his name with this cosmological conundrum.

Later students of this puzzle included Lord Kelvin, who speculated that the extra light may be absorbed by dust. This is no solution to the problem either because, while dust may initially simply absorb optical light, it would soon heat up and re-radiate the energy at infra-red wavelengths. There would still be a problem with the total amount of electromagnetic radiation reaching an observer. To be fair to Kelvin, however, at the time of his writing it was not known that heat and light were both forms of the same kind of energy and it was not obvious that they could be transformed into each other in this way.

To show how widely Olbers’ paradox was known in the nineteenth Century, it is worth also mentioning that Friedrich Engels, Manchester factory owner and co-author with Karl Marx of the Communist Manifesto also considered it in his book The Dialectics of Nature. In this discussion he singles out Kelvin for particular criticism, mainly for the reason that Kelvin was a member of the aristocracy.

In fact, probably the first inklings of a correct resolution of the Olbers’ Paradox were contained not in a dry scientific paper, but in a prose poem entitled Eureka published in 1848 by Edgar Allan Poe. Poe’s astonishingly prescient argument is based on the realization that light travels with a finite speed. This in itself was not a new idea, as it was certainly known to Newton almost two centuries earlier. But Poe did understand its relevance to Olbers’ Paradox.  Light just arriving from distant sources must have set out a very long time ago; in order to receive light from them now, therefore, they had to be burning in the distant past. If the Universe has only lasted for a finite time then one can’t add shells out to infinite distances, but only as far as the distance given by the speed of light multiplied by the age of the Universe. In the days before scientific cosmology, many believed that the Universe had to be very young: the biblical account of the creation made it only a few thousand years old, so the problem was definitely avoided.

Of course, we are now familiar with the ideas that the Universe is expanding (and that light is consequently redshifted), that it may not be infinite, and that space may not be Euclidean. All these factors have to be taken into account when one calculates the brightness of the sky in different cosmological models. But the fundamental reason why the paradox is not a paradox does boil down to the finite lifetime, not necessarily of the Universe, but of the individual structures that can produce light. According to the theory Special Relativity, mass and energy are equivalent. If the density of matter is finite, so therefore is the amount of energy it can produce by nuclear reactions. Any object that burns matter to produce light can therefore only burn for a finite time before it fizzles out.

Imagine that the Universe really is infinite. For all the light from all the sources to arrive at an observer at the same time (i.e now) they would have to have been switched on at different times – those furthest away sending their light towards us long before those nearby had switched on. To make this work we would have to be in the centre of a carefully orchestrated series of luminous shells switching on an off in sequence in such a way that their light all reached us at the same time. This would not only put us  in a very special place in the Universe but also require the whole complicated scheme to be contrived to make our past light cone behave in this peculiar way.

With the advent of the Big Bang theory, cosmologists got used to the idea that all of matter was created at a finite time in the past anyway, so  Olber’s Paradox receives a decisive knockout blow, but it was already on the ropes long before the Big Bang came on the scene.

As a final remark, it is worth mentioning that although Olbers’ Paradox no longer stands as a paradox, the ideas behind it still form the basis of important cosmological tests. The brightness of the night sky may no longer be feared infinite, but there is still expected to be a measurable glow of background light produced by distant sources too faint to be seen individually. In principle,  in a given cosmological model and for given assumptions about how structure formation proceeded, one can calculate the integrated flux of light from all the sources that can be observed at the present time, taking into account the effects of redshift, spatial geometry and the formation history of sources. Once this is done, one can compare predicted light levels with observational limits on the background glow in certain wavebands which are now quite strict .

A Potted Prehistory of Cosmology

Posted in History, The Universe and Stuff with tags , , , , , , , , , , , , , , , , , , , , , on January 26, 2012 by telescoper

A few years ago I was asked to provide a short description of the history of cosmology, from the dawn of civilisation up to the establishment of the Big Bang model, in less than 1200 words. This is what I came up with. Who and what have I left out that you would have included?

–0–

 Is the Universe infinite? What is it made of? Has it been around forever?  Will it all come to an end? Since prehistoric times, humans have sought to build some kind of conceptual framework for answering questions such as these. The first such theories were myths. But however naïve or meaningless they may seem to us now, these speculations demonstrate the importance that we as a species have always attached to thinking about life, the Universe and everything.

