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    galactic disks Basic idea Density wave propogates thru disk at a constant p pattern speed r is not constant for much of disk DIFFERENTIAL ROTATION At one point r p corotation radius Q How is r determined Inside corotation p Stars gas overtake spiral pattern increases gas piles up on inside of spiral arm compression of gas star formation OB stars in spiral arms Time for stars to form Blue stars downstream from gas dust but not much Older stars further along arms less distinct See M51 See also Figure 23 18 in text for density wave diagram Consider motion of star in galactic potential Orbits not quite circular Due to gravitational perturbations Can model this motion as the combination of Circular motion ang velocity of a point period 2 Epicyclic motion ang vel κ of object about this point period 2 κ Note motion on epicycle is retrograde NOTE also have vertical oscillations throught galaxy midplane for same reasons In general unless and κ are commensurate the orbit is not closed to an outside inertial reference frame rosette pattern If we choose a non inertial i e rotating reference frame with ang velocity lp can make r κ r commensurate lp local pattern Over some time t t κ 2 m t lp 2 n where m epicycles are completed in n circular orbits Closed orbits in reference frame of lp Now consider the global pattern speed p of a density wave If result is a fixed spiral pattern in frame rotating at p For n m 1 2 2 fold symmetry 2 armed spiral Get closed ellipses in p frame If ellipses are twisted and p constant Fixed spiral pattern in p frame Packed gas stars dust density waves Q Why are more stars in spiral arms Resonances

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/spstructure.html (2016-02-13)
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    the chain rule Therefore Taking the logarithm of A m Since log N const 0 6 m then log A m constant 0 6 m Differential counts have the same dependence on m as the integrted counts but the constants differ In practice we don t observe a slope of 0 6 when counting stars The actual counts tend to flatten off due to extinction Extinction The extinction A V is obtained from studies of interstellar reddening A V 3 E B V From the above equation the apparent distance is Thus objects will appear to be much further than they really are From the solar neighbourhood where a 1 6 mag kpc Star Count Analysis Luminosity and Density Functions There is a continuous distribution of stars over luminosity The space density of stars is r l b M S the number of stars per unit volume per unit absolute magnitude at distance r in direction l b and of spectral type S Suppress the variables l and b they only define direction and adopt a separable function for 1 S M can be regarded as giving the number of stars per unit absolute magnitude per unit volume in which case the density function is a dimensionless i e normalized function of distance 2 S M dM can be regarded as giving the probability of finding a star of absolute magnitude M in the interval dM i e we normalize so that In this case r gives the relative number of stars per unit volume of all M S Quite often S M dM is represented as a Gaussian with some small dispersion representing cosmic scatter General star counts over all S A m is observed If M is known then this integral equation can be solved to obtain r This is a hard problem Modern approach parameterized models for r and use the star counts to constrain these parameters Malmquist Bias General Star Counts over all S Adopt functional form for the luminosity function M as a Gaussian distribution Note Gaussian distributions Normal Probability Distribution FWHM 2 354 1 68 of distribution 2 95 3 99 4 event unlikely event x x o 4 M m mean absolute magnitude of the stars in the sample that have apparent magnitude m or A m is based on magnitude limited samples that consist of all objects brighter than a limiting magnitude m 1 that lie within a specified area of the sky Malmquist bias luminous objects are over represented in a magnitude limited sample mag mean absolute magnitude observed in a magnitude limited sample vol true mean absolute magnitude of volume limited sample mag will be brighter than the true mean absolute magnitude of the population as a whole because volume within which we can see the most luminous objects is larger than that within which we can also see the faintest objects there are more intrinsically brighter more distinct objects that scatter into the sample than there are intrinsically fainter

