Astr 5460 Mon. Sep. 13, 2004

Today: Kinematics and Masses

Unless noted, all figs and equations from Combes et al. or Longair’s Galaxy Formation.

WIRO…

Radio vs. Optical

What are the relative advantages/disadvantages of each?

Radio vs. Optical

What are the relative advantages/disadvantages of each?

Radio has great velocity resolution.

Radio sensitive to cool gas, beyond stars.

Radio can map whole galaxy at once.

Optical work used to be easier before VLA

Optical work has, usually, better spatial resolution.

Radio: The “Spider” Isovelocity Diagram, interpreting HI maps

Isovelocity plots

Galaxy Masses: Analytical

Spirals -- no perfect solution with so much dark mass with somewhat unknown distributions!

Elliptical/spheroidal components treated, at least initially here, using the virial theorem. Follow Longair here, more detail. Also no perfect solution.

Order of magnitude, really. Dark matter dominates.

Galaxy Masses: Ellipticals

Virial Theorem: A relationship between gravitational potential energy and velocities for a dynamically relaxed and bound system.

Ellipticals not necessarily rotating.

T = ½ |U|, where T is the total kinetic energy and U is the potential energy.

So, for a cluster of stars or a cluster of galaxies, measuring T (by measuring velocities) can give U and therefore M.

Galaxy Masses

Virial Theorem: T = ½ |U|

You do need to worry about the conditions of the theorem in an astrophysical context. For instance, comparing crossing times with the relevant timescale. Text examples are the sun’s orbital period and galaxies in Coma.

Galaxy Masses

Galaxy Masses

Astronomical context more complex. Cannot in general get all the 3D velocities. In exgal context, uncertain cosmology can translate into uncertain spatial dimensions. Usually only have position on sky plus radial velocities. Must make assumptions about velocity distribution to apply virial theorem.

Galaxy Masses

Isotropic case: <v²> = 3<v_r²> (why?)

If velocity dispersion independent of masses:

T = 3/2 M <v_r²>, where M is total mass

More complex if the above is not true. Assuming spherical symmetry and an observed surface distribution, get a weighted mean separation R_cl:

Galaxy Masses

For the specific case of a sphere of total mass M, size R, and constant density, the potential energy U = -3/5 (GM²/R). Thus the virial theorem says T = ½ U, so

(3/2) M <v_r²> = (3/5) GM²/R

M_virial = 5σ_r²R/G, where σ_r is the radial velocity dispersion

Works for elliptical galaxies and yields mass to light ratios of 10-20 in solar units.

Galaxy Masses: Spirals

Textbook (Combes et al.) goes through the old approach of flattened spheroids with a particular mass density as a function of radius. This is perhaps useful review for qualifying exam as an exercise in gravitational physics.

Key is to use the rotation curve.

Solutions are appropriately simple for flat curves:

M(total) = V²_rot R/G (spherical dark matter)

M(total) = (2/π) V²_rot R/G (disk dark matter)

Where the disk is cut off at r=R, and V = constant (flat curve) inside r < R.

Mass increases linearly with r, for V = constant

Mass/Light ratio increases faster

Visible Mass ratios vs. Luminosity

Normalized mass distributions based on average rotation curves for different “types.” Sc is type I, Sa and Sb are of all three types.

The famous Tully-Fisher law (basically L ~ V4). Mike Pierce is an expert. Why is H-band better and why is this so important in extragalactic astronomy?

What is its origin?

Significance of Tully-Fischer (Spirals)

M(total) = V²_rot R/G

Recall M/L = constant (at least within class – and is more true across classes for H-band)

L ~ V²_rot R/G

Also, surface brightness, μ, L/R² ~ constant (“Freeman’s Law,” μ(r) = μ₀exp(-r/r₀))

Spiral central surface density = 21.65 B-mag arcsec^-2

Together these give L ~ V⁴, or in terms of Absolute magnitude, M = -10logV + constant

Next chapter, Faber-Jackson equivalent for ellipticals

Longair Chapter 4: Galaxy Clusters

Large Scale Distribution of Clusters

Galaxy Distribution in Clusters

Dark Matter in Clusters

Forms of Dark Matter

(Combes et al. covers this way too briefly in ch. 11, so something of an aside now.)

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…! /sarcasm

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.

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

Dark Matter in Galaxy Clusters

How do we know it is there?

Dynamical estimates of cluster masses

X-ray emission/masses

(Sunyaev-Zeldovich Effect)

(Gravitational lensing)

What is the dark matter???

Baryons vs. non-Baryons

Dark Matter in Galaxy Clusters

Dynamical estimates of cluster masses

Virial Theorem as we have discussed, but…

Very few clusters exist that can be well done!

