A galaxy's rotation curve is a measure of the orbital speed of stars as a function of distance from the galaxy's centre. The fact that rotation curves are primarily flat at large galactocen- tric distances (vrot(r) ~ constant) is the most common example of why astronomer's believe dark matter exists. Let's work out why! Assuming that each star in a given galaxy has a circular orbit, we know that the accelera- tion due to gravity felt by each star is due to the mass enclosed within its orbital radius r and equal to v/r. Here, ve is the circular orbit velocity of the star. (a) Show that the expected relationship between ve and r due to the stellar halo (p(r) x r-3.5) does not produce a flat rotation curve. (b) Show that a p(r) x r-² density profile successfully produces a flat ro- tation curve and must therefore be the general profile that dark matter follows in our galaxy.

A galaxy's rotation curve is a measure of the orbital speed of stars as a function of distance from the galaxy's centre. The fact that rotation curves are primarily flat at large galactocen- tric distances (vrot(r) ~ constant) is the most common example of why astronomer's believe dark matter exists. Let's work out why! Assuming that each star in a given galaxy has a circular orbit, we know that the accelera- tion due to gravity felt by each star is due to the mass enclosed within its orbital radius r and equal to v/r. Here, ve is the circular orbit velocity of the star. (a) Show that the expected relationship between ve and r due to the stellar halo (p(r) x r-3.5) does not produce a flat rotation curve. (b) Show that a p(r) x r-² density profile successfully produces a flat ro- tation curve and must therefore be the general profile that dark matter follows in our galaxy.