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Dark Matter Observation Nearby and in Galaxies

Dark Matter Observation Nearby and in Galaxies

Dark and the Topic 2 Observation Nearby and in Why don’t Galaxies fly apart and how does this tell us that Dark Matter exists? Contents of Topic 2 Following our introduction to the composition of the Universe and with some basic tools of in hand from Topic 1, we begin a survey of the observational evidence for Dark Matter, starting with the local neighbourhood and in galaxies. ‣ Dark Matter observation on different scales ‣ Quantifying dark matter using the Mass to Light ratio ‣ Dark Matter near the Solar System and the Oort technique ‣ Measuring Rotation Curves of Galaxies ‣ The observation of flat rotation curves and implications ‣ Derivation of the mass distribution in galaxies ‣ The isothermal sphere model of galaxies halos ‣ Modified Newtonian Dynamics (MOND) and dark matter ‣ structures, N-body simulations and the Dark Matter Scale and Observation ‣ We have seen from Topic 1 that our current view of the Universe is that it mostly contains Dark Matter and but what is the real observational evidence for the Dark Matter part and how widely is it spread? ‣ It turns out that Dark Matter is found on a vast range of scales in the Universe - from the Solar System, Milky Way and Galaxies (covered here in Topic 2) to Galaxy Clusters, Super Clusters and beyond (covered in Topic 3). ‣ By definition Dark Matter is dark, so studying its presence usually comes from observing its gravitational effects. ‣ There are basically four techniques that can be used: (1) The Oort Technique (2) From Galaxy Rotation Curves (3) Use of the Virial Theorem (4) Gravitational Lensing ‣ This Topic focusses on techniques (1) & (2). Dark Matter & the Mass to Light Ratio ‣ In general we look to measure the mass around using one or other of these techniques, then compare the value with the mass inferred from the observed luminous material. ‣ A good way to quantify this, as used by astronomers, is simply to determine the Mass to Light () ratio:

M M⊙ = η L L⊙

‣ Here M⊙ and L⊙ refer to the mass and luminosity of the Sun. Thus the Mass to Light ratio M/L for any system being studied (e.g. galaxy, cluster etc) is simply a number η in units of M⊙/L⊙. By definition η = 1.0 for the Sun.

‣ The actual value of M⊙/L⊙ (for the Sun) is 5133 kg/W. Dark Matter & the Mass to Light Ratio ‣ Measurement of the Mass to Light ratio M/L can tell is if there might be hidden dark matter around, for instance: (1) M/L < 1 likely implies we might have a system of luminous main sequence only. (2) M/L > 1 likely implies a system with hidden matter. ‣ How you measure M/L depends where you are measuring. ‣ e.g. near our Solar System you might count up the luminosity of nearby stars and objects and compare with the mass inferred from their motions (the Oort technique). ‣ e.g. in a you might measure the total luminosity of the galaxies and use the Virial Theorem to estimate the total mass of the cluster (covered in Topic 3). Dark Matter & the Mass to Light Ratio ‣ It is important to distinguish stars or objects which are intrinsically so dim that we can not see them from those which we can not see just because they are far away. ‣ i.e. there will always be stars that are intrinsically faint and so “hidden” just because of the luminosity rule: observed intensity of object distance to object

2 ls = Ls/4πd

intrinsic luminosity of object ‣ Examples of faint astronomical objects are: Brown Dwarfs, White Dwarfs, Black Holes and Neutron Stars. Dark Matter Nearby ‣ As we found in Topic 1 with the mass of the Earth problem, the idea of “hidden” matter revealed by its gravitational effect is not new. Examples in and around the Solar System are:

(1) Observation of the anomalous orbit of Uranus due to “hidden” matter led to discovery of Neptune by Galle (1846).

