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This is the Past Exam of Mathematical Tripos which includes Class Field Theory, Artin Map, Abelian Extension of Number Fields, Decomposition Group, Inertia Group, Factorisation of Prime Ideals, Version of Hensel’s Lemma, Hilbert Norm Residue Symbol etc. Key important points are: Accretion Discs, Keplerian Accretion Disc, Negligible Radiation Pressure, Ordinary Differential Equations, Zero Boundary Conditions, Kinematic Viscosity, Dimensionless Constant, Fractional Error
Typology: Exams
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Wednesday 8 June, 2005 1.30 to 3.
Attempt TWO questions.
There are THREE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 The vertical structure of a thin, Keplerian accretion disc with constant (Thomson) opacity and negligible radiation pressure is governed by the equations
∂p ∂z
= −ρΩ^2 z, (1) ∂F ∂z
αΩp, (2)
∂T ∂z
3 κρF 16 σT 3
p =
kρT μmmp
(i) Explain briefly the physical meaning of each equation.
(ii) Show that the problem of the vertical structure of such a disc can be reduced to a universal system of dimensionless ordinary differential equations and boundary conditions by a suitable rescaling of the variables. You may assume that the viscosity parameter α and the mean molecular weight are independent of z and that the ‘zero boundary conditions’ apply at the surfaces of the disc.
(iii) Assuming that the dimensionless system has a unique solution, deduce that the density-weighted mean kinematic viscosity of the disc is of the form
ν¯ = Cα^4 /^3 (GM )−^1 /^3
( (^) κ
σ
) 1 / 3 ( (^) μ mmp k
rΣ^2 /^3 ,
where Σ is the surface density of the disc at a distance r from the central mass M , and C is a dimensionless constant, which need not be determined.
(iv) Equations (1) and (2) involve several approximations that are valid only in the limit of a thin disc. Estimate the order of magnitude of the terms neglected in these equations, and show that the fractional error in each equation is of the order of (H/r)^2 , where H is the semithickness of the disc.
Paper 73
