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Probing Velocity Structures of Protostellar Envelopes: Infalling and Rotating Envelopes within Turbulent Dense Cores

We have observed the three low-mass protostars, IRAS 15398$-$3359, L1527 IRS and TMC-1A, with the ALMA 12-m array, the ACA 7-m array, and the IRAM-30m and APEX telescopes in the C$^{18}$O $J=2$-1 emission. Overall, the C$^{18}$O emission shows clear velocity gradients at radii of $\sim$100-1000 au, which likely originate from rotation of envelopes, while velocity gradients are less clear and velocity structures are more perturbed on scales of $\sim$1000-10,000 au. IRAS 15398$-$3359 and L1527 IRS show a break at radii of $\sim$1200 and $\sim$1700 au in the radial profile of the peak velocity, respectively. The peak velocity is proportional to $r^{-1.38}$ or $r^{-1.7}$ within the break radius, which can be interpreted as indicating a rotational motion of the envelope with a degree of contamination of gas motions on larger spatial scales. The peak velocity follows $v_\mathrm{peak} \propto r^{0.68}$ or $v_\mathrm{peak} \propto r^{0.46}$ outside the break radius, which is similar to the $J/M$-$R$ relation of dense cores. TMC-1A exhibits the radial profile of the peak velocity not consistent with the rotational motion of the envelope nor the $J/M$-$R$ relation. The origin of the relation of $v_\mathrm{peak} \propto r^{0.46\operatorname{-}0.68}$ is investigated by examining correlations of the velocity deviation ($δv$) and the spatial scale ($τ$) in the two sources. Obtained spatial correlations, $δv \propto τ^{\sim0.6}$, are consistent with the scaling law predicted by turbulence models, which may suggest the large-scale velocity structures originate from turbulence.