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The Pioneer anomaly became a problem! Not that it shouldn’t be one, but it has a new twist: one cannot say if it is a fundamental problem or just a routine one. Some say it can be routinely explained. This idea has two streamlines: first there is a possibility of explanation by on-board energy produced by the devices of the spacecraft. This is mainly pursued for the purpose of eliminating the biasing factors. Then, there is a routine explanation according to accepted theories. On the other hand, there are others who advance the idea of some “new physics” involved here. In order to make up one’s mind about the problem, it is quite sufficient to read the very last productions of the members of the team closely involved in this problem (Anderson, Nieto, 2009; Turyshev, Toth, 2009). From these one can trace back the whole history of the problem of Pioneer anomaly, as well as some other unsolved problems. These can cast doubt on the thoroughness of our theoretical knowledge, and one side of this issue will be discussed here.

In short the Pioneer anomaly consists of a constant rate decrease of the difference between the measured and calculated Doppler Effect for the frequencies of communication with Pioneer spacecrafts. This is translated into a constant acceleration of (8.74±1.33)·10^-10 m/s² present in the data from both Pioneer spacecrafts, after the flybys that launched them into the deep space. This acceleration points somewhere in the region of the inner solar system.

Assuming that the on-board effects of the functioning devices are eliminated, and one can assign it to some physical cause, the tiny difference really challenges our theoretical capability to know the Nature. There are many estimations based on usual theoretical wisdom and just as many proposals based on… proposed theoretical wisdom. None of them is apparently convincing enough in order to be universally accepted, in spite of the quantitative arguments, perhaps because they are so many or perhaps because they are just… quantitative arguments.

This is one illustrative case when the men of science are, as Ernest Becker would say, “choking on truth”. There are so many possible explanations, yet no one is convincing, and the quest for truth still goes on, because of the very nature of human being. There is a secret hope that this effect would lead us to the discovery of a new type of fundamental force, perhaps in the same way in which at the beginning of the previous century the analysis of Newtonian force led Einstein to the construction of general relativity.

But the situation has changed dramatically since the beginning of the last century when Eddington helped establish the truth of Einstein’s theory. In the first place, in that historical case Eddington had strong logical reasons to believe that Einstein’s proposal is the legitimate continuation of Newton’s ideas. Only after that came he to sustain his own conviction by presenting the celebrated quantitative arguments related to the deviation of light rays. Even if these arguments would be criticizable as such, because such arguments usually are, the theory would succeed anyway, because it had strong roots. And this was indeed the case, as anyone can witness today. Eddington had therefore the natural certainty that any other scientist can eventually understand Einstein arguments and see that they make sense, because they have, indeed, solid ontological, as well as gnoseological, roots. In understanding a theory the scientist is alone in front of the whole human knowledge, just as a faithful Christian is alone with God.

Today, as a general rule, the scientific community cannot understand anything anymore, because there are no roots, or the roots are rather superficial. So, what can one do? Simply apply a political strategy: form committees, debate, propose and vote. Whatever the public likes best becomes truth. The scientist cannot judge by himself, because he/she is trapped in the correctness of quantitative evaluation which in turn asks for already existing explanations. In the words of Max Jammer

In our present age of rapid technological progress the frightening discrepancy between our technical “know-how” and our philosophical incomprehension, in general, of basic scientific conceptions seriously endangers the integrity of our intellectual outlook. The cogitative activity of the modern scientist, who is more a technician than a philosopher, is strained to its utmost limits by the necessity for digesting the swiftly accumulating information in his specific field of research. He has little opportunity to indulge in the fundamental problems relating to the very concept which he applies. Moreover, in our present system of academic instruction a thorough and critical discussion of fundamental and apparently simple concepts in science is consciously omitted (relegating them to a stage where the student’s mind is still too immature to understand their true meaning). The nuclear physicist, for example, who works on exchange forces (of the Majorana, Bartlett, or Heisenberg type) and who discusses noncentral tensor forces, rarely has ever analyzed the concept of force in general, a concept that is absolutely fundamental to his work. A historico-critical analysis of the basic conceptions of science is therefore of paramount importance, not merely for the professional philosopher or historian of science. (Jammer, 1957; our italics)

Incidentally, the last two sentences of this excerpt from Jammer are clearly applicable to the problem at hand. Indeed, in no proposal whatsoever from among those presented by the scientific community in order to explain the Pioneer anomaly, is the very classical concept of force taken under scrutiny. Yet, it is historically quite clear that we have to make a case for this concept in the first place, due to minuteness of the anomaly.

