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PH607/14/test3 UNIVERSITY OF KENT PH607: GALAXIES Test 3, Wednesday Week 24, 09:00 . FORMULA AND DATA ARE PROVIDED OVERLEAF. This is a 50 minute test. Please solve ALL questions. Students with an ILP that gives them more time in exams, please indicate on the answer sheet and do NOT solve Question 5. 1. An E3 galaxy at a redshift of 0.04 is measured to have a radius of 2 arcseconds. Apply Hubble’s Law to estimate its radius in kiloparsecs (hint: convert redshift into distance and arcseconds into radians). The redshift 0.04 correspo9nds to a recession speed of v = 0.04c. According to Hubble’s law, its distance is D = v / Ho = (0.04 x 3 x 105 ) / 70 Mpc = 1200/7 = 170 Mpc Angle subtended by galaxy = 2 arcseconds = 2 (1 AU/ 1 pc) radians = 2 x (1.5 ×1011 m / 3 ×1016 m ) = 10-5 radians Galaxy size = angle subtende x distance = 10-5 x 170 Mpc = 170 x 10-5 Mpc = 1.7 kpc – a smallish galaxy 2. State the two major observational differences between a Seyfert 1 and a Seyfert 2 galaxy. State briefly the major reason for both the differences. 2 o o Seyfert 1 galaxies are bright in the optical, and have both broad and narrow lines in their spectra. Seyfert 2 galaxies are fainter in the optical (but the same as Seyfert 1s in the infrared), and they only have narrow lines in their spectra. Seyfert 2s are obscured Seyfert 1s; a dust torus hides the inner broad line region and nucleus but is too small to hide the narrow line region. Hence it is a an orientation effect. 3. Calculate the Schwarzschild radius in units of the Sun’s radius for a black hole with the Sun’s mass and for a super-massive black hole of mass 108 M . Using ( ds = cdt 1-2GM/rc 2 2 ) 2 æ ö dr -ç - r 2 ( dq 2 + sin 2 q dj 2 ) ÷ è 1-2GM/rc2 ø 2 The equation indicates a singularity occurring at the Schwarzschild radius which is 2GM/c2 = 2 x 6.67 × 10-11 x 2.0 × 1030 / (3 × 108 ) 2 = 3.000 m = 3,000 / 7.0 ×108 = 4.3 x 10 -6 R The radius is proportional to the mass, hence for the SMBH, the Schwarzschikd radius is 430 R 4 The rate of expansion of the Universe can be described by 3 1 dR 2 8 1 2 2 2 G c R kc , 3 3 R dt where R is the scale factor, is the density, is the cosmological constant and k is the curvature. Define the Hubble parameter, H(t), in both words and an equation, and the Hubble constant, Ho, in words. H(t) is the Hubble parameter, the rate of expansion of the universe at any time: H= 1 dR R dt The Hubble constant is the present value of the Hubble parameter. 5. Taking a Hubble constant of Ho = 70 km s-1 Mpc-1, and = 0 and k = 0, determine the critical density of the Universe in units of solar masses per cubic megaparsec. Taking and k = 0, and re-arranging, setting H = Ho , yields the critical density c = 3Ho2/(8G). Ho = 70 x 1000 / (106 x 3.086 x 1016 ) s-1 = 2.26831e-18 Substitute for Ho and G yields: c = 9.208 x 10-27 kg m-3 M = 1.989 x 1030 kg Mpc = 3.086 x 1022 m c = 1.36 x 10-7 M /Mpc3 turn over 4 In spherical coordinates, the Schwarzschild metric is ( ds = cdt 1-2GM/rc 2 2 ) 2 æ ö dr -ç - r 2 ( dq 2 + sin 2 q dj 2 ) ÷ è 1-2GM/rc2 ø . Ho = 70 (km/s)/Mpc G = 6.67 × 10-11 N m2 kg-2, 1 M = 2.0 × 1030 kg , 1 R = 7.0 ×108 m, c = 3 × 108 m s-1, 1 pc = 3 ×1016 m, 1 AU = 1.5 ×1011 m 1 radian = (1 pc / 1 AU) arcseconds 2