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Galaxies: Structure, Dynamics, and Evolution Galaxies: Structure, Dynamics, and Evolution Spiral Galaxies/Disk Galaxies (III): Spiral Structure Review: Collisionless Boltzmann Equation and Collisionless Dynamics Layout of the Course Feb 5: Review: Galaxies and Cosmology Feb 12: Review: Disk Galaxies and Galaxy Formation Basics Feb 19: Disk Galaxies (I) Feb 26: Disk Galaxies (II) Mar 5: Disk Galaxies (III) / Review: Vlasov Equations this lecture Mar 12: Elliptical Galaxies (I) Mar 19: Elliptical Galaxies (II) Mar 26: Elliptical Galaxies (III) Apr 2: (No Class) Apr 9: Dark Matter Halos Apr 16: Large Scale Structure Apr 23: (No Class) Apr 30: Analysis of Galaxy Stellar Populations May 7: Lessons from Large Galaxy Samples at z<0.2 May 14: (No Class) May 21: Evolution of Galaxies with Redshift May 28: Galaxy Evolution at z>1.5 / Review for Final Exam June 4: Final Exam You have a homework assignment that is due on Monday, Mar 9, before noon There will be a new homework assignment that will be due on Monday, Mar 16, before noon First, let’s review the important material from last week Multiple arm spiral Grand design spiral How doNGC 6946the arms in spiral galaxies evolve with time? 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-3 Most spiral3-4-12see arms http://www.strw.leidenuniv.nl/˜ are This could franx/college/galaxbe determinedies12 by 12-c02-4 looking at Flocculent spiral Most spiralfound arms toare trailingbe trailing. reddening in globular clusters / novae globular clusters seen around disk galaxy. amount of reddening indicated by whether circles are solid or open allows us to determine which way a spiral galaxy is 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-3 3-4-12see http://www.strw.leidenuniv.nl/˜tilted. franx/college/galaxies12 12-c02-4 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-3 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-4 What is nature of the arms in spiral arms? Do the spiral arms travelWinding at the same problem speed as the stars? If spiral arms did,one would predict that the spiral arms in a galaxy would wind up very quickly. Winding problem 12 The predicted outcome is in contrast to what is observed! The problem: most spiral galaxies would be tightly wound by now, which is inconsistent with observations. Spiral arms cannot be a static structure (i.e. at di↵erent times, arms must be made of di↵erent stars) 13 How can we solve the winding problem? Density Wave Theory Lin & Shu (1964-1966) The spiral arms in disk galaxies are not fixed structures that rotate around the center of disk galaxies, but rather density waves. These density waves can move at a different speed than the stars within the galaxy itself. The speed at which the spiral density waves propagate around the disk of a spiral galaxy is called the pattern speed Ωp. 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-7 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-8 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-7 3-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-8 Epicycle Approximation IV An important question is: “When is the epicycle approximation valid?” First consider the z-motion: The equation of motion, z¨ = ν2z implies a − constant density in the z-direction. Hence, the epicycle approximation is valid as long as ρ(z) is roughly constant. This is only approximately true very close to equatorial plane. In general, however, epicycle approx. is poor Starsfor motion inin zSpiral-direction. Galaxies are on Epicyclic orbits In the radial direction, we have to realize that the Taylor expansion is only accurate sufficiently close to R = Rg. Hence, the epicycle approximation is Theonly motionvalid for small canlibrations be approximatelyaround the guiding describedcenter; i.e., for asorbits thewith combination of an angular momentum that is close to that of the corresponding circular orbitalorbit. motion around a disk galaxy and an epicyclic motion in radius: Epicyclic Motion Orbital Frequency Frequency of epicyclic around Disk Galaxy = motion = 2π/κ 2π/Ω The orbital frequency of a star Ω(R) can be written as follows Meanwhile, the frequency of epicyclic motion κ(R) can be written as follows: The frequency of epicycle motion is very similar to the orbital frequency: In general, Ω < κ < 2 Ω Near the solar system, the epicycle frequency κ ~ 1.3 Ω NOW new material for this week Epicyclic orbits For the case of a point mass (Ω ∝ R-3/2), e.g., solar system, the epicyclical time perfectly matches the rotation time around the central body so that orbits close on each