PHY323:Lecture 6 Observational Evidence for Dark Matter from Galaxy Clusters
PHY323:Lecture 6 Observational 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 How to Contact Me
PHY 323 Neil Spooner n.spooner@sheffield.ac.uk office: E23 , extension 2-4422
Lecture PDFs http://www.shef.ac.uk/physics/teaching/phy323/index.html MOND - Modifying Newton’s Laws 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 GMm μ = a/a0 F = = maµ r2 which in the low acceleration limit (large r, a ≪ a0) yields GMa a = 0 € r Equating this with the centrifugal acceleration associated with a circular orbit, we arrive at a0 ~ 1.2 x 10-10 m s-2
€ GMa v 2 note error in 0 = ⇒ v = (GMa )1/ 4 r r 0 last lecture
€ Worked Example Prove that under Newtons law this concept of combining masses in N-body simulations is alright. First consider a particle of mass 4M, what is the acceleration of a test mass at distance r? 4M test mass G(4M) 4GM a = = r r2 r2
now repeat for this scenario..and dr 2dr compare the result (use x ≡ << 1 ) € r M M test mass M r €
ANS: given in lecture. M Notes How does Barnes Hut Work?
Start by bisecting the simulation volume along all axes. Subdivide again each cell that contains more than 1 particle. Repeat subdivision until you have one or zero particles in the each ‘leaf’ - smallest subdivision 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. How does Barnes Hut Work? For each particle, Descend the tree of ever smaller blocks, starting with the big blocks that may contain more than one subblocks and more than 1 particle. Compute the ratio: distance from ‘red’ particle to centre of mass of block distance from center of mass of block to block edge. If this ratio exceeds some pre-set parameter, the block is treated as a single particle with its total mass at the centre of mass. Tree descendency is an Nlog2N process, so everything is of order Nlog2N. Repeat for each particle, still overall Nlog2N NlogN CPU time 9 9 9 10 9 2 For instance for 10 points, 10 log2(10 ) is 3.10 c.f. (10 ) 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 body simulation final state. e.g. 50MPc on a side, that means 34 kpc elements
BUT - structure is seen on all scales - worrying for ‘smooth’ models of DM www-hpcc.astro.washington.edu/ picture Summary of N-body Predictions Here are some predictions made by many N-body simulations of 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 Matter It is not obvious that the mechanisms giving rise to large structures in these simulations are correct! For example: (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.
(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%. Notes What do dark matter halos look like in practice and why? Dark Matter in Clusters of Galaxies So far, we have considered evidence for dark matter in individual galaxies. There are two categories of evidence 1. Dynamics within our own galaxy. Data available first, but problems with reliability of results. 2. Dynamics of other galaxies. More reliable data, gathered more 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 new technology - 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 Basis of the Virial theorem In mechanics the Virial theorem gives a general equation relating the average total kinetic energy of a stable system over time T , bound by potential forces, with that of the total
potential energy Vtot N 2 T = "$ Fk # rk k=1 ! where Fk represents the force on the kth particle, which is located at position rk If the force between any two particles of the system results from a ! n potential energy V(r) = ar proportional to some power n of the inter-particle distance r, the virial theorem adopts a simple form 2 T = n V 2 T V ! tot = " GPE for gravity of course n = -1
! ! Basis of the Virial theorem 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.
2 T = " VGPE
! Basis of the Virial theorem 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
related to the motion of 2 T V depends on the mass of the the bodies = " GPE whole system 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... Virial Theorem for Gravity - Galaxy Cluster e.g. if individual galaxies in the cluster have a velocity v and the total cluster mass is M then: T = 1/2M
These individual measurements can also be used to determine the mass of the cluster by using the virial theorem : mean mean gravitational potential kenetic 2 T V energy of a bound system in energy = " GPE equilibrium
mean 1 1 2 kinetic T = " VGPE T = M v energy 2 2
! 2 2 " VGPE = 2 T = # Mivi = # Mi vi i i ! where vi is the velocity of an !individual galaxy of the cluster of mass Mi ! Worked Example - what Fritz Zwicky did 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 3GM V = " GPE 5R Zwicky’s value for R was 2 x 106 light years (613 kpc) indicates simplify previous equation: M v 2 = M v 2 the average " i i taken over ! i both time and mass 5R v 2 thus the total mass of the cluster M = ! 3G This equation now only depends on the velocities of each galaxy
! Worked Example - what Fritz Zwicky did 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
Zwicky’s value for v = 236 km / s (changed from printed notes)
sum done in lecture....
5R v 2 M = 3G
ANS: M > 7.8 x 1043 kg
13 ! = 3.9 x 10 M0 Worked Example - what Fritz Zwicky did
results for the total mass for the cluster of: = 3.9 x 1013 M0 so the average mass of the ~1000 nebulae (that he observed):
> 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. Notes Check the M/L calculation for Coma, what is the result if the velocity v = 300 km/s? Other Results on Coma Cluster
plot shown in lecture
M/L ~ 351 15 Total Mass = 1.4 x 10 M0 (Geller et al. 1999) 12 Total mass of galaxies ~ 4 x 10 M0
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. Application of Virial in Practice There are some problems with using the Virial technique: (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
2) Gravitational forces due to other matter in the neighbourhood of the galaxy.
other galaxy
us hubble ‘peculiar expansion velocity’ Measuring peculiar velocities could tell us where the mass is Hubble Expansion or Peculiar Velocity? Suppose you use doppler shift to measure a recession velocity ,and in fact the source has peculier velocity
other galaxy
us If you only know the redshift, it us is impossible to
disentangle vH
from vP We need a redshift-independent measure of the galaxy distance. Luminosity as a Measure of Distance The flux of light from a galaxy received on Earth (neglecting extinction of light along the line of sight) is related to the galaxies 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 instance the rotation velocity of the stars. Perhaps a relationship can be 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 whose distances are well known - those in nearby clusters. (absolute magnitude) [BRIGHTER->] LUMINOSITY LINE WIDTH Using the Hydrogen Linewidth - Luminosity Relation Now pick a galaxy cluster for which the distance is not so well known, eg, VIRGO CLUSTER MAGNITUDE ARENT APP (measure of brightness at earth)
HI Linewidth The Tully Fisher Linewidth Luminosity Relation
is a constant, the luminosity of a ‘standard’ galaxy
The distance to the Virgo cluster is adjusted until the data fit the Tully Fisher relation. Notes Determining Galaxy Peculiar Velocities 1. Measure the width of the HI line in the galaxy 2. 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.
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. Results from Real Peculiar Velocity Surveys http://www.solstation.com/ x-objects/great2at.jpg
plot given in lecture or see web site
The GREAT ATTRACTOR, is a region of high density, with no obvious visible counterpart, whose existence has been inferred from measurements of peculiar velocities. Typical velocities towards the great attractor, 600 - 1000km/s. Inferred great attractor mass: