Today in Astronomy 142: the Milky Way s disk

Transcription

1 Today in Astonomy 14: the Milky Way s disk Moe on stas as a gas: stella elaxation time, equilibium Diffeential otation of the stas in the disk The local standad of est Rotation cuves and the distibution of mass The otation cuve of the Galaxy Figue: spial stuctue in the fist Galactic quadant, deduced fom CO obsevations. (Clemens, Sandes, Scoville 1988) 1 Mach 013 Astonomy 14, Sping 013 1

2 Stella encountes: elaxation time of a stella cluste In ode to behave like a gas, as we assumed last time, stas have to collide elastically enough times fo thei andom kinetic enegy to be shaed in a themal fashion. But stella encountes, even distant ones, ae ae on human time scales. How long does it take a cluste of stas to themalize? One chaacteistic time: the time between stella elastic encountes, called the elaxation time. If a gavitationally bound cluste is a lot olde than its elaxation time, then the stas will be descibable as a gas (the sta system has tempeatue, pessue, etc.). 1 Mach 013 Astonomy 14, Sping 013

3 Stella encountes: elaxation time of a stella cluste (continued) Suppose a sta has a gavitational sphee of influence with adius (>>R, the adius of the sta), and moves at speed v between encountes, with its sphee of influence sweeping out a cylinde as it does: v V = π vt vt If the numbe density of stas (stas pe unit volume) is n, then thee will be exactly one sta in the cylinde if 1 nv = nπ vtc = 1 tc = Relaxation time nπ v 1 Mach 013 Astonomy 14, Sping 013 3

4 Stella encountes: elaxation time of a stella cluste (continued) What is the appopiate adius,? Choose that fo which the gavitational potential enegy is equal in magnitude to the aveage stella kinetic enegy. Gm 3 1 v = mv tc = 4π Gmn Done in moe detail (Astonomy 3-level): fo a spheical cluste with a coe adius R, it can be shown that 3 v 1 tc =. 4πG mnln ( R) Not that fa fom ou ough estimate, as the logaithm is a vey slow function. 1 Mach 013 Astonomy 14, Sping 013 4

5 Stella encountes: elaxation time of a stella cluste (continued) In this week s homewok, you will show among othe things that fo such a cluste of N stas, with coe adius R and typical stella mass m, G( N 1) m Gm G( N 1) m v = and = R 4R Assume N >> 1 and substitute these into the expession fo elaxation time: R N tc v N 4ln The time tx = Rvis called the cossing time; it s the time it takes a sta moving at the mean speed v to tavese the coe of the cluste (diamete R) if it doesn t collide. 1 Mach 013 Astonomy 14, Sping 013 5

6 Themal equilibium A handy way fo bookkeeping andom motions in themal equilibium is the viial theoem: In an isolated system of paticles that exet foces on each othe descibable by scala potentials, the system s moment of inetia I, total kinetic enegy K, total potential enegy U and total mechanical enegy E ae elated by d I dt = K + U = K + E. In many cases (see, e.g., this week s ecitation), fo which K, U and E ae elated by 1 K = U = E. d I dt = 0, It is often easy to calculate U and the systematic-motion pat of K; thus we can get the andom-motion pat of K. 1 Mach 013 Astonomy 14, Sping 013 6

7 Themal equilibium (continued) Fo example: suppose a unifom-density sta cluste N stas of mass m, Nm = M has adius R and otates like a solid body at angula speed Ω. What is the andom speed v of a typical sta in this cluste? Obviously d I dt = 0, since the cluste s stuctue is constant, so 1 = 1 + Ω 1 K mv i i I = U i N 1 13 mv + MR Ω = GM v = R Ω 5 R 5 GM R 1 Mach 013 Astonomy 14, Sping 013 7

8 Themal equilibium (continued) We will esist the temptation to pove the viial theoem hee; though the poof isn t difficult, it s long and complicated. It will be poven fo you in PHY 35 and AST 3. If you can t wait, see FA, pp Caveats: The viial theoem applies only to themal equilibium o steady-state motion. Thus, befoe evey use, one should check to see whethe the system on which one s using it has been aound long enough to be in themal equilibium. Long enough means the system has been togethe fo a time much longe than the elaxation time. 1 Mach 013 Astonomy 14, Sping 013 8

9 Rotation of the stella population Aveaging ove the andom motions, one can detect diffeential otation in the disk of the galaxy, fom the adial velocities of neaby stas. The otation is diffeential in the sense that diffeent adii have diffeent angula velocities. The angula velocity deceases monotonically as adius fom the Galactic cente inceases. Measuement of aveage stella motions along the line of sight and pependicula to the line of sight can be used to detemine the local angula velocity. Unfotunately, motions pependicula to the line of sight (pope motions) ae cuently had to measue fo enough stas, and stas that ae fa away. 1 Mach 013 Astonomy 14, Sping 013 9

10 Rotation of the stella population, and Oot s constants Oot s constants, defined: dω 1 d A= B= Ω d d ( ) whence Ω= A B In tems of the aveage adial velocities and aveage pope motions: v = Ad sin v = Ad cos + Bd In the absence of pope motions (v t ), B is usually obtained less diectly fom the statistics of andom motions, with the esult 1 A/ B = 1.6. t v t v d v v t Sun 1 Mach 013 Astonomy 14, Sping

