Erratum: Simulation of a compact object with outflows moving through a gaseous background

We have detected a bug in the code used for the simulation of the isotropic wind (section 3) in the original paper (Li et al. 2020, hereafter L+20). This bug caused the actual mass loss rate implemented for the isotropic wind simulation to be |$\dot{m}_w=0.5\, \text{M}_{\odot }$|/yr, half the value stated in L+20. In figs 1, 2, 4, and 5 of L+20, the analytical calculations are calculated following Gruzinov, Levin & Matzner (2020) with |$\dot{m}_w=1\, \text{M}_{\odot }$|/yr. We have checked that the qualitative conclusion of negative dynamical friction still holds. However, the agreement between analytical values and simulation results require modification. After correcting for the bug of mass loss rate, Fig. 1 shows the gas density slice in the central x − y plane at t = 630tB with |$\dot{m}_w=1\, \text{M}_{\odot }$|/yr and Vw = 3000 km s−1. All other parameters are the same as fig. 1 in the original paper. The colour shows the gas density and the blue streamline follows the gas velocity. The white line in Fig. 1 shows the the analytical solution given by Wilkin (1996). Fig. 2 shows the density, velocity and pressure profiles along the x-axis at the center of the computational box. The dashed line indicates R0 from the analytical solution by Wilkin (1996) calculated from the pressure balance between the ourflow and the incoming gas. The position of pressure balance corresponds to the contact discontinuity, and the density is accumulated at the bow shock front observed to be ∼2R0 away from the central object. We also used a smaller time step in the new simulation which reduces the oscillation observed in fig. 2 of the original paper. The oscillation resides in the low density and does not affect the gravitational force. Slice of gas density in the central x − y plane with isotropic wind at t = 630tB. The streamline shows the gas velocity and the white line is the analytical solution from Wilkin (1996). Slice of gas density in the central x − y plane with isotropic wind at t = 630tB. The streamline shows the gas velocity and the white line is the analytical solution from Wilkin (1996). Gas profile along the x-axis at the center. Upper panel: gas density, Middle panel: velocity parallel to the x-axis, Bottom panel: pressure. The dashed line is the analytical shock position from Wilkin (1996). Gas profile along the x-axis at the center. Upper panel: gas density, Middle panel: velocity parallel to the x-axis, Bottom panel: pressure. The dashed line is the analytical shock position from Wilkin (1996). Fig. 3 shows the evolution of the gravitational acceleration for the compact object in the x direction from the ambient gas. Fig. 4 shows the gravitational acceleration for different values of u (varying Vw while holding V* constant) with all other parameters kept fixed. Black dots are simulation results which is the average acceleration after a steady state is reached. The value of negative dynamical friction is significantly larger than the analytical calculations (equation 11) shown by the red dots and the asymptotic expression for u ≪ 1 (equation 12) shown by the blue dots. The orange is 1.5 times the analytical calculations and shows good agreement with the simulation results. Evolution for the magnitude of the gravitational acceleration of the compact object in the x direction from the ambient gas. The acceleration points in the −x direction and has negative values. The dotted line is the average acceleration for t > 100tB and the dashed line is the theoretical calculation from equation (11) of L+20 and the dash-dotted line is 1.5 times that value. Evolution for the magnitude of the gravitational acceleration of the compact object in the x direction from the ambient gas. The acceleration points in the −x direction and has negative values. The dotted line is the average acceleration for t > 100tB and the dashed line is the theoretical calculation from equation (11) of L+20 and the dash-dotted line is 1.5 times that value. The dependence of acceleration on the wind speed Vw. Black dots are simulation results. Red dots are analytical results (L+20 equation 11), blue dots are the asymptotic expression for u ≪ 1 (L+20 equation 12) and orange dots are 1.5 times the values given by L+20 (equation 11). The dependence of acceleration on the wind speed Vw. Black dots are simulation results. Red dots are analytical results (L+20 equation 11), blue dots are the asymptotic expression for u ≪ 1 (L+20 equation 12) and orange dots are 1.5 times the values given by L+20 (equation 11). Gruzinov et al. (2020) uses the bow shock solution given by Wilkin (1996) and assumes the shock is infinitesimally thin. In reality, the solution by Wilkin (1996) describes the position of the contact discontinuity well, but the bow shock has finite thickness ∼R0 and leads to a total acceleration |${\sim}50{{\ \rm per\ cent}}$| larger making the effect of negative dynamical friction more prominent. Still, the qualitative conclusion for the phenomena of negative dynamical friction and analytical scaling with outflow velocity Vw remain valid. The authors are grateful to Lile Wang for pointing out the bug in our code used for the original paper.

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