The micro-canonical phase-space volume for the three-body problem is a topic of intrinsic interest. Within the flux-based statistical theory, it provides a means to predict the scale of disintegration times for non-hierarchical systems. While the bare phase-volume diverges, Dandekar et al. (Celest Mech Dyn Astron 134(6): 55, 2022. <a href="https://doi.org/10.1007/s10569-022-10108-1">https://doi.org/10.1007/s10569-022-10108-1</a>) (Paper I) showed that a regularized version can be defined. Building on Paper I, which determined the regularized phase-volume for a given energy <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2025_10262_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">${{\bar{\sigma }}}(E)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, this paper extends the analysis to its distribution over angular momentum, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2025_10262_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">${{\bar{\sigma }}}(E,L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Through analytical integrations, we reduce the problem to a 3d numerical integration, a step-up in complexity from the 2d integration required for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2025_10262_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">${{\bar{\sigma }}}(E)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We provide regularized phase-volume values for several mass sets across a range of E and L, validated through an L-integration test. Notably, the values remain positive for all tested parameters, lending further support to the validity of the chosen regularization procedure. For high values of L at fixed masses and E, we observe a strong suppression of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2025_10262_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">${{\bar{\sigma }}}(E,L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Distribution of regularized three-body phase-volume

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2025年10月1日

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Yogesh Dandekar,
Barak Kol

摘要

The micro-canonical phase-space volume for the three-body problem is a topic of intrinsic interest. Within the flux-based statistical theory, it provides a means to predict the scale of disintegration times for non-hierarchical systems. While the bare phase-volume diverges, Dandekar et al. (Celest Mech Dyn Astron 134(6): 55, 2022. https://doi.org/10.1007/s10569-022-10108-1) (Paper I) showed that a regularized version can be defined. Building on Paper I, which determined the regularized phase-volume for a given energy ${{\bar{\sigma }}}(E)$ $\bar{σ} (E)$ , this paper extends the analysis to its distribution over angular momentum, ${{\bar{\sigma }}}(E,L)$ $\bar{σ} (E, L)$ . Through analytical integrations, we reduce the problem to a 3d numerical integration, a step-up in complexity from the 2d integration required for ${{\bar{\sigma }}}(E)$ $\bar{σ} (E)$ . We provide regularized phase-volume values for several mass sets across a range of E and L, validated through an L-integration test. Notably, the values remain positive for all tested parameters, lending further support to the validity of the chosen regularization procedure. For high values of L at fixed masses and E, we observe a strong suppression of ${{\bar{\sigma }}}(E,L)$ $\bar{σ} (E, L)$ .