# Talk:Galactic dynamics

## (New) Reviewer A

**Comments as of 28 Mar 2011**

Galactic Dynamics is a vast subject involving several areas of mathematics and physics. The authors have chosen to illustrate these matters from a particular perspective, that of stellar dynamics, which, even if not embracing every aspects involved in the field, is surely wide enough to shed light on many of the building blocks of the physics of large self-gravitating systems. On this basis, the article provides a thorough illustration of the main results so far obtained and of the ideas and tools to obtain them. These ideas and results are also the starting point of many research programs concerning issues that are still to be understood.

The authors faced a difficult task because the body of mathematical gear needed to deepen the subject is vast and complex. On this side I think that they should do an extra effort in order to clarify some statements and to fill some gaps in the presentation of the results. In the following I provide some examples to illustrate what I think about:

1. Introduction and basic concepts: the generic notion of "disc galaxy" appears somewhere but is not explained its relation with the Hubble classification. This remark is useful also to justify the distinction between the contents of section 2.3, 4 and 5. In introducing relaxation, it is useful to refer to the discussion in section 6.1.

2. Orbits and integrals: the Hamiltonian pops out from nothing and the
same for the notion of *formal* integral and that of Poisson
brackets. The hints on "KAM theory", "overlapping of resonance" and
"Lyapunov exponents" is so sketchy that it is difficult to think it
can be useful to the inexpert reader. Since it is clear this is not
the place to fully introduce these issues, appropriate references seem
mandatory.

3. Construction of equilibria: Concerning the Jeans equation, some words of warning could be useful about problems with possibly unphysical required distribution functions. Some illustration of the construction of more refined analytic self-consistent models can be useful, for example models with distribution function depending also on the angular momentum.

4. Gas dynamics: In equation 38 concerning collisional Boltzmann equation, the index "c" in the right hand side is probably referred to the "collisions"; it seems useful to give a hint of these effects, the time-scale and a comparison with the stellar relaxation time.

5. Spiral structure: This section is quite detailed and clear.

6. N-body systems: Some warning concerning the present status of the theory of "Violent Relaxation" is mandatory, since it is not clear how to reconcile almost trivial and rough theoretical results with generic numeric simulations of collapses.

**Reply by the authors**

We thank the new reviewer for his comments, which were all taken into account. In particular:

1. We added a sentence on the meaning of "disc galaxies" in the second paragraph of the introduction. We also added a sentence at the end of the fifth paragraph were we refer to section 6 for a more detailed discussion of relaxation.

2. We extended the introductory paragraph of section 2 so as to go more gently into the definitions of concepts like formal integrals, KAM theory, Lyapunov characteristic numbers etc. We then suggest the book of Contopoulos (2004) for further reading. In the first paragraph of section 2.1 we better explain how the Hamiltonian (8) is derived. Finally, we rephrased the sentence before Eq.(13) so as to render clear that the formal integrals series are actually defined by the requirement of having a vanishing Poisson bracket with the Hamiltonian.

3. In the second paragraph of section 3 we give a warning on possibly unphysical distribution functions arising by solutions of Jeans' equation, mentioning explicitly the cases where the d.f. may turn to be negative in some parts of the modelled galaxy or the velocity dispersion curve may have an unphysical form. We think that these cases are the most characteristic. Finally, we wrote a new paragraph on inversion formulae for spherical models depending on the angular momentum or axisymmetric models depending on the z-component of the angular momentum.

4. We explained the meaning of the subscript "c" in Eq.(38) and we speak about the relevant phenomena immediately after Eq.(38).

5. No change

6. We revised the whole subsection 6.2 in order i) to give a more fair account, and ii) better explain our own point on this matter. Briefly, this can be summarized as follows: while we certainly agree that Lynden-Bell's formula and the whole theory around it are far from describing realistic galactic systems, we think that this theory continues to play in important role today in our understanding of the structure of galaxies. This is because violent relaxation has offered a unique up to the present "paradigm" of theoretical framework showing how can statistical mechanics be applied even in principle in collisionless systems. In fact, we think that the main contribution of this theory is to explain how can relaxation be at all possible in galaxies over a timescale as short as the dynamical time. We hope that the new text conveys this message while providing a fair account.