If the anomaly is indeed acceleration, and if a force acts in order to produce it, that force is by default accepted to be a central force. There is even an opinion that finding the true direction of the acceleration vector (Sun, Earth, spin axis etc) would help finding its cause. The finding itself is relegated to the analysis of the huge amount of data of the Pioneer mission which has been saved. This might not be that bad, but the concept of central force is not quite so carefully considered. When it is limited to forces depending only on coordinates, in most cases it means central forces, but this concept has some strings attached to it.

The case is not isolated in the history of science. In fact it is quite similar with another case of tiny effects that imposed a reconsideration of the way one has to take the principles of Newtonian mechanics. That is the case of binary stars. These systems would just have started being figured out in the times of Newton, so that they didn’t have any impact in the formulation or consideration of the principles of classical mechanics. However, they surely testify on the dependability of the Newtonian approach of the natural philosophy.

The binary stars are systems of two stars, similar to Kepler type systems, except for the fact that the main star is not in the focus of the ellipse, but can very well be in any position in the plane of motion of its satellite. Thus, the satellite star seems to be acted upon by a force that, if central, points not towards the focus of its trajectory, as the Newtonian gravitational force, but to some other point of the plane of motion. In this respect, and only in this, a binary star is not a Kepler system. However, it is a brilliant confirmation of the Newtonian philosophy per se.

The first reckonable theoretical work - from a classical dynamical point of view, of course - on binary systems seems to be that of Yvon Villarceau (Villarceau, 1850), who struggled to prove that the asserted universality of the Newton’s law of force is not too much upset by the astronomical facts regarding the binary stars already accumulated at that time. Starting from the second law of Newton, and using the information that the trajectory is a conic, Villarceau set out to find the analytical form of force, assumed to depend only on the coordinates of the orbiting body. He was assisted by the fact that the observational data confirmed the second law of Kepler; therefore the forces acting in binary systems are definitely central forces.

Villarceau’s result was indeed astounding, and could very well upset academic circles, which explains why he took very much care in presenting it. The truth of the matter stays in the fact that central forces were taken at that time, as they are still taken today, with a fringe benefit, apparently out of any question: they have also magnitude depending only on the distance between materials points they are called to correlate. Or, by strictly applying the Newtonian philosophy of deducing the force from the apparent motion Villarceau came to an unorthodox conclusion. Namely the force responsible for the motion of a companion in the binary system has indeed the magnitude dependent on the distance, but it also depends on coordinates separately in a manner reflected in the equation of the orbit.

Villarceau might have upset the academic circles after all, once this problem was apparently buried for some three decades. It seems, however, that it needed a time to hatch, for it is not quite so easy to give up a dogma taken as a natural law. Indeed, it appeared again, this time right at the very summit of academy, in the form of a problem formulated by Joseph Bertrand, who was apparently obsessed with the Kepler problem

Knowing that the planets describe conic sections and assuming nothing more, find the expression of the components of force soliciting them as a function of the coordinates of its point of application (Bertrand, 1877a)

Gaston Darboux (Darboux, 1877) and George-Henri Halphen (Halphen, 1877) solved it right away, with results different from that of Villarceau but, as far as we can judge today, pertaining to the same geometrical family. These solutions, as well as his own, gave Bertrand the opportunity of a roundup of the problem, where he acknowledges for the first time the sound results of Villarceau. It seems like giving his blessings and asking the academy to sanction them:

Our colleague M. Yvon Villarceau, in his wonderful researches on the double stars, has been led to search for the expression of the central force that, directed towards a given point of the plane, can make possible an ellipse, and the expression which he has obtained, and which is inserted in the Additions to Connaissance des Temps for 1852, coincides precisely with one of the solutions of the problem I proposed (Bertrand, 1877b)

Once the names of Bertrand, Darboux and Halphen were involved, the events started picking momentum. Other schools of thought pitched in with very important results, on which it is not the case to insist right now. Over a period of two-three decades the problem was hot. However, again as far as we can judge today, it has none of the gnoseological consequences one can think it was called to trigger. The Pioneer anomaly seems a good opportunity to review at least a few of those consequences.

