We present high spatial resolution (12pc) Atacama Large Millimeter/sub-millimeter Array CO(J=3-2) observations of the nearby lenticular galaxy NGC4429. We identify 217 giant molecular clouds within the 450pc radius molecular gas disc. The clouds generally have smaller sizes and masses but higher surface densities and observed linewidths than those of Milky Way disc clouds. An unusually steep size - line width relation and large cloud internal velocity gradients (0.05 - 0.91 km s^-1 pc^-1) and observed Virial parameters (alpha_obs,vir = 4.0) are found, that appear due to internal rotation driven by the background galactic gravitational potential. Removing this rotation, an internal Virial equilibrium appears to be established between the self-gravitational (Usg) and turbulent kinetic (Eturb) energies of each cloud, i.e. alpha_sg,vir=Usg/Eturb = 1.3. However, to properly account for both self and external gravity (shear and tidal forces), we formulate a modified Virial theorem and define an effective Virial parameter alpha_eff,vir = alpha_sg,vir + Usg/Eext (and associated effective velocity dispersion). The NGC4429 clouds then appear to be in a critical state in which the self-gravitational energy and the contribution of external gravity to the clouds energy budget (Eext) are approximately equal, i.e. Eext/Usg~1. As such, alpha_eff,vir = 2.2 and most clouds are not virialised but remain marginally gravitationally bound. We show this is consistent with the clouds having sizes similar to their tidal radii and being generally radially elongated. External gravity is thus as important as self-gravity to regulate the clouds of NGC4429.
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