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Félix Tisserand's report on the work of Édouard Roche
Édouard Roche had been made a corresponding member of the Paris Academy of Sciences in December 1873 but when Joseph Liouville died in September 1882, a vacancy occurred in the Astronomical Section of the Academy. Félix Tisserand proposed Roche to fill the vacancy and an election was held on 16 April 1883. Tisserand wrote a report on the work of Roche as part of his proposal and, after the death of Roche only two days after the vote to fill the vacancy, it was decided to print Tisserand's report in Comptes Rendus.
Memoirs and Communications from the Members and Correspondents of the Academy.
The Permanent Secretary announces to the Academy the painful loss it has just suffered in the person of M Roche, Correspondent for the Astronomy Section, who died in Montpellier on 18 April 1883. To pay a final tribute to the memory of its eminent Correspondent, the Academy decides that the Report recently written by M Tisserand on the work of M Roche will be included in the Proceedings.
Report on the work of M Roche, Professor of Astronomy at the Faculty of Sciences of Montpellier; by M F Tisserand.
M Roche's most remarkable works relate to the figure of celestial bodies (planets and comets) and to Laplace's cosmogonical theory; these Memoirs are intimately linked to one another, and it is by following the natural sequence of his ideas that the author has successfully developed, over the last forty years or so, various important chapters of celestial mechanics.
1st. Miscellaneous Memoirs on the Equilibrium of a Homogeneous Fluid Mass Subject to Certain Conditions.
We know the fine research of geometers on the equilibrium of a homogeneous fluid mass, animated by a rotational movement around its axis, and whose molecules attract each other according to Newton's law.
M Roche set out to determine this equilibrium figure, taking into account a new force, the attraction exerted by a centre located at a great distance. He assumed that this centre rotates with an angular velocity equal to that of the rotational movement of the fluid mass; this is not a problem of pure curiosity; the Moon is precisely in this case, and photometric experiments show that the same is true for the satellites of Jupiter and Saturn; it seems that the equality of the translational and revolution movements of the satellites is a general law of our planetary system.
The problem had been addressed by Laplace, in the special case where the mass of the satellite is very small compared to that of the planet, and where the shape of the satellite is assumed to differ very little from that of a sphere. Laplace had found that, under these conditions, there exists only one equilibrium shape: it is an ellipsoid rotating around its smallest axis, and whose largest axis is directed towards the planet.
M Roche wanted to free himself from the two restrictive conditions mentioned above and, assuming that the equilibrium shape is that of an ellipsoid, he sought the lengths of the axes of this ellipsoid.
Here are the most important results to which he was led by a remarkable analysis:
The ellipsoids serving as equilibrium figures are of two kinds: they are elongated, some toward the attracting point, others in a perpendicular direction, and, consequently, present an unstable equilibrium; it is therefore sufficient to deal with the former. If the rotation speed is smaller than a certain limit, these ellipsoids are two in number; when the speed increases, the two figures approach each other, and cease to exist when the speed reaches the indicated limit.
When the speed is very low, one of the ellipsoids differs very little from a sphere; the other is a sort of needle excessively elongated toward the planet.
M Roche has reduced the calculations of these equilibrium figures to those of transcendentals depending on elliptic functions, and he has given for this numerical tables that leave nothing to be desired.
He has made interesting applications of his formulas, first to the Moon, then to the satellites of Jupiter and Saturn; he finds for the latter much more pronounced flattenings than in the case of the Moon.
In another Memoir, M Roche takes up the problem again, without admitting a priori that the equilibrium figure is that of an ellipsoid; he only supposes it to be little different from a sphere, but he admits at the same time that the speeds of translation and rotation are different. It is true that, in this case, the fluid will not have, strictly speaking, a permanent equilibrium figure; but, if we suppose that this fluid takes at each instant the form with which it would be in equilibrium under the action of the various forces, it will only be necessary to take into account a sort of tide acting on its mass.
The author believes that, in the evolution of the solar system, this case must have preceded and led to the one currently presented to us by the satellites, which are today susceptible to a permanent equilibrium figure.
He also examines what would happen if the fluid were also subjected to the action of an attractive force following Newton's law and having its seat at the centre of gravity; this assumption includes the case where the fluid covered a sphere of different density. Finally, he assumes that the spheroid is formed of an infinity of layers of variable densities.
In all cases, he finds as a solution an ellipsoid flattened at the poles and elongated towards the exterior body; it only happens that this figure differs more from the sphere than in the case of homogeneity.
