I want to study the orbital resonance type 3:1 between an asteroid and Jupiter. For this purpose, I found the expansion of the disturbing function, $R$, in Celletti A., Stability and Chaos in Celestial Mechanics (Springer-Praxis, 2010), but I do not understand how I could determine the expression of the factor $R_{13}$. I mention that, below, $L$, $G$, $l$ and $g$ are the Delaunay variables:

$$\begin{aligned} R &=R_{00}(L, G)+R_{10}(L, G) \cos \ell+R_{11}(L, G) \cos (\ell+g) \\ &+R_{12}(L, G) \cos (\ell+2 g)+R_{22}(L, G) \cos (2 \ell+2 g) \\ &+R_{32}(L, G) \cos (3 \ell+2 g)+R_{33}(L, G) \cos (3 \ell+3 g) \\ &+R_{44}(L, G) \cos (4 \ell+4 g)+R_{55}(L, G) \cos (5 \ell+5 g)+\ldots \end{aligned} $$ where the coefficients $R_{i j}$ are given by the following expressions: $$\begin{array}{ll} R_{00}=-\frac{L^{4}}{4}\left(1+\frac{9}{16} L^{4}+\frac{3}{2} \mathrm{e}^{2}\right)+\ldots, & R_{10}=\frac{L^{4} \mathrm{e}}{2}\left(1+\frac{9}{8} L^{4}\right)+\ldots \\ R_{11}=-\frac{3}{8} L^{6}\left(1+\frac{5}{8} L^{4}\right)+\ldots, & R_{12}=\frac{L^{4} \mathrm{e}}{4}\left(9+5 L^{4}\right)+\ldots \\ R_{22}=-\frac{L^{4}}{4}\left(3+\frac{5}{4} L^{4}\right)+\ldots, & R_{32}=-\frac{3}{4} L^{4} \mathrm{e}+\ldots \\ R_{33}=-\frac{5}{8} L^{6}\left(1+\frac{7}{16} L^{4}\right)+\ldots, & R_{44}=-\frac{35}{64} L^{8}+\ldots \\ R_{55}=-\frac{63}{128} L^{10}+\ldots \end{array} $$

Please, could anyone to help me with an answer?

If none is forthcoming, are there alternative texts or papers that might shed more light on this?

  • 1
    $\begingroup$ Book recommendations aren't really what we are about, take a look at How to Ask. However over on Physics there is a suggestion : "For the construction and properties of Delaunay variables, see the excellent textbook of Morbidelli (2002) - Modern Celestial Mechanics (this is also a good resource for perturbation theory mentioned earlier)." $\endgroup$
    – James K
    Commented Jun 5, 2021 at 9:49
  • $\begingroup$ $R_{13}$ doesn't appear to be in the expression... $\endgroup$
    – James K
    Commented Jun 5, 2021 at 9:50
  • 1
    $\begingroup$ For the study of 2: 1 resonance, for example, I considered the expression of the perturbing function as the sum of the secular term R_ {00} and the resonant term R_ {12}, then I obtained the Hamiltonian of this problem, and from here, the system formed by the equations of Hamilton. I thought I should do the same in the case of 3: 1 resonance. If I'm wrong, please tell me. $\endgroup$
    – Augustin
    Commented Jun 5, 2021 at 10:00
  • 1
    $\begingroup$ Welcome to astronomy SE, please try to edit your question so it better fits to the guidelines How to Ask - for instance by editing in your comment(s) into your question. $\endgroup$
    – B--rian
    Commented Jun 5, 2021 at 13:35
  • 1
    $\begingroup$ @JamesK I've added the resource-request tag which already had 43 other questions. The original question was not a "Book recommendation" and requests for better resources in well-written, on-topic questions is certainly one thing that this community has decided are on-topic. The language "what we are about" is problematic for the you .vs. us aspect (all users are community members, including new ones) and there is no "us" identity in SE that any one user can deem to speak for. A single user shouldn't deem to say "We don't allow..." or "We don't think that..." or "What we're about..." $\endgroup$
    – uhoh
    Commented Jun 6, 2021 at 0:12

1 Answer 1


Finally, I found out that, within the restricted coplanar circular problem of the three bodies, the respective coefficient, $R_{13}$ is 0.


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