1. With measurements of true Distance, plus recessional velocity, can infer mass concentration within a given volume [us to GA] M/L ≈ 500-1000 M O /L O Can also derive an  m, from GA work derive  m ≈ 0.3 ( about right) Looking ahead: derive M/L for objects Compare with total light in a volume divided into  m. Before lambda ≠ 0, assumed  m =1, then derived “this way” M/L ≈ 3 times bigger than Clusters (lead to fudge concept of bias)

2. b = bias parameter now is about 1.07 with  m = 0.3,   =0.7 (  ) galaxies = b(  ) matter

3. Accurate distance and velocity measurements can be used to infer the existence of dark matter Question: How is CDM distributed compared to galaxies and clusters of galaxies. Now maybe close, but except for GA before Lambda ≠ 0, had to have “bais”, galaxies more concentrated.

4. How to measure  L How to measure “local” values? Local = near the sun in our galaxy and our galaxy on average Measure the motions perpendicular to the galactic plane of bunch of stars Assume the max we see is the true max. See how far off the plane we see objects that we think are undergoing motion perpendicular to the galactic plane.

5. [(1/2) x m(V max ) 2 = GMm/R max ] => measure v max and R max, then infer M. V max we see in the plane; R max is the maximum height above we stars

6. To measure  L Find M/L 5M O /L O.. This is the “local” value = within 100 pc of the sun. Side viewn of our galaxy “Bulge” ~ 2 kpc diameter x We are about here, ~ 8 kpc from center “gas & star disk,~ 200 pc thick Star motion up & down

7. For spiral galaxies we can use 21 cm, so called rotation curve. For our galaxy, since we are inside it, this is difficult, but as well as we can tell we are imbedded in a “halo” and the M/L for the Milky Way Galaxy is between 10 and 30 => take 20M O /L O as a good value...

8. What data look like Black hole From http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

9. External Galaxies: The Cosmic Conspiracy http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

10. What’s the problem? What would we expect to see if there were no problem? If “uniform disk” If “Keplerian” “Just right” = the problem

11. Now the math Basic concepts: centrifugal force balances gravitational force If have uniform mass distribution, only the mass inside a give radius matters (Gauss’ Law), which enables us to write simple equations. For a uniform disk with a uniform mass/unit area (call it  ), we have mv 2 /r= G(  r 2  )m/r 2 Or v goes as the sqrt(r)

12. More math If the amount of mass increase is negligible, then mv 2 /r= GMm/r 2 ; where now the mass (M) is effectively constant with increasing radius. Then, we find v goes a 1/sqrt(r), this is called “Keplerian” Return to the model and then the data

13. What’s the problem? What would we expect to see if there were no problem? If “uniform disk” If “Keplerian” “Just right” = the problem

14. What data look like Black hole From http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

15. External Galaxies: The Cosmic Conspiracy http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

16. QSO 3C273+ jet LMCNGC 3310 M85, S0M87 Ellip M87 jet