A POSSIBLE RELATION BETWEEN PLANETARY DISTANCES
AND THE 160-MINUTE SOLAR PULSATION
V. A. Kotov and S. Koutchmy*
Izvestiya Kryniskoi Astrofizicheskoi Observatorii,
Vol. 72, pp. 199–208, 1985
The discovery of global pulsations on the Sun with period Po = 160 min  enables us to consider a characteristic wavelength for the solar system L =cP0=19,24 a.u., where c is the velocity of light. The planetary distances show a statistically significant quasicommensurability between L and 2pai for the inner planets or between 2ai and L for the outer ones (ai is the major semiaxis of the orbit). This L commensurability leads to a new approach to the Titius-Bode planetary distance law. The physical mechanism responsible for this L commensurability in the solar system is evidently related to gravitational waves from an external source of unknown nature.
It is generally recognized that the distribution of the planets is not random and provides information on the formation mechanism and evolution of the solar system [1,2]. In many theories of the origin of the solar system, attempts are made to derive a planetary-distance law analogous to Bodes law, but none of these formulations can be taken as satisfactory. Alven and Arrhenius  criticized Bodes law but at the same time recognized that the solar system has regular structure and dynamics; they consider that resonant phenomena must play an important part in establishing the regularity, which is evidently also reflected in an exponential law of Bodes type, as well as the commensurability of many motions within the solar system.
Two average motions are taken as commensurable, or two orbital periods, if the ratio of them can be represented as the ratio of two quite small integers [1,2]. In the discussion of the reasons for commensurabi1ity, the main hypothesis has been that it is due to the direct gravitational interaction of the bodies during the evolution of the solar system or a result of tides arising on planets and satellites. Nieto  gives strong arguments for dividing the entire search for a physical basis for the law into two parts: on the one hand, one has to explain the geometrical progression in the planetary distances, and on the other,the commensurability. Nieto , Dermott , and Ovenden  concluded that the tidal theory has no basis in explaining the regularity in the planetary distances, since it requires time to establish commensurability much greater than the time for which the planetary system has existed. In  it is shown that Bodes law reflects the principle of minimal interaction and is simply a result of the gravitational attraction between the planets, so it provides no information on the formation conditions. In turn, Hills numerical calculations  imply that commensurability in the periods of rotation around the central body is the final and most important feature of a system of strongly interacting bodies, and a state with commensurability is the most stable state out of all the possible configurations for the orbits; therefore, Bodes law is a natural expression of the commensurability. However, Hills approach involves the following difficulty: relaxation in a system of gravitating bodies in general inevitably leads to unbounded increase in the orbital eccentricities, which is unrealistic.
According to Molchanov , the planetary-distance law follows from simple linear (resonant) relations between the frequencies of rotation of the planets. However, later  it was shown that the set of relationships (correlations) considered in  is statistically unreliable.
*Astrophysics Institute, Paris, France.
The partial solutions for resonance and commensurability are based on the fruitful idea of tidal interaction but do not give an exhaustive answer to the origin of quasicommensurability, traces of which can be found virtually throughout the solar system.
The discovery of the 160 min pulsations on the Sun [10,11] in our view provides a new and very attractive means of relating the structure of the planetary system to a detailed phenomenon actually observed: the periodic pulsation in the central body.
1. SOLAR PULSATIONS
In 1974, Severnyi et al.  observed periodic oscillations in the solar photosphere, which extended to virtually all the visible hemisphere of the Sun and had a period of about 160 min. This was very shortly afterwards confirmed by others [12-14]. Observations over nine years in the Crimea  refined the value of the period, which is
P0 = 160m,0101±0m,0007. (1)
The nature of this oscillations is a difficult problem for the theory. It is possible to explain it as resonance between gravitational (nonradial) g modes and certain combinations of them of gi- gk type for a solar model  with low initial heavy-element content Z0 = 0.001 .
In [17,18], it has been suggested that the 160 min period on the Sun is a relict phenomenon, i.e., has existed for 108-109 yr, namely over a time-scale comparable with the time of substantial evolution in the solar system. In that case, it is quite reasonable to suppose that this period may play a considerable part in establishing the regularity and commensurability in our system. As regards the physical mechanism, one can evidently envisage periodic fluctuations in the gravitational field near the Sun accompanying the 160-min pulsations.
There have been many suggestions that one could observe gravitational perturbations (waves) from studies on the entire solar system or some parts of it acting as trial bodies, in particular from Weber , Braginskii , and Dicke .
