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In astrophysics, the gravitational stability of a self-gravitating polytropic fluid sphere is an intriguing subject, especially when trying to comprehend the genesis and development of celestial bodies like planets and stars. This stability is the sphere's capacity to stay in balance in the face of disruptions. We utilize fractional calculus to explore self-gravitating, hydrostatic spheres governed by a polytropic equation of state \P=K\rho^{1+1/n}. We focus on structures with polytropic indices ranging from 1 to 3 and consider relativistic and fractional parameters, denoted by \sigma and \alpha, respectively. The stability of these relativistic polytropes is evaluated using the critical point method, which is associated with the energetic principles developed in 1964 by Tooper. This approach enables us to pinpoint the critical mass and radius at which where polytropic spheres shift from stable to unstable states. The results highlight the critical relativistic parameter where the polytrope's mass peaks, signaling the onset of radial instability. For polytropic indices of 1, 1.5, 2, and 3 with a fractional parameter \alpha, we observe stable relativistic polytropes for \sigma values below the critical thresholds of \sigma= 0.42, 0.20, 0.10, and 0.0, respectively. Conversely, instability emerges as \sigma surpasses these critical values. Our comprehensive calculations reveal that the critical relativistic value (\sigma_{CR}) for the onset of instability tends to increase as the fractional parameter {\alpha} decreases.

Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman-Oppenheimer-Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Rieman Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions to solve the fractional TOV equations yields a limited physical range to the convergent power series solution. Therefore, the two techniques of Euler-Abel transformation and Pade approximation have been combined to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters (\sigma), we calculated twenty fractional gas models for the polytropic indexes n=0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n=0.5 are that the sphere's volume and mass decrease with increasing \sigma and the fractional parameter (\alpha). For n=1, the volume decreases when \sigma=0.1 and then increases when \sigma=0.2 and 0.3. The volume of the sphere reduces as both \sigma and \alpha increase for n=1.5 and n=2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n=3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters (\alpha=1, \sigma=0.001). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit=1.63348 M at \alpha=0.95 and \sigma=0.001.

Monte Carlo (MC) method played an essential role in many areas of human activity and has found application in many branches of physical science. This paper proposes a computational technique based on MC algorithms to solve Lane-Emden (LE) type equations. We analyze four LE equations arising in astrophysics: the positive and negative indices of the polytropic gas spheres, the isothermal gas sphere, and the white dwarf equation. We calculated eleven models (i.e., eleven LE equations) of the positive index polytropes, nine for the negative index polytrope, the isothermal gas sphere, and the white dwarf equation. Comparing the MC and numerical/analytical models gives good agreement for the four LE equations under study.

We present BVR observations and low-resolution spectra collected by the Kottamia Astronomical Observatory 1.88 m telescope (KAO) for the new pulsating star ASAS J063309+1810.8 (hereafter it will be called ASAS06+18). The photometric analysis revealed that the star is a $\delta$ Scuti star with low amplitude (a=0.054-0.099 in V mag.) and a short period (102.604 min). Fourier analysis of the light curves reveals the fundamental mode with two harmonics. The photometric analysis yielded a new value of the updated frequency of 13.0035232 cd-1 with an amplitude of 49.93 mmag at phases 0.326 and S/N 21.75 and to two frequencies (20.2099237cd-1, 5.9130945cd-1). Given the available data, 37 new times of maximum light are presented, and an updated ephemeris for the star and its O-C data. Assuming its period decreases and changes smoothly, a new value of (1/P)dP/dt is determined. We calculated the effective temperature and surface gravity as Teff=7125+- 250 K and log g=4.0+-0.2 dex from model atmosphere analysis of the star's spectra at different phases. The bolometric magnitude Mbol=2.798 (0.016), radius R=1.577(0.077) R_sun, luminosity L=5.714(1.066) L_sun, the mass is M=1.595 M_sun and pulsation constant Q=0{\^m}.0338(0.0003). The star's locations in the evolutionary mass-luminosity and mass-radius relationships are discussed.

