Paper72 - MATHEMATICAL TRIPOS Tuesday 14 June 2005 Part III 1.30 to 4.30 PAPER 72 GALAXIES AND DARK MATTER Attempt THREE questions There are FOUR

Paper72 - MATHEMATICAL TRIPOS Tuesday 14 June 2005 Part III...

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MATHEMATICAL TRIPOS Part III Tuesday 14 June, 2005 1.30 to 4.30 PAPER 72 GALAXIES AND DARK MATTER Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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2 1 (1) We may study the vertical structure of a thin axisymmetric disk by neglecting all radial derivatives and adopting the form f = f ( E z ) for the distribution function, where E z 1 2 v 2 z + Φ( z ). Show that if f = ρ 0 (2 πσ 2 z ) - 1 / 2 exp( - E z 2 z ), the approximate form of Poisson’s equation may be written: 2 d 2 φ 2 = e - φ , where φ Φ σ 2 z , ζ z z 0 and z 0 σ z 8 πGρ 0 . By solving this equation subject to the boundary conditions φ (0) = dφ/dζ | 0 = 0 , show that the density ρ in the disk is given by ρ ( z ) = ρ 0 sech 2 ± z 2 z 0 ² . (2) Consider a spherically-symmetric stellar-dynamical system with distribution function f ( E ) = ρ 1 (2 πσ 2 ) 3 / 2 exp[ E/σ 2 ] , where E = Ψ - 1 2 v 2 is the relative energy, where Ψ is the potential, v
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