(2) Observation of the anomalous motion of the Sirius (a near neighbour of the Sun at 8.6 ly away) led to discovery of a companion star Sirius B by Clark (1862). The Oort Technique ‣ One of the first techniques used to examine hidden mass in the Galaxy close to us was by (1932). The idea is to use the motion of nearby stars at right angles to the plane of the Galaxy to derive the total mass near the Sun.

z direction motion of stars caused by the gravitational potential of the galactic disk is an up and down motion ‣ This “vertical” motion of stars (let’s say in the z-direction) is expected to be like that of a Harmonic Oscillator with form F (= ma) = -kz where k is the restoring force constant due here to the gravitational potential of the galactic disk. Thus: d2z 2 = -4πρ0Gz dt density of acceleration material in disk The Oort Technique ‣ Integration gives the velocity and the Vertical Frequency λ: dz = vz = vz,0 cos λt dt 1/2 λ = (4πρ0G) ‣ So by measuring the Vertical Velocity of stars it is possible to determine ρ0. However, this is very hard in practice and you have to assume that vz depends only on ρ0. ‣ Better is to assume that the number of oscillations made is much greater than the age of the galaxy. Then examine the distribution of stars away from the . ‣ Oort did such measurements of observed vertical distances of stars and derived ρ0 ~ 0.2 M⊙/pc-3. ‣ This is about twice the visible matter near the Sun, hence the Mass to Light Ratio is η(near the Sun) = ~2. Mass to Light Ratio on Bigger Scales ‣ The Oort measurement is hard and controversial and specific to our neighbourhood, near the Sun. The technique is also only really looking at mass in the disc of our galaxy. ‣ However, it was a first hint from near the Galactic Centre that there is more matter than can be accounted for by the stars alone. Obviously this still could be gas and dust (at the time cold hydrogen could not be seen). ‣ But we now know things get far more obvious as we move to larger scales and use new techniques, in summary: η(near Sun) = ~2 η(our Galaxy, inner part) = ~6 η(our Galaxy, outer part) = ~40 η(galaxy pairs) = ~80 η(galaxy ) = ~160 η(galaxy clusters) = ~500 η(Abell clusters) = ~500 Mass to Light Ratio on Bigger Scales ‣ Here is some data showing the Mass to Light Ratio for various objects going to greater and greater scale, out to galaxy clusters. ‣ Note the lines marking the equivalent value of Ω and how at the largest scale the value appears to saturate at ~ Ω = 0.3. ‣ This is suggestive that there is not enough matter in the Universe to give us a flat geometry with Ω = 1.0. The missing factor will eventually turn out to be Dark Energy. Dark Matter in Whole Galaxies ‣ A big advance came in the 1970’s with the advent of Radio . In particular, observations of the velocity of HI Hydrogen Clouds orbiting galaxies, started by . ‣ This is her paper and an early result.

‣ This allowed production of Galaxy Rotation Curves - plots of circular radial velocities vc in the plane of the disc of a galaxy vs. distance r from the centre at distances beyond the visible. Basics of Rotation Curves ‣ The concept and importance of measurements is illustrated below. Here we see a satellite HI cloud of mass m orbiting a galaxy of mass M at velocity vc with distance r from the centre.

vc M m

r mass m in orbit with circular velocity vc

‣ Newton’s laws of should tell us the relationship between the masses and the velocity. But first let’s consider how the measurement itself is possible. Rotation Curve Measurements ‣ The measurement is possible because HI (atomic hydrogen) emits 21cm Wavelength Radio radiation (or 1.42 GHz). ‣ This radiation arrises because the hydrogen 1s ground energy state is slightly split by the interaction between the electron spin and nuclear spin. ‣ A 21cm photon is emitted when there is a transition from the state with nuclear and electron spin aligned parallel (the so- called F = 1 Hyperfine State) to the state where the nuclear and electron spins are antiparallel (F = 0 Hyperfine State).

