The 'Ledoux Criterion' on the stability of the inner layers of stars

"In this paper the effect of a discontinuity in the mean molecular weight in stellar models is examined": in one sentence of the preamble of one of his most famous articles (1), published in 1947 in The Astrophysical Journal, Paul Ledoux summarizes a whole part of his research.

Before reaching this point, however, his life was as eventful and unstable as that of the stars he studied. As with all researchers of his generation - but for him more than for others - Ledoux's working conditions were indeed dependent on the events that shook the world at the time. But in spite of all these vicissitudes, Paul Ledoux remained faithful to one line of research, present from his first publication and which would not cease to assert itself over the following years: to study the stability of stars and thus their evolution over time. Where did this passion come from? "Eric Gosset and Jean-Pierre Swings, astrophysicists, respectively Maître de recherches FRS-FNRS and Honorary Professor, explain that reading Arthur Eddington's The Internal Constitution of stars during his studies of physics at the University of Liège was a determining factor. The book was published in 1926, a time when the understanding of what stars really are was very sketchy. Where does their fabulous energy come from? For Kelvin and others, it is from the collapse of the star on itself, but Eddington suggests on the contrary that nuclear reactions - particularly the fusion of hydrogen atoms to give helium - are at work in the center of stars. He is also the first to determine a relationship between mass and luminosity and to explain the pulsation of variable stars. Next to Eddington, whom Ledoux seems never to have met, two personalities emerged during the 1930s in the study of stars; and these two names were to be important for Paul Ledoux: Svein Rosseland (1894-1985) and Subrahmanyan Chandrasekhar (1910-1995).

Svein Rosseland is a Norwegian astrophysicist who, a little older than Paul Ledoux, will guide him in his first research. Thanks to Professor Polydore Swings with whom he worked as a young graduate in physics, Paul Ledoux arrived at the Institute of Theoretical Astrophysics in Oslo in 1939 to study the structure of stars and their stability, Rosseland's specialty. The year, one guesses, is not favourable to fundamental research on stars: the important will soon take place on the battlefields. In spite of this, as we shall see, ledoux's stay in Oslo was beneficial. During the invasion of Norway, Paul ledoux started a rather incredible journey (see his biography ) which led him from Oslo to the United States, more precisely to the Yerkes Observatory of the University of Chicago. There he met a brilliant young Indian astrophysicist of the same generation as him, Subrahmanyan Chandrasekhar, future Nobel Prize winner in physics, with whom he worked and became friends. Paul Ledoux stayed at Yerkes from December 1940 to September 1941 before putting his knowledge at the service of the Allies on the meteorological front... meteorology. " As eventful as it was, this first period of Paul Ledoux's career was fruitful, explain Eric Gosset and Jean-Pierre Swings. The proof is an article published in 1941 (2). Ledoux starts from Rosseland's work and applies for the first time the theory of stability to stars." Paul Ledoux is thus the first to conceive the existence of a maximum mass of stable stars of the main sequence (3) and to calculate it! The verdict: a star with a mass greater than 100 solar masses cannot be stable. He did not stop there and despite the world conflict and his occupations in the Royal Air Force, he continued his research and published another article, very technical, on the same subject in 1945 (4). an article that has a history. "Chandrasekhar had been interested in Ledoux's first article as well as in his later work, which remained in the form of unpublished notes. He wanted to publish his own work, but he was ethically embarrassed because he should have referred to the unpublished work of his colleague. Chandra, as he was called, and had it published! Without co-signing it, an elegant gesture on the part of the future Nobel Prize winner, " an article whose signature is unique in the annals of the Astrophysical Journal (and probably of all scientific journals): P. Ledoux, F/L.R.A.F., Stanleyville, Belgian Congo!

The 1947 paper already mentioned (1) and the following ones will lead to a theoretical formulation of stellar evolution. Back at the University of Liège, Paul Ledoux was one of the first to build stellar models based on a convective nucleus that became depleted as the hydrogen atoms disappeared. convection was one of his favorite subjects, " explain Eric Gosset and Jean-Pierre Swings. The progressive disappearance of hydrogen within a star leads to a discontinuity in what Ledoux calls the average molecular weight - even though there are no molecules in the strict sense - and he studied the effect of this discontinuity on the structure of the star. to explain it, he introduced the notion of semi-convection. Recall that in the core of an average star, the nuclear fusion energy is transmitted by radiation (this is the radiative zone of the star). Around this zone, there is the convective zone where the energy is transmitted by the movements of the matter according to a cycle: the hot matter rises, expands and eventually heats the surrounding matter, cools down, and then descends towards the base of the convective zone, and so on. This is the phenomenon of convection dear to Paul Ledoux. As early as 1906, the German astrophysicist Karl Schwarzschild (1873-1916) had established a criterion based on the temperature gradient to determine if a zone of the star is stable or not (no movement of matter or movements, especially radiax, of it). To explain the shape of the molecular weight discontinuity resulting from the transformation of hydrogen into helium at the edge of the convective zone, he postulated the existence of a semi-convective zone. he also defined a stability criterion for the convective zone that took into account the variation of the average molecular weight, thus considerably refining the Schwarzschild criterion. "This criterion - now called the Ledoux criterion - is still used to develop models of stars. Thanks to this, we can determine where the convective zones are located in a star."

Among the major publications of Paul Ledoux, one cannot ignore the two articles (5) that he published in the Handbuch der Physik of 1958. The first one, written in collaboration with the Dutch astronomer Theodore Walraven (1916-2008) deals with variable stars. The second, of which he is the only signatory, deals with stellar stability. "These two articles, and particularly the second one, have become the bible of all those who have started to do theoretical work on stellar stability and evolution. One can even say that the second one is considered as the founding article of asteroseismology (6). a reference, in particular for those who were (and are) part of the so-called liège school of theoretical astrophysics that Paul ledoux initiated.

An article written by Henri Dupuis, with the help of Eric Gosset and Jean-Pierre Swings

Read the biography of Paul Ledoux 

Scientific references

(1) Stellar models with convection and with discontinuity of the mean molecular weight, P. Ledoux, The Astrophysical journal, vol. 105, 1947.

(2) On the vibriational stability of gaseous stars, P. Ledoux, The Astrophysical journal, vol. 94, 1941.

(3) To make it short and simple, the main sequence stars are those whose energy comes from the fusion of hydrogen into helium. This is the case for the majority of stars. This is because this phase of a star's life is by far the longest and most stable.

(4) On the radial pulsation of gaseous stars, P. ledoux, The Astrophysical Journal, vol. 102, 1945.

(5) Variable Stars, Ledoux, Paul; Walraven, Theodoor and Stellar Stability, ledoux, Paul. Handbuch der Physik, volume 51, 1958.

(6) This discipline studies the seismic movements in stars (their oscillations, pulsations). These movements are either self-sustained or due to violent convection movements that occur in stars. The study of these seismic movements allows to determine the internal structure of stars. It is therefore understandable that Ledoux's work on the stability of stars. It is therefore understandable that Ledoux's work on the stability of stars, and more particularly on the phenomena at work in the convection zone, made the Liege astrophysicist a precursor of modern asteroseismology.

Documentary sources

A. Noels, In Memoriam : Paul Ledoux, Ciel et Terre, Vol 105, 1989.

Robert Halleux, Geert Vanpaemel, Jan Vandersmissen en Andrée Despy-Meyer (éds.), Histoire des sciences en Belgique, 1815-2000, bruxelles : Dexia/La Renaissance du livre, 2001.