Comptes Rendus Physique

. In his 1758 book, Lambert (1728–1777) established the most compact expression of the refraction integral, and developed it in a power series of sine or tangent of the zenithal distance. He then examined the geodetic consequences of the curvature of the light rays. Mots-clés. Lambert (séries de), Intégrale de réfraction, Coe ffi cient de réfraction, Dépression de l’horizon, Distance de l’horizon


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This volume devoted to astronomical and atmospheric refraction is singular in several ways.
First, by the apparent simplicity of the phenomenon discussed here, the refraction of light, taught in high schools with the famous Snell-Descartes formula found at the beginning of the xvii th century, and whose study, going back to the x th century with Ibn Sahl, did not cease to fascinate the astronomers in front of the difficulty of observing the stars after the travel of their light in the Earth's atmosphere.
Then, by the concern of many of the present authors to follow the historical evolution of the problem in referring explicitly to some of the most important original articles that have marked this research until the advent of satellites.
Finally, by the number and the content of articles written in French in order to allow easy access to teachers, in particular of the first cycle of higher education, and to invite them to show to their students the genesis of a scientific approach of almost four centuries, carried out between mathematics, astronomy and physics.
This volume is an educational issue of history of science, an exemplary illustration of the story of great scientific adventures, which we hope will be the first of its kind but not the unique one. Perceiving the progress of science in its historical dimension is an essential approach to practice in our institutions to understand not only the world around us but also the way it has gradually opened up to us. Exhibiting research on astronomical refraction provides a great opportunity to do so.
Its main concern is the effects -excluding those of extinction -produced on the observation of the stars by the propagation of their light in the Earth's atmosphere, at the wavelengths of the visible, later extended to the infrared, radiations: the celestial bodies are seen in an apparent direction, different from their real direction, and more or less distorted. The study of these phenomena has a long and rich history, particularly since the xviii th century, in connection with developments not only in optics, but also in mathematics, physics of gases and the atmosphere, and geodesy. This gave rise to successive mathematical models and approximations interesting to examine in their historical development. Indeed, astronomical refraction calculations progressed hand in hand with those of differential geometry, integral calculus and series expansions, which they stimulated and used very early on. Obtaining good tables of astronomical refraction was a major issue for navigation. At a time of intense competition in the development of overseas empires, scientific interest met geopolitical interests. At the same time, Newton stressed the fundamental role of these tables for astronomy. Indeed, the accuracy of astrometric measurements had long been dependent on the careful consideration of astronomical refraction and its modeling, until the advent of space-based observations.
The inverse problem, i.e. the determination of properties of the Earth's atmosphere from observations of refraction, was also the subject of much work, until it was recognized that the problem was ill-posed and had no unique solution. The refraction of light in the atmospheres of other bodies in the solar system -which can be seen in particular during stellar occultations or transits -led to the discovery of an atmosphere around the planet Venus in the xviii th century. This method is still used today to probe the atmospheres of solar system bodies from a distance even if they are tenuous, and to follow their temporal evolution.
In addition, to improve the altitude calculations of geodetic points on land, it appeared necessary to study what is usually called "terrestrial" or "atmospheric" refraction. Moreover, it is involved in the precise knowledge of the apparent height of the sea horizon, and therefore in the practice of astronomical navigation. This second aspect of the effects of refraction in the Earth's atmosphere, which includes various categories of particular and spectacular phenomena, such as mirages or Fata Morgana, also aroused curiosity and then quantitative explanations.
Beyond the large-scale refractive properties of the atmosphere, its local random fluctuations, which are caused by small-scale turbulence, were for a very long time a fundamental obstacle to the high angular resolution of astronomical images in the visible and infrared domains, until the development of contemporary adaptive optics methods, which are now used in all large ground-based telescopes. In parallel to this adaptive correction, the chromatic dispersion occurring in the atmosphere must be compensated for, so that multi-telescope optical interferometry, which is now rich in discoveries, is fully operational. The major importance of these two methods in contemporary astronomy justifies addressing them in this thematic issue dominated by refraction.
These developments, starting from the origin of the study of astronomical and atmospheric refractions, continuing with the exposition of theoretical properties and the analysis of various observations, and finally concluding with various consequences on astronomical observation using today's instruments, constitute a first part of twelve articles. The second part of this special issue wishes to shed light on several remarkable pieces of work from scientists of the past centuries, and to connect these with today's presentations. It includes seven more properly historical articles, including six commentaries on major texts, which opened up fruitful avenues of study. They are placed in their historical and scientific context, with their repercussions. Extensive excerpts from the original editions bring readers into direct contact with the source of notions that have become essential. In order to overcome the difficulty of getting to the heart of an ancient text, a table of correspondences is proposed, linking the ancient notations to the current ones, as used in the two articles at the beginning of the first part. It is recommended to read these articles before tackling these historical comments. The two parts, both for the concepts and for the notations, thus retain an overall coherency, beyond the diversity of the articles.
