Astr 5460 Wed., Feb. 26, 2003

Today: Reminders/Assignments

Longair, Chapter 4, Clusters

Unless noted, all figs and eqs from Longair.

Reminders/Preliminaries

Astro-ph preprints on Friday:

http://xxx.lanl.gov/

Galaxy Spectra/Modeling Assignment – deadline extended to Friday. L

Reading Bennett et al. 2003 (MAP) paper

WIRO possible on Saturday (WEBDA)

Galaxy Spectra assignment

The textbook is rather weak when it comes to observational properties like spectra – as budding young observers you need to know more!

Find and download the galaxy spectra templates of Kinney et al. (1996) – and read the paper!

Find and download the spectral synthesis population models of Bruzual and Charlot.

“Fit” the elliptical template and one spiral galaxy.

Show some plots indicating how broad-band colors change with redshift assuming not evolution (up to z=2).

Write up your results like you would for publication with clarity, citations, etc.

Chapter 4: Galaxy Clusters

Large Scale Distribution of Clusters

Galaxy Distribution in Clusters

Dark Matter in Clusters

Forms of Dark Matter

Cluster Catalogs

Palomar Sky Survey using 48 inch Schmidt telescope (1950s)

Abell (1958) cataloged “rich” clusters – a famous work and worth a look

Abell, Corwin, & Olowin (1989) did the same for the south using similar plates

All original work was by visual inspection

Pavo Cluster

Cluster Selection Criteria (Abell)

Richness Criterion: 50 members brighter than 2 magnitudes fainter than the third brightest member. Richness classes are defined by the number in this range:

Cluster Selection Criteria (Abell)

Compactness Criterion: Only galaxies within an angular radius of 1.7/z arcmin get counted. That corresponds to a physical radius of 1.5 h^-1 Mpc. The redshifts are (were) estimated based on the apparent magnitude of the 10^th brightest cluster member.

Cluster Selection Criteria (Abell)

Distance Criteria: Lower redshift limit (z = 0.02) to force clusters onto 1 plate. Upper limit due to mag limit of POSS, which matches z of about 0.2. Distance classes based on magnitude of 10^th member:

More on Abell Clusters

Complete Northern Sample:

1682 Clusters of richness 1-5, distance 1-6.

Counts in Table 4.2 follow:

This is consistent with a uniform distribution*.

Space Density of Abell Clusters richer than 1:

For uniform distribution, cluster centers would be 50 h^-1 Mpc apart, a factor of ten larger than that of mean galaxies.

Clusters of Clusters

Based on Abell’s Northern Sample:

Spatial 2-point correlation function (Bahcall):

Scale at which cluster-cluster correlation function has a value of unity is 5 times greater than that for the galaxy-galaxy correlation function.

Clusters of Clusters

Peebles (1980) schematic picture:

Cloud of galaxies is basic unit, scale of 50 h^-1 Mpc

About 25% of galaxies in these clouds

All Abell Clusters are members of clouds (with about 2 per cloud), and contain about 25% of the galaxies in a cloud are in Abell Clusters (superclusters occur when several AC combine)

Remaining 75% follow galaxy-galaxy function

In terms of larger structures, galaxies hug the walls of the voids, clusters at the intersections of the cell walls.

Galaxies within Clusters

A range of structural types (Abell)

Regular indicates cluster is circular, centrally concentrated (cf. Globular clusters), and has mostly elliptical and S0 galaxies. Can be very rich with > 1000 galaxies. Coma is regular.

All others are irregular (e.g., Virgo).

I don’t know why he didn’t just call them type 1 and type 2…!

Galaxies within Clusters

A range of structural types (Oemler 1974)

cD clusters have 1 or 2 central dominant cD galaxies, and no more than about 20% spirals, with a E: S0: S ratio of 3: 4: 2.

Spiral-rich clusters have E : S0 : S ratios more like 1: 2: 3 – about half spirals.

Remainder are spiral-poor clusters. No dominant cD galaxy and typical ratio of 1: 2: 1.

Galaxies within Clusters

Galaxies differ in these types (Abell)

In cD clusters galaxy distribution is very similar to star distribution in globular clusters.

Spiral-rich clusters and irregular clusters tend not to be symmetric or concentrated.

Spiral-poor clusters are intermediate cf. above.

In spiral rich clusters, all galaxy types similarly distributed and no mass segregation, but in cD and spiral-poor clusters, you don’t see spirals in the central regions where the most massive galaxies reside.

cD Galaxies

Kormendy (1982) distinguishes these from being merely giant ellipticals.

