The Equilibrium of Distorted Polytropes: (I). The Rotational Problem

Chandrasekhar, S.; Milne, E. A.

doi:10.1093/mnras/93.5.390

Cited by 227 publications

(212 citation statements)

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“…It is clear that for a given central density the value of the total mass is larger in the case of a rotating object than for a static body. This is in accordance with the physical expectations based upon other alternative studies [18][19][20][21][22]. A similar behavior takes place when we explore the mass as a function of the equatorial radius, as shown in Fig.…”

Section: An Example: White Dwarfssupporting

confidence: 92%

Hartle formalism for rotating Newtonian configurations

et al. 2016

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We apply the Hartle formalism to study equilibrium configurations in the framework of Newtonian gravity. This approach allows one to study in a simple manner the properties of the interior gravitational field in the case of static as well as stationary rotating stars in hydrostatic equilibrium. It is shown that the gravitational equilibrium conditions reduce to a system of ordinary differential equations which can be integrated numerically. We derive all the relevant equations up to the second order in the angular velocity. Moreover, we find explicitly the total mass, the moment of inertia, the quadrupole moment, the polar and equatorial radii, and the eccentricity of the rotating body. We also present the procedure to calculate the gravitational Love number. We test the formalism in the case of white dwarfs and show its compatibility with the known results in the literature.

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Section: An Example: White Dwarfssupporting

confidence: 92%

Hartle formalism for rotating Newtonian configurations

et al. 2016

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“…17. Its structure is then similar to a distorted polytrope of index n = 3/2 as computed by Chandrasekhar (1933) in the limit of slow rotation.…”

Section: Rotating Self-gravitating Fermionsmentioning

confidence: 84%

“…The functions ψ j (ξ) have been defined and tabulated by Chandrasekhar (1933). The inner and outer potentials are given by…”

Section: Resultsmentioning

confidence: 99%

“…Now, following a procedure that dates back to Milne (1923) and Chandrasekhar (1933), we shall assume for Θ the following form…”

Section: The Rotating Isothermal Spherementioning

confidence: 99%

“…However, we shall restrict ourselves in the present paper to the case of slowly rotating systems for which the problem can be tackled analytically. We shall adapt to the case of isothermal spheres the classical procedure developed by Chandrasekhar (1933) for distorted polytropes, i.e. we shall expand the solutions of the Boltzmann-Poisson equation in terms of Legendre polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Gravitational instability of slowly rotating isothermal spheres

Chavanis

2002

A&A

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Abstract. We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plane is deformed by rotation. We find that gravitational instability occurs sooner in the microcanonical ensemble and later in the canonical ensemble. According to standard turning point arguments, the onset of the collapse coincides with the minimum energy or minimum temperature state in the series of equilibria. Interestingly, it happens to be close to the point of maximum flattening. We generalize the singular isothermal solution to the case of a slowly rotating configuration. We also consider slowly rotating configurations of the self-gravitating Fermi gas at non-zero temperature.

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Integralgleichungstheorie des Sternaufbaus IV. Über rotierende Polytropen

Bucerius¹

1947

Astron Nachr

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Integralgleichungstheorie des Sternaufbaus IV. Ober rotierende Polytropen ( V e r o f f e n t l i c h u n g e n d e r S t e r n w a r t e M u n c h e n B d . 3 N r . 4) Von H. BUCERIUS, Munchen Mit 2 Abbildungen (Eingegangen 1947 April 15)Das Figurenproblem gravitierender Massen, deren innerer Aufbau einer polytropen Druck-Dichte-Beziehung genugt, unter Einwirkung zeitlich konstanter, von einem Potential ableitbarer aufierer Krafte wird als Randwertproblem der Potentialtheorie aufgefaot, wobei der Rand, d. h. die Oberflache der Gleichgewichtsfigur Als Niveaufliiche des Gesamtpotrntials, noch von der gesuchten Potentialfunktion abhiingig ist. I m I . Abschnitt werden zunachst die klassischen Ergebnisse iiber inkompressible Flussigkeiten in den allgemeinen Zusammenhang eingeordnet ; anschliefiend wird die rotierende LAPLACESChe Polytrope behandelt. Ini 3. Abschnitt wird der Grund gelegt fur eine Behandlung des allgemeinen Problems der rotierenden Polytropen voni Integralgleichungs-Standpunkt. Der Anhang bringt eine Anwendung der Methode auf die Frage nach rotationssymmetrischen Verzweigungen der i\IACLAURINSChen Ellipsoide. Es ergeben sich unendlich vide Verzweigungsmoglichkeiten, die dem Ast der stark abgeplatteten Ellipsoide angehoren, deren erste ausfuhrlich untersucht wird und eine wulstformige Gleichgewichtsfigur als ubergangsforni zur Ringablosung darstellt. definiert ist Dieser Kern K (r, r'; 0, 0') hat an der Stelle Y = r', 0 = 0' eine logarithmische Singularitat. Der Ansatz der Rotationsspmetrie ist natiirlich nur statthaft, wenn das Potential der aufgezwungenen Kraft auch VOXI rp unabhangig ist, so daB ein R (0) aus 0 ( R , 0) = const. folgt.Analog GI. (2) besteht fur den Kern der Integralgleichung folgende Entwicklungsmoglichkeit nach Kugelfunkt ionen: K (r, r'; 0, 0'; 9, v') = wobei wir noch von dem Additionstheorem der Kugelfunktionen ( n -m ) ! P, (cos y ) = P, (cos 0) P, (cos 0') + 2 2 PT (cos 0) P r (cos 0') * cos m (p1-9') m = l (n + m) Gebrauch machen. Beim rotationssymmetrischen Kern hat man die Entwicklung K ( Y , Y ' ; 0, 0') = ._ -Setzt man G1. (9) in (8) ein, so ergibt sich 4*

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The Equilibrium of Distorted Polytropes: (I). The Rotational Problem

Cited by 227 publications

References 0 publications

Hartle formalism for rotating Newtonian configurations

Hartle formalism for rotating Newtonian configurations

Gravitational instability of slowly rotating isothermal spheres

Integralgleichungstheorie des Sternaufbaus IV. Über rotierende Polytropen

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