Homework 6 Solutions

Evaluate 0 and plot the density on a logarithmic

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Unformatted text preview: e integral is simple power law for xα+1, which yields: MD 2 2π Σ 0 R 0 (Rout /R0 )α+2 − (Rin /R0 )α+2 = α+2 moving terms over and pulling out the (Rout/R0)α+2 factor in the brackets: Σ0 = MD (α + 2) 2 2π R 0 α+2 R0 Rout α+2 −1 Rin Rout 1− switching out two powers of R0/Rout will give you the expression above. (c) [10 pts] Combine these expressions to obtain a single expression for the density of the disk as a function of r and z: ρ(r, z ) = ρ0 r Rout α−β − e z2 H ( r )2 and derive an expression for the constant ρ0. This is just a matter of combining equations: MD (α + 2) ρ(r, 0) = 3/2 2 2 π R 0 H0 R0 Rout α+2 1− Rin Rout α+2 −1 r R0 α−β Putting this into the form above, it’s clear that MD (α + 2) ρ0 = 2 2 π 3 / 2 R 0 H0 R0 Rout α+2 1− Rin Rout α+2 −1 (d) [10 pts] Consider a disk with total mass 0.1 M¤, Rin = 0.1 AU, Rout = 300 AU, α =  ­1.5, β = 0.25 and H(Rout) = 300 AU (this is known as a “bow tie” disk) around a star of mass...
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