Galactic Astronomy Chapter 6...2 Milky Way ISM...2 Kinematics of differential rotation...2 lv, plot...2 The naive ( ) Radii and distances from ( lv, ) plot...7 Non circular motion...8 Observations of 21-cm line emission...12 Observations of CO line emission...14 MW circular-speed curve...16 Radial distribution...19 Evidence of spiral arms...2 Vertical distribution of HI and CO...22 Middle disk 3 kpc < R < R...23 Outer disk...25 Central disk...26 21-cm observations...26 Observation in CO and CS...29 Dynamical model (Binney et al. 1991)...31
Chapter 6 Milky Way ISM Problems: 1. Wealth of information ==> too much details cannot see general trends; 2. Location of the Sun not ideal and distances difficult to estimate; Kinematics of differential rotation Velocities relative to Local Standard of Rest (LSR): current velocity of a fictional particle that moves around the plane of MW on closed orbit in the plane that passes through the present location of the Sun conversion velocity heliocentric to LSR is trivial; Assumption: MW is axisymmetric orbits are circular otherwise oval; The naive ( lv, ) plot Consider ( lv, ) plot observed if all the gas were in circular rotation; o R position vector of material with respect to Galaxy center; o Ω ( R ) angular velocity of rotation of this material; o v c velocity of this material vc =Ω R; 2
o When observed from the particle which define the LSR, whose rotation velocity is v Ω R the line of sight velocity v los of this material is the projection of the velocity difference v c v on the vector ( ) R R that runs from the Sun to the material: R R Eq 5.1 vlos = Ω ( R) R Ω ( R) R R R ; Using the vector identities: a ( a b) = ; and a ( b c) = b ( c a) Eq. 5.2 v los = Ω( R) Ω( R) ( R R) ; R R o From figure above: R ˆ R = RR sin α n perpendicular to the disk; where ˆn is the unit vector that points o Also by the law of sinus: sinα sinl = ; R R R o MW rotates in sense shown in figure (viewed from northern hemisphere): Ω= Ω ˆn ; Eq. 5.3 ( ) = Ω( ) Ω( ) v l sin los R R R l ; Assume all the gas in MW concentrated into a ring of radius R eq. 5.3 states that along any line of sight that intersects the ring we will detect material at just one velocity and more over that this velocity will be proportional to l; 3
Traces in ( lv, ) plot of 3 rays each a section a sin curve; o RR 1 section small and nearly straight runs from l ( RR) = arcsin, half the angular extend of the ring; max o RR 1 section distinctly curve; o For any value of l = lmax to l lmax = ; RR slope of the trace at the origin proportional to ( R) ( R ) For any plausible circular-speed curve Ω ( R ) increases inwards ==> slope Ω Ω ; largest for smallest rings; As RR increases from small values towards unity, the slope continuously declines and is zero for RR = 1 ==> Point on solar circle do not move in relation to LSR; o RR > 1 slope of the sine curve near origin opposite to that for RR < 1 and absolute value of the slope increases steadily as RR increases; For any plausible circular-speed curve Ω( R) as R, from eq. 5.3, the amplitude of the sine curve tends in the limit R to minus the circular speed R Ω R ; at the solar circle ( ) 4
MW = infinite number of rings ==> any observed ( lv, ) plot is occupied; o Material interior to solar circle occupy 2 quarters: < l < 9, vlos (top left) and 9 l, vlos (bottom right); gas on outside the solar circle occupy remaining 2 quarters; Theoretical division of ( lv, ) plot only partially confirmed by observations; o Material outside quarters = forbidden velocities; o Central slope of curve associated with a ring of radius r never greater than R Ω ( R ) for r > R; For l > velocities less than RΩ ( R) sinl are strongly forbidden; Material should not be found at such velocity (same on other side of Galactic center); Magnitude of optical depth τ of 21-cm line in the ( ), lv plane for a model disk in which all material moves on perfectly circular orbits; o τ non zero only in 2 broad arcs that are bounded on one side by section of a sine curve; 5
