Boundary layer

The standard model of accretion disc is derived under assumption of an inner boundary, where $\mathrm{d}\Omega ∕\mathrm{d}R=0$. Note that this condition is not possible for purely Keplerian rotation of the fluid in Newtonian gravity, however, it may fullfilled in realistic situations in (at least) two ways. First, in the regime of strong gravity, i.e., when the central object is very compact, the generalised Newtonian rotation profile does fullfill the above given condition at the last marginally stable circular orbit. Second possibility is presence of an additional force which supports matter against gravity together with the centrifugal force. This section discusses briefly the second option. Section 4.5 of the textbook by Vladimir Karas is the main study material; my hand-notes probably do not provide anythig substantial in addition.

The most important take-off from this lesson is supposed to be the qualitative logical consideration which leads to the conclusion that it is the gradient of pressure that mostly ballances the gravitational pull of the central body (star). Let me briefly sketch the argumentation here:

• the fluid is supposed to slow down roughly to corotation at the surface of the star
• angular velocity of the stellar surface has to be sub-Keplerian (otherwise, the centrifugal force would tear the star apart)
• at some radius above the surface, where the accreting fluid rotation is nearly Keplerian, its angular velocity is larger than that of the stellar surface
• in between these radii, the rotation profile is supposed to be continuous which implies the shape shown in Fig. 10 of the textbook
• radial component of the (stationary) Euler equation states that four terms are in ballance. Gravitational force acts always towards the centre, while the centrifugal one points outwards. In the case of purely Keplerian rotation, these two are in ballance and the other two terms are negligible. In the boundary layer, however, the angular velocity of the fluid is smaller than Keplerian and, consequently, the centrifugal force cannot ballance the gravitational pull. For simplicity, we further assume that the centrifugal force is negligible. Hence, we are left with two other terms in the Euler equation that may ballance the gravitational force – the radial velocity gradient and the pressure gradient
• the assumption made about the continuous shape of the rotation profile of the fluid says that the fluid elements above the stellar surface are aware of that surface. The information about the presence of the surface spreads to the higher layers approximately with the speed of sound
• if the radial velocity gradient and, consequently, the radial velocity magnitude were larger than the speed of sound, the information about the stellar surface could not reach the outer part of the boundary layer, i.e., it has to be the pressure gradient term which dominates over the velocity gradient term in the Euler equation.