## Authors replies to earlier comments by reviewers

Report on "Galactic dynamics", by G. Contopoulos and C. Efthymiopoulos

edited by reviewers "A" and "B":
reply by the authors

A: I have read the article "Galactic dynamics" and found it useful and interesting, but I have a number of comments that I hope the authors will be able to consider.

B: Given the title "Galactic Dynamics", it's obligatory to compare this article with the excellent textbook of the same title by Binney and Tremaine (2008, Princeton University Press. Essentially, a html version of that book would be a VERY useful resource, for everybody who could possibly be interested in reading this scholarpedia article. I suspect that this is a general feature/problem for many scholarpedia articles). Clearly, this article has no chance in a such comparison: it fails in width (too specialised on a few subjects), clarity, and up-to-dateness; its only advantage naturally lies in brevity and availability.

Authors' Reply:

While we thank both reviewers for their constructive comments, we have to stress that the spirit of a scholarpedia article is to present only very basic notions on a subject within an extent of ~5000 words. It is not reasonable to compare this to the scope of a textbook like Binney and Tremaine, which is larger than the present article by a factor 100, or other books listed in our reference section. The present article has not even the form of a review article. In fact, our choice of topics is based on the sole criterion of outlining a set of notions which are an absolute must for a first step in the study of galactic dynamics. Even so, we had to limit ourselves to a a severely restricted list of topics which is obviously far from exhaustive. On the other hand, we intend to continuously create new electronic links to some article keywords, as an on-going "curator" task that will render the article more useful.

Major points

(1) A: The title appears to be discordant with the content of the paper. Galactic dynamics encompasses a number of astrophysical issues (such as processes of galaxy formation and evolution, and, in particular, the dynamics of the dissipative components) that are basically ignored in this review, which covers mostly the dynamics of stellar orbits. In my opinion, a more appropriate title would be "Stellar dynamics" or "Orbits in stellar systems". The use of the word "galaxies" is sometimes unjustified. The sections "Orbits in axisymmetric galaxies" and the following "Orbits in triaxial galaxies" are actually a discussion of orbits in axisymmetric or triaxial models inspired by galaxies; the potential described by Eq. 18 is interesting and useful, but should not be considered an approximation of the potential present in elliptical galaxies, because it has been found to be too shallow and associated with unrealistic density distributions.

B: I sort-of agree with the first reviewer. This article is meant to be theoretical and traditionally galactic dynamics has been concerned with the motion of stars (and dark-matter) only. Galactic gas dynamics is still not very well developed theoretically and hence hard to review. I think it helpful if this article were accompanied by or appended with an observational review about galaxies and their properties, similar to the book "Galactic Astronomy" by Binney & Merrifield.

B: However, I also think that even in the limited realm of stellar dynamics the article is too narrow, mainly dealing with orbits in static potentials and density-wave theory in disc galaxies (which are, I think, the main areas of expertise of the authors). However, galaxy dynamics is MUCH richer than this, just look at the table of contents of "Galactic Dynamics" by Binney & Tremaine.

Authors' reply (1):

We emphasize stellar dynamics because it is our basic thesis that stellar dynamics is the most important ingredient of galactic dynamics. We believe that this viewpoint is shared by most leading experts in the field. One referee invokes Binney and Tremaine's book. But in the first paragraph of the "Preface" of this book it is clearly stated that "the principal tool for answering these questions" (regarding the formation, structure, and evolution of galaxies) "is stellar dynamics". This is repeated in the first two paragraphs of the "Introduction" of the same book.

Dissipative processes due to gas play, however, also a role in galaxies, in particular as regards phenomena of secular evolution. In order not to leave this aspect completely ignored, we added a new section 3 entitled "gas dynamics", where we briefly discuss how stellar dynamics should be modified and what sort of new phenomena could arise due to the gas.