Radio telescope pointing in direction l 1 = 2 (vertical dotted line in figure 9.3): first quadrant < l < 9 1 runs through ( ) v = v R sinl below to: los c 1 Eq. 5.4 ( t ) ( ) ( ) v l = sin los 1 Ω r Ω R l t above; 1 where r t = radius of the smallest ring (largest value of Ω ) that can be seen along line of sight. Line of sight tangent to this ring: Eq. 5.5 rt = Rsin l1; The point at which the line of sight touch the ring of radius r t = tangent point and terminal velocity; ( t) v = los Empirically determined by examining observed analog of figure 9.4: it is the velocity of abrupt decrease in emission at v > for l > and v < for l < ; Ex. in figure 9.4 ~.6v c ( R ); Combining eq. 5.4 and 5.5: Eq. 5.6 ( ) Since c ( ) ( t) los ( t ) ( ) ( ) v r = v l + v R sinl ; c t los 1 c 1 v determined for any value of 1 v R is known; data for l < and Determining ( ) l between and 9 o ==> vc ( R ) for R R l > yield independent determination of ( ) < provided v R ; vc R extremely difficult amplitude of sine curve that forms one boundary of occupied band in figure 9.3; c 6
Radii and distances from ( lv, ) plot From eq. 5.6 determine vc ( R ) without prior knowledge of distances to material filling ( lv, ) plot; after circular-speed curve Ω ( R ) determined ==> radius of material at any value of v los solving eq. 5.3; Ω ( R ) monotonic ==> radius uniquely determined by ( lv ) ; Once radius determined ==> distance of observed material from the Sun; o any given line of sight intersect ring outside solar circle just once ==> d follows from r; o line of sight inside solar circle intersect twice or not at all ==> 2 values of d possible; ambiguity resolved observing extension in b nearer value show higher solid angle (Schmidt 1957);, los 7
Non circular motion Observed plots deviate significantly from naive ( lv, ) plot ==> more complex motion; Possibilities: 1. Axisymmetric radial expansion; 2. Oval distortions; 3. Spiral structures; 4. Random motion; Axisymmetric radial expansion: o Ring expand radially as well as rotating; expansion perpendicular to line of sight at tangent point ==> v los unaffected; o Line of sight through Galactic center: expansion parallel ==> section of sine curve open up into curve loops; 8
Oval distortion: o Ellipticity of rings changes the angle it subtends on the sky ==> changes in velocity; o Dynamical theory needed to predict changes ==> Weak bar theory: Φ R, φ in which object orbit is the sum of 2 terms: Gravitational potential ( ) Φ ( R) and weaker bar ( R) cos( 2[ φ t] ) Φ Ω, where Φ b is the force of the bar b and Ω b pattern speed of the bar (angular velocity of bar in rotation like solid body); b Orbit is compound of clockwise motion around a circle and anti clockwise motion about epicycle centered on guiding center and has principal axes aligned with local radial and tangential directions; o Orbit is at pericenter (closest to Galactic center) when epicycle displaces object inwards and at apocenter (furthest from Galactic center) when epicycle displaces the object outwards; o Contribution to v φ from circle and epicycle add at pericenter and subtract at apocenter; 9
Effect of epicycle upon ( lv, ) plot depends on φ - angle between sun center line and major axis of orbit; in general traces becomes closed curve reminiscent of cross section through aerofoil; Spiral structure: o Spiral form if stars and gas clouds move on elliptical orbits whose major axis position vary smoothly as a function of radius; o Disk made of number of ovals - each oval uniformly populated with gas clouds; because long axes of ovals rotate more than 54 o from smallest to largest ovals ==> oval in some places are closer together ==> surface density of clouds higher in these regions ==> crowded parts form 2 spiral arms; 1
