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Astrophysics 2

by: Stephan Kuvalis

Astrophysics 2 ASTR 3830

Stephan Kuvalis

GPA 3.89


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This 34 page Class Notes was uploaded by Stephan Kuvalis on Thursday October 29, 2015. The Class Notes belongs to ASTR 3830 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/231965/astr-3830-university-of-colorado-at-boulder in Astronomy at University of Colorado at Boulder.


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Date Created: 10/29/15
Photometric properties of galaxies Empirically the surface brightness declines with distance from the center of the galaxy in a characteristic way for spiral and elliptical galaxies For spiral galaxies need first to correct for Inclination of the disk Dust obscuration Average over spiral arms to obtain a mean profile Corrected disk surface brightness drops off as 1R 10 e39RhR where O is the central surface brightness of the disk and hR is a characteristic scale length ASTR 3830 Spring 2004 In practice surface brightness at the center of many spiral galaxies is dominated by stars in a central bulge or spheroid Central surface brightness of disk must be estimated by extrapolating inward from larger radii us an 7m x smsnmm owPuNwr surface brightness rad US ASTH 3830 Spring 2004 Typical values for the scale length are 1kpC lt hR lt10 kpC In many but not all spiral galaxies the exponential part of the disk seems to end at some radius Rmax which is typically 3 5 hR Beyond Rmax the surface brightness of the stars decreases more rapidly edge of the optically visible galaxy The central surface brightness of many spirals is constant irrespective of the absolute magnitude of the galaxy I B O z 2165 mag arcsec392 Empirical laws presumably arise from physics of galaxy and or star formation ASTR 3830 Spring 2004 Elliptical galaxies Surface brightness of elliptical galaxies falls off smoothly with radius Measured for example along the major axis of the galaxy the profile is normally well represented by the R14 or de Vaucouleurs law IR 10 e kR where k is a constant This can be rewritten as 767 RRe 025 1 IR 16 e where Re is the effective radius the radius of the isophote containing half of the total luminosity e is the surface brightness at the effective radius Typically the effective radius of an elliptical galaxy is a few kpc ASTR 3830 Spring 2004 Profile of elliptical galaxies can deviate from the R14 law at both small and large radius Close to the center Some galaxies have cores region where the surface brightness flattens and is constant Other galaxies have cusps surface brightness rises steeply as a powerlaw right to the center A cuspy galaxy might appear to have a core if the very bright center is blurred out by atmospheric seeing HST essential to studies of galactic nuclei ASTR 3830 Spring 2004 surface brightness v mag mmquot Results from HST SJ Hm 8 ym I mum VVHHH yw rrm 12 739 Cores Powerlaws 14 16 18 I Huml yum VWWHIH Hum 1 1 0 100 Mean Radius no radius mml 1000 mom ASTR 3830 Spring 2004 Surveys of elliptical galaxies show that o The most luminous ellipticals have cores typically a slope in the surface brightness distribution R3903 or flatter o Low luminosity ellipticals have power law cusps extending inward as far as can be seen o At intermediate luminosities mixture of cores and cusps Unknown why elliptical galaxies split into two families ASTR 3830 Spring 2004 In giant elliptical galaxies CD galaxies found at centers of some galaxy clusters surface brightness at large radius may exceed that suggested by Rquot4 law HST image of galaxy cluster Abell 2218 Lens the cluster is at a redshift z03 Galaxies behind are at redshifts up to 558 and are strongly magnified and distorted ASTFi 3830 Spring 2004 Expansion of the Universe Expansion of Universe cannot Homogeneity isotropy alter the relative orientations of galaxies expanding with the Universe Means that if the present separation between two galaxies is do then the separation at time t can be written as d d0at at is the scale factor it is dimensionless and depends upon time but not on position Relative velocity of the two galaxies is o a v d d0at d a ASTR 3830 Spring 2004 Definition of the Hubble parameter is v H x d so H a H is a function of time present value is denoted HO Sometime useful to define comoving coordinates If we divide distances by at then two galaxies which simply recede from each other due to the Hubble expansion always have the same separation in comoving coordinates ASTR 3830 Spring 2004 Can derive the evolution of at using mostly Newtonian mechanics provided we accept two results from General Relativity 1 Birkhoff s theorem this states in part that for a spherically symmetric system the force due to gravity at radius r is determined only by the mass interior to that radius 2 Energy contributes to the gravitating mass density which equals u 9 energy density pm 6 2 ergs cm393 of radiation and density of matter relatIVIstIc particles ASTR 3830 Spring 2004 Consider the evolution of a spherical volume of the Universe