IS SOLAR SYSTEM STABLE? - CiteSeerX

IS SOLAR SYSTEM STABLE? A REMARK Vladik Kreinovich, Andrew Bernat Computer Science Department, University of Texas at El Paso, El Paso, TX 79968, USA Abstract. It is not yet known (1994) whether the Solar System is stable or not. Common belief is that the Solar System is stable if and only if itP is not a resonant P system, i.e., whenever its orbital frequencies ωi satisfy an inequality | ni ωi | ≤ ε for i |ni | ≤ N , a similar inequality is true for randomly chosen frequencies. In this paper, we show that the Solar system does not have such resonances, and therefore (if the above-mentioned belief is correct), we have strong reasons to believe that it is stable. What do we mean by stable? During the thousands of years that we humans have been observing the Solar system, it has been stable in the sense that planets did not fly away and did not collide. However, computer experiments show that systems described by equations of celestial mechanics can be unstable: planets can all of a sudden collide and/or fly away. Will it ever happen to the Solar system? In other words, is the Solar system stable in this sense? Theoretical results on stability. The only cases when stability or instability have been theoretically proven are some simple 3–body systems [M73], [M74], [M78], [LR87], [LR90], [LR91] (the last three papers use interval mathematics).PSuch systems turn out to be: • unstable if there is aPresonance, i.e., a relationship i ni ωi = 0, where ni are integers with a small sum i |ni |, and ωi = 2π/Ti are frequencies of the planets’ orbital motions (Ti is the period of i−th planet); • stablePif there is no such resonance. The sum i |ni | is called an order of this resonance. Experimental data on stability. For more complicated systems, we only have computer experiments to judge whether a system is stable, or collisons and/or fly-aways will occur. It turns out (see, e.g., [BC61], [D75]) that if we start with a system of almost circular orbits, and choose ωi at random, then this system is usually stable (at least long computer simulations do not lead to any collisions or fly-aways). If, on the other hand, P we start with a system in which the orbital frequencies are close to a resonance (e.g., i ni ωi ≈ 0 for some small ni ), then this system is usually unstable. The main idea of checking stability: informal description. With this empirical fact in mind, in order to check whether the Solar system is stable or not, we must check whether its set of orbital frequencies is close to a resonance. How can we express this “closeness” in precise mathematical terms? We can say that a system is close to a resonance if there existP such ε > 0, k, andPN that: • the observed frequencies ωi satisfy k relation of the type | i ni ωi | ≤ ε with i |ni | ≤ N , and • randomly chosen frequencies (in some reasonable sense) do not have that many ε−approximate resonances of order ≤ N . 1

Molchanov’s results. For 9 planetary frequencies of the Solar system, Molchanov has found eight ≈ 1.5%−approximate resonances of order ≤ 9 [M66]: –1 1 2 1 0 0 0 0 0 0 –1 0 3 0 1 0 0 0 0 0 –1 2 –1 1 –1 0 0 0 0 0 1 –6 0 –2 0 0 0 0 0 0 –2 5 0 0 0 0 0 0 0 –1 0 7 0 0 0 0 0 0 0 0 –1 2 0 0 0 0 0 0 0 –1 0 3 For example, the first line represents the relation −ω1 + ω2 + 2ω3 + ω4 ≈ 0. Comment. Since we have 8 linear equations that relate 9 frequencies, this list of resonances is complete: indeed, if we had a 9-th relation that was independent from these 8, then we would have 9 independent linear equations with 9 unknowns ωi , and thus, we would have ωi = 0 for all i. Similar approximate resonances have been discovered for satellite systems of big planets [M66]. Molchanov claims ([M66], [D75]) that the probability of these particular relations happening for random data is ≤ 10−13 and therefore, our Solar system is unstable. A problem with Molchanov’s results. Molchanov’s arguments are based on the following idea: he fixes the values ni from the above table, and estimates the probability that randomly chosen frequencies satisfy these 8 relations. There is a flaw in this argument: yes, randomly chosen frequencies do not satisfy these approximate resonance relations for values ni from the above table, but maybe, they satisfy similar relations with other (still small) values of ni ? What are we planning to do? In this paper, we plan to show that Molchanov’s approximate resonances are typical. Therefore, based on the above-described empirical stability criterion, we can conclude that the Solar system is stable. Comment. These results were partially announced in [K80] and [K89]. The problem that we want to solve is typical for interval data processing. In many real-life situations, we want to know whether a certain relation holds. For example, if we analyze a picture with one object on front of several others, then we would like to know whether two lines coming behind the front object are actually one line interrupted by this front object, or they are different lines. If we had the exact straight lines, and if all the measurements were absolutely precise, then it would be easy to solve this problem: we just check whether the coefficients of these two straight lines coincide. In reality, measurements are not precise, and lines are not strictly linear. Therefore, we get only approximate values of the coefficients. In this case, our answer depends on how close these values are: • If the difference between the coefficients is so small that it cannot be explained as a random coincidence, then we conclude that this is one and the same line. • If, however, the difference is such that in a random picture with several random lines, two of them may have their parameters as close to each other as the two given lines, 2

