Numerical simulations of self-gravitating accretion discs have shown that the evolution of such systems depends strongly on the rate at which it cools. In this work, we study the vertical structure of the self-gravitating accretion discs and also investigate the effect of the cooling rate on the latitudinal structure of such accretion discs. In the spherical coordinates, we write the hydrodynamics equations and simplify the basic equations based on the assumptions of axisymmetric and steady state. We use the self-similar method for solving the equations in the radial direction and we find proper boundary conditions. We find inflow–outflow solutions by considering the meridional component of the velocity field. In order to formulate the cooling term in energy equation, we introduce the new parameter β as a free constant that is the cooling time-scale in units of the dynamical time-scale. Our numerical solutions show that the thickness of the disc decreases with smaller β (or increasing the cooling term in energy equation) and it makes the disc colder and outflows form in the regions with lower latitude. So by increasing the cooling rate in the disc, the regions which belong to inflow decrease.

中文翻译：

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流出时自重力吸积盘的结构

自引力吸积盘的数值模拟表明，此类系统的演变很大程度上取决于其冷却速率。在这项工作中，我们研究了自重力吸积盘的垂直结构，并研究了冷却速度对此类吸积盘的横向结构的影响。在球坐标系中，我们编写了流体动力学方程，并基于轴对称和稳态的假设简化了基本方程。我们使用自相似方法在径向上求解方程，并找到合适的边界条件。我们通过考虑速度场的子午线分量来找到流入-流出的解决方案。为了用能量方程式来表示冷却项，我们引入新参数β作为自由常数，它是冷却时间尺度，以动态时间尺度为单位。我们的数值解法表明，圆盘的厚度随着β的减小而减小（或增加能量方程中的冷却项），并且使圆盘变冷，并且在低纬度区域形成流出。因此，通过提高圆盘中的冷却速度，属于流入的区域会减少。