‣ This transition is sometimes called the Spin-flip Transition. Rotation Curve Measurements The 21cm Radio Line from HI is ‣ Messier 83: 21-cm and Optical very useful because there is lots of HI rotating it at the edges of galaxies. towards us ‣ The 21cm Radio Line also penetrates well through dust. ‣ The figure here shows an example measurement from M83. Note how HI rotating away the radio emission extends to greater radius than the optical (lower image on the same scale). ‣ Use of the Doppler Shift allows the velocities of the HI to be measured. (the blue in the top image signifies gas moving towards us, the red is moving away). Rotation Curve Measurements ‣ The cartoon here illustrates how the Doppler Shift works to allow rotation of a Galaxy to be observed. 21cm Radio from the Milky Way ‣ Below shows an image of 21cm Radio Line emission from our own Galaxy. Here we are looking through the disc. Caution Measuring Rotation Curves ‣ Although the 21cm Line can be used in principle to measure the circular velocity of orbiting clouds via the Doppler Effect in practice caution is needed for several reasons: (1) There can be random Peculiar Motions of the clouds and the galaxy observed could also be moving away/towards us as a whole. These motions must be considered. (2) The easiest case is if the galaxy is seen edge on (as in the cartoon above). If it’s face on it can’t be done. If it’s at an angle then it can because the ratio of the elliptical axes gives us the angle and hence the fraction of rotational velocity we would expect to see in our direction. (3) Even for an edge on galaxy it is complicated because we see light from different radii on the same line of sight. This can usually be accounted for. Limitations of Radio Telescopes ‣ Another factor of concern is the practical limitations of radio telescopes, specifically the Resolving Power θmin - the minimum angular distance between two objects in the sky where they can still be separately resolved. ‣ If L is the longest dimension of the objective lens, mirror, or for a radio telescope the dish, and λ the wavelength (21 cm in our case), then: λ θmin ≃ L

‣ To probe a galaxy rotation, we need many measurements of the rotation velocity of the gas at different radii. Say we need 10 measurements out to 30 kpc. This means that we need to resolve measurements separated by an angle of 300 microradians. So large radio dishes are needed. What Rotation might we Expect? ‣ So how might we expect the rotation to behave, i.e. the function for vc vs. r? ‣ The simplest form would be that of a Solid Disc Rotation:

Rotation of a solid disc or wheel.

‣ Here we have vc ∝ r or vc = const × r. ‣ Obviously although galaxies look disc-shaped they are not solid so we don’t expect this behaviour. What Rotation might we Expect? ‣ More realistic is to use Newton’s Laws applied to the orbit of a mass m around a central mass M as illustrated in the galaxy picture above. Use equations for the force and solve for v: 2 GMm mvc F = F = hence vc r-1/2 r2 r ∝ ‣ Notice that the orbital speed falls as you go to greater radii. This is a Keplerian Rotation Curve - the type of rotation that follows Kepler's 3rd Law e.g. as obeyed by the planets

moving around the Sun. Keplerian, planet-like or differential rotation. Keplerian Rotation Curve ‣ This type of Rotational Curve, applicable to the distribution of velocities of planets in the Solar System, led to the discovery of the Newton Gravitational Law. Here is that distribution for the planets, shown as an example:

vc ∝ r-1/2

‣ Note: an Astronomical Unit (AU) is the Earth-Sun distance = 1.5 x 1011 m So What is actually Observed? ‣ So based on the presence of visible matter alone and assuming the Keplerian form we might expect the rotation curve to have form as shown in blue in the following plot: (1) At moderate to large radii the velocity decreases as r-1/2. (2) The velocity is assumed zero at the centre but must then rise to meet the Keplerian form at some intermediate r. (3) Between these extremes we would expect a turn-over where the velocity peaks and begins to drop. ‣ However, this is not what it observed. Instead the HI measurements show a so-called Flat Rotation Curve. ‣ The plot shows real HI radio (and other) data for the example NGC3198 in the Ursa Major constellation at distance 9.1Mpc from us. Rotation Curve for NGC3198

as measured, looks like vc = const

additional -1/2 Keplerian vc ∝r expectation velocity needed to match the data

allow for central bulge where velocity must drop to zero

‣ The black line indicates the additional velocity that must be added to the expected form to get a match with the data. Question ‣ Based on the above which of these representations of a galaxy is the more realistic?