Observed ( lv, ) plot of MW ==> crowded regions identified to spiral arms; transformation to real space ==> needed to be modeled dynamically; Random motion: o clouds do not move on perfectly symmetric orbits at any given position velocities spread over a few km/s; o Within clouds spread in velocities of individual atoms; lv, plot = smoothing; o Effect on ( ) 11
Observations of 21-cm line emission Many surveys using single dish (25 meters) at resolution ~ 1 o contain in excess of ten thousands resolution elements ==> observations over one year period; ( lv, ) plot very much similar to theoretical one: 12
Sharply discontinuous behavior near l = ; horns at v 14 km/s near l = 15 and v 15 km/s near l = 15, as expected based on theoretical model; Boundary at l >, v < and l <, v >, approximated by sine curve; Differences: o Fainter horns at smaller value of l ==> very little HI at R 3 kpc, l < 2 ; o Amplitude of sine curves fitting the boundary 6% instead of 85% based on model ==> sensitivity or negligible emission beyond sine curve amplitude ~13 km/s; Knapp, Tremaine & Gunn (1978) MW HI do not extend beyond 21 kpc; Ridges along v ~at 9 < l < 9 : from model, associated to a jump in optical depth for < l < 9 two identical points contribute to emission at v >, whereas only one contribute for v < ==> T B depends on optical depth τ ; in most direction, the disk is not very optically thick at 21-cm; 13
Disk is not axysymmetric from figure, curve for l < displaced to the left; o Blitz and Spergel (1991a): the outer MW is oval and LSR currently receding from Galactic center; o Kuijken & Tremaine (1994): outer part of the disk is lopsided; Observations of CO line emission CO = most important tracer of molecular gas; Velocity resolution and sensitivity ~ 21-cm, but resolution significantly higher ==> too much resolution elements, no survey cover more than small fraction of the sky; 14
CO more strongly confined to the plane; More patchy than 21-cm emission, but patterns quite similar; Envelopes at ( l >, v> ) and ( l, v ) < < decline to the origin from prominent horns; horns at larger longitudes l 25 and smaller velocities v 13 km/s than 21-cm line ==> molecular ring of radius R sin25 3.6 kpc ; At positive longitude, terminal velocity curve seems to fall to zero before l = 9 (not the same for 9 < l < 18, v< and l < ) ==> very little emission in regions ( ) ( 18 l < < 9, v> ) associated with gas beyond solar circle, although emission at v = at all longitude comes only from solar circle and solar neighborhood; Emission at R > R associated to ridges of intense emission at 21-cm ==> spiral arms; Digel et al. (1996): intensity of CO 25 times smaller between spiral arms than in arms; 15
MW circular-speed curve 21-cm and CO line data = key to determine v ( ) c R and radial distribution of mass; For R R <, v ( ) c R estimated from terminal velocity ( t ) v and (, ) los lv plot, using eq. 5.6; Good general agreement in observations: o Small scale variations ~7 km/s ==> effect of spiral structure on orbital velocities; ( t) o Differences between v los for HI and CO at fixed l follow Gaussian distribution centered on zero with dispersion 4.1 km/s (Burton 1992) ==> dispersion in random tangential velocity of ISM; 16