Finally, in a third Memoir, M Roche considers the equilibrium figure of an immobile fluid mass, all of whose parts attract each other, subject in addition to the attraction of a distant centre. He finds in this case two ellipsoids of revolution around the axis directed towards the attracting point. If we assume that the distance from the centre of attraction is decreasing, it may happen that the two ellipsoidal figures cease to exist; all other things being equal, this will happen all the more quickly the smaller the density of the fluid.
Thus, if we imagine a comet falling in a straight line towards the Sun, its shape, initially spherical, will become ellipsoidal, will elongate more and more towards the centre of attraction, and it may happen that the ellipsoidal shape ceases to exist and the mass of the comet divides into several fragments, each falling on its own side towards the Sun.
Here, M Roche found himself naturally led to the study of the shape of comets; we will say later what progress he has made in this theory.
2nd. Memoirs on the Physical Constitution of the Terrestrial Globe (1848).
We know the average density of the globe and the average density of the continents; the first is approximately double the second; the Earth is therefore not homogeneous; the specific weight of the interior layers must be much greater than that of the superficial layers; it must increase from the surface to the centre and in a roughly regular manner. Knowledge of this law of variation would be very important. Assuming the interior of the Earth to be fluid, the simplest hypothesis that presents itself is that the ratio of the increase in pressure to the increase in density is proportional to the density: this is Legendre's famous hypothesis; it makes integrations easy.
M Roche was led to imagine another law, according to which the decrease in density would be proportional to the square of the distance from the centre; this hypothesis is as likely as Legendre's; the author concludes that the density at the centre of the Earth would be roughly double the average density. It is up to experiment to decide between the two hypotheses; now, M Airy made an important observation in 1854, by swinging a pendulum at the bottom of a 385m mine shaft; he noted an increase in the intensity of gravity; M Roche's law represents, very approximately, the result of the observation, while Legendre's law differs significantly from it.
Although a single experiment cannot be decisive in such a delicate matter, it nevertheless follows that M Roche's hypothesis is today more likely than Legendre's.
3rd. Memoir on the Interior State of the Terrestrial Globe (1881).
About thirty years after the previous work, M Roche published another, much more important one. We will try to give an idea of it.
It is generally accepted that the Earth is entirely fluid in its interior, with the exception of a very thin superficial crust; most mathematical studies of the shape and interior constitution of the Earth take this hypothesis as their starting point.
In recent years, this assumption has been hotly contested, particularly by our illustrious Foreign Associate, Sir W Thomson, who believes that the tides that would necessarily occur in this fluid mass, under the influence of the Sun and the Moon, would be such that no envelope could resist them.
It was therefore interesting to resume mathematical studies on the physical constitution of the globe, under a different hypothesis, for example, by assuming the terrestrial globe to be formed of a solid core, covered with a less dense layer, which may be partially fluid at a certain depth.
To decide between the two systems, we have three data from experiment, namely:
The average density;
The flattening at the surface, determined by geodetic measurements;
A certain constant depending on the moments of inertia, given by the precession of the equinoxes.
By admitting Legendre's law, or his own, M Roche showed that the hypothesis of complete fluidity cannot represent both, within the limits of observational errors, the flattening at the surface, as it results from the most precise measurements, and the constant of the precession of the equinoxes.
He concluded that complete fluidity is impossible.
He then proposed to see if, with a solid central core, we can achieve the agreement that has just eluded us.
He found that we can reconcile everything by assuming the density of the inner core equal to approximately 7, that of the outer layer equal to 3, and its thickness equal to one-sixth of the radius. The inner terrestrial block would therefore, in terms of specific weight, be analogous to meteoric iron; while the layer that envelops it would be comparable to aerolites of a stony nature, where iron only enters in a small proportion.
4th. Miscellaneous Memoirs on the Shape of Comets.
We have seen that M Roche, after having dealt with the shape of satellites, found himself led quite naturally to the theory of the shape of comets. He proposed to study the phenomena which must occur in the atmosphere of a comet, taking into account only the mechanical forces due to the attractions of the Sun and the comet itself.
M Roche started from a work by Laplace, on the shape of atmospheres of celestial bodies. Laplace had in mind above all the atmosphere of the Sun; in this case he had given the general equation of level surfaces, but he had not highlighted a curious property of these surfaces; M Roche has deepened this research and drawn from it some very interesting results.
He considers a comet in the form of a nucleus in which the greater part of the comet's mass is concentrated; this nucleus is surrounded by an atmosphere whose various points are attracted both by the Sun and by the centre of gravity of the nucleus.
He shows that the level layers of this atmosphere, which will be appreciably spherical if the comet is very far from the Sun, must elongate towards this star if the comet approaches it.
The atmosphere cannot extend indefinitely; it is limited to the points where the Sun's attraction balances that of the central core. There therefore exists a final level surface where the atmosphere necessarily ends; everything beyond must abandon the comet and disperse into space.
