lecture-14 - These lecture notes were prepared for Rutgers...

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These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c ± 2011 Lecture 14: Spiral Galaxy Rotation Curves I. Disk Rotation Curves The orbits of stars in disk galaxies are close to circular, which is convenient. There are so many stars that, even though they are all moving, the average mass distribution hardly changes with time. So for now we can imagine that the mass distribution is fixed in time, which means that the gravitational force as a function of radius is also fixed in time. This is essentially the one-body problem again, except with an extended mass distribution. I.1. 3-d spherical mass distribution Let’s start with the spherical case. Suppose the density as a function of radius is ρ ( r ). Then the mass enclosed by radius r is M ( r ) = 4 π Z r 0 ρ ( u ) u 2 du ρ ( r ) = 1 4 πr 2 dM ( r ) dr Recall from Newton that for a spherical mass distribution we can consider: outside a spherical shell, the gravity is the same as if all the mass of the shell were concentrated at the center inside a spherical shell, there is no gravity from that shell So the net gravitational force at radius r depends only on the mass enclosed by radius r , F ( r ) = - GM ( r ) m r 2 For circular orbits, the force must provide the centripetal acceleration a = - v ( r ) 2 /r , so GM ( r ) m r 2 = m v ( r ) 2 r v ( r ) = ± GM ( r ) r ² 1 / 2 and M ( r ) = r v ( r ) 2 G The second equation is useful if we suppose that we know the mass distribution and want to compute the corresponding rotation curve. (The second equation should look familiar: you 1
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r/h R Figure 1: Rotation curve for an exponential disk. derived it by dimensional analysis on PS#1. It turns out that there are no dimensionless factors, provided M ( r ) is the enclosed mass.) The third equation is useful if we measure the motion and want to infer the mass. Specifically, if we measure the rotation speed at one radius, we can automatically compute the mass enclosed by that radius. If we measure the full rotation curve, we can determine the mass profile curve, which in turn gives us the full density profile. As always, motion mass! I.2. 2-d circular mass distribution The analysis of a thin disk is more involved. Since a disk is not spherically symmetric, you have to evaluate some moderately sophisticated integrals to determine the gravitational force. (See Galactic Dynamics disk we are interested in is the exponential disk, let me just quote the rotation curve: v ( r ) 2 = πG Σ 0 r 2 h R ± I 0 ² r 2 h R ³ K 0 ² r 2 h R ³ - I 1 ² r 2 h R ³ K 1 ² r 2 h R ³´ where I 0 , K 0 , I 1 , and K 1 are special functions called modified Bessel functions. This is plotted in Figure 1. Let’s look at the general features. Quantitatively, the rotation curve peaks at r max = 2 . 15 h R v max = 1 . 56( G Σ 0 h R ) 1 / 2 Qualitatively, the important feature is that the rotation curve declines as you go to large
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This note was uploaded on 10/20/2011 for the course PH 341 taught by Professor Gawiser during the Fall '11 term at Rutgers.

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lecture-14 - These lecture notes were prepared for Rutgers...

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