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Unformatted text preview: Phys 341 Final Exam: Solutions 1. Consider a galaxy with a flat rotation curve with rotation speed v g . There is a dwarf galaxy orbiting the big galaxy in a circular orbit with radius R g . The dwarf galaxy has a flat rotation curve with rotation velocity v d . (a) The tidal force from the big galaxy strips stars out of the “front” and “back” of the dwarf galaxy. We can estimate the tidal radius of the dwarf galaxy to be the distance from the center of the dwarf galaxy where the tidal force pulling “out” on a star equals the gravity from the dwarf galaxy pulling back “in.” Derive an expression for the tidal radius in terms of R g , v g , and v d . We say that the tidal radius is the radius r T at which the tidal force from the big galaxy equals the gravity from the dwarf galaxy: 2 GM g ( R g ) r T R 3 g = GM d ( r T ) r 2 T If the dwarf galaxy has a flat rotation curve, the mass inside r T is M d ( r T ) = r T v 2 d /G . As for M g ( R g ), we know from Newton that for a spherical mass distribu- tion the only mass that matters is the mass inside the orbit. Assuming the dark matter halo of the big galaxy extends at least as far as the dwarf galaxy’s orbit, we have M g ( R g ) = R g v 2 g /G . (You may wonder whether we should use the mass within R g- r T rather than R g itself. That leads to a relatively small correction, which we will ignore.) Putting these masses in the equation above, we get 2 v 2 g r T R 2 g = v 2 d r T Solving for r T gives r T = R G √ 2 v d v g (b) The Large Magellanic Cloud is a dwarf galaxy orbiting about 50 kpc . from the center of the Milky Way. It rotates at about 50 km s- 1 . Recall that the Milky Way’s rotation speed is about 220 km s- 1 . Compute the tidal radius r T of the LMC (in kpc). Plugging in numbers, R g = 50 kpc, v g = 220 km s- 1 , and v d = 50 km s- 1 , we have r T = 50 kpc √ 2 50 km s- 1 220 km s- 1 = 8 . 0 kpc 1 (c) Roughly speaking, only matter within the tidal radius is gravitationally bound to the dwarf galaxy. What is the total mass (in M ) bound to the LMC?) bound to the LMC?...
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This note was uploaded on 04/04/2008 for the course PHYSICS 341 taught by Professor Keeton during the Fall '08 term at Rutgers.
- Fall '08