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 depends on the mass of the the bodies 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 2 T = " VGPE 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<v2> For a spherical system of radius R the gravitational energy is: VGPE = -αGM2/R where α depends on how the mass is distributed but is typically 1/2-2 from Virial theorem 2T = -VGPE 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 Notes Write out the simple derivation of the Virial theorem as applied to a cluster of galaxies... Worked Example Calculate the Mass to Light ratio for the Coma cluster using the Virial theorem Worked Example - what Fritz Zwicky did These individual measurements can also be used to determine the mass of the cluster by using the virial theorem : mean mean gravitational potential kenetic energy of a bound system in energy 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 2 T = " VGPE ! ! 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...