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 spiralgalaxies (Sc)

• Dwarf ellipticals - convincing if anisotropies aren’taffecting the results

• Inflationary Model - mostly convincing in light of theWilkinson 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, rLM,

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 I0 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.

An Example: M87

• For M87,

T(r) = constant = 107 K.

Thus,

• For an X-ray Halo size of 200 kpc, the M(r) traced by the

halo is 1.5x1013 Msun.

• The B-band luminosity of M87 us LB = 6.2x1010 Lsun, and

Thus M / LB ~ 250 Msun / Lsun.

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 1.7x106 ly = 1.6x1022m, 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 0.08-2x109 Lsun,

thus M / L ~ 500 Msun / Lsun.

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 Vr = -125 km/s.

Assumption: MW & M31 formed near each other at not

much greater than the distant they are apart now.

Thus, during ~ 1010 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 DiskVertical 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 KErotation/|W|(>0.14), are unstable to bar formation, but only 50% of spiralgalaxies have bars

• Solution: there exists a massive spherical halo that controls,in part, the potential well & much of the self-gravity. I.e., thehalo increases the KErandom, 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

KErotation/|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 Msun / Lsun.

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 oftotal 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 H0 = 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, 4He, 3He, Deuterium (D), &

Lithium (7Li) 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 aconstraint on what $baryon can be

Before tuniverse ~ 1 second (T > 1010 K)

The following reactions occured

The ratio of p to n around this time was, 1.5x1010 K

Also note that the D abundance was kept low #

Photodissociation rate via blackbody photons > formation

rate via p + n.

At T < 1010 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 # 3He, 4He, 3H, & 7Li

Primordial Abundance Models

Baryon

density

Primordial Abundance Models (cont’)

Baryons/Photon

Number density

Ratio

For Sun, ( ~ 100

For SN core, ( ~ few

Helium

fraction

4He Abundance

• Because np ~ 7 nn

• And 2p + 2n # 4He

• This is similar to the present day abundance of 4He in

the universe.

• Most 4He was created in the first 100 seconds of the

Universe

Constraining Abundances

•D & 7Li measured in

atmosphere of Jupiter, in Halo

stars, & in gas at cosmological

distances

Abundance of Light Elements are Used toConstrain $baryon

• The best estimate based on these analyses yield,

• From dynamical estimates of dark matter in galaxies,

• Thus, ~10% of DM is probably baryonic in nature, and~ 90% is something else

$baryon ~ 0.04

Baryonic Dark Matter Candidates

• Stellar remnants (white dwarfs, black holes, neutralstars)

# Would overproduce metals, inconsistent with MACHO

• Brown dwarfs or Jupiters

# Inconsistent with MACHO

• Intermediate mass black holes (102-6 Msun)

# Possible candidate (high mass/low density), but X-rayand MACHO could eventually rule this out

• Large black holes (>106 Msun)

# Would heat the disk too much

• Note that Microlensing toward LMC, SMC, M31 havetentatively concluded that ~20% of halo DM is in 0.5 Msun

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.