Cosmology began to emerge as a recognisable scientific discipline with the Greeks, notably Thales (625-547 BC) and Anaximander (610-540 BC). The word itself is derived from the Greek “cosmos”, meaning the world as an ordered system or whole. In Greek, the opposite of “cosmos” is “chaos”. The Pythagoreans of the 6th century BC regarded numbers and geometry as the basis of all natural things. The advent of mathematical reasoning, and the idea that one can learn about the physical world using logic and reason marked the beginning of the scientific era. Plato (427-348 BC) expounded a complete account of the creation of the Universe, in which a divine Demiurge creates, in the physical world, imperfect representations of the structures of pure being that exist only in the world of ideas. The physical world is subject to change, whereas the world of ideas is eternal and immutable. Aristotle (384-322 BC), a pupil of Plato, built on these ideas to present a picture of the world in which the distant stars and planets execute perfect circular motions, circles being a manifestation of “divine” geometry. Aristotle’s Universe is a sphere centred on the Earth. The part of this sphere that extends as far as the Moon is the domain of change, the imperfect reality of Plato, but beyond this the heavenly bodies execute their idealised circular motions. This view of the Universe was to dominate western European thought throughout the Middle Ages, but its perfect circular motions did not match the growing quantities of astronomical data being gathered by the Greeks from the astronomical archives made by the Babylonians and Egyptians. Although Aristotle had emphasised the possibility of learning about the Universe by observation as well as pure thought, it was not until Ptolemy’s Almagest, compiled in the 2nd Century AD, that a complete mathematical model for the Universe was assembled that agreed with all the data available.

Much of the knowledge acquired by the Greeks was lost to Christian culture during the dark ages, but it survived in the Islamic world. As a result, cosmological thinking during the Middle Ages of Europe was rather backward. Thomas Aquinas (1225-74) seized on Aristotle’s ideas, which were available in Latin translation at the time while the Almagest was not, to forge a synthesis of pagan cosmology with Christian theology which was to dominated Western thought until the 16th and 17th centuries.

The dismantling of the Aristotelian world view is usually credited to Nicolaus Copernicus (1473-1543).  Ptolemy’s Almagest  was a complete theory, but it involved applying a different mathematical formula for the motion of each planet and therefore did not really represent an overall unifying system. In a sense, it described the phenomena of heavenly motion but did not explain them. Copernicus wanted to derive a single universal theory that treated everything on the same footing. He achieved this only partially, but did succeed in displacing the Earth from the centre of the scheme of things. It was not until Johannes Kepler (1571-1630) that a completely successful demolition of the Aristotelian system was achieved. Driven by the need to explain the highly accurate observations of planetary motion made by Tycho Brahe (1546-1601), Kepler replaced Aristotle’s divine circular orbits with more mundane ellipses.

The next great development on the road to modern cosmological thinking was the arrival on the scene of Isaac Newton (1642-1727). Newton was able to show, in his monumental Principia (1687), that the elliptical motions devised by Kepler were the natural outcome of a universal law of gravitation. Newton therefore re-established a kind of Platonic level on reality, the idealised world of universal laws of motion. The Universe, in Newton’s picture, behaves as a giant machine, enacting the regular motions demanded by the divine Creator and both time and space are absolute manifestations of an internal and omnipresent God.

Newton’s ideas dominated scientific thinking until the beginning of the 20th century, but by the 19th century the cosmic machine had developed imperfections. The mechanistic world-view had emerged alongside the first stirrings of technology. During the subsequent Industrial Revolution scientists had become preoccupied with the theory of engines and heat. These laws of thermodynamics had shown that no engine could work perfectly forever without running down. In this time there arose a widespread belief in the “Heat Death of the Universe”, the idea that the cosmos as a whole would eventually fizzle out just as a bouncing ball gradually dissipates its energy and comes to rest.

