(3a) Use the model equation for the radial density distribution of the dark-matter halo of the Milky Way Co p(r) a2 + 2

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3a Use The Model Equation For The Radial Density Distribution Of The Dark Matter Halo Of The Milky Way Co P R A2 2 1 (116.06 KiB) Viewed 30 times

(3a) Use the model equation for the radial density distribution of the dark-matter halo of the Milky Way Co p(r) a2 + 2 to show that the amount of dark matter interior to a radius r is given by the expression [8 marks] Mamy = 47C. (r – a arctan] (3b) Show that the mass of dark matter interior to a radius r can be approximated Mam, 41 Cor when r » a. Show that a reasonable estimate of C, is Co = 9 x 108 M./kpc, based on a reasonable estimate of the mass of dark matter in the Galaxy within a radius of 50 kpc, given that the characteristic rotational velocity in the flat part of the Milky Way's rotation curve is 235 km/s. State clearly any assumptions you have made. Estimate the mass-to-light ratio for matter within this radius. [8 marks] (3c) Show that the mass of dark matter interior to a radius r can be approximated Mam, dr 41C.(3/3a) when r <a. Show that a reasonable estimate of the mass of dark matter in the Galaxy within a radius of 1 kpc, crudely assuming r <a, yields a ~1.8 kpc. The characteristic rotational velocity at this radius is 220 km/s. Estimate the mass-to-light ratio within a radius of 1 kpc. [8 marks] (3d) Briefly explain what is meant by "dark energy”. What percent of the universe is made up of baryonic matter, dark matter, and dark energy? [6 marks]