Dynamics: Note 12

Circular Motion Application: Dark Matter Dark matter is the proposed solution for a phenomenon witnessed within galaxies. Background Using circular motion, we observe the planets in our solar system to have an inverse relationship when we compare their orbital speeds, and their distances away from the sun.

This graph shows that as we increase our distance from the sun, the orbital speed of each planet is drastically reduced. This can be shown using our formula for centripetal force: Fnet = ma

mv 2 r 2 GMm / m /v = r/ r 2/ GM v= r This formula will produce the relationship above. In theory, this should apply to all orbiting bodies, even for stars orbiting around the edge of a galaxy. The Problem We can make additional observations about the velocity of bodies around a central object. We can apply our knowledge of Doppler shift to make these measurements. Remember, Doppler shift is the apparent change in wave frequency due to the relative motion between objects. If an ambulance is speeding toward you, it will appear that its siren will have a higher pitch than if it was at rest. If it was speeding away, it would have a lower pitch. The same principal is true for light waves. If a star was travelling toward you, its light would appear to be at a higher frequency, thus it will appear a little blue. If it is moving away, it would look slightly red. Fnet =

Dynamics: Note 12 When looking at the stars, we can measure their blue/red shifts, and then calculate their orbital velocities (as they orbit around the centre of a galaxy). We observe the following data:

The red line illustrates what we expect to see when we measure the orbital speeds of stars around a galaxy, but the green line is what we actually observe. How is this possible? This means that galaxies rotate faster than expected! There must be some additional mass in the galaxy to increase centripetal force (more mass than what we can observe). If physicists calculate the mass of a galaxy (by counting up all the stars), we obtain values much lower than what this graph predicts. This means that there is mass that is unaccounted for. We call this mystery mass: DARK MATTER. Dark matter does not interact with light or radiation, thus it cannot be seen. Since dark matter has mass, it effects our perceptions of space-time and its warping. We observe this through the effect of gravitational lensing (as described in the special relativity unit). By making observations with Hubble and other land-based telescopes, we can make approximations about the distribution of dark matter throughout a galaxy: