Components of Galaxies: Dark Matter Dark Matter: Any Form of matter whose existence is inferred solely through its gravitational effects. -B&T, pg 590 Nature of Major Component of Universe Galaxy Formation Fate of the Universe
Evidence for Dark Matter Rotation curves of spiral galaxies - convincing X-ray halos in elliptical galaxies - convincing Clusters of galaxies - convincing Local group infall - cute, but not very convincing Vertical velocities in galactic disk - not very convincing (LM = DM) Disk stability - convincing only for a subset of spiral galaxies (Sc) Dwarf ellipticals - convincing if anisotropies aren t affecting the results Inflationary Model - mostly convincing in light of the Wilkinson Microwave Anisotropy Probe (WMAP)
Evidence for Dark Matter: 1. Rotation Curves of Spiral Galaxies HI Rotation Curves do not turn over at Large Radii (where there is negligible luminous matter)
Spiral Galaxies Rotation Curves (cont ) Rotation curves: v(r) ~ r at small radii v(r) ~ constant at large radii Central Bulge: distribution of stars is homogeneous & spherical. For a spherical distribution of mass of density!(r), Since the distribution is homogeneous, then,
Thus, Substituting in the following equation for M(r), leads to,
Beyond the radius where luminous matter is seen, r LM, the rotation curve should fall off like a Keplerian rotation curve, But what is seen is V(r) = constant. The discrepancy is believe to be due to a significant amount of dark matter in the outer part of spiral galaxies.
How much Dark Matter is there in Spiral Galaxies Relative to Light Matter? In terms of the critical density of the universe of the universe,! crit, and
Density Profile of Dark Matter V(r) = constant. Thus, Can be differentiated with respect to r, Substituting the above differential equation into, the density of the dark matter halo must be,
Many groups have tried to deconvolve the contribution to the rotation curve of the halo & disk components by modeling the disk as a constant M/L exponential disk, where I 0 is the central intensity, and by modeling the halo by using the (physically unmotivated) fitting function, where! halo (0) is the central density of the halo component.
In the example to the right, " ~ 2.1 2.25. For r >> a, which is slightly different from r -2, but is due to the non-negligible disk contribution at larger r. (Van Albada et al. 1985 ApJ, 295, 305)
2. X-ray Halos in Elliptical Galaxies Elliptical Galaxies and Galaxy Clusters have X-ray Halos The equation of hydrostatic equilibrium + perfect gas law # M(r) The equation of hydrostatic equilibrium is, From the ideal gas law, Taking the equation of H.E. & substituting in for P,
Solving for M(r) yields, Note that for stellar systems that do not rotate, Thus, I.e., velocity dispersion & gas temperature are equivalent.
For M87, An Example: M87 T(r) = constant = 10 7 K. Thus, For an X-ray Halo size of 200 kpc, the M(r) traced by the halo is 1.5x10 13 M sun. The B-band luminosity of M87 us L B = 6.2x10 10 L sun, and Thus M / L B ~ 250 M sun / L sun.
3. Clusters of Galaxies For a cluster of galaxies, the virial theorem can be written, This method typically yields values of
An Example Early determination of the mass of the Coma cluster For 21 galaxies, The velocity dispersion is thus, Given a cluster size of 1.7x10 6 ly = 1.6x10 22 m, the total mass is The total # of Brightest Cluster Members (BCMs) is 670, thus Each BCM has a mass of The luminosity range of the BCDs is 0.08-2x10 9 L sun, thus M / L ~ 500 M sun / L sun.
Possible Sources of Error in Determining Cluster Masses Assuming the cluster is virialized when it s not Non-cluster members affecting the determination of the cluster velocity dispersion Also Note: One has to keep in mind scale of matter that is being sampled! E.g., only 10% of galaxies are in clusters, so the $ of clusters may not be representative of the density of DM in the universe.
4. Local Group Timing Argument The MW & the M31 are observed to be approaching each other at V r = -125 km/s. Assumption: MW & M31 formed near each other at not much greater than the distant they are apart now. Thus, during ~ 10 10 years, they must have completed a substantial fraction of one orbit. The orbit is thus less than 15 billion years. The Period, P, of the Local Group is taken to be, where a = semi-major axis radius, and M * is the reduced mass of the MW & M31, i.e.,
Assuming no angular momentum (i.e. to provide the smallest minimum mass), the total energy of the LG is where, D = present MW-M31 separation (480 kpc) KE = kinetic energy per unit mass Solving for reduced mass yields, Which is six times larger than the reduced mass of MW & M31.
5. Galactic Disk Vertical velocity dispersion % z of stars & to the disk determines how high star can climb out of the disk % z depends on vertical gas density of the disk + putative dark matter halo component, Scale height Result: there is as much DM as LM. Caution: measurements of % z require use of halo stars, Which are very distant and thus faint z
6. Stability of Galactic Disks vs. Bar Instabilities Self-gravitating cold disks, i.e., disk with high KE rotation / W (>0.14), are unstable to bar formation, but only 50% of spiral galaxies have bars Solution: there exists a massive spherical halo that controls, in part, the potential well & much of the self-gravity. I.e., the halo increases the KE random, which heats the disk.