‣ A: left image ‣ B: right image The Rotation Curve Bombshell! ‣ This result, and many like it, is completely astonishing! -1/2 ‣ We expected vc∝ r , but we observe vc = const. ‣ It shows that galaxies are rotating far faster than expected. So fast that they should fly apart! ‣ Here is another example, NGC2403. The blue line shows the expected Keplerian form, red shows the extra needed to get a fit to the data. ‣ The simplest way to resolve this discrepancy is to postulate that there is a large distribution of unseen Dark Matter material in the form of a Spherical Halo. ‣ It is the gravitational effect of this that holds the luminous matter together despite the rotation speed. Dark Matter in Galaxies ‣ So, as illustrated below, we image galaxies are composed of a luminous central part, the stars mainly in a disc, plus a Spherical Halo distribution of non-luminous Dark Matter.

‣ Astonishingly, for this to work must mean that: galaxies are ~90% Dark Matter! ‣ In terms of Mass to Light Ratio we have: η = ~40 The Isothermal Sphere ‣ But what do we mean by a spherical distribution in the form of a Dark Matter Halo? It’s just the simplest density distribution we can assume from which we can calculate rotation curves that fit the data reasonable well. ‣ More accurately we describe the distribution of dark matter as having the characteristics of an Isothermal Sphere. ‣ This is the type of distribution that would be followed by a stable cloud of gas of a given temperature. ‣ That is we have a spherically symmetric density distribution, centred on the centre of the galaxy, with a single uniform velocity for the dark matter particles, with a Maxwellian Distribution. The Isothermal Sphere ‣ In summary we are making three assumptions about the dark matter distribution here, that it is: (1) Spherical - the halo density is spherically symmetric about the centre of the galaxy. (2) Smoothly distributed - the matter is smoothly distributed on all scales in the halo. (3) Isothermal - as described above. ‣ Inside the Dark Matter Sphere we also dark matter halo have a core, the visible ‘bulge’ and disc at M(r) the centre containing stars, gas & dust. ρ(r) r Effectively, material outside the core can be treated as in orbit around the core. visible part ‣ We can define the total mass and total density within radius r as M(r) and ρ(r) respectively. What M(r) and ρ(r) fits the Data? ‣ Using the assumption that dark matter dominates and has a smooth Isothermal Sphere Distribution combined with the observation of a flat rotation curve in turns out that:

M(r) ∝ r ρ(r) ∝ r-2 total mass total density within radius r within radius r

‣ We can prove this as follows: Derivation of M(r) for Flat Rotation ‣ Start by equating the centripetal force needed to maintain a small satellite mass in circular orbit with the gravitational force due to mass within that orbit:

‣ The small mass m cancels, and we should write the M as a function of r as in this model the satellite gas is within a mass distribution that continues outside the orbit, so:

‣ The data says we have a flat rotation curve, i.e. the rotation velocity is independent of radius, so vc = constant. ‣ Thus we see M(r), the whole mass enclosed in orbit of radius r, rises linearly with radius. ‣ However, there must be a cut off at the edge of the halo. Derivation of ρ(r) for Flat Rotation ‣ Now we can derive the equation for the density as follows:

The Dark Matter density at dark matter halo radius r is denoted by shell thickness dr M(r) A shell thickness dr at ρ(r) r radius r has volume It contains a mass of matter

Therefore between radius r and radius r+dr, the total mass enclosed changes by dM(r), where:

or Derivation of ρ(r) for Flat Rotation

Now, we know that:

Differentiate this with respect to r gives:

Equate the two expressions for dM/dr:

Rearrange to get an expression for the density at radius r:

