31:[{"id":"175a564cac34977969d2d55c8f25f15ae5c7842eb659e170c52a70a53589bdc96b2750a1be5397f7668967f8954e72271bf26d992c469a8102dee020eb490b4a","href":"https://static.observableusercontent.com/files/175a564cac34977969d2d55c8f25f15ae5c7842eb659e170c52a70a53589bdc96b2750a1be5397f7668967f8954e72271bf26d992c469a8102dee020eb490b4a","name":"Laplace_coefficients_Traite_mecanique_celeste_III_p66.png","type":"image/png","lastModified":1662405021652.158,"size":410626}] 2f:{"title":"Laplace coefficients","cells":[{"id":0,"value":"# Laplace coefficients\n## (secular perturbation theory)","pinned":false,"hidden":false,"mode":"md"},{"id":27,"value":"introduction = md `

Laplace coefficients (also known as Legendre coefficients) are essential coefficients that intervene in celestial mechanics calculations.^{[[1](#Murray_Dermott),[2](#Fitzpatrick)]} In the secular perturbation theory, they are used in the development of the planetary disturbing function. To evaluate this disturbing function, one needs first to calculate the Fourier cosine expansion of a function ${tex`b_s^{(m)}(\\alpha)`} that depends on:

${tex`\\alpha = \\frac{\\textrm{min}\\{a,a'\\}}{\\textrm{max}\\{a,a'\\}}`}, the ratio of the semi-major axes of the disturbed and disturbing planets (${tex`a`} and ${tex`a'`}, respectively),
the angular separation ${tex`\\psi`} of their orbital positions.

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":670,"value":"
\n

The Laplace coefficient, ${tex`b_s^{(m)}(\\alpha)`}, is defined as follows:

","pinned":false,"hidden":false,"mode":"html"},{"id":12,"value":"tex.block`\nb_s^{(m)}(\\alpha) = \\frac{1}{\\pi} \\int_{0}^{2\\pi} \\frac{\\cos(m\\psi)}{ \\big( 1 - 2\\alpha\\cos(\\psi) + \\alpha^2 \\big)^s} \\,d\\psi \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":37,"value":"where ${tex`m`} is a natural number, \n
${tex`s`} is a half-integer,\n
and ${tex`\\alpha`} is a positive real number < 1:","pinned":false,"hidden":false,"mode":"html"},{"id":29,"value":"tex.block`m \\in \\N_0 =\\{0,1,2,...\\}, \\\\\ns \\in \\Z + \\frac{1}{2}, \\\\\n\\alpha \\in \\R_+ \\; : \\; 0 < \\alpha < 1. \\\\\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"hypergeometric_series = md `

This function can be represented by hypergeometric series. The series expansion of ${tex`b_s^{(m)}(\\alpha)`} is ^{[[3](#Izsak1963),[4](#Mardling2013)]}:

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":116,"value":"tex.block`\nb_s^{(m)}(\\alpha) = 2\\alpha^m \\sum_{p=0}^{\\infin} \\; \\frac{\\Gamma(s+p)\\;\\Gamma(s+m+p)}{[\\Gamma(s)]^2 \\; p! \\; (m+p)!} \\; \\alpha^{2p}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":686,"value":"

Note here that the Gamma function (represented by the Greek capital letter ${tex`\\Gamma`}) is an extension of the factorial function to complex numbers (${tex`\\Complex`}).

","pinned":false,"hidden":false,"mode":"html"},{"id":577,"value":"

Calculation

Click on the sliders or enter new values below.

