PHY323:Lecture 6Observational Evidence for Dark Matter

from Galaxy Clusters

•Conclusion on galaxy halos and in-fall•Galaxy clusters•The Virial theorum•Super-clusters and beyond

Experimental Evidence for Dark Matter IV

PHY 323Neil Spooner

n.spooner@sheffield.ac.ukoffice: E23 , extension 2-4422

How to Contact Me

Lecture PDFs http://www.shef.ac.uk/physics/teaching/phy323/index.html

Applying the modified form of Newton’s second law to the gravitational force acting on a star outside of a galaxy of mass M leads us to

which in the low acceleration limit (large r, a ≪ a0) yields

Equating this with the centrifugal acceleration associated with a circular orbit, we arrive at

MOND - Modifying Newton’s Laws

€

F =GMmr2

= maµμ = a/a0

€

a =GMa0r

€

GMa0r

=v 2

r⇒ v = (GMa0)

1/ 4

a0 ~ 1.2 x 10-10 m s-2

note error in last lecture

Worked Example

First consider a particle of mass 4M, what is the acceleration of a test mass at distance r?

Prove that under Newtons law this concept of combining masses in N-body simulations is alright.

€

a =G(4M)r2

=4GMr2

4Mr

test mass

ANS: given in lecture.

M

M2dr

r

test mass

M

M

now repeat for this scenario..and compare the result (use )

€

x ≡ drr

<<1

Notes

How does Barnes Hut Work?Start by bisecting the simulationvolume along all axes.

Subdivide again each cell thatcontains more than 1 particle.

Repeat subdivision until youhave one or zero particles inthe each ‘leaf’ - smallestsubdivision of the volume.

There are about Nlog2N smallest blocks. For each block,calculate the mass and the position of the centre of mass.This is about Nlog2N steps.

For each particle, Descend thetree of ever smaller blocks,starting with the big blocks thatmay contain more than onesubblocks and more than 1 particle. Compute the ratio:distance from ‘red’ particleto centre of mass of block distance from center of massof block to block edge.

If this ratio exceeds some pre-set parameter, the block istreated as a single particle with its total mass at the centreof mass. Tree descendency is an Nlog2N process, so everythingis of order Nlog2N. Repeat for each particle, still overall Nlog2N

How does Barnes Hut Work?

NlogN CPU time For instance for 109 points, 109log2(109) is 3.1010 c.f. (109)2

This would typically be 3.1013 floating point operations. On a current supercomputer about 3 seconds (c.f. 3.2 years!).

3 billion particle N bodysimulation final state. e.g. 50MPc on a side, that means 34 kpc elements

BUT - structure is seenon all scales - worryingfor ‘smooth’ models of DMwww-hpcc.astro.washington.edu/picture

Summary of N-body PredictionsHere are some predictions made by many N-body simulationsof dark matter halos:

(1) Halos form from the ‘bottom up’. The first structure seen is on the smallest distance scales, then small clusters, sometimes called ‘subhalos’ merge to form larger structures.

(2) Any ‘smooth background’ results from tidal stripping of subhalos. Thus N-body simulations typically predict a lower density of smoothly distributed matter (Moore, 0.18-0.3GeV/cc)

(3) The velocity dispersion of the matter in the simulated halos is not that of a Maxwell gas. Typical predictions are that trajectories of particles tend to be radially biased.

(4) Most particles are attached to subhalos, and hence the velocity dispersion here at Earth depends on the subhalos in our vicinity.

Summary of N-body Predictions

early stage example showing remnants of merging of sub-halo structures

Warnings about the Predictions of N-Body Simulations for Halo Dark MatterIt is not obvious that the mechanisms giving rise to largestructures in these simulations are correct!

(1) The simulations themselves suggest that the first signs of structure are at the smallest distance scales, with ‘fractal’, self similar structure formation at ever smaller distance scales. Do N body simulations correctly model formation of structure at small distance scales?

(2) There is still not agreement within the simulation community about the correct way to apply boundary conditions at the edges of the mass distribution.

For example:

(3) The simulations tend to predict very high dark matter densities at the centres of galaxies, which appears to contradict data from, for example, rotation curves.

Clumping - the Nemesis of Halo Dark Matter Experiments

Is there any scale on which the dark matter distribution is smooth?

N-body simulations exhibit matter concentrations at all scales, seemingly limited only by the number of particles in the simulation.

Could actual dark matter halos be like this? If so, it might be bad for dark matter search experiments.

If there is dark matter clumping at the scale of our solar system, for example, Earth might be in a dark matter void, in which case Earth-based direct dark matter searches would be a challenge.

Many in the field feel the ‘fractal structure’ scenario is likely, however the degree of “clumpiness” in a typical halo looks on average to be 10-20%.

NotesWhat do dark matter halos look like in practice and why?

