Geometry and the Cosmos (3) : from Ptolemy’s circles to Inflationary Cosmology

Sequel of the previous post Geometry and the Cosmos (2): From the Pre-Socratic Universe to Aristotle’s Two Worlds

Ptolemy’s Circles

Any model of the universe must incorporate the mechanisms determining the motion of the planets and other celestial bodies. From Plato and Aristotle to Kepler, astronomers could not imagine the universe governed by shapes other than circles and spheres, the only geometric forms that could possibly represent divine perfection. This constraint forced them to devise extremely complex systems which would “fit the facts”, in other words account for the apparent movements of the planets and stars as observed from the earth while conforming to the ideological demands of the concept of universal harmony.

Despite the ingenuity of astronomers like Euxodus (see previous post), their circular systems did not accurately describe the complex movements they had observed: the planets accelerated and decelerated and even occasionally went back the way they had come. Moreover, they did not account for the changes in brightness of the planets, which suggested variations in their distance from the earth that were incompatible with the idea that they travelled in circles centred on or near the earth.

How could Aristotelian cosmology be reconciled with astronomical observation? The most elaborate attempt to do so was made by Ptolemy (Claudius Ptolemaeus) in the second century AD. In his Syntaxis Mathematicae, better known by its Latin title Almagest, the Alexandrian thinker succeeded in explaining the motion of each celestial body by a system of extremely elaborate mathematical constructs.

Left : a depiction of Ptolemy at work in a medieval manuscript. Right : A Greek manuscript of the Almagest, IXth century.

Ptolemy adopted the concept of a stationary earth and celestial bodies which could move only in circles. But he multiplied the number of circles and offset them one against the other, proposing complex and ingenious interactions between them. The circle in which a planet moves, called its epicycle, no longer had the earth at its centre as in Eudoxus’ theory, but itself revolved around another circle, called the deferent (or eccentric circle if its own centre is offset from the earth’s position). This theory enabled Ptolemy to “fit the facts” without departing too far from Aristotelian philosophical principles and it survived for 1,500 years — longer than any other idea in the history of science – until the discovery of elliptical orbits by Kepler.

The Motion of the Outer Planets According to Ptolemy. Ptolemy’s theory of planetary morion was first mentioned in his Planetary Hypotheses, which survives only in an Arabic translation, and fully developed in his Almagest. The original Greek title, Syntaxis, means “compendium”, but the work seems to have been known as megiste, “the greatest”, whence the Arabic al-Majesti and subsequently the Latin title Almagestum. The most influential Latin translation was made in 1175 by Gerard of Cremona. The page above, from a later edition of Book X, shows a kinematic model of the motion of the outer planets – Mars, Jupiter and Saturn. The earth remains still while the planets move in a regular pattern relative to an equant point offset from the centre of the planetary sphere. Claude Ptolemy, Almagestum, translated into Latin by Gerard of Cremona, 13th century. Vatican, Biblioteca Apostolica Vaticana, Lat. 2057
Ptolemy’s Epicycles.
The Ptolemaic system was based on three geometric patterns: the epicycle, the eccentric circle and the equant. The epicycle had been invented in the third century BC by Apollonius of Perga, a brilliant mathematician whose most famous work is a treatise on conical sections (ellipses, parabolas and hyperbolas), and developed by Hipparchus a century later.
(a) Epicycle : a planet P rotates in a small circle (epicycle) whose centre (C) is simultaneously moving along the circumference of a large circle, known as the deferent, with the earth (T) at its centre.
(b) Eccentric Circle : the earth is offset from the centre (O) of the deferent. This model breaks the Aristotelian rule which states that the earth must be at the centre of the cosmos.
(c) Equant : despite its complexity, the eccentric circle model does not provide a sufficiently accurate explanation of the apparent motion of the planets. Ptolemy therefore postulated an equant point (E) about which the centre of the epicycle (C) rotates. Both the geometric centre of the deferent (O) and the centre of motion are now offset from the earth’s position (T).
(d) Final Model : the Ptolemaic model plots the motion of the planets according to this sytem, but it was so complicated that it was not fully understood by Western civilisation until the 15th century.
Ptolemy proudly defended its complexity: “We must as far as possible apply the simplest hypotheses to the movements of celestial bodies but, if these are inadequate, we must find others which explain them better.” (Almagest, XII, 2) – a statement which placed him firmly in the vanguard of modern scientific thinking.

 Nevertheless, the system of epicycles and eccentric circles suggested that the earth was not exactly at the centre of the cosmos and Islamic astronomers raised several objections to this infringement of Aristotelian harmony. It was the existence of an equant point offset from the earth that particularly preoccupied later scientists. Copernicus, for example, in his De Revolutionibus announced his intention to rid the celestial model of this “monstrosity”. Continue reading

45 Years of Black Hole Imaging (1): Early work 1972-1988


Black holes are to many the most mysterious objects in space. According to the laws of General Relativity, they are by themselves invisible. Contrarily to non-collapsed celestial bodies, their surface is neither solid nor gaseous ; it is an immaterial border called the event horizon, beyond which gravity is so strong that nothing can escape, not even light.