Savin  suggested an important role for a 160 min period in the structure and evolution of the solar system long before the actual discovery  of the global pulsation on the Sun; in 1946 he said that the period of the natural vibrations of the Sun, so to say, the period of its infrasound (1/9 day), plays an important part in the distribution of the outer planets. It is true that now it is almost impossible to establish how he derived this conclusion on the period of the Sun of 1/9 day (160 min);it seems that it was an intuitive guess supported by analysis of the periods of rotation of the planets.
Another important point is the statistically significant commensurability in the mean periods of axial rotation of the planets and asteroids, period 160 min . This forces one not only to see a physical mechanism for this preferred synchronization but also other possible signs of the fundamental periodicity P0within the solar system.
A discussion of the possible relationship between this 160 min period and the structure of the system as a whole leaves on one side the nature of P0 and the physical process that determined the commensurability (of the axial rotation speeds ) with P0. The treatment is basically statistical and involves the following three major postulates:
1) the period P0 = 160.010 min of the Sun is a relict phenomenon;
2) periodic perturbations in the gravitational field are, or have been, related to this period; and
3) these perturbations have a wavelength L determined by P0 and the propagation speed c, which is equal to the speed of light (L = cPo).
The theory of gravitation and gravitational radiation implies that the equations for weak gravitational fields are analogous to Maxwells wave equations for the electromagnetic field. Einstein postulated that gravitational waves propagate with the speed of light; since then, the speed of light c as a universal constant has formed the basis of the special theory of relativity and the theory of gravitation. Correspondingly, we introduce the wavelength
We further suppose that L is a certain characteristic scale that may be important in establishing the final dimensions of the system. It is then logical to compare L with the actual distances within the planetary system, which are comparable with L as regards order of magnitude.
2. COMPARABILITY OF L AND THE ORBIT DIMENSIONS
A circular orbit is the limiting case of an ellipse with eccentricity e tending to zero. Therefore, the motion of the planets around the central body in eccentric orbits can be represented as equivalent circular motions occurring in a certain preferred plane and perturbed by radial oscillations with period Tn, where Tn is the period of rotation of planet number n around the Sun. The perturbations that transform a circular orbit into an elliptical one are characterized by the length zn ~ 2pan(an is the major semiaxis of the orbit). On the basis of resonance, it is reasonable to assume that zn may be commensurable with L , as may be the radius an and the diameter of the orbit 2an.
We now enumerate all combinations of ratios of the type x1/x2 between these three orbit parameters (an, 2an, and zn) and L :
To establish whether x1 and x2 are comparable on average for the 10 objects in the solar system (the nine planets and the asteroid belt), we impose three conditions:
where N is the number of objects.
The summation may be taken directly for all 10 objects at once or separately for the five inner planets (including the asteroid belt) and the five outer ones. The division of the objects into two groups is quite natural by virtue of the marked difference in physical characteristics between the planets of the terrestrial group and the giant ones, as well as the differences in evolutionary history.
Condition (4) is obvious, since we are interested in x1/x2 being close to integers; (5) restricts the order of the commensurability, while (6) is an obvious requirement for the mean multiplicity (commensurability) between x1 and x2, where Zn are integers close to (Xl/X2)n. It is evident from (3) that L does not show preferred quasicommensurability with any of the three orbital parameters (an,2an, and 2pan) for all objects taken simultaneously in accordance with (4)-(6); instead, there is quasicommensurability only between L and 2pan for the five inner planets or between L and 2an for five outer ones; the values of y are correspondingly 0.118 and 0.217.
The quasicommensurability is illustrated in more detail in Table 1, which gives not only the mean distances from the Sun (the an of ) for the 10 main objects but also the L /2pan for the inner planets and 2an/L for the outer ones. All the values are close to small integers (less than 10): C(x1/x2)n- Zn C<1/4, apart from Jupiter, for which 2a6 /L y1/2.
Instead of the major semiaxis an, we can consider the so-called equivalent radius of the orbit rn, which by definition is equal to the radius of a circle whose area is equal to that enclosed by the elliptical orbit:
rn = an (1-e2)1/4 (7)
The average commensurability of 2prn or 2rn with L is then substantially higher than that in Table 1, mainly on account of the orbital parameters of Mercury and Pluto, which are the two planets with the largest eccentricities (these correspondingly give L /2pr1 = 7,996, 2r10/L = 4,034).