Extensive stellar envelope up to 100 kpc

Only in regions of enhanced galaxy density (a factor of 100 denser than the average)

Mutiple nuclei in 25-50% of cDs (a very rare thing)

Regular cD clusters are systems that have relaxed into dynamical equilibirum.

Isothermal Gas Spheres

Clusters appear to have relaxed to stationary dynamical states similar to that seen in globular clusters, which can be represented by the mass distribution of so-called isothermal gas spheres.

Mean kinetic energy constant throughout a cluster – that is, velocity distribution is Maxwellian everywhere.

Isothermal Gas Spheres

Starting point is the Lane-Emden equation, which assumes spherical symmetry and hydrostatic equilibrium:

Isothermal Gas Spheres

Simple case, for which ideal gas law holds at all radii, p = ρkT/mass. In thermal equilibrium, 3/2 kT = ½ mass <v²>, therefore:

Non-linear, only has an analytical solution for large r using a power-series expansion and applying some reasonable boundary conditions.

Isothermal Gas Spheres

Boundary conditions:

Smoluchowski’s Envelope at large radii where density, timescales go to extremes.

In astrophysical systems, outmost radii are subject to interactions with other clusters, hence a tidal radius.

Isothermal Gas Spheres

Then recast our equation as:

Where we have the dimensionless quantities x=r/α and ρ = ρ₀y, & A, structural length α:

Isothermal Gas Spheres

As usual in real astronomy things are complicated because we only see a projection rather than a 3D distribution. The projection surface density distribution onto a plane is:

Where now q is the projected distance from the center.

Isothermal Gas Spheres

Tabular solution:

Isothermal Gas Spheres

Graphical solution:

Isothermal Gas Spheres

Fit the function to an observed distribution.

N(q) = ½ at q=3, or core radius R_1/2 =3 α.

To get a central mass density then also need to measure the velocity distribution. From equipartition ½ m<v²>=3/2 kT. So:

King Profiles

More complex version based on Fokker-Planck equation by King (1966). Assumes no particles present with escape velocity+.

Structures of Regular Clusters

Bahcall (1977) describes distributions as truncated isothermal distributions:

Where f(r) is the projected distribution normalized to 1 at r=0, and C is a constant that makes N(r) = 0 at some radius. Results in steepening distribution in outer regions vs. pure isothermal soultion.

R_1/2 = 150-400 kpc (220 kpc for Coma)

Structures of Regular Clusters

In central regions King profiles work well:

For these distributions N₀ = 2R_cρ₀.

De Vaucouleur’s law can also work.

Problem is observations do not constrain things quite tightly enough.

Rich Cluster Summary


	Today: Reminders/Assignments
	Longair, Chapter 4, Clusters



	Unless noted, all figs and eqs from Longair.


	Astro-ph preprints on Friday:
		http://xxx.lanl.gov/

	Galaxy Spectra/Modeling Assignment – deadline extended to Friday. L

	Reading Bennett et al. 2003 (MAP) paper

	WIRO possible on Saturday (WEBDA)


	The textbook is rather weak when it comes to observational properties like spectra – as budding young observers you need to know more!
	Find and download the galaxy spectra templates of Kinney et al. (1996) – and read the paper!
	Find and download the spectral synthesis population models of Bruzual and Charlot.
	“Fit” the elliptical template and one spiral galaxy.
	Show some plots indicating how broad-band colors change with redshift assuming not evolution (up to z=2).
	Write up your results like you would for publication with clarity, citations, etc.


	Large Scale Distribution of Clusters
	Galaxy Distribution in Clusters
	Dark Matter in Clusters
	Forms of Dark Matter


	Palomar Sky Survey using 48 inch Schmidt telescope (1950s)
	Abell (1958) cataloged “rich” clusters – a famous work and worth a look
	Abell, Corwin, & Olowin (1989) did the same for the south using similar plates
	All original work was by visual inspection


	Richness Criterion: 50 members brighter than 2 magnitudes fainter than the third brightest member. Richness classes are defined by the number in this range:


	Compactness Criterion: Only galaxies within an angular radius of 1.7/z arcmin get counted. That corresponds to a physical radius of 1.5 h^-1 Mpc. The redshifts are (were) estimated based on the apparent magnitude of the 10^th brightest cluster member.