Physical conditions deep within planets and exoplanets have yet to be measured directly, but indirect methods can calculate them. The polytropic models are one possible solution to this problem. In the present paper, we assume that the interiors of planets follow a polytropic equation of state. Hydrostatic equilibrium conditions are used to determine the overall structural properties of the constituent matter. In the frame of the conformable fractional derivatives, we use polytropic gas spheres to model the density profiles, pressure profiles, temperature distributions, and the mass-radius relations for the interiors of the initial stage of exoplanets. Planets of single chemical composition were used to study the behavior of the mass-radius relation, pressure distributions, and temperature distribution variation with the fractional parameter. We calculated 72 fractional models for the mass of protoplanets of 1MJ, 3MJ, and 10MJ (MJ is the mass of Jupiter), and the values of the polytropic index are n=0, 0.5, 1, 1.5, and the fractional parameter rang 0.75-1., Comment: 19 pages; 6 figures; 2 tables

A matter bouncing entropy-corrected cosmological model has been suggested. The model allows only positive curvature with negative pressure and no violation of the null energy condition. The result obtained in this paper is supported by some recent theoretical works where the combination of positive spatial curvature and vacuum energy leads to non-singular bounces with no violation of the null energy condition. An important feature of the current model is that evolutions of the cosmic pressure, energy density and equation of state parameter are independent of the values of the prefactors $\alpha$ and $\beta$ in the corrected entropy-area relation. The validity of the classical and the new nonlinear energy conditions has been discussed. The cosmographic parameters have been analyzed, Comment: http://www.astroscu.unam.mx/RMxAA/abstracts/PDF/RMxAA..58-2_nahmed-VII.pdf , https://www.astroscu.unam.mx/RMxAA/accepted.html

In the present paper, we report the photometric and spectroscopic observations obtained by the 1.88 m telescope at the Kottamia astronomical observatory of the pulsating star BL Cam. Fourier analysis of the light curves reveals that the fundamental mode has two harmonics. The O-C method is used to establish the period changes. So far, the analysis has been very successful in mapping the pulsation amplitude of the star across the instability strip. By using the formalism of Eddington and Plakidis (1929), we found significant results and strong indications of the evolutionary period change. A total of 55 new maximum light timings are reported. New values of (1/P) dP/dt are estimated using the O-C diagram based on all newly obtained times of maximum light combined with those taken from the literature, assuming the periods are decreasing and changing smoothly. To compute the effective temperature and surface gravity of the star, we performed model atmosphere analysis on its spectra. The physical parameters of the star are calculated and compared with the evolutionary models.

The isothermal gas sphere is a particular type of Lane-Emden equation and is used widely to model many problems in astrophysics like stars, star clusters, and the formation of galaxies. In this paper, we present a computational scheme to simulate the conformable fractional isothermal gas sphere using an artificial neural network (ANN) technique and compare the obtained results with the analytical solution deduced using the Taylor series. We performed our calculations, trained the ANN, and tested it using a wide range of the fractional parameter. Besides the Emden functions, we calculated the mass-radius relations and the density profiles of the fractional isothermal gas spheres. The results obtained provided that ANN could perfectly simulate the conformable fractional isothermal gas spheres., Comment: 23 pages, 10 figures

In this paper, the hyperbolic tangent function method is applied for constructing exact solutions for space-time conformal fractional Burgers equation. Furthermore, the space-time conformal fractional Burgers equation is tested for the Painleve property and consequently new numerous exact solutions are generated via Backlund transform., Comment: 20 pages

The orbits of visual binary systems still attract many of the working groups in astronomy. These orbits are the most important and reliable source of the stellar masse. In the present paper, we are going to compute orbits and dynamical masses of some visual binaries using an independent code. We used Kowalsky method to compute the geometrical elements. The dynamical elements (the period and the time of the periastron passage) are computed by implementing the double areal constant. We used the developed code to calculate the orbits for the four visual binaries, WDS J02262+3428, WDS J14310-0548, WDS J17466-0354, and WDS J12422+2622. We introduced a new orbit of the neglected visual binary WDS J17466-0354 and modified orbits for the rest three binaries. Using the Gaia DR2 parallaxes we computed the total masses of the systems. Comparing the adopted total masses with those derived from the spectral type relation revealed good agreement., Comment: 14 pages, 4 figures, 5 tables

In this paper, an algorithm for the ephemerides of visual binaries in near-parabolic orbits for either elliptic or hyperbolic case has been developed. The algorithm uses the Gauss method for determining both the true anomaly and the radial distance. Applications of the algorithm are presented for the binaries ADS 11632, WDS 19553-0644, and WDS 01243-0655. The result obtained shows a good agreement and in some cases more accurate when compared with the observations as well as the predicted ephemerides., Comment: 8 pages

