PaperIII_63 (3) - MATHEMATICAL TRIPOS Part III Thursday 9...

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MATHEMATICAL TRIPOS Part III Thursday, 9 June, 2011 9:00 am to 11:00 am PAPER 63 DYNAMICS OF ASTROPHYSICAL DISCS Attempt no more than TWO questions. There are THREE 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. 2 1 (a) Accretion on to a black hole can be modelled within Newtonian dynamics by using the modified gravitational potential Φ = − GM R − Rs , where R = (r2 +z2)1/2 is the spherical radial coordinate in terms of cylindrical polar coordinates (r, φ, z), and Rs is the radius of the event horizon of the black hole. By constructing an effective potential governing the r and z components of the equation of motion, or otherwise, determine the orbital frequency Ω(r), the epicyclic frequency κ(r) and the vertical frequency Ωz (r) of circular orbits of radius r > 3Rs in the plane z = 0, in this potential. Explain the significance of the result Ωz
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