then probably these two given lines are of different origin (and the closeness of their coefficients is a random coincidence). Unfortunately, although this general problem is typical in applications, no general methods have been proposed to solve it, and moreover, no general mathematical formulation has been proposed. In this paper, we propose a formulation and solution for one particular problem (the problem of stability of solar system) in the hopes that eventually, a general formulation and a general solution will appear. Basic astronomical data. Before we can formulate the problem in mathematical terms, let us recall the basic formulas of celestial mechanics: 3/2 • According to Kepler’s law, the period Ti of an orbital notion is proportional to ri , where ri is the distance between the Sun and i−th planet. Therefore, ωi = 2π/Ti ∼ −3/2 . ri • According to Bode’s law (see, e.g., [K74]), ri ≈ ai for some a ≈ 2. Historical comment. Bode’s law was first discovered for the planets. It works well if we place a hypothetic additional planet in between Mars and Jupiter. This result prompted the search for this extra planet; in its place, “small planets” (asteroids) were discovered. When later on satellites of the big planets were discovered, Bode’s law turned out to be an excellent fit for them as well. From Bode’s and Kepler’s laws, we conclude that ωi ∼ (a3/2 )−i , and therefore, √ ωi /ωi+1 ≈ a3/2 . For a = 2, this ratio is 2 2 ≈ 2.8. For the majority of consequent pairs, this ratio is between 2 and 3. Now, we are ready to formulate the main result. PROPOSITION. If ω1 > ω2 > ω3 > ω4 > ω5 > 0,P and 2 ≤ ωi /ωi+1 P < 3 for i = 1, 2, 3, 4, then there exist integers n1 , ..., n5 such that n1 6= 0, i |ni | ≤ 9, and | i ni ωi | ≤ 0.017 ω1 . Comment. In other words, for every five consequent planets from a typical planet system, there exists a resonance relation of order ≤ 9 that is satisfied with accuracy 1.7% (≈ 1.5%). This means that the existence of such resonances is a typical rule, not an exception, and the fact that these resonances (and not those of smaller order) are satisfied by the Solar System means that our Solar System is typical, not an exceptionally resonant one. The proof of the Proposition is based on the following lemma: LEMMA. Assume that ω1 > ω2 > .. > ωN > 0, N > 2, are positive real numbers. Then, there exist integers n1 , ..., nM such that n1 6= 0, |

