We present new spectral line observations of the CH3CN molecule in the accretion disk around the massive protostar IRAS 20126+4104 with the Submillimeter Array that for the first time measure the disk density, temperature, and rotational velocity with sufficient resolution (0.37, equivalent to ~600 AU) to assess the gravitational stability of the disk through the Toomre-Q parameter. Our observations resolve the central 2000 AU region that shows steeper velocity gradients with increasing upper state energy, indicating an increase in the rotational velocity of the hotter gas nearer the star. Such spin-up motions are characteristics of an accretion flow in a rotationally supported disk. We compare the observed data with synthetic image cubes produced by three-dimensional radiative transfer models describing a thin flared disk in Keplerian motion enveloped within the centrifugal radius of an angular-momentum-conserving accretion flow. Given a luminosity of 1.3x10^4 Lsun, the optimized model gives a disk mass of 1.5 Msun and a radius of 858 AU rotating about a 12.0 Msun protostar with a disk mass accretion rate of 3.9x10^{-5} Msun/yr. Our study finds that, in contrast to some theoretical expectations, the disk is hot and stable to fragmentation with Q > 2.8 at all radii which permits a smooth accretion flow. These results put forward the first constraints on gravitational instabilities in massive protostellar disks, which are closely connected to the formation of companion stars and planetary systems by fragmentation.
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