We present new spectral line observations of the \mathrm { CH _ { 3 } CN } 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 \farcs 37 , equivalent to \sim 600 \mathrm { AU } ) to assess the gravitational stability of the disk through the Toomre- Q parameter .
Our observations resolve the central 2000 \mathrm { 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 1.3 \times 10 ^ { 4 } L _ { \odot } , the optimized model gives a disk mass of 1.5 M _ { \odot } and a radius of 858 \mathrm { AU } rotating about a 12.0 M _ { \odot } protostar with a disk mass accretion rate of 3.9 \times 10 ^ { -5 } M _ { \odot } \mathrm { yr ^ { -1 } } .
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 .