11 Oot s constants (continued) The measuement of the Oot constants thus equies good adial velocity measuements (v ), pope motions (v t ), and distances (d) ove a wide ange of distance. These days one can t measue vey well fo stas vey fa away, but this will impove dastically with the upcoming ESA Gaia mission. Figues: v /d (uppe) and v t /d (lowe) fo classical Cepheid vaiable stas. Fom FA. v t /d, km/sec v /d, km/sec B A A A 1 Mach 013 Astonomy 14, Sping

12 The local standad of est Fom A and B we get the aveage otational motion of the Sun s obit, called the local standad of est (LSR): 16-1 Ω= 9.8 ± adians s v φ ( ) ( ) 6-1 = 54 ± 16 km s = 8.4 ± 0.6 kpc P = 00 ± 0 10 yeas The sola system actually moves slightly with espect to the LSR, at about 7 km/s. Fom the motion of the LSR, the Galaxy within = 8.4 kpc can be weighed: vφ 11 M = = ( 1.3 ± 0.) 10 M. G (Reid et al. 009.) 1 Mach 013 Astonomy 14, Sping 013 1

13 Rotation cuves The aveage obits in the disk of the Galaxy seem to be cicula, centeed on the Galactic cente. A measuement of aveage angula velocity at any adius allows a detemination of the mass within that adius of the Galactic cente. Done as a function of adius: otation cuve Enables detemination of enclosed mass, and in tun the density, as a function of. Intestella gas has fa smalle andom motions than stas, is widespead, and detectable thoughout the galaxy; atomic (e.g. H I 1 cm) and molecula (e.g. CO.6 mm) lines ae the best to use fo detemination of the Galactic otation cuve. 1 Mach 013 Astonomy 14, Sping

14 Example otation cuves 1. Point mass, M: F = GMm mv v () GM = = Kepleian motion v deceases with inceasing. Constant density, spheically symmetic: 4π 3 M( ) = ρ Mach 013 Astonomy 14, Sping Gm Gm 4π 3 mv M( ) = ρ 0 = 3 4πGρ0 Solid-body otation v () = 3 v inceases linealy with inceasing

15 Example otation cuves (continued) 3. Spheical symmety, 1/ density distibution: ρ = ρ ( ) ( ) ( ) As we will see, many otation cuves of disk galaxies, including ous, look like this one. ( : coe adius of galaxy) = ρ 4π = 4πρ = 4πρ00 ( ) = 4πρ 00 = M d d GmM Gm mv 00 v = 4 πgρ = constant Flat otation cuve 1 Mach 013 Astonomy 14, Sping

16 Measuement of Galaxy s otation cuve fom H I and CO line pofiles Wavelength o fequency shift and adial velocity: the Dopple effect. λ λ0 λ ν = = = λ λ ν Along a given line of sight though the plane, maximum adial velocity must come fom obit tangent to line of sight: motion paallel to line of sight, cosφ = 1. Thus distance and otational motion of tangent points vey well detemined. thee is a distance ambiguity: fo lines of sight towad the inne galaxy (fist and fouth quadant), thee ae two locations with the same adial velocity. 1 Mach 013 Astonomy 14, Sping v c

17 Intepetation of H I line pofiles Sun 1 1 cm line intensity 5 4 1, v Radial velocity ( ) 1 Mach 013 Astonomy 14, Sping

18 Measuement of Galaxy s otation cuve fom H I and CO line pofiles (continued) Resolution of the ambiguity usually involves infomation othe than velocities: association o lack theeof with visible-wavelength nebulosity (less extinguished = neae). cloud angula size (bigge ones tend to be neae by). height above Galactic plane (clouds that appea highe would be neae by). In the oute galaxy it is much hade to detemine the distance to clouds, so the uncetainties ae lage. Best method so fa: association of clouds with H II egions o sta clustes; cluste distances detemined by mainsequence fitting. 1 Mach 013 Astonomy 14, Sping

19 Measuement of the (inne) Galaxy s otation cuve Sun v Fo the H I cloud at,: note that, fom the law of sines, sin sin sin so its velocity elative to us is v sin sin v,max sin, max and its speed in obit is v v,max sin,, as you will show in Homewok #7. 1 Mach 013 Astonomy 14, Sping

20 Results, fom CO obsevations Clemens Mach 013 Astonomy 14, Sping

21 Notable featues of the Galaxy s otation cuve Cental egion has v inceasing linealy with inceasing, as in solid body otation. (Constant density if spheical.) Most of the disk has a athe flat otation cuve (i.e. diffeential otation), meaning that the enclosed mass inceases linealy with inceasing adius - as if the mass wee dominated by a spheical, 1/ density out to the lagest adii at which intestella gas is detected. This is the case in spite of the fact that the obseved stella density (sta counts) deceases moe shaply than 1/. Kepleian otation is expected eventually, at lage enough distances, but is not seen. Dak matte again? (Yes, as we will see next time.) 1 Mach 013 Astonomy 14, Sping 013 0