One of the most renowned classical experiments that verifies the curved nature of space-time geometry is the bending of light. This phenomenon is extensively discussed in most textbooks on general relativity, with a clear and comprehensive explanation provided. In this study, we employ the material medium approach to determine the refractive index associated with the gravitational field of a compact object with a quadrupole moment. Our research presents a method for calculating the gravitational deflection angle for compact objects by utilizing the refractive index and the GBT theorem for an isotropic metric. This method is particularly important because it allows for the calculation of the deflection angle for both light and relativistic particles. The material medium approach enables us to consider the compact object's gravitational field as a medium with a refractive index. By applying this approach, we establish a relationship between the refractive index and the quadrupole moment of the compact object. We then utilize this relationship to calculate the deflection angle of light and relativistic particles. Одним из самых известных классических экспериментов, подтверждающих искривленную природу геометрии пространства – времени, является искривление света. Обычное объяснение отклонения света простое. Когда присутствует массивный объект, световой луч изгибается под углом, который прямо пропорционален замкнутой массе системы в определенной области, обычно называемой прицельным параметром. Существует новый подход, предлагающий уникальный взгляд на отклонение света в присутствии статического сферически-симметричного гравитационного поля. Этот метод подчеркивает значение топологии траектории световых лучей. Для расчета угла отклонения этот метод использует применение теоремы Гаусса-Бонне. В данной работе мы используем подход материальной среды для определения показателя преломления, связанного с гравитационным полем компактного объекта с квадрупольным моментом. В нашем исследовании представлен метод расчета угла гравитационного отклонения для компактных объектов с использованием показателя преломления и теоремы Гаусса-Бонне для изотропной метрики. Этот метод особенно важен, поскольку позволяет вычислить угол отклонения как для легких, так и для релятивистских частиц. Подход материальной среды позволяет рассматривать гравитационное поле компактного объекта как среду с показателем преломления. Применяя этот подход, мы устанавливаем связь между показателем преломления и квадрупольным моментом компактного объекта. Затем мы используем это соотношение для расчета угла отклонения света и релятивистских частиц. Кеңістік-уақыт геометриясының қисық сипатын растайтын ең әйгілі классикалық тәжірибелердің бірі – жарықтың қисықтығы. Жарықтың ауытқуының әдеттегі түсіндірмесі қарапайым. Массивтік нысан болған кезде, жарық сәулесі әдетте әсер ету параметрі деп аталатын белгілі бір аймақтағы жүйенің жабық массасына тура пропорционал бұрышпен иіледі. Статикалық сфералық симметриялы гравитациялық өріс болған кезде жарықтың ауытқуының бірегей перспективасын ұсынатын жаңа көзқарас бар. Бұл әдіс жарық сәулелерінің траекториясының топологиясының маңыздылығын көрсетеді. Бұл әдіс ауытқу бұрышын есептеу үшін Гаусс-Бонне теоремасын қолдануды пайдаланады. Бұл жұмыста төртполюсті моменті бар ықшам нысанның гравитациялық өрісімен байланысты сыну көрсеткішін анықтау үшін материалдық орта тәсілін қолданамыз. Біздің зерттеуіміз сыну көрсеткішін және изотропты метрика үшін Гаусс-Бонне теоремасын пайдалана отырып, жинақы объектілер үшін гравитациялық ауытқу бұрышын есептеу әдісін ұсынады. Бұл әдіс әсіресе маңызды, себебі ол жеңіл және релятивистік бөлшектер үшін ауытқу бұрышын есептеуге мүмкіндік...

In this work, we study shadows from the naked singularity spacetime. The most analytical solutions of black hole shadows only investigated the case that the geodesic equations for photons can separate variables. We review the spherical null naked singularity metric and this spherically symmetric naked singularity spacetime metric is the solution of Einstein equations with an anisotropic fluid source which has no photon sphere. We also review a static, axially-symmetric singular solution of the vacuum Einstein’s equations without an event horizon which is can be used to describe the exterior gravitational field of a mass distribution with quadrupole moment. Moreover, the corresponding spacetime is characterized by the presence of naked singularities. It is theoretically known that not only a black hole can cast shadow, but other compact objects such as naked singularities, gravastar or boson stars can also cast shadows. We present the analytical calculation of shadows for both naked singularities spacetime and compare with the shadow of Schwarzschild static black hole, we show that this can serve as a black hole mimicker. Keywords: compact object, naked singularity, shadow.

