High-Order Analytic Expansion of Disturbing Function for Doubly Averaged Circular Restricted Three-Body Problem

<jats:p>Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> between perturbed and perturbing bodies in the inner case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, and up to the fifteenth order in the outer case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>></mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the doubly averaged disturbing function yields a result that agrees well with the numerical quadrature and with the numerical integration. Local extremums of the doubly averaged disturbing function are quantitatively reproduced by the high-order analytic expansion even when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math> is large. Although the analytic expansion is not applicable in some circumstances such as when orbits of perturbed and perturbing bodies cross or when strong mean motion resonance is at work, our expansion result will be useful for analytically understanding the long-term dynamical behavior of perturbed bodies in circular restricted three-body systems.</jats:p>