In this article, a mathematical model describing the unsteady adiabatic flow of spherical shock waves in a self-gravitating, non-ideal radiating gas under the influence of an azimuthal magnetic field is formulated and similarity solutions are obtained. The ambient medium is assumed to be at rest with uniform density. The effect of thermal radiation under an optically thin limit is included in the energy equation of the governing system. By applying the Lie invariance method, the system of PDEs governing the flow in the said medium is transformed into a system of non-linear ODEs via similarity variables. All the four possible cases of similarity solution are obtained by selecting different values for the arbitrary constants involved in the generators. Among these four cases, only two possess similarity solutions, one by assuming the power-law shock path and other by exponential-law shock path. The set of non-linear ODEs obtained in the case of the power-law shock path is solved numerically using the Runge–Kutta method of 4th order in the MATLAB software. The effects of variation of various parameters such as non-ideal parameter \((\overline{b })\), adiabatic index of the gas \((\gamma )\), Alfven-Mach number (\({M}_{a}^{-2}\)), ambient magnetic field variation index \((\phi )\), and gravitational parameter \(({G}_{0})\) on the flow quantities are discussed in detail and various results are portrayed in the figures. Furthermore, the article includes a detailed comparison made between the solutions obtained for cases with and without gravitational effects in the presence of magnetic field.

在本文中，建立了一个数学模型，描述在方位角磁场影响下自引力非理想辐射气体中球形冲击波的非定常绝热流动，并获得了相似解。假设环境介质处于静止状态且密度均匀。光学薄极限下的热辐射效应包含在控制系统的能量方程中。通过应用李不变性方法，通过相似变量将控制所述介质中流动的偏微分方程组转换为非线性常微分方程组。相似性解的所有四种可能情况都是通过为生成器中涉及的任意常数选择不同的值来获得的。在这四种情况中，只有两种具有相似解，一种采用幂律冲击路径，另一种采用指数律冲击路径。在 MATLAB 软件中使用四阶龙格-库塔法对幂律冲击路径情况下获得的非线性常微分方程组进行数值求解。各种参数变化的影响，例如非理想参数\((\overline{b })\)、气体绝热指数\((\gamma )\)、阿尔文马赫数 ( \({M}_详细讨论了{a}^{-2}\) )、环境磁场变化指数\((\phi )\)和重力参数\(({G}_{0})\)对流量的影响图中描绘了各种结果。此外，本文还对存在磁场的情况下有和没有重力效应的情况下获得的解决方案进行了详细比较。