Talk:Stability of the solar system

Short report about the contribution “Stability of the solar system” by Jacques Laskar

Part of the article collects material coming from some previous papers of the Author and some of his beautiful talks. This is particularly interesting, because those works also contain nice historical overviews. Everything is reorganized in a coherent way, by focusing the discussion on the stability of the solar system. First, the article describes the historical evolution of the thinking about such a problem and, then, the main results in the most recent years.

I truly think that the community reading Scholarpedia is greatly appreciating the article. In the following, a list of corrections and some remarks is reported; this will hopefully improve the quality of such a nice contribution. Since it seems to me that the layout of the printed pages does depend on the browser, in the following I will not refer to the numbers of both pages and lines, as it is usual (and comfortable).

List of corrections and remarks

(A) About the first displayed formula in the third section, that is called “The problem of the eccentricities”. I think that the k elements of the vector $(z_1 , . . . , z_k )^T$ are missing; indeed, that vector should be replaced by $(z_1 , . . . , z_k , \zeta_1 , . . . , \zeta_k )^T$ as in formula (3) of Laskar, J., Cel.Mech. & Dyn. Astr., 64, 115–162.

(B) Nearly at the end of the third section, that is called “The problem of the eccentricities”. “(see Laskar, 1992 for more details on this point)” should be replaced by “(see Laskar, 1992a for more details on this point)”.

(C) In Figure 2. The unit of measure of the inclination, i.e., the symbol ◦ , is not displayed correctly.

(D) Starting from the beginning of the fourth section, that is called “Chaos in the Solar System”. “Ma” (that should mean “millions d’ann ́ees”, if I’m right) should be replaced everywhere by “Myr”. Analogously, “Ga” should be replaced by “Gyr”.

(E) At the beginning of the fifth section, that is called “Evolution of planetary orbits”. “Laplace−−Lagrange” should be replaced by “Laplace–Lagrange”.

(F) In the fifth section, that is called “Evolution of planetary orbits”. When the effective stability of the orbital secular evolution of the major planets is com- mented (by referring to Figure 4), in my opinion, it would be interesting to add the information that they also are weakly chaotic, if also the mean motion angles are involved in the equations of motion. This is in a nice agreement with the vision of “marginal stabil- ity” of our Solar System (that has been introduced by the Author two decades ago) and it could be done by shortly (but adequately) commenting the papers by Murray & Holman (Science, 283, 1999) and Hayes (Nature Physics, 3, 2007 and MNRAS, 386, 2008).

(G) At the end of the caption of Figure 4. “chaotic zone. (Laskar, 1994)” should be replaced by “chaotic zone (Laskar, 1994).”, because the final dot is misplaced.

(H) Nearly at the beginning of the seventh section, that is called “Planetary collisions in the Solar System”. “over 5 Gyr.This” should be replaced by “over 5 Gyr. This”.

(J) In the eighth section, that is called “Collisions of Mercury, Mars and Venus with the Earth”, in the paragraph starting with the words “With the JADE machnine, . . .”. I think it would be interesting to add a few sentences clarifying how the initial conditions have been selected and how much they differ each other (if it is possible doing so rather shortly).

Second reviewer's report about the contribution “Stability of the solar system” by Jacques Laskar

This is a nice review of a fascinating subject, the long-term stability of the solar system, which lies in the core of developments in the whole discipline of celestial mechanics in the last three centuries. Jacques Laskar presents the history of the subject, as well as the latest numerical results, many resulting from his own leading research in the field since decades. These results indicate that the classical theory of Laplace and Lagrange, which predicts that the eccentricities and inclinations (and their related canonical arguments) undergo only quasi-periodic long term variations, is essentially applicable only to the outer giant planets in our solar system. On the other hand, the inner planets exhibit chaotic variations of their eccentricity and inclination vectors. Laskar explains the source of such chaotic behavior, which is related to the existence of resonances between the secular frequencies of the inner and the outer planets. These resonances limit the applicability of series expansions. The chaotic variations of the eccentricity (mainly of Mercury and Mars) result in a non-negligible probability of planetary orbit crossings, implying, in turn, a non-zero probability of destabilization of the inner solar system over a timescale of some billion years. It is also of great interest to see how general relativistic effects actually help to reduce this probability from the level of 50% to about 1%, by influencing the inner planets' secular frequencies and effectively "detuning" the most dangerous resonance conditions.

Overall, the article is very interesting and I am sure it will be received with great interest by scholarpedia readers. On the other hand, I believe there is room for several technical improvements, and/or corrections. I attach below a list of comments and suggestions, following the order of appearance of the related parts in the text. Apart from corrections on obvious errors, I let to the author's (and editors') judgement whether or not to adopt them.

