AST 1420 Galactic Structure and Dynamics Today: galactic rotation • Brief overview of observations: velocity fields and rotation curves • Quantitative understanding of velocity fields • Rotation curves —> dark matter • Gas rotation in the Milky Way • Local observations of differential rotation Galactic rotation: observations • Gas: assumed to be on non-crossing, closed orbits —> trace circular(-ish) orbits —> trace galactic potential • Different setups: • Long-slit spectra: spectrum of galaxy at all points along a 1D slice (typically major axis) —> rotation curve along this axis • Optical gas emission lines like Hα, [NII] • Observations of 2D velocity field: spectrum at each point of galaxy • Radio observations (1970s onwards) • Currently also possible in optical with IFUs • Important to take into account the beam (radio) or PSF when measuring velocity fields! Long-slit spectra Rubin et al. (1980) 2D velocity fields (radio) Bosma (1978) Walter et al. (2008) Forster-Schreiber et al. (2008) IFU 2D velocity fields Anatomy of 2D velocity fields • Just from looking at the contours, we can see that this galaxy has • a rising rotation curve at small radii • and a flat rotation curve at larger radii • We’ll learn why in the next slides! 2D velocity fields • Consider velocity field V(x,y): • Center of galaxy at (x,y) = (0,0) • Major-axis along y=0 • Peak recession at positive x • Can rotate any galaxy’s observations to satisfy this • Two planes: • Sky plane: (x,y): observed position on the plane of the sky • galaxy plane: (x’,y’) observed position in the galaxy disk, seen face- on • Related by the inclination i: i=0 (edge-on) to i=90 (face-on) Sky and galaxy planes sky galaxy Observed velocity field for circular rotation • rhat: line-of-sight direction • Rhat:from center of galaxy to observed (x’,y’) • nhat: perpendicular to galaxy • khat: perpendicular to rhat and nhat (rhat x nhat) +systemic motion V0 Examples • Solid-body rotation: vc(R) = Ω R • x = R cos θ Examples • Flat rotation: vc(R) = v0 • Only depends on y/x —> straight lines with intercept 0 Examples • Rotation curve with peak: • At y=0: V(x,y) = vc(R) sin i —> velocities near the peak attained at two x • For this value, go to y > 0 • Get same V(x,y) from R closer to peak of the rotation curve —> still two x • At some y, require peak vc to keep following the contour —> no solutions for larger y • Contours therefore close Examples Examples: disk rotation curves Example: rising then flat rotation curve Reading velocity fields From velocity fields to rotation curves • Long-slit spectra: • 2D velocity fields: tilted-ring models Rotation curves Rubin et al. (1980) Do Rubin’s flat rotation curves imply the existence of dark matter? • Optical rotation curves typically get close to the ‘optical radius’, the radius which contains most of the light • If the disks were exponential, we expect a peak at R ~ 2.15 Rd < optical radius • However, disks are not all exponential and a somewhat shallower radial profile could keep the rotation curve flat to the optical radius • Question: given surface photometry, can we fit the Rubin rotation curves with the rotation curve implied by the light profile and M/L that fits the inner part? Rotation curve for general bulge+disk light distribution • Bulge-disk decomposition of light: • Use results from last few weeks’ classes to calculate the rotation curve of the disk and bulge components • Bulge: assume spherical, 3D density from Abel inversion like two weeks ago Rotation curve for general bulge+disk light distribution • For spherical mass distribution, circular velocity determined by enclosed mass profile, so we calculate the enclosed light profile • vc(r) follows from M/L assumption (constant) Rotation curve for general bulge+disk light distribution • For the disk we start from the general expression for a razor-thin disk from last week: • Result is: Maximum-disk fits • We can obtain a fit to the rotation curve that contains as much (bulge+disk) matter as allowed as follows: • Compute the rotation curves from the bulge and disk components • Adjust the bulge and disk M/L such that the combined (bulge+disk) rotation curve does not go above the observed rotation velocity (in the center) • Because this fit has as much mass in the (bulge+) disk as allowed, these are known as maximum disk fits Kent maximum-disk fits to Rubin et al. data • Kent (1980) obtained good photometry for galaxies whose rotation curves were obtained by Rubin et al. • Many galaxies actually well represented by max-disk hypothesis • But last few vc(R) points typically somewhat high • Not all Rubin et al. optical rotation curves require large amount of dark matter Rotation curves from radio velocity fields • Radio observations typically extend well outside the optical radius (~2x optical