other. There can be no more than 6 integrals of motion. Typically there is 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-17 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-18 at least one integral of motion (energy). A general 3-dimensional potential A Simple recipe to build galaxies Here are some examples of orbits where the phase space is only Schwarzschild’s method: • Define density ρ 12-10-07 see http://www.strw.leidenuniv.nl/˜incompletely franx/college/ filled: mf-sts-07-c4-3 2 2 12-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-4 St¨ackel potential( ρ =1/(1 + m ) ) • Calculate potential, forces Stacker potential: triaxial potential In general,typical orbit inthis spherical is not potential true, forms however. a Orbits regress and one finds a • Integrate orbits, find orbital densities ρi Typical orbitplanar in arosette spherical potential is a planarthat admits rosette three integrals of motion • Calculate weights wi > 0 such that planar rosette. ⃗¨r = −Ω2⃗r ρ = ρiwi or in cartesian coordinates x, y ! x¨ = −Ω2x y¨ = −Ω2y Examples: build a 2D galaxy in a logarithmic potential 2 2 Hence solutions are Φ =ln(1+x + y /a). • As we saw, box orbits void the outer x-axis x = X cos(Ωt + c•xAs) we saw,y = loopY orbitscos( voidΩt + thec innery) x-axis → both box and loop orbits are needed. where X, Y, cx and cy are arbitracy constants. Hence, BT 3.1: page 106 Suppose we have constructed a model. Angle ∆ψ between successive apocenter passages de- even though energy and angular momentum restrict orbit to a “rosetta”, these• What orbits kind of are rotation even can more we expect special: ? pendsin most ongeneral mass case, distribution: the orbit is not closed box orbits: no net rotation they do not fill the area betweenloop orbits: the can rotate minimum either way: and positive, nega- and will fill entire area between rmin and rmax maximum radius, but are always closed ! π < ∆ψ < 2π tive, or “neutral”. BT 3.4: page 155 The same holds for Kepler potential. But beware, for Hence: a maximum rotation is defined if all loop the homogeneous sphereorbits the particlerotate the same does way. two The radial rotation can vary homogeneous sphere point mass excursions per cycle aroundbetween the zero, center, and this for maximum the Kepler rotation potential, it does one radial excursion per angular cy- cle. Special cases We now wish to “classify” orbits and their density dis- 2 tribution in a systematic way. For that we use Integrals v⊥ dΦ GM(r) rmin = rmax circular orbit = = of motion. r dr r2 1 2 L =0 ⇒ radial orbit 2 r˙ = E − Φ(R) Homogeneous sphere 1 2 2 Φ(r)= 2 Ω r + Constant In radial coordinates Epicyclic orbits Using the measured values for κ and Ω at the radial position of the sun in our galaxy is as follows: κ = 1.3 Ω similar to the case for an isothermal sphere... Period for orbit around galaxy = 2π/Ω Period for epicyclic orbit = 2π/κ Which resonances drive spiral density wave growth? Now let us now consider a possible spiral density wave in the disk of a galaxy: Rotational Frequency of Spiral Density Wave = Ωp In these illustrations, let’s adopt the most common type of “grand design” spiral galaxy where we just have 2 arms (rotational symmetry = 180 degrees) Which resonances drive spiral density wave growth? When might we expect growth of a spiral density wave? Rotational Frequency of Spiral Density Wave = Ωp We might expect such if a star completes one period of epicyclic motion every time it encounters the spiral density wave in its orbit around the galaxy. Which resonances drive spiral density wave growth? Let us consider a few examples of the orbit of stars that would finish a complete epicyclic orbit in the spiral density wave itself: Example #1: The star is moving at the same speed as the spiral density in orbiting around the center of a galaxy. Let’s consider snapshots in time where the star completes an entire epicyclic orbit. Typically a star must complete 70% of a revolution around a galaxy before this happens. spiral arm spiral star arm spiral spiral arm arm one epicyclic orbit two epicyclic orbits time = 2π/κ time = 4π/κ three epicyclic orbits spiral pattern ~ 250 deg spiral pattern ~ 500 deg time = 6π/κ star orbitted ~ 250 deg star orbitted ~ 500 deg spiral pattern ~ 750 deg star orbitted ~ 750 deg Which resonances drive spiral density wave growth? Let us consider a few examples of the orbit of stars that would finish a complete epicyclic orbit in the spiral density wave itself: Example #2: The star is traveling much faster than the speed of the spiral density wave.
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