## Problems

For all problems, unless otherwise stated, use H0 = 70 km/s/Mpc.

18.1. For the cluster in Example 18.2, what is the total kinetic energy?

18.2. For some cluster of galaxies, the radius is 500 kpc, and the rms radial velocity is 300 km/s. What is the mass of the cluster?

18.3. Rewrite equation (18.1) so that if velocities are entered in km/s and distances in Mpc, the mass results in solar masses.

18.4. Suppose that one-half of the mass of the cluster in Example 18.2 is in the form of hot intergalactic gas, spread out uniformly over nature of the dark matter. No matter what we try, no single theory (or type of dark matter) can explain the large- and small-scale structures that we see, but CDM appears to be doing a better job.

18.11. What makes us think that there is dark matter in the Virgo cluster?

18.12. Why do we expect a cluster of galaxies to obey the virial theorem but not a supercluster?

18.13. In adding up the 'visible' mass in clusters, what is the problem in accounting for the masses of the galaxies that we can see?

18.14. Why would we expect intergalactic gas to be able to emit X-rays?

18.15. What are the current possibilities for the dark matter in clusters?

18.16. If we cannot see the dark matter in clusters of galaxies, how do we know that it is there?

18.17. Explain how we might determine the mass of a binary galaxy system. Why is this important, since we already have masses determined from rotation curves?

18.18. Why are redshift surveys important in studying clustering of galaxies?

18.19. How are hot and dark cold matter related to various scenarios of galaxy formation?

18.20. What type of structure is hot dark matter best at explaining? What types for cold dark matter?

the whole cluster. What density of gas would this require?

18.5. For the galaxies represented in Fig. 18.11, draw a graph of the length of the red line vs. distance. Do this three times, each time using a different galaxy as your reference point.

18.6. Find a relationship between the Hubble constant, expressed in km/s/Mpc, and the Hubble time, expressed in years. Use this to find the Hubble times corresponding to Hubble constants of 50, 65, 100 and

18.7. Suppose we have a universe whose size increases by 1% in 1 Gyr. Show that the average rate of separation between any two points is proportional to their distance, and find the proportionality constant.

18.8. For some galaxy, we measure a recession velocity of 2000 km/s. How far away is the galaxy?

18.9. Rewrite Hubble's law so that you can put in recession velocities in km/s and get distances in Mpc.

18.10. If the typical random velocity in a cluster is 300 km/s, how far out must we go before this is only 1% of the expansion speed at that distance?

18.11. Suppose that we can use supernovae to measure distances out to 200 Mpc. What is the recession speed at that distance?

18.12. What is the crossing time for a cluster moving at 103 km/s through a typical superclus-ter-sized object?

*18.13. What are the Jeans mass and length if we have 1016 M0 worth of H spread out over 10 Mpc, with random internal motions of 1000 km/s?

18.14. What is the density of galaxies (galaxies/ly3) in the local supercluster?

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