Journal of Physical Studies 26(1), Article 1902 [24 pages] (2022)
DOI: https://doi.org/10.30970/jps.26.1902

METHOD OF INTEGRAL EQUATIONS IN THE THEORY OF ROTATIONAL WHITE DWARFS

M. Vavrukh , S. Smerechynskyi , D. Dzikovskyi 

Ivan Franko National University of Lviv,
8, Kyrylo & Methodiy St., Lviv, UA–79005, Ukraine
e-mail: mvavrukh@gmail.com

Received 26 October 2021; in final form 18 January 2022; accepted 22 February 2022; published online 29 March 2022

We propose a new analytical approach to the calculation of a massive white dwarf's characteristics with axial rotation based on the Chandrasekhar model. The defining feature of our approach is the simultaneous usage of the differential and integral forms of the equilibrium equation. In the dimensionless form, the differential equilibrium equation is a strongly non-linear inhomogeneous equation of the second order in partial derivatives, which contains two dimensionless parameters – the relativistic parameter in stellar center $x_0$ and dimensionless angular velocity $Ω$. In the dwarf's interior, the rotation is taken into account as perturbation in the linear approximation for $Ω^2$. In the peripheral region, the rotation is considered to be the main factor. The use of the integral equation ensures correct calculation of integration constants. The dwarf's mass, moment of inertia relative to the axis of rotation, equatorial and polar radii, the acceleration of free fall at the equator in the region $1≤ x_0≤24$, $0≤Ω<Ω_{\rm max}(x_0)$ were calculated based on the solution of the equilibrium equation. For the first time, the total energy of the dwarf was calculated as a function of these parameters. Using the extrapolation method, we calculated the maximal values $Ω_{\rm max}(x_0)$, as well as the angular velocity $ω_{\rm max}(x_0)$. The calculation results for the mass and the equatorial radius are in good agreement with the results of the work [R. A. James, Astrophys. J. 140, 552 (1964)], which are based on the numerical calculation of the equilibrium equation for dwarfs of small and intermediate masses $(0.5≤ x_0≤6.245)$. Figures \ref{fig_05wd} and \ref{fig_06wd} illustrate the ranges for the mass of a rotating dwarf and its total energy. The influence of the Coulomb interparticle interactions causes a decrease in the pressure of the degenerate electron subsystem and, consequently, a decrease in the maximal mass of the dwarf without rotation (see [I. W. Roxburgh, Z. Astrophys. 62, 134 (1965)]). It means that the massive dwarfs in binary systems rotate with velocities close to $ω_{\rm max}(x_0)$. Among the field white dwarfs of intermediate masses, there are also some where rotation plays a significant role. The proposed approach allows the estimation of the angular velocity for specific observed white dwarfs according to their masses and radii. However, it requires a more accurate description of the white dwarfs without rotation in a zero approximation model; instead of the Chandrasekhar model, another model should be used, which takes into account interparticle interactions.

Key words: white dwarfs, axial rotation, mechanical equilibrium equation, relativistic parameter, interparticle interactions.

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