2025ApJ...989...94J

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 $\beta \equiv 2G\bar{\rho }{r}_{{\rm{out}}}^{2}/{a}_{{\rm{out}}}^{2}$ that characterizes the strength of self-gravity, where $\bar{\rho }$ and r_out are the mean density and outer radius of the flow, respectively, and a_out is the adiabatic sound speed of the external medium. For practical estimation, $\bar{\rho }$ may be approximated by the external medium density ρ_out. We identify an upper limit for β, 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 γ increases. For γ = 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 disks, where self-gravity is non-negligible under certain conditions and should be evaluated using β.