In section 2, we prefer to keep the word "galaxies" rather than "models inspired by galaxies" in all subsection titles. However, we mention in the text that this is an idealization. Regarding Staeckel potentials (Eq.21), their discussion is necessary, because the nomeclature for orbits found in such potentials (Box, ILAT, OLAT SAT) has become standard in the characterization of orbits in generic triaxial potentials. In the new version we stress that box orbits are unlikely to survive in cuspy potentials, being replaced, instead, by chaotic orbits.

Reviewer B characterizes this article as "too narrow". We can hardly see how to substantially widen the article's content without exceeding by far the length standards of a scholarpedia article (which have already been exceeded in the first article's version). Besides the new section 3, the only marginal additions that we found possible to accomodate were: i) a reference to virial equilibria and to the concept of rotationally vs. pressure supported systems, ii) possible forms of distribution functions and how these affect the form of velocity ellipsoids in galaxies, and iii) add more to the section on density wave theory (see below).

We finally agree that a separate lemma on "Galaxies: structure and evolution" would be useful.

(2) A: (Possibly related to the previous point.) Some statements do not do justice to the subject. In particular, "The shapes of the galaxies are governed by their stars, that form their main bodies. The gas forms a relatively small proportion..." is incorrect and misleading. As proved by near-infrared (K-band) observations, and as expected from the analysis of Jeans instability, gas plays an important role in determining the morphology of spiral galaxies.

B: I don't think that gas-dynamics is very important. Even in spiral galaxies, the gas contributes only about ~10% (in the Milky Way) to the baryonic mass. Spirals are seen in the old stars (K-band) of most disc galaxies, are the natural modes of discs, and do not require gas dynamics, though dissipative processes may play some role.

Authors' reply (2):

While we now discuss the role of gas (section 3), we should point out that the argument based on K-band observations is not correct. These observations yield the distribution of matter as deduced mainly by old stars (red giants) rather than the gas. Such a distribution is in general characterized by smooth and well-defined spiral arms. In fact, the reason why such observations have been used extensively in recent years in reconstructions of the gravitational potential of disc galaxies lies, precisely, on the consideration that old stars are better tracers of the overall mass distribution because they have more time to `respond' to the total potential, hence settling to a smooth matter distribution. On the contrary, the distribution of young stars, molecular clouds, or OB associations exhibits large and non-smooth variations. This remark notwithstanding, we followed the reviewer's suggestion to rephrase our sentence in a more balanced way regarding the role of all components (dark matter, stars, gas) in the shapes of galaxies.

(3) A: The general set-up of the basic equations is inadequate. The set of Eqs 1- 4 is practically inapplicable, even when we assume that the interstellar medium can be neglected. The main reason is that it is now established that galaxies are embedded in halos of dark matter and that dark matter generally has a density distribution different from that of visible matter. This requires the inclusion of an external component in Eq. 4, the symmetry of which has to be discussed separately, with important consequences on dynamics, and in particular on the structure of stellar orbits.

B: Essentially, only equation (4) needs to be amended by adding the dark-matter potential. Alternatively, one may extend the formalism to include dark-matter and stars on an equal footing. However, I think one should try to keep these equations as simple as possible. An important point is that the dark-matter potential must not be considered as a god-given external potential, but as an independent player. From a dynamical point of view, stars and dark-matter are governed by the same equations: perturbations affect the dark-matter as well as stras, dark-matter exchanges energy and angular momentum with spiral and bar modes (rather than just providing an additional static background potential).

Authors' reply (3):

We essentially agree with reviewer B. The set of equations (1-4) is valid to a very good approximation if $f$ represents the distribution function of all collisionless matter, i.e. the stars and the dark matter. Also, it is valid for describing the luminous matter alone, if the dark matter is considered to yield a fixed external potential. On the other hand, we may consider the dark halo as `live', i.e. responsive to the luminous matter, in which case the study of the problem becomes more complex. These facts are mentioned in the revised text.

