AST 1420 Galactic Structure and Dynamics

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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(

• Similarly, for razor-thin disk vc(R) —> �(R) —> M(

• Enclosed mass proﬁle differs by a few tens of percent, but overall trend the same

• Flat rotation curves imply rising mass M( 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 (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 = 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 out motion of guiding center, motion is ellipse: epicycle

• Axis ratio Oort constants and the epicycle approximation

• Can express the epicycle frequency in terms of the Oort constants

• From the measurements of the Oort constant we then get Random velocities and the Oort constants • So far related the mean velocity of local stars to the Oort constants

• Can we relate the velocity dispersion of local stars to the Oort constants?

• Velocity of star currently at R = R0

• Each star currently at R=R0 has its own guiding-center radius Rg Random velocities and the Oort constants

• Observed velocity = velocity - Sun’s velocity

• Expand Ω(Rg)-Ω(R0) in terms of (Rg-R0) and replace (Rg- R0) with its epicycle approximation Random velocities and the Oort constants • Can then write relative velocities in terms of the Oort constants

• Averaging the squared velocities assuming that the phases are random gives

• And the ratio is directly set by the Oort constants Random velocities and the Oort constants • Epicycle amplitudes for stars near the Sun:

• radial dispersion ~ 30 km/s

• Ratio of velocity dispersions from measured Oort constants: ~2/3

• Measurement from Hipparcos agrees with this (Dehnen & Binney 1998) [but somewhat of a coincidence, because corrections to the epicycle approx. are large for the observed sample]