	Distance Criteria: Lower redshift limit (z = 0.02) to force clusters onto 1 plate. Upper limit due to mag limit of POSS, which matches z of about 0.2. Distance classes based on magnitude of 10^th member:


	Complete Northern Sample:
		1682 Clusters of richness 1-5, distance 1-6.
		Counts in Table 4.2 follow:

		This is consistent with a uniform distribution*.
		Space Density of Abell Clusters richer than 1:

		For uniform distribution, cluster centers would be 50 h^-1 Mpc apart, a factor of ten larger than that of mean galaxies.


	Based on Abell’s Northern Sample:
		Spatial 2-point correlation function (Bahcall):


		Scale at which cluster-cluster correlation function has a value of unity is 5 times greater than that for the galaxy-galaxy correlation function.


	Peebles (1980) schematic picture:
		Cloud of galaxies is basic unit, scale of 50 h^-1 Mpc
		About 25% of galaxies in these clouds
		All Abell Clusters are members of clouds (with about 2 per cloud), and contain about 25% of the galaxies in a cloud are in Abell Clusters (superclusters occur when several AC combine)
		Remaining 75% follow galaxy-galaxy function
		In terms of larger structures, galaxies hug the walls of the voids, clusters at the intersections of the cell walls.


	A range of structural types (Abell)
		Regular indicates cluster is circular, centrally concentrated (cf. Globular clusters), and has mostly elliptical and S0 galaxies. Can be very rich with > 1000 galaxies. Coma is regular.
		All others are irregular (e.g., Virgo).
		I don’t know why he didn’t just call them type 1 and type 2…!


	A range of structural types (Oemler 1974)
		cD clusters have 1 or 2 central dominant cD galaxies, and no more than about 20% spirals, with a E: S0: S ratio of 3: 4: 2.
		Spiral-rich clusters have E : S0 : S ratios more like 1: 2: 3 – about half spirals.
		Remainder are spiral-poor clusters. No dominant cD galaxy and typical ratio of 1: 2: 1.


	Galaxies differ in these types (Abell)
		In cD clusters galaxy distribution is very similar to star distribution in globular clusters.
		Spiral-rich clusters and irregular clusters tend not to be symmetric or concentrated.
		Spiral-poor clusters are intermediate cf. above.
		In spiral rich clusters, all galaxy types similarly distributed and no mass segregation, but in cD and spiral-poor clusters, you don’t see spirals in the central regions where the most massive galaxies reside.


	Kormendy (1982) distinguishes these from being merely giant ellipticals.
		Extensive stellar envelope up to 100 kpc
		Only in regions of enhanced galaxy density (a factor of 100 denser than the average)
		Mutiple nuclei in 25-50% of cDs (a very rare thing)

	Regular cD clusters are systems that have relaxed into dynamical equilibirum.


	Clusters appear to have relaxed to stationary dynamical states similar to that seen in globular clusters, which can be represented by the mass distribution of so-called isothermal gas spheres.
	Mean kinetic energy constant throughout a cluster – that is, velocity distribution is Maxwellian everywhere.


	Starting point is the Lane-Emden equation, which assumes spherical symmetry and hydrostatic equilibrium:


	Simple case, for which ideal gas law holds at all radii, p = ρkT/mass. In thermal equilibrium, 3/2 kT = ½ mass <v²>, therefore:



	Non-linear, only has an analytical solution for large r using a power-series expansion and applying some reasonable boundary conditions.


	Boundary conditions:
		Smoluchowski’s Envelope at large radii where density, timescales go to extremes.
		In astrophysical systems, outmost radii are subject to interactions with other clusters, hence a tidal radius.


	Then recast our equation as:


	Where we have the dimensionless quantities x=r/α and ρ = ρ₀y, & A, structural length α:


	As usual in real astronomy things are complicated because we only see a projection rather than a 3D distribution. The projection surface density distribution onto a plane is:



	Where now q is the projected distance from the center.


	Fit the function to an observed distribution.
		N(q) = ½ at q=3, or core radius R_1/2 =3 α.
	To get a central mass density then also need to measure the velocity distribution. From equipartition ½ m<v²>=3/2 kT. So:


	More complex version based on Fokker-Planck equation by King (1966). Assumes no particles present with escape velocity+.


	Bahcall (1977) describes distributions as truncated isothermal distributions:

	Where f(r) is the projected distribution normalized to 1 at r=0, and C is a constant that makes N(r) = 0 at some radius. Results in steepening distribution in outer regions vs. pure isothermal soultion.
	R_1/2 = 150-400 kpc (220 kpc for Coma)


	In central regions King profiles work well:




	For these distributions N₀ = 2R_cρ₀.
	De Vaucouleur’s law can also work.
	Problem is observations do not constrain things quite tightly enough.