![]() And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function (CHF), we find two linearly dependent solutions corresponding to the same eigenstate, from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant. We first convert the angular Teukolsky equation under the special condition of τ ≠ 0, s ≠ 0, m = 0 into a confluent Heun differential equation (CHDE) by taking different function transformation and variable substitution. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving ( a / r L ) ² and ( a / R ) ² where a is the stellar oblateness or prolateness, R the smallest star radius, and r L the light-cylinder radius. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Particular emphasis is put on the magnetic dipole radiation. The solutions are expanded in series of radial and angular spheroidal wave functions. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations.Īims. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. $latex $ and this does not cover the four quadrants of the Cartesian plane.Context. The x and y coordinates form the legs of the triangle and r forms the hypotenuse. We can use a right triangle and the Pythagorean theorem to find the value of r in terms of x and y. ![]()
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