1	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…
2	Radio vs. Optical What are the relative advantages/disadvantages of each?
3	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.
4	Radio: The “Spider” Isovelocity Diagram, interpreting HI maps
5	Radio: The “Spider” Isovelocity Diagram, interpreting HI maps
6	Isovelocity plots
7	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.
8	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.
9	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.
10	Galaxy Masses
11	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.
12	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:
13	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.
14	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
15	Visible Mass ratios vs. Luminosity
16	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?
17	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
18	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.)
19	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
20	Pavo Cluster
21	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:
22	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.
23	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:
24	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.
25	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.
26	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.
27	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
28	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.
29	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.
30	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)
31	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.
32	Rich Cluster Summary
33	Rich Cluster Summary
34	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
35	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…
36	Dynamic Properties of Coma
37	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?
38	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
39	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:
40	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).
41	X-ray Masses for Clusters Fabricant, Lecar, and Gorenstein: Bremsstrahlung spectral emissivity: Gaunt factor can be approximated:
42	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):
43	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.
44	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.
45	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.
46	Forms of Dark Matter??? We’re certain it is present. Some is baryonic. More is non-baryonic.
47	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.
48	Baryonic Dark Matter Black holes constrained by lensing effects (or lack thereof). MACHOs (Alcock et al. 1993):
49	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???
50	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.