M Roche shows that this limiting surface offers two singular points, where it merges with a cone. Within this limiting surface, the level surfaces are closed; beyond, they open in the vicinity of these singular points to develop into infinite sheets. It follows that if, for any reason, the cometary fluid comes to exceed the limiting surface, it will flow through the two singular points and disseminate into space.
According to this theory, every comet should have two tails, one directed toward the Sun, the other away, which is contrary to observation. The theory was therefore flawed, or it was incomplete.
It was at this time that our colleague, M Faye, introduced into science the notion of a repulsive force, emanating from the Sun and acting mainly on low-density materials, such as the parts that constitute the atmosphere of comets; M Faye succeeded in explaining, with the help of this force, all the peculiarities presented by Donati's comet.
M Roche thus found himself led to resume his research on the figure of comets, introducing M Faye's force.
The study of the new level surfaces showed him that they were profoundly modified; one of the singular points of the limiting surface was eliminated: it was the one which was turned towards the Sun; therefore, in the new hypothesis, comets should no longer have a tail opposite the Sun.
M Roche was able to compare down to the smallest details the results of his theory with those that observation had provided for Donati's comet; this is how he explained the black line that was found in the middle of the tail, the changes in curvature presented by certain lines of the tail.
M Roche was able to make, in another order of ideas, an interesting application of his ideas by fixing an upper limit on the mass of comets. Laplace had proven that the mass of Lexell's comet, which passed very close to the Earth, was at most part of the mass of the Earth; this was the only information we had for a long time on the masses of comets. M Roche deduced from his theory a relationship between the mass of a comet, the diameter of its atmosphere, and its distance from the Sun; it is undoubtedly difficult to fix by observation the diameter of a comet's atmosphere; however, M Roche was able to show in a plausible manner that the mass of Donati's comet was less than part of the mass of the Earth.
5th. Essay on the Constitution and Origin of the Solar System.
M Roche's studies on the shape of celestial bodies and the arrangement of level layers in the atmospheres surrounding them have led him to interesting results, applicable to the Sun, its atmosphere, and the nebula which, by progressive condensation, produced the various members of the solar system.
M Roche was thus led to develop Laplace's beautiful cosmogonical theory, to clarify it on certain points, and to modify it on others.
Here again, it is the geometric study of level surfaces that leads him to new and interesting results
Following Laplace's theory step by step, we encounter a first difficulty concerning the Moon; at a certain time, the terrestrial nebula must have extended beyond the Moon; now, by calculating, according to Laplace's ideas, the greatest distance that the Earth's atmosphere has ever reached, we find that this distance would be only three-quarters of the distance from the Earth to the Moon.
A difficulty of the same order arises concerning Saturn's ring, which is considered by Laplace as still-existing proof, a witness to the phases through which the zones abandoned by the Sun before transforming into planets. Saturn's rings are half-enclosed in a region where it would be impossible for Saturn's atmosphere to have abandoned these materials.
M Roche clarified these delicate points, and many other similar ones in the solar system, with the help of an ingenious concept. Laplace had considered only the parts abandoned by the solar nebula in the region of the equator. M Roche showed that the position of the solar nebula, having become free, does not come only from the equator, but from a superficial sheet which extends much further towards the two poles and which begins to flow towards the equatorial opening.
He was thus led to admit the formation of rings inside the nebula, which later become free, but under conditions other than the outer rings of Laplace; it is in these inner rings that he finds a plausible explanation of phenomena which escaped Laplace's cosmogonic theory.
In this brief exposition, we have left aside some important work, which we will say a word about in closing.
M Roche applied the method of variation of arbitrary constants, recommended by Poisson, to the determination of the parallactic inequality of the motion of the Moon; Poisson had made this calculation himself, but incompletely; he had omitted, in fact, several sensitive terms, which M Roche took into account.
We will also mention his Memoirs on the Light of the Sun; his historical research on the obfuscations of the Sun, on the old observatory of Montpellier; his work on the current climate of Montpellier, compared to the observations of the last century.
Finally, in another order of ideas, M Roche found for the Taylor series a form of the remainder, which appears today in all Treatises on Differential Calculus.
Through the variety, scope, and importance of his astronomical research, M Roche has earned a special place for himself; the Astronomy Section wished to give him a testimony of high esteem by placing him on the list of candidates for the position left vacant by the death of M Liouville.
***
Less than two weeks after reading this Report, which it had fully approved, the Academy learned of M Roche's death; it wished to honour the memory of an eminent scholar who, for forty years, occupied with rare distinction the Chair of Astronomy at the Montpellier Faculty of Sciences, and it decided to print this Report in the Comptes Rendus.
Last Updated June 2025