Triggered by the works of the French school, was an article of James Whitbread Lee Glaisher (Glaisher, 1878) whose essential task was to put in analytical form some consequences of Newton’s Proposition VII, Corollary 3 (Newton, 1995, p. 48). Just by doing this Glaisher was able to recover all the results of Villarceau, Bertrand, Darboux, Halphen and also a previous result known today as the theorem of Hamilton, with which all the others are equivalent. Careful, like Villarceau three decades before, in order not to raise susceptibilities, but somehow ironic, Glaisher makes a subtle observation

It is not unlikely that Sir W. R. Hamilton deduced his law from Newton’s Prop. xvii., and it is curious that the general formulae which are deducible from the translation of Newton’s results into analysis should not have been examined (Glaisher, 1878, p. 85)

Fact is that in such occurrences the science appears to be a pure tautology: nothing has been discovered over what Newton had already analyzed; the things are just expressed some other way. However, bearing in mind that Bertrand, Darboux and Halphen started from the second principle of dynamics, this occurrence may be taken as a proof of consistency of the Newtonian philosophy as a whole. It also shows that Newton was the only one to correctly grasp the whole implication of his “invention” of the centripetal forces. Once, however, the analysis undertook the concept, the imagination was unleashed and the roots have been easily forgotten. The force with magnitude inversely proportional with the square of distance became the “force of nature”. Denying it in any way was a dangerous thing to do for anyone who would think to pursue an academic career. Even Newton himself was, at times, overwhelmed by the misinterpretations of those promoting his very ideas. One exceedingly good thing though, is that the concept of “force of nature” eventually produced the general relativity.

Coming back to the problem at hand, we need to make a point: why inventing new explanations, while we did not understand thoroughly how the old explanations really work?! Case in point: the Pioneer spacecrafts have reached their escape velocities by a kind of “sling shot” effect, i.e. assisted by the gravitational attractions of Jupiter and Saturn, which projected them in the deep space. Assuming that the only effect having influence on the trajectory of spacecrafts is that of the last encounter from the solar system, allows us to reformulate the Corollary 3 of the Proposition vii from the first book of Principia, for instance in the form

The force which attracts the spacecraft to the Sun is to the force which attracts the spacecraft to Saturn (or Jupiter) etc. etc…

Then the force of attraction towards the last planet appears naturally as a perturbation, even a periodic perturbation, of the main force which is directed towards the Sun. Obviously if the main acceleration points towards the Sun, the excess acceleration will point in the same general direction, and we may even be able to find its periodicity. In that case we can very well check if the above reformulation of the Newton’s Corollary 3 for the problem of Pioneers is correct, i.e. it refers indeed to the San and Saturn or Jupiter, for the periodicity must follow closely the motion of the source of perturbation.

While we are at it, let’s notice that another anomaly might have its natural explanation: there is a tiny but consistent difference between the velocities pre- and post-perigee of the flyby spacecrafts (Anderson, Nieto, 2009). Because it is always assumed that the motion in gravitational field must satisfy the conservation of energy, this is indeed an anomaly. However, if we follow correctly the Newton’s Corollary 3 of the Proposition vii, the resulting force is not conservative but in special situations in which the perigee and apogee play an important part. Thus, the tiny deference in velocities of the spacecraft, while reflecting the non-conservation of energy for the motion in gravitational field, appears, in this old light, as quite natural.

References

Anderson, J. D. , Nieto, M. M. (2009): Astrometric Solar-System Anomalies, arxiv: 0907.2469 v1

Bertrand, J. (1877a): Sur la Possibilité de Déduire d’une Seule des Lois de Kepler le Principe de l’Attraction, Comptes Rendus de l’Académie des Sciences Paris, Vol. 84, pp. 671 - 674

Bertrand, J. (1877b): Note sur un Problème de Mécanique, Comptes Rendus de l’Académie des Sciences Paris, Vol. 84, pp. 731 - 732

Darboux, G. (1877): Recherche de la Loi que Doit Suivre une Force Centrale pour que la Trajectoire qu’elle Détermine Soit Toujours une Conique, Comptes Rendus de l’Académie des Sciences Paris, Vol. 84, pp. 760 - 762; 936 - 938

Glaisher, J. W. L. (1878): On the Law of Force to any Point in the Plane of Motion, in order that the Orbit may be always a Conic, Monthly Notices of the Royal Astronomical Society, Vol. 39, pp. 77-91

Halphen, G-H. (1877): Sur les Lois de Kepler, Comptes Rendus de l’Académie des Sciences, Paris, Vol. 84, pp. 939-941

Jammer, M. (1957): Concepts of Force, Harvard University Press, Cambridge, Massachusetts

Newton, I. (1995): The Principia, Prometheus Books, Amherst, New York

Turyshev, S. G., Toth, V. T. (2009): The Pioneer Anomaly in the Light of New Data, arxiv: 0906.0399 v1

Villarceau, Y. (1850): Memoires et Notes sur les Etoiles Doubles, Paris, Bachelier