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It is likely that young protostellar disks undergo a self-gravitating phase.Such systems are characterized by the presence of a spiral pattern that can beeither in a quasi-steady state or in a nonlinear unstable condition. Thisspiral wave affects both the gas dynamics and kinematics, resulting indeviations from the Keplerian rotation. Recently, a lot of attention has beendevoted to kinematic studies of planet-forming environments, and we are nowable to measure even small perturbations of velocity field ($\lesssim 1 \%$ ofthe Keplerian speed) thanks to high spatial and spectral resolutionobservations of protostellar disks. In this work, we investigate the kinematicsignatures of gravitational instability: we perform an analytical study of thelinear response of a self-gravitating disk to a spiral-like perturbation,focusing our attention on the velocity field perturbations. We show thatunstable disks have clear kinematic imprints into the gas component across theentire disk extent, due to the GI spiral wave perturbation, resulting indeviations from Keplerian rotation. The shape of these signatures depends onseveral parameters, but they are significantly affected by the cooling factor:by detecting these features, we can put constraints on protoplanetary diskcooling.

Investigating Protoplanetary Disk Cooling through Kinematics: Analytical GI Wiggle