Beyond vc R : o Usual 2 2 2 determine distance d of object from the Sun ==> R = R + d 2Rdcosl; R, 2 alternate ways to determine ( ) d MS fitting method (Cepheid variable, PN or Carbon stars; v los from radio observation; v los Define eq. 5.7 W = = R Ω( R) Ω( R ) ; where v los is the component of sin l the relative velocity between the LSR at the Sun and at location of a tracer at distance d from the sun at longitude l (and using eq. 5.3); ==> simple multiple of the difference between the circular frequency at estimated radius R of the tracer and at the Sun. 17
Upper panel: W falls steeply at R < R 1.2R then falls more gradually or not at all; Lower panel: v c, as deduced from eq. 5.7, steadily rising for R > 1.2R, after falling for R 1.2R ; Implausible dynamically (Binney & Dehnen 1997): due to errors, most of the points at R > 1.25R really are at R 1.6R ==> lower v c ; v R R Keplerian behavior ruled out: ( ) and 1.6R ; this is not observed! c 12 at R R > ==> falls by > 5 km/s between R 18
Radial distribution Interior to R ~4 kpc density falls from wide plateau to near zero inside R ~1.5 kpc; beyond R, density remains large to beyond R ; R ~16 kpc ==> 8% of the 9 ~4.3 1 M of Galactic HI lies Molecular hydrogen (dashed curve) ==> H 2 almost entirely confined to R < R, 77% of the 9 ~1 M of molecular material lies inside R ; At l > a large fraction of H 2 appears to be concentrated into a ring r ~4.5 kpc with a bimodal distribution peaking at 2.5 kpc and 6 kpc; Alternatively (Dame 1993) distribution traces 2 giants spiral arms connected by central bar; 19
Evidence of spiral arms Clearest evidence in ( lv, ) plot of CO; assuming mapping highly incomplete; v = 22 km/s and R = 8.5 kpc ; Problems = c Map suggests elongated structures: o At R ~12R elliptical ridge form by ½ of Molecular clouds ==> ring (or spiral arms?); because of possible bar clouds at R ~12R non circular orbits; o Sun near outer edge of spiral arm logarithmic spiral with pitch angle 1 o Inside R and for l > Sagittarius arm; Inside R and for l < Carina arm; Les conspicuous (incomplete) at bottom left: Orion-Cygnus arm; 2
Trace of spiral arms at R > R difficult due to scarcity of molecular clouds ==> HI emission, terminal velocity curve reminiscent of spiral structures; ridge lines of spiral arms in HI ( ), lv plot ==> local maximum in B T ; mapped in real space (assuming circular orbits) ==> trailing spiral arms; o Perseus arm ~ 2 kpc from the Sun; 21
Vertical distribution of HI and CO Let ( R,, Z) φ be a system of Galactocentric cylindrical coordinate oriented such that plane at Z = identical with b = vertical distribution determined by: o R, φ ; Z distance where density of ISM peak for same values of ( ) c o Z 12 distance where density falls 12 its value at Z c ; Since HI and CO form a thin disk mean velocity of ISM at ( R,, Z) at ( R, φ,) (Lockmam 1984); o Every features in ( lv, ) plane generalized to features in (,, ) o Terminal velocity curve terminal velocity surface; φ lies near circular speed lvb data cube; 22
Middle disk 3 kpc < R < R Contains most of the molecular gas and SF; Z (, ) c R φ defines a surface nearly a plane: o Values oscillate around zero with rms amplitude ~3 pc, less than 1% of R ==> extremely flat; o Oscillations are coherent adjacent values highly correlated and two layers oscillate in phase ==> dynamical process corrugate the disk (Spicker & Feitzinger 1986); 23
Thickness of CO increases slowly from 45 pc at R = 3 kpc to 7 pc at R = R ; HI 3 times as thick at given radius; Thickness = result of force equilibrium: gravity vs energy densities of IS magnetic field + cosmic rays + random motions of clouds; Malhotra (1995) σ HI = 9± 1 km/s independent of R for.3 RR< 1: o calculation suggest this is not enough to explain thickness ==> underline importance of other mechanism (in particular magnetic fields); o similarity with external galaxies (transverse not radial) ==> dispersion is isotropic; σco 7 km/s ==> thickness of clouds proportional to 2 σ, in agreement with theory; 24