By gathering here such a set of articles around refraction in the terrestrial and celestial atmospheres, we wish to show to the reader, researcher, professor or student, the richness and the variety of a subject which could seem almost trivial today, and to propose to him a tool which will be able to renew in depth his vision of a "simple", most often geometrical, problem of optics. Many eminent scientists have dealt with it, as it has played and still plays a major role in astronomical observation and in the study of the atmospheres of celestial bodies.
The first three texts of Part I are by the same author and split for clarity. The first one ("Panorama historique. . . " [1]) presents a detailed history of the study of astronomical refraction, without limiting itself to general ideas. It goes into technical details, as is nowadays the practice in the history of mathematical sciences at large, and transcribes the outstanding formulas obtained in the past, accompanied by numerous quotations. The second article ("Propriétés remarquables de la réfraction astronomique. . . " [2]) exposes the main mathematical results, whether they take the form of rigorous theorems or more or less approximate expressions, in the case of an idealized, spherically symmetric atmosphere. The last article of this methodological presentation ("Phénomènes de réfraction atmosphérique terrestre" [3]) deals with terrestrial refraction and its various manifestations, of which it presents an attempt at a reasoned classification but nonexhaustive since some mysterious observations remain unclassifiable today.
Nine articles then discuss a variety of remarkable situations in which the interpretation or quality of the observation depends on refraction. Hence, article 4 ("Les tables de réfraction astronomique" [4]) presents the historical development of astronomical refraction tables. Article 5 ("Relations among atmospheric structure. . . " [5]) explores efforts to deduce the properties of the Earth's atmosphere from refraction. With paper 6 (" Study of atmospheres in the solar system. . . " [6]), the richness of the methods used is revealed in the contemporary study of the atmospheres of solar system bodies, as it was in the discovery of an atmosphere around the planet Venus, presented in paper 7 ("The Lomonossov arc. . . " [7]), with interesting modern extensions. In articles 8 ("Les effets optiques de la turbulence atmosphérique. . . " [8]), 9 ("Optique adaptative. . . " [9]) and 10 ("Correction de la dispersion atmosphérique. . . " [10]), contemporary developments in adaptive optics and optical interferometry in the presence of atmospheric turbulence are analyzed. The reader's curiosity will be nourished by articles 11 ("Novaya Zemlya effect and Fata Morgana" [11]) and 12 ("Images du Soleil et de la Lune. . . " [12]), which explore some consequences of unusual refractions: the Novaya Zemlya effect (1597), where the Sun is seen when it is significantly below the horizon; the distortion of the aspects of the Sun and the Moon at the horizon, seen from the ground or the International Space Station.
The second part, as announced, deals with historical studies, without however neglecting the theoretical and quantitative aspects. Six short articles by the same author each focus on a major text by a different scholar, systematically giving the context, then a summary with comments, and finally the most important derivations and results, reproducing certain essential original passages. This second part, with its precise analysis, along with a choice to go in depth into the presentation of these works, is clearly aimed at physicists as well as historians of science. Three texts are dedicated to the xviii th century. Bouguer's seminal work ("L'invariant de Bouguer et ses conséquences" [13]) explores the curvature of light rays. Shortly afterwards, Lambert ("Les développements de Lambert" [14]) corrects the summit altitudes calculated by the Cassinis who had neglected terrestrial refraction, and proposes various series expansions to express the astronomical refraction angle. The last text ("Did Monge really explain inferior mirages?" [15]) shows that the claim about Monge, who after Bonaparte's expedition to Egypt is often credited with explaining ordinary mirages, is neither historically nor scientifically valid.
Three scholars of the xix th century follow: Biot, author of a remarkable theorem and method of calculation concerning the integral of refraction ("Le théorème de Biot. . . " [16]); Kummer, applying a classical method of discussion in mechanics to the study of the vision of the horizon, which makes it possible to expose the whole of the corrections required at the time of the astronomical point at sea ("La discussion par Kummer d'une quadrature. . . " [17]); and finally, Radau, synthesizing the studies of the time -largely based on series expansions of which the divergence was then little known ("Les développements de Radau et leur divergence" [18]).
For the xx th century, we have chosen the remarkable works of Link, who laid the foundations of a theoretical photometric study of total lunar eclipses and showed that they can be indirectly used to probe the upper atmosphere of the Earth ("La photométrie des éclipses de Lune. . . " [19]).
We hope that this issue, with its wealth of historical references and the talent of its authors, will contribute to the work of researchers and historians and inspire many teachers to go beyond teaching science and tell its wonderful story!