(4) A: The section on "Orbits in disc (spiral - barred) galaxies" is confused, not only because of the general difficulties noted in point 1 above. The reader is not informed of the assumptions made for the description of the non-axisymmetric part of the potential. In particular, in applications to real galaxies one has to justify to what extent such non-axisymmetric part should be treated as stationary (time independent) and bisymmetric.

B: I think the simplified picture presented is useful and helps understanding the situation with real galaxies. As usual in science (especially in physics), simplifying assumptions allow for analytic insight and hence understanding of much more complicated systems. It is important, though, to emphasise this and clearly state which assumptions are made and how good they are.

Authors' reply (4):

We essentially agree with the comment of reviewer B. We added some sentences on the importance of both even and odd components of the Fourier trasform of the potential, but we emphasize that the most basic theory of orbits in disc galaxies stems from considering that the m=2 component of the non-axisymmetric perturbation is the most important one.

(5) A: The section "Density wave theory" is poor, because (i) key issues at the basis of the density wave theory (basic issues raised by the observations, such as the winding dilemma or the preference, but non-universality, of trailing bisymmetric structures; dynamical issues such as collective instabilities, from Jeans instability and beyond, wave propagation and dynamics, self-regulation,overreflection, light disks vs. heavy disks, role of dark halos; observational tests, i.e. what do the observations tell us in favor or against the alternative scenarios that may be proposed) are ignored; (ii) the description of the theory is improperly reduced to a statement of Fourier analysis (Eq. 31); and (iii) the impression left by the presentation and by sentences such as "The density wave theory of spiral arms was developed first by Lindblad, and later by Lin, Shu, Kalnajs, Toomre, Lynden-Bell, etc." is that the theory has remained frozen at the stage of its initial formulation in the '60s. (The justification of the "Nonlinear theory" given in this review is misleading: as for many other problems in hydrodynamics or plasma physics, although it is fair to say that nonlinear effects are interesting and likely to be important at resonances, the linear theory can make and has made adequate predictions on the role of resonances.) The book "Spiral structure in galaxies: a density wave theory, Publisher: Cambridge, MA MIT Press, 1996 (http://adsabs.harvard.edu/abs/1996ssgd.book.....B)" by G. Bertin & C.C. Lin is fairly complete (in terms of references and topics covered) up to the mid '90s; additional interesting work has been done and published in the last 15 years, after the publication of that book.

B: I am not an expert on density-wave theory, but thought that this theory is dead: it cannot explain spirals galaxies. Why is swing amplification not mentioned? As I understand it, this is thought to be essential for understanding the formation of spirals and resolving the anti-spiral theorem. I think that chapter 6 of "Galactic Dynamics" (Binney & Tremaine) is a much better account of our current knowledge in this area.

Authors' reply (5):

It is hard to compromise the two reviewers' opinions on this point. One reviewer wants us to add more on new developments of spiral density wave theory beyond Lindblad, Lin, Shu, Kalnajs and Toomre, while the other thinks the theory is "dead". Our own viewpoint is that the most important, up to the present, contribution of the density wave theory lies in what was the central idea of its founders, namely, that this theory provides a "paradigm" of dynamical mechanism which describes spiral structure as a wave propagating in the galactic disc with angular speed different than that of most stars or gas going in and out of it. This paradigm is not "dead" but it remains today the most basic description of rotating spiral arms in galaxies. The main open questions are rather different, focusing mainly on the degree of quasi-stationarity and/or secular evolution of such a structure. Namely, it is still unknown whether it is more consistent to approximate spiral structure i) by quasi-stationary solutions to Boltzmann's equation supporting a unique pattern speed, or ii) by relatively short-lived, recurrently generated spiral arms formed by instabilities in the disc, which support multiple pattern speeds (e.g. of the bar and spiral arms).