Lane-Emden differential equations describe different physical and astrophysical phenomena that include forms of stellar structure, isothermal gas spheres, gas spherical cloud thermal history, and thermionic currents. This paper presents a computational approach to solve the problems related to fractional Lane-Emden differential equations based on neural networks. Such a solution will help solve the fractional polytropic gas spheres problems which have different applications in physics, astrophysics, engineering, and several real-life issues. We used Artificial Neural Network (ANN) framework in its feedforward back propagation learning scheme. The efficiency and accuracy of the presented algorithm are checked by testing it on four fractional Lane-Emden equations and compared with the exact solutions for the polytopic indices n=0,1,5 and those of the series expansions for the polytropic index n=3. The results we obtained prove that using the ANN method is feasible, accurate, and may outperform other methods., Comment: 29 pages, 11 pages, 5 tables

The isothermal gas sphere is well known as a powerful tool to model many problems in astrophysics, physics, chemistry, and engineering. This singular differential equation has not an exact solution and solved only by numerical and approximate methods. In the present paper and within the framework of the Newtonian hydrostatic equilibrium, we have developed general analytical formulations for the fractional isothermal gas sphere. To obtain analytical expressions for mass, radius, and density, besides the fractional isothermal gas sphere, we used the conformable fractional calculus. Using the series expansion method, we obtained a general recurrences relation, which allows us to determine the series coefficients. The comparison of the series solution with the numerical ones for the fractional parameter \alpha=1 is good for dimensional parameters up to x=3.2, beyond this value, the series diverges. We applied a combination of Euler-Abel and Pade techniques to accelerate the series, therefore accelerated series converge to the numerical desired value. We analyzed some physical parameters of a typical model of the neutron stars such as the mass-radius relation, density, and pressure ratio for different models. We found that the current models of the conformable neutron stars had smaller volumes and masses than both stars in the context of modified Rienmann-Liouville derivatives as well as the integer one., Comment: 15 pages, 4 figures, 2 tables

Lane Emden differential equation of the polytropic gas sphere could be used to construct simple models of stellar structures, star clusters and many configurations in astrophysics. This differential equation suffers from the singularity at the center and has an exact solution only for the polytropic index n=0,1 and 5. In the present paper, we present an analytical solution to the fractional polytropic gas sphere via accelerated series expansion. The solution is performed in the frame of conformable fractional derivatives. The calculated models recover the well-known series of solutions when \alpha=1. Physical parameters such as mass-radius relation, density ratio, pressure ratio and temperature ratio for different fractional models have been calculated and investigated. We found that the present models of the conformable fractional stars have smaller volume and mass than that of both the integer star and fractional models performed in the frame of modified Rienmann Liouville derivatives., Comment: 14 pages, 4 figures

In the present paper, we aim to determine the kinematical properties, velocity ellipsoid, and Oort constants using a sample of halo red giants. The study is based mainly on the space and radial velocities of about 1583 red giant stars collected from the SEGUE-1 and SEGUE-2 surveys. We divided the sample into three subsamples, the inner halo, the outer halo and the stars near the galactic plane. The fitting of the radial velocity equation gives a the mean of Oort constants, A = 15.6 +- 1.6 km s-1 kpc-1 and B= -13.9 +- 1.8 km s-1 kpc-1, the angular velocity |A-B|= 29.5 +- 0.2 km s-1 kpc-1 implying a rotational velocity of 221.25 +- 26.66 km s-1 if we take the distance to the Galactic center as 7.5 kpc., Comment: 15 pages, 3 figures. arXiv admin note: text overlap with arXiv:0706.3005 by other authors

Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple- {\alpha} process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann-Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the abundances calculated at the fractional parameter \alpha=1 , which represents the integer solution, with the numerical solution revealed a good agreement with maximum error \epsilon =0.003. The product abundances are calculated at\alpha=0.25, 0.5, 0.75 to declare the effects of changing the fractional parameters on the calculations., Comment: 12 pages

Lane-Emden equation could be used to model stellar interiors, star clusters and many configurations in astrophysics. Unfortunately, there is an exact solution only for the polytropic index and . In the present paper, a series solution for the fractional Lane-Emden equation is presented. The solution is performed in the frame of modified Rienmann Liouville derivatives. The obtained results recover the well-known series solutions when . Fractional model of n=3 has been calculated and mass-radius relation, density ratio, pressure ratio and temperature ratio have been investigated. We found that the fractional star has a smaller volume and mass than that of the integer star., Comment: 18 pages, 7 figures and 1 table