标题： 球对称吸积与自引力：解析公式和数值验证 显示英文标题

标题： Spherically Symmetric Accretion with Self-Gravity: Analytical Formulae and Numerical Validation

摘要： 球对称吸积包含自引力构成一个三点边值问题（TPBVP），由外边界、声点和吸积体表面的约束条件控制。 以前的研究有两个局限性：要么在解析处理中使用了错误的自引力势公式，要么在数值实现中引入了额外的输入参数以避免求解完整的TPBVP。 为了解决这些问题，我们提出了一种自洽的TPBVP公式，并使用松弛法进行求解。 我们还推导了近似解析公式，以快速估计自引力效应。 我们的分析确定了一个无量纲参数$\beta \equiv 2G \bar{\rho} r_\mathrm{out}^2/a_\mathrm{out}^2$，它表征自引力的强度，其中$\bar{\rho}$和$r_\mathrm{out}$分别是流的平均密度和外半径，而$a_\mathrm{out}$是外部介质的绝热声速。 对于实际估算，$\bar{\rho}$可以近似为外部介质密度$\rho_\mathrm{out}$。 我们确定了$\beta$的上限，超过该上限后稳态吸积将不可持续 —— 这一行为与经典引力不稳定性一致，而之前的研究未能捕捉到这一现象。 吸积率增强随着绝热指数$\gamma$的增加而单调减少。 对于$\gamma=5/3$，自引力不再增强吸积率。 这些理论预测得到了我们的数值解的验证。 我们进一步将结果应用于两种天体物理情景：早期宇宙中超大质量黑洞种子的超爱丁顿吸积，在这种情况下自引力是显著的；以及嵌入在活动星系核（AGN）盘中的恒星级物体的吸积，在特定条件下自引力是不可忽略的，应使用$\beta$进行评估。 显示英文摘要

摘要： Spherically symmetric accretion incorporating self-gravity constitutes a three-point boundary value problem (TPBVP) governed by constraints at the outer boundary, sonic point, and accretor surface. Previous studies have two limitations: either employing an incorrect formula for self-gravity potential in analytical treatments, or introducing additional input parameters in numerical implementations to circumvent solving the full TPBVP. To address these issues, we present a self-consistent TPBVP formulation, solved using the relaxation method. We also derive approximate analytical formulae that enable rapid estimates of self-gravity effects. Our analysis identifies a dimensionless parameter $\beta \equiv 2G \bar{\rho} r_\mathrm{out}^2/a_\mathrm{out}^2$ that characterizes the strength of self-gravity, where $\bar{\rho}$ and $r_\mathrm{out}$ are the mean density and outer radius of the flow, respectively, and $a_\mathrm{out}$ is the adiabatic sound speed of the external medium. For practical estimation, $\bar{\rho}$ may be approximated by the external medium density $\rho_\mathrm{out}$. We identify an upper limit for $\beta$, beyond which steady accretion becomes unsustainable -- a behavior consistent with classical gravitational instability that previous studies failed to capture. The accretion rate enhancement decreases monotonically as the adiabatic index $\gamma$ increases. For $\gamma=5/3$, self-gravity ceases to augment the accretion rate. These theoretical predictions are validated by our numerical solutions. We further apply our results to two astrophysical scenarios: hyper-Eddington accretion onto supermassive black hole seeds in the early Universe, where self-gravity is significant; and accretion onto stellar-mass objects embedded in active galactic nuclei (AGN) disks, where self-gravity is non-negligible under certain conditions and should be evaluated using $\beta$.