1

There is a very important component that cannot be seen at all

How do we know it is there ? Almost all galaxies have a dark halo .... in other spirals, we see its gravitationaleffect on their rotation curves ...

For example

Rotation at large radiiis much faster than canbe understood from thegravitational field of thestars and gas alone.

the dark halo

2

Dark Matter in Galaxies

We believe that galaxies formed through a hierarchyof merging. The merging elements were a mixture ofbaryonic and dark matter.

The dark matter settled into a partially virializedspheroidal halo while the baryons (in disk galaxies)settled into a rotating disk and bulge

What can we learn about the properties of darkhalos ? Do the properties of dark halos predicted bysimulations correspond to what is inferred fromobservational studies ?

3Rotation of the Galaxy: Merrifield (1992)

4

M(R) R is not what we would expect for a gravitating

system of stars : we would expect the Keplerian M(R) R 1/2

Is this evidence for a dark halo ? Not necessarily : it depends on how far the rotation curve extends.

Most spirals have a light distribution that is roughly exponential:

I(R) = Io exp (- R / h)

(for a large galaxy like the Milky Way, the scale length h ≈ 3-4 kpc)

V(R) flat the enclosed mass M(R) R

5

Optical rotation curves, measured from the spectra of ionized gas,typically extend to R ≈ 3 h

Now assume that the surface density distribution

of stars in our disk galaxy follows the optical surface brightness distribution.

Can this surface density distribution, with its gravitationalpotential (R) explain the observed rotation curve V(R) ?

The answer to this question is yes and no

6

Yes, for optical rotation curves extending out to about3 disk scale lengths:

In the next slide, the points are the observations and the curve is the expected rotation curve

Despite the very different shapes of the two rotationcurves, the light distribution can explain the observed

optical rotation curves out to about 3 scale lengths

Optical rotation curves out to ~ 3 scale lengths do not tell us very much

about dark matter in spirals

7

Two very different rotation curves. The points are therotation data and the curves are as expected if

mass follows light. The only scaling is through the adopted M/L ratio

8

No for galaxies with 21 cm HI rotation curves that extend farout, to R >> 3 h.

Begeman 1987

eg maximum disk decomposition for NGC 3198: M/LB = 3.8 for disk

observed

9

For NGC 3198 the HI rotation curve extends out to 11 h.(for some galaxies the rotation data go to more than 20 h)

The expected V(R) from stars and gas falls well below theobserved rotation curve in the outer region of the galaxy.

This is seen for almost all spirals with rotation curves that extend out to many scale lengths.

We conclude that the luminous matter dominates the radialprotential gradient ∂∂R for R < 3h but beyond thisradius the dark halo becomes progressively more important.

10

Typically, out to the radius where the HI data ends, the ratio of dark to luminous mass is 3 to 5:

values up to about 10 are found in a few examples.

This is a lower limit on the ratio of(total dark mass) to luminous mass

11

For the decomposition of NGC 3198, the stellar M/L ratiowas taken to be as large as possible without leading to

a hollow dark halo - this is a maximum disk (minimum halo)decomposition.

Many galaxies have been analysed in this way - the decomposition often works out as for NGC 3198, with

comparable peak circular velocity contributions from disk and dark halo

Some people believe that maximum disk decompositions

are not correct

12

The maximum disk question is important for us here,because inferences about the properties of dark halosfrom rotation curves depend so much on the correctness of the maximum disk interpretation.

eg if the maximum disk decompositions are correct,

then the dark halos haveapproximately uniform density cores

which are much larger than the scale length of the disk

13

halo

14

In contrast, the halos that form in cosmological simulations

have steeply cusped inner halos with r -1 or even steeper

near the center

15

Optical rotation curves favor the maximum disk interpretation. In the inner regions of the disks of larger spirals, the rotation curves are well fit by assuming that mass follows light.

eg Buchhorn (1991) analysed 552 galaxies with optical V(R)and I-band surface photometry, and a wide range of rotationcurve morphology. He was able to match the rotation curves well for 97% of hissample, with realisticM/L ratios (includingthe kinks and bumpsas in the right panel)

16

The implication is that

either

the stellar disk dominates the gravitational field in the inner parts of the disk (R < 3h)

or

the potential gradient of the halo faithfully mimics thepotential gradient of the disk in almost every spiral,

which seems very unlikely

stop

17

How to model the dark halo

The goal is to estimate typical parameters for dark halos (density, scale length, velocity dispersion, shape) to compare with halo properties from simulations

Since about 1985, observers have used model dark halos withconstant density cores to interpret rotation curves

Carignan & Freeman (1985) made the first rotation curvedecomposition into contributions from the stellar disk,gas and dark halo.