So far, we have considered evidence for dark matter inindividual galaxies. There are two categories of evidence1. Dynamics within our own galaxy. Data available first,but problems with reliability of results.2. Dynamics of other galaxies. More reliable data, gatheredmore recently using new experimental technologies.

Now we will consider evidence for dark matter in the spaces between the galaxies of galaxy clusters. Again, there are two classes of evidence.

1. Dynamics within our own local group of galaxies. Again,data is older, but the measurements were very difficult.2. Data from other galaxy clusters, gathered using a newtechnology - GRAVITATIONAL LENSING

Dark Matter in Clusters of Galaxies

The members of a cluster of galaxies move because of their mutual gravitational attraction

In most cases the velocity of the cluster galaxies is much higher than can be accounted for from the individual galaxy masses

The result is there must be an unseen core of dark matter attracting the galaxies with more gravity and therefore more velocity

Dark Matter in Clusters of Galaxies

Basis of the Virial theorem

where Fk represents the force on the kth particle, which is located at position rk

!

2 T = " Fk# r

k

k=1

N

$

In mechanics the Virial theorem gives a general equation relating the average total kinetic energy of a stable system over time , bound by potential forces, with that of the total potential energy

!

T

!

Vtot

!

V (r) = arn

If the force between any two particles of the system results from a potential energy

!

2 T = n Vtot

proportional to some power n of the inter-particle distance r, the virial theorem adopts a simple form

for gravity of course n = -1

!

2 T = " VGPE

In any system of bodies what determines if it growing or shrinking is the balance between gravitational attraction and the motions of the bodies.

If its to be in a steady state (i.e. the time taken for an object to move across is much less than the lifetime of the assembly) then the speed of objects must be comparable to the escape velocity.

If it is much greater then the system will fly apart, if less then it will collapse.

The criteria is that the kinetic energy be equal in magnitude to the gravitation potential (binding) energy.

Basis of the Virial theorem

!

2 T = " VGPE

So the Virial theorem for a system of bodies in a gravitationally bound system postulates a simple relationship between the average kinetic energy and and average gravitational potential energy of the bodies, e.g. a cluster

Basis of the Virial theorem

!

2 T = " VGPE

depends on the mass of the whole system

related to the motion of the bodies

Assumptions in calculations:(1) system must be equilibrium or the method fails(2) the measurements span a representative sample of bodies(3) all bodies are the same mass (or use a fudge)(4) the velocity distribution is isotropic

hence applying Virial to galaxy clusters is not exact...

e.g. if individual galaxies in the cluster have a velocity v and the total cluster mass is M then:

T = 1/2M<v2>For a spherical system of radius R the gravitational energy is:

VGPE = -αGM2/Rwhere α depends on how the mass is distributed but is typically 1/2-2

Virial Theorem for Gravity - Galaxy Cluster

thus we get the mass of a cluster is: M = <v2>R/αG and can be found by measuring v and R

We can find the “luminous mass” and obtain the mass to light ratio M/L

2T = -VGPEfrom Virial theorem

NotesWrite out the simple derivation of the Virial theorem as applied to a cluster of galaxies...

Worked ExampleCalculate the Mass to Light ratio for the Coma cluster using the Virial theorem

These individual measurements can also be used to determine the mass of the cluster by using the virial theorem :

mean kinetic energy

where vi is the velocity of an individual galaxy of the cluster of mass Mi

!

T =1

2M v

2

!

" VGPE

= 2 T = Mivi

2

i

# = Mivi

2

i

#

mean gravitational potential energy of a bound system in equilibrium

Worked Example - what Fritz Zwicky did

!

2 T = " VGPE

!

T = "1

2VGPE

mean kenetic energy

Zwicky then assumed the galaxies are evenly distributed within a sphere of radius R, and calculates the gravitational potential as

gravitational constant total mass of the cluster

Zwicky’s value for R was 2 x 106 light years (613 kpc)

!

VGPE

= "3GM

5R

simplify previous equation: indicates the average taken over both time and mass

thus the total mass of the cluster

This equation now only depends on the velocities of each galaxy!

Mivi

2

i

" = M v2

!

M =5R v

2

3G

Worked Example - what Fritz Zwicky did

Zwicky’s value for v = 236 km / s (changed from printed notes)

The values he measured were along the line of sight, not the radial velocities, but by assuming spherical symmetry he used the relation

measured line of sight velocity

Worked Example - what Fritz Zwicky did

sum done in lecture....

!

M =5R v

2

3G

ANS: M > 7.8 x 1043 kg= 3.9 x 1013 M0

so the average mass of the ~1000 nebulae (that he observed):

Worked Example - what Fritz Zwicky did= 3.9 x 1013 M0 results for the total mass for the cluster of:

> 3.9 x 1010 M0 Zwicky’s value for the luminosity of an average nebula was:

8.5 x 107 L0

a mass to light ratio M0/L0 η > 459!Compared to ~3 in the local solar area, a huge discrepancy

Zwicky tried to account for this by changing his assumptions, (e.g. assuming the system was not in equilibrium), but at best could only modify his mass measurement by a factor of two or so. Changing other assumptions led to scenarios where galaxy clusters could not be formed at all.

