sess10 - 10 Accretion disks Reading Shu Vol.2 Ch 7 10.1...

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10. Accretion disks Reading: Shu, Vol.2, Ch. 7 10.1 Viscosity Experience shows that we have overlooked some interactions, when we wrote down the force terms in the hydrodynamic equations. What is missing is the ability of gases and fluids to sustain viscous stress. In homework we saw that for an isotropic distribution function, e.g. as established in LTE, the kinetic pressure tensor collapses to a scalar isotropic pressure. However, we must add stress terms on account of viscosity. π ij = P δ ij + P ij (10 . 1) with P ij = - μ D ij - β ( ~ ∇ · ~ V ) δ ij D ij = ∂V i ∂x j + ∂V j ∂x i - 2 3 ( ~ ∇ · ~ V ) δ ij (10 . 2) where μ and β are respectively called the shear and bulk coefficients of viscosity. The shear effect corresponds to a randomization of bulk velocity gradients, i.e. the transfer of relative bulk kinetic energy to thermal energy. Viscosity plays an important role in contractive flows around central masses, such as the accretion of gas by galaxies, galaxy clusters, stars or black holes. Normally any mass point, that is falling toward a massive central object, will carry a non- vanishing angular momentum, ~ l = m~r × ~ V , with respect to the central mass. The centrifugal force then scales as ~ F | ~ l | 2 m r 3 V φ | ~ l | m r (10 . 3) Two conclusions can be drawn: The bulk velocity of the material will soon be very high, so the material could be radially supported by centrifugal forces. Support in other directions must be effected by pressure, which would be weaker if v th < v φ . This explains why many structures have the geometry of a disk. The material must loose angular momentum or it can not reach the central object. 10.2 Angular momentum transport Let us study a cold accretion flow around a gravitationally dominant central object in cylin- drical coordinate ( r, φ, z ), for it will have the shape of a thin disk. As useful quantity is the surface mass density, σ , that we can obtain by integration of the volume mass density. For an axisymmetric system σ ( r, t ) = Z -∞ dz ρ ( r, z, t ) (10 . 4) In the thin disk the velocity will depend, if at all, only weakly on z . We can therefore integrate the fluid equations over z and only use the mid-plane value of the velocity ~ V = ~ V ( z = 0).
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