E.g, which are cluster members?

Must measure many velocities

Case of Coma

Regular rich cluster, looks like isothermal sphere

Crossing time arguments OK

Virial mass issue for Coma first by Zwicky (1937)

Surface distribution, velocities in next figure…

Dynamic Properties of Coma

Dynamic Masses for Coma

Merritt (1987) analysis:

Assuming constant M/L ratio, then mass is 1.79 x 10¹⁵h^-1 solar masses, and M/L is 350 h^-1 in solar units (think about that!).

Typical M/L for ellipticals is 15 in solar units.

Differ by a factor of 20

Why should the dark matter have the same distribution as the light?

Why should the velocities even be isotropic?

X-ray Masses for Clusters

UHURU in 1970s:

Rich clusters very bright in X-rays!

Bremsstrahlung emission of hot intercluster gas

Very hot gas requires large potential to hold

Can use to estimate the cluster mass

X-ray Masses for Clusters

Fabricant, Lecar, and Gorenstein (1980, ApJ 241, 552):

Assume spherical symmetry (as usual!)

Assume hydrostatic equilibrium (yes!):

Perfect gas law:

X-ray Masses for Clusters

Fabricant, Lecar, and Gorenstein:

For ionized gas, cosmic abundances, μ = 0.6

Differentiating the gas law, and inserting into the hydrostatic equation:

So, the mass distribution can be found if the variation of pressure and temperature with radius are known (measured).

X-ray Masses for Clusters

Fabricant, Lecar, and Gorenstein:

Bremsstrahlung spectral emissivity:

Gaunt factor can be approximated:

X-ray Masses for Clusters

Fabricant, Lecar, and Gorenstein:

Bremsstrahlung spectrum is roughly flat up to X-ray energies, above which it cuts off exponentially. Cut-off is related to temperature. E = hν ~ k_BT. The measurement is a projection onto 2D space. Integrating emissivity and converting to intensity (surface brightness at the projected radius a):

X-ray Masses for Clusters

Simplified procedure:

Luminosity Density = total energy per second per unit volume integrating over all frequencies, for a fully ionized hydrogen plasma:

L_vol = 1.42 x 10^-27n_e²T^1/2ergs/s/cm³

Sometimes it can be assumed that you have an isothermal sphere (which is the case when the dark matter and the gas have the same radial dependence). More next week on this.

X-ray Masses for Clusters

So, from the spectrum we can get the temperature as a function of radius, and from the intensity we can get the emissivity and hence the particle density.

Chandra is great for this type of observation:

http://chandra.harvard.edu/photo/2002/0146/

http://chandra.harvard.edu/photo/0087/

http://www1.msfc.nasa.gov/NEWSROOM/news/photos/2002/photos02-037.html

Cooling flows are one possible complication.

X-ray Masses for Clusters

ROSAT pictures: old and busted.

Chandra images: The New Hotness.

Important result is that the dark matter does follow the galaxies.

Typical masses then are 5x10^14-15 solar masses, only 5% visible light, 10-30% hot gas, rest is DM.

Forms of Dark Matter???

We’re certain it is present.

Some is baryonic.

More is non-baryonic.

Baryonic Dark Matter

Protons, Neutrons, electrons (include black holes here too).

“Bricks?”

Brown dwarfs and the like.

BB nucleosynthesis constrains baryons to less than 0.036 h^-2 of closure density.

Baryonic Dark Matter

Black holes constrained by lensing effects (or lack thereof).

MACHOs (Alcock et al. 1993):

Baryonic Dark Matter

MACHOs:

http://www.owlnet.rice.edu/~spac250/coco/spac.html

MACHOs are rather massive, around half a solar mass, and can contribute up to half of the dark halo mass.

White dwarfs???

Non-Baryonic Dark Matter

I’m no expert on this stuff (and in some sense NO ONE is). Particle physicists play in this area more than astronomers.

Leading candidates include

Axions. Cold, low mass, avoid strong CP violation.

Neutrinos. Hot, low mass (getting better constrained), lots of them. SN helps.

WIMPs. Gravitino, photino, etc.

Mirror Matter.


	Today: Kinematics and Masses

	Unless noted, all figs and equations from Combes et al. or Longair’s Galaxy Formation.

	WIRO…


	What are the relative advantages/disadvantages of each?
	Radio has great velocity resolution.
	Radio sensitive to cool gas, beyond stars.
	Radio can map whole galaxy at once.
	Optical work used to be easier before VLA
	Optical work has, usually, better spatial resolution.


	Spirals -- no perfect solution with so much dark mass with somewhat unknown distributions!