‣ So IF the Dark Matter dominates the density, and IF the halo is spherically symmetric and smooth, a 1/r2 density distribution of matter fits the “Flat Rotation Curve” data. Alternative Density Distributions ‣ So the simplest density distribution implied by a flat rotation curve is: some constant ‣ The velocity is constant so we can write: ‣ But note there is clearly a problem here because this equation implies infinite density at the centre. ρ r2 (r) 0 0 ‣ A better form that avoids this issue would be: ρ = 2 2 r + r0 where ρ0 and r0 are constants ( ) ‣ Note how for small radii this form reduces to give a constant density of . € ‣ Note also that more complex equations for the density distribution are sometimes used. For instance to account for the possibility that in reality the halo is not spherical. Implications of the Halo Model ‣ The simple Halo Model appears to work well at explaining the observations, particularly the Flat Rotation Curve. ‣ But it implies that the distribution of Dark Matter and visible matter in galaxies are very different, notably as follows: Visible Part of Galaxies ‣ Gives off lots of light, very luminous ‣ Distributed mainly in a compact flat disc ‣ Rotating at high velocity ~200 kms-1 Dark Part of Galaxies ‣ Gives off no light, revealed by gravity ‣ Distributed in a large isothermal sphere ‣ Not rotating, no net rotational velocity ‣ What can this tell us already about the likely nature of Dark Matter and can we try to explain why it behaves so differently from the visible matter? Behaviour of Galactic Visible Matter ‣ We know the visible matter in early prototypical galaxies is largely hot ionised Baryons. Such particles can undergo Inelastic Collisions whereby some of their kinetic energy is converted to heat or photons that escape. The matter thus slows down and collapses under the influence of gravity.

INELASTIC COLLISIONS

‣ Any early perturbation likely will cause rotation. This will be accentuated during the collapse as Angular Momentum must be conserved. The result is formation of a disc.

CONSERVATION OF ANGULAR MOMENTUM Behaviour of Galactic Dark Matter ‣ If we imagine the Dark Matter as particles then we know that any inelastic couplings of it to ordinary matter must be feeble or we would have detected it in the lab. Thus dark matter- dark matter Inelastic Self-couplings are also likely feeble. ‣ The strongest self-couplings between dark halo particles are then gravitational. But gravitational interactions are essentially elastic. Elastic Collisions conserve energy, if the matter is initially roughly uniformly distributed, it stays that way and maintains the same density. ‣ So dark matter tends NOT to collapse to a disk nor rotate. Dark Matter in the Milky Way ‣ It is harder to map the Dark Matter in our own Galaxy as we have to look through the disc. But here is the conclusion:

10-20 LUMINOUS MATTER -21 10 actual density (Galactic disk) 10-22 mean spherical density -3 10-23 ~0.3 GeV cm 10-24 DARK MATTER

mean 10-25 mean spherical density density 10-26 -3 (g cm ) 10-27 10-28 LMC 10-29 Sun ~30 kpc

0 1.1023 2.1023 distance from galactic centre (cm) Dark Matter in the Milky Way ‣ There are a few points to note from the previous plot: (1) Note the Sun’s position, 8 kpc from the Galactic Centre. The Large Magellanic Cloud (LMC) galaxy is at 48.5 kpc. (2) Note that at our position at the Sun we are moving through the Dark Matter Halo at ~230 kms-1. (3) There are two lines marked for the luminous matter: the top one shows the actual density in the disc, the lower one shows what this density looks like if spread into a sphere, to allow comparison with the Dark Matter Distribution. (4) The Dark Matter appears as a smooth distribution much greater than the visible at large distances but comparable to the mean spherical density of the visible near the Sun. (5) The Dark Matter Density near us is ~ 5 x 10-25 gcm-3 or 0.3 GeVcm-3, like about 1/3rd of a proton/cm3. The Extent of Galactic Dark Matter ‣ To what radius does the mass in a Galaxy extend? ‣ At least as large as the largest radius at which gas rotational velocities can probe the matter distribution. It’s safe to say that the halo has to be at least as large as the gas disk, which may well extend to several times larger than the visible disk. Recent measurements have shown Dark Matter to very large distances from the centres of Galaxies.