We used the real non-singular isothermal sphere (computed from Poisson's equation) to model the dark halo. It has a flat core, a well defined core radius and central density, and has r -2 at large radius, so V(r) ~ constant, as is usually observed.

18

V(R)

R

A simple analytic form is the pseudo-isothermal sphere

o {1 + (r / rc ) 2 } -1

which again has a flat core, a well defined core radius and central density, and has ~ r - 2 at large radius.

Although they have similar asymptotic behaviour, the rotation curves of the real and pseudo isothermal models are different and will give different halo parameters when used inrotation curve decompositions

real

pseudo

19

Using the pseudo isothermal model for the dark halo of large galaxies like the Milky Way, we find that o ~ 0.01 M pc -3

and rc ~ 10 kpc

(for comparison, the stellar density of ourGalaxy near the sun is about 0. 1 M pc -3 )

We will see later that the values of o and rc dependstrongly on the luminosity of the galaxy.

20

Another model with a flat core that is sometimesused in rotation curve decompositions is Burkert's (1995) model (r) = B (1 + r/rB) -1 {1 + (r/rB)2} -1

Its (r) ~ r -3 at large r, like the NFW model (later)

21

These models have flat centralcores : why were they used ?

I think it was because

• rotation curves of spirals do appear to have an inner solid-body component which indicates a core of roughly constant density

• hot stellar systems like globular clusters were successfully modelled by King models, which are modified isothermal spheres with flat central cores

V

R

NGC 6822

22

On the other hand ....

CDM simulations consistently produce halos that are cusped at the center. This has been known since the 1980’s, and has been popularized by Navarro et al 1996 with the NFW density distribution which parameterizes the CDM halos

(r / rs ) - 1 {1 + (r/rs)} - 2

These are cusped at the center, with r - 1

23

We have suffered a long controversy over the last decade aboutwhether the rotation curves imply cusped or cored dark halos.

This continues to be very illuminating

Galaxies of low surface brightness are important in thisdebate.

The normal or high surface brightness spirals have a fairly well defined characteristic surface brightness scale(central surface brightness around 21.5 B mag arcsec -2)

In the LSB galaxies the disk density can be more than 10 x lower than in the normal spirals.

These LSB disks are fairly clearly sub-maximal and the rotation curve is dominated everywhere by the dark halo.

24

.

The observational problem is to determine the shape of the rotation curve near the center of the galaxies - a cored halo gives a solid body rotation curve near the center, while a cusped halo has a steep slope at small r

Observationally it is not easy to tell.

HI rotation curves have limited spatial resolution so the beam smearing can mask the effects of a possible cusp.

2D optical rotation data (Fabry-Perot) have much better resolution - current data favor a cored halo.

Example of NGC 6822, a Local Group LSB galaxy

25

Local Group LSB galaxy in near-IR.

HI map has 20 pc resolution

NGC 6822

26

High spatialresolution HIobservations

of Local GroupLSB galaxyNGC 6822

Weldrake et al 2002

pseudoisothermal

halo

min disk min disk + gas

SPS M/L max diska real

isothermalwould be

better

27

High spatialresolution HIobservations

of Local GroupLSB galaxyNGC 6822

NFW halo

Weldrake et al 2002

Not as good as the isothermal

28

Distribution of inner slope of density ~ r

Sample of about60 LSB galaxies

optical rotation curvesgive inner slope ofdensity distribution

NFW halos have = -1

Flat cores have = 0

de Blok et al 2002

NFW

29

How large and massive are the dark halos of large spirals like the Milky Way ?

