Velocity Gradient in the Presence of Self-Gravity: Identifying
  Gravity-induced Inflow and Determining Collapsing Stage
  Understanding how star formation is regulated requires studying the energy
balance between turbulence, magnetic fields, stellar feedback, and gravity
within molecular clouds. However, identifying the transition region where the
gravity takes over remains elusive. Recent studies of the Velocity Gradient
Technique (VGT), which is an advanced tool for magnetic field studies, reveal
that the gradients of spectroscopic observables change their directions by 90
degrees with respect to the magnetic fields in the regions of gravitational
collapse. In this study, we perform 3D MHD numerical simulations. We observe
that star formation successfully proceeds in strongly magnetized and fully
ionized media. We confirm that the self-gravity induces the change of
gradients' orientation and high gradients' amplitude. We explore two ways of
identifying collapsing self-gravitating regions through the double-peak feature
in the histogram of gradients' orientation and the curvature of gradients. We
show that velocity gradients' morphology and amplitude can be synthetically
used to trace the convergent inflows. By comparing with the column density
Probability Density Functions (N-PDFs) method, we show that VGT is a powerful
new tool for studying the gas dynamics and tracing magnetic field in
star-forming regions. By analogy with VGT, we extend the Intensity Gradient
Technique (IGT) to locate the gravitational collapsing region and shocks. We
demonstrate the synergy of VGT and IGT can determine the collapsing stages in a
star-forming region.