Rotation curves for the Milky Way (left) and other example galaxies (right) MOND - an Alternative to Dark Matter ‣ It is natural to contemplate whether, rather than being indications of dark matter’s existence, the observations might instead be revealing departures from the laws of gravity. ‣ Since proposed by Milgrom (1983), efforts have been made to explain the observed galactic rotation curves without dark matter within the context of a phenomenological model known as Modified Newtonian Dynamics, or MOND. MOdified Newtonian Dynamics ‣ The idea of MOND is that the acceleration of bodies orbiting the galaxy is very small, much smaller than accelerations measurable in Earth-based experiments. At these small accelerations perhaps Newton’s 2nd Law is modified. ‣ Specifically F = ma, becomes F = ma × μ(a), where μ = 1 except in the case of very small accelerations, for which μ behaves as μ = a/a0. MOND Predicts a Flat Rotation Curve ‣ Applying the modified form of Newton’s law to the gravitational force acting on a mass m outside of a galaxy of mass M leads us to: GMm F = = maµ r2 where μ = a/a0 at very small accelerations

‣ So in the low acceleration limit (large r, a ≪ a0) this yields: GMa a = 0 € r ‣ Equating this with the Centrifugal Acceleration associated with a circular orbit, we arrive at:

2 this is a € GMa0 v 1/ 4 = ⇒ v = (GMa0 ) constant r r velocity!

€ MOND in Action ‣ So MOND predicts that Galactic Rotation Curves should become flat (independent of r) for sufficiently large orbits.

‣ The constant a0 can be determined from data on rotation curves typically yielding a0 ~ 1.2 x 10-10 m s-2, in good agreement with galaxy-scale observations.

PHOTO CREDIT: ‣ An example is NGC6946: STACY MCGAUGH Spiral galaxy NGC 6946 with rotation curve velocity V vs. R (blue circles); curve computed with Newtonian gravity (green) and prediction of MOND (gold). MOND provides a good description of the data with no free parameters.

‣ For the value of a0 here, the effects of MOND are too small to be seen in laboratory or Solar System scale experiments. MOND Success and Failure ‣ However, MOND is not as successful at explaining the other evidence for Dark Matter: (1) MOND fails to explain Rotation Curves at very large r where the velocity is seen eventually to drop off. (2) It fails to explain all details of Rotation Curves (next box). (3) It fails to describe observed features of Galaxy Clusters such as Galaxy Mergers (see next box). (4) The Cosmic Microwave Background Anisotropies and large scale structure, are not addressed by MOND. (5) It can not explain Gravitational Lensing of light observed from distant galaxies thought to arise from Dark Matter. (6) MOND is a Phenomenological Model of Newtons laws only, with no firm theoretical basis. The theory would require the re-writing of Big Bang Cosmology. Example Evidence against MOND ‣ Recent evidence against MOND comes from observation of merging galaxies like the Bullet Cluster. Simulations show that the nuclei of merging galaxies can only spiral together in the time available if they can surrender their energy and Angular Momentum to Dark Halos. If we banish halos by using MOND, the Galactic Nuclei take too long to merge.

Purple is the mass distribution from gravitational lensing - the total matter density vs. position. Red is x ray emission from ionised hydrogen in the region where the clusters collided. ‣ The mass has followed the feebly interacting galaxies, even though the baryonic matter is dominated by hydrogen. This is

http://en.wikipedia.org/wiki/Bullet_cluster strong evidence for feebly interacting matter. Galaxy Halo Structures in Detail ‣ Note the Bumps and Wiggles in the detailed Rotation Curve plots above. We know that the baryonic matter (stars etc) are clumped (for instance we see spiral structures). Evidence here suggests structure in the halo as well. ‣ To study this we can use N-body Simulations of galaxy formation and indeed this reveals that we expect some Substructure compatible with the observations, including:

(1) Clumps and sub-halos (large and small) (2) Distorted halo shapes, triaxial, oblate, prolate etc. (3) Caustics (discontinuities) due to galaxy mergers (4) Some rotations of the halo or velocity anisotropy

