AST 1420 Galactic Structure and Dynamics Today: galactic rotation - - PowerPoint PPT Presentation

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!

Rubin et al. (1980)

Long-slit spectra

Bosma (1978)

2D velocity fields (radio)

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-
• n
• 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

• btained 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

Bosma (1978)

• Radio
• bservations

typically extend well outside the

• ptical radius (~2x
• ptical radius)
• No good

photometry available at the time, so Kent-style forward analysis not possible

Enclosed mass implied by rotation curves

Bosma (1978)

• For spherical mass

distribution vc(r) —> M(<r)

• Similarly, for razor-thin disk

vc(R) —> 𝛵(R) —> M(<r) [but more difficult!]

• Enclosed mass profile differs

by a few tens of percent, but

• verall trend the same
• Flat rotation curves imply

rising mass M(<r) ~ r out to twice the optical radius! —> dark matter

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 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

• Observed velocity = velocity - Sun’s velocity

Random velocities and the Oort constants

• 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]