Flat rotation curves => M(r) ~ r, like the isothermalsphere : ~ r-2

This cannot go on for ever - the halo mass would be infinite.Halos must have a finite extent and mass, and their densitydistribution must truncate or be steeper than ~ r-3 at very large r.

eg NFW with(r / rs ) - 1 {1 + (r/rs)} - 2

has ~ r-3 at large r.

30

Tracers of dark matter in the Galaxy (rotation curve to ~ 20 kpc, kinematics of metal poor stars, globular clusters and satellites out to ~ 50 kpc) indicate that the halo mass M(r) = r(kpc) x 1010 solar masses.

Again, this is what we expect if ~ r-2 ie the rotation curve stays approximately flat at 220 km/s out to 50 kpc.

How large are dark halos - how far in radius do they extend ?

31

M31 (Andromeda) is now approaching the Galaxy at 118 km s-1. Its distance is about 750 kpc. Assuming their initial separation

was small and the age of the universe is say 18 Gyr, we can estimate a lower limit on the total mass of the

Andromeda + Galaxy system.

The Galaxy’s share of this mass is (13 2) x 1011 solarmasses.

A similar argument using the Leo I dwarf at a distance of about230 kpc gives (12 2) x 1011 solar masses.

Timing argument

32

The relation for the mass of the galactic halo

M(r) = r (kpc) x 1010 solar masses

out to r ~ 50 kpc then indicates that the dark halo extends out beyond r = 120 kpc

if the rotation curve remains flat ie if (r) ~ r -2

and possibly much further than 120 kpc if the density distribution declines more rapidly at large radius

33

This radius is much larger than the extent of any directly measured rotation curves, so this

“timing argument” gives a realistic lower limit onthe total mass of the dark halo.

For our Galaxy, the luminous mass (disk + bulge) is about 6 x 1010 solar masses

The luminosity is about 2 x 1010 solar luminosities

The ratio of total dark mass to stellar mass is then at least 120/6 = 20 and

the total M/L ratio is at least 60

34

Satellites of disk galaxies can also be used to estimatethe total mass and extent of the dark halos of otherbright spirals

Individual galaxies have only a few observable satellites each, but we can make a super-system by combining observations ofmany satellite systems and so get a measure of the mass of atypical dark halo.

eg Prada et al (2003) looked at the kinematics of about 3000satellites around about 1000 galaxies

35

With a careful treatment of interlopers, they find that the velocity dispersion of the super-satellite-system decreases slowly with radius

The halos typically extend out to about 300 kpc but the density distribution at large radius is steeperthan the isothermal: (r) ~ r -3, like most cosmologicalmodels including NFW

The total M/L ratios are typically 100-150, compared with the lower limit from the timing argument of 60 forour Galaxy. (The Prada galaxies are bright systems, comparable to the Galaxy)

36

Prada et al 2003

Velocities |V| of 3000 satellites relative to their

parent galaxy

error bars show the velocity dispersion decreasing withradius out to ~ 300 kpc !

37

Wilkinson & Evans (1999) used motions of 27 globularclusters and satellites : made a dynamical model of thehalo and estimated the total mass to be

Mtotal = 1.9 +3.6 - 1.7 x 1012 M

or

M(50) = 5.4 +0.2- 3.6 x 1011 M out to 50 kpc

consistent again with M(r) = r (kpc) x 1010 M

Back to the mass of the Galaxy ....

38

For comparison, from least action arguments, the likely mass of the local group is 4-8 x 1012 M

( Peebles 1996, Schmoldt & Saha 1998)

Within the uncertainties, most of the mass in the Local Groupcould be in the two large spirals

M31 has a similar rotation amplitude so its total massmay be similar to the total mass of the Galaxy.