Example N-body simulation ‣ Such structure also can not be explained by MOND. N-Body Simulation Basics ‣ So-called N-body Computer Simulations are a powerful tool in Dark Matter and studies. ‣ We start with an initially uniform distribution of masses and then introduce a small Perturbation. ‣ Gravity is attractive, so even small density inhomogeneities tend to grow with time - dense regions get denser, sparse regions get sparser. This is Gravitational Instability. ‣ Computation is slow because if there are n bodies in the simulation, there are n(n-1)/2 (=~n2) Inter-body Separations to calculate in every time interval. ‣ Fortunately the masses evolve into Clumps and this can simplify the calculations since n can gradually be reduced. MOND and N-Body Simulations ‣ So the trick in the simulation is that once a clump of masses is sufficiently small, then we can treat it as one mass. ‣ Interestingly this approximation does not work for MOND, proved as follows: Consider a mass 4M as a single mass or divided into 4 parts separated by dr. Compare the acceleration at distance r assuming dr/r <<1. For normal gravity the calculation G(4M) 4GM ‣ a = = for both gives the same result. r2 r2 ‣ But for MOND the results differ by a factor x2! ‣ This does not mean that MOND is necessarily wrong, just that the normal N-body simplification€ can not be used. Summary of Topic 2 A guide for the exam ‣ Dark Matter exists on all scales - understand the concept of Mass to Light ratio and its application. ‣ Understand the principles and limitations of using the Oort technique for studying dark matter nearby. ‣ Know about galactic rotation curve measurements, the types of possible curve, the radio measurements involved. ‣ Understand the isothermal sphere model, be able to derive the equations for mass and density in galaxies given a flat rotation curve and know the relevant galactic dynamics. ‣ Know about dark matter in the Milky Way and how MOND works to be an alternative explanation of flat rotation curves. ‣ Understand the basics of N-body simulations and why they are important in understanding dark matter. Terms to know from Topic 2 A guide for the exam ‣ Galaxy Clusters, Galaxy Super Clusters, Galaxy Mergers ‣ Oort Technique, Galaxy Rotation Curve, Virial Theorem ‣ Mass to Light (Luminosity) Ratio ‣ Brown Dwarf, White Dwarf, Black Holes, Neutron Stars (more later) ‣ Jan Oort, Harmonic Oscillator, , Hydrogen Clouds, ‣ 21cm Line, F=1 (0) Hyperfine State, Spin-flip Transition ‣ Peculiar Motions, Resolving Power, Astronomical Unit (AU) ‣ Solid Disc Rotation, Keplerian Rotation Curve ‣ Spherical Dark Matter Halo, Isothermal Sphere, Halo Model ‣ Maxwellian Distribution, Flat Rotation Curve ‣ Baryons, Inelastic Collisions, Elastic Collisions, Angular Momentum ‣ Modified Newtonian Dynamics, MOND, Phenomenological Model ‣ Bullet Cluster, Galactic Nuclei, N-Body Simulations, Gravitational Instability Equations from Topic 2 Equation reminders for the exam

GMm λ F = = maµ θmin ≃ 2 L r

GMm€ GMa v 2 F = = maµ 0 = ⇒ v = (GMa )1/ 4 r2 r r 0

€ € Questions on Topic 2 to help with exam revision ‣ Describe and draw the Oort technique 3 3 ‣ Convert GeV/m to Msun/Mpc. How much DM is near us in GeV/cm ? ‣ How does η change with scale, and why? ‣ Describe 3 different forms of vcirc(R) that might be present in galaxies and why? What do we mean by dark matter halo? ‣ How do HI distributions look c.f. DM? ‣ Draw a plot showing the matter distribution in the Milky Way, show units and labels ‣ Prove that the MOND assumption that F = ma × μ(a) where μ behaves as μ = a/a0 for very small a, results in a flat rotation curve. What would 2 happen if μ = a /a0? ‣ Derive v(r) for a Keplerian system and for MOND. ‣ What is a typical value for a0 ‣ Give some examples of why MOND is probably wrong..

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