Dynamical friction and the evolution of black holes in cosmological simulations: A new implementation in OpenGadget3

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Volume 692 (December 2024)

A&A, 692 (2024) A81

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Table 2

Relevant equations and parameters that characterise the sub-resolution model of BH evolution and AGN feedback.

Resolution	Seeding	Accretion	Merger	Feedback
$\begin{array}{l} Mass \\ m_{DM} = 4.69 \times 10^{7} M_{⊙} \\ m_{gas} = 8.67 \times 10^{6} M_{⊙} \\ m_{} = 8.7 \times 10^{6} M_{⊙} \end{array}$ $\eqalign{ & \,\,\,\,\,\,\,\,\,{\rm{Mass }} \cr & {m_{{\rm{DM}}}} = 4.69 \times {10^7}{{\rm{M}}_ \odot } \cr & {m_{{\rm{gas }}}} = 8.67 \times {10^6}{{\rm{M}}_ \odot } \cr & \,\,\,\,\,{m_} = 8.7 \times {10^6}{{\rm{M}}_ \odot } \cr}$	$M_{BH,seed} = M_{0} \frac{M_{, h}}{f_{} M_{DM,seed}}$ ${M_{{\rm{BH,seed}}}} = {M_0}{{{M_{ * ,{\rm{h}}}}} \over {{f_ * }{M_{{\rm{DM,seed}}}}}}$	$\begin{array}{l} \min ({\dot{M}}_{Bondi}, {\dot{M}}_{Edd}) \\ {\dot{M}}_{Bondi} = \frac{4 π G^{2} M_{BH}^{2} α ρ}{{(c_{s}^{2} + v^{2})}^{3 / 2}} \\ {\dot{M}}_{Edd} = \frac{4 π G M_{BH} m_{p}}{η_{r} σ_{T} c} \\ α = 10 / 100 \\ η_{r} = 0.1 \end{array}$ $\eqalign{ & \min \left( {{{\dot M}_{{\rm{Bondi}}}},{{\dot M}_{{\rm{Edd}}}}} \right) \cr & {{\dot M}_{{\rm{Bondi}}}} = {{4\pi {G^2}M_{{\rm{BH}}}^2\alpha } \over {{{\left( {c_F^2 + {\nu ^2}} \right)}^{3/2}}}} \cr & {{\dot M}_{{\rm{Edd}}}} = {{4\pi G{M_{{\rm{BH}}}}{m_p}} \over {{\eta _r}{\sigma _T}c}} \cr & \alpha = 10/100 \cr & {\eta _r} = 0.1 \cr}$
	M₀ = 2.7 × 10⁵ M_⊙		$\begin{array}{l} Threshold distance: \\ Δ r_{BH} < d_{merg} \\ d_{merg} = 5 \cdot ϵ_{BH} \end{array}$ $\eqalign{ & {\rm{ Threshold distance: }} \cr & \,\,\,\,\,\Delta {r_{{\rm{BH}}}} < {d_{{\rm{merg}}}} \cr & \,\,\,\,{d_{{\rm{merg}}}} = 5 \cdot {_{{\rm{BH}}}} \cr}$	${\dot{E}}_{feed} = η_{r} η_{f} {\dot{M}}_{BH} c^{2}$ ${\dot E_{{\rm{feed }}}} = {\eta _{\rm{r}}}{\eta _{\rm{f}}}{\dot M_{{\rm{BH}}}}{c^2}$
	M_DM,seed = 6.94 × 10¹⁰ M_⊙			η_r = 0.1
$\begin{array}{l} Force softening \\ ϵ_{DM} = 4.13 ckpc / 1.38 kpc \\ ϵ_{gas} = 1.38 ckpc \\ ϵ_{} = 0.35 ckpc \\ ϵ_{BH} = 0.35 ckpc \end{array}$ $\eqalign{ & \,\,{\rm{ Force softening }} \cr & {_{{\rm{DM}}}} = 4.13{\rm{ckpc}}/1.38{\rm{kpc}} \cr & \,\,\,\,\,\,\,{_{{\rm{gas}}}} = 1.38{\rm{ckpc}} \cr & \,\,\,\,\,\,\,{_} = 0.35{\rm{ckpc}} \cr & \,\,\,\,{_{{\rm{BH}}}} = 0.35{\rm{ckpc}} \cr}$	$\frac{M_{, h}}{M_{h}} = 2 %$ ${{{M_{ ,{\rm{h}}}}} \over {{M_{\rm{h}}}}} = 2{\rm{\% }}$		$\begin{array}{l} Gravitationally bound: \\ v_{rel} < 0.5 \cdot c_{s} \\ \frac{\| Φ_{{BH}_{1}} - Φ_{{BH}_{2}} \|}{a} < 0.5 \cdot c_{s}^{2} - v_{rel}^{2} \end{array}$ $\eqalign{ & {\rm{ Gravitationally bound: }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,{v_{{\rm{rel}}}} < 0.5 \cdot {c_s} \cr & {{\left\| {{\Phi _{{\rm{B}}{{\rm{H}}_1}}} - {\Phi _{{\rm{B}}{{\rm{H}}_2}}}} \right\|} \over a} < 0.5 \cdot c_{\rm{s}}^2 - v_{{\rm{rel}}}^2 \cr}$	Quasar : η_f = 0.05
	$\frac{M_{, h}}{M_{DM, h}} = 5 %$ ${{{M_{,{\rm{h}}}}} \over {{M_{{\rm{DM}},{\rm{h}}}}}} = 5\%$			Radio: η_f = 0.02
	$\frac{M_{gas, h}}{M_{, h}} = 10 %$ ${{{M_{{\rm{gas}},{\rm{h}}}}} \over {{M_{ ,h}}}} = 10{\rm{\% }}$

Notes. Column 1: values of the masses of the different particle species in the high-resolution regions of Dianoga clusters and Cosmobox: m_DM, m_gas and m_* are the masses of DM, gas and star particles, respectively; ϵ_DM, ϵ_gas, ϵ_*, and ϵ_BH are the Plummer-equivalent softening lengths of the different particle species. For BH, gas and star particles, the softenings are fixed in comoving units. For DM particles, we fix the softening in comoving units until z = 2, and then in physical units until z = 0. Column 2: criteria for BH seeding: M_BH,seed is the seeding mass of a BH; M₀ is the minimum BH mass seed; M_DM,h, M_gas,h and M_*,h are the halo masses in the DM, gaseous and stellar components; M_h is the total halo mass and M_h,seed is the minimum halo mass required to seed a BH. Column 3: BH mass accretion rate: Ṁ_Bondi and Ṁ_Edd are defined in Eqs. (11), (12). Column 4: criteria for BH-BH merging: ∆r_BH is the distance between the merging BHs, d_merg is the minimum threshold distance required for merging, $Φ_{{BH}_{1}}$ ${{\rm{\Phi }}_{{\rm{B}}{{\rm{H}}_1}}}$ and $Φ_{{BH}_{2}}$ ${{\rm{\Phi }}_{{\rm{B}}{{\rm{H}}_2}}}$ are the values of the gravitational potentials of the two BHs, υ_rel the magnitude of their relative velocity, c_s the local sound speed and a is the scale factor. Column 5: expression of the rate of the feedback energy released by the BH and thermally coupled to the surrounding gas; c is the light speed, η_r and η_f are the radiative and coupling efficiency, respectively. See Sect. 3 for additional details.

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