Evans & Wilkinson (2000) used satellites and GCs in M31 to derive a lower mass of 1.2+3.6

-1.7 x 1012 M for M31 - similar to the Galaxy, within the uncertainties

So the total mass of MW + M31 ~ 3 x 1012 M

39

Timing M31-MW (eg Zaritsky 1999 ) gives 3-4 x 1012 M for MW and M31 together, ignoring tranverse motions and growth of halo with time and overlap of halos at early times (all of these would increase the mass)

Least action arguments like Schmoldt & Saha use moregalaxies and give similar result

Kochanek (1996) put together timing, velocity dispersionand escape velocity arguments and derived M(200kpc) = 0.5-2 x 1012 M

Zaritsky concludes that the data are consistent with anisothermal halo with Vc ~ 180-220 km/s, extending out toabout 200 kpc, with M > 1012 M - see fig on next slide

40

Zaritsky 1999

all more or less consistent with

M(r) = r (kpc) x 1010 M

out to about 200 kpc

41

More recent estimates give similar MW halo properties

Sakamoto et al 2003: 11 satellites, 137 GCs and 413 HB stars outto 10 kpc from the sun. To keep these objects bound in a flat rotation curve potential, the mass is about (2 +0.5

-1.0) x 1012 M.Out to 50 kpc, M = 5.5 x 1011 M (This is a widely agreed number)

Battaglia et al 2005: 240 halo objects with distances and radialvelocities: satellites, GCs, HB stars, halo giants (spaghetti)to derive the (r) relation. The (r) is constant at 120 km/sout to 30 kpc, then drops to 50 km/s at 120 kpc. Adopting the tracer density distribution (r) ~ r -3.5, the halomass is (1.2 +1.8

-0.5) x 1012 M on a scale of ~ 120 kpc

42Battaglia et al 2005

Halo tracers

(r)

(r)

43

Battaglia et al 2005

Best fit: truncated flat rotation curve halo M ~ 1.2 x 1012 M :constant anisotropy, scale ~ 120 kpc

44

Dehnen (2006) argued that the Battaglia tracers are consistent with a non-truncated isothermal halo if the tracer population truncates at r ~ 160 kpc, again assuming constant anisotropy.Finds a similar mass to Battaglia's.

A non-kinematical estimate: Bellazini (2004) used tidal radii of outer GC in the Galaxy to derive the total mass: finds (1.3 +2.9 -1.0) x 1012 M out to 90 kpcAgrees well with kinematical estimates

45

Conclusion

The total mass of the Milky Way is ~ 1.5 x 1012 M

The MW is one of the few galaxies for which we have anestimate of the total mass, rather than just the mass outto the end of a rotation curve.

The stellar mass is about 6 x 1010 M

So the stellar baryons are only about 4% of the total mass

Compare this with the universal baryon / matter = 15%

46

Dark halo of M31

Little recent work onrotation curves of M31 and M33

Large spiral withprominent bulge.

47

Carignan et al 2006(HI : GBT & Effelsberg)

Out to 35 kpc, M* = 2.3 x 1011 M, Mhalo = 1.1 x 1011 M, sostellar mass dominates. M(35 kpc) = 3.4 x 1011 M: comparewith Milky Way M(50 kpc) = 5 x 1011 M

48

Evans & Wilkinson (2000) used satellites and GCs in M31 to derive a lower limit on the mass of

1.2+3.6 -1.7 x 1012 M

for M31 - similar to the Galaxy, within the uncertainties

49

Dark halo of M33

Almost pure diskgalaxy, normalsurface brightness

50

gas

disk

halo

Corbelli & Salucci 2000

Mhalo > 5 x 10 10 M, Mhalo / Mbar = 5 out to R = 15 kpc

V(R) to 13scale lengths

Halo dominatesfor R > 3 kpc≈ 3 scale lengths

51

Dwarf spheroidal galaxies

Faint satellites of our GalaxyMV down to -8Very low surface brightnessTotal masses ~ 107 solar masses

Radial velocities of individual stars in several of these dSph galaxies show that their M/L ratios can be very high: the fainter ones have M/L ratios > 100

52

Velocity dispersion of the Fornax dSph galaxy - approximatelyconstant with radius (Mateo 1997). Fornax is the brightest ofthe galactic dSph galaxies: its M/LV 10 (expect M/LV = 2from its stellar content alone)

expected for constant M/L

53

M/L ratios for dSph galaxies. Some have M/L > 100. The curve is for a luminous component with M/L = 5 plus a halo with M = 2.5 x 107 M. Mateo 1997

54

The lack of tidal extensions in Dra, Sex, Scl and UMifound by many authors supports the view that thedSph galaxies are immersed in large extended darkhalos with masses ~ 109 M.