3. COMMENSURABILITY SPECTRUM
The main difficulty in demonstrating any planetary distance law lies in determining the statistical significance of the approximation for the observed an. We have 10 values of an and can obtain almost any regularity with the probability practically equal to one. The values of L /2pan and 2an/L given in Table 1 readily give us the probability of obtaining this close correspondence to integers by accident:
In all previous studies on the planetary distances, it has been usual to find the best approximation to 10 or less values of an with a parametric function containing two or more constants, whose form is not known a priori; this is dependent on several untestable and usually unreliable assumptions. An advantage of our argument is that this commensurability study is based on the advance assumption of a unique geometrical scale L , and it will be seen from what follows that this enables us to evaluate the reliability of the result more or less rigorously, namely the quasicommensurability of the orbital parameters with L .
Table 1 shows that 2an and 2apn tend to be commensurable with L , so we are justified in expecting a maximum commensurability at the frequency f0 = c/L = P0-1. One can calculate the commensurability function  in order to determine whether f0 in fact corresponds to the best commensurability (i.e., the turning point in a certain function) and to determine the level of statistical significance for the maximum.
We assume that the commensurability of the an with L (Table 1) is accidental. Then one should assume that in a reasonably chosen range of wavelengths l(or frequencies f = c/l), there exists certain values l‘that show average quasicommensurability with the 10 values of an better than the L = 19,24 a.u. (frequency f0 = l04.l6 mHz).
We have introduced  the commensurability function F(f) for certain constants kn with the variable f , which shows for what values f ‘ there is preferred commensurability for the entire set of kn, i.e., scope for approximating kn/f by integers more or less satisfactorily. The meaning of F(f) is entirely analogous to that of the power spectrum of a time-varying signal, so F(f) is naturally called the commensurability spectrum.
We construct the following two F(f) separately for the planets of the terrestrial group (n 1,…, 5) and the outer ones (n = 6,…, 10):
Here the Zn are integers closest to the x1/x2, while for Jupiter, to retain the generality, we take 2a6instead of a6(see below on the same on replacing the real an by random numbers); l = c / f , where f is the traveling frequency and b = (12)- 1/2 = 0,2887 .
Figure 1 shows F1calculated in the range of periods from about 10 to 460 min with a step DP = 2.5 min; there are two peaks with identical amplitudes A = 0.173 y 2.8s , which correspond to periods of about 159 min and about 305 min (±4 min, the error corresponding to the peak width at the 2s level). The value of s is determined here, as everywhere, by calculating F( f ) for uniformly distributed random numbers Rn replacing the actual an.
Figure 2 shows F2for the five outer planets; frequency step D f = 2 mHz. The largest peak with amplitude about 3.5s corresponds to a period of 163 min (±3 min), while the second peak for P y 82 min has an amplitude less than 3s and is an artefact of the quasicommensurability for P y 163 min (see also ).
The two figures show that there is a single period of about 160 min in the entire frequency range that gives a dominant and statistically significant quasicommensurability with the parameters 2pan(n = 1,…,5) and 2an(n = 6,…,10) simultaneously. As the amplitudes A of the peaks (about 160 min) in Figs. 1 and 2 are >2.3s and >3.2s, we get the probability of accidentally finding two peaks at the same frequency simultaneously as
Interest also attaches to the form of the average function F = (F1 + F2)/2, which is shown in Fig. 3. The calculations were made for periods from about 56 min to about 1440 min, and for P P 470 min, where F1is not defined, we took only F2, i.e., here we assumed F = F2. The F(f) is then dominated by quasicommensurability for the period 160 min ( 3 min), peak amplitude about 4.3s and probability of accidental occurrence at that frequency about l0-5.
To illustrate the preferred commensurability with L for 2prn and 2rn. where rn are the equivalent orbital radii, we calculated a function for all 10 planets:
(for Jupiter, we took 2r6 instead of r6). Figure 14 shows the spectrum, where the largest peak has amplitude A(160m)y4,3 .
4. PLANETARY – DISTANCE REGULARITY
The fullest survey of the topic is given by Nieto , who gives the history of Bodes law and subsequent modifications, as well as theories of the origin of the solar system. He shows convincingly that none of the proposed theories gives an exhaustive explanation of the geometrical progression in Bodes law, although many of them are more or less compatible with a progression. In it is emphasized that none of the formulations of this law has any physical significance. Nevertheless, the distribution of the planets is one of the major features of the solar system, since for example there is a clear-cut linear relationship between the logarithm of the specific momentum in the planets and the logarithm of the distance from the Sun; there is also a close correlation between the mean densities of the planets and the gravitational potential energies GM@ / an, where M@ is the mass of the Sun and G is the gravitational constant .