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/count.html (2016-02-13)
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    halo no DM M L 1 M L For the central bulge M L 3 M L Note Galactic Disk 3D Observations of external galaxies MW Spheroid Bulge Visible below and above the Galactic plane Visual extinction A V 20 mags in most l o s Baade s Window discovered in 1944 towards N6522 located in bulge RR Lyrae stars are visible beyond the GC At l b 1 3 9 Offers close look at bulge closest to center 540 pc from GC in the optical disk meets bulge at 1kpc bulge independent component of Galaxy scale height changes from 325 pc for thin disk r 1kpc to 400 pc for r 1kpc measured at 1 2 3 4 m by COBE once thought to be spheroid with r 1 4 profile observations of Mira variables reveal peanut shaped bar elongated bulge confirmed in COBE picture bar would be inclined by 45 to our l o s ratio of b a for bulge is 0 6 Is our bulge deVauc i e No Kent Dawe Fazio 1996 use Space Shuttle IR telescope and find exponential disk with scale height increasing gently outwards SB profile of bulge also exponential with sharp cutoff bar and scale length 500 pc Freudeurich 1998 L band from COBE Disk has hole at center Type II disk Bulge is triaxial and peanut shaped probably barred Extragalactic perspective 2 3 of all spirals have bars most spirals have exponential bulge plus exponential disk bulge metallicities unknown Metallicity Varies from quite metal poor to very metal rich 1 Fe H 1 with a mean near 0 3 On average Fe H has twice the solar value suggests that some fraction of the bulge is young perhaps only 10 billion years old this conclusion supported by Mira variables Miras

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/comp.html (2016-02-13)
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    type metal rich Fe H 0 8 flattened rotating v 190 km s concentrated at center associated with thick disk same vertical scale height as thick disk 1 4 kpc Halo population F type metal poor Fe H 0 8 spheroidal distance slowly rotating v 40 km s The GC system is flattened close to the Galactic center and becomes spherical beyond 2 3 kpc For R 3 kpc R R 3 5 Show Fig 22 12 Notable exception 47 Tuc youngest GC Fe H 0 64 Most GCs found withing 33 kpc a few discovered between 66 kpc r 100 kpc captured by Milky Way Halo Stars Structure of halo best revealed by RR Lyrae pop II variables like metal poor Cepheids R similar to that of GCs Kinematics GCs and field stars have large velocity components perpendicular to the Galactic plane field stars high velocity stars plunging orbits Eg bright giant Arcturus is just a halo star passing through Mass density Total stellar mass density in solar neighbourhood is 0 05M pc 3 of which high velocity halo stars contribute 0 2 Total estimated mass of stellar halo is 10 9 Fraction contribution 1 GCs 99 field stars Add up all disk stars HI dust halo stars total mass of luminous matter 9 10 10 M dynamical estimates suggest 10 times more mass in the Galaxy Why do we study globular clusters Galaxy Formation GCs among first objects to form in Galaxy tracer of galaxy formation and early evolution chemical evolution ages of clusters formation timescale Galactic Halo few visible objects in halo GCs are the most luminous objects in halo studied in distant galaxies Age of the Universe look at the age of the oldest objects cosmological significance Where are the GCs located in the halos of galaxies Milky Way halo 150 GCs M31 halo 400 GCs some giant elliptical galaxies 5000 20 000 spherically distributed around Galactic center Shapley distance to Galactic center using GCs distribution R 3 5 same as halo field stars some GCs located 50 kpc from Galactic center 2 distinct sub populations metal poor low Fe H population dominant component more extended halo population metal rich high Fe H population flattened related to Galactic bulge Colour Magnitude Diagrams CMDs cluster stars at same distance CMD HR Diagram magnitude colour luminosity temperature CMDs of globular clusters GCs are OLD all hot high L high mass stars have died off lower L low mass stars left Some differences in CMDs and GCs more metal rich clusters have redder CMDs G type F type redder horizontal branch stars both due to line blanketing Fe H is the first parameter in what CMDs look like but some clusters of same Fe H show different HBs 2nd parameter not clear what it is yet Age of globular clusters How can we tell position of the main sequence turnoff MSTO older populations fainter MSTOs absolute ages of GCs fitting of theoretical isochrones to data yield ages 11 15

