With measurements of true Distance, plus recessional velocity, can infer mass concentration within a given volume [us to GA] M/L ≈ 500-1000 MO/LO

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)

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b = bias parameter now is about 1.07 with m= 0.3, =0.7

()galaxies = b()matter

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.

How to measure LHow 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.

[(1/2) x m(Vmax)2 = GMm/Rmax] => measure vmax and Rmax , then infer M. Vmax we see in the plane; Rmax is the maximum height above we stars

To measure L≈Find M/L 5MO/LO . .

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 thickStar motion up & down

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 20MO/LO as a good value.

. .

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

External Galaxies: The Cosmic Conspiracy

http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

What’s the problem?What would we expect to see if there were no problem?

If “uniform disk”

If “Keplerian”

“Just right” = the problem

Now the mathBasic 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 mv2/r= G(r2)m/r2

Or v goes as the sqrt(r)

More math

If the amount of mass increase is negligible, then

mv2/r= GMm/r2; 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

What’s the problem?What would we expect to see if there were no problem?

If “uniform disk”

If “Keplerian”

“Just right” = the problem

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

External Galaxies: The Cosmic Conspiracy

http://www.ioa.s.u-tokyo.ac.jp/~sofue/h-rot.htm

QSO 3C273+ jet LMC NGC 3310

M85, S0 M87 Ellip M87 jet