The result of Figs. 3 and 4 emphasizes the importance and unique character of L and thus of the period of 160 min for the planetary distribution in our system. From Table 1 we propose the following formula for the planetary distances:
(for the terrestrial-group planets at the top and for the outer planets at the bottom, beginning with Jupiter). The formula represents the actual an with a cross-correlation coefficient of 0.9998 and a relative standard deviation of 4%. It has an undoubted advantage over all other formulations of Bodes law, since there is no adjustable parameter, and it follows from the natural and statistically sound requirement of quasicommensurability with a characteristic scale known in advance.
5. DISCUSSION OF THE RESULTS
It has been shown that the 160 min pulsation on the Sun [10,11] is important not only to the distribution of the axial periods of rotation for the planets and asteroids  but also for the planetary distance sequence; the analysis is based on the requirement of quasicommensurability for the observed distances with the wavelength L = c.P0. The gravitational field is the only physical field whose perturbations propagate with the speed of light and which in our view could lead to such commensurability within the evolution time of the planetary system (about l08-l09 yr).
The law of (13) differs favorably from other such laws; it contains the observed period of oscillation of the Sun P0and the fundamental velocity c. The formulation is based also on the hypothesis of average quasicommensurability, which gives us the numbers Zn automatically (see Table 1 and (13)) without making any other assumptions; the enumeration of the objects by the Zn is objective and based on the quasicommensurability itself, while in ordinary formulations of Bode’s law, the orbits are usually enumerated by the natural number series n = 1,2,…;insome formulations, however, deviations from this sequence of n are allowed, such as omitting certain numbers or even a reverse enumeration order.
If we express the an in terms of the mean orbital velocities vn and the frequencies of rotation around the Sun Fn = Tn-1, the conditions of (13) of the inner and outer planets become
where F0 = P0-1 is the frequency of the Suns oscillation, while n and Zn are the numbers given in Table 1.
We therefore have to seek a common cause for this commensurability, namely of P0with the periods of rotation of the various bodies in the solar system about their axes  and commensurability with L = c.P0 for the orbital parameters, which we see in gravitational waves. It has previously been pointed out repeatedly [19-21] that gravitational waves could have an appreciable effect on the solar system if they are of sufficient amplitude.
Here we may note that the prediction from the general theory of relativity that gravitational waves may be radiated has received some confirmation in l974, when it was observed that the orbital period of the pulsar PSR 1913+16 was gradually decreasing, this being part of a binary system, and which occurred on account of loss of gravitational radiation .
The source of the possible gravitational waves might be the Sun itself (which we consider unlikely) or some external object near the solar system such as a massive binary system having a period of rotation 2 P0 y 320 min, angle of inclination of the plane of the orbit i K 0, and sufficiently close to the Sun (it was first suggested in [17,18] that there is a possible evolutionary relation between the 160 min oscillation of the Sun and perturbations of the gravitational field near it).
Many calculations have been performed [19-21,25,26] on the energy flux incident on the solar system from a binary one. Consider, for example, a binary system composed of two massive collapsed objects with masses m1and m2. If the system has a circular orbit with radius r and angular velocity of one component around the other w, the energy flux from gravitational waves near the Sun is 
where R is the distance from the binary system to the Sun. Taking m1~m2~M@, w = p/P0 ~ 3,3•104 sec-1 and R y 10 parsec, we get r ~(GM@ / w2 )1/3 ~ 1011 cm, and for I we get the very small value O2.4.l0-10 erg.cm-2.sec-1. If we assume that the components in the system are black holes (F. Delache, 1983, personal communication) with masses about l03 M@ each, then we get the dimensions of such a system as r y 1012 cm and I O2.4 erg.cm-2.sec-1, i.e., comparatively large. The energy losses by gravitational radiation are DE = 4pR2I ~ 3.1040 erg.sec-1, and the lifetime of the system is t O2.103 M@c2/DE~4.109 yr, which is comparable with the time of existence of the solar system. This problem requires a special discussion, which should cover the possibility of such a massive binary system existing at all near the Sun, its real lifetime (<t) and the identification with some peculiar object.
If there is an external gravitational-radiation source, the observed oscillation with P0 = 160.010 min is to be considered as a forced oscillation, which is the more interesting because the region of periods around 160 min is according to Severnyi et al.  the most preferable for the Sun for resonant interaction and excitation of certain gravitational g modes. Resonant energy transfer from one mode to another may occur in the presence of a weak but permanently acting external perturbation at frequency P0-1, which would lead to appreciable amplification in this oscillation by comparison with all other g modes.
The authors are indebted to A.B. Severnyi for some critical comments, which led to substantial improvements in the paper, and we are also indebted to A. Wittman (Gottingen) for a discussion of the statistical significance of the commensurability found in the solar system, to A. G. Kosovichev for valuable comments, and to F. Delache (Nice), who provided some of his results before publication.
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