In the new version we i) discuss to some extent the winding dilemma and the problem of preference of trailing waves, ii) only marginally refer to most other topics mentioned by the reviewers, since their treatment in reasonable length is practically impossible, iii) added the book of Lin and Bertin in the bibliography, iv) mention alternative scenaria of spiral structure considered as a reccurent disc instability.

(6) A: Given the importance of "N-body systems" and "Violent relaxation - collective instabilities" in galactic dynamics, the description given at the end of this review is too short and out-of-balance with respect to the details on stellar orbits provided in the main body of the article; in addition, the statements given on "N-body simulations" (the text around Eqs. 32-34) appear to be restricted to stellar systems with little or no rotation (except for the reference to instabilities in disks, at the very end).

B: I completely agree with the first reviewer.

Authors' reply (6):

We enriched the text, adding some reference to N-Body simulations of galactic discs, and of other interesting phenomena like interacting or merging galaxies. But there is no real need for further additions at this point, since we found a separate scholarpedia article specialising on "N-Body simulations (gravitational): curators: M. Trenti and P. Hut", which is very specific and informative. We thus added a link to that article.

Minor points (reviewer A)

7. The asymptotic expansion at the basis of Eq. 11 and the physical meaning of the expansion parameter in the context of galaxy models should be explained better.

8. The distinction between "smooth core" and "core with a central point mass" (see the discussion illustrated by Fig. 4) should be explained better. What about a cuspy core, without a central point mass?

9. After Eqs. 18-21, the attention moves to a polynomial potential (22). Could a classification diagram like Fig. 7 be drawn also for orbits in Staeckel potentials?

10. (see also point 4 above) Before Eq. 23, "Disc galaxies and their patterns rotate in general with large angular velocities \Omega_s" is a very strange and ambiguous statement. The assumptions on the potential, the question of drawing orbits in an inertial or rotating frame, the symmetry break, and the location of L_1, L_2, L_4, L_5 with respect to potential minima and maxima could be explained much better. The current text is confused and some figures are difficult to understand (in particular, Fig. 11). The assumption of stationarity of the symmetry-breaking perturbation in a suitable rotating frame is crucial and not given in a sufficiently explicit way (especially if the authors believe that the models studied are to be applicable to observed galaxies).

11. The factor (1/2) in the definition of the radial wavenumber (after Eq. 31) should be dropped.

12. It would be nice to explain better how and why in general the Fourier analysis associated with Eq. 31 leads to an integral equation with eigenvalues \omega. Then to explain how in the initial stages (until the mid '70s) the theory could only handle, through an approximate dispersion relation, the local properties of density waves. And finally to show what mechanisms are thought to be at the basis of the excitation of a discrete spectrum of global modes and how the problem has been addressed analytically and numerically.

13. The infinities at turning points and resonances that appear in algebraic dispersion relations reflect a wrong choice of representation of the perturbation and are removed in more general analyses of the linear theory.

14. The use of "collision time" as a synonym of "dynamical time" (after Eq. 32) is misleading.

Authors' reply:

7) We added a sentence clarifying the physical meaning of small parameter in galaxies.

8) We now clarify that the distinction is between i) "smooth core" and ii) "cuspy core or core with a central point mass".

9) The answer is no. The periodic orbits in Staeckel potentials can only have neutral stability since the potentials are integrable. However, Fig.7 is important because it represents the generic case, i.e. when periodic orbits of all four types of (in-)stability co-exist.

10) We thank the reviewer for this point. We rephrased the statement in a hopefully satisfactory way. As regards Fig.11 we think that it is important to give some examples of characteristics of periodic orbits.

11) We dropped 1/2.

12) The mathematical derivation of the dispersion formula is too lengthy to present in all detail, thus we just sketch it by words. We furthermore clarify that this is a local theory and mention also modal theory.