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/halo.html (2016-02-13)
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    circular velocity at R o 8 0 kpc is o 220 km s The period of the Sun around the galactic center is P 2 R o o 2 3 10 8 yr This is called the Galactic Year The age of the Sun is about 4 8 10 9 yr which is about 20 Galactic years Mass estimate The above equation is equivalent to Kepler s 3rd Law The LHS is the gravitational force on the Sun due to all the mass interior to the Solar distance M G R o and the RHS is the centripedal or inertial force The above equation can be rearranged to give GM G R o o 2 R o Since o 220 km s and R o 8 0 kpc then M G R o 8 8 10 10 M This is close to the mass that is estimated from the amount of light emitted by the stars However the flat rotation curve beyond R o implies M G R o R and the resulting high M L values imply large amounts of dark matter in the outer Galaxy Images Table of proper motions in right ascension Net motion in right ascension Observed distributions of and Z Table of proper motions in declination Net motion in declination Solar motion with respect to disk and spheroidal component objects Velocity ellipsoid parameters for various types of stars Velocity vectors relative to the LSR and their distribution in Galactic longitude Distribution of high velocity stars in and The density of stars near the Sun in velocity space Dehnen 1998 AJ 155 238 Asymmetric drift adapted from Carroll Ostlie 1996 ELS diagram Eggen Lynden Bell Sandage 1962 ApJ 136 748 The relation between rotation velocity and metallicy Gilmore Wyse Kuijken 1989 ARA A 27 555 Table of scale heights and surface densities Table of parameters for photometric models of the MW Space density as a function of distance z from the Galactic plane List of likely metal weak thick disk stars Chiba Beers 2000 AJ 119 2843 The relation between metallicity and eccentricity and mean eccentricities Ibid Distribution of mean rotational velocities vs Fe H Ibid Overall and close up views of the disk and halo and the age metallicity relation in the MW Analysis Oort Constants The radial and transverse velocities are see text p 949 for derivation v r o R o sin l v t o R o cos l d observed velocity components corrected for peculiar motions where R and o o R o is the angular velocity of rotation The above are exact general equations valid for objects at any distance d Oort derived an approximation for nearby stars d R o useful for interpreting observations Now o R R o Set R R o R with R R o Then Taylor expand R ignoring the higher order terms Thus Since R apply the chain rule for differentiation Thus Recall that for d R o R R o d cos l

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/kin.html (2016-02-13)
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    energy Their presence is revealed by gravitational effects on nearby matter Black holes can accrete matter The kinetic energy is released as gas collides with other matter near the BH The energy dissipates into heat and radiation Accreted material will have some angular momentum Accretion disk hot and radiates energy viscous forces in the disk can scrub off kinetic energy of rotation and cause material to slowly move inward to event horizon releases potential energy as heat Accretion rate Some images HST image of the Galactic center IR image of the non thermal filaments surrounding Sgr A Wide Field Radio Image of the Galactic Center Map showing the movement of stars around Sgr A Orbital acceleration of 3 stars located 0 005pc 1000 AU from Sgr A Ghez et al 2000 astro ph 0009339 Interior mass function for the central 1 kpc of the MW Carroll Ostlie 1996 Kinematics in the nucleus of the MW Genzel et al 1997 MNRAS 291 219 see also http www mpifr bonn mpg de staff hfalcke bh Rotation dispersion and M L ratio in M31 Kormendy Richstone 1995 ARA A 33 581 Line of sight velocity distribution and velocity profile in the nucleus of

