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

UDC 523.9--1/8;523.214:530.12

The discovery of global pulsations on the Sun with period P_o = 160 min [10] 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 2pa_i for the inner planets or between 2a_i and L for the outer ones (a_i 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 [3] 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 [4] 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 [4], Dermott [5], and Ovenden [6] 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 [6] 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 [7] 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 [8], the planetary-distance law follows from simple linear (resonant) relations between the frequencies of rotation of the planets. However, later [9] it was shown that the set of relationships (correlations) considered in [8] is statistically unreliable.

*Astrophysics Institute, Paris, France.

© 1987 by Allerton Press, Inc.

1. SOLAR PULSATIONS

In 1974, Severnyi et al. [10] 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 [15] refined the value of the period, which is

P₀ = 160^m,0101±0^m,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 g_i- g_k type for a solar model [16] with low initial heavy-element content Z₀ = 0.001 [15].

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 [19], Braginskii [20], and Dicke [21].

Savin [22] suggested an important role for a 160 min period in the structure and evolution of the solar system long before the actual discovery [10] 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 [18]. This forces one not only to see a physical mechanism for this preferred synchronization but also other possible signs of the fundamental periodicity P₀ within 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 [18]) 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 = cP_o).

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

(2)

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 T_n, where T_n 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 z_n ~ 2pa_n(a_n is the major semiaxis of the orbit). On the basis of resonance, it is reasonable to assume that z_n may be commensurable with L , as may be the radius a_n and the diameter of the orbit 2a_n.

We now enumerate all combinations of ratios of the type x₁/x₂ between these three orbit parameters (a_n, 2a_n, and z_n) and L :

(3)


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 Z_n are integers close to (X_l/X₂)_n. It is evident from (3) that L   does not show preferred quasicommensurability with any of the three orbital parameters (a_n,2a_n, and 2pa_n) for all objects taken simultaneously in accordance with (4)-(6); instead, there is quasicommensurability only between L   and 2pa_n for the five inner planets or between L   and 2a_n 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 a_n of [23]) for the 10 main objects but also the L /2pa_n for the inner planets and 2a_n/L  for the outer ones. All the values are close to small integers (less than 10): C(x₁/x₂)_n- Z_n C<¹/₄, apart from Jupiter, for which 2a₆ /L  ^y1/₂.

Instead of the major semiaxis a_n, we can consider the so-called equivalent radius of the orbit r_n, which by definition is equal to the radius of a circle whose area is equal to that enclosed by the elliptical orbit:

r_n = a_n (1-e²)¹/₄             (7)

The average commensurability of 2pr_n or 2r_n 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 a_n. We have 10 values of a_n and can obtain almost any regularity with the probability practically equal to one. The values of L  /2pa_n and 2a_n/L  given in Table 1 readily give us the probability of obtaining this close correspondence to integers by accident:

           (8)

whereis the number of combinations of 10 elements taken seven at a time.

In all previous studies on the planetary distances, it has been usual to find the best approximation to 10 or less values of a_n 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 2a_n and 2ap_n 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 [18] 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 a_n 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 a_n better than the L = 19,24 a.u. (frequency f₀ = l04.l6 mHz).

We have introduced [18] the commensurability function F(f) for certain constants k_n with the variable f , which shows for what values f ' there is preferred commensurability for the entire set of k_n, i.e., scope for approximating k_n/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 Z_n are integers closest to the x₁/x₂, while for Jupiter, to retain the generality, we take 2a₆ instead of a₆ (see below on the same on replacing the real a_n by random numbers); l = c / f , where f is the traveling frequency and b = (12)- ^1/₂ = 0,2887 [18].


Figure 1 shows F₁ calculated 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 R_n replacing the actual a_n.



Figure 2 shows F₂ for 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 [18]).



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 2pa_n(n = 1,...,5) and 2a_n(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

                     (11)

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 F₁ is 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  2pr_n and 2r_n. where r_n 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(160^m)y4,3 .

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 a_n 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 [18] 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 P₀ and the fundamental velocity c. The formulation is based also on the hypothesis of average quasicommensurability, which gives us the numbers Z_n automatically (see Table 1 and (13)) without making any other assumptions; the enumeration of the objects by the Z_n 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 a_n in terms of the mean orbital velocities v_n and the frequencies of rotation around the Sun F_n = T_n^-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 Z_n are the numbers given in Table 1.

We therefore have to seek a common cause for this commensurability, namely of P₀ with the periods of rotation of the various bodies in the solar system about their axes [18] 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.

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 m₁ and 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 [26]

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. [15] 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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11 November 1983

*Bulletin of the Crimean Astrophysical Observatory (USSR Academy of Sciences). Available from Allerton Press, Inc., 150 Fifth Avenue, New York, NY 10011; (212) 924-3950.