13) We agree with the reviewer's comment that there are methods to take care of the problem of infinities at resonances. In fact, one of the authors (G.C.) has been a main proponent of resonant theory in discs along the lines mentioned by the reviewer. We doubt, however, that we should call such a theory `linear'. At least it is not linear in the same way as the Lin and Shu theory, because the formal scheme yielding the corrections for the distribution function near resonances is different from the formal scheme used by Lin and Shu. Thus, we prefer to keep this point unaltered, i.e. as part of nonlinear theory.

14) We removed "collision time".

01-11-2010

remark. The email alert triggered by me was errorneous (bug in scholarpedia, I suppose), as it said "Reviewer A alerts ...", rather than reviewer B. Also, I have not been informed about the decision by reviewer A --- has he accepted, so that now there is only one of us, and hence I default to be "A" ??? this is very confusing.

Reviewer A:

I have read the revised article and the authors replies. The article is somewhat improved, but on the whole the changes are minor and rather disappointing.

My main criticism is that the article is still quite badly balanced with way too much emphasis on orbits and density waves--perhaps the authors should consider to write an article about these subjects instead. Other important topics are almost entirely missing, such as equilibriums systems other than discs and their stability properties (a lot has been learned from spherical models); methods to construct equilibria; interactions, dynamical friction, tides, and mergers; dynamics in the vicinity of a dominating point mass (supermassive black hole); formation/dynamics of bars and warps to name a few.

It is certainly impossible to write a scholarpedia article (with the imposed limit on its size) on galactic dynamics to the same depth as the existing section on density waves. However, the solution then is NOT to pick just one or two (density waves and orbits) of many subjects and expand on them, rather all subjects should be mentioned at the same level. Naturally, there may be further articles about each of these more specialised subjects. Moreover, the article merely describes methods and tools, but I am missing the greater picture: what is the goal/purpose of galactic dynamics? what are the most pressing open questions? what are its successes? How does it fit into astrophysics as a whole?

%------------------------------------------------------------------------ Authors reply:

We have already agreed with the reviewer that this article cannot present all aspects of galactic dynamics. The section on density wave theory was expanded because the other reviewer requested a number of additions in his/her original report. On the other hand, with all these additions this section deals now with the general problem of spiral structure rather than pure density wave theory. We thus changed its title to "Spiral Structure", while we did some reduction by merging some subsections (doing some re-ordering) and by removing some paragraphs. It now occupies about 15% of the article.

On the other hand, we think that the section on `orbits' should be kept essentially as it now stands (apart from one paragraph on the dynamics in the corotation region, which we removed as too technical). This section occupies about 30% of the article, and we think that this percentage should not be considered as too high. In fact, our experience is that some topics like e.g. the form and bifurcations of periodic orbits in rotating galaxies are less known than they should be, and thus it is useful and informative to present them in some depth. Finally, it should be noted that the article appears presently under the category "Celestial Mechanics" , where orbital dynamics is a key issue (although we believe that the article should eventually be moved to an independent category on "Galactic Dynamics).

Apart from the previous changes:

i) We added a sentence in the introduction stating what is the focus of the article.

ii) We added a section on `construction of equilibria' where we speak about fixed models and methods of construction (Schwarszchild).

iii) we expanded the last subsection on collective instabilities, which is now an independent sub-section.

iv) Responding to some comments of the other reviewer (who has now accepted the article), we added a reference to Jeans instability in "Gas dynamics", and we discuss some differences between gas and stellar dynamics.

v) We give replies to the specific comments of Reviewer A (see below).

It has to be understood that by this process the article is now well beyond the limits of an encyclopedic introduction. In particular, there is no way to seriously go into questions like 'how does' galactic dynamics 'fit into astrophysics as a whole'. Even the question what are 'the most pressing questions' of galactic dynamics is rather subjective. We thus prefer to be concise rather than going into such subjective discussions. %-------------------------------------------------------------------------

Reviewer A:

In addition, I identify the following issues with the existing text (but this list is very likely incomplete).

