A Magnetic α ω Dynamo in Active Galactic Nuclei Disks - NMT Physics

– 28 –When the disk becomes self gravitating, it may become subject to gravitational instability.Let us check that by calculating Toomre parameter To = κc s(e.g., Binney & Tremaine 1994).πGΣEpicyclic frequency κ is equal to its value for the point mass M located at the position of theblack hole κ = Ω K = (GM) 1/2 /r 3/2 , since the mass of the disk is small compared to the mass ofthe black hole. Sound speed is equal to c 2 s = kT mpdin zone (b) and c 2 sm = P rin the zone (a) (apρcoefficient close to 1 is neglected). Substituting appropriate expressions we obtain for the soundspeed in zone (b)in zone (a)( c s = 5.37 · 10 7 cm s −1 α) ( )ss−1/10 0.1 l 1/5E×0.01 ǫ 0.1( ) rc2 −9/20( √ ) 1/5M −1/10 3rg8 1 − , (A28)GMrc s = 3.5 · 10 10 cm s −1 l (E ǫ) ( ) −1 rc2 −3/2(1 −0.1 0.1 GMThe Toomre parameter becomes in zone (a)To = 8.33 · 10 11 α (ss lE0.01 0.1= 0.77and in the zone (b)( αss0.01) 4/7M−10/78) 2 ( ǫ) ( −2 rc20.1 GM(lE0.1) −9/2M −18√ )3rg. (A29)r() −10/7 ( ǫ) ( ) 10/7 r −9/20.1 r ab1 −√3rgr) 2(A30)(To = 2.97 · 10 3 α) ( )ss7/10 0.1 l −2/5 ( )EM −13/10 rc2 −27/2080.01 ǫ 0.1GM( ) r −27/20 (= 0.73 M −10/7 αss) ( ) 4/7 lE 0.1 −10/78(A31)r ab 0.01 0.1 ǫ(= 9.7 · 10 −2 α) ( )ss7/10 lE 0.1 −2/5 ( )M 1/20 r −27/208.0.01 0.1 ǫ0.01pcGravitational instability develops if To < 1. One can see from expressions (A30) and (A31) that Tostrongly declines with increasing the radius. The disk has a well defined outer radius of gravitationalstability r T such that To(r T ) = 1. For our fiducial parameters, r T is close to the r ab . At the outeredge of the zone (b) one hasTo(r bc ) = 2.0 · 10 −2 ( α ss0.01) ( 7/10−13/10 lEM 80.1)0.1 −13/10. (A32)ǫLarge values of α ss , small masses of the central black hole, and low accretion rates cause the To toincrease and can cause the radius r T to become larger than r ab . As follows from expression (A30)