\n","pinned":false,"hidden":false,"mode":"html"},{"id":543,"value":"viewof m = Inputs.range([0, 10], {value: 0, step: 1, label: \"m = \"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":315,"value":"viewof n = Inputs.range([-10, 10], {value: -1, step: 1, label: \"s = n + ½, with n = \"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":302,"value":"viewof alpha = Inputs.range([0.00000001, 0.99999999], {value: 0.53516076, step: 0.00000001, label: \"α = \"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":590,"value":"viewof p_max = Inputs.range([20, 200], {value: 200, step: 1, label: \"max {p} = \"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":575,"value":"
","pinned":false,"hidden":false,"mode":"html"},{"id":300,"value":"tex.block`\n\n\\begin{array}{rl}\n & m = ${m}, \\quad s = ${math.sign(2*n+1)=='-1'? '-':''}\\frac{${math.abs(2*n+1)}}{2}, \\quad \\alpha = ${alpha} \\\\ \\\\ \\\\\n b_{${2*n+1}/2}^{(${m})}(${alpha}) &\n = \\quad ${b_m_s_alpha(m,n+0.5,alpha,p_max)[0]} \\qquad (p = 0 \\,... \\, ${b_m_s_alpha(m,n+0.5,alpha,p_max)[1]}) \\\\ \\\\\n \\textcolor{#808080}{\n b_{${2*n+1}/2}^{(${m})}(${alpha})} &\n \\textcolor{#808080}{\n = \\quad ${b_m_s_alpha(m,n+0.5,alpha,p_last-1)[0]} \\qquad (p = 0 \\,... \\, ${b_m_s_alpha(m,n+0.5,alpha,p_last-1)[1]})} \\\\ \\\\\n \\textcolor{#C0C0C0}{\n b_{${2*n+1}/2}^{(${m})}(${alpha})} &\n \\textcolor{#C0C0C0}{\n = \\quad ${b_m_s_alpha(m,n+0.5,alpha,p_last-2)[0]} \\qquad (p = 0 \\,... \\, ${b_m_s_alpha(m,n+0.5,alpha,p_last-2)[1]})}\n \n\\end{array}\n\n \n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":711,"value":"examples = md `

Examples

Hereafter, we show examples of calculations found in Laplace's \"Traité de mécanique céleste\" (Tome III, page 66).^{[[5](#Laplace1802)]}

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":720,"value":"Laplace_coefficient = html`\n

P. S. Laplace, \"*Traité de mécanique céleste*,\" Tome III, p. 66, 1802. \n
[ Source: Gallica archive ]

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":573,"value":"
","pinned":false,"hidden":false,"mode":"html"},{"id":143,"value":"## References","pinned":false,"hidden":false,"mode":"md"},{"id":626,"value":"Murray_Dermott = reference(\"introduction\", 1, \"C. D. Murray & S. F. Dermott, \\\"Solar System Dynamics,\\\" Cambridge University Press, 2000. DOI: 10.1017/CBO9781139174817\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":636,"value":"Fitzpatrick = reference(\"introduction\", 2, \"R. Fitzpatrick, \\\"An Introduction to Celestial Mechanics,\\\" Cambridge University Press, 2012. DOI: 10.1017/CBO9781139152310\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":152,"value":"Izsak1963 = reference(\"hypergeometric_series\", 3, \"I. G. Izsak, \\\"Laplace coefficients and their Newcomb derivatives,\\\" SAO Special Report 129, 1963. [PDF]\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":183,"value":"Mardling2013 = reference(\"hypergeometric_series\", 4, \"R. A. Mardling, \\\"New developments for modern celestial mechanics — I. General coplanar three-body systems. Application to exoplanets,\\\" Monthly Notices of the Royal Astronomical Society 435 (3), pp. 2187–2226, November 2013. DOI: 10.1093/mnras/stt1438\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":704,"value":"Laplace1802= reference(\"example\", 5, \"P. S. Laplace, \\\"Traité de mécanique céleste,\\\" Tome III, 1802. [Gallica archive]\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":10,"value":"---\n## Appendices","pinned":false,"hidden":false,"mode":"md"},{"id":199,"value":"p_last = b_m_s_alpha(m,n+1/2,alpha,p_max)[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":197,"value":"function b_m_s_alpha (m,s,alpha, P_max = 100){ // P_max is an optional parameter\n let b_m_s_alpha = 0;\n let sum = 0;\n let sum_test = 0;\n let p = 0;\n for(p; p <= P_max && Number.isFinite(sum_test); p++){\n sum_test = math.gamma(s+p)/math.pow(math.gamma(s),2); \n sum_test /= math.factorial(p);\n sum_test *= math.gamma(s+m+p)*math.pow(alpha,2*p);\n sum_test /= math.factorial(m+p);\n Number.isFinite(sum_test)? sum += sum_test : null; // Check if the calculated value is a finite number\n }\n b_m_s_alpha = 2*math.pow(alpha,m)*sum;\n return [b_m_s_alpha,p-1];\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"math = require('mathjs@6.2.1/dist/math.js')","pinned":false,"hidden":false,"mode":"ojs"},{"id":147,"value":"// https://observablehq.com/@jashkenas/markdown-style-footnotes\nreference = (referencingCellName, number, text) => {\n return html`

^ ${text}

`;\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}