Introduction

Fourth paragraph: please add a reference regarding the historical remark about Newton's idea of the need for "divine intervention" in order to restore the planets in stable orbits.

Fourth paragraph: "Morover ... present status,...", I suggest to write "present status of acceptance" (otherwise it is unspecified what "status" refers to).

Fifth paragraph: "By crudely ...from the Sun". Please rephrase this sentence. As now written, it sounds as if the concluded event, of a nearly equal distance of Saturn and Jupiter from the Sun about six million years ago, was a true fact. I would write: "By naively extrapolating these observations, one would have been lead to the intriguing conclusion that about only six million years ago Jupiter and Saturn should have been at the same distance from the Sun."

Fifth paragraph, next two sentences, "...Laplace took up these observations ...". The remark on Laplace's quote is somehow left unfinished. One takes up something in order to do something else. Please complete the sentence, by explaining what was the goal of Laplace. Otherwise, the quote remains in the air.

Laplace - Lagrange stability of the Solar System

End of first paragraph: "calculated secular ...other planets". I would rephrase: "Calculated secular (i.e. long term) variations in the planet's ..."

Second paragraph: "exlpains" ---> "explains". Overall, this paragraph is somewhat unclear. The resonant terms in the disturbing function are of the form cos(m\lambda_Jupiter - n\lambda_Saturn + ...), with amplitudes depending on the major semi-axes, eccentricities or inclinations of the planets. In what sense does the amplitude of a resonant term get expressed in arcminutes "in Saturn's longitude"? Maybe the meaning here is that, due to the great inequality, the critical argument 2\lambda_J - 5 \lambda_S is not exactly constant, but has a variation, whose amplitude is of a certain size. Please explain and/or better rephrase this sentence.

Fourth paragraph: I would rephrase "This result of Laplace and Lagrange, namely that the planets' semi-major axes undergo only small oscillations, and are not subject to secular variations, was the first ...". (Please note that a semi-major axis cannot "have" secular terms, it is the time evolution that can be governed or not by such terms.) Also: "...Newton's law as the universally accepted explanation ...".

The problem of eccentricities

First paragraph: "insure" ---> "ensure", "inclination" ---> "inclinations".

Second paragraph: remove the period before the equation.

Half of the secular equations are missing (those referring to the \zeta), please correct the equations.

I would write (k \times k) instead of (k,k).

"and (...) *are the* classical elliptical elements".

Third paragraph, last line, add a comma in between the expressions for z_i and \zeta_i.

A general remark here: After referring to Laplace-Lagrange, the author jumps immediately to the solutions given by Laskar (e.g. Table 1) in the 90's. It would show a good style to give here some reference to previous works improving the solutions of Laplace-Lagrange, such as, for example, Brouwer and van Woerkom, Bretagnon, Duriez etc. Readers will welcome finding such references showing the historical evolution of the subject and the analytical approaches profiting from the advent of modern computers.

Second paragraph: the statement that the eccentricities and inclinations are subject "to only small variations about their mean values" is misleading. Just check figure 3. The eccentricity of the Earth varies from 0 to 0.06, with a mean around 0.03. Thus, the amplitude of the variation is about the same as the mean value, and it is by no means "small" around the mean value. Same for Mars. In fact, the amplitudes of oscillation are only determined by the initial conditions. What the author wants probably to say here is that the eccentricities and inclinations are subject "to only *quasi-periodic* variations about their mean values", and hence, that the eccentricities and inclinations remain bounded forever, due to their regular variation, which cannot be destabilized in the Laplace-Lagrange approach. Please rephrase.

Sixth paragraph: I would carefully rephrase the reference to the work of Poincar\'{e}. It is more precise to state that the work of Poincar\'{e} shows that there exist no integrals analytic in the small parameter (here, the planetary masses) other than the trivial ones. However, about the existence of convergent series *solutions* to the equations of motion, the issue is rather tricky. For example, in the three-body problem the well known Sundman series provide such solutions for every initial condition except the ones leading to a triple collision (see, for example, the scholarpedia article "Three Body Problem" by A. Chenciner, section "Collisions").

Seventh paragraph: the reference to the inapplicability of the KAM theorem is not updated, since there are the works by Celletti and Chierchia - exactly on Henon's model - who obtained results for mass ratios equal to 0.001, and by Giorgilli, Locatelli and Sansottera for the planetary problem, where they showed that, at least for the outer planets, solutions lying on invariant tori are possible. Reference to such developments is recommended.