Outer disk Warp first discovered in Galaxy (Burke 1957; Ken 1957; Westerhout 1957); Intersection of HI with 4 cylinders that have R equal to 12, 16, 2 and 24 kpc; Warp just discernable at R = 12 kpc; Pronounced by R = 16 kpc; For R 17 kpc, simple sine curve; Beyond R = 17 kpc pronounced asymmetry; Sun happens to lies near warp s line of nodes the line in which the warped outer disk intersects the plane of the inner Galaxy; Warp traced in: CO similar than in HI (Grabelsky et al. 1987; Digel et al. 1991); OB stars (Miyamoto et al. 1988; Smart & Lattanzi 1996) RG (Carney & Seitzer 1993) IRAS sources (Djorgosky & Sosin 1989); There is still no generally accepted explanation for the cause of the warp (Binney 1992); 25
Thickness: increase with R continues and gather pace; Z 12 for HI rises from 25 pc at R to 6 pc at 2 R and even larger further out; Olling & Merrifield 1998: combining variation of mass with radius + how much mass lies close to the plane ==> slightly oblate ( q =.8) massive disk halo necessary; Central disk ISM inside R 3 kpc ==> l 2, very complex structure and dynamics; 21-cm observations Problem: o Great column of gas in middle disk moving on circular orbit ==> strong absorption in interval (,3sin l ) km/s; o Diffuse 21-cm continuum emission ==> line appears weaker; In figure 9.13: envelope of 21-cm emission at ( l, v ) ( l = 15, v= 14 km/s ) > > pronounced horn falls steeply towards center ==> hole in HI. 26
(, ) lv plot for ( l < 12, b <.5 ), with grayscale much lower brightness than in figure 9.13 ==> central disk; l >, v> envelope sweep up from v 18 km/s at l = 12 to o At ( ) v 27 km/s at l = 3 ; o Rounds up a horn and plunges steeply but crosses l = with v 2 km/s ; circular rotation model: for l small and v > velocities highly forbidden observed: upper envelope reaches l 5 before dropping below v 1 km/s ==> orbit non circular; Model of bar (oval orbit) ==> explains passage l = with v ; o Observational evidence: pattern must be symmetric on inversion of origin; o In figure 9.33, symmetry observed, but absence of emission with forbidden velocities ==> orbit circular; 27
Solution of the Paradox (Burton & Litz 1978; Burton 1992): o Point of disk in front of Galactic center masked by absorption; o Central disk not in the plane b = ; Absorption for v < (right panel) not observed for v > ; near side of the disk is approaching; Also ridge line at l >, v > runs straight along b = ; for l < runs.5 o above b = ; for v < l > below b = while l < above b = ; Angle between line of sight and long axes ~2 o explains passage of l = with v ; both θ n ax ax bmax 1 and φ n small and φ n < ; 2 and since sinθ n < ==> θ n < 7 ; a a l 4 y y max 28
Observation in CO and CS In figure 9.15, great spikes at l 4 v > 18 km/s ; Band of strong emission runs horizontally ==> gas outside 7 pc; central material v 3 km/s ; ( lv, ) = ( 3,1 km/s ) ==> emission from gas confined to.6 (8 pc) = Bania s clump 2; ( lv, ) = (.66,55 km/s) 6 3 1 M (Bally et al. 1988); ==> Sgr B2 complex; angular diameter ~.25 o (35 pc) and mass 29
Upper panel: emission CS J = 2 1 (98 GHz); o Sgr B2 lies along ridge of intense molecular emission; density of colisional de-excitation 12 3 1 m ==> densest molecular clouds; n H o Hot spot at ( lv, ) = (.1,55 km/s) ==> Sgr A molecular complexes; o 34 of molecular emission at Galactic center lies at l > ; 3
Dynamical model (Binney et al. 1991) Simple rotating barred potential: o x 1 orbits: largest radii elliptical orbit becoming more elongated going inward ==> cusped orbits; o x 2 orbits: inside cusped orbits elongated oval perpendicular to cusped orbit becoming rounder going inward; o Transition from x 1 to x 2 = inner Lindblad resonance; 31
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