This study examines the equilibrium structure and stability of white dwarfs, incorporating both isotropic and anisotropic pressure distributions. The Tolman–Oppenheimer–Volkoff (TOV) equation is numerically solved using the Chandrasekhar equation of state (EoS) to analyze the effects of pressure anisotropy. A general anisotropy function is introduced to close and solve the system of differential equations. The results indicate that anisotropy remains negligible at the center and increases toward the stellar surface. Stability is assessed using the speed of sound criterion, $v_s^2=dp/d\rho$, and the Buchdahl bound, $2M/R<8/9$, confirming that white dwarfs remain within stability limits. We performed a sensitivity analysis to examine how variations in the anisotropy parameter ${\alpha }_0$ and central density affect the mass, radius, and compactness of white dwarfs. Additionally, we calculated the gravitational redshift at the stellar surface and found that it varies with anisotropy, ranging from $z_s\sim 3.15\times {10}^{-3}$ in isotropic cases to $z_s\sim 0.2\times {10}^{-3}$ in highly anisotropic models. These results link anisotropy to potentially observable features. The findings suggest that while anisotropy does not significantly affect the overall equilibrium structure, it may play a role in astrophysical scenarios involving strong magnetic fields, rotational deformations, or accretion processes in binary systems.

We observed top 10 X-class solar flares registered in the period November 1997 - September 2017. We measure physical parameters of 10 solar flares, such as the temporal scale, size, and magnetic flux density, and find that the sizes of flares tend to be distributed more broadly as the GOES class becomes weaker and that there is a lower limit of magnetic flux density that depends on the GOES class. We also made a brief analysis of solar flares registered in these days, also has shown the duration of time and peak of solar flares in Universal time. We have identified several physical quantities of solar flares and estimated reconnection rate of solar flares. To determine the physical parameters we used images taken with the AIA instrument on board SDO satellite at wavelengths 131 Å, 174 Å, 193 Å, 211 Å, 335 Å, 1600 Å, 1700 Å, 4500 Å, SXT - pictures, HMI Magnetogram, SOLIS Chromospheric Magnetogram, GOES XRT-data. We estimate reconnection inflow velocity, coronal Alfven velocity, and reconnection rate using the observed values. The inflow velocities are distributed from a few km s-1 to several tens of km s-1, and the Alfven velocities in the corona are in the range from 103 to 104 km s-1. Hence, the reconnection rate is 10-3. We find that the reconnection rate in a flare tends to decrease as the GOES class of the flare increases.

In this study, we examine how variations in local pressure anisotropy affect the internal structure and equilibrium of white dwarfs. A generalized anisotropy model is developed, defined by three parameters: the amplitude coefficient α₀ and the shape exponents l and k. This formulation ensures that the anisotropic pressure continuously vanishes at both the center and surface of the star while reaching a single peak in the intermediate region. By applying appropriate boundary and regularity conditions, the model allows us to determine physically stable parameter domains consistent with realistic stellar configurations. Our analysis shows that even a small degree of anisotropy can have a measurable effect on the mass–radius relation and overall compactness of white dwarfs, which may help explain the origin of super-Chandrasekhar systems observed in astrophysical data. This modeling approach provides a clear and flexible way to describe compact stars with anisotropic pressures and can also be applied to neutron and quark stars. Keywords: compact stars, white dwarfs, anisotropic pressure, stability analysis, generalized anisotropic factor

We investigate the motion of massive test particles close to the equatorial plane of a weakly deformed q-metric spacetime. Working to first order in the dimensionless quadrupole parameter q, we derive the orbital angular velocity and the vertical epicyclic frequency for equatorial circular orbits while keeping the physical monopole mass fixed. The Schwarzschild limit is recovered at q = 0, whereas a nonzero quadrupole removes the degeneracy between the vertical and azimuthal frequencies already at O(q). A small phenomenological damping term is then introduced into the linearized vertical equation to describe the gradual approach of a perturbed orbit toward the equatorial plane. In the conservative case, the particle continues to oscillate about this plane without any decrease in amplitude. With damping, the amplitude decays exponentially. The quadrupolar deformation changes the vertical oscillation frequency and, consequently, the number of cycles completed before a specified settling threshold is reached. It does not, however, set the decay timescale, which follows from the adopted damping prescription. Since the coordinate-time damping coefficient is kept fixed in our numerical examples, the trajectories share the same exponential envelope but oscillate at different frequencies. The comparison with the exact q-metric also provides a direct check of the firstorder approximation. Over the sampled domain |q| ≤ 0.1 and 6 ≤ r/M ≤ 20, the relative error in the vertical frequency remains below 0.34%, while the corresponding frequency ratio is reproduced to better than 0.10%. These results show explicitly how the quadrupolar geometry modifies vertical orbital dynamics within a model in which dissipative settling is introduced independently.