Kinematics and Collimation of the Two-Sided Jets in NGC 4261: VLBI Study on Sub-parsec Scales

We report multi-frequency VLBI studies of the sub-parsec scale structure of the two-sided jet in the nearby radio galaxy NGC 4261. Our analyses include new observations using the Source Frequency Phase Referencing technique with the Very Long Baseline Array at 44 and 88 GHz, as well as archival data at 15 and 43 GHz. Our results show an extended double-sided structure at 43/44 GHz and provide a clear image of the nuclear region at 88 GHz, showing a core size of $\sim$0.09 mas and a brightness temperature of $\sim1.3\times10^{9}$ K. Proper motions are measured for the first time in the two-sided jet, with apparent speeds ranging from $0.31\pm0.14\,c$ to $0.59\pm0.40\,c$ in the approaching jet and $0.32\pm0.14\,c$ in the receding jet. The jet-to-counter-jet brightness ratio allows us to constrain the viewing angle to between $\sim54^{\circ}$ and $84^{\circ}$ and the intrinsic speed to between $\sim0.30\,c$ and $0.55\,c$. We confirm the parabolic shape of the upstream jet on both sides of the central engine, with a power-law index of $0.56\pm0.07$. Notably, the jet collimation is found to be already completed at sub-parsec scales, with a transition location of about 0.61 pc, which is significantly smaller than the Bondi radius of 99.2 pc. This behavior can be interpreted as the initial confinement of the jet by external pressure from either the geometrically thick, optically thin advection-dominated accretion flows (ADAF) or the disk wind launched from it. Alternatively, the shape transition may also be explained by the internal flow transition from a magnetically dominated to a particle-dominated regime.