Spherically Symmetric Accretion with Self-Gravity: Analytical Formulae and Numerical Validation
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908431956312064 |
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| author | Jiao, Cheng-Liang Zhao, Er-gang Zhu, Liying Shi, Xiang-dong |
| author_facet | Jiao, Cheng-Liang Zhao, Er-gang Zhu, Liying Shi, Xiang-dong |
| contents | 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 $β\equiv 2G \barρ r_\mathrm{out}^2/a_\mathrm{out}^2$ that characterizes the strength of self-gravity, where $\barρ$ 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, $\barρ$ may be approximated by the external medium density $ρ_\mathrm{out}$. 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 (AGN) disks, where self-gravity is non-negligible under certain conditions and should be evaluated using $β$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_02621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spherically Symmetric Accretion with Self-Gravity: Analytical Formulae and Numerical Validation Jiao, Cheng-Liang Zhao, Er-gang Zhu, Liying Shi, Xiang-dong High Energy Astrophysical Phenomena Astrophysics of Galaxies Solar and Stellar Astrophysics 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 $β\equiv 2G \barρ r_\mathrm{out}^2/a_\mathrm{out}^2$ that characterizes the strength of self-gravity, where $\barρ$ 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, $\barρ$ may be approximated by the external medium density $ρ_\mathrm{out}$. 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 (AGN) disks, where self-gravity is non-negligible under certain conditions and should be evaluated using $β$. |
| title | Spherically Symmetric Accretion with Self-Gravity: Analytical Formulae and Numerical Validation |
| topic | High Energy Astrophysical Phenomena Astrophysics of Galaxies Solar and Stellar Astrophysics |
| url | https://arxiv.org/abs/2507.02621 |