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AST 3003S: Galactic and Extragalactic Astronomy Tutorial 1: Due: Aug 7 Question 1: Show that the law of sines in spherical trigonometry reduces to the planar formula when we use the small angle approximations. Do the same for the law of cosines for angles (ignore all quadratic and higher-order terms). Question 2: Star A has coordinates a1 = 10h, d1 = 70° and Star B has coordinates a2 = 11h, d2 = 80°. Calculate their angular separation if we treat them as points on a planar triangle. Now calculate their separation treating them as points on a sphere. Under which conditions would the two answers be closest? Question 3: Derive the transformation equations for converting from Galactic to Supergalactic coordinates and vice-versa. The north SG pole (SGB=90º) lies at Galactic coordinates (l=47.37º, b=+6.32º). The zero point (SGB=0º, SGL=0º) lies at (l=137.37º, b=0º). Make a sketch of the 2 coordinate systems and clearly mark the spherical triangle and any angles or sides you use for your derivations of the transformation equations. Determine the Supergalactic coordinates for the Galactic coordinates of M87 (l,b = 283.78, 74.49), the central galaxy of the Virgo cluster) using your equations. Question 4: The Sun has an apparent magnitude of -26. What is the optical thickness, tau, of a layer of fog if the Sun seen through it appears as bright as the full Moon (apparent magnitude of -12.5)? Question 5: The apparent magnitudes of two stars in binary system are 7 and 11 respectively. What is the apparent magnitude of the system? Question 6: Two open clusters, which are seen near each other in the Galactic plane, have angular diameters a and 3a, and distance moduli 16.0 and 11.0, respectively. Assuming their actual diameters are equal, find their distances and the interstellar extinction coefficient a. Question 7: The V magnitude of a star is 15.1, B-V = 1.6 and Mv = 1.3. The extinction in the direction of the star (in the visual band) is Av = 1 mag kpc-1. What is the intrinsic colour of the star? Question 8: Determine the foreground and internal extinction towards a galaxy Go to NED, the NASA/IPAC Extragalactic Database (http://ned.ipac.caltech.edu/) and find the coordinates of 3 galaxies, one far from the Galactic Plane, i.e. M87, the central galaxy of the Virgo cluster; , one deep in the extinction layer of the Milky Way, i.e. HIPASS J0836-43, the most HImassive galaxy in the nearby Universe (fully obscured in optical) and your favorite galaxy, as well as the B-band magnitude, morphological type and axis ratio. Determine the total foreground extinction in the B- and K-bands towards these objects using the methods discussed in class (partially available through the extinction calculator on same (NED) website, i.e. the (a) the cosecant law by Sandage based on galaxy counts (b) The Galactic HI-column density (use the HI-column density calculator on http://www.astro.uni-bonn.de/hisurvey/profile) (c) the NED Extinction calculator which is based on the DIRBE/COBE extinction maps. What are the values? How do the respective determinations compare? (d) Do any of the magnitudes of these 3 galaxies need correction for internal extinction as well. If so, how large would that correction be (use the Sandage and Tammann 1981) approximations listed in course notes?)