(To Be Discussed Wed, March 22) 1. the Virial Theorem in 1933, Fritz

Experimental astroparticle physics, PHY465 Exercise sheet 4 (to be discussed Wed, March 22)

1. The virial theorem In 1933, Fritz Zwicky observed the radial velocities of galaxies in the Coma cluster and used the virial theorem to calculate the mass of the cluster. The state of the system is often quoted in terms of the virial ratio:

hT i Q = − (1) vir hΩi

A system which is in virial equilibrium has Qvir = 0.5, where Ω is the potential energy and T is the kinetic energy.

(a) Derive an expression for the total mass of a gravitationally bound system as a function of the dispersion velocity hv2i. (b) Zwicky estimated that the Coma cluster contains approximately 1000 galaxies with a total radius of R = 3 Mpc. From measuring the line-of-sight velocities of several galaxies in the cluster, he estimated the dispersion in the radial velocity to be hσi = 900 9 km/s. If one assumes an average luminosity of L = 8 · 10 L per galaxy, calculate the mass to light ratio for the Coma cluster in terms of M . L (c) List the main assumptions that were involved in Zwicky’s mea- surement. If one observes a mass-luminosity ratio on the order of 4 M from, e.g., cluster main sequence fitting, how can one inter- L pret Zwicky’s result from the virial theorem?

2. Rotation velocity and dark matter The classical evidence for dark matter comes from the measurement of the rotation curves of velocity versus radial distance for stars and gas in spiral galaxies. This has given strong, if indirect, indications for the existence of missing mass, in the form of non-luminous matter. Ob- served rotation curves of galaxies increased with r for small r, but then they flatten out becoming v(r) ≈ constant up to very large distances.

1 (a) Under the assumption that the density profile of a galaxy decreases as power law ρ ∝ r−n, find the value of n which implies constant velocity at large r. 5 (b) At r = 10 light years, a galaxy has a measured velocity of vmeas = 225 km/s and an expected velocity of vexp = 15 km/s. Calculate the visible as well as the true mass of the galaxy. What is the percentage of dark matter in the galaxy? How high is the average dark matter mass density?

3. Gravitational lensing The Large Magellanic Cloud hosts R136a1, the most massive star known with M = 256 M . Consider a case of a point-like object with such mass in the local galaxy situated midway between an observer on the Earth and a light source at distance 2 pc. Estimate the resolution needed to observe separated images of individual stars with an optical telescope. Can one observe it with the Hubble Space Telescope which has a resolution of 0.1 arc sec? Discuss the result.

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