N X

ni ωi | ≤

i=1

and

N X

1 ωN , 2

N X ¥ ωi−1 ¦ |ni | ≤ 1 + . ω i i=1 i=2

3

Comment. Here, bxc denotes an integer part of a real number x. Proof of the Lemma. 1. Let us denote pi = bωi−1 /ωi c. Let us consider the sums n2 ω2 +n3 ω3 +...+nN ωN , where 0 ≤ ni ≤ pi for all i. To these sums, we will also add a term (p2 + 1)ω2 (that corresponds to n2 = p2 + 1, n3 = ... = nN = 0). 2. Let us order these sums in lexicographic order of the sequences n2 , ..., nN . In other words, let’s first take the sums that correspond to (0, ..., 0, 0), (0, ..., 0, 1), ..., (0, ..., 0, pN ), then (0, ..., 0, 1, 0), ..., (0, ..., 0, 1, pN ), (0, ..., 0, 2, 0), etc, with (p2 , ..., pN ) and (p2 +1, 0, ..., 0) as the last terms. 3. Let us show that the difference between each 1 , ..., nN ) and the next sum P sum (nP (m1 , ..., mN ) in this sequence is ≤ ωN , i.e., that i mi ωi − i ni ωi ≤ ωN . Indeed, if nN < pN , then the next sum is obtained by adding 1 to nN , and therefore, the difference is equal exactly to ωN . If nN = pN and nN −1 < pN −1 , then the next sum corresponds to (n1 , ..., nN −2 , nN −1 + 1, 0), and the difference is equal to ωN −1 − pN ωN . Since pN is defined as an integer part of the ratio ωN −1 /ωN , we can conclude that ωN −1 < (pN + 1)ωN , and therefore, ωN −1 − pN ωN < (p + 1)ωN − pN ωN = ωN . If nN = pN , nN −1 = pN −1 , and nN −2 < pN −2 , then the next sum corresponds to (n1 , ..., nN −3 , nN −2 + 1, 0, 0), and the desired inequality can be reduced to ωN −2 − pN −1 ωN −1 − pN ωN ≤ ωN . To prove this inequality, it is sufficient to use the fact that ωN −1 < (pN −1 + 1)ωN −1 . Therefore, ωN −2 − pN −1 ωN −1 − pN ωN < (p + 1)ωN −1 − pN −1 ωN −1 − pN ωN = ωN −1 − pN ωN , and we have already proved that this expression is ≤ ωN . Similarly, we can prove this statement for all possible cases. 4. Let us now prove that for every x ∈ [0, (p2 + 1)ω2 ], there exists a sum s = which |x − s| ≤ 0.5 ωN .

P

ni ωi for

If x ≤ 0.5ωN , then the desired inequality is true for s = 0 = 0 · ω2 + ... + 0 · ωN . So, it is sufficient to consider only the case when x > 0.5 ωN . In this case, for the first sum s = 0, we have x − s > 0.5 ωN . Let us compare x with all the sums s(1) = 0, s(2) , ..., s(S) = (p2 + 1)ω2 in the above-described lexicographic order (here, S denotes the total number of such sums). We already have x > s(1) + 0.5 ωN . Since we assumed that x ∈ [0, (p2 + 1)ω2 ], we have x ≤ s(S) = (p2 + 1)ω2 . So, we cannot have x > s(S) + 0.5 ωN . Let us denote by k, the ordinal number of the first sum for which x 6> s(k) + 0.5 ωN (i.e., x ≤ s(k) + 0.5ωN ). Since k is the first such number, we have x > s(k−1) + 0.5 ωN . According to 3., s(k) < s(k−1) + ωN . 4

Therefore, s(k−1) > s(k) − ωN , and from x > s(k−1) + 0.5 ωN , we can conclude that x > s(k) − ωN + 0.5 ωN = s(k) − 0.5 ωN . So, s(k) − 0.5 ωN < x ≤ s(k) + 0.5ωN , and |x − s(k) | ≤ 0.5 ωN . 5. Now, we are ready to prove the Lemma. Similarly to 3., we can prove that ω1 ≤ (p+2)ω2 . Therefore, if we take x = ω1 in 4., we PN can conclude that |−ω1 + i=2 ni ωi | ≤ 0.5 ωN , where either ni ≤ pi for all i, or n2 = p2 +1 and n3 = ... = nN P = 0. In other P words, we have a resonance-type relation with n1 = −1. Let us show that i |ni | ≤ 1 + pi . Indeed, ifPni ≤ pi for all i, then this inequality is P n evidently true. If n2 = p2 + 1, n3 = ... = 0, then i=2 ni = p2 + 1 ≤ p2 + p3 ≤ i pi . The Lemma is proven. Proof of the Proposition. Now,Pwe can provePour Proposition. Indeed, in this case, P p2 = ... = p5 = 2, so pi = 8, and i |ni | ≤ 1 + pi = 9. Now, since ωi+1 ≤ (1/2)ωi , we conclude that ωi ≤ 2−i ω1 . In particular, ω5 ≤ (1/32)ω1 . So, 0.5 ω5 ≤ (1/64)ω1 < 0.017ω1 . Q.E.D. Conclusion. It is not yet known (1994) whether the Solar System is stable or not. Common belief is that the Solar System is stable if and only P if it is not a resonant P system, i.e., whenever its orbital frequencies ωi satisfy inequalities | ni ωi | ≤ ε for i |ni | ≤ N , a similar inequality is true for “randomly” chosen frequencies. Some empirically discovered inequalities between the planetary frequencies in the Solar system have prompted researchers to conclude that the Solar system is unstable. We have shown that similar approximate resonances hold for an arbitrary (“random”) set of frequencies, and therefore, the Solar system seems to be stable. Acknowledgments. This work was sponsored by NSF grant No. CDA-9015006, NASA Research Grants No. 9-482 and No. NAG 9-757, and a Grant No. PF90–018 from the General Services Administration (GSA), administered by the Materials Research Institute. One of the authors (V.K.) is greatly thankful to K. V. Kholshevnikov, S. Yu. Maslov, Yu. Matiyasevich, and N. A. Shanin (St. Petersburg, Russia) for valuable discussions, to R. de la Llave for encouraging reprints and preprints, and to the anonymous referees for their valuable suggestions. REFERENCES [BC61] D. Brouwer, G. M. Clemence, Methods of celestial mechanics, Academic Press, N.Y., 1961. [D75] V. G. Demin, The fate of the Solar System, Nauka, Moscow, 1975 (in Russian). [K80] V. Kreinovich, A review of [D75], Zentralbl¨att f¨ ur Mathematik, 1980, Vol. 404, pp. 426–427, review No. 70–007. [K89] V. Kreinovich, Is the Solar System a resonance?, Leningrad, Center for New Informational Technology “Informatika”, Technical Report, 1989 (in Russian). 5

[LR87] R. de la Llave, D. Rana, Accurate strategies for small divisor problems. In: M. Mebkhout, R. Seneor (eds.), Lecture given at the VIII International Congress on Mathematical Physics, World Scientific, Singapore, 1987. [LR90] R. de la Llave, D. Rana, Accurate strategies for small divisor problems, Bulletin of the American Mathematical Society, 1990, Vol. 22, No. 1, pp. 85-90. [LR91] R. de la Llave, D. Rana, Accurate strategies for K.A.M. bounds and their implementation. In: K. Meyer, D. Schmidt (eds.), Computer Aided Proofs in Analysis, SpringerVerlag, N.Y., 1991, pp. 127–146. [M66] A. M. Molchanov, Resonances in mutifrequency oscillations, Soviet Mathematics (Doklady), Vol. 7, No. 3, pp. 636–639. [M73] J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973 (Annals of Mathematics Studies, Vol. 77). [M74] J. Moser, Stability theory in celestial mechanics, In: Y. Kozai (ed.), The stability of the Solar System and of small stellar systems, 1974, D. Reidel, Dordrecht, 1974, pp. 1–9. [M78] J. Moser, Is the Solar System stable?, Mathematical Intelligencer, 1978, Vol. 1, pp. 65–71.

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