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Alpha Disks Consider a disk having a dimensionless viscosity The disk accretes at a steady rate M The disk cools radiatively Neglect the difference between the effective temperature of the disk

Alpha Disks Consider a disk having a dimensionless viscosity . The disk accretes at a steady rate ?M . The disk cools radiatively. Neglect the difference between the effective temperature of the disk Teff (which is nothing more than a convenient way of stating what the emitted flux is) and the actual gas kinetic temperature T. Take the gas to have sound speed cs and angular frequency , both of which vary with disk radius r. (a) Find how h/r scales with r, where h is the disk vertical scale height. Sketch how the disk looks. (b) Find how  scales with r. (c) Find how the disk midplane density  scales with r. (d) Find an approximate expression for how long it takes a pressure disturbance to equilibrate away. Call this time tz, and express it as simply as possible. (e) Find an approximate expression for how long it takes a temperature disturbance to equilibrate away. Call this time tcool, and express in terms of and . (f) Find an approximate expression for how long it takes a mass disturbance (say, a local bunching of material) to viscously diffuse away. Call this time tvisc, and express in terms of , , and h/r. Arrange tz, tcool and tvisc in increasing order. (g) Find an expression for the critical radius rcrit beyond which Toomre’s Q < 1. Alpha disks are generically unstable at large radii, leading some to surmise that the outer peripheries of quasar 2 accretion disks / protostellar accretion disks are fertile breeding grounds for starbursts / binary companion stars or brown dwarfs.

Jun 19 2020 View more View Less

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