radius) • No good photometry available at the time, so Kent-style forward analysis not possible Bosma (1978) Enclosed mass implied by rotation curves • For spherical mass distribution vc(r) —> M(<r) • Similarly, for razor-thin disk vc(R) —> �(R) —> M(<r) [but more difficult!] • Enclosed mass profile differs by a few tens of percent, but overall trend the same • Flat rotation curves imply rising mass M(<r) ~ r out to twice the optical radius! —> dark matter Bosma (1978) NGC 3198 • Poster child for flat rotation curves • Disk scale length ~2.7 kpc • Optical radius ~10 kpc • Rotation curve flat at ~11x disk scale length! de Blok et al. (2008) Kinematics of the Milky Way’s interstellar medium Phase-space distribution of gas • Want to use gas to measure Milky Way’s rotation, but difficult to obtain distances to gas, so interpreting the velocity of the ISM in terms of vc(R) is difficult • For gas orbiting in a plane, phase-space is four- dimensional (x,y,vx,vy) • Because gas orbits cannot cross, at each (x,y) there can only be a single velocity [vx,vy](x,y) • Thus, the phase-space distribution of the ISM is only two dimensional The longitude-velocity diagram • ISM phase-space distribution is 2D, so if we can measure two (independent) phase-space dimensions, we can fully map its phase-space DF • We can take spectra (e.g., 21cm, CO) that show the distribution of vlos at each Galactic longitude l • 2D distribution of (l,vlos) == direct phase-space map! [up to some degeneracies] The longitude-velocity diagram: HI Sparke & Gallagher (2007) The longitude-velocity diagram: CO Making sense of the longitude-velocity diagram • How does circular rotation vc(R) map onto (l,v)? • Makes sense: • For disk in solid-body rotation relative distance between any two points remains the same —> vlos = 0 • Dependence on sin l gives correct v=0 at l=0,180 • Must have minus the local circular velocity (relative to LSR/ Sun) Ring of gas in the longitude- velocity diagram • Ring at R < R0 subtends -asin(R/R0) < l < asin(R/R0) • vlos(l) ~ sin l between these limits, with amplitude depending on [Ω(R)-Ω(R0)] • Ring at R > R0 spans entire -180 < l < 180, also sinusoidal Ring of gas in the longitude- velocity diagram Molecular ring Circular velocity from (l,v)? • Observed vlos only depends on difference in rotation rates • Therefore, to derive vc(R) from vlos(l) we need to assume vc(R0) • If we assume that Ω(R) —> 0 as R goes to infinity then vlos —> -Ω(R0)R0 sin l = vc(R0) sin l • Unfortunately, need to go to large R and very little gas exists at large R! Terminal velocity • For 0 < l < 90: distribution of vlos terminates at positive value, because Ω(R) monotonically decreases with R (at -90 < l < 0; vlos terminates at same negative value) • Termination at given l is at largest ring at R < R0 that reaches l • At this ring • Can thus map [Ω(R)-Ω(R0)] by tracing the terminal velocity curve Predicted terminal velocity curve for different rotation curves Oort constants Local velocity distribution • We can observe velocities for large samples of local stars —> galactic rotation ? • First discovery of differential rotation based on local stars • Consider mean velocity field near the Sun <v>(x) • Can Taylor expand this wrt distance from the Sun Local velocity distribution • In cartesian Galactic coordinates • After subtracting the mean motion, can write • We observe vlos and the proper motion Local velocity distribution • Proper motion • Thus, we can measure A,B,C,K from measurements of vlos(D,l) and μl(D,l) • But what are A,B,C,K? Oort constants: C and K • For axisymmetric galaxy <vR> = 0 • At the Sun, vx = -vR and X = R-R0 • d vx / d x should thus be zero —> K+C = 0 • Similarly, vy = vc (for circular rotation) and y is parallel to the phi direction —> d vy / d y = 0 —> K-C=0 • K=C=0 • Deviations from zero for either of C or K —> Milky Way is non-axisymmetric Oort constants: A and B • Similarly, for an axisymmetric Galaxy we have that • Thus, A and B measure (a) local derivative of vc(R), (b) local angular frequency (necessary for [l,v]!) Oort’s measurement • For axisymmetric galaxy (ignoring Sun’s peculiar motion): • Like before, solid-body rotation —> vlos = 0 —> A = 0 • A therefore indicative of differential shear (aka azimuthal shear) • Oort measured A =/= 0 —> differential rotation (measured A = 31 km/s/kpc, which is quite far off though!) Modern measurements Bovy (2017) Modern measurements • C and K are both significantly non-zero (but < A-B) —> importance of non-axisymmetry • A+B = d vc / d R ~ -3 km/s/kpc —> slightly falling rotation curve Oort constants and the epicycle approximation Radial motion • Radial oscillation around guiding-center radius: radius of circular orbit with angular momentum • Azimuthal motion from conservation of angular momentum • Subtracting