Seen in projection on a sky background, the event horizon would have the aspect of a perfectly circular black disk if the black hole is static (the so-called Schwarzschild solution) or of a slightly distorted one if it is in rotation (the Kerr solution). Due to strong gravitational lensing, such a « bare » black hole could leave an observable imprint on a starry background. However, in typical astrophysical conditions, whatever its size and mass (ranging from stellar to galactic scales) a black hole is rarely bare but is dressed in gaseous material. Swirling in a spiral motion, the gas forms a hot accretion disk within which it emits a characteristic spectrum of electromagnetic radiation. Giant black holes, such as those currently lurking at the centers of galaxies, can be also surrounded by a stellar cluster, whose orbital dynamics is strongly influenced. As a matter of fact, , if a black hole remains by itself invisible, it “switches on” in its characteristic way the materials it attracts, and distorts the background starry field by gravitational lensing.

Thus, as soon as the basics of  black holes astrophysics developed in the 1970’s,  the scientists logically wondered what could look like a black hole. Many of you certainly saw didactic or artistic representations of a black hole in popular science magazines, in the form of a black sphere floating in the middle of a circular whirlwind of brilliant gas. So striking they are, these images fail to report the astrophysical reality. This one can be correctly described by means of numerical simulations, taking into account the complex distortions that the strong gravitational field prints in spacetime and light rays trajectories.

Since the first numerical simulations performed 40 years ago, tantalizing progress has been done to detect black holes through electromagnetic radiation from infalling matter or gravitational waves. The first telescopic image by the Event Horizon Telescope of the nearest giant black hole SgrA*, lurking at he center of our Milky Way galaxy, is expected for 2018.

The aim of this series of posts is to retrace the rich history of black hole imaging.

Preliminary steps

Black hole imaging started in 1972 at a Summer school in Les Houches (France). James Bardeen, building on earlier analytical work of Brandon Carter, initiated research on gravitational lensing by spinning black holes. Bardeen gave a thorough analysis of null geodesics (light-ray propagation) around a Kerr black hole. The Kerr solution had been discovered in 1962 by the New Zealand physicist Roy Kerr and since then focused the attention of many searchers in General Relativity, because it represents the most general state of equilibitum of an astrophysical black hole.

The Kerr spacetime’s metric depends on two parameters : the black hole mass M and its normalized angular momentum a. An important difference with usual stars, which are in differential rotation, is that Kerr black holes are rotating with perfect rigidity : all the points on their event horizon move with the same angular velocity. There is however a critical angular momentum, given by  a = M (in units where G=c=1) above which the event horizon would « break up » : this limit corresponds to the horizon having a spin velocity equal to the speed of light. For such a black hole, called « extreme », the gravitational field at the event horizon would cancel, because the inward pull of gravity would be compensated by huge repulsive centrifugal forces.

James Bardeen computed how the black hole’s rotation would affect the shape of the shadow that the event horizon casts on light from a background star field. For a black hole spinning close to the maximum angular momentum, the result is a D-shaped shadow.

Apparent shape of an extreme Kerr black hole as seen by a distant observer in the equatorial plane, if the black hole is in front of a source of illumination with an angular size larger than that of the black hole. The shadow bulges out on the side of the hole moving away from the observer (at right) and squeezes inward and flattens on the side moving toward the observer (at left).

The reference is Bardeen, J. M. 1973, Timelike and null geodesics in the Kerr metric, in Black Holes (Les Astres Occlus), ed. C. Dewitt & B. S. Dewitt, (New York: Gordon and Breach) p.215–239

At the time, C.T. Cunningham was preparing a PhD thesis at the University of Washington in Seattle, under the supervision of Bardeen. He began to calculate the optical appearance of a star in circular orbit in the equatorial plane of an extreme Kerr black hole, taking account of the Doppler effect due to relativistic motion of the star, and pointed out the corresponding amplification of the star’s luminosity. He gave formulas but did not produced any image.
The reference is Cunningham, C.T. and Bardeen J.M., The optical appearance of a star orbiting an extreme Kerr black hole, ApJ 173 L137-142 (1972).

One year later Cunningham and Bardeen published a more complete article with the same title. For the first time a picture was shown of the primary and secundary images of a point source moving in a circular orbit in the equatorial plane of an extreme Kerr  black hole. They calculated as functions of time the apparent position and the energy flux of the point source as seen by distant observers.

Apparent positions of the two brightest images as functions of time for two orbital radii and an observer art a polar angle 84°.024. The small, dashed circle in each plot gives the scale of the plot in units of M. The direct image moves along the solid line, the secundary image along the dashed line. Ticks mark the positions of the images at 10 equally spaced times.

In the upper diagram showing the distorted image of a circle of radius  20M, we clearly see that, whatever the observer’s inclination angle, the black hole cannot mask any part of the circle behind. We also see that the black hole’s spin hardly affects the symmetry of the primary image (although the asymmetry is stronger for the secundary image).
The exact reference is Cunningham, C.T. and Bardeen J.M., The optical appearance of a star orbiting an extreme Kerr black hole, 1973, ApJ, 183, 237. The article can be uploaded here.

In 1975, Cunningham calculated the effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole. He gave formulas and drawed graphics but no image.
The reference is  Cunningham, C. T., The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole, ApJ, 202, 788 (1975)

In 1978 Leigh Palmer, Maurice Pryce and William Unruh carried out,  for pedagogical purpose, a simulation of starlight lensed by a camera orbiting a Schwarzschild black hole, using an Edwards and Sutherland Vector graphics display at Simon Fraser University. They showed a film clip in a number of lectures in that period, but unfortunately they did not publish their simulation, so that I can’t reproduce here any image.

First calculations for a black hole accretion disk

The same year and quite independently, as a young researcher at Paris-Meudon Observatory specialized in the mathematics of General Relativity, I wondered what could be the aspect of a Schwarzshild black hole surrounded by a luminous accretion disk. Continue reading