radius L Sphere expands with the Universe so L L0at Since expansion is described entirely by at can consider any size sphere we want if L is small reasonable to assume that space is approximately Euclidean Expansion of the sphere will slow due to the gravitational force of the matter energy inside dZL GM dt2 L2 ASTR 3830 Spring 2004 Note no pressure forces because Universe is homogenous Contributions to the gravitating mass come from matter plus energy density from radiation Matter density pm Radiation with energy density u has pressure Plu 3 gravitating mass density is 3P P 1 2 C Mass within sphere radius L is M pV EL3p ASTR 3830 Spring 2004 Substitute into acceleration equation d2L G 4 3 gtlt ytL d L2 3 0 Since L L0at with LO a constant can write this as an equation for the evolution of the scale factor at also substituting for p in the above expression Matter density pmgt O Pressure of radiation is also positive RHS of the equation is always negative Impossible to have a static Universe ASTR 3830 Spring 2004 Lack of static solutions is not a problem Universe is expanding But this was not known in 1917 Einstein therefore modified the equations of General Relativity so the equation becomes A is the cosmological constant the factor 3 is just convention A positive cosmological constant tends to accelerate the expansion ie as if the Universe is filled with material with a negative pressure Is a static solution stable ASTR 3830 Spring 2004 Properties of the cosmological constant Cosmological constant is assumed to be a smooth component ie it does not cluster or clump together in the same way as ordinary matter Original cosmological constant was constant in time This is just an assumption however models in which the vacuum energy varies with time are called quintessence For A to be important today it must have a value comparable to the first term in the equation A 47tGpm 10 36s392 for p 103930 g cm393 Fundamental unit of time is the Planck time tplmk 1 G f 10 43 s C Might guess that A 2an bad guess by factor 10120 Cosmological constant problem ASTR 3830 Spring 2004 Which terms are most important o Early times energy density of radiation is large compared to the energy density of matter Later matter dominates o Finally if A is nonzero eventually it dominates Radiation dominated N Each of these changes Matter dominated in different way as gt Universe expands Cosmological constant F39Stht expans39on dominated aWS J ASTR 3830 Spring 2004 Dark matter in the Galactic Halo Rotation curve ie the orbital velocity V of stars and gas as a function of distance to the Galactic Center r of the disk of the Milky Way is measured for the inner Galaxy by looking at the Doppler shift of 21 cm emission from hydrogen for the outer Galaxy by looking at the velocity of star clusters relative to the Sun details of these methods are given in Section 23 of Sparke amp Gallagher Fact that Vr constant at large radius implies that the Galaxy contains more mass than just the visible stars and gas Extra mass the dark matter normally assumed to reside in an extended roughly spherical halo around the Galaxy ASTR 3830 Spring 2004 Possibilities for dark matter include o molecular hydrogen gas clouds baryonic dark o very low mass stars brown dwarfs matter made stellar remnants white dwarfs originaHY from neutron stars black holes ordinary 935 o primordial black holes elementary particles probably nonbaryonic dark matter currently unknown The Milky Way halo probably contains some baryonic dark matter brown dwarfs stellar remnants accompanying the known population of low mass stars This uncontroversial component of dark matter is not enough is the remainder baryonic or nonbaryonic ASTR 3830 Spring 2004 On the largest scales galaxy clusters and larger strong evidence that the dark matter has to be nonbaryonic Abundances of light elements hydrogen helium and lithium formed in the Big Bang depend on how many baryons protons neutrons there were light element abundances theory allow a measurement of the number of baryons observations of dark matter in galaxy clusters suggest there is too much dark matter for it all to be baryons must be largely nonbaryonic On galaxy scales no such simple argument exists Individual types of dark matter can be constrained using various indirect arguments but only direct probe is via gravitational lensing ASTR 3830 Spring 2004 Gravitational lensing Photons are deflected by gravitational fields hence images of background objects are distorted if there is a massive foreground object along the line of sight Bending of light is similar to deflection of massive particles eg if a star passes by a massive body at velocity V with an impact parameter distance of closest approach b its path is deflected V impact 1 w parameter b mass M deflection velocity Fairly easy to show that the transverse velocity imparted due to the gravitational acceleration during the flyby is AVL ZS VM textbook 322 ASTR 3830 Spring 2004 Am ZGM V by2 The deflection angle is a General relativity predicts that for photons the bending is exactly twice the Newtonian