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/gc.html (2016-02-13)
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    monotonic increase in the abundance of any robust element with time Therefore the Age abundance relation Images Metallicity vs age for nearby F and G stars Distribution function of oxygen abundances in G dwarfs Test prediction in high z absorption line systems on the line of sight to quasars damped Ly systems believed to represent an early form of disk galaxies Scatter at any redshift too large Assumptions The system is isolated with a constant total mass ie no inflows or outflows g t s t M const g gas s stars The system is well mixed at all times ie Z Z t is the abundance of any element s in the gas and in newly born stars The system stars as pure gas with primordial abundances ie g 0 M s 0 Z 0 0 The IMF and nucleosynthetic yields of stars with given initial mass are unchanging as far as primary elements such as oxygen are concerned In galaxy look at disk stars in solar neighbourhood Instantaneous simple model assume first generation of stars born without metals Z o 0 and chemical evolution occurs in a closed box Statistical correlations exist match simple model Scatter due to local inhomogeneities and or mixing of stellar populations from different parts of Galaxy Problem calculation predicts too many stars of low metallicity G dwarf problem Suggested solutions Prompt initial enhancement assume Galactic disk formed with Z o 0 This could occur if the heavy element enrichment of the ISM resulted from rapidly evolving massive stars before the gas and dust settled into the disk Mass accretion substantial infall of metal poor material onto the Galactic disk has occured since the initial formation as gas enters the system it mixes with the metal enriched ISM Lower initial mass density pre accretion fewer stars formed early on fewer metal poor stars observed today IMF metallicity mass dependence IMF produced a larger fraction of more massive stars early fewer low mass stars too Massive stars are short lived fewer metal poor stars today Galaxy Formation Models Eggen Lynden Bell Sandage 1962 ELS Important early attempt to model evolution of our Galaxy MW formed from rapid collapse of a large proto Galactic nebula due to gravitational instability akin to models of formation of Solar System Oldest stars formed early in collapse while still on radial plugging orbits metal poor pop II stars formed out of primordial ISM may not be consistent with G dwarf problem ISM evolves chemically over time proto Galactic nebula collapses adiabatically no heat inflow or outflow slowing down due to collisions between particles and clouds kinetic energy of infall is dissipated converted into thermal energy of random motion Angular momentum conservation spin up mvr constant r enriched gas stars dissipate into flattened rotating disk If M initial 10 12 M and r initial 100 kpc then t ff 10 9 yrs How long can MW sustain SF If M HI 5 10 9 M then Prolonged by adding new material

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/gce.html (2016-02-13)
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    v is all you can get Fundamental cosmological parameter H o present expansion rate of the universe Can estimate age What is H o Final result for many extragalactic distance determinations Revised heavily in early years changes in distance scale 1960 s H o 50 km s Mpc Sandage H o 100 km s Mpc de Vaucouleurs Many values between 50 and 100 km s Mpc since then Much work on techniques systematics 1990 s H o 60 80 km s Mpc for the most part HST Key Project H o 71 6 km s Mpc Most results in last 5 years H o 65 80 km s Mpc Why is it so difficult to get H o Systematic errors and changes to distance scale Nearby galaxies v contaminated by relative motion of Local Group towards Virgo Significant peculiar velocities of nearby galaxies Need distance to clusters where Hubble flow dominates Usually Coma Cluster v r 7000 km s d 100 Mpc Hubble anisotropy due to other clusters bulk flows Smooth Hubble expansion perturbed by large scale mass fluctuations A typical way to get H o Derive m M Virgo d Virgo Cepheids or TRGB or PNLF or Find relative distance from Virgo Coma D n GCLF or SBF Get d Coma know v get H o OR get d to more distant objects TF D n Some Very Basic Cosmology Expansion Rate Consider a dimensionless scale factor R t Stretching of due to cosmological redshift Eg at z 1 observable universe 1 2 its present size change in not due to Doppler effect H t expansion rate We usually define R now R o 1 A Newtonian look at an expanding universe see text 27 1 Consider an expanding shell of mass m radius r Energy of the shell

    Original URL path: http://www.astro.queensu.ca/~courteau/Phys216/ho.html (2016-02-13)
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