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Dark Matter and Rotational Motion e.t. revised feb 09 What is the Velocity of Stars that Revolve around the Center of a Galaxy? From Newtonian dynamics, the centripetal force acting on an orbiting star of mass m is the gravitational force pulling the star toward the center of the galaxy: 2 v m r Orbital velocity v GM m r2 GM r • Here r and v are the star’s average radial distance from the center of the galaxy and its orbital velocity. G is the universal gravity constant and M is the mass of the galaxy within radius r. • NOTE: only mass lying within a star's orbit affects the star's motion. So we need to consider the mass distribution in the galaxy. How does the star’s orbital velocity depend on the distribution of the galaxy mass? • Simplifying assumption: Spherical galaxy of mass Mtotal, radius R, and uniform density r Outside of galaxy: all galaxy mass Mtotal within the orbit radius r v GM tot r G and Mtot are constants Speed Distance from center of galaxy How does the star’s orbital velocity depend on the distribution of the galaxy mass? Inside galaxy: v velocity GM r v uniform density r M total 4 R3 3 r R 4 G r r3 3 r 4 G r r 3 Speed Distance from center of galaxy How does the star’s orbital velocity depend on the distribution of the galaxy mass? • For spherical galaxy of mass Mtotal, radius R, and uniform density r inside outside r R Speed R Distance r from center of galaxy What is the actual distribution of visible matter in a galaxy? • Galaxies do not have uniform distribution of matter. NGC 3198 • Most mass at small radius but mass drops more gradually with increasing radius. http://www.astro.princeton.edu/~frei/Gcat_htm/Ca talog/Jpeg/n3198_pg.jpg Copyright © 1999 by Princeton University Press. Freeman, K. Dark Matter in Galaxies, in Encyclopedia of Astronomy and Astrophysics, November 2000 Astronomers measure the luminosity from various regions in the galaxy and can relate luminosity to mass. How do measured galaxy rotation curves compare to predictions? A halo of dark matter is needed. Freeman, K. Dark Matter in Galaxies, in Encyclopedia of Astronomy and Astrophysics, November 2000 The orbital velocity of stars can be measured using the Doppler shift. Additional Remarks • Dark matter is overall estimated to be about 6 times more abundant than ordinary matter. • Similar orbital analyses done for galaxy clusters show similar need for dark matter. • Dark matter has yet to be directly detected and we do not know what it is. However, several completely independent indirect measurements support it’s existence.