Last sentence: I would rephrase: "However, the results obtained through numerical integration in the past two decades show the contrary."

Chaos in the solar system

First paragraph: I would rephrase: "..., the study of Solar System stability..." (instead of "problem", sounds weird to say that a problem was "advanced"). Also, I would replace "analytic" with "computer-algebraic" calculations. Finally "...over time scales" (remove "model").

Third paragraph, first line: "...consists of ...". Please change "Ma" to "Myr". In fact, the symbols "Ma" and "Ga" appear throughout the text and in some figure captions. Please change to "Myr" and "Gyr". As a SI notation it is correct, but most people use "Ma" and "Ga" to indicate time measured in the negative axis starting from the present (i.e. million or billion years ago).

Fourth paragraph: " ...originates from ...". Since the symbols g_i, s_i were explained before, I would remove the parenthesis "(the g_i ... nodes)" and substitute by "(see Table 1)".

Evolution of the planetary orbits

(please add "the" and remove period in the end)

A general comment: in this section I have the impression that the presentation appears somewhat more detailed than needed. Many numbers are given regarding, e.g. how many orbits were integrated, for how long, in which papers, how much was the variation of the eccentricity of the Earth, or Venus, or Mars etc. Since more recent simulations established the validity of the whole approach, I would suggest to abbreviate this section which contains older results, emphasizing the most basic results. In fact, even a brief reference to the results of Figure 4 would suffice to show the impressive final result. At any rate, I still make some paragraph by paragraph comments, letting the author to decide whether or not to shorten this part.

First paragraph, middle: I would rephrase "...of the Earth, which is, in turn, at the origin of ..."

Second paragraph: I would rephrase: "...between the secular motions of the inner planets. This is not possible to obtain by using the classical ...". Later on: "...Fourier techniques, after numerically integrating the averaged equations of motion..."

Third paragraph: I would change "...provide a very clear vision..." to "...provide a clear picture ...". The author uses the word "vision" in several other places in the text, but, in english, I think the word "picture" fits better, "vision" sounds more like endeavor or ambitious plan. Another possibility is "viewpoint".

Seventh paragraph: "...the maximum eccentricity reached ... *is* about 0.08".

Then the author says: "...its current variations are approximately 0.06". But "current" here can be easily misunderstood to mean short term variations at the present epoch. In fact, these variations are still over quite long periods (of the order of 10^5 years). Please rephrase to avoid a misunderstanding.

Tenth paragraph: I would rephrase: "...the general aspect of the solutions remains the same." I would personally avoid using the word "undoubtedly" in any scientific text whatsoever.

I would remove all together the paragraph "It should be noted ... possible collision.". It is like questioning the whole approach. A word of caution about the limitations of the averaged equations of motion could be made, instead, at the beginning of the section.

Last paragraph: I would rephrase: "...most unstable orbit is Mars, whose eccentricity can, by this same method, reach approximately ... barely reaches 0.1".

Marginal stability of the Solar system

(remove period in the end)

First paragraph: I would change the statement that "the Solar System interns is "full"" as: "the inner Solar System is dynamically compact". The term "dynamically compact" has been used in literature about extrasolar planetary systems, in a similar sense like here.

Second paragraph: "... that it always was *so* for the Solar System..."

Second and fifth paragraphs: the author writes repeatedly that the solar system is in a state of "marginal stability", and, even more, that in the planetesimal formation scenario, "the planetary systems will always be in a state of marginal stability". I believe both these claims are exaggerated. I cannot see any evidence in the author's plots or arguments that necessarily implies that losing one planet leads the system again to a state of marginal stability, let alone that this should be the endstate of a system of planets formed by planetesimal accretion. This may be possible in some cases. However, the issue of the endstates of planet-forming systems, or systems losing one or more planets, is so complex, and influenced by so many dynamical parameters, that the author's thesis here sounds no more than an non-substantiated claim. I would avoid this claim all together. It adds nothing essential to the text.

Fourth paragraph: "Recent numerical simulations show that ...", please add a reference.

Planetary collisions in the Solar System

Second pragraph: "...problem, Laskar (2008) carried ..."

Last paragraph: "These results *are* still incomplete."

Collisions of Mercury, Mars, and Venus with the Earth

First paragraph: I would rephrase"...planetary ephemeris developed in past years" (avoid use of "we").

A final general suggestion: albeit not strictly an error, I would replace everywhere the word "movement" with "motion". Please note that the french word "mouvement" has, as a rule, the same use as the english word "motion", while the french word "motion" rather corresponds to the english word "movement" (e.g. political movement).