Dark Matter and Rotational Motion e.t. revised feb 09

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: 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. What is the Velocity of Stars that Revolve around the Center of a Galaxy? Orbital velocity

How does the star’s orbital velocity depend on the distribution of the galaxy mass? Simplifying assumption: Spherical galaxy of mass M total, radius R, and uniform density   Outside of galaxy: all galaxy mass M total within the orbit radius r Distance from center of galaxy Speed G and M tot are constants

How does the star’s orbital velocity depend on the distribution of the galaxy mass?  Inside galaxy: Distance from center of galaxy Speed r R

How does the star’s orbital velocity depend on the distribution of the galaxy mass? For spherical galaxy of mass M total, radius R, and uniform density  Distance r from center of galaxy Speed r R R inside outside

What is the actual distribution of visible matter in a galaxy? Galaxies do not have uniform distribution of matter. Most mass at small radius but mass drops more gradually with increasing radius. talog/Jpeg/n3198_pg.jpg Copyright © 1999 by Princeton University Press. NGC 3198 Astronomers measure the luminosity from various regions in the galaxy and can relate luminosity to mass. Freeman, K. Dark Matter in Galaxies, in Encyclopedia of Astronomy and Astrophysics, November 2000

How do measured galaxy rotation curves compare to predictions? The orbital velocity of stars can be measured using the Doppler shift. Freeman, K. Dark Matter in Galaxies, in Encyclopedia of Astronomy and Astrophysics, November 2000 A halo of dark matter is needed.

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.