Describe what is meant by dark matter.

The distribution of mass in a spherical system is such that the density ρ varies with distance r from the centre as

ρ = \(\frac{k}{{{r^2}}}\)

where k is a constant.

Show that the rotation curve of this system is described by

v = constant.

Curve A shows the actual rotation curve of a nearby galaxy. Curve B shows the predicted rotation curve based on the visible stars in the galaxy.

Explain how curve A provides evidence for dark matter.

dark matter is invisible/cannot be seen directly
OR
does not interact with EM force/radiate light/reflect light

interacts with gravitational force
OR
accounts for galactic rotation curves
OR
accounts for some of the “missing” mass/energy of galaxies/the universe

OWTTE

[6 marks]

«from data booklet formula» \(v = \sqrt {\frac{{4\pi G\rho }}{3}} r\) substitute to get \(v = \sqrt {\frac{{4\pi Gk}}{3}} \)

Substitution of ρ must be seen.

[1 mark]

curve A shows that the outer regions of the galaxy are rotating faster than predicted

this suggests that there is more mass in the outer regions that is not visible
OR
more mass in the form of dark matter

OWTTE

[2 marks]