NotesCheck the M/L calculation for Coma, what is the result if the velocity v = 300 km/s?

M/L ~ 351Total Mass = 1.4 x 1015 M0 (Geller et al. 1999)Total mass of galaxies ~ 4 x 1012 M0

Other Results on Coma Cluster

The total mass has been measured to be 1.4 · 1015 M0 and the cluster has a mass to light ratio of 351. Although the new values for the radius and mass are quite a bit higher, the result is the same — this ratio does not account for the majority of the mass; 85% of the mass of the Coma cluster is dark.

plot shown in lecture

There are some problems with using the Virial technique: Application of Virial in Practice

(1) we need to be sure which galaxies are actually in the cluster - i.e. look at red shift (v = H0r) in fact the different zs is what gives us <v> (2) we may accidentally include fast moving galaxies that can skew the value of <v> and hence the mass(3) we must deal with the possibility of two clusters moving near each other, do you combine them or try to treat separately?,

(4) we can only measure v towards us or away so we must multiply our observed <v> by 3 because v2 = v12 + v22 + v32 (this assumes a spherical cluster),(5) we need to assume that it is a cluster and not chance coincidence and that the cluster is neither contracting or flying apart.

Hubble Expansion or Peculiar Velocity?What makes galaxies move relative to us?

1) The expansion of the universe.

2) Gravitational forces due to other matter in theneighbourhood of the galaxy.

us hubbleexpansion

‘peculiarvelocity’

other galaxy

Measuring peculiar velocities could tell us where the mass is

Suppose you use doppler shift to measure a recessionvelocity ,and in fact the source has peculier velocity

We need a redshift-independent measure of the galaxy distance.

us

other galaxy

us

If you only knowthe redshift, itis impossible todisentangle vHfrom vP

Hubble Expansion or Peculiar Velocity?

Luminosity as a Measure of DistanceThe flux of light from a galaxy received on Earth (neglectingextinction of light along the line of sight) is related to thegalaxies intrinsic luminosity L and its distance r by:

(flux is the power into a detector per unit detector area)

However, until the mid 1970s there was no reliable way to determine the luminosity of a galaxy without knowing how far away it is.

Idea: a galaxy has many measurable parameters, for instancethe rotation velocity of the stars. Perhaps a relationship canbe found between one of these numbers and the luminosity.

The Tully Fisher Hydrogen Linewidth - Luminosity Relation

Tully and Fisher (1977 - Astronomy & Astrophysics vol 54, 661 - 673)

Noted a correlation between the WIDTH of the H-I line in a galaxy emmission spectrum and the luminosity of a galaxy.

How do you find this out ? First, plot galaxy luminosity against hydrogen line width for galaxies whosedistances are well known - those in nearby clusters.

LINE WIDTH

LUM

INO

SITY

(abs

olut

e m

agni

tude

) [BR

IGHT

ER->

]

Using the Hydrogen Linewidth - Luminosity Relation

Now pick a galaxy cluster for which the distance is not sowell known, eg, VIRGO CLUSTER

APPA

RENT

MAG

NITU

DE(m

easu

re o

f brig

htne

ss a

t ear

th)

HI Linewidth

The Tully Fisher Linewidth Luminosity Relation

is a constant, theluminosity of a ‘standard’ galaxy

The distance to theVirgo cluster isadjusted until the datafit the Tully Fisherrelation.

Notes

The peculiar velocities of galaxies are due to their response to the local gravitational potential due to other matter in their neighbourhood. Knowing the peculiar velocities allows us to determine the mass density, and it turns out that the mass densities are inconsistent with the mass observed through light on a cluster scale.

Determining Galaxy Peculiar Velocities1. Measure the width of the HI line in the galaxy2. Using the Tully Fisher relation, determine the luminosity.3. Using the brightness of the galaxy, determine its distance.4. Using Hubble’s law, predict the galaxy recession velocity.5. Measure the recession velocity the galaxy actually has.6. The DIFFERENCE between the measured recession velocity and that predicted from the distance and Hubble’s law is the peculiar velocity of the galaxy, resolved along the line of sight to the galaxy.

Results from Real Peculiar Velocity Surveys

The GREAT ATTRACTOR, is a region of high density, withno obvious visible counterpart, whose existence has beeninferred from measurements of peculiar velocities.

http://www.solstation.com/x-objects/great2at.jpg

Typical velocities towards the great attractor, 600 - 1000km/s.Inferred great attractor mass:

plot given in lecture or see web site