(1) The idea/concept of a "rigid halo" is not a useful one, even though it has been used a lot in the past. Essentially, this concept is at best dangerous as the effect of a rigid halo can be opposite to that of an active halo, e.g. a halo may suppress the bar instability by reducing the importance of self-gravity in the disc, but a life halo can also absorb angular-momentum and thus promote the growth of a bar once it has become non-linear.

%------------------------------------------------------------------------- Authors reply (1):

While it is true that a rigid halo can suppress the bar instability and lead to other spurious phenomena, we do not agree that the concept "is not a useful one". In fact, whenever one is interested in the orbits on the disc plane, it is useful to assume a fixed halo potential, at least temporarily. For example, when we attempt to estimate the total gravitational potential from K-band observations, our only way to fix the mass-to-light ratio is by adopting an ansatz on how much the halo contributes to the total rotation curve in the inner or outer parts of the disc (e.g. to choose a maximal or sub-maximal disc solution). The same thing we practically do with the bulge (this was noted by reviewer B). The correct attitude is then that a rigid halo model can be useful in some contexts while it is not so in other contexts. The rephrasal we did in the text is, we think, fair. %-------------------------------------------------------------------------

Reviewer A:

(2) the concept of "approximate" third integrals is confusing and should be avoided. An orbit is either regular or not, it either admits a third integral or not. Perhaps what the authors meant is whether the third integral is universal and can be given in closed form for all orbits or whether it only holds for an orbital family at a time and no closed form can be given. In the latter case, the third integral is not approximate. Another option is to give a closed form (e.g. derived using perturbation theory) which is not exactly but approximately conserved and can thus be used effectively as a third integral in some approximate modelling--in this sense the angular momentum w.r.t. the short axis is an approximate integral for short-axis tubes in triaxial potentials.

%---------------------------------------------------------------------------- Authors reply (2):

The referee is not right at this point and this remark actually reveals a common confusion about the concepts "regular orbit" and integrals that can be "local", "formal" or "approximative". The confusion stems from the following point: the word "integral" in general means a function $I(q,p)$ of the phase space coordinates which has zero Poisson bracket with the Hamiltonian i.e. ${I,H}=0$ (in authonomous systems). But in order that such a bracket can be defined in the first place, we should be able to define the function $I$ in an {\it open domain} of the phase-space. To give an example: if we have a nearly-integrable system, most motions lie on invariant tori whose existence is guaranteed by the Kolmogorov - Arnold - Moser theorem. However, a careful study of the proof of the KAM theorem reveals that there is no function $I$, either global, or local on whatever small open domain, that represents such tori, because these tori are defined on a Cantor set of values in the action space which is not an open set. Of course, this does not mean that the orbits on these tori are not regular. They are regular because they are quasi-periodic and they have zero Lyapunov exponents. But contrary to the reviewer's claim, these completely regular orbits do NOT 'admit' any exact `third integral'.

On the other hand, a 'formal integral' is a formal series whose Poisson bracket with the Hamiltonian vanishes. A series does not represent in general a quantity in 'closed form'. Thus integrals in 'closed form' are not in general 'given by perturbation theory'. in fact, the formal series are not convergent in general. However, a proper truncation creates an 'approximate integral'. This quantity is defined in open domains of the phase space and it is therefore useful in practice, since the remainder can be exponentially small according to Nekhoroshev theorem. In fact, this seems to be a very useful quantity in Galactic Dynamics, because it bounds the motions of {\it both regular and weakly chaotic orbits} for times surppassing by far the Hubble time.

Finally, the fact that the formal integrals (and their truncations) take different forms at different resonances is well known, but it is irrelevant in this discussion.

These facts are well known to nonlinear dynamists, but not so well known to dynamical astronomers and one often sees a confusion in the literature stemming from inappropriate use of the various concepts. In our text we have been very careful in giving the right definitions and we think the text should be left as it now stands.

%------------------------------------------------------------------------

Reviewer A:

(3) in the virial theorem section, both T and K are called kinetic energy. I think T should not be called kinetic energy, as it is only the component of ordered motion to the kinetic energy.

