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Rarely, if ever, is it mentioned in the scientific literature, the fact that Kepler ellipse has ever had a challenger, in spite of the fact that, when inventing the centripetal forces Newton himself took the precaution of such an alternative. Indeed, the phrase “any orbit whatsoever” is repeatedly used in Section ii of the first Book of Principia, where Newton describes his invention. And yet, there was once one challenger for that ellipse, proposed by the great astronomer J-D Cassini, in the form of what is now known as Cassini ovals. Unlike the ellipsis, which is defined by the sum of the distances of its points to the foci, the Cassini ovals are defined by the product of those distances. Analytically these figures are represented therefore by quartic equations. A geometrical analysis conducted in the modern spirit of analysis (Sivardière, 1994) shows that the Cassini oval has just about the same position with respect to the ellipse that this one has to the circle. Therefore it cannot be considered among the geometrical figures able to represent a planet’s orbit, and this is the main reason for their elimination.

Sivardière concludes his analysis with the following words:

The strange suggestion of Cassini is not acceptable from an experimental point of view since the difference between an oval and an ellipse having a common focus and a common axis a is of the same order as the difference between the ellipse and a circle of radius a. Of course, this suggestion was eliminated when the dynamical theory presented by Newton explained the observations by Kepler. Using the Binet’s formula, which gives the acceleration in a central motion knowing the polar equation of the orbit, also shows that no central force can be responsible for the motion of a planet on a Cassini oval. (Our italics)

It is the object of the present essay to show that, in order to be completely true, this last sentence asks for a further specification of force: not only it has to be central but also needs to have magnitude depending exclusively on the distance between planet and Sun. Otherwise, from the point of view of Newtonian dynamics embodied in Binet’s formula, the Cassini oval might be a real threat for the Kepler ellipse. Here is the proof:

Binet’s formula, in its most general form, can be written as

where the discussion proceeds in plane polar coordinates (r, θ) and the acceleration of magnitude f(r, θ) is supposed to be generated by a force acting centrally. The accent refers to derivative upon angle. Now, in the case where forces are of the Jacobi type (Appell, 1893), the acceleration can be written in the form

so that the Binet’s formula reduces to

The general solution of this equation is of the form

where a₁ and a₂ are constants to be determined by initial conditions, and Φ(θ) is a particular solution of equation (3).

Now, all the Newtonian forces give accelerations of the form

in case the orbiting body follows a conic section but the force does not point towards the center or a focus, or the orbiting body follows some other path. The Cassini oval is a quartic, and here is an example of central force for which the orbit of revolving planet is a quartic:

Here is an arbitrary vector, whose direction is chosen as origin for the polar angle. This force is of Jacobi type, with

One can convince oneself that in this case F(q) from equation (4) can be taken as

so that the polar equation of the orbit is

Reestablishing here the Cartesian coordinates, gives the following quartic equation:

Consequently, it is not true that no central force can generate a Cassini oval. It is certainly true, however, that no central force with magnitude depending only on the distance r can generate a Cassini oval.

Contemporary of Newton, Jean-Dominique Cassini seemed to have been in possession of more accurate data referring mostly to the motion of the Moon. He even formulated three laws specifically for this motion (Colombo, 1966), laws which have been lately a subject of deep theoretical investigation (Peale, 1969). It is therefore appropriate to think that his studies of the motion of the Moon were those which inspired his proposal for the ovals. However, the Newtonian forces help us understand even such a “strange suggestion”. Indeed, the system Earth-Moon is special among the celestial systems available to us for a closer study, for these two celestial bodies are very close to one another. Thus, the material point approximation, which is the principal property of the Newtonian forces with magnitude depending only on distance, is not likely to work well here, because we don’t know what distance may mean. However, if a central force like that from equation (6) works here to a certain extent, then it asks for further qualifications which are in line with the theoretical studies today. Indeed, one of the most important properties of such forces is their non-conservativity. And the Cassini laws, when applied to close systems like that of Moon and Earth or Mercury and Sun, seem to point out towards such a conclusion (Ward, 1975). Indeed, in such systems the energy dissipation by tidal friction is the main feature upon which the theoretical physics insists mostly.

REFERENCES

Appell, P. (1893): Traité de Mécanique Rationnelle, Gauthier-Villars et Fils, Paris, Tome I

Colombo, G. (1966): Cassini’s Second and Third Laws, The Astronomical Journal, Vol. 71, pp. 891 – 896

Peale, S. J. (1969): Generalized Cassini’s Laws, The Astronomical Journal, Vol. 74, pp. 483 – 489

Sivardière, J. (1994): Kepler Ellipse or Cassini Oval? European Journal of Physics, Vol. 15, pp. 62 – 64

Ward, W. R. (1975): Tidal Friction and Generalized Cassini’s Laws in the Solar System, The Astronomical Journal, Vol. 80, pp. 63 – 70