	Elliptical/spheroidal components treated, at least initially here, using the virial theorem. Follow Longair here, more detail. Also no perfect solution.

	Order of magnitude, really. Dark matter dominates.


	Virial Theorem: A relationship between gravitational potential energy and velocities for a dynamically relaxed and bound system.
	Ellipticals not necessarily rotating.
	T = ½ \|U\|, where T is the total kinetic energy and U is the potential energy.
	So, for a cluster of stars or a cluster of galaxies, measuring T (by measuring velocities) can give U and therefore M.


	Virial Theorem: T = ½ \|U\|
	You do need to worry about the conditions of the theorem in an astrophysical context. For instance, comparing crossing times with the relevant timescale. Text examples are the sun’s orbital period and galaxies in Coma.


	Astronomical context more complex. Cannot in general get all the 3D velocities. In exgal context, uncertain cosmology can translate into uncertain spatial dimensions. Usually only have position on sky plus radial velocities. Must make assumptions about velocity distribution to apply virial theorem.


	Isotropic case: <v²> = 3<v_r²> (why?)

	If velocity dispersion independent of masses:
		T = 3/2 M <v_r²>, where M is total mass

	More complex if the above is not true. Assuming spherical symmetry and an observed surface distribution, get a weighted mean separation R_cl:


	For the specific case of a sphere of total mass M, size R, and constant density, the potential energy U = -3/5 (GM²/R). Thus the virial theorem says T = ½ U, so
		(3/2) M <v_r²> = (3/5) GM²/R
		M_virial = 5σ_r²R/G, where σ_r is the radial velocity dispersion


	Works for elliptical galaxies and yields mass to light ratios of 10-20 in solar units.


Textbook (Combes et al.) goes through the old approach of flattened spheroids with a particular mass density as a function of radius. This is perhaps useful review for qualifying exam as an exercise in gravitational physics.
Key is to use the rotation curve.
Solutions are appropriately simple for flat curves:
	M(total) = V²_rot R/G (spherical dark matter)
	M(total) = (2/π) V²_rot R/G (disk dark matter)
		Where the disk is cut off at r=R, and V = constant (flat curve) inside r < R.
		Mass increases linearly with r, for V = constant
		Mass/Light ratio increases faster


	The famous Tully-Fisher law (basically L ~ V4). Mike Pierce is an expert. Why is H-band better and why is this so important in extragalactic astronomy?
	What is its origin?


	M(total) = V²_rot R/G
	Recall M/L = constant (at least within class – and is more true across classes for H-band)
	L ~ V²_rot R/G
	Also, surface brightness, μ, L/R² ~ constant (“Freeman’s Law,” μ(r) = μ₀exp(-r/r₀))
	Spiral central surface density = 21.65 B-mag arcsec^-2
	Together these give L ~ V⁴, or in terms of Absolute magnitude, M = -10logV + constant
	Next chapter, Faber-Jackson equivalent for ellipticals


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

	(Combes et al. covers this way too briefly in ch. 11, so something of an aside now.)


	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…! /sarcasm


	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.


	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.


	How do we know it is there?
		Dynamical estimates of cluster masses
		X-ray emission/masses
		(Sunyaev-Zeldovich Effect)
		(Gravitational lensing)

	What is the dark matter???
		Baryons vs. non-Baryons


Dynamical estimates of cluster masses
	Virial Theorem as we have discussed, but…
		Very few clusters exist that can be well done!
		E.g, which are cluster members?
		Must measure many velocities

	Case of Coma
		Regular rich cluster, looks like isothermal sphere
		Crossing time arguments OK
		Virial mass issue for Coma first by Zwicky (1937)
		Surface distribution, velocities in next figure…


	Merritt (1987) analysis:
		Assuming constant M/L ratio, then mass is 1.79 x 10¹⁵h^-1 solar masses, and M/L is 350 h^-1 in solar units (think about that!).
		Typical M/L for ellipticals is 15 in solar units.
		Differ by a factor of 20

	Why should the dark matter have the same distribution as the light?
	Why should the velocities even be isotropic?


	UHURU in 1970s:
		Rich clusters very bright in X-rays!
		Bremsstrahlung emission of hot intercluster gas
		Very hot gas requires large potential to hold
		Can use to estimate the cluster mass


	Fabricant, Lecar, and Gorenstein (1980, ApJ 241, 552):
		Assume spherical symmetry (as usual!)
		Assume hydrostatic equilibrium (yes!):



		Perfect gas law:


	Fabricant, Lecar, and Gorenstein:
		For ionized gas, cosmic abundances, μ = 0.6
		Differentiating the gas law, and inserting into the hydrostatic equation:




		So, the mass distribution can be found if the variation of pressure and temperature with radius are known (measured).