%------------------------------------------------------------------------ Authors reply (3):

We changed the terminology by properly defining the kinetic energy tensor. %------------------------------------------------------------------------

Reviewer A:

(4) N-body force solvers that scale as O(N) are not available for "rough models" only, but are as general as the tree method (see Trenti & Hut).

%------------------------------------------------------------------------- Authors reply (4):

To our knowledge, there are two types of O(N) N-Body algorithms: (i) the so-called "self-consistent field" method, pioneered by Clutton-Brock, uses basis functions to develop the galactic potential. This has led to useful applications (e.g. the Hernquist-Ostriker approach), which, however, are limited to isolated systems and seem to hardly be able to treat more complex systems such as merging galaxies, spiral structure etc. Furthermore, recent research by many groups (including some works of ours) shows that the choice of a basis set must by very finely tailored to the specific morphological details of a system under study in order to correctly represent the dynamics. (ii) Improvements of the tree method (e.g. Dehnen 2000) claim to scale linearly with N rather than O(NlogN). However, Trenti & Hut (recalled by the referee) are clearly very sceptic about that. In our article, we rephrased this sentence mentioning in more detail the self-consistent field method. %-------------------------------------------------------------------------

Reviewer A:

(5) The Lynden-Bell statistic is presumably incorrect (in a more recent paper co-authored by Lynden-Bell, the authors admit that Lynden-Bell's original method was inconsistent), and is, as far as I am aware, hardly used at all. Moreover, there is more recent work on violent relaxation (e.g. Tremaine et al 1986 and Dehnen 2005).

%------------------------------------------------------------------------- Authors reply (5):

This is a highly controversial issue. The referee may wish to consult section 11.3 of a tutorial article of one of us (C.E. et al. Lect. Notes in Physics 729, 297 (2007)), where the questions raised by the referee are reviewed in detail (in particular see subsection 11.3.6). Briefly, the work of Tremaine et al. has itself been largely criticised on various grounds (i.e. on that the choice of entropy functional cannot be arbitrary or that even Tolman's theorem may not be consistent with their approach). Clearly, we cannot here present all the new developments. Lynden-Bell's original distribution deserves, in our opinion, particular reference because it opened the way to a whole new branch of statistical mechanics, namely the statistical mechanics of collisionless gravitating systems. We added, however, one paragraph speaking about the various criticisms and deferring to the tutorial of C.E. et al for more information.

%-------------------------------------------------------------------------

Reviewer A (on 13-Jan-2011):

Sorry for this delay (I missed the email on 25-Nov and only got a second one today). I have read the article and the authors reply. I am not happy with this article, as detailed below. However, I have the feeling the authors are not prepared to meet with me on this and therefore decided to quit without accepting. For the benefit of any future reviewer, I leave some comments as follows.

1 The main criticism is that this article does not encompass "Galactic dynamics" in its whole width. Some subjects are laid out it great detail (the section on Orbits and Integrals takes about 40% of the whole article), while others are left out altogether (see my previous review). Perhaps a solution is to split this into two scholarpedia articles on "Spiral Structure" and "Stellar Orbits", respectively.

2 In answering many of my minor comments the authors have sometimes demonstrated a rather narrow and opinionated view: they present their own occasionally ill-informed opinion rather than seeking to present a fair account. An example is their account of violent relaxation, which is certainly one-sided (the authors admit that this is a controversial issue).

3 The reply to my previous point (2) about regular orbits should be incorporated into the article either directly or via a suitable reference.

4 There is only one type of O(N) force solvers: the FMM (fast-multipole method). The so-called "SCF" methods are not O(N) if one increases the force resolution with N, as one should and does with any other method (in fact, in this case they become O(N^2). The FMM, on the other hand are O(N), though most implementations have failed to reach this goal with the exception of Dehnen's (2002) algorithm, which he demonstrated to have complexity O(N).