液态Lane-Emden星的线性稳定性

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
K. Lam
{"title":"液态Lane-Emden星的线性稳定性","authors":"K. Lam","doi":"10.1090/qam/1677","DOIUrl":null,"url":null,"abstract":"<p>We establish various qualitative properties of liquid Lane-Emden stars in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, including bounds for its density profile <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho\">\n <mml:semantics>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\rho</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and radius <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than-or-equal-to 2 left-parenthesis d minus 1 right-parenthesis slash d\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma \\geq 2(d-1)/d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>; linearly stable when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than 2 left-parenthesis d minus 1 right-parenthesis slash d\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma >2(d-1)/d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for stars with small relative central density <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho left-parenthesis 0 right-parenthesis minus rho left-parenthesis upper R right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>ρ<!-- ρ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\rho (0)-\\rho (R)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>; and linearly unstable when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than 2 left-parenthesis d minus 1 right-parenthesis slash d\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma >2(d-1)/d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.</p>","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Linear stability of liquid Lane-Emden stars\",\"authors\":\"K. Lam\",\"doi\":\"10.1090/qam/1677\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We establish various qualitative properties of liquid Lane-Emden stars in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript d\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>d</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}^d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, including bounds for its density profile <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"rho\\\">\\n <mml:semantics>\\n <mml:mi>ρ<!-- ρ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\rho</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and radius <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R\\\">\\n <mml:semantics>\\n <mml:mi>R</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">R</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than-or-equal-to 2 left-parenthesis d minus 1 right-parenthesis slash d\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mi>d</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma \\\\geq 2(d-1)/d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>; linearly stable when <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than 2 left-parenthesis d minus 1 right-parenthesis slash d\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mi>d</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma >2(d-1)/d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for stars with small relative central density <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"rho left-parenthesis 0 right-parenthesis minus rho left-parenthesis upper R right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ρ<!-- ρ --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi>ρ<!-- ρ --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>R</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\rho (0)-\\\\rho (R)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>; and linearly unstable when <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than 2 left-parenthesis d minus 1 right-parenthesis slash d\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mi>d</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma >2(d-1)/d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.</p>\",\"PeriodicalId\":20964,\"journal\":{\"name\":\"Quarterly of Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-08-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quarterly of Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/qam/1677\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly of Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/qam/1677","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2

摘要

我们建立了Rd中液态Lane-Emden星的各种定性性质,包括其密度分布ρ\rho和半径R R的边界。利用它们,我们证明了当γ≥2(d−1)/d\gamma\geq2(d-1)/d时,对于径向扰动,液态Lane-Emden星是线性稳定的;对于相对中心密度ρ(0)−ρ(R)\rho(0)-\rho(R)较小的恒星,当γ>2(d−1)/d\gamma>2(d-1)/d时线性稳定;当γ>2(d-1)/d时,中心密度较大的恒星线性不稳定。这种对中心密度的依赖性在气态的莱恩-埃姆登恒星中是看不到的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Linear stability of liquid Lane-Emden stars

We establish various qualitative properties of liquid Lane-Emden stars in R d \mathbb {R}^d , including bounds for its density profile ρ \rho and radius R R . Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when γ 2 ( d 1 ) / d \gamma \geq 2(d-1)/d ; linearly stable when γ > 2 ( d 1 ) / d \gamma >2(d-1)/d for stars with small relative central density ρ ( 0 ) ρ ( R ) \rho (0)-\rho (R) ; and linearly unstable when γ > 2 ( d 1 ) / d \gamma >2(d-1)/d for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.

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来源期刊
Quarterly of Applied Mathematics
Quarterly of Applied Mathematics 数学-应用数学
CiteScore
1.90
自引率
12.50%
发文量
31
审稿时长
>12 weeks
期刊介绍: The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume. This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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