Another spanner was thrown into the works of Newton’s cosmic engine by Heinrich Olbers (1758-1840), who formulated in 1826 a paradox that still bears his name, although it was discussed by many before him, including Kepler. Olbers’ Paradox emerges from considering why the night sky is dark. In an infinite and unchanging Universe, every line of sight from an observer should hit a star, in much the same way as a line of sight through an infinite forest will eventually hit a tree. The consequence of this is that the night sky should be as bright as a typical star. The observed darkness at night is sufficient to prove the Universe cannot both infinite and eternal.

Whether the Universe is infinite or not, the part of it accessible to rational explanation has steadily increased. For Aristotle, the Moon’s orbit (a mere 400,000 km) marked a fundamental barrier, to Copernicus and Kepler the limit was the edge of the Solar System (billions of kilometres away). In the 18th and 19th centuries, it was being suggested that the Milky Way (a structure now known to be at least a billion times larger than the Solar System) to be was the entire Universe. Now it is known, thanks largely to Edwin Hubble (1889-1953), that the Milky Way is only one among hundreds of billions of similar galaxies.

The modern era of cosmology began in the early years of the 20th century, with a complete re-write of the laws of Nature. Albert Einstein (1879-1955) introduced the principle of relativity in 1905 and thus demolished Newton’s conception of space and time. Later, his general theory of relativity, also supplanted Newton’s law of universal gravitation. The first great works on relativistic cosmology by Alexander Friedmann (1888-1925), George Lemaître (1894-1966) and Wilhem de Sitter (1872-1934) formulated a new and complex language for the mathematical description of the Universe.

But while these conceptual developments paved the way, the final steps towards the modern era were taken by observers, not theorists. In 1929, Edwin Hubble, who had only recently shown that the Universe contained many galaxies like the Milky way, published the observations that led to the realisation that our Universe is expanding. That left the field open for two rival theories, one (“The Steady State”, with no beginning and no end)  in which matter is continuously created to fill in the gaps caused by the cosmic expansion and the other in which the whole shebang was created, in one go, in a primordial fireball we now call the Big Bang.

Eventually, in 1965, Arno Penzias and Robert  Wilson discovered the cosmic microwave background radiation, proof (or as near to proof as you’re likely to see) that our Universe began in a  Big Bang…

The Next Three Weeks

Posted in Biographical, The Universe and Stuff with tags , , , on April 11, 2010 by telescoper

Busy day today, getting ready for tomorrow’s return to teaching. The year’s second semester is always a strangely fragmented affair because of the Easter hiatus. We teach for eight weeks from late January until late March, have three weeks off for Easter, and then return to teach another three weeks before a brief revision period and, then, the examinations. It’s an awkward business, that gap.  There’s also quite a danger of missing lectures later on, if you happen to be teaching on a Monday, owing to the Spring Bank Holiday. I lose a lecture in that way,  for my first year module Astrophysical Concepts, although it’s only in revision week so I’m not going to be struggling for time. I hope.

I’ve organized my first year lectures (if “organized” is the right word!) in four sections and managed to make sure I finished three of them, representing areas covered by three of the four questions in the forthcoming examination, before the break. Now I just have half-a-dozen  lectures on cosmology to get through, so this bit should be reasonably self-contained and it won’t matter too much if the students have forgotten the other three parts I did before Easter.

I’ve also got my third-year particle physics lectures to finish off in this period, so it’s going to be quite a busy three weeks. Still, I’ll have plenty to distract me from the General Election campaign which will cover pretty much the same period. Polling day is May 6th, and my last (revision) lecture will be on May 7th.

Another curiosity about Cardiff’s calendar is that we only get three weeks for Easter. I seem to remember it’s usually been four weeks in the other places I’ve worked. One of the downsides of this is that we’re back to term-time while the annual National Astronomy Meeting is going on. This moves around from year to year, and this time is in the splendid city of Glasgow. I’d like to have gone, and would have done if I hadn’t had so much teaching concentrated in this period. Regrettably I’ll have to give it a miss this year.

Anyway, I was getting my notes together this afternoon, sitting in the April sunshine among the new flowers and listening to the birds singing. Completely by accident I came across this little quote from Johannes Kepler, translated from the Mysterium Cosmographicum, which I thought I’d share with you…

We do not ask what useful purpose the birds do sing, for song is their pleasure since they were created for singing.  Similarly, we ought not to ask why the human mind troubles to fathom the secrets of the heavens…  The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment.

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