Assume that most matter in the Galaxy is distributed spherically symmetrically and that the Sun travels in a circular or

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Assume That Most Matter In The Galaxy Is Distributed Spherically Symmetrically And That The Sun Travels In A Circular Or 1 (130.21 KiB) Viewed 21 times

Assume that most matter in the Galaxy is distributed spherically symmetrically and that the Sun travels in a circular orbit about the centre of the Galaxy in accordance with Newtonian motion. You will be considering only motions in the solar neighbourhood, so that you can treat the mass of the Galaxy interior to the solar orbit as a constant, M. (3a) Derive a simple expression for O(R) in terms of M, and R, where R is the distance from the Galactic centre and is the rotational velocity about the Galactic centre. [10 marks] (3b) The definitions of the Oort constants are A= 1 , dᎾ R. 2'dR Ꮎo R. Өот 1 , dᎾ R. + Ro 2'dR R. BE where R, and e, are the Sun's distance from the Galactic centre and the Sun's rotational velocity about the Galactic centre. Use your answer from (a) to show that the Oort constants A and B can be described by the following simple expressions in this approximation: (10 marks] 30. 10. A= B 4R 4R (3c) Determine numerical values for A and B in the solar neighbourhood (in km s- kpc) based on the observational estimates , = 220 km/s and R. = 8.5 kpc. (4 marks] (3d) Suppose we are given instead the observational estimates of the Oort constants Apbs 14.4 km s- kpc and Bobs = –12.0 km s- kpc-7. Estimate de/dR in the solar neighbour- hood using your values for A and B and then using the values Aobs and Bobs. How does the observed behaviour of the Galactic rotation curve in the solar neighbourhood compare with that expected for the assumptions made here? [6 marks]