tokens: SMBH. The forward shock region is shown as the purple area. The cocoon is not depicted. On the other hand, within the duration of these two short-term processes, e.g., t m and t vis , the disk wind radius may reach v d ( t vis − t m ) ∼ 10 14 − 10 16 cm above the disk, where v d is the disk wind velocity that is of the order of the escape velocity, for the circumbinary disk, and is the radius of the circumbinary disk at the decoupling. In reality, not only the circumbinary disk but also mini-disks around two SMBHs contribute, which would make the wind bubble more complicated. For simplicity, we assume the density profile of the winds at the decoupling to obtain the density distribution of the wind bubble at larger distances,where is the mass accretion rate onto the binary system, χ ∼ 1 is introduced to take into account the contribution of mini-disks, and η w represents the fraction of accreted mass converted to the disk wind. According to the simulations, for SANE (Standard And Normal Evolution) models, the parameter η w may vary from 10 −4 to 10 −1 (Jiang et al. 2019b , 2019a ; Ohsuga et al. 2009 ) when the mass accretion rate changes from sub-Eddington to super-Eddington. In MAD (Magnetically Arrested Disk) models, η w can reach 10 −2 to 10 −1 (Akiyama et al. 2019 ). With v d ∼ v esc ( R d ,dec ), we have , where , β d ,−1 ≡ v d /(0.1 c ), the parameter is defined as the ratio of , and the Eddington value (assuming a radiation efficiency of 0.1). After the coalescence, a powerful jet driven by the spin energy of the newly formed SMBH can appear, subsequently propagating in the pre-merger wind bubble. Considering a sub-Eddington accretion rate with the MAD configuration, we estimate the jet kinetic luminosity to bewhere η j ∼ 0.3–1 is the ratio of the accretion energy converted to the jet energy (Tchekhovskoy et al. 2011 ). Following the standard jet propagation theory (e.g., Bromberg et al. 2011 ; Mizuta Ioka 2013 ), we write down the dimensionless parameter that represents the ratio of the energy density of the jet and the rest-mass energy density of the surrounding mediumwhere θ j is the jet opening angle, is the isotropic-equivalent luminosity. Since the quantity lies in the regime , we expect that the jet is “uncollimated” for our fiducial parameters. 6 6 However, jet collimation, which was assumed in Yuan et al. ( 2020 ), would be achievable for the super-Eddington accretion accompanied by disk winds with η w ∼ 0.1–0.3. This situation is similar to that in choked jet propagation in the circumstellar material (Senno et al. 2016 ; Nakar 2015 ), and β h is evaluated from . In the relativistic limit, the jet head Lorentz factor is (Senno et al. 2016 ), and we have Γ h ∼ 2 in our fiducial case with . Note that the jet head radius is , and is introduced to represent time measured in the central engine frame, which can be converted to the observation time T via (that is in the relativistic limit) for on-axis observers. Furthermore, to ensure
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