Dr G. I. Ogilvie Lent Term 2005

Transcription

1 Aretion Diss Mathematial Tripos, Part III Dr G. I. Ogilvie Lent Term Visous evolution of an aretion dis Introdution The evolution of an aretion dis is regulated by two onservation laws: onservation of mass, and onservation of angular momentum. These laws are embodied in the threedimensional equations of fluid dynamis, but we must first redue them to one-dimensional versions. The equation of mass onservation is where ρ is the density and ρ t + (ρ ) = 0, the veloity of the fluid. The equation of motion is ρ D Dt = ρ Φ p + T, where D Dt = t + is the Lagrangian time-derivative following the fluid flow, Φ is the gravitational potential, p is the pressure and T is the stress tensor. We adopt ylindrial polar oordinates (r, φ, z) for all alulations, suh that the entral mass is at r = z = 0 and the mid-plane of the dis is z = 0. The origin of the stress T is a matter of entral importane in aretion dis theory. It may be a turbulent stress, most likely involving tangled magneti fields. There may also be ontributions from large-sale magneti fields, or from waves in the dis. These possibilities will be explored later in the ourse. In the lassial theory, the stress is parametrized as a visous stress Conservation of mass Define the surfae density Σ(r, t) aording to Σ = 1 2π ρ dz dφ. 2π 0 The mass ontained between r 1 and r 2 is then r 2 GIO r 1 Σ 2πr dr.

2 The equation of mass onservation is ρ t + 1 r r (rρu r) + 1 r φ (ρu φ) + z (ρu z) = 0. Integrate with respet to φ and z, over the full extent of the dis, to obtain 2π Σ t + 1 r r = 0, where (r, t) is the radial mass flux, defined by = 2π 0 rρu r dz dφ, and we assume that there is no vertial loss of mass to z =. One an also define a (density-weighted) mean radial veloity ū r (r, t) aording to = 2πrΣū r. We then have Σ t + 1 r r (rσū r) = 0, (1) whih expresses the onservation of mass in one dimension Conservation of angular momentum The azimuthal omponent of the equation of motion is ρ Du φ Dt + u ru φ r = ρ Φ r φ 1 r p φ + 1 r 2 r (r2 T rφ ) + 1 r In the ase of an axisymmetri potential, it follows that ρ D Dt (ru φ) = 1 r T φφ φ r (r2 T rφ ) + 1 r φ ( rp + rt φφ) + z (rt φz). + T φz z. Therefore angular momentum is onserved, but is transported radially outwards by a negative shear stress T rφ. Suh a stress is neessary for aretion. Now assume that the azimuthal veloity in the dis is given by u φ = rω, where Ω(r) is the angular veloity of irular orbits in the potential Φ. approximation later. Then ρu r dh dr = 1 r r (r2 T rφ ) + 1 r φ ( rp + rt φφ) + z (rt φz), We return to examine this where h = r 2 Ω is the speifi orbital angular momentum. Multiply by r and integrate with respet to φ and z to obtain dh dr = r, GIO

3 where (r, t) is the visous torque, defined by 2π = 0 r 2 T rφ dz dφ, and we assume that there is no vertial loss of angular momentum to z =. In astrophysial diss the moleular visosity is muh too small to aount for the torque. Nevertheless, it is onventional to parametrize the torque in terms of an effetive visosity. Aording to the Navier Stokes equation, we have a visous stress T = µ + ( ) T + (µ b 2 3 µ)( where µ is the visosity and µ b the bulk visosity. (Reall that the dynami visosity µ and the kinemati visosity ν are related by µ = ρν.) In the ase of irular orbital motion, the only stress omponent is T rφ = T φr = µr dω dr, i.e. the visosity multiplied by the shear rate. Define the (density-weighted) mean kinemati visosity ν(r, t) aording to νσ = 1 2π µ dz dφ. 2π 0 Then we find = 2π νσr 3 dω dr. Although this form is derived from a Navier Stokes visosity, the torque an always be parametrized in this form for some suitable funtion ν(r, t). We then have dh Σū r dr = 1 r r )1, νσr 3 dω, (2) dr whih expresses the onservation of angular momentum in one dimension Diffusion equation for surfae density Equations (1) and (2) may be ombined to eliminate ū r, leading to Σ t + 1 r r dh dr 1 r For a point-mass potential (Keplerian dis), we have Ω = GM r 3 1/2 GIO νσr 3 dω dr, h = (GMr) 1/2, = 0.

4 and so Σ t = 3 r r r 1/2 r r 1/2 νσ. This has the harater of a diffusion equation for the surfae density. This may be interpreted as follows. Visous torques ause the redistribution of angular momentum, and therefore there is a visous spreading or diffusion. Most of the mass goes to smaller radii to be areted by the entral objet, but some goes to larger radii in order to take up the angular momentum that is transported there. An alternative form, more obviously related to the lassial diffusion equation, is dh dω = νr2 t dr dr 2 h 2, whih desribes the torque diffusing in the spae of speifi angular momentum. This form is valid only if ν is independent of t Analysis of the diffusion equation Inner boundary ondition The inner boundary ondition depends on the nature of the entral objet. There are three important possibilities. Weakly magnetized star If the entral objet is a non-magneti (or weakly magnetized) star, the dis may extend to the stellar surfae. Usually the star rotates at only a fration of the Keplerian angular veloity at its surfae, i.e. 1/2 GM Ω < R 3. This is beause the star is mainly supported by pressure, not by the entrifugal fore. The angular veloity of the fluid makes a rapid transition from the Keplerian value to the stellar value, in a visous boundary layer. Somewhere in the boundary layer is a radius r in at whih the shear rate vanishes, and therefore the visous torque = 0. This may be regarded as the inner radius of the aretion dis. To a good approximation, r in R. Blak hole If the entral objet is a blak hole, the dis does not extend to the event horizon. This is beause of the existene of a marginally stable irular orbit at r = r ms. For r < r ms, irular orbits are unstable and the gas spirals rapidly into the blak hole without need for a visous torque. GIO

5 Angular veloity profile for aretion from a dis on to a star Aretion from a dis on to a blak hole For a non-rotating (Shwarzshild) blak hole, the marginally stable orbit is at r ms = 6GM/ 2, while the event horizon is at r S = 2GM/ 2. To obtain this result properly, of ourse, requires a relativisti treatment of orbital motion. In order to onserve mass, the surfae density Σ dereases very rapidly just inside r ms as the gas aelerates into the hole. The visous stress is then essentially zero at r ms. GIO

6 Strongly magnetized star If the entral objet is a strongly magnetized star, the inner part of the dis may be disrupted by the strong magneti field. The dis terminates at a magnetospheri radius whih depends on ompliated (and ontroversial) physis. The aretion flow is then hannelled along the magneti field lines on to the magneti poles of the star. Aretion from a dis on to a strongly magnetized star In summary, the inner boundary ondition an usually be onsidered to be = 0 at r = r in. If the angular veloity is treated as Keplerian throughout the dis, this implies r 1/2 νσ = 0 at r = r in. GIO