<p>Observations of stars other than the Sun are sensitive to oscillations of only low degree. Many are high-order acoustic modes. Acoustic frequencies of main-sequence stars, for example, satisfy a well-known pattern, which some astronomers have adopted even for red-giant stars. That is not wise, because the internal structures of these stars can be quite different from those on the Main Sequence, which is populated by stars whose structure is regular. Here I report on pondering this matter, and point out two fundamental deviations from the commonly adopted relation. There are aspects of the regular relation that are connected in a simple way to gross properties of the star, such as the dependence of the eigenfrequencies on the linear combination <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10509_2025_4409_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mi>l</mi> </math></EquationSource> <EquationSource Format="TEX">$n+\textstyle {\frac{1}{2}}l$</EquationSource> </InlineEquation> of the order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10509_2025_4409_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation> and degree <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10509_2025_4409_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>l</mi> </math></EquationSource> <EquationSource Format="TEX">$l$</EquationSource> </InlineEquation>, which is characteristic of a regular spherical acoustic cavity. That is not a feature of red-giant frequencies, because, as experienced by the waves, red-giant stars appear to have (phantom) singular centres, which substantially modify the propagation of waves. That requires a generalization of the eigenfrequency relation, which I present here. When fitted to the observed frequencies of the Sun, the outcome is consistent with the Sun being round, with no singularity in the core. That is hardly novel, but at least it provides some assurance that our understanding of stellar acoustic wave dynamics is on a sound footing.</p>

Some musings on erythrogigantoacoustics

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2025年2月27日

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Douglas Gough

摘要

Observations of stars other than the Sun are sensitive to oscillations of only low degree. Many are high-order acoustic modes. Acoustic frequencies of main-sequence stars, for example, satisfy a well-known pattern, which some astronomers have adopted even for red-giant stars. That is not wise, because the internal structures of these stars can be quite different from those on the Main Sequence, which is populated by stars whose structure is regular. Here I report on pondering this matter, and point out two fundamental deviations from the commonly adopted relation. There are aspects of the regular relation that are connected in a simple way to gross properties of the star, such as the dependence of the eigenfrequencies on the linear combination $n + \frac{1}{2} l$ $n+\textstyle {\frac{1}{2}}l$ of the order $n$ $n$ and degree $l$ $l$ , which is characteristic of a regular spherical acoustic cavity. That is not a feature of red-giant frequencies, because, as experienced by the waves, red-giant stars appear to have (phantom) singular centres, which substantially modify the propagation of waves. That requires a generalization of the eigenfrequency relation, which I present here. When fitted to the observed frequencies of the Sun, the outcome is consistent with the Sun being round, with no singularity in the core. That is hardly novel, but at least it provides some assurance that our understanding of stellar acoustic wave dynamics is on a sound footing.