Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n = 0 and n = 1

dc.citation.epage358
dc.citation.issue2
dc.citation.spage338
dc.contributor.affiliationЛьвівський національний університет імені Івана Франка
dc.contributor.affiliationIvan Franko National University of Lviv
dc.contributor.authorВ Ваврух, М.
dc.contributor.authorДзіковський, Д. В.
dc.contributor.authorVavrukh, M. V.
dc.contributor.authorDzikovskyi, D. V.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2023-10-24T07:21:49Z
dc.date.available2023-10-24T07:21:49Z
dc.date.created2021-03-01
dc.date.issued2021-03-01
dc.description.abstractРозрахунки характеристик зір з осьовим обертанням у рамках політропної моделі грунтуються на розв’язку рівняння рівноваги — диференціального рівняння другого порядку в частинних похідних. Різні варіанти наближеного визначення сталих інтегрування засновані на традиційному в теорії зоряної поверхні наближенні, а саме: умові неперервності гравітаційного потенціалу в околі поверхні. Нами запропоновано новий підхід, в якому одночасно використовуються диференціальна та інтегральна форми рівняння рівноваги. Ця замкнута система дозволяє самоузгоджено визначити сталі інтегрування, форму поверхні політропи та розподіл речовини за об’ємом зорі. На прикладі політропи n = 0 і n = 1 встановлено існування двох режимів обертання (з малими та великими ексцентриситетами). У випадку n = 0 доведено, що поверхня політропи є поверхнею однорідного еліпсоїда обертання. Розраховано характеристики політропи n = 1 у різних наближеннях як функції кутової швидкості. Вперше розраховано відхилення поверхні політропи при заданому значенні кутової швидкості від поверхні асоційованого еліпсоїда обертання.
dc.description.abstractCalculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives. Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity. We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations. This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star. With the examples of polytropes n = 0 and n = 1, we established the existence of two rotation modes (with small and large eccentricities). It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n = 0. The polytrope characteristics with n = 1 in different approximations were calculated as the functions of angular velocity. For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.
dc.format.extent338-358
dc.format.pages21
dc.identifier.citationVavrukh M. V. Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n = 0 and n = 1 / M. V. Vavrukh, D. V. Dzikovskyi // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 2. — P. 338–358.
dc.identifier.citationenVavrukh M. V. Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n = 0 and n = 1 / M. V. Vavrukh, D. V. Dzikovskyi // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2021. — Vol 8. — No 2. — P. 338–358.
dc.identifier.doidoi.org/10.23939/mmc2021.02.338
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/60387
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofMathematical Modeling and Computing, 2 (8), 2021
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dc.relation.referencesen[2] Smith F. G. Pulsars. Cambridge University Press (1977).
dc.relation.referencesen[3] Lane H. On the Theoretical Temperature of the Sun under the Hypothesis of a gaseous mass Maintining Its Volume by Its Internal Heat and Depending on the Law of Gases Known to Terrestrial Experiment. American Journal of Science. s2-50 (148), 57–74 (1870).
dc.relation.referencesen[4] Emden R. Gaskugeln: Anwendungen der mechanischen W¨armetheorie auf kosmologische und meteorologische Probleme. Leipzig, Berlin (1907), (in German).
dc.relation.referencesen[5] Fowler R. H. Emden’s equation: The solutions of Emden’s and similar differential equations. MNRAS. 91 (1), 63–91 (1930).
dc.relation.referencesen[6] Eddington A. S. The Internal Constitution of the Stars. Cambridge University Press (1926).
dc.relation.referencesen[7] Milne E. A. The equilibrium of a rotating star. MNRAS. 83 (3), 118–147 (1923).
dc.relation.referencesen[8] Chandrasekhar S. The Equilibrium of Distorted Polytropes. I. The Rotational Problem. MNRAS. 93 (5), 390–406 (1933).
dc.relation.referencesen[9] James R. A. The Structure and Stability of Rotating Gas Masses. Astrophysical Journal. 140, 552–582 (1964).
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dc.relation.referencesen[12] Monaghan J. J., Roxburgh I. W. The Structure of Rapidly Rotating Polytropes. MNRAS. 131 (1), 13–22 (1965).
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dc.relation.referencesen[17] Vavrukh M. V., Smerechynskyi S. V., Tyshko N. L. The microscopic parameters and the macroscopic characteristics of real degenerate dwarfs. Journal of Physical Studies. 14 (4), 4901-1–4901-16 (2010).
dc.relation.referencesen[18] Vavrukh M. V., Dzikovskyi D. V., Smerechynskyi S. V. Consideration of the competing factors in calculations of the characteristics of non-magnetic degenerate dwarfs. Ukr. J. Phys. 63 (9), 777–789 (2018).
dc.relation.referencesen[19] Kong D., Zhang K., Schubert G. An exact solution for arbitrarily rotating gaseous polytropes with index unity. MNRAS. 448 (1), 456–463 (2015).
dc.relation.referencesen[20] Knopik J., Mach P., Odrzywo lek A. The shape of a rapidly rotating polytrope with index unity. MNRAS. 467 (4), 4965–4969 (2017).
dc.relation.referencesen[21] Shapiro S. L., Teukolsky S. A. Black Holes, White Dwarfs and Neutron Stars. Cornell University, Ithaca, New York (1983).
dc.relation.referencesen[22] Chandrasekhar S. An Introduction to the Study of Stellar Structure. University of Chicago Press, Chicago (1939).
dc.relation.referencesen[23] Lyttleton R. A. The Stability of Rotating Liquid Masses. Cambridge University Press, Cambridge (1953).
dc.relation.referencesen[24] Abramowitz M., Stegun I. A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Government Printing Office Washington (1972).
dc.relation.referencesen[25] Duboshin G. N. Celestial Mechanics, Basic Problems and Methods. Nauka, Moscow (1968).
dc.rights.holder© Національний університет “Львівська політехніка”, 2021
dc.subjectзорі-політропи
dc.subjectнеоднорідні еліпсоїди
dc.subjectосьове обертання
dc.subjectрівняння механічної рівноваги
dc.subjectстабільність зір
dc.subjectpolytropic stars
dc.subjectheterogeneous ellipsoids
dc.subjectaxial rotation
dc.subjectmechanical equilibrium equation
dc.subjectstability of stars
dc.titleMethod of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n = 0 and n = 1
dc.title.alternativeМетод інтегральних рівнянь у політропній теорії зір з осьовим обертанням. І. Політропи n = 0 і n = 1
dc.typeArticle

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