We study the dynamical structure of a self-gravitating disc with corona around a super-massive black hole .
Assuming that the magneto-rotational-instability ( MRI ) responsible for generating the turbulent stresses inside the disc is also the source for a magnetically dominated corona , a fraction of the power released when the disc matter accretes is transported to and dissipated in the corona .
This has major effect on the structure of the disc and its gravitational ( in ) stability according to our analytical and self-consistent solutions .
We determine the radius where the disc crosses the inner radius of gravitational instability and forms the first stars .
Not only the location of this radius which may extend to very large distances from the central black hole , but also the mass of the first stars highly depends on the input parameters , notably the viscous coefficient , mass of the central object and the accretion rate .
For accretion discs around quasi-stellar objects ( QSOs ) and the Galactic center , we determine the self-gravitating radius and the mass of the first clumps .
Comparing the cases with a corona and without a corona for typical discs around QSOs or the Galactic center , when the viscosity coefficient is around 0.3 , we show that the self-gravitating radius decreases by a factor of approximately 2 , but the mass of the fragments increases with more or less the same factor .
Existence of a corona implies a more gravitationally unstable disc according to our results .
The effect of a corona on the instability of the disc is more effective when the viscosity coefficient increases .