<p>Finally, I found out that, within the restricted coplanar circular problem of the three bodies, the respective coefficient, <span class="math-container">$R_{13}$</span> is 0.</p>

CC BY-SA 4.0
2021-06-24T18:55:08.977Z
2021-06-24T18:55:08.977Z
Augustin
<p>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, <span class="math-container">$R$</span>, in Celletti A., <a href="https://www.springer.com/gp/book/9783540851455" rel="nofollow noreferrer">Stability and Chaos in Celestial Mechanics (Springer-Praxis, 2010)</a>, but I do not understand how I could determine the expression of the factor <span class="math-container">$R_{13}$</span>. I mention that, below, <span class="math-container">$L$</span>, <span class="math-container">$G$</span>, <span class="math-container">$l$</span> and <span class="math-container">$g$</span> are the Delaunay variables:</p>
<blockquote>
<p><span class="math-container">$$\begin{aligned} R &amp;=R_{00}(L, G)+R_{10}(L, G) \cos \ell+R_{11}(L, G) \cos (\ell+g) \\ &amp;+R_{12}(L, G) \cos (\ell+2 g)+R_{22}(L, G) \cos (2 \ell+2 g) \\ &amp;+R_{32}(L, G) \cos (3 \ell+2 g)+R_{33}(L, G) \cos (3 \ell+3 g) \\ &amp;+R_{44}(L, G) \cos (4 \ell+4 g)+R_{55}(L, G) \cos (5 \ell+5 g)+\ldots \end{aligned} $$</span> where the coefficients <span class="math-container">$R_{i j}$</span> are given by the following expressions: <span class="math-container">$$\begin{array}{ll} R_{00}=-\frac{L^{4}}{4}\left(1+\frac{9}{16} L^{4}+\frac{3}{2} \mathrm{e}^{2}\right)+\ldots, &amp; R_{10}=\frac{L^{4} \mathrm{e}}{2}\left(1+\frac{9}{8} L^{4}\right)+\ldots \\ R_{11}=-\frac{3}{8} L^{6}\left(1+\frac{5}{8} L^{4}\right)+\ldots, &amp; R_{12}=\frac{L^{4} \mathrm{e}}{4}\left(9+5 L^{4}\right)+\ldots \\ R_{22}=-\frac{L^{4}}{4}\left(3+\frac{5}{4} L^{4}\right)+\ldots, &amp; R_{32}=-\frac{3}{4} L^{4} \mathrm{e}+\ldots \\ R_{33}=-\frac{5}{8} L^{6}\left(1+\frac{7}{16} L^{4}\right)+\ldots, &amp; R_{44}=-\frac{35}{64} L^{8}+\ldots \\ R_{55}=-\frac{63}{128} L^{10}+\ldots \end{array} $$</span></p>
</blockquote>
<p>Please, could anyone to help me with an answer?</p>
<p>If none is forthcoming, are there alternative texts or papers that might shed more light on this?</p>

CC BY-SA 4.0
2021-06-05T09:49:34.483Z
Book recommendations aren't really what we are about, take a look at [ask].  However over on [physics.se]  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)."
James K
CC BY-SA 4.0
2021-06-05T09:50:54.020Z
$R_{13}$ doesn't appear to be in the expression...
James K
CC BY-SA 4.0
2021-06-05T09:55:17.780Z
Thank you for your advice and patience, Mr. James K.
Augustin
CC BY-SA 4.0
2021-06-05T10:00:48.343Z
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.
Augustin
CC BY-SA 4.0
2021-06-05T13:35:13.083Z
Welcome to astronomy SE, please try to edit your question so it better fits to the guidelines [ask] - for instance by editing in your comment(s) into your question.
B--rian
CC BY-SA 4.0
2021-06-06T00:12:40.357Z
@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..."
uhoh
CC BY-SA 4.0
2021-06-06T03:00:46.240Z
@uhoh you berate jamesk that he shall not speak as if he represents the community, yet you assume the role to speak for the community? Hypocrisy at its best! Please stop making this an unwelcoming place
planetmaker
CC BY-SA 4.0
2021-06-06T04:37:22.230Z
@planetmaker I'm just describing how SE works so well. There is no you .vs. us, there is no "we", there are site policies in the help center and some levels of consensus in meta, it's best to stick to those. And since there IS a resource-request tag with many well-received questions, that goes to show how things can go sideways when one user deems to speak for all rather than cites specific established  conventions or levels of concensus established in meta. So your equivalence can't really be supported.
uhoh
<data-analysis><celestial-mechanics><mathematics><resource-request><orbital-resonance>
Orbital resonances - expansion of disturbing function
Augustin
# Orbital resonances - expansion of disturbing function

(Asked by: username_0 on 2021-06-05)

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)](https://www.springer.com/gp/book/9783540851455), 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?

**Tags:** data-analysis, celestial-mechanics, mathematics, resource-request, orbital-resonance

> **Comments:**
> **username_1**: Book recommendations aren't really what we are about, take a look at [ask].  However over on [physics.se]  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)."  
> **username_1**: $R_{13}$ doesn't appear to be in the expression...  
> **username_0**: Thank you for your advice and patience, Mr. username_1.  
> **username_0**: 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.  
> **username_2**: Welcome to astronomy SE, please try to edit your question so it better fits to the guidelines [ask] - for instance by editing in your comment(s) into your question.  
> **username_3**: @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..."  
> **username_4**: @username_3 you berate jamesk that he shall not speak as if he represents the community, yet you assume the role to speak for the community? Hypocrisy at its best! Please stop making this an unwelcoming place  
> **username_3**: @username_4 I'm just describing how SE works so well. There is no you .vs. us, there is no "we", there are site policies in the help center and some levels of consensus in meta, it's best to stick to those. And since there IS a resource-request tag with many well-received questions, that goes to show how things can go sideways when one user deems to speak for all rather than cites specific established  conventions or levels of concensus established in meta. So your equivalence can't really be supported.  

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## 1 answers

### Answer by username_0 on 2021-06-24. Score: 0 | Accepted answer

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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