Why is Rotation Speed Proportional to Mass?

mv2/R = GMm/R2;

centrifugal force = gravitational force due to the mass “M” within the radius “R.”

=> Measure v and you produce a curve that describes v versus M

Jackson-FaberSimilar to Tully-Fisher, except for

• Elliptical galaxies

• width of visible [star] light lines are the measured

Refined does even better.

• The “fundamental plane.”

• Adds surface brightness of the galaxy.

• Plane formed from=>

Fundamental Plane

Line width

Luminosity

Surface brightness

3 points determine a plane

Line width results from?? =>

• The lines come from individual stars.

• The stars are orbiting the galaxy center that when we add up the light from all the stars

Ways to make lines broad:

• Intrinsic: transition rate

• Thermal: motion of atoms due to heat

•Bulk motior (needed for Jackson-Faber)

=> To detect it, we must have bulk motion be large enough to see “on top” of other effects

• Luminosity = total intrinsic light output

• Surface brightness = Flux/(“solid angle”),

• Flux = L/4D2

• Solid angle r2/D2, where

• D = the distance to the object

• r = the radius of the object

• Surface brightness is independent of distance (ignoring relativity and GR)

≅

• Measuring radius of a “fuzzy” object at a given surface brightness level good

• For the same objects at different distances gives the same “true” radius

Surface Brightness (the math) =

(L/4D2) x 1/(r2/D2) =L

4πD2 xD2

πr 2 =L

4π 2 r2

Concept different from luminosity. The key is the concept of “solid angle”

Sky patch

Angle of cone

=> Angle area of cone = ( ≡ Solid angle

r2/D2 = solid angle = (r/D)2=

(/2)2

r

D

Flux x 1/solid angle

=> Physical concept is

• Higher the surface brightness

• The better a object stands out against the background

For our small angles (less then about 1 degree), = r/D where r = the physical [actual] radius of the object’s projected image on the sky and D = the distance to the object

r

D/2

r/D = tan(/2) ≅ sin(/2)

≅/2 in radians

1 radian = 57.2957795 degrees

Example:

tan(1 degree) = 0.01745506

sin(1 degree) = 0.01745240

1./57.2957795 = 0.01745329

=> Good to 4 decimal places, which is good enough

If you don’t have a calculator handy, use 1 radian 60 degrees.≅

Fundamental Plane

Example of cross calibration:

FLUX

Sur. Br x Line Width*

Measured

Then measure D by Cepheids. =>*Real formula is more complex.

Fundamental Plane

Example of cross calibration:

Luminosity

Sur. Br x Line Width

Derived

Now luminosity; with L = F x 4D2

Fundamental Plane

Measure surface brightness and line width,

Derive L

Also measure F

then derive D or

D=L

4πF

Other methods:

Other (besides Cepheids and SNeIa)

standard candles

• Globular Clusters and Brightest Cluster Galaxy use two concepts:

There is maximum size to things:

• The largest globular cluster in a galaxy

or

• The largest galaxy in a cluster

• Are always about the same size.

• Total mass and total light (luminosity) are directly related.

Standard Candle

What is a globular cluster?

M2 has a diameter of about 140 light-years (47 pc), contains about 150,000 stars,

(You already know about clusters of galaxies)

Other methods:

(1) Gravitational Lenses:

• Delay in the travel time over one path vs another

• Dependent on the true path length

•Measure distances this way.

(2) The S-Z (Sunyaev-Zel’dovich Effect):

• Interaction of the hot gas with the CMB

• Effect depends on the true size of the cluster

• Plus model of the gas temperature profile.

GR lens Concept:“false” =

“lensed image”

True image

If “ a” varies we’ll see “b” vary exactly the same way later.

Measure time delay = = /c where = ()- , use trig. to calculate measure v, derive H0

a

b

observer

Problems with Grav. lens

• hard to correlate the variability of the images since the object varies randomly

• we need to model the mass distribution of the lens to derive the path length

•(c ) there aren’t many of these systems, and they are difficult to identify.

Identification requires demonstrating the two (really many, but we’re keeping it simple) images you detect are from the same “true” object and not a chance “coincidence”

Typical separations are a few arc seconds. Remember 1 degree = 60 arc minutes; 1 arc minute = 60 arc seconds => 1 degree = 3600 arc seconds.

An abnormally high gravitational field can cause a path deflection that corresponds to an arc minute or more=> About 15 years ago, astronomers in a prestigious eastern school found two images that appeared to come from the same object and were separated by nearly 2 arc minutes=>

They concluded that they’d detected something unusual called a “cosmic” string” =>

15 minutes of fame

Until an astronomer at another [competing] prestigious eastern university showed that this was a chance coincidence of two similar looking objects

If we ever discover a cosmic string this will be very exciting. This discovery would show that “defects in space” can really exist!