value 4GM2amp W 7 where FlS is the Schwarzschild radius of a body of mass M Formula is valid provided that b gtgt RS Not valid very close to a black hole or neutron star o Valid everywhere else lmplies that deflection angle or will be small ASTR 3830 Spring 2004 Geometry for gravitational lensing Consider sources at distance dS from the observer 0 A point mass lens L is at distance dL from the observer Observer Observer sees the image of the source S at an angle 6 from line of sight to the lens In the absence of deflection would have deduced an angle 3 ASTR 3330 Spring 2004 Recall that all the angles or 3 E are small ii 31 0536 31 dL dS dS dLS Substitute these angles into expression for deflection angle x y 4GM dLS bc2 4GM eds ds Wags 6 9 l 4GM dm 6 c2 deL HJ Geometric factors Quadratic equation for the apparent position of the image 6 given the true position 3 and knowledge of the mass of the lens and the various distances ASTR 3830 Spring 2004 Simplify this equation by defining an angle 8E the Einstein radius of the lens 2 GMdLS 9f 7 L S Equation for the apparent position then becomes 62 36 6 0 Solutions are 6 phifwd i 2 For a source exactly behind the lens 3 O Source appears as an Einstein ring on the sky with radius 0E For 3 gt 0 get two images one inside and one outside the Einstein ring radius ASTR 3830 Spring 2004 Dynamics of elliptical galaxies Galaxies that appear elliptical on the sky may be intrinsically oblate prolate or triaxial depending upon their symmetries Oblate T Prolate For an individual galaxy can t determine from an image what the intrinsic shape of the galaxy is Orbits of stars differ substantially in different types ASTB 3830 SpringzaM Orbits in elliptical galaxies Classification of stellar orbits in elliptical galaxies is much more complicated than for disk galaxies Most important distinction is between axisymmetric galaxies prolate or oblate and triaxial galaxies In an axisymmetric galaxy there is a plane perpendicular to the symmetry axis in which gravitational force is central No azimuthal force so component of angular momentum about symmetry axis say zaxis is conserved LZ mr gtlt VZ 9 Only stars with LZO can reach center other stars must avoid the center ASTR 3830 Spring 2004 Orbits in elliptical galaxies Triaxial potential energy is conserved but not LZ Simple example is the potential X 100293 002 2 002 2 2 x yy ZZ which is the potential inside a uniform density ellipsoid 00X etc are constants Star in this potential follows harmonic motion in each of the xyz directions independently Unless 00X by and 002 are rational multiples of each other eg 127 orbits never close star completely fills in a rectangular volume of space in the galaxy Example of a box orbit ASTR 3830 Spring 2004 Orbits in elliptical galaxies Numerical examples of orbits from Josh Barnes ztube orbit bogtlt orbit Orbit loops around the minor Main orbit family in triagtltia agtltis only orbit family in potentials note orbit does oblate potential not avoid the center A SH 3530 Spring 2004 Fine structure in elliptical galaxies Contours of constant surface brightness often depart slightly from true ellipses Twists major axis of the isophotes changes angle going from the center of the galaxy to the edge This can be interpreted as a projection effect of a triagtltia galaxy in which the ellipticity changes with radius Wm wa on flat surface See apparent from lower twist in the left isophotes Asm 3930 Spring 2004 Fine structure in elliptical galaxies Surface brightness distribution can also depart slightly from empses Boxy isoghotes Disky isoghotes Normally subtle distortions Luminous ellipticals are often boxy midsized ellipticals disky Could classify ellipticals based on their degree of boxiness diskyness 803 would then be continuation of a trend to increasing diskyness ASTR 3830 Spring 2004 FaberJackson relation Analog of the TullyFisher relation for spiral galaxies Instead of the peak rotation speed Vmax measure the velocity dispersion along the line of sight 0 This is correlated with the total luminosity 4 LV 2 x10101 Lm 200 kms Can be used as a not very precise distance indicator ASTR 3830 Spring 2004 Fundamental plane Recall that for an elliptical galaxy we can define an effective radius Re radius of a circle which contains half of the total light in the galaxy Measure three apparently independent properties o The effective radius Re o The central velocity dispersion o The surface brightness at the effective radius leIRe Plot these quantities in three dimensions find that the points all lie close to a single plane Called the fundamental plane ASTR 3830 Spring 2004 Fundamental plane 124mg a 4 D zlog lt1 5 e A a o 5 1 l5 log I kPC log a 2 2 5 1 lug r gazing ltlgt Jorgensen et al 1996 Plots show edgeon views of the fundamental plane for observed elliptical galaxies in a galaxy cluster Approximately 124 7082 Re cc a 2 Measure the quantities on the right hand side then compare apparent size with Fle to get distance Origin of the fundamental plane unknown ASTB 3830 Spring 2004


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