Growing evidence that these systems have flat darkmatter cores - eg Goertz et al (2006) argument about survival of the globular clusters near center of the Fornax dSph against dynamical friction

55

Wilkinson et al (2004) : (R) for Dra, UMi

Falling at edge is probably associated with the edge of the stellar distribution.

A steep gradient in the tracerdensity goes with a fall invelocity dispersion.

This does not mean that the edge of the halo has been reached

56

Stellar surface density profile for Draco (Segall et al 2006)Steep gradient where velocity dispersion falls

57

Kravtsov et al (2004) : why don't most of the subhalos have stars ? In their simulations, find that about 10% of halos that are small now (Vc < 30 km/s) were much largerat z > 2, but suffer tidal stripping in the hierarchical mergingprocess. Propose that the MW dSph formed in such objectswith M > 109 M, so were able to build up some stellar mass and survive reionization despite their present shallowpotential wells

Kazantzidis et al (2004) on same problem : match the (r)profiles for tidally stripped subhalos to observed (r) forDraco and Fornax. Find that the Vc values for Fornax andDraco are around 20-35 km/s now. (These are somewhatlarger than suggested by the velocity dispersions alone).

More on evolution of subhalos and dSph galaxies

58

Taffoni et al (2003) : looked at interplay of dynamical friction and tidal mass loss on subhalo evolution.

* Massive subhalos (~ 0.1 Mparent) merge with host.

* Intermediate mass satellites (~ 0.02 Mparent ) lose mass tidally, reduced to 1-10% of initial mass, but mostly survive.

* lower mass subhalos are not much affected

Surviving remnants have masses ~ 0.001 Mparent after a Hubble time

59

Taffoni et al 2003

Evolution of sub-halos:dynamicalfriction andtidal mass

loss

Intermediatemass subhalos

surviveMore massive

subhalosperish

60

Hayashi et al (2003): looked at the tidal evolution of sub-halos as they lose mass tidally and are tidally heated. Match the tidallystripped systems to surface brightness profiles and velocity dispersions of Carina and Draco dSph galaxies.

Argue that peak circular velocities of the dSph halos are much higher than expected from the stellar velocity dispersions. The true tidal radii are much larger than those given by star counts,which are just some feature of the stellar distribution.

eg Carina has a velocity dispersion of 7 km/s, and its halo has a peak Vc ~ 55 km/s : its true tidal radius is about 20 times larger than the apparent stellar tidal radius.

In their picture, only sub-halos with Vc < 35 km/s lack visible counterparts.

61Hayashi 2003

apparent rt

from stellardistribution

true rt

beforestripping

afterstripping

62

Flaring of HI layer in the Galaxy:

The HI layer has an approximately isothermal velocity dispersion(~ 8 km s -1 ). In a spherical dark halo, the outer HI layer willflare vertically more than in a flattened dark halo.

For the Galaxy, Olling & Merrifield (2000) find that the axial ratio q 0.8 from the flaring of the HI layer.

HI flaring is a powerful technique - it provides a measure of∂/∂z to combine with ∂/∂z from rotation data. Can potentiallysort out the maximum disk controversy because ∂/∂z nearthe plane comes mainly from the stellar disk in the inner Galaxy.

HI flaring will be used more in future as good high resolutionHI data comes in for edge-on galaxies.

63Olling & Merrifield 2000

Flaring of the Galactic HI layer givesconstraint on flattening of the dark halo

64

Final slide: how much galactic dark matter is there ?

Current estimate of matter = 0.3, so galactic dark matter isabout 10 to 30% of the total matter content of the Universe

stars + cold gas 0.004 BBNS 0.04so luminous mass in galaxies is about 10% of the expectedbaryon mass - rest currently believed to be hot gas somewhere

If (luminous/dark) mass ratio in galaxies is 10 to 20, thendark halos 0.03 to 0.06, fairly similar to the baryon BBNS

Weak lensing estimate of ~ 0.11 (Hoekstra 2003 ? )