Disk Stability(cont ) Note that other things can heat the disk as well, thus stabilizing it against bar formation: large bulges can stabilize disks large random motions (velocity dispersion), which lowers KE rotation / W Thus, DM is needed for Sc galaxies But not S0 & Sa galaxies (see Ostriker & Peebles 1973 ApJ, 186, 467 Sellwood 1981 A&A, 99, 362)
7. Dwarf Elliptical Galaxies The velocity dispersion of dwarf elliptical galaxies have been used to determine M / L. M / L = 10 100 M sun / L sun. We ll talk more about these in a few weeks
8. The Inflationary Universe Many believe that the universe went through a period of exponential expansion. Consider the Newtonian approximation of the expanding universe. Let the universe be an expanding sphere of total mass M, radius R, velocity R, & density!. The total energy of the universe is, Substituting the following equation for mass,
The energy equation becomes, Setting H 0 = R / R and solving for density yields, For E = 0, the universe has critical density, In terms of the critical density, $ can be expressed as,
If the universe underwent a period of exponential expansion, then, and thus, Based on Big Bang Nucleosynthesis, WMAP observations of the angular size of the background fluctuations, & high-z supernovae measurements, the present thinking is that -
The Inflationary Universe theory solved a couple of problems. Among them, 1) Flatness Problem: why $ is so close to 1. 2) Isotropy Problem: why the universe looks the same in every direction. The luminosity density of the universe is Thus, for $ = 1, With a cosmological constant, the value is 30% of this number.
Summary of $ Determinations Based on Various Techniques WMAP
What is the Dark Matter? Non-baryonic &/or baryonic The value of $ is important for making this distinction.
Present Day Abundances of Light Elements In the present universe, 4 He, 3 He, Deuterium (D), & Lithium ( 7 Li) have observed abundances that cannot be accounted for by stellar nucleosynthesis Helium fraction, Y ~ 0.25 0.3, but only 'Y ~ 0.04 can be due to stellar nucleosynthesis. Thus, most He must have a primordial origin.
Big Bang Nucleosynthesis (see Kolb & Turner: The Early Universe) First 100 seconds: conditions suitable for fusing the above mentioned nuclei out of available n & p Important: expansion rate must be less than Rx rate Rx rate = (cross section)(number density)(velocity [T]) Thus determining primordial abundances puts a constraint on what $ baryon can be
Before t universe ~ 1 second (T > 10 10 K) The following reactions occured The ratio of p to n around this time was, 1.5x10 10 K Also note that the D abundance was kept low # Photodissociation rate via blackbody photons > formation rate via p + n.
At T < 10 10 K The above listed Rx could only go into one direction. Result: Neutrinos decouple from Baryonic matter n / p ratio freezes out Blackbody photons energy < D binding energy Thus, D # 3 He, 4 He, 3 H, & 7 Li
Primordial Abundance Models Baryon density
Primordial Abundance Models (cont ) Helium fraction Baryons/Photon Number density Ratio For Sun, ( ~ 100 For SN core, ( ~ few
4 He Abundance Because n p ~ 7 n n And 2p + 2n # 4 He This is similar to the present day abundance of 4 He in the universe. Most 4 He was created in the first 100 seconds of the Universe
Constraining Abundances D & 7 Li measured in atmosphere of Jupiter, in Halo stars, & in gas at cosmological distances
Abundance of Light Elements are Used to Constrain $ baryon The best estimate based on these analyses yield, $ baryon ~ 0.04 From dynamical estimates of dark matter in galaxies, Thus, ~10% of DM is probably baryonic in nature, and ~ 90% is something else
Baryonic Dark Matter Candidates Stellar remnants (white dwarfs, black holes, neutral stars) # Would overproduce metals, inconsistent with MACHO Brown dwarfs or Jupiters # Inconsistent with MACHO Intermediate mass black holes (10 2-6 M sun) # Possible candidate (high mass/low density), but X-ray and MACHO could eventually rule this out Large black holes (>10 6 M sun ) # Would heat the disk too much Note that Microlensing toward LMC, SMC, M31 have tentatively concluded that ~20% of halo DM is in 0.5 M sun objects. Better data will yield a tighter constraint
Cold dark matter The rest of the DM is Cold Dark Matter. We have no clue what it might be.
Hot dark matter Note that for a long time it was speculated that Hot Dark Matter (neutrinos) might make up the bulk of DM. Two conditions that must be true are that 1) Neutrinos have mass, and 2) Structures form in a top-down manner: Large scale structures (superclusters) would form first because neutrinos, which are weakly interacting and initially possess random relativistic velocities, would stream out of high density